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UCRL-52863 Rev. 1Distribution Category UC-38

UCRL—52863-Rev.lDE87 012387

Conduction Heat TransferSolutions

James H. VanSant

M anu scrip t date: Au gust 1983

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neith er the United States Government nor any agency thereof, nor any of theiremployees.^ makes, any warranty, express ox impUed, <w ass um es any legal liability os responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constitute or imply its endorsement, iecom-mendation, or favoring by the United States Government or any agency thereof. The viewsand opinions of authors expressed herein do not necessarily state or reflect those of theUnited Slates Government or any agency thereof.

LAWRENCE LIVERMORE NATIONAL LABORATORYUniversity of California • Livermore, California • 94550I,Available from: National Technical Information Service • U.S. Department of Commerce5285 Port Roval Ro ad • Springfield. VA 22161 • M l. 50 per copy • (Microfiche M .50 )

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CONTENTS

Pre face v

Nomenclature vxi

I n t r o d u c t i o n 1

S t e a d y - S t a t e S o l u t i o n s

1. Plan e Su rface - S teady S ta te

1.1 S o lid s Bounded by Pla ne Su rfac es 1-1

1.2 S o lid s Bounded by Pla ne Surf ace s

—W ith In te rn al He at ing 1-27

2. C y l i n d r i c a l S u rf ac e - S t e a d y S t a t e

2 .1 So l id s Bounded by C y l i n d r i ca l Surfa ces—No In te r na l Hea t ing 2 - i

2 .2 S ol id s Bounded by C y l i n d r i ca l Surfac es—With In te rn a l Hea t ing 2 -333 . Sp he r i ca l Sur face - S teady S ta te

3 .1 S ol id s Bounded by S ph er ic a l Surfaces—No In te rn a l Hea t ing 3 -1

3.2 So l id s Bounded by S ph er ic a l Surfaces—W ith In te rn al Hea t ing 3-10

4 . Tr ave l ing Heat Sources

4. 1 Tr av el in g Heat Sources 4-1

5 . Extended Surface - S teady S ta te5 .1 Extended Surfaces—No I n te rn a l Heat ing 5-1

5 .2 Ex tended Sur faces—W ith In te rn a l Hea t ing 5 -31

Tr a n s i e n t S o l u t i o n s

6 . I n f i n i t e S o l i d s - Tr a n s i e n t

6 . 1 I n f i n i t e S o li ds — N o I n t e r n a l H e a ti n g 6 - 1

6 . 2 I n f i n i t e S o l id s — Wi t h I n t e r n a l H e a t in g 6 -2 2

7 . S e m i - I n f i n i t e S o l i d s - Tr a n s i e n t

7 . 1 S e m i - I n f i n i t e S o l id s — N o I n t e r n a l H e a t i n g 7 - 17.2 - i m i - I n f in i t e S olid s— W ith I n te r n a l H ea tin g . . . . 7-22

8 . P lane Sur fac e - Tra ns ie n t

8.1 S ol id s Bounded by Pl an e Su rfa ce s—No In te rn a l Hea t ing 8 -1

8.2 S o li ds Bounded by Pla ne Su rfa ces—With In te rn a l Hea t ing 8 -52

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9 . C y l i n d r i c a l S u rf ac e - Tr a n s i e n t9 .1 So l id s Bounded by C y l i n d r i ca l Surfa ces

—No In te rn a l Hea t ing 9 -19.2 S ol id s Bounded by C y l i n d r i ca l Surfac es

—With In te rn a l Hea t ing 9 -24

10. Sp he r i ca l Sur face - Tr an s ie n t

10 .1 So l id s Bounded by S ph er ic a l S urfaces—No In te rn al Heat ing 10-1

10.2 So l id s Bounded by S ph er ica l S urfaces—W ith In te rn a l Hea t ing 10-19

11 . Change of Ph ase

11 .1 Change of Phase— Plane In te r f ac e 11-1

11. 2 Change of Phase— Nonplanar In te rf ac e 11-13

12. Tra ve l in g Boundar ie s

12 .1 Tra ve l in g Boundar ie s 12-1

F igures and Tab les fo r So lu t ions F - l

Misce l l aneous Da ta

13 . M athemat i ca l Func t ions 13-1

14. Roo ts o f Some C h ar ac te r i s t i c Equa t ions 14-1

1 5 . Co ns tan t s and Convers ion Fa c to r s 15-1

16 . Convec t ion Co eff i c i en t s 16-1

17 . C o n t a c t C o e f f i c i e n t s 1 7 - 1

18. Thermal P ro pe r t i e s 18-1

References R- l

i v

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PREFACE

This text is a collection of solutions to a variety of heat conductionproblems found in numerous publications, such as textbooks, handboo ks,

journals, reports, etc. Its purpose is to assemble these solutions into one

source that can facilitate the search for a particular problem solution.

Generally, it is intended to be a handbook on the subject of heat conduction.

Engineers, scientist s, technologists, and designers of all disciplines

should find this material useful, particularly those who design thermal sys

tems or estimate temperatures and heat transfer rates in stru ctu res. More

than 500 problem solutions and relevant data are tabulated for easy retrieval.Having this kind of material available can save time and effort in reaching

design decisions.

There are twelve sections of solutions which correspond with the class of

problems found in each. Geometry, state, boundary conditions, and other cate

gories are used to classify the problems . A case number is assigned to each

problem for cross-referencing, and also for future reference. Each problem

is concisely described by geometry and condition statements , and many times a

descriptive sketch is also included. At least one source reference is given

so that the user can review the methods used to derive the solutions . Problem

solutions are given in the form of equati ons, graphs, and tables of data, all

of which are also identified by problem case numbers and source references.

The introduction presents a synopsis on the theory, differential equa

tions, and boundary conditions for conduction heat transfer. Some discussion

is given on the use and interpretation of solutions. Als o, some example prob

lem solutions are included. This material may give the user a review, or even

some insight, on the phenomenology of heat conduction and its applicability to

specific problem s.

Supplementary da ta such as mathematical functions, convection correla

tions, and thermal properties are included for aiding the user in computing

numerical values from the solutions. Property data were taken from some of

the latest publications relating to the particular properties listed. Only

the international system of units (SI) is used.

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Consistency in nomenclature and terminology is used thr oughout, making t his

text more readable than a collection of different reference s. Also, dimension-

less parameters are frequently used to generalize the applicability of the

solutions and to permit easier evaluation of the effects of problem conditions.

Even though some of the equational solutions are lengthy and include

several different mathematical funct ions, this should not pose a formidable

task for most use rs. Modern computers can make complicated calculations easy

to perform. Even many electronic calculators can be used to compute complex

functions. If, howeve r, these tools are not available, one can resort to hand

computing meth ods. The table of mathematical functions and constants would be

useful in Chat cas e.

Heat conduction has been studied extensively, and the number of published

solutions is larg e. In fact, there are man y solutions that are not included

in this text. For example, some solutions are found by a specific computa

tional process that cannot be described briefly. More over, new solutions are

constantly appearing in technical journals and reports. Never thele ss, this

collection contains most of the published solutions.

The differential equations and boundary-condition e quations for heat flow

are identical in form to those for other phenomena such as electrical fields,

fluid flow, and mass diffusion. This similarity gives additional utility to

the heat conduction solutions. The user needs only identify equivalence of

conditions and terms when selecting a proper solution. Thi s practice is pre

scribed in many texts on applied mathe mati cs, electrical the ory, heat transfer,

and mass transfer.

A search for particular solutions has frequently been a tedious and dif

ficult task. Too ofte n, countless hours have been spent in searching for a

problem solution. Locating and obtainin g a proper reference can require con

siderable effort. Al so , it is frequently necessary to study a theoretical

development in order to find the applicable solution. In so doing, thsre are

sometimes misinterpretations which lead to erroneous results . This text

should help alleviate some of these prob lems .

Science gives us information for reaching new frontiers in technology.

It is, thus, appropriate to give something back. I hope this text is at least

a small contribution.

James H. VanSant

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NOMENCLATURE

2A = Area, m

b = Time constant, s

c = Specific heat , J/kg' C

C = Circumference, m

d, D = Diameter, depth, m, 2 o

h = Heat transfer coefficient, W/m * C

k = Thermal conductivity, W/m- C-1m = V hC/kA, m

I, L = Length, m2

q = Heat flux rate, W/m

q = Heat flux in x, y, z directi ons, W/m'3q " ' = Volumetric heating rate, W/m'

Q = Heat transfer rate, W

r, R = Radius, mo

t, T = Temperature, C, K

V = Velocity, m/s

w = W id th , m

x , y, z , = C a r t e s i a n c o o r d i n a t e s , m

2a = T h e rm a l d i f f u s i v i t y , k / p c , m / sS = Te m p e ra t u re c o e f f i c i e n t , c

Y = H e a t o f e v a p o r a t i o n , J / k g

y = L a t e n t h e a t o f f u s i o n , J / k g

A = D i f f e r e n c e

e = E m i s s i v i t y f o r t h e r m a l r a d i a t i o n

„ ,_. ( a c t u a l h e a t t r a n s f e r r e d )H = F i n e f f e c t i v e n e s s , -*r— r—r 2 3—TTC—T~ci— T

( h e a t t r a n s f e r r e d w i t h o u t f i n s )

( a c t u a l h e a t t r a n s f e r r e d> = F i n e f f e c t i v e n e s s ,3

p = D e n s i t y, k g / m

a = 2Vox, m—2 —4a = S t e f a n - B o l t z r a a n n c o n s t a n t , W/m "K

T = Time , s

dT = Radiation configuration—emissivity factor

(he^t transferred from infinite conductivity fins)

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DIMENSIONLESS GROUPS

B i j , B i = B io t mo du lu s = h8 ,/k

Bf „ , Bf = B i Fo = hi /p cS ,2

F o „ , F o = F o u r i e r m o d u l u s = CIT/S.

F o „ , F o = M o d i f i e d F o u r i e r m o d u l u s = 1 / ( 2 V Fo)3 2G r „ , Gr = G ra sh of number = gBAt£ /V

K i „ , K i = K i r p i c h e v n u m be r = q J l / k A t

N u , , Nu = N u s s e l t number = hd /kd 2

Pd„, Pd = P re dv od ite le v modulus = hi. /a

P o . , Po = Pomerantsev modulus = q ' ' ' J . /kAt

Pr = P ra nd tl number = v/ a

R = Rad ius r a t i o = r / ro

Re,, Re = Reynol ds number = vd/v

X = Length ratio = x/l

Y = Width ratio = y/w

Z = Length ratio = z/2.MATHEMATICAL FUNCTIONS

exp = Exponential function

Ei = Exponential integral

erf = Error function

erfc = Complem entary error function

i erfc = Comple menta ry error function integral

I = Modified Bess el function of the first kindnJ = Bessel function of the first kind

nK = Modified Bes sel function of the second kindn&n = Natural log

Y = Bessel function of the second kindn

P = Legendre polynomial of the first kind

T = Gamma function

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INTRODUCTION

1 . HEAT CONDUCTION

Energy in th e form of he at h as been used by man ever s in ce he began

walk ing on th i s ea r th . Moreover, t he t r an s f e r o f hea t i s e s s en t i a l t o our ve ry

ex is te n c e . Not only do our own ph ys io lo gi ca l fun ct ion s re qu ire some form of

hea t t r an s f e r , bu t so do mos t l i f e - s u s t a i n in g p roce sses o f na tu re and many man-

c o n t r o l l e d a c t i v i t i e s . The i m po rt an c e o f t h e t h er m a l s c i e n c e s i n t h e t o t a lsph ere o f sc i en ce can , th u s , ha rd ly be d i s pu ted .

Conduct ion i s one of the thr ee p r i n c ip a l heat t ran sfe r modes , the o t he rs

be ing convec t ion and ra d ia t i o n . I t i s cus tom ar i ly d i s t ing u i s he d as be ing an

energy d i ffus ion p roc ess in m a te r i a l s which do no t con ta in molecu la r convec t ion .K in e t i c energy i s exchanged between molecu les r e su l t i ng in a ne t t r a ns fe r

between regio ns of d i f fe re n t energy l e v e l s , th es e energy le v e ls ar e commonly

c a l l e d t e m p e r a t u r e . P a r t i c u l a r l y , h ea t c o n d u c t i o n i n m e t a l s i s m a i n l y a t t r i

bu ted to the mot ion o f f r ee e l ec t ron s and in so l i d e l e c t r i c a l in su la to r s to the

l o n g i t u d i n a l o s c i l l a t i o n s of a to m s . I n f l u i d s , t h e e l a s t i c i m p ac t o f m o l e cu l es

i s cons ide red as the hea t conduc t ion p rocess .

The pro cess of he at t r an sf er in m a te r i a l s has been s tu die d for many

c en tu r i e s . Even ea r ly Greek ph i los op he rs , such as Lu cre t iu s ( c . 98 -55 B . C . ) ,

m ed i t a t ed on the su b j ec t and recorded th e i r co nc l us ion s . Much l a t e r , t he

famous mathem at ica l p h y s ic is t , Joseph B. J . F ou r ier (1768-1830) , developed a

mathem at i ca l exp ress ion th a t became the ba s i s of p ra c t i ca l l y a l l hea t conduc

t i o n s o l u t i o n s . He p o s t u l a t e d t h a t a l o c a l h e a t f l u x r a t e i n a m a t e r i a l i s

p r op or t io na l to the lo ca l t empera tu re g r ad ie n t in the d i r ec t io n o f hea t flow:

where a i s the hea t f low in the x - d i re c t io n pe r un i t a rea a s i l l u s t r a t e d in

F i g . l a . M a t e r i a l p r o p e r t i e s a r e a c c ou n t ed f o r by i n c l u d i n g a p r o p o r t i o n a l i t y

c o n s t a n t :

V - k S ' ( 2 )

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t ,

J3t

* q x

t

• 3x .. X

Q

(a)

*«-Ax-»x

(b)

FIG. 1. Illustration of heat flow and temperature grad ient.

where the constant k is called thermal conductiv ity. (The minus sign must be

included to satisfy the second law of thermodynamics.) This equat ion is

called Fourier's law for heterogeneous isotropic continua.

A simpler form of Fourier's law is for homogeneous isotropic continua.

For example , consider a plate of this type mate rial having isothermal surfaces

and insulated edges as shown in Fig. 1. The plate has a width Ax, surface

area A, and thermal conductivit y k. The heat flow in the plate is expressed as

<t.-kA

VAx

(3)

Th is expr essio n becomes Eq. (2) when Ax di mini sh es t o z er o .

fx lim At _ . a tA " \ " ~ K Ax+0 Ax " " K 3x

(4)

The heat f lux q i s presumed t o have bot h magnitude and d i r e c t i o n . Thus,

i t can be given as a ve ct o r , q which is normal to an iso th er mal su r f ac e . For

example, in Carte sian coo rdi nat es

q = V + V + q

zk (5)

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where i , j , k a re un i t ve c to r s in une x - , y -r and z - d i r e c t i o n s , r e s p e c t i v e l y.

Since Eq. (4) defines qx = - k 3 t / 3 x , and s i m i l a r l y qy = - k 3 t / 3 y , qz = - k 3 t / 3 z ,

we can s ta te

q = -k ( i3 t /3 x + j 3 t / 3 y + k3t /3z) (6)

q = -kV t . (7)

In anisotropic continua t he direction o f the heat flux vector i s not

necessari ly normal to an isothermal surface. Example materials a r e c rys ta ls ,

l ami na t e s , an d oriented fiber composites. I n such materials w e m a y assume

each component of the heat flux vector to be l inearly dependent o n a l l com

ponen t s o f the temperature gradient at a po in t . T h e vector form o f Four ier ' s

law fo r heterogeneous anisotr opic continua becom es

i = -K • Vt , (8)

where K i s t he conduc t iv i ty ten so r ; the components of th is tens or a re ca l le d

t h e c o n d u c t i v i ty c o e f f i c i e n t s .

In C ar te si an form, Eq. (8) is

"^11 3x + k ^ + k ^12 3y K13 3z,

S = "(k21 M + k22 U + k23 ft ) ( 9 )

_ _/. at at 3 t \q z ~ ^ 3 1 3x + k 3 2 3y + k3 3 3z / '

To compute heat f low by Fo ur ier ' s law, a thermal con du ctiv ity v alu e i s

needed. I t can be es t im ated from th eo re t ic a l pre dic t ion s for some id ea l

m at er ia ls , but m ostly , i t i s determined by measurement and F ou ri er 's law,Eq. (2 ) . As i l lu s t ra te d in Fig . 2 , thermal con du ct iv i ty can have a larg e

ran ge, which depends on m at er ia ls and tem pera ture. For example, copper a t

20 K has a thermal c on d u ct iv ity of app roxim ately 1000 W/m'K and diatom aceous

ea rt h at 200 K has a co n d u ct iv it y of 0.05 W/m*K. Co nseq uen tly, h ea t flow i n

m at er ia ls can have a very lar ge range, depending on a combined ef f e c t of

temperature grad ient and m ate ria l prop erty.

Thermal pr op ert ies of some se lec te d ma te ria ls are given i n Sec t ion 18 .

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"i 1 r -] r

0 .01 ' I i L200 400

-Tef lon

Copper

LeadIron

. Fused qu ar u

• Diatomaceous earth

±

•

J i_600 800

Temperature — K

1000 1200 1400

FIG. 2. Thermal conductivity of some selected soli ds.

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2. DIFFERENTIAL EQUATIONS OF HEAT CONDUCTION

Solutions to heat conduction problems are usually found by so me mathe

matical technique which begins with a differential equation of the temperature

field. The appropriate equation should include all energy sources and sinks

pertinent to a particular problem. Also, the equation should be expressed in

terms of a convenient coordinate system such as rectangular, cylindrical, or

spher ical. Then analytica l or differencing method s can be used to solve for

temperature or heat flow.

A common method for deriving the generalized differential equation for

heat conduction is to apply the first law of thermodynamics (conservation of

ener gy) to a volume element in a selected coordinate system. By accounting

for all the thermal energy transferred through the element faces, the change

of internal energy and thermal sources or sinks in the element, and by letting

the element dimensions approach zero, the differential equation can be derived.

This procedure is typified by the following heat energy accounting of the

rectangular solid element shown in Fig. 3. The net heat flow though its six

faces is

netQ . + Q*x+Ax *y Q . + Q*y+Ay z •^z+Az

(10)

FIG. 3. Coordinate systems for heat conduction equations: (a) rectan gular ,

(b) cylindrical, (c) spherical.

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w h e r e t y p i c a l l y

Q = -AyAzx ML

Q =-AxAZ(k g ) ,

Q = - A x Ay

and k , k , and k a r e d i r e c t i o n a l c o n d u c t i v i t i e s .x' y ' zAn in c r ea se in in t e rn a l ene rgy of the e lement i s t ep res en ted by

A I = AxAyAzpc | ^ , (11)

where t i s th e mean temp erature of the e lemen t , p i s th e m a te r i a l d en s i ty ,

and c i s i t s s p e c i f i c h e a t .In te rn a l ene rgy sources can be express ed as

Q U I _ q i i i A x A y A z f ( 1 2 )

where q " ' i s the un i t volume source r a t e . Examples o f in t e rn a l hea t ing in

m a t e r i a l s a r e j o u l e , n u c l e a r, o r r a d i a t i o n h e a t i n g . Summing t h e s e e n e rg i e s i n

accordance wi th the ene rgy conse rva t ion l aw y ie lds

A- W. ^ ^ ^ + ^ y ^ + ^ ^ + * ' " - C 1 3 ,

For the l i m it s A x, Ay, Az + 0, we o b ta in

at , , , B qx i 3 q «

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where

*x =

"k

x 8 i >

q y ~ " k y 3y '

q = - k •5— .^Z Z d z

Using Bq. (14) as a gen era l d i f fe re n t i a l eq ua t ion , we can der ive the

f o l l o w i n g s p e c i f i c e q u a t i o n s .

2.1 Re ctan gula r Coo rdinate System

For i so t ropic heterogeneous media

„ at a /.. a t\ . 3 /,. 3 t\ . 3 A, a t\p c 3? " 3x" \ k 37) + 37 \ k 3y"j + 37 [k 37j + q" (15)

For i s o tr o p ic homogeneous media th i s becomes

J L t _ k _3 T ~ pc

3 2 t + 3 2 t + 3 2 t

3 x 2 3 y 2 3 z 2

J 1 * * 2+ - 1 T — - aV t +c pc (16)

When q " ' = 0 , Eq. (16) becomes F o u r i e r ' s eq ua t io n .

In s te ad y -s ta te co nd i t io ns , 3 t /3 x = 0 and Eq. (16) becomes the Po isson

equa t ion .When q " ' = 3t / 3x • 0, Eq. (16) redu ces to the Lap lace equa t ion .

N o n i s o t r o p i c m a t e r i a l s , s u ch a s l a m i n a t e s , ca n h av e d i r e c t i o n a l l ys e n s i t i v e p r o p e r t i e s . F or s u ch m a t e r i a l s t h e c o n d u c ti o n d i f f e r e n t i a l e q u a t i o nin two dime nsions i s expressed i n the fol low ing form:

pc | | = ( kc cos 2B + kn s i n 2 a ) ^ - f + (k? s i n 2 B + k.) c o s 2 B ) ~

2

(17)

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FIG. 4. Coordinate sys tem for a no ni so t ro pi c medium.

where k r and k are d i r e c t i o n a l t he rm a l c o n d u c t i v i t i e s , and B is the ang le

o f l a m i n a t i o n s as i n d i c a t e d in F i g . 4. When the geo me t r i ca l axes of then o n i s o t r o p i c m a t e r i a l are o r i e n t e d w i th t h e p r i n c i p a l a x es of the t he rm al

c o n d u c t i v i t i e s , t h en E q. (17) s i m p l i f ie s to the form of Eq. (14)

p c f i = k $ | + k & + q >3T X . . . 2 y g y 2ax'

(18)

2 .2 C y l in dr i c a l Coord ina te System

Rec tangu la r coord ina tes can be t ra n s fo r m e d i n t o c y l i n d r i c a l c o o r d i n a t e s

by t h e r e l a t i o n s x = r cos 9, y = r sin 9, and z = z. The p a r t i a l d i f f e r e n t i a l

e q u a t i o n s (15) and (16) tran sfo rm ed to c y l i n d r i c a l c o o r d i n a t e s a r e t h u s

at . i a_ f rk a t\ i s_ / at\ a_/ at\ ,3x r 3r \ I K 3r ) r 89 \* 38 ^ + 3z \ K Zz ) + q (19)

2 t „ a / l t i a t + j L 3 t + l t \ a i i i* W r 8 r r 2 3 9 2 3 z 2 / ' C

(20)

F o r n o n i s o t r o p i c m a t e r i a l s w i t h the cond ucc iv i ty and geomet ry axes a l ig ne d as

in Eq. (18) the d i f f e r e n t i a l e q u a t io n is

3 t r 3 / 3 t \ ^ k 8 3 t , 3 tp c a T = T a7 r 17 + T 73 + k

z 77+ q

r 38 3z

(21)

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2.3 Spherical Coor dinat e System

A transformation from rectangular to spherical coordinates can be accom

plished by substituting the relations x = r sin \ \ > cos if), y = r sin I)J sin $ and

z = cos \|) into Eqs . (15) and (16), which yield the partial differential

equations for isotropic heterogeneous and homogeneous mat eri als, respectively.

3 t 1 3 / 2 . 3 t \ x 1 3 / . 3 t \

1 32 • ir s i n i|)

h (k s i n* f t ) +< J " " <2 2>

at _„/i t 2 3t i aft i i t \ + al A ( 2 3 )

\ 3 r r si n i|) 36 r tan \ \ i T /

The d i f f e r e n t i a l e q u a ti o n f o r n o n i s o t r o p i c m a t e r i a l s w i t h a l i g n e d

con du c t iv i ty and geom et r i c axes i s

„ 3 t k r 3 2 t , *& 82 t , *S|) 3 , l h 3t . „ , , , , , . ,p c a 7 = Trr + 2 : 2 , 7 7 + 2 . . 1+ 8 l n * i * + * • ( 2 4 )

3 r r s i n ip 3ij) r s i n V

3 . SPECIAL DIFFERENTIAL EQUATIONS

Some de fin in g e qu ati on s can have im plie d assum ptions and boundary

co nd i t io ns . They a re usua l ly employed in spe c ia l cases to s im pl i fy th e me thod

of so lu t io n . However, F ou r i e r ' s law i s the ba s i s fo r de r iv in g thes e spe c ia l

e q u a t i o n s .

3. 1 Combined Co ndu ction-C onv ection

T h in m a t e r i a l s h a v in g r e l a t i v e l y h i g h t h er m a l c o n d u c t i v i t y h a ve v e ry

s m a l l l a t e r a l t e m p e r a t u r e g r a d i e n t s . I f t h e s u r f a c e s a r e c o n v e c t i v e l y h e a t e d

or coo led , the co nve ct ion co nd i t ion becomes pa r t of a he at acc oun t ing on a

d i f fe re n t i a l e l em en t . Refe r r ing t o F i g . 3 , l e t Az be the th i ck ne ss b o f a

th in so l i d . On the s urf ace z and z + Az, the so l i d has a con vec t ion boundary

des cr ib ed by q = h ( t - t ) , where t . i s the convect ion f l u id tem pe ra tu re .

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Applying the s ame p r inc ip les used to develop th e g e n e r a l e q u a t i o n s for

r e c t a n g u l a r c o o r d i n a t e s y s t e m s s h o u l d r e s u l t in

p c f f + 2 ^ t - t f , = q " - + f c ( k f i ) + f 7 ( k g ) . ,25,

I f the geometry i s a t h i n rod of c i r cumference C, the a p p r o p r i a t e

e q u a t i o n is

pof7 + i r (t " t £ ) - q ' " + t b ( k f f ) - (26)t j c3T A

3.2 Moving he at sou rce s

The ge ne ra l he at conduct ion Eq. (15) can a l s o be used for moving heats o u r c e s , but a si m p le r q u a s i - s t e a d y - s t a t e e q u a ti o n can be de r ived by

c o o r d i n a t e t r a n s f o r m a t i o n . If the c o o r d i n a t e s are r e l a t i v e to the t r a v e l i n g

s o u r c e , the te m p e r a t u re d i s t r i b u t i o n s a p pe ar to be s t a t i o n a r y . For example ,

i f a po in t sou rce of s t r e n g t h Q i s moving at a v e l o c i t y U p a r a l l e l to the

x - a x i s , th e t r ans fo rmat ion would be x = x' + UT, where x' is the x - d i r e c t i o n

dis tance f rom the s o u r c e . By s u b s t i t u t i o n in Eq. (16) we can ob ta in

f c - ( f e - ) * f e ( - e ) * f e ( » e ) * « - £ • • > • • • - • •The app l i cab le equa t ion for a moving source in a t h i n rod t h a t i s

convec t ive ly coo led i s

M k H O * * " ? ? + * • • • - * £ ( t - v • ( 2 8 )

T he m o v in g s o u r c e s t r e n g t h i s a c c o u n t e d for in the b o u n d a r y c o n d i t i o n s

f o r a p a r t i c u l a r p ro b l em s o l u t i o n .

4 . BOUNDARY COND ITIONS

S o l u t i o n s t o h e a t c o n d u c t i o n p r o b l e m s r e q u i r e s t a t e m e n t s of c o n d i t i o n s .

F o r g e n e r a l s o l u t i o n s t h e r e m u s t be ; i v e n a t l e a s t a d e f i n i t i o n of the s o l u

t i o n r e g i o n , s u ch a s i n f i n i t e , s e m i - i n f i n i t e , q u a r t e r - i n f i n i t e , f i n i t e , e t c .

A d d i t i o n a l l y , l i m i t s can be s p e c i f i e d f o r any of t h e s e r e g i o n s .

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If specific solutions are needed, then complete conditions must be

defined. They could include , for example, i n i t i a l , in t e r na l, and surface

con dit io ns. Other conditio ns might include proper ty de f in it io ns . Whether

der iving a so lut io n or searching for e xi st in g so lu t io ns, one must decide which

co ndi ti on s are a pp li ca bl e t o the problem and how they can be s ui t ab ly

expressed.

4.1 Initial Condition

Unsteady-state problems must have an initial condition defined. Mostly,

this implies a temperature distribution at T = 0 but, also, internal or

surface conditions could have initial values. A problem solution for T > 0

depends, of course, on whatever is specified at T = 0.

4.2 Surface Conditions

The most commonly employed surface conditions in heat conduction problems

are prescribed surface convection, temperature, heat flux, or radiation. It is

even acceptable to prescribe two of these for the same s urfa ce, such as com

bined radiation and convection. Other surface conditions could include phase

change, ablation, chemical reactions, or mas s transfer from a porous solid.

4.2.1 Convectio n boundary

Conduction and convection heat transfer rates on a surface are equated to

satisfy continuity of heat flux according to Newton's law:

3t G'H T)-k ^ — = h [ t ( x b , T) - t f ] , (29)

3x

where t is the temperature of the convection fluid, and x. is the boundary

location. The convectio n coefficient h must be determined from suitable

sources that give predicted values satisfying the conditions of the fluid.

(Some correlations of convection coefficie nts are given in Section 16.) The

method for defining h can vary depending on the type of convection or the

methods prescribed by those researchers who have supplied valu es for the

coefficient. However, h is usually define d as

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TABLE 1. Sample co nv ect ion co ef fi c ie n t v a lu e s .

F lu id Condi t ion h, W/nT-K

Air F ree convec t ion on v e r t i c a l p l a t e s

Air Forced conv ect ion on p la te s

Air Forced f low in tu be sSteam Forc ed flow in tu be s

Oil Forced f low in tub es

Water Forced f low in tu be s

Water Nu clea te bo i l in g

Liquid helium Nu cleate bo i l in g

Steam Film con den sation

Liquid me ta l Forced f low in tu be s

Steam Dropwise con den sati onWater Forced convect ion b o i l in g

10

100

200300

500

2,000

5,000

8 ,000

10,000

20,000

50,000100,000

h = t ( x b , x) - t f ' (30)

where t can be giv en as

t f = e[t (V T, - t j + t m ( (31)

and where 3 £ 1, and t^ is the temperature outside the the rmal boundary layer

of the fluid. In this respect, one mu st take care to use the proper fluid

temperature and convection coefficient.

Some typical order-of-magnitude values for the convection coefficient h

are given in Table 1.

4.2.2 Surface temperature

Of all boundary conditions, this is probably the simplest in a

mathematical sen se. It can be variable or constant with respect to position

and time. In the real sense, it is very difficult to achieve a prescribed

surface temperature, but it can be closely approached by imposing a relatively

high convection r ate.

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4. 2 . 3 Heat f lux

Four ier ' s law def ines the f lux on a boundary by

3 t ( x , T )

q = -k — ^ • (32)

An ad ia ba t ic su rfa ce can be def ined by e i t h e r se t t in g q in Eq. (32) or h in

Eq . (29) to ze r o . In ve r s e ly , if t he s o l i d ' s t empera tu re d i s t r ib u t i o n has been

so lve d, then Eq. (32) can be used to determ ine the sur fac e he at f l ux .

Un ders t andab ly, hea t f lux can be , in p a r t i c u la r c ase s , t ime and pos i t io n

dependen t .

4 .2 .4 Thermal r a d i a t io n

Heat t ra ns fe r f rom an opaque su rfa ce by ra di a t io n can be expre ssed a s

r , .- l 3 t ( X . , T )

a j r [ r \ v T ) - T J = - k — ^ , (33)

where a i s the Stefan-Bol tzmann r a d ia t i o n co n st an t , <#"is the combined

c o n f i g u r a t i o n - e m i s s i v i t y f a c t o r fo r m u i l t i p l e - s u r f a c e r a d i a t i o n e x ch a ng e , and

T is the s ink or sou rce temp era ture for ra d ia t i o n . Because Eq. (33) i s a

n o n l i n e a r e x p r e s s i o n , i t i s f r e q u e n t l y d i f f i c u l t t o f i n d e x a c t s o l u t i o n s t o

problems wi th th i s co nd i t io n .

A common method for dea l ing wi th r ad ia t i o n problems is to t r e a t the

ra d ia t i o n boundary as a convec t ion bou ndary . According to Eq. (29) we can

w r i t e

3 t( x , T)

~k

— 3 * = h

r [t (

V * ~ fc

s] • ( 3 4

>where

h r = OJT[-T{xb, x ) + T ] [ r2 ( x b , T) + T^J

Using th is method means th a t T(x . , T) must f i r s t be es t im ate d in order to

compute a va lue of h . After a va lu e fo r T(x. , T) has been computed from

the problem s o lu t i o n , then the es t im ate d value for h can be improved.

T h i s , o f c o u r s e , b eco me s an i t e r a t i v e p r o c e s s .

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4 . 3 I n t e r f a c e C o n d i t i o n s

4 .3 .1 Con tac t

Two contacting soli ds, either similar of dissimilar, will almost always

have some interface thermal resistance to heat flow between the m. Themagnitude of this resistance can depend greatly on the condition of the two

contacting surf aces . Properties that can effect the surface condition include

cleanliness, roughn ess, waviness, yield strengt h, contact pres sure, and the

thermal conductivities of the solids and interstitual fluid. Since there

are so many influences on the contact thermal.resistance, it is difficult to

theoretically pred ict its value. Consequ ently, experimental results are

frequently used. Som e representative valu es of the inverse thermal contact

resistance, commonly referred to as the thermal contact coeff icient, are givenin Section 17.

The generally accepted definition of the contact coefficient is

h c = A E . ' i 3 5 )

w he re q i s t h e s t e a d y h e a t f l u x c o r r e s p o n d i n g t o a f i c t i t i o u s i n t e r f a c e

tempera tu re d rop o f At . de f ined by ex t r ap o l a t in g the v i r tu a l l i ne a r

t e m p e r a tu r e g r a d i e n t i n e ac h s o l i d t o t h e c o n t a c t c e n t e r l i n e . T h i s t e m p e r a tu r e

drop , which i s i l l u s t r a t e d in F ig . 5 , wou ld d imin i sh to ze ro i f t he in t e r fac es

were in pe r fe c t co n t ac t .

4. 3. 2 Phase change

Other in t e r f ac e c ond i t ions inc lud e tho se caused by endo the rmic r ea c t io ns

s u c h a s m e l t i n g , s o l i d i f i c a t i o n , s u b l i m a t i o n , v a p o r i z a t i o n , an d c h em i ca l

d i s s o c i a t i o n .A s t a t eme nt o f an in t e r f ac e r e ac t io n con d i t io n de f in es th e d i ff e r enc e o f

hea t f lux ac ross the in te r f ac e . I f t he in t e r f ac e which se pa ra te s two phases

o t n m at er ia l i s lo ca te d a t x = x . , the h ea t ba lance for a pha se change

react ion i s g iven in the form

k 2 3x " k l 3x " 2 dx * t 3 6 ]

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FIG. 5. Illustration of interface contact between solid s.

where y is the latent heat or chemical heat ca pacity, and subscripts 1 and 2

refer to the two phas es.

5.0 Solutions

5.1 Extending solutions

A solution can be retrieved after identifying a problem by boundary

conditions, geometry, and other pertinent data. Usually, a temperature

solution is given, but heat flow can be derived from the temperature

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d i s t r i b u t i o n by u s i n g F o u r i e r ' s l aw, i . e . E q . ( 2 ) . I f c u m u l a t i v e h e a t flo w i s

req u i r ed , a t ime and su r fa ce in te g r a t io n o f loc a l hea t f lux i s nec essa ry.

- / . £ * & * « - •where n i s the d i r e c t i o n normal to the sur fa ce s .

S t e a d y - s t a t e s o l u t i o n s c an be c o n s i d e r e d a s t h e i n f i n i t e - t i m e c o n d i t i o n f o r

un s te ad y- s t a t e s o lu t i o n s . Tha t i s , p rob lems which have a t ime -asy mp to t i c

s o l u t i o n e x h i b i t s t e a d y - s t a t e s o l u t i o n s f o r x •> °°. T h u s, s t e a d y - s t a t e s o l u t i o n s

can be de r ived from t r an s i en t so lu t io ns .

A s teady su rfa ce temp era ture c on di t ion can be impl ied from a conv ect ion

boundary co nd i t io n . For h + °°, the su rfa ce tem pera tu re appro ache s the f l u i d

tem pera tu re . Th ere fo r e , a so lu t io n which inc lud es a convec t ion boundary can bet rans fo rmed in t o a co ns tan t t empera tu re boundary so lu t io n by so lv in g fo r the

i m p l i e d l i m i t i n g c a s e .

5 .2 Dimens ion less pa ram ete r s

Group ing p a r t i c u la r v a r i ab les y ie l ds d im ens ion less numbers th a t can be

u se fu l . Sy mb ol ica l ly, they can sho r t en an eq ua t io na l ex pre ss i on . Bu t , t hey can

a l s o g ive ins igh t to the behav io r o f hea t t r ans f . e r in a p a r t i c u la r p rob lem.One very us ef ul p aram eter is th e B io t number, Bi = hj , /k, which re s u lt s from

conv ec t ion boundary co nd i t i on s . Th i s pa ram ete r i s p r op or t io na l to the r a t i o o f

the conduc t ion re s i s t an ce to the convec t ion re s i s t a n c e . Thus , we cou ld say th a t

for

Bi > 1, conduction is highest resistance to heat transfer ,

Bi < 1, convection is highest resistance to heat transfe r,

Bi « 1, the solid behaves like k = ™.2

Another dimensionless parameter is the Fourier number, Fo = ai/l , which

is found in cransient solutions. This number is a dimensionless time value,

but it is also considered an indicator of the degree of thermal penetration2

into a solid. Since ax/SI = (kxAt/U)/(pc£At) , it is proportional to the

ratio of conduction heat transferred to thermal capacity. Thus , an increasing

Fo value implies approaching thermal equilibrium.

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The product of Bi and Fo numbers yields the parameter Bf « hx/pdf. which

occurs in transient problems having a convection boundary. This is also a

dimensionless time parameter, but it is based on convection heat transfer

instead of conduction ?s in the Fourier number.

Solutions to problems having an internal heat source q ' " usually have a

dimensionless heating parameter called the Pomerantsev modulus

Po = q"''2, 2/kAt. This number is a ratio of internal heating to heat

conduct ion rates. Large values of Pr> imply large temperature differences will

occur in the solid.

The parameter Fo = X,/2 rr is a form of the reciprocal of the Fourier

number and occurs in many solutions for transient temperatures in

semi-infinite solids.

When time dependent boundary conditions have a time constant, the

solution will frequently include a dimensionless group called the

Predvoditelev modulus , Pd = bi /d, where b is the inverse time constant.

Sma ll values of Pd imply a slow changing cond ition . It signifies the ratio of

the change rate of the boundary condition to the change rate of the solid

temperature.

5.3 Example Problems

5.3.1 Steady heat-transfer in a pipe wall

Hot water flows at 0.5 m/s in a 2.5 cm i.d., 2.66 cm o.d. smooth copper

pipe . The pipe is horizontal in still air and covered with a 1-cm layer of

polystyrene foam insula tion. For a 65°C water temperature and 20°C air

temperature, estimate the heat loss rate per unit length. The solution given

in case 2.1.2 is

2n (t^ - t 2 )q =

1 „ r 2 . 1 . r 3 ^ 1 _, 1r— In — + 7 — An — + — r - + — r -k l E l k 2 r 2 r l n l r 3 h 3

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From t h e p r o b le m d e s c r i p t i o n

fclfc4

o= 65 C

= 20°C

r 1 = 1. 25 cm

r

2= 1 .33 cm

*-> = 2 .33 cm

k = 400 W/m" C ( f rom T a b le 18 .1)

k 2 = 0 .0 38 W/m-°C ( f rom Ta b l e 1 8 .2 )

h d / k = 0 . 0 1 55 P r ° "5 R e ° " 9 ( fr o m S e c t . 1 6 . 1 )

h l = ( k w a t e r /2 r l > ( ° - 0 1 5 5 ^ ° ' 5 ^ ° ' 9 )k u = 0 . 65 9 W/m-°C (a t 65°C)

w a t e r v

Pr = 2 .73

R e = 2 p v r l / ] ip = 980 kg/m

v = 0 .5 m /s

U = 4 . 3 x 10 kg /m "s

2 ( 9 8 0 ) ( 0 . 5 ) ( 0 . 0 1 2 5 )e =4.3 x 10

- 4= 28 488

( 0 . 6 5 9 / 0 . 0 2 5 ) ( 0 . 0 1 5 5 ) ( 2 . 7 3 ) ° "5 ( 2 8 4 8 8 )0 , 9 = 6895 W/(m'2 o,'Oh3 " ' W 2 ^ 1 C ( G r d P r >k

a i r = Q

"0 2 5 w

/m

" °c

P r = 0 . 7 1

G r = g B ( t ,

( fr om S e c t . 1 6 . 8 )

t 3 ) ( 2 r 3 ) 3 / \ ) 2

g - 9 . 8 V s

B = 1 /T4 = 1 /293 K- 1

V = 1 6 . 5 5 x l O- 6 m 2 / s

= ( 9 . 8 ) ( 0 . 0 4 6 6 ) ( 0 . 7 1 ) = ^ ( _ (

( 29 3) ( 1 6 . 5 5 x 1 0 "b r 4

C = 1 . 14 , m = 1 /7 ( f rom Ta b l e 1 6 .3 )

h 3 = ( 0 . 0 2 5 /0 . 0 4 6 6 ) ( 1 . 1 4 ) ( 8 7 7 4 )1 / 7 = 2 . 2 4 W / m2 ' °C

1°C)

q =400 i n IS) 0 . 0 3 8 In

2ir (65-20

( 0 . 0 1 2 5 ) ( 6 8 9 5 ) ( 0 . 2 3 3 ) ( 2 . 2 4 )

2w(45)

1 .5 5 x 10 + 14 .76 + 0 . 01 2 + 19 .16= 8 .3 3 W/m

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2 i r r 3 h 3

8 .33271(0.0233) (2 .24)

= 25 .4"C

U s i n g t h i s new e s t i m a t e o f ( t - t ) , we c a n r e c a l c u l a t e h .

h 3 = 2 .2 4 ( 2 5 . 4 )1 / 7 = 3 . 5 6 W / m2 ' ° C ,

q = 10 .5 4 W/m.

Additional i terat ions on h would little improv e this result.

Note that the copper tube and water film have a small effect on the

results because they present little resistance t o heat transfer b y comparison

to t h e insulation and ai r film.

5.3.2 Transient heat conduct ion in a slab

A billet of 304 s tainless steel measuring 2 x 2 x 0.1 m thick and

having a uniform temperature of 30 C is heated by sudden immer sion into a

45 0 C molten salt bath. T h e mean convection coeff icien t i s 350 W/ m " C .

Determine the time required for the center tempera ture of the b i l le t to reach

4 0 0 ° C . T h e solution is found in Section 8.1 (case 8.1.7 and F ig . 8 .4a):

400 450t.

l

30 450 0.119

From Table 18.1

k = 21 W/m*° C #

-6 2 ,a = 7 x 10 m / s ,

k_ = 21hi, (350) (0.05 )

1.2 .

-h.t.

From Fig. 8.4a

„2QLT/i, 3.4,

3.4Jt 2 _ ( 3 . 4 H 0 . 0 5 ) 2

7 x 10 61214 s

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5.3.3 Trans ient heat conduction in a semi-infinite plate

For the conditions given in 5.3.2, find the billet center temperature at

0.05 m from the edges and sides of the billet.

The solution is found in case 7.1.21 and Fig. 9.4a for a semi-infinite

plate and expressed as T = (P)(Fo)'S(X). Values of P(Fo) are given in Fig. 8.4

and values of S(X) are given in Fig. 7.2.

hVaxBiVfo

Fo

o\l -6.350\(7 x 10 ) (1214)

21 1.54V V

2Vaf r

0 05

2V(7 x 10 6 ) (1214)

0.27

If (T = (t - t £ ) / ( t . - t f ) ,

S(X) = 1 - 0.45 = 0.55 (from Fig. 7.2).

The P(Fo) value is given in 5.3.2. Thus , the solution is

I

t ~ t £ P(Fo) S(X) = (0.119)(0.55) = 0.066

t = (0.066)(30 - 450) + 450 = 423°C

5.3.4 Extended surface steady-state heat transfer

A 160 °C uniform-temperature copper plate has a long rectangular rib

brazed to it. All surfaces are convectively cooled by 30 C air having a

convection coefficient of 53 W/(m • C ) . The rib is yell ow brass extending

4 cm from the flat surface and 2 cm wid e. Estimate the additional heat loss

from the flat surface caused by the rib.

The solution for temperature distribution in the rib is given in

Case 1.1.17 for f(x) = t., w = t and 1 = a.

V \ T

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* - * £~ Bi cos (Xn X ) xn c o s h [ Xn ( B - Y ) l + B i s i n h [ Xn ( B - Y) l I

1 " tf £-\ co s (X ) ( B i2 + X2 + B i) fx co sh (X B) + Bi s in h (X B) ln= l n n L n n n J

X t a n (X ) = B i ( c h a r a c t e r i s t i c e q u a t i o n )

B = b /a = 4 /1 = 4X = x / a , Y = y / a

B i = -r2- = (5 3 > ( Q -0 1 ) = o .o o4 (k va lu e f rom Ta b le 1 8 . 1 )k 130

From Ta b le 1 4 .1

\ x = 0 . 0 6 3 2 , X2 = 3 . 1 4 2 9 , X3 = 6 . 2 8 3 8 , X4 = 9 . 4 2 5 2 , X5 = 1 2 . 5 6 6 7 .

H e a t l o s s f or m t h e c o n v e c t i v e l y c o o l e d s u r f a c e s i s d e t e r m i n e d b y E q . 3 2

ap p l i ed to th e boundary y = 0 .

_ , . f 3 t ( x , o )q = 2k / — dxo 3y

°° Bi ta n (X )[X si n h (X B) + B i co sh (X B )]= 4k (t . - tf ) £ " " S «—

n= l (B i + X'' + B i) [X c os h (X B)+ Bi s in h (X B) ]n n n n

= 666 W/m

Heat loss with out the rib attached w ould be

q = hA (t± - t f ) = 53 (0.02)(160 - 30) = 138 W/m .

The additional heat loss is thus

Aq = 666 - 138 = 528 W/m ,

5.3.5 Rectang ular fin heat transfer

Use the straight rectangular fin solution to estimate heat loss from the

rib described in 5.3.4.

The solution is found in Case 5.1.4 and Fig. 5.2.

/ —Vka / 5 3

= 6 .385 m 1/ —Vka V ( 1 3 0 ) ( 0 . 0 1 ) = 6 .385 m

1 =c

0.1 54 + 0 .0 1 = 0 .0 5m

mAc = 0 3193

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5 . 3 . 6 S e m i - i n f i n i t e p l a t e h e a t t r a n s f e r

Find an eq ua t ion for the hea t t r an sf er ra te throu gh the edge of the sem i-

in f i n i t e p la t e d esc r ibed in case 1 .1 .5 wi th f (x ) = t .

Using the give n temp eratu re s o lu ti o n and Eqs. (2) and (37) we can f in d

the heat t r an sf er in the fo l low ing m anner :

S = ~k I ? = I ( t

2 V S ' w h e r e T = T7=\

H | Y = 0 = -2 ]£ s in (m tX )[l - cos (nit)]

n=l

00

Q

Y | Y = 0 = l l q

Y | Y = 0 ^ = 2 k ( t 2 ~ V / S S i n ^ ^ C

1 _ c o s (

n 1 ,

) ]d x

n=l

GO 00

= 2 k ( t2 - t x ) 2L E l - c o s ( m r ) 32 = f * ( t2 - t > 2 , i , n = 1, 3, 5,n=l mr n= l

22

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Ifin»a.

R E P R O D U C E D F R O MBEST AVAILABLE COPY

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Section 1.1. Solids Bounded by Plane Surfa ces— No Internal Heat ing.

Case No. References Description Solution

1.1.1 19, Convectively heated andp. 3-103 cooled plate.

W Jy t 2

q =" i ( t i - V

Bi x + 1 + i\/^ 2)

t - t. B ^ + 1

fc 2 " *1 ~ i l 1 + ( W

^ 1.1.2 19 , The composite p late,p. 3-103

( t 0 " Vn

£ (W./ki + 1/h^ + l /h Q

k , h_4 h n- 2 h n-1

V h H

l /»<«*^>v">^

k- U-\-* n

i-1

Temp in the j th la ye r:

j - lS (S,./k i + 1/ IK ) + (Xj/kj) + ( l/ h 0 )

^ («,./k i + 1/h.) + (l /h Q )

i=l

, J > 1

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Section 1.1. Solids Bounded by Plane Sur fac es— No Internal Heating.

Case No. References Description S o l u t i o n

1 .1 .3 1 , p . 138 P la te wi th tem pera tu red e p e n d e n t c o n d u c t i v i t y,k = k^ + S ( t - tx).

M

q = Kfci - to

1 v 2m J,

t - t1 B \ 1+ 2 e k

m( t

2 - V f - X

k = (k. + k_) /2in 1 2

i H . 4 2, p . 221 Porous plate with internalfluid flow,t = t x , x = 0.

t = t_ , x = 6.

P a P o r o s i t y.

u . t 0 , k , , p , . c

P = porosity

V P ~-*2

t ~ t,

tt

r - ^ - = expl"- E 5 (1 - x/6)"l , 0 < x < 62 "O

t - t 0 e x p ( 5f x )

t 2 - t l exppp6 - 1]

Mean temp:

, -<*> < x < 0

^S— = ^ [1 - e*P< V ] ' ° 1 X i 6

c„ =p f u c

' 5- =p f u c

' p k p ( l - P) ' •'f k f (1 - P)

(See P ig . 1.1)

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Case No . References Description Solution

1.1.5 2, p. 122 Semi-Infinite plate.9 , p . 164 t = t x , x = 0 , A, y >. 0.

t - f (x ), 0 > x > A, y - 0. t - t = 2 2^ exp (-im Y) s i n (rarX)

n=l

ix J l"f (X) - t i " | s i n (mrX) dX

For f (x ) = t 2 :

V t = f<x)- — = — / — exp (-m rY) si n (mr X)f l - co s (nir)"]t a — t 1 IT fc-» n — -1

n= l

1 . 1 . 6 3 , p . 2 50 R e c t a n g u l a r s e m i - i n f i n i t e r o d .t = t , x « 0.t = t _ on o t h e r s u r f a c e s .

2 "1 "2 - 2 2on co m+n

(-1) exp • ( x n x / w )2 + am x / f c 2

n*0 m=0X Xn m

x co s (X ) co s (X )n m

X = (2n + \) \ , \ = (2m + l ) fn 2 m *•

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Section 1.1. Solids Bounded by Plane Surf ace s—N o Internal Heating.

Case No . References Description S o l u t i o n

1 . 1 . 7 2, p . 130 R e c t a n g u l a r i n f i n i t e r o d .t = F ( x ) , 0 < x < I, y =

t = F ( x ) , 0 < x < I, y = w.

i

t m h + fc n + 'in + fciv

t = G ( x ) , x = I, 0 < y < w

c = r- \x.) , u •> x *,, y = w. ^t » G (x ) , x = 0 , 0 < y < w. fc I = 2 Z

1 n=l

^ . sinh [ ^ ( l - Y)l

s lS S/ff s i n <n X L> f *t W s i n <n 7 r x>d x

t = G 1 W > -

t = F, ( x )

1

t F 2(x)

s i n h ( m r / L ) s i n ( r f l , X L > / F 2 ( X > s i n ( m r X ) d X

n = l

- t G 2W) n = l

= 2 J^j2_LniLii_z_xLL s i n ( f f l r y ) , G < i ( y ) s i n ( n i r y ) d y

"IV " ^ s i n h (niTL)n = l

L = %/M

ii

) f G l (

J0

*-h ,f"4f(i--)1 . (m ix ) ,2 1 nc o sh I

F x ( x ) = F 2 ( x ) = t 2 , G 2 ( y ) = G x ( y ) = t

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1 . 1 . 8 2 , p . 1 47 T h i n r e c t a n g u l a r p l a t e .

t = t Q , x = 0, 0 < y < x .

t « t , 0 < x < 4 , y = 0 .

t » t , 0 < x < %, y = w.

t - G (y ), x - A, 0 < y < w.

h i r t 0 a t z « 6.

h 2 , t 0 a t z » 0 .

y h v t 0 to

im

t = G(y)

*o h 2 ' *o «•

t - t Q = 2 ISn= l

nh U + n W ) * x J s . n m

s i nh |_(Bi + n IT L ) \ |

f [G<Y) - tQ ] sin (MTY) dY

Bi - (h + h2 ) i / k 6 , L = l/v

1 . 1 . 9 4 , p . 4 1 I n f i n i t e r e c t a n g u l a r r o d i na s e m i - i n f i n i t e s o l i d .

/7777W77777777777777777777777

h- fco

n 1 „i L r 3 . S ( d + K / h ) 1R ~ k ( 5 . 7 + f _ ) ta[.0.25b0.75 J

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Section 1.1. Solids Bounded by Plane Surfa ces —No Internal Heating.


1.1.10 4, p. 41 Infinitely long thin plate Vertical plate :in a semi-infinite solid. , ,. . .

k ( t i - V d

n t

Q * 7 - — T O T ' o .5 < - 

1.1.11 4, p. 43 Thin rectangular plate on the kwn(t - t )surface of a semi-infinite Q =solid. *(%)

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1.1.12 4, p . 44 Thin rectangular plate in an 2irwk(t. - t_)infinite solid . Q «- —

In (1)

V 1.1.13 5, p. 54 Rectangular parallelpiped*"* with wall thickn ess of 6. Q t = [ $ ( d w + db + wb) + 2.16{d + w + b) + 1.261 (t - t

= total heat flow through six walls

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1.1.14 4, p. 37 Infinite hollow square rod.

T lW I

q »2TTk(t 2 - t x)

J S _ + i n i^>8w + J k _h l C 0 2 r 0 2 h 2 W

, — h 2 , t 2

1.1.15 9, p. 166 Rectangular infinite rod.t - f ( x ) , 0 < x < a, y = 0.

t - t = V A n s in (mrx) si nh [{1 - Y) ( ^f ) 1 cose ch 0 ^ \

n=l

1

A = 2 J [f(X) - O s i n (miX)dX , L = Jt/w

Fo r: f(X) = t 2"

t - t

tprr = J 2 K s i n <n irx) s i n h [( 1 Y) ft)] o s e c h ft)'n=l n = 1. 3, 5, 7

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1.1.16 9, p. 167 Rectangular infinite rod.t = f ( x ) , 0 < x < H, y = 0.

q = 0, 0 < x < H, y = w.

a = 0, x = 0, 0 < y < w.

~n- n , t f

t = f(x)

^ , M a i2 + X 2 J c o s {X X) c o s h ["(1 - Y)WX 1

^ 2 n = l [ (B i 2 + Xn ) + B i] c o s h (V"

x / Tf(X ) - tf ] cos (XnX)dX

X ta n (X ) = Bi , Bi = hSL/k , W = w/X.n n

F o r : f ( x ) = t

fc- *ffcl" fcf

= 2 Bi 1n = l

co s ( XnX) cosh [ ( 1 - 5 f )wXn]

K * >i) B l co s (X ) co sh (X W)n n

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Section 1.1. Solids Bounded by Plane Surf aces —No Internal Heat ing.


1.1.17 9, p. 168 Case 1.1.16 with q = 0,

y = w, 0 < x < fc i s re pl ac e dby convection boundary h , t - .

J v ( B i 2 + X 2 ) co s (X X){X co sh Tx (W - y)~| + Bi s inh Tx (W - Y)1}. _ . - T > \ , " ' n n L n J *- n ^_

f , i f B i 2 + X 2 J+ Bil{ X cosh (X W) + Bi s in h (X W)}n=l L\ n / J n n n

JL • • '0a

I [f(X) - t f~j cos (XnX)<

X tan (X ) = Bi , Bi = hA/k , W = w/X.n n

For: f(x) = t , :

t - t_ J . Bi cos (XX) {X cos h fX (W - y) ] + Bi si nh [\ (W - Y )l }t _ _ x n n *- n _ *- n -*fcl ~ fc f , cos (X )| ( B i 2 + X 2 J + Bi lr X cosh (X W) + Bi si nh (X W)l

n= l n LA n / j L n n n J

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Section 1.1. Solids Bounded by Plane Su rfa ce s—N o Internal Heating.


1.1.18 9, p. 168 Case 1.1.16 with t = t f ,

y = w, 0 < x < it.

J , ( B i 2 + X 2 ) co s (X X) s in h fX (W - Y)"| /-1

t - t . = 2 > ^ 1-7 - ^ =i — ~ x I Tf (X) - t . 1 co s (X X)dX

' n=l L (B i + Xn ) + B i] S i n h < V > l l ^X tan (X ) = Bi , Bi = hA /k, W = v/ln n

For : f (x ) = t :

i

M t - t . __ co s (X X) s in h TX (W - Yf|t__ __ . X n L n J: x. = 2 B i y n L n

fcl " fcf ^ Y B i2 + \ 2 \ + B i l c os (X ) s in= l LV n / J n

nh (X W)n

1.1.19 9, p. 169 Case 1.1.16 witht = t x , y = 0, 0 < x < % .

t = t , y = w, 0 < x < A.

t - t

i^k »lcos (XnX ) { s i n h [ *n (W - Y) ] - s i n h (XnY ) ( t 2 - t f ) / ( t x - t f ) }

j f B i + X2 j + Bi co s (X ) s i nh (X Wf . | [ B i + X_ ) + B i ] c o s (X_) s in h (X_W)

X ta n (X ) = Bi , W = w/SL

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Section 1.1. Solids Bounded by Plane Sur fa ces —N o Internal Heating.


t - t„1.1.20 9, p. 178 Rectangular parallelpiped.

t = t r x = 0 r 0 < y < w , " " U0 _ 1 6 V V [s in h (L - LX) + T si nh (LX)] sin (CTY ) s in (nnrZ). . ^ , t , - t„ = _ 2 Z - Z , (nm) si nh (L)

n=l m=l

n = 1, 3, 5,

0 < z < d "0 TT

t = t 2 , x = i , 0 < y < w ,

0 < z < d .

Remaining surfaces at t n .

(run) sinh (L)

, m = 1, 3, 5,

L 2 = ( rnrVw) 2 + ( imrVd) 2 , z = z/d

i - 1

i

*S 1

c—-r^c—-r^0 / \ X

h e

T = ( t2 - t 0 ) / ( t l - t 0 )

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Section 1.1. Solids Bounded by Plane Sur fa ces —No Internal Heating.


1.1.21 9, p. 179 Rectangular parallelpiped.t = t y , x = 0 ,

-w < y < +w , -d < z < +d .

t = t 2 , x = i. ,

-w < y < +w f -d < z < +d .

Remaining surfaces convection boundary with h,t, .

00 COt - -fc. - , * — * - > Tsinh (L - X) + T sinh (LX)] cos (X Y) cos (B Z)

— = 4 Bl D > > j--= 5 =y-z = z rc l ' f , . c os (X ) co s (B ) U + B i + Bi [B + Bi D + Bi D ls in h (L)

n = i m=± n m |_ n Jv in /

X ta n (X ) = B i , B ta n (B ) = Bi D, Bi = hw/kn n m m

L 2 = x V / w 2 + Bm J l 2 / d 2 , D = d/w , X = x/SL, Y = y/ w

Z = z /d , T = ( t - t ) / ( t - t )

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Section 1.1. solids Bounded by Plane Surfaces—N o Internal Heating.


1.1.22 9, p. 180 Case 1.1.21 except t = t ,x = 0 , - w < y < + w ,-c < z < +c .Remaining faces arecorrection boundarieswith h,t_.

t - t _ „ [AB i sinh (L - LX) + L cosh (L - LX Q cos (X Y) cos (6 Z)

t x - t f ~ B l D 2-> 2* ~ [A Bi sinh (L) + L cosh (L)] NM cos (X ) cos (f$ m)

n*l m=l

A = l/v , N = X 2 + B i 2 + Bi , M " S 2 + Bi 2 C 2 + BiC.n m

X , fi , L , D , X , Y , Z, and Bi are defined inn in

case 1.1.21 .

1.1.23 2, p. 175 Infinite plate with Case 2.2.10cylindrical heat source.

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1.1.24 9, p. 428 Infinite strip withstepped temp boundary,t - t 0 . y « 0 .-oo < x < + 0 0 .

t = t l f y = w , x > 0 .t = t 2 , y = w , x < 0 .

k " k 1 , 0 < y < w , x < 0

k » k _ , 0 < y < w , x > 0

s in*2

L X iV ^

fc- fco= y 2 [<V V - l]k2 f (-1)"

* ( k l + V n=l n '

t _iiio_ T l 21>- <yy>i y (-n°t 2 - t x II ( k r + k 2 ) ^ n

s i n (mtY)exp(-mrX ) , X > 0

s i n (mnf)exp(nirX) , X < 0

X = x/w , Y = y/w

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1.1.25 9, p. 452 Heated planes on asemi-infinite medium.t = t , x < - & , y = 0 .

t = t 2 , x ^ + A , y = 0 .

q = 0 , -i. < x < +1 ,

y - 0 .

Q = — cosh *Gk t 2 ) , -x x < x < -i.

x i • y l 2

i+e

1.1.26 9, p. 453 Heated parallel planes inan infinite medium.t » t , x > _ 0 , y « s .

t - t, , -<*• < x + « .

Y

Heat flow from bottom side of semi-infinite pla ne:

Q = k Rxj/s ) + U A ) ] ( t L - t 2 ) , 0 < x < x x .

Heat flow from to p sid e of sem i- in fi ni te plant

Q = £ Jin[(xrxj/s) + l j i ^ - t 2 ) , 0 < x < x x .

H 1 h—

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Section 1.1. Solids Bounded by Plane Surfaces —No Internal Heating.


1.1.27 9, p. 454 Infinite right-angle corner.t « t j , x > 0 , y = 0 .

t « t 1 , x = 0 , y > 0 .

t = t 2 , x > v 1 , y = w 2 .

t • t 2 , x • w x , y > w .

Q = k w 2 W ; L T T W2 \ « J T, \ 4 W l w 2 /

f ^ ' t a n _ 1 ( ^ ) j ( t i " V ' ° * x * x i ' ° K Y * y iX I * 1

I r

w 2 I *y

. w l 4-

J4

For w x = w 2 = w , x 1 = yj = x:

i * 0 0 Q = k{2(I<x/w) - 1 ] + 0.559} ( t x - t 2 )

1.1.28 9, p . 462 A wedge with s tepped surfacetemp. t - t x ± x ./*" sin X[i,n(r 0/r)] cc

t 2 - t L

= I + it J Q A cosh (A6 Q )cosh(X6)dX

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Section 1.1. Solids Bounded by Plane S urf aces —No internal Heating.


1.1.29 15 Semi-infinite strip withconvection boundary.t = t , x « 0 , w , y =

Convection boundary at

y = 0 .y

exp(-mrY) s in (mrx)2 2rnr + (n it /Bi)

1, 3, 5,f 1 n=l

Heat transfer into strip at y - 0:

CO

ki t - t 1 = F " 2 ' n = 1 ' 3 ' 5 ' " *M r l V " n=l n + (nîr/Bi)

See Tables 1.2a and 1.2b.

1.1.30 8827

Periodic strip heatedpla te q^Wjy) = 0 ,

on 2b wide strips.t(w ,y) •» t , on (2a - b) wide

strips spaced 2a on centers.

t f ,h-

See Fig. 1.6a and b for values of maximum differenceson y = 0 surface, i.e. Ct(0,a)and heat transfer.

t<0,0)] = A t m ,

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Section 1.1. Solids Bounded by Plane Sur face s—No Internal Heating.


1.1.31 28 Case 1.1.30 exceptQ = surface heat fluxon (2a - b) wide stripsspaced 2a on centers.

tfx, y) - t

QaH - = | j - (1 + BiX) + 2

%_T

Bisin (mrB) cos (mry) cosh (nirx) + — sinh (mrX)J

n=l .»[,inh (mrw) + - ^ cosh (mrW)

B = b/a , Bi = ha/k , W = w/a , X = x/a , Y = y/a .See Fig. 1.7 for values of T = t(0 , a) - t(0 , 0) .

1.1.32 29 Spot insulated infiniteplate with constant tempon one face and convectionboundary with an insulatingspot on the other.

See Fig. 1.9 for values of 6(r/w,z/w) =

Tt(r , z) - t f J/rt(» , w) - t f"j at r = 0 , z = w .

M«MZLW/A I

\* T

"7"w

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Case No . References Descr iption Solution

1.1.33 32 Infinite plate containingan insulating strip.t « t 1 , - a > < x < ° 0 , y = w .

t - t f l , -«" < x < «> , y - -w

0 f 0 < x < S , r y = 0 .sm

IO

t ( x ' y» - fto 1 1 - i— rr. — - 1 + w c o s

{[p2

+ G 2

+ l + V ( F2 F

+ G 2

+ l , 2 - 4 F 2 j 4f V

2yv p99WW9v9 • •

2 ( l + c o sh 1> (2X - L)H c o s r 2Tr Y) ]}1 + co s h (TTL)

2 s i n h [ i t (2X - L) ] s in (27TY)1 + co sh (TIL)

F o r L •*• < • :

F = 2 c o s (2mf)exp(-2TTX) - 1 , G = 2 s i n ( 2 l tY ) e xp< -2 l tX ) - l ,

X • x/w , Y - y /w , L - «,/wI n f i n i t e t h i n p l a t e w i t h t - t^, K0(Br/<S).1 .34 1 9 ,

p . 3 - 11 0 h e a t e d c i r c u l a r h o l e .t " fcl ' c " c l

fc3 " «*» VB r l / 6 ) , r > r ,

" r ' l

cf

B = V B i j + B i j , t r o = ( t x + H t 2 ) / ( 1 + H ) , H = B i ^ B i j

h 2 , t 2

• h

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Sec t ion 1 .1 . So l id s Bounded by Plane Surfaces—No In tern a l Heat ing .

Case No. R eferen ces Desc r ip t i on So l u t i o n

1 .1 . 35 19 ,p . 3-110

Case 1.1 .34 with treplaced by a heatsource of strength q.

k 6 ( t - t j K0(Br /6)q = 2Tr{Bri/ (5) K^ Brj /6 ) ' r > r

x

B and t defined in case 1.1.34 .

1.1.36 1 9 ,p . 3-111

Case 1.1.34 with h = 0. t - t. I„ (Br/6)t 3 - t x 1^8^/6) ' t > r 1

B given in case 1.1.34 .

1.1.37 60 Infinite plate with wall19 , cuts as shown. Heat flowp . 3-123 normal to cuts,

q

See Table 1.3 for conductance data K/K

K ,. = ka/X.uncut

Q = K t l - t 2

uncut

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1.1.37.1 84 Infinite medium withsingle and multipleinsulating cuts.

See Ref. 84 for temperature and heat flow solutions.

1.1.38 61

19.p. 3-124

Case 1.1.37, cut (c),

with only one cut.

Conductance:

[<«, - d)/a ] + (d/b) + ( 4 / T I ) Sin {sec (it/2) CI - <b/a)]f

Q = K t.

1.1.39 6219,

Infinite rib on aninfinite plate with

p . 3-126 convectio n.t - t w , x - ±6/2 , y > i .

t = t w , x > |6/2 | , y - H .

Con vection boundary at y = 0

±

8IA/WI

y

Tvn

See F ig . 1.10 for temp at x = 0 , y > 0

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Sect ion 1. 1. Sol ids Bounded by Plane Surfaces—No In te rn al Heating.


1.1.40 79 Finite plate withcentered ho le. Mt. V

TTd

2w + S,nl

2TTM

C 4 } _ J ZgiT«

F1 . 1 . 4 1 79 Tube centered in a

finite plate.

4

1f e d 2 r

| 1 t 1

2itk(t.

q = VX-n (i) r < 10

w/d1.00 0.16581.25 0.07931.50 0.03562.00 0.0075

2.50 0.00163.004.00

0.00031.4 x 10~

CD 0

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Secti on 1. 1. So li ds Bounded by Plane Surfaces—No Internal Heating.


1.1.42 79 Tube in a raulti-sidedinfinite solid.

2 1 1 , 5 ( t l " V

l n ^ - C ( n ,

n = No. sidesr l < r 2 y ' 1 0 f o r n = 3

n3456

789

10

C(n)0.56960.27080.16060.1067

0.07610.05700.04420.0354

0

1.1.43 79 Infinite square pipe.

Tw

1

2TTk(t i - t 2 )

0.93 In (w/d) - 0.0502

2ttk(t.

, w/d > 1.4

V0.785 An (w/d) w/d < 1.4

1.1.44 87 Partially adiabaticrectangular rod with

an isothermal hole.

q = Sk( t x - t Q )

See Fig . 1.11 for values of S.

/"'""'

*-J&.2b

T2a

1

I

^r77777777~ 2 b

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i

m

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Sect ion 1.1. Sol ids Bounded by Plane Sur fac es— No Internal Heat ing.


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Section 1.2. Solid s Bounded by Plane Surfac es— Wit h Internal Heating.


1.2.1 1, Infinite plate,p. 169 t = t , x = 0 .

t = t 2 , x = A .fc 2~ fc l

X + Po X U - X) /2

h-H1.2.2 4, p. 50 Infinite plate with

convection boundaries.

V h i —-h,.t,

tj _ - t 2 1 + B i 2 + H + B i 2

2 U * '

B i ^ l + Po( l /Bi 2 + 1/2)1(1 - X)

1 + Bi, + H

H = h 2 / h 1

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Section 1.2. Solids Bounded by Plane Surf aces— With Internal Heating.


1.2.3 3,p . 130

k

Infinite plate withtemperature dependentconductivity.k = Jc f + 6( t - t f ) .

(t - t,)k c r • — T . ,

^ = f[-l + Vl + 2 B ( l - X2 ) ]q " 1 .

t f . h _k(t)

B = 6 q ' " L 2 / 2 ^

See Fig. 1.2.

-h,t.

1.2.4p . 215

Inf in i te p la te wi thtemperature dependentinternal heating.

(t - t Q ) k

t = t Q , x ±9 . . % ••I 2 ^B

cos CVPo Q X)

cos (VPO£)

q , , . „ q „ . + B ( t - t ( J ) P o e - 3*Vk

h—

q'"(t)

k I

Mean temp:

< t m - y k

% l 2 Po372 t a n l V F 5 3 )

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Section 1.2. Solids Bounded by Plane Surfaces—With Internal Heating.


1.2.5 2, Infinite plate with

ito

p . 217 r ad i a t i on hea t i ng .= 0= t Q , x = 0 ,

q ' " • qyeg <* radiation energy flux.Y " mean radiationabsorption coeff.

H«H

(t - t 0)Jqr= 1 - yxe -Y*- _ e - Y *

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Section 1.2. Solids Bounded by Plane Surface s—Wi th Internal Heating.


1.2.6 2,p . 223

Porous plate withinternal fluid flow.P = Porosity.

u.x 0.k.p f.c

q'".k p ,P

—I

t - t- = | - [_1 - e p J + X , 0 < _ X < 1

C I

(See F ig 1 .3 )

fc " fc0 1 ( 1P) Vc e,

Mean temp:

X < 0

t m - fc0 l / F + "V| + l

^p k p ( l - P) ' ^f K £ ( 1 - P) ' ^ " p f u c

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Section 1.2. Solids Bounded by Plane Surf aces —With Internal Heating.

Case No. References Description So l u t i o n

1 .2 .7 2 , p . 227 Porous p la te wi th in t e rn a l D Xf l u id f low and temperature S t + 1 _ e 1 ( 6 t l + 1 D 2dependent in t e rn a l h ea t in g .t = t 2 , x = I,

q ' " = q j " < l + B t ) ,P = Po ros i t y.

B t . + 1 E \ B t , + 1

* °2 D l

E = e - e

e - 1

D 2X

U - V k f « r - C

q*"(t)

D i - V 2 + [ ( V 2 ) 2 P OB ]

['„ /2 - | ( ? „ / 2 )2 - POgl'5

B t x =( D 1 - D2> ( B t 2 + 1 } - ^ p - E

*E - EC

* D D p uci.E = D l e - D2e , Sp - k ( 1 . p ) . *° = q»'Bl-A p

2

1.2.8 3, p . 220 Rectangular rod.

I t - t - j K i 2t = t n , x = ±b , - a < y < + a. ^ = r- (1 - O - 20 - I ' l ' a ^ '

t = t , - b < x < +b, y = ±a.q ' - ' a

V I b

\ q'" a• • r

X

• * —

X

^ ( - 1 ) " c o sh | ( 2 n + l ) j x | c o s | ( 2 n + 1 ) | Y |

n=0 | (2n + l ) j j cosh | (2n + 1 ) ?B1

See Fi g . 1 .4 , Table 1 . 1 .Y = y / a , X = x /a , B = b /a

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Section 1.2 . Solids Bounded by Plane Sur faces —With Internal Heating.


1 . 2 . 9 3 , p . 469 In f in i te t r ian gular bar.t = t Q , x = ±y.t - t 0 , x = «..

- ° - - fe < *2 - W * )< •>t 2

y/l

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Section 1.2. Solids Bounded by Plane Surf aces —Wit h Internal Heating.


1.2.10 2, p. 197 Infinite rectangular rodwith temperature dependentinternal heating.

q ' " = q ^ " ( l + S t ) .

t = t , x = ±b, -a < y < +at = t Q, -b < x < +b, y = ±a

Vt b

oo oo n+m(t - t Q) %-* •%- (-1) cos U n X ) cos (X mY)

4Po„ 2 , Z/B + t n 6 n=l m=l X X [ x 2 A + ( X 2 / A \ - PoJn m|_ n \ m / 3J

Nf

iu>

q ' ( t )

X = 27T{2n - 1) , X = 2Tl(2ni - 1 )n in

X = x / b , Y = y / a , P o & = q^ • • a b 6 / k , A = a / b

S e e F i g 1 . 5 .

a

*• x

^1.2.11 9, p. 171 Infinite rectangular rod (t - t^Jk

with convectionq

tangular rod (t - t )k , îon boundary. — = ^~ - + ±-(1 x) - 4 £ivity in x-dir. q' a / 1 n=l= conductivity

k = conductivity in y-dir. s i n (X ) c o s (X X) co sh (X AY)n n n

X n [ 2 X n + sin (2X)] [ ( ^ ) sinh (AXJ + c o s h ( A Xn ) ]

X n tan ( A [ ] ) = B i r B i x = h i a / k x , B i 2 = h 2 a / k x

A = r-t f , h .

| V iT/ iT" , X = x / b , Y = y / a , B = b / a

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1.2.12 9, p. 423 Infinite plate with pointsource.t = t Q , z = 0 , «,.

Source of strength Q islocated at r. , 9 , b.

(t - t 0 ) W

Q

R

n=lsin (nirz) sin (rnrB)K fl (nitR)

Ja[r 2*'rJ - 2 ^ 0 0 . (8-8^]

i

w

•HB = b/i, , Z = z/l

1.2.13 29 Spot insulated infiniteplate.Insulated on one side andan insulating spot withconvection boundary on the

other.

M f

mkrm \r J

////////////A'/////////////

i Z

Solutions of |"t(r

at r » 0 , z = 0 ,' Z ) " t f ] /[ t ( °° ' W ) " fcf] = e ( r / w ' Z / W )

w are given in Fig . 1. 8.

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S e c t i o n 1 . 2 . S o l i d s B o un d ed b y P l a n e S u r f a c e s — Wi t h I n t e r n a l H e a t i n g .

C a s e N o . R e f e r e n c e s D e s c r i p t i o n S o l u t i o n

i<J1

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« « 1... — — » « - « - — « • * • — B " " " a -


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t J

t/5

I

R E P R O D U C E D F R O MBEST AVAILABLE CO PY

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Section 2.1. Solids Bounded by Cylindrical Sur fac es—N o Internal Heating.


2.1.1 4, p. 37 Infinite hollow cylinder. 2Trk(t Q - t ±)

In ( r Q / r i ) + (1/fci^ + (l/Bi Q )

t - t„ An ( r /r Q )

t. - t Q In ( r . / r 0 ) + (l /Bi Q + ( i / fey

B i . = h. r. /k , Bi Q = h Q r 0 / k

2.1.2 19,p . 3-107

The composite cylinder. 2TT{t - t . )n 1__

n- 1 . . n

Temp in the j t h la ye r :

^n-Vj - 1

Ii=l

k i \ r i / r i h i

n^l

k. I r. I r. h.3 \ 3 / 3 3

- . J > 1

£ * i \ ' i ; 4 ^

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Section 2.1. Solids Bounded by Cylindrical Su rfa ces —No Internal Heating


2.1.3 3, p. 121 Insulated tubes. 2irk(t. V

teRn

See Fig. 2.1

Max heat loss occurs when r n = k/h.

ww 2.1.4 1 , p. 138 Infinite cylinder with

temperature dependent. thermal cond uct ivit y,

k = k fl + B(t - t 0 ) .

k = k Q at r Q .

k = k. a t r . .l I

^ y t j - vIn ( r ( J / r i )

( t ~ V g

= ^ , 2 B km fc t l< V ^k

0 l n

< VE

i »k m = (k Q + k.,/2

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Section 2,1. Solids Bounded by Cylindrical Su rfa ces —N o Internal Heating.


2 . 1 . 7 3 , p . 2 28 I n f i n i t e c y l i n d e r w i t hs p e c i f i e d s u r f a c e h e a tf l u x ,q = f (<f>) , r = rn .

q = f (0)

I

fc " fc0 = a 0 + 7_ R a co s (rap) + b s i n (nd>)n = l L n n J

2m= i f e / f l * ) d * ' a n = TThd + n/ Bi ) /

27T

—1 f(1 + n/Bi) J

f(ip) cos (n(p)dip

b n = n h f(<p) si n (nip)dip

For q =q Q s i n (cp) , 0 < tp < n

0 , u < (p < 2ir

(t - t0 ) h 1 R s i n (tt>) 2 V 2n co s (2mr)

" * 2 C 1 + {1/Bi)1

' * £ (4n2

- l)[l + ( 2 n / S i) ]

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Section 2.1. Solids Bounded by Cylindrical Surf ace s—N o Internal Heating.


2.1.8 2, p. 133 Infinite half-cylinderwith specified surfacetemperature,t = t n , <ji = 0 and iu

ao -IT

- t = - A R n si n (n<>) I ff m - t I s i n (n<>)d<>0 *• £ 1 J o

t = f m , r - r 0"For f(<>) f t ,

t = mr

fc - fc 0 4 y i nt, - t„ TT * - n

sin (n(())-1 "0 " n=l

For f (40 = t x , s i n (4>)

it * ^ n— ? _ = R s i n <*) - 7 L tR" sin < " * >fcl " fc 0 * n=l

2.1. 9 3, p. 230 Cyli ndr ical sh el l sect ionwith spe cif ie d surface heatflux and temperature.t = t . , r = r . .

I I

q = f(<>) , r - r n .

q = f(0>

- t . - a 0 An (R) + £ a ^ j ^ " - R 0

n j cos (X n *)

n=l

I \ -X \ 2 r n A

X n = rni/$ 0 , R = c/r i

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Section 7..1. Solids Bounded by Cylindrical Sur fac es— No Internal Heating.


2 . 1 . 1 0 2 , p . 148 I n f i n i t e q u a r t e r c y l i n d e r ,t = t , <> = 0 an d i t / 2 .

t = f (<t>) , r = r n .

t = f<0)

<" Tl /2

t ~ c o £ 2 R 2 n s i n ( Z n ^ / tf ' ~ ' o J s i n (2n<1' )<

n=l JQF o r f(<ji) = t :

fc ' fc0 2 , - 1— = — t a n

fcl " fc0 *

2 R 2 s i n ( 2* )

1 - R *

2.1.11 2, p. 133 Finite cylinder with twosurface temperatures,t = t x , z = 0.

t - t t , c - c„.

t = t 2 , z = A.

t - t,. . „ sinh (X Z)J (X R)1 = 9 \ n 0 n

t, - t. L X sinh (XL)J (X n)2 1 , n n l nn=l

J o ( V = ° ' z = z / r o ' R = r / r o ' L = £ / r oFo r t 2 = f (r )

s i n h (X Z )J (X R), ^ s i n h (A Z ) j (A R) f

~ ^ * * ~7t^Jt7 \ R^ (R ) ^ o ' V *r n . s i nh (X L) J , ( X_ ) J n

r „ , s i nh (X ^ ,u , i„ ,0 n = l n I n

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2.1.12 3, p. 253 Case 2.1.11 witht = t , r = O r r = Z .

t = t Q , r = r Q , n < <t> < 2n.

t - t„-_~_-0_ _ 2 V W * s i n ( V >t, - t„ ~ Ti Z n I n ( A „ R n )

n=l

„ _ T- fcJ) i n

tX

„z

) i n

W>o_ x x m n n2 Z Z mnl (XE.n=l m=l

A = nrr , R = r/JL , z = z/in

n l n 0'

2.1.13 3, p. 234 Semi-infinite rod withvariable end temperature,t = f (r) , z = 0.fc " fco ' r " V

t = f(r)

n= l

fc " fc0 = 2 Vo ( X n R ) eXp (-V>= 1

1

2 [apm -t o p o (X n R) SB.

A =n J i < V

Z = Z / r Q , J 0 ( X n ) = 0 , Xn > 0

I f f ( r ) = t ,

exp ( -Xn Z ) J Q ( X n R )fci - fcn Z X J (X )

1 0 __ . n 1 nn = l

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Section 2.1. Solids Bounded by Cylindrical Su rfac es—N o Internal Heat ing.


2.1.14 3, p. 234 Case 2.1.13 witht = t , 2 = 0 , and

convection boundary h , t c

at r = r„

t , - t f

" Bi J 0 f t n R) exp <-X nZ)

n=l K + * i2> WX n W = B i W

2.1.15 3, p. 238 Infinite rod with a travelingboundary between twotemperature zon es,t = t L , r = r Q , z < 0.

toI

CD

t = t 2 , r V z °Velocity of boundary(z- = 0) = v.

t, - t„ x Z-n=l

[l + ( X n /P ) 2 ] 1/2W>Vl'V

exp[( l + [l + ( V P ) 2 ] 1 / 2 ) P * ] ' Z < °

3 l

fc - c 2 _ Vt, - t, Z

n=l

1 +[1 + ( A / o , 2 ] 1 ^

x exp [| l - [ l + ^ n / P ) 2 ] 1 / 2 ) p z ] , Z > 0

Z = z / V P = v r Q / a r J 0 ( X n ) = 0

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Section 2.1. Solids Bounded by Cylindrical Su rf ac es— No Internal Heating.


2.1.16 3, p. 238 Finite rod with band heating.q = 0, z = + b.^z —Convection boundary at r = r ,

-b < z < +b.Surface heati

-a < z < +1 .

Surface heating (q ) at r = r ,

-2b-

Ih-2\Z- {

r r

n =izri

Heating band

t - t

q,

s i n (X L )I (X pR) c o s (X Z)

r . / k Bi Z X J" Bi I (X p) + (X p ) I (X p)" |0 _ nl_ i) n n x n Jn= l

X = rm , Bi = h rr t / l c , L = l/b, Z = z / bn u

R = r / r Q , p = r Q / b

2 . 1 . 1 7 4 , p . 3 7 E c c e n t r i c h o l l o w c y l i n d e r ,

t = t 2 , r = r 2 .

t = V r = r r

2TTk( t2 - t x )

D 2 -In

R

V(R + l )2 - S 2 + V(R - D 2 - - s 2

In

R

_V(R + l )2 - S 2 - V(R -

= r 2/r 1> S = s/r.^

D 2 - - s 2

See Ref. 80 for other solutions to eccentric cylinder.

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Section 2.1. Solids Bounded by Cylindrical S urfa ces— No Internal Heating.


2.1.18 4, p. 39 Pipe in semi-infinite solid.

d > 4 r r

2 l T k l < t 2 - t l )

,x i \ r i / K L B i 2 ji

h a . t j

innii)ii>MHin>»>»»i)»»»i»>iiHi

i

h i r i h 2 d k 2 dB i l = T f ' B i 2 = ^ K = k7' D = r7

2.1.19 4, p. 40 Row of pi pes in semi-infini te solid.

2TT k( t2 - tj_)

rr- + «-n .^«-H°^])], for one pipe,

/Z>?zmz7zzpmz?zmzm7777??77d

1k „ . V l „. h 2 d

n dB l i = T - ' B l 2 • — ' D " I

^;v viV vi-v<H

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2.1.20 4, p. 40 Row of pipes in a wal l .

yyT7777777T77777777Ty777777777

q =4 i r k ( t 2 - t x )

l i L i V L B l :

• h

lr

l h

2d

dB 1 i = — • B 1 2 = - r - D = i

, f o r e a c h p i p e ,

yyyyyyyyyyyyyyyyy/y/yyyyyyyyy/

2.1.21 4, p. 38 Two pipes in a semi-infinite solid. q l =

2 i r k ( t 1 - t Q ) z±

d' J k1 r 2 / 6 1 C 0 ^ + ( d x - d 2 ) Z

/2d \ 2d , / s2 + (d + a. ) \ 2

For q_, interchange indices 1 and 2

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Section 2.1. Solids Bounded by Cylindrical Sur fac es— No Internal Heating.


2.1.22 4, p. 43 Circular disk on the q = 4 r 0 M t 2 - c^)9, p. 215 surface of a semi-infinite

solid.t = t L , 2 t - t,

t 2 - t j L 7T

2 . - i f= — sin ;i [ ( R - l ,

2

+ Z

2

]1 / 2

+ [(R + l )2

+ Z

2

]1 / 2

.

'W/////MWW 'k

zT.Z = z/r„

2.1.23 4, p. 44 Circular disk in aninfinite solid.t = t , z + ± < > .

q = 8r QK ( t 2 - 4 )

r2I 1

l

*1 1k

2.1.24 4, p. 42 Circular ring in a semi-infinite solid.

M0

t k i1- 2 r „ V

4Tf r 2 ( t

In (^j + Jin

I " V" 4 r 2 "

4Tf r 2 ( t

In (^j + Jin

A - 4, r x « d « r 2

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Section 2.1. Solids Bounded by Cylindr' "\1 Sur fac es— No Internal Heating.


2.1.25 4, p. 42 Vertical cylinder in aserai-infinite s olid.

h , t n

q =2D B l ,

\ln

//////////JWA

d

1k

M"tf.+ T K k | t i - V

k D = d/r„

<*> 2.1.26 9, p. 216 Two semi-infinite regionsof different conductivities

t - t„ 2k „

c o n n e c t e d b y a c i r c u l a r d i s k . fcl ~ fc0 ^ l + k 2 }

t = t , z + +» .

t = t 1 # z -»• -<=.

q z = 0, r > cQ, z = 0.x s in

| [ ( R - D :+ Z Z ] l / 2

+ [(R + l , 2

+ Z

2 ] 1 / 2, Z < 0

'1'///////nsm mm*//////

t - t„fcl " fc0

2k,= 1 -

T T l ^ + k 2 )

x s i n1 ( R - l ) 2

+ Z 2 ] 1 / 2

+ [(R + l ) 2

+ Z 2 ] 1 / 2

q = [ 4 r 0 k l V ( k l + k 2 ' ] ( t 0 - V ' * = 2 / r 0

, Z < 0

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Section 2.1. Solids Bounded by Cylindrical Su rfaces—No Internal Heating.


2.1.27 9, p. 218, F in it e cyli nder with69 J n (R XJ sin h ([ 1 - ZjLX n)t = £ ( r ) , z = 0, 0 < r < r . - V r ° n

on- - r ~ 0 ~ <£ n sinh (LX )t = t Q, z - a, 0 < r < r Q. ^ n

Convection boundary atr V °with h, t.

r = r Q , 0 < z < &,2X 2

nn ( B i 2 + X 2 ) J 2 ' (X )

D 0 n

T R[f(R) - t 0 ] j Q ( RX n )dR

Jo

I

L = J.A 0 , Bi = h r n A , X n J- (X n) + Bi J Q (X n ) = 0, X n > 0

For f( r) = t 1"

t - t J 0 (RX n ) sinh ([_ - Z]LX n)

t . ° - 2 B I y 7-2— 1\1

° n=l (B i +

VW s i n h

<L

V2.1.28 9, p. 219 Case 2.1.27 with t = f ( r ) ,

z = 0, 0 < r < r Q . t - t Q - J V o ( 1 VRemaining surfaces convectionboundaries h, t..

n=l

X cosh (£1 - z ]LX ) + Bi s in h (LI - Z3U )n n nX. cosh (LX ) + Bi s in h (LX )n n n

C , X , L, and Bi are def ined in cas e 2 .1 .2 7.n n

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Section 2 d . Solids Bounded by Cylindrical Surfac es—N o Internal Heating.


2.1.29 9, p. 220 Case 2.1.27 with t = f ( z ) . °° x ( m

r = r , 0 < z < H. t - t Q = 2

t = t Q , z = 0, I, 0 < r < r Q . n=l

_ I Q (mrR) r

2 USE) s i n < ™z ) / [ f z ) " co] i n (n1 ,Z)dz

R = r/S.

2.1.30 9, p. 221 Case 2.1.27 with t = f(z ),K C O S a n 2 ) * BH Si " ( XnZ ) I 0( XnR )]

r = r Q , 0 < z < I. t - tQ = 2 ^ p- 2 2 ,

Remainin g surfaces convection n=l (, BlA + X n + 2 B l l J I 0 { X n R 0 >

boundaries h, t .1

/f(Z)["X cos (\ nZ) + Bi^ sin (X nZ)] dz

•'o

tan (X ) = (2X Bi„ )/( X 2 + B i 2 ) , X > 0, R = i/%n n Jo n v, n

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2 . 1 . 3 1 9 , p . 2 20 F i n i t e c y l i n d e r w i t h s t r i ph e a t i n g a n d c o o l i n g .

q f = - q ^ r = r f l I 5, > z > «, - b .

q = 0 , r = r „ , S, - b > z > - £ + b .T: 0q = +q , c = C , -8, + b > z > - 8 , .

q z = 0 , z = ±i., r = r Q .

t = t Q , z = 0 , 0 < r < r Q .

( t - t n ) k D ^ ( - l ) " l0 ( m i T R )V " _ 8_ V,5- ~ 2 Z,

s i n (imrb/J,) s i n (nmZ)TT- n = Q m i ^ i r n i R ^

m = 2n + 1 , R = r /2 « , , Z = z /2« . , B = b /2J ,

q

t ///////////// ////////q

j I^ ro I1 t1 hf/}///////)T

X "

2 . 1 . 3 2 9 , p . 2 20 F i n i t e h o l l o w c y l i n d e r w i t h :

t - t Q = 2 ^ ^ s in (nnz) / [ f (Z ) - tQ ] s in (mrZ)dZ

n = l

F Q = I0(nTTR) K0 ( m r R 0 ) - K0(n i rR} I0 ( m t R 0 )

P l = I i ( n 1 T R i ) K

1 ( n l r R

0 ) " ^ ( m r R ^ I ^ m r R )

R = l/l

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2.1.33 9, p. 221 Case 2.1.32 with q = f (z), <» „ r l

t - t Q = - ||- V -—- sin (mrz) / f (Z) sin (nirZ)dZ= r., 0 < z < l .

Remaining surfaces at t . n=l 0

F defined in case 2.1.32

P x = i^ni rR ^ K 0 (wiR 0 ) + K 1 (mrR i ) i 0 (miR 0 )

For f (z ) = q , w < z < (£ - w)

i = o , 0 < < w and (I - w) < z < l :

(t - t n ) k ^ F,

qV k v Fo

TS = > —5— cos (rnrw) s in (nuz) , n = 1,3, 5 ,1 ~ n F,=l 1

W = w/Jl

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Section 2.1. Solids Bounded by Cylindrical Su rf ace s—N o Internal Heating.


2.1.34 9, p. 222 Case 2.1.32 with t = f( r) ,

z = 0, i t < t < t Q .

Remaining surfaces at t

z = 0 , r . < r < rQ . t - t Q

__.y ^ 'V W s i n h [ ( 1 - z ) U n ]2 n=l [ J 0 ( V - J0(Vo>] S i n h ( L V

1x f [Rf (R) - tQ ] O 0 (R Xn )dR

J0

U„(X ) = 0 , X > 0 , R = r / r . , Z = _ /« . , L = l/i.O n n l l

r U 0 ( R V • J0 (R An> V X n V ~ W V Y 0 ( R Vi-"CO — — — — — — — — — — — — - — — - — - — — _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ — _ _ _ — _ _ ^ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

2.1.35 9, p. 222 Cas e 2.1.32 with t = f ( z ) , ™ - - . I - / . ,A COS (A Z) + Bl Sin (A Z)IG(R,n)

• • _• , | X 2 + B i 0 + 2Bi„|G{r.,n)Rema inin g surf aces conve ction n=l |_ n x. S.J 0boundary with h, . . .

= r i , 0 < z < «.. t - tf = 2 Yn=]

x j [t (Z) - tfJj"X_ cos (XnZ) + B i£ s i n (AnZ)]-X)

tan (XJ = 2X^1^1 - B iJ ) . X_ > 0

G(H,n> = V ^ n ' t W W -B 1 l K O f Vo ' ]

dZ

+ I f l < t t n ) [ V l ( W + B i VW]

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Section 2.1. Solids Bounded by Cylindrical Sur face s—No Internal Heating.


2.1.36 9, p. 223 Sem i-in fini te cylinder with:Jn^J

t = f ( z > , r = V z > 0. t - t Q = ^ — j - 5 - / [ f(Z) - t Q ] exp (-X^ + X^)

t = t n , z = 0, 0 < r < r . n=l J0

i

exp (-X Z + X Z)dZv

n nJ n (X ) = 0, X > 0, Z = z/ r

u n n u

2. 1.3 7 9, p. 223 Case 2. 1. 36 with

t = f ( z ) , r = r 0 , z > 0.

Convection cooling at

z =» 0 with h, t f .

t - exp (-X Z + X Z)n n„ J n R A J r _ ,

n=l 1 n J 0

J 0 ( X J = 0, X n > 0

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Section 2.1. Solids Bounded by Cylindrical Surfa ces— No Internal Heating.


2.1.38 9, p. 223 Finite cylinder with heateddisks on (t „ - t_ .)k I, (mm..)

q z = +9 , z = 0, Jl, 0 < r < r L < V IT 2 n ^ d W H , , ) L l u A

- K^mrRg) i^mrR^] , n = l, 3, 5,Remaining surfaces insulated

i i

^

-— Tr i lro

io £-

R = z/l

t . = itm,0

t „ = mean temp, over region 0 < r < r.. , z = £•

t . = mean t emp, ove r region 0 < r < r,, z = 0m, 0 1

2.1.39 9 , p . 2 24 Case 2.1.32 with t = t ^

z = 0 and l, t . < r < r Q .

t = f ( z ) , r = r Q , 0 < z < A.

q = 0, r = r. r 0 < z < l .

/-t=Hz) z = I

* i.

r1 >;/////;///;/;//;/;/ , r ot v//////////////////,

t „ ^ " ^ - t n

t - t 0 = 2

°°2 * ^ s i n (n'flZ) / f"f(Z) - t0"l sin (mrZ)dZ

n = l 2 -T)

F l = I i < n 7 r R i J K 0 ( n 1 T R ) + ^( ff lT R^ IQ(n7TR)

F 2 = I 1 ( m r R . ) K0(n7TR0) + K ^ m r R ^ I n (n7TR 0)

R = r / fc , Z = z / d

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Section 2.1. Solids Bounded by Cylindrical Su rfaces—N o Internal Heating.


2.1.40 9, p. 426 Infinite cylinder in aninfinite medium having alinear temp gradient.At large distances fromcylinder, t = t Q z.

Cylinder, conductivity = k .

Medium conductivity = k .

z = *A„

2.1 .41 9, p. 434 Sector of a c y l i n d e r ,

t = t x, 0

e = o, e n

t = t 1 # 0 < r < r Q ,

fc = V r • V0 < 9 < 9„ .

fc-fcl— tan

si n ( T T 9 / 90)

Isinh K-©]

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2.1.42 9, p. 451 Two infinite cylinders inan infinite medium.

airkC^ - t 2 )

cosh ( * ^ )

R = r ] / r 2 ' s = s / r

2

w 2.1.43 9, p. 462 Spot heated semi-infinite solid.

K q z = q 0 , 0 < r < r Q f z = 0 .

q z = 0 , r > rQ , z = 0 .t = t r r > 0 , z -* =° .

( t " U kt j k rro qo i

exp (-\z)jQaR)J1a)^

%

H<H

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Section 2.1. Solids Bounded by Cylindrical Su rfac es—N o Internal Heating.


2.1.44 73 Infinite cylinder with twointernal holes.

t m V r " r

0 •t = t., hole surface .

hole radius = A. .

q = Sk(t fl - t.)

S = shape factor

See Fig. 2.6

2.1.45 79 Infinite cylinder withmultiple internal holes.

2nk{t, V

®-i -® 'n

n = No. hol es

r 2 < r x , n > 1

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2.1.46 79 Ellipsoidal pipe. 2irk(t x - t 2 )

Jin( ^ )

2.1.47 79 Ell ips oid with a constanttemperature s l ot .

2TTk{t. VJin (*f*)

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Case Ho. References Description Solution

2.1.48 79 Off-center tube row in aninfinite plate. q =

%<5 © © ©~ 1

-»|2rf«- «-2s-» ^ 7

2Ttk(t1 - t 2 )/ 6 2 ( a / 2 d ) \ ( , & ,

CO

9 (a/d) = 2 , exp i i rs/n + |Y /2 d cos [(2n + l )n a/2 d]n=l

e l ( o) = i re2 e 3e 0

OO

8 2(0 ) = ] £ exp kits (n + |)2 / 2 dn=l

9 = 1 + V e x P ( i t sn2 /2d)

n=l

8 = 1 t 2 2 (- •)" e x P (ittsn2/2d)n=l

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Section 2.1. Solids Bounded by Cylindrical Surfa ce s—N o Internal Heating.


2.1.49 79 Single off-center tube inan infinite plate. q =

2 * 1 ^ - t 2 )

H-P^D' - f lS)T~T

t 2_/ t , y

2.1.50 73 Tube centered in a finiteplate. q =

2TTk{t1 - t 2 )

An

Y////////A(—)*¥•\Ttt / 2w

W////M

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Section 2.1. Solids Bounded by Cylindrical Surfaces—No Internal Heating.


2.1.51 79 Heat flow between two pipes

in an insulated infinite plate. q =2nk(t 1 - t 2 )

w \nr /

2.1.52 79i

Tube in a semi-infiniteplate.

'zzm -zz*-AW/MUW

q =2irk(t 1 - t 2 )

2a + an (-*)2w \Tr)

2.1.53 79 Ellipsoid in an infinitemedium.Q -

4TTVI - a 2 / b 2 k{t x - t 2 )

^ — T T T :rctanh (\ l - a / b )

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Section 2.1. Solids Bounded by Cylindrical Su rfa ces —No Internal Heating.

Case No. References <?c r ip t i on S o l u t i o n

2 . 1 . 5 4 7 9

«oI

oo 2 . 1 . 55 79

E l l i p s o i d a l s h e l l . airkf^ - t 2 )

a r c t a n h ( c / b ) - a r c t a n h ( c / b . )

c = yb^ - 3] = Vb2 - a

R od i n a n i n f i n i t e m e di um ,t = t 2 ; x , y , z + °° .

4 t f k < ^ - V r

An (28,/r ) ' I < 0 .1

,-4 ±8- -<T

2 . 1 . 5 6 7 9 S h o r t c y l i n d e r i n a n i n f i n i t e Q = c k ( t , - t2 )zned. jnut = t 2 , - x , y, z + oo .

^ 12r

_ ^ £0 80 . 2 5 1 0 . 4 20 . 5 1 2 . 111 .0 14 .972 .0 1 9 . 8 74 .0 2 7 . 8 4

-28-

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Section 2.1. Solids Bounded by Cylindrical Su rfa ces —No Internal Beating.


2.1.57 79 Toroidal ring in an infinite a-n^k(t- - t- )medium. 1 2t = t 2 . x , y , z - ~ . Q ~ In (Sr^/r.,) ' l / I : 2 * Z U

2.1.58 79 Thin flat ring in an infinitemedium,t = t 2 . x ,y,z -»» .

4ir'k(t - t )Q = to a6 t l / r 2 ) ' V r 2 > 0

2.1.59 79 Two parallel rods in aninfinite medium.

I^

26

Ty

4 7 r k ( t 1 - t 2 )

^ E Z H ]s > 5ra » r

For parallel strips of width 2w, use r = w/2

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Section 2.1. Solids Bounded by Cylindrical Su rf ac es —N o Internal Heating.

Case No . References Description Solu t ion

2.1 .60 79

, ~

Two alig ne d rods in anin f in i t e med ium.

T

28 -

V,

4 i rk ( t 1 - t 2 )

'[* f fW J-H- , s - 21 > 5r

For aligned strips of width 2w, use r = w/2

2 .1 .61 79 Pa ra l le l d isks in a iiin f in i t e med ium.

1 1M-H

1 1s

1

Q =iitktt^ - t 2 )

' . I- - arc tan ( r /s ), s > 5r

2.1 .62 86 I n f i n i t e c y l i n d e r w i t hsymmetr ic isothermal caps .

kTTt^ - t 2 )q ~ 2 Hn [ 2 ( 1 + c o s 9 ) / s i n 8 ]

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M

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toi

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Section 2.2. Solids Bounded by Cylindrical Surfaces—With Internal Heating.


2.2.1 4, 51 Infinite cylinder withconvection boundary.

tn,h.

( t - t Q ) k

§ i - J

2 . 2 . 2 4 , 5 1

Iu >ui

H o l l o w i n f i n i t e c y l i n d e r w i t h ( t - tF ) kn o x i o w i n r i n i c e c y x i n a e r w i c n ( c - C _ ; K . i . r _ -i _ _ iw i t h c o n v e c t i o n b o u n d a r y o n — = - r r v l - R , + 1 - R + 2 R . Jin ( R ) lo u t a i d e s u r f a c e . q " ' r „ « IB 1 L J-J i Iq r - 0, r = i L

c/cit B i = h rQ / k

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Section 2.2. Solids Bounded by Cylindrical Surfac es—Wi th Internal Heating.


2.2.3 4, 51 Hollow infinite cylinder with (t - t f ) k .. , . p , -i „ , iconvection cooled inside — = ~ | — R Q- 1 + 1 - R + 2R In (R)|

oI

surface.q r = 0. r = r 0

q-" r :

R = r/r , Bi = hr./k

2.2.4 2, p. 189 Infinite cylinder with

temperature dependent heat

t = t Q , r = r Q

q ' " = q ^ " d + 3t)

J (VPS: R)

s H r i - § — • o R - ten"* '*B t 0 + X J n(VPo7) B ° °

IJ o l " " S '

B t m + l

Mean temp: -g2 J

1

( V ? 5 i )

Max temp:

0 t° ^ V ^0t + l

max _ 16 to '^(Gfy

See Fig. 2.2

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2.2.5 2, p. 189 Case 2.2.4 with:

toI

j (VPO" R) J (VPO~ R>

t = t„, r - r , 0 < $ <ir. t ' Vt = 2 ~ ~ * 7 > „ ZZT" '

q. . . = «.(i + 6t)n = 1, 3, 5, ...

Pojj-B^'rjA

2.2.6 2, pg 191 Case 2.2.4 with convection

boundary.

-O8t i- l0 t f + l

3 o ^ R >VPOT

' o 1 * 5 5 ^ - ~ i i J i ( V K i )

, P og = B q ' - ' r ^ A

Mean temp:2 J l( V 5 o T )

See Ref. 85 for nonuniform convection boundary solution.

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Section 2.2. Solids Bounded by Cylindrical Surfac es— With Internal Heating.


2.2.7 2, p. 193 Infinite hollow cylinder with „

temperature dependent heat Q

source.

fc

" Vr = r

0 •t •> t i # r = r i .

q ' " - q n " ( l + St) .

t . + 1 \

(0 t „ + 1)

V ^ ' ( B t . + i) '0 „ (V fSlRj.)

( 3 t 0 + l)

(Bt. + D V ^ B V " J o ( V ^ )

a0(VPoTR)

Y 0(VPO^R)

MI

B =J 0 ( R . V P O ^ ) YQ ( V P V " J o ( V 5 ° B ' Y o ( R i V 5 ° B )

p

° B = B W k ' R = r / r o

2.2.8 2, p. 194 Infinite hollow cylinder

with temperature dependent

thermal conductivity,

t - t 0 , r - r..

fc = V r = Vk = k Q [l + B<t = t n ) ] .

( t - t Q ) k

. „ 2 2PoD| P oB 6(1 - R 2 ) (' - 0 n (1/R)

Zn

(k)

1/2

Po - , » T 0 B A 0 , K - r / r Q

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Section 2.2. Solids Bounded by Cylindrical Sur fac es— Wit h Internal Heating.


2.2.9 6, p. 272 Electrically heated wire withtemperature dependent thermaland electrical conductivities.

t " V r r o •k t A t Q -.l + 3 t ( t - t Q) .k e / k e O = 1 + B e ( t - t n ) .

B = ^ & , . -X-^3k t o L

I(Jj

- J

k = thermal conductivity

k = electrical conductivitye J

E = voltage drop over length L

2.2.10 2 , p. 175 Cylindrical heat sourcein an infinite plate.

t = t . r

—VASA c 1-V/# ^v.

TL

2 , n 2

t ~ > V ™0

q'/kW 2n Bk ^B ]

See Fig . 2 .3

B = r 0^h x + h 2 ) / k w

h ^ + h 2 t 2

^ " h 1 + h 2

q' = heating rate in cylindrical source

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Section 2.2. Solids Bounded by Cylindrical Surf aces —Wit h Internal Heating.


2 .2 .U 9 , p . 224 F i n i t e cy l inde r wi th s t eadysurface temperature .t = t , z = 0 and SL , 0 < r < r .t » t Q f r = r Q, 0 < z < I.

l t ' V * z - z2 I„(nTrR)

q " ' * , 2

4 y *o*3 £ i n 3V ™ V

sin (nuZ) ,

iCO

2^ N q"

n = 1 , 3 , 5 , . . .

Z - z/l, R = c/SL

2 .2 .1 2 9 , p . 224 F in i t e ho l low cy l in de r . (t - t„) k „ „20 Z - Z

q'"Si

i _ V si n (mrZ) . _1

n = 1, 3, 5,

F = ijtniT R.jK (nirR) + I (mrR)K (m m .)

F 2 = ij^tnirRjJKjjtmrRjj) + I0 ( m r R0 ) X 1 ( m r Ri )

Z = z/fc, R = r/S,

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Section 2.2. Solids Bounded by Cylindrical Surf aces —Wi th Internal Heating.


2.2.13 9, p. 224 Case 2.2.11 with

q ' " = q^"[ i + Pit - t Q)]

( t " V k _ c o s [(VFo72 - Y P Q " Z J _ 1_

q ' " H Po co s (VPo/2) P o

. r- . I . ( X R) s i n (n i rz )4_ X u n

" ^n T l V Wn X

n

, n = 1 , 3 , 5 ,

X 2 = ( 2n + 1)2 T T2 - P o , P o = q ' " B J .2 A , Z = z/l, R = r / J l

2.2.14 9, p. 423 Finite cylinder with line heat

t = t Q, r = r Q , 0 < z < «..

t = t Q, 0 < r < r Q , z = 0, SL.

q'

unit length is located at r , 8 ,

Line source of strength g' per

unit length i

b < z — c o s (rairB) s i n (imrZ)

m=l

I (mTTR)? [I (nnTR )K (n mK .) - K (mpR ) I {nfflR j]~ i„ (nnrR„) L n ° n l n ° n 1 J

n=0 V ^ V

x E n c o s [ n ( 9 - ei ) l , 0 < R < R

E = 1 i f n = 0 , E = 2 i f n > 0 , m = 2 n i - ln n

B = b/SL, R = r A , Z = z/l

F o r R < R < R i n t e r c h a n g e R f o r R . .

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I



2.2 .15 9, p. 423 Case 2. 2. 14 wit h a po int . . . „ °°(t - t Q)KJ(, L _

source of strength Q located = — y si n (mirz) sin (mirz..)a t r. , 9 . , z . . m=l

_ I (mnR)

* Z E a H 5 i j [ I n < M B l l n ( l r t l l ' " * n < n i m ) ' n ( B , , V ]n=0 n °

x cos["h(9 - 9 )~| i 0 < R < R

For R < R < R , interchange R for R n .

R = r/SL, Z = z/SL, E = 1 if n = 0, E = 2 if n > 0n n

2.2.16 9, p. 423 Infinite cylinder with ,. =>~ 0 0 1 V

point source of strength Q. g = 2? A c o s [ n '® ~ ®i']fc = fc . r = r . n~~°°

0 0ource located a t r l f 6 ^ ^ exp( -X | z | ) Jn & n R ) J

n^ n

R{>

X>0

4 % C - — f J n ( X n ) = 0 , Z = z / r0 , R = r / r Q

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Case No. References Descript ion Solution

2.2.17 16

I

Hollow cylinder with

temperature dependent

heating and thermal

conductivity.t V r Vq t = 0, r = r Q .

V r

q'

k

r = 0.

k Q + a ( t 0 - t ) ,

['" = 1'" + b(t . t ) ,

Solution by numerical method .

See Fig. 2.4 for values of t at r = r. .l

P i = I - ( r. / r Q) , K = k 0 / a t 0 , R = a t y b r ^ , G = g - • ' /b t Q

For a = b = 0: .

( t n - t , )k n

— — - ( l - p.) + - n/3-^-J - -

*o 0

(Eg. (5) in Fig. 2.4)

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2.2.18 64 Triangular arrayed cylindricalcooling surfaces in an infinites o l i d .t = t Q , r = r 0 .

< m o ,.. .at?

t t - t „)ko ' K V 5• it 2 TT^ to <R)

2 £ (f) V (;f] - •<«See Table 2.1 for values of &. .

See Fig. 2.5 for t values ,maxS - s / V R = r/r Q

Equilateral triangular array

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2.2.19 64 Case 2.2.18 with a squarearray of holes.

i

O Ox0

o m o

Square array

<fc - V k 2,,,2 = ? £ n ( R )

q' ' s

j=i

See Table 2.2 for values of 6..

See Fig. 2.5 for t values.max

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Section 2.2. Solids Bounded by Cylindrical Su rface s—Wit h Internal Heating.


2.2.20 67 Eccentric, hollow, infinite Se e Ref. 67 for equational solution,cylinder.

K>i

2.2.21 73 Cylinder cooled b y ring of See Ref. 73 for max temperatures,internal holes.

Insu/ated surface

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Case Ho . References Description Solution

tot

• >v n

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Case No . Heferences Description Solution

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<*>

REPRODUCED FROM S

BEST AVAILABLE COPY

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Section 3.1. Solids Bounded by Spherical Su rf ac es— No Internal Heating.


3.1.1. 4, p. 37 Spherical shell. 4n r 0k(t. - t Q)

<r 0/r.) - 1 + (r 0/ r. ) (k/h. r. ) + k / h ^

t - t„ (r QA) - 1fc i " fc 0 ( r 0 / r i » " * + ( r 0 / r i > ( k / r i h l » + k / r 0 h 0

i

3.1.2 92 Composite sphere. 4 i r ( t i " Vn-1

?. 'i ('i rLl) + ? . r2hi = l i = l I

*i-*i i[^( ' i" r ^ ) + ^ J + ^(^'")n-l n

i- 1 1 V 4 1 + l / i - 1 h i r i

n 1- J > 1

t. = local temperature in j layer

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Section 3.1. Solids Bounded by Spherical Surf ace s— Ho Internal Heating.


3.1.3 1, p. 139 Sphere with temperature-

dependent thermal conductivity,

t • t. , r • r. .

t - t Q, c - t Q .

k - k 0 + B(t - t Q) .

4Trr.k (t. - t„)0 m i 0

( r ^ r. )

* m " k 0 V / 2

k - k 0 . t - t fl

k = k., t = t A

v 6 v

3.1.4 2, p. 137 Sphere with variable

surface temperature,

t » f ( 9 ) , r - r n .

t • ^ a r n P (cos 8)

n=0

f ( 9 )

a - * &n 2

£{9) P (cos 9) sin 9 d9n

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Section 3.1. Solids Bounded by Spherical Su rf ac es —No Internal Heating.


3.1.5 2, p. 137 Case 3.1.4 wit h: t 1 , 3 D D . „„ Q , 7 _,3 _ , „„ „,

t = t 0 , 0 < 8 < IT/2 . c 0 Z * * i t > J

t = Q, n/2 < 8 < it . . t i i R

5 Pfcos 8) . . .32 5

3.1.6 2, p. 137 Spherical shell with specified » n 2 n + 1 _ n 2 n + 1

surface temperatures. fc ~ fc 0 = X ( n + 2/—/"' 2n+l \ — ( l ) p

n ( c ° s 9 )

t « t„ , r - r„ . n=0 \, R0 " 7

t = f ( 9 ) , r = tL

f< I P (cos 8)

f(cos 8) - 1] sin 6 d6

R = r/r.I

t = W )

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Section 3.1. Solids Bounded by Spherical Su rf ac es —N o Internal Heating.


3.1.7 4, p. 41 Sphere in a semi-infinit e 4irr k(t - t )

solid. Q = i1 +

2 [ ( d / r Q) + ( l /B i ) " ]

'/A\V////////////A B i = h r ° A

I

3.1 .8 i, p . 43 Hemisphere in the su rfa ce Q = 2rrr k{t - t )of a sem i- inf in i te so l id .

J2v

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Section 3.1. Solids Bounded by Spherical Sur fa ce s—N o Internal Heating.


3.1.9 9, p. 423 Sphere in an infinite medium

having linear temp gradient.

For large distances from sphere

t = t_z. Sphere conductivity

is k, and medium conductivity is r—

k °K 0 .

t 3 k Q

S + ( R ) (2k° + kJ ) J cos 9, R > 1

3.1.10 5919,p. 3-111

Irradiated spherical thin shel l. See Fig. 3.1 for T/T .

q = source heat flux .sT = sink temp .

a = absorptivity . T =a

/ E I / E 2 + 1/4V

1 \ l / e 2 + X I

1/4

î.

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Section 3.1. Solids Bounded by Spherical Sur fac es— No Internal Heating.


3.1.11 79 Sphere in an infinite medium. Q = 4T T r Qk(t., - t„)t = t 2 , r + » .

3.1. 12 79 Two spheres separated a la rg e

distance in an infinite medium.

4irrk(t - t )Q m 2(1 - r/s) ' s > 5 r

t.

2r

for s > 2r, error = 1%

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Section 3.1. Solids Bounded by Spherical S urf ace s—N o Internal Heating.


3.1.13 79 Two spheres separated a small . ._ ... ,. .Hi L / , , / * 2 , . . . 3 , - , . 'Q =2rrrk(t 1 - t ? ) U + n/s + (r/s) + (r/s) + 2 (r/s)

distance (see case 3.1.12) in

an infinite medium.+ 3( r /s) + -], 2r < s < 5s

3.1.14 79 Two sphe res of dif fe re nt r ad ii

in an in f in it e medium.

4 T r r 2 k ( t l - t 2 )

/

3©r l

\ ( V S ) 4

r l 1 - ( r 2 / s ) 2

- 2r /s , s > 5 r 2

2r,

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Section 3.1. Solids Bounded by Spherical Sur fac es— Ho Internal Heating.


i

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• i <*„r£aces—No Internal Heatingection 3.1. Solids Bounded by Spherical Surfaces

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Section- 3.2. Solids Bounded by Spherical Surfaces—W ith Internal Heating.


3.2.1 Spherical shell with specified

inside surface heat flux.fc " V r = C 0 •q r - q r r = r. .

<t - t Q)k q , , ' r i

q i r i **<

R = r/r.

2 < R -

0 ' 2 2R* + R 0 ' R

R o - R

RR~

3.2.2 1, p. 190 Sphere with temperature

dependent heating.

, 1 1 1 _ — I I I + 0(t - t n )

( t " V g _ 1 sin (K VPO)

"*0

t - t,

sin (VPO)1 , Po = 0rV k

max 0

t - t,

0 2= 1 - R , Po = 0

0 sin (TTR) ,/-—

max 0

( t m a x - V fV P O

sin (VPO)- 1

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Section 3.2. Solids Bounded by Spherical Sur fac es— Wit h Internal Heating.


3.2.3 6, p. 274 Sphere with nuclear type heating. (t - t.Jk

fc " V r Vq 1 " = q u " [ i - 3 < r / r

0

) 2 ]^ • - * [ « - • ' - » « - ' ' ]

*0 0

3.2.4 9, p. 232 Solid sphere with convection

boundary.

(t - t f )k= i[L - R 2 + (2/Bi)]

See Ref. 85 for nonuniform convection boundary

solution.

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Section 3.2. Solids Bounded by Spherical Surf aces —Wit h Internal Heating.


3 . 2 . 5 9, p . 232 So l id sphere in an in f i n i t emedium.k = k Q f 0 <. r < r Q .It = kj_, r > r Q .q ' " = q j " , 0 < r < r Q .q ' " = 0 , r > r Q .h = con ta c t coe ff i c i en t a t t = f .t = t^ , r > «. .

<fc " V Q 1 22 — = pi - R + 2 /B i + 2 ^ / ^ ) , 0 < R < 1

o' ' l t ~0 r 0

( t - t j k = 1/3R, r > r% X0

Bi = h r 0 / k Q

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Section 3.2. Solids Bounded by Spherical Surf aces —Wit h Internal Heating.

Case No . References Descciptioii Solution

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Section 3.2. Solids Bounded by Spherical Surface s—Wi th Internal Heating.


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I

I

REPRODUCED FROMBEST AVAILABLE CO PY

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Section 4.1. Traveling Heat Source s.

Case No . Refe rences Descr ip t ion Solution

4.1 .1 2, p. 189 Traveling li ne heat source in9, p. 267 an infi nit e sol id .

t = t r x •»• « ° , y •*• ± ° ° .

• • V

» » v

(t - t„)k0 ) l t 1 / vx\„ /v V 2 __ 2 \' — = 2 ? e x P ( - 2 E T M t oV x + y )

y = ve loc i ty of line sourc e

g' = heat ing rate in line p e r

unit length

4.1.2 2, p . 291 Travelin g l ine heat source in

an infini te plate .

t = t Q , x •* °°, y + ±°°.

to.h,

I Iy- "o-'M

£L .

t Q ' h 2 ^

(t - t 0 ) w 1

q'A 2n exp ( - t > . [ A ^ * & ) ^v = ve loc i ty o f line source

q' = heat ing rate of line

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Section 4.1. Traveling Heat Sources.


4.1.3 2, p. 286 Traveling plane heat source

in an infinite medium,

t = t Q, x + <*>.

(t - t Q)kv

(t - t 0)kv

exp (-S)

1, x « 0

, x > 0

q " a

q " = heating rate of the plane

4.1.4 1, p. 352 Traveling point heat source t - tA _ nee 5L — =—9, p. 266 in an infinite solid.

t = t Q, r + ».

q'/kr 4n " 4 " 2a 'j

q' = heating rate of the point

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Sec t ion 4 . 1 . Trave l ing Hea t Sources .

C a s e N o . Re fe rences Desc r ip t ion So lu t ion

4.1.5 9 , p . 268 Travel ing plane source in

an inf ini te r o d .

t = t Q , x + ±» .

to.h-

X V

(t - t Q ) k C

q~i= <*£ (vx - |x|u\

1/2U = (v 2 + 4 a 2 hC/kA)

v = velocity of source

q' = heating rate of source

C = circumference of rod

A = cross-sectional area

4.1.6 9, p . 268 Traveling point source on the

surface of an infinite plate

and no surface loss.

t = t Q , i •*• =>.

n /Mr* <%

C O ' 1/21

n=l

x cos (•?) xp(vx /2a )

2 2 ^ 2r = x + yv = source ve loc i ty i n x-direct ion

q' = heat ing rate o f point source

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4.1.7 9, p. 268 Traveling line source on the

surface of an infinite pl ate

with no surface losses,

t = t Q , x + ±«.

(t - tQ) v w k1 + N — cos(rmz/w ) (1 - N) exp (vx /2a)

n=l

/ j

I -J/<WAN = Cl + ( 2 a i m / w » >2 ] 1 / 2

v = source velocity in the

x-direction

q' = heating rate of source

4.1.8 9, p. 269 Traveling infinite strip

source on the surface of a

semi-infinite solid with no

surface losses,

t = t fl , x •* ±» .

( t • V * _ 2 ^ + B

j ' a ir /"'X-B

exp (X)K0[{ Z2 + X 2 ) I / 2 ] d X

X = vx/2a, z = vz/2a, B = bv/2a

See Fig. 4.1

v = strip velocity in thex-direction

q' = heating rate of strip

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4 .1 .9 30 Tr a v e l i n g i n f i n i t e s t r i p s o u r c e

on a s e m i - i n f i n i t e s o l i d w i t h

convec t ion boundary,

t = t f , x + ±°°, and z .

"" /-Source = q s

( t (x ,z ) - tf)k i rvX+L

•LOO

/

V

m V Z

2

+ n , 2

, 2 - 2 .TTH ex p (HZ) J T e x p ( H ' T ~ ) e r f c (Z/2T + HT)

0

x {erf [ (X + D/ 2T + t ] - e r f [ (X - L) / 2r + i :}}*:

v = v e l o c i t y of s t r i pH = 2ah /kv , L = vJ l /2a , X = vx/2a , z = vz/2 a

See F i g . 4 . 2 .

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4 . 1 . 1 1 34 I n f i n i t e c y l i n d e r w i t h

m o v i n g r i n g h e a t s o u r c e

on i t s s u r f a c e .

(t

J

- t f ) 2 n r 0 k " X 2 J Q ( X n R ). 1 . 1 1 34 I n f i n i t e c y l i n d e r w i t h

m o v i n g r i n g h e a t s o u r c e

on i t s s u r f a c e .

(t

J

Q 0 " A , (N u2/4) + X Wn=l n

/ h . t , ^ t ( z , r . t )

(t

J

exp( f ±Vfce 2 /16) + x;; C

^(p^/ie) + xj

Nu = 2 h r 0 / k , Pe = 2 v r Q / a , 5 = - * OT

Nu J n ( X ) = 2X j (X )O n n 1 n

+ f o r 5 < 0- f o r £ > 0

i/J-'^ ~-Ring source |

t °0 \

(t

J

exp( f ±Vfce 2 /16) + x;; C

^(p^/ie) + xj

Nu = 2 h r 0 / k , Pe = 2 v r Q / a , 5 = - * OT

Nu J n ( X ) = 2X j (X )O n n 1 n

k- •- ) •

exp( f ±Vfce 2 /16) + x;; C

^(p^/ie) + xj

Nu = 2 h r 0 / k , Pe = 2 v r Q / a , 5 = - * OT

Nu J n ( X ) = 2X j (X )O n n 1 n

' R = r / r 0) •

exp( f ±Vfce 2 /16) + x;; C

^(p^/ie) + xj

Nu = 2 h r 0 / k , Pe = 2 v r Q / a , 5 = - * OT

Nu J n ( X ) = 2X j (X )O n n 1 n

See F igs . 4.4 a and b.

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4 .1 .12 34

I

I n f i n i t e c y l i n d e r w i t h

a moving rin g heat sour ce on

i t s inne r su r face .

A t ,

(t - tf)2TvrjkC - I sn = l

J n ( x R) W Y n (^ R)0 l n* ' Y,{X ) 0 ' n1 n

Xn[WV W ~ W W o>]= f N"[VXnV W - ^< V VX nV ] .

C / 2

• n " ^(Pe /16) +\'ex p I (Pe/4) ± \ ( P e2 / 1 6 ) + \ M ,

+ for? < 0- for? > 0

[ww - ww ] wBn • "I { [ W o - W - W V W ] *

+ [WWV " WWo n}

-f[WW " WWTR = r / r. , Nu = 2hr . /k , Pe = 2v r. / a , £

l l l

See Fi gs . 4.5 a and b .

Z + vr

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4.1.13 70 Infinite cylinder traveling Temperature solution given in source Ref. 70.

through temperature zones.

^ , , y- h 2-t 2

1 r v )C

— z — ~ < iZone boundary

Traveling plane source in a Temperature solution given in source Ref. 71.

thin rod with change of phase.

* i f ' iMelt zone — ' IQ = Source

4.1.15 71 Traveling plane source in an Temperature solution given in source Ref. 71.

infinite medium with change of

phase.

4.1.14 71

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iO

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<J\

|

REPRODUCED FROMBEST AVAILABLE CO PY

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Section 5.1. Extended Su rf ac es— No Internal Heating.


5.1.3 7, p. 43 Finite rod with end loss,

t = t Q , x = 0.

mkAQ =

ft i2/mk)+ ta n (m&f| ( tQ - t f )

1 + (h /mk ) t a n (mJl)

h i - t f-h„t,V ^ " H

P

t - t co sh QnU - x) ] + (h /mk) si nh [m(£. - x)]

co sh (md) + (h /ink) s in h {ml)

iu = kA

5.1.4 7, p. 44 Straight rectangular fin.

<

h . t ,

°7Eu

Use m = •Wrr in cas e 5.1.3 .

tanh (ml ) .

— a -1

- - *c =* +

See Fig. 5.2

Recommended design criteria: 2k/hb > 5 .

Minimum weight criteri a: 2X,/b = 1.419 V2k/hb

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Section 5.1. Extended Sur fac es— No Internal Heating.


5.1.5 7, p. 52 Straig ht triangular fin.

r ^bf^-o

q = Y2hkbI ± (2>p)

I Q (2-vfI)

2hHkb

t - t f I 0 ( 2 ^ x )fc0 " fcf I (2V0I)

± I 1 (2TÎ)See Fig . 5. 2,

Optimum %/h r at io : Vb = 0.655-j | jr .

5.1.6 1, p . 228 St r a igh t f in of minimum mass, t - t " ^n ~ fcp' x

fc 0 fc fi q'

q- = total heat loss.

Profi le:

y = 2k ( X <* \ 2

y = 2k ( X ( t Q - vf q '

- h(t Q - V

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Section 5.1. Extended Su rf ac es —N o Internal Heating.


5.1.7 10, p. 95 Straight fin of convex

parabolic profile.t - 1

t o - fc f W

m = V2h/kb

q = kbm(t 0 - t f )

wi_,„(f^V / 4 )1/3V

211

-1/3

(M

(H

(H1 I 2 / 3 ( ? m i l )

-1/3

, See Fig . 5.2

5.1.8 10 , p. 175 Straight f i n of trapezoidal t - t g

parabolic profile. t - t , ~ \Jl /

f W » ' R - i + i" 2 + 2 VI + 4 m'Jl „2

^ ( t ,- T" (t oI

3 + 1 '

V

See F i g . 5.2

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5.1 .9 10, p. 175 Str aig ht fin of trapezo ida l

prof i le .

t - t . K1«WB> + WV B )

fc0 " fc f = WW + W WKl ((y W - W WS c ( t 0 V

H 2 cb s G _ 1 e 0 c 1 e 0 c .

B K (BJ 1, (3J - 1,(0JK (0 )c 1 e 1 c 1 e 1 c

2H* W W + WW

0 = 2H

2H

\ x +b (1 - ta n 9)

e _2 tan 0

= 2H M - ta n 9)

2 tan 9

VI +b (1 - ta n 9)e

2 tan 9 , H = Vh/k s i n 9 ,

S L = £ + —C 2

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Section 5.1. Extended Su rfaces—No Internal Heating.

Case Mo. References Description Solution

5.1.10 7, p. 54 Cylindrical fin—rectangular

profile.

I (x m)K (x m) - K (x a (x m)_ _ „ Mrmffr ^ \ I D 1 e 1 D

y - <«rx bDKmtt 0 - t f j x m ) K ( x m ) + K ( x m ) T ( x m )e ' <T b l l e ' ( K b

«WuwJ ^-h,t,

m = Y2h/kb

t - t K. ( x m) I <rm) + I {x m)K n (rm)t_ _ 1 e 0 1 e 0t Q - t f - K l ( x e m ) l 0 ( x b m ) + I ^ B O K ^ X J B . )

f, , / ,21 I . (x,m) + BK„(x.i

' I ^ x ^ ) - BK x m)

,m), See Fig . 5.3

8 = I , ( x / x ) / K (x / x. )1 e o 1 e b

Minimum material: use Fig. 5. 4.

5.1.11 7, p. 58 Cyli ndrical fin—triangul ar91 „ . n

p r o f i l e . x b m[l - ( x e / x b ) 2 ]

- I - 2 / 3 ( 6 ) + g I 2 / 3 ( 5 )

i 1 / 3 ( 6 ) + Bl _ 1/ 3 ((S)

m = V2h7kb, 6= - x b m, - I _ 2 / 3 fe)Se e Fig. 5.5.

2/3 i,3/2-

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5.1.12 10 , p. 106 Cylindrical f i n —

hyperbolic profile .b x b

Q 2 . W * , , , t0 t f ) [ l _ a / 3 ( B 8 ) I _ l y 3 ( B b ) - I 2 / 3 ( B e ) I 1 / 3 ( B b ) ]

(t - t f )( t 0 " V

(x_\1/2 [ I 2 / 3 ( B e ) I l / 3 ( B ) " J ,^ b / [^'V1 1/3 <V

M ' V ' - l f l1 " ]I - 2 / 3 ( B e ) I - l / 3 ( B b » ]

1 / 4 0 ( 1 - 0 ) \1 / 2 [ I 2 / 3 { B e > I - 2 / 3 " ' - 2 / 3 ( Be » * 2 / 3 < V ]n \ d + P)^n (1/pJ P ^ ' V ^ ' V " '2/3 <e> *l/3 ( V ]

M = Y2hAbx B = | M x3 / 2 , B = f MX3, 7 2, B = | M xV 2

b e 3 e b 3 b 3

, 3 / 2D = <xe - *h) ' M/Vin l / p , p = xf a / x e

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Section 5.1. Extended Sur fac es —No Internal Heating.


5.1 .13 1 , p . 234 C yl in dr ic a l f i n of minimum

m a t e r i a l .

UIi

hux 'Q = -j± ( x + 2 ) ( x e / ^ - 1 ) ( tQ - t £ )

P r o f i l e :

y W " ^ ( x j " 2 x e

+ 6 x

5 . 1 . 1 4 1 0 , p . 114 P i n f i n s — c y l i n d r i c a l t y p e .

^ h . t ,

t o — - * - — t , — ^

Q = ~ kd2m ta n h (mK.) ( tQ - t f )

<) = • t a n h J m a ) , "I = > W k d , See Fi g . 5 .1

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5.1.15 1 0 , p. 115 Pin f ins—rectan gular type.

Depth of fin = b.Same as for case 5.1.14 except I —d I is replaced by (bd)(H

.1 C

<,-/F r^5.1 .16 10, p. 117 Pin fi ns—c oni cal ty pe. , . 2„ I„ (M)

U Bl I^M) l C 0 V

t - t f ft \ (M>Â)t Q - t f \ x I l ( M)

, See Fig . 5.1

4I 2 (M)

Ml] ")", M = 2V2 ml, m = V2h/kb

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5 . 1 . 1 7 10, p. 119 P i n f i n s— c o n c a v e p a r a b o l i cs = W-3]<t 0 - V

t - t

^ f

_ i+ W w2

<>1 + (I + | M ^ ) 1/2

, M = V 4 h / k b I , S e e P i g . 5.1

5 . 1 . 1 8 10, p. 121 P i n f i n s — c o n v e x p a r a b o l i c

t y p e .Q jAh t _ t )u 16A I Q ( M ) ( t o c r

« . b r ' * \ , / 2

t - t f I 0 l l M ( x / J l ) ]3 / 4

I 0 ( M ) ' M " 3 V kb

2 V M >M IQ (M)

, S e e F i g . 5. 1

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Section 5.1. Extended Surf ace s—N o Internal Heating.


5.1.19 10 , p. 134 Infinite fin heated by square See Fig. 5.6

arrayed round rods.

— Sn

5.1.20 10, p . 135 In fin ite f in heated bye q u i l a t e r a l - t r i a n g u l a rarrayed round rods,

h.t.

E ~Pk-«n

See Fi g. 5.7

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5.1.21 10, p. 185 Straig ht rectangular fin

with nonuniform cooling.

h(x) " + (iy-h = av coeff.a

/h(x).t,

It

t - t. i&Mt Q - t f 2 l / ( Y + 2 >

1 / (Y+2)I Xil <V

1 i <> + i J + 2 , tu D ) J i

* L ( m * ,2

^+ 1

> J

Y+2

V(Y+2) r i

Y+ 2 '

(U)

( Y + D / ( Y + 2 ) ( V1 (Y+D/(Y+2) ( V

V+2

L Y+2.

2 V Y + T r - Y / 2 v ( Y + 2 ) / 2Y + 2

U „ = *• L ml, m = V2h / kbJ, Y + 2 a

.1.22 10, p. 191 Same conditions as case

5.1.21 with:

h(x ) = h.["(1 + a) (x/fc +

{(1 - = ) a + 1 " «cTa+l

n| I { 0)1. „(U 0 ) - I „(U)I , ( 0 0 )/ u \ " n ' ^ l - n ' ^ ' *- n 1 " ' n - l 1 " ^

W I . W W V * I

-B<D

0) I

»-l( D

.2(1 - n)

T / , 2( l- nn"oL1 ( W J

V l ' V W V " H -nô'Vi 'VWWV WVWV J

- - K^)/2 ( x / & + c) 1/n

(1 + c ) a - n ) / n _ c ( l - n ) / nml

u, x = 0; \i Q = u, x = l , n = — , m = V2h / kb2 a '

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5.1.23 10, p. 187 Same conditions as case 5.1.22 2m£ _ / xc\n = — ' U = n V 5 e X P ( - 2l)with: x '

[ , , ... _ i|) = nVa exp(c), 8 = nVa

1 - (a/c) p. - exp (-c)J J If n is an integer:

0 = . 3f ^1 ~ h2v 2m£(l - (a/c)Cl - exp(-c)]) A 3 - A 4

\ = [*„_ <*> - V l 1 * 1 ] [ J n-1 (B > " V l ( S > ]

A2 " [ J n - 1 " W " * ] [ V l ( B ) " W e > ]

A3 = J n ( B ) [ V l ^ - V l ' *1 ]

A 4 = Y m ( e , [ J n - l W " Jn +l ' *>]

If n is not an integer:

. c . A l " A 2

2(raH)2(l - (a/c)Cl - exp(-c)]) A 3 A 4

A l = [" nJ -n ( l | ; : " ^ l - n W i r- " ^ ' 6 ' + 0 J n - l ( e ) ]

A2 = [-J n(*> + * Jn_ 1(W][-nJ .n(B ) - B J ^ t B ) ]

A3 = J n ( e ) [ ~ n J - n ( , , , , " ^ l - n ^ ' l

A4 = J - n

( 0 ) [ ~ n J n ( , t ' ) + ^ n - l W ]

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Section 5.1. Extended Surfaces —No Internal Heating.


5.1.24 10 , p. 205 Straight fin of rectangular

profile radiating to free

space.

t = t 0 , x = 0.

T = T , X = l .e

Space temp = O K .

OK

\>.r^-

Z^

4 0oeT_

kbB(0.3, 0.5) - B (0.3, 0.5)

B = complete beta functionB = incomplete beta function

Q = k b \ f | h ( To " e ) ' s e e F i g s - 5 , B a n d 5 - 9

2 >l/z 3 - L/z 8

=

**' 20o« /kb ' V e

2 3Optimum length: b/i. = 2.486aeT A

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5.1.25 10 , p. 211 Straight fin of rectangular

profile radiating to

nonfree space.

T = T , x = 0.

T = T , x = I.

K = incident radiationabsorbed from surroundings.

K 2

2 K „ T :_ f l L . .l p L g v r - [kbT0 \ 5kbZ

Ql/kb1Q

2ae L2Tpkb

4 3 2 5 Z \Z + Z + Z + Z + 1 —K 1 T 0

1 / 2

, K = 2a e , z = T / T1 u e

S e e F i g s . 5 . 1 0 a n d 5 . 11

O p t i m u m l e n g t h : b/SL2 = 2 .365aeT:? /k

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5.1.26 10 , p. 218 Straight trapezoidal fin

radiating to nonfree space.

T = T , x = 0.

K = incident radiationabsorbed from surroundings.

Se e Figs. 5.12a, 5.12b, 5.12c, and 5.12d.

\ = b / b . = 0.75, 0.50, 0.25, and 0 (triangular fin)e 0

K = 2 ere.

5.1.27 10, p. 228 Straight fin of least material Profile:

radiating to free space.

T = 0, x = 0.f(x) = _JSL.

2kaeT;

7/2

( ) ' •=

3Q4

2aeT„

OK

fc-(fi1/2

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5.1.28 10, p. 230 Straight fin with constant Profile:

temperature gradient

radiating to nonfree space

T - TdT 0 e

dx =

SL

10kZ 3 (Z - 1 ) Z < L *• J KjT* *-*• Jl

K S,T 4(Z 5 - 1)Q = 5(Z - 1) K2l

Z = T Q / T e

K = 2 ere

z 5 - i5Z 5 (Z - 1) KjT*

K_ = incident radiation absorbed

T = ( K , / K , )1 / 4

e,rain 2 1 t See Figs. 5.13 and 5.14

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Section 5.1. Extended Su rf ac es —N o Internal Hea<-i


5.1.29 1 0 ,

p . 247

Radial fin of trapezoidal

profile radiating to nonfree

space.

T = T„ , r = r„ .0 0

Fin efficiency: (See Figs. 5.15a-5.15£)

X = b /b„ = 0, 0.5, 0.75, 1.0e o

K

2/ K

1T

0 =

° ' ° "2

' °- 4

Prof i le No. = K^r j j /kbj j

K. = oz , K_ = incident radiation absorbed

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5.1.30 1 0 ,

p. 275

The capped cylinder,

convection boundary,

t = t Q , x = I.

Insulated inside surfaces.

M f - i 5 r„

- H I — S t.

1

Side:

t - t . cosh (mx) + sinh (mx) fl, (mr )/ l 0(mr )~\£_ LI e 0 e -1

t„ - t ~ co sh (mi) + si nh (mS,) r i , (mr )/ I„ (m r )1O r L l e Q e J

si nh (rnS.) + [c o sh (mS.) - l ] [ l (mr ) / I (mr >]

^ =

mS ,|cosh (mX.) + s in h (mS.) [ l1 ( m r e ) / I ( ) { m r e ) ] }

Top:

t - t . I 0 ( m r ) / I0 ( m r e )

t . - t , co sh (m£) + si nh (m£) fl., (mr )/ I„ (m r )"lO r L l e O e J

2 I 1 ( m r e ) / I 0 ( m r e )mr jc o sh (mS.) + s in h (mJl) [ I (rar ) / I (rar )1 I

m = "Vh/kfi, See F ig s . 5. 16 a, and 5.16b

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Section 5.1. Extended Suc Ca ces —No Internal Heating.


5.1.31 10 , Doubly heated straight,

p . 407 rectangular fin.

t = t Q , x = 0.

t = t & , x = I.

fc - "f

t o - t f= exp(mx) - 2B sinh (mx)

q Q = kbm(2B - 1) (t Q - t f )

r'' i0

q fc = kbmJexp{mJl) - 2B co sh (mJl) ( t - tf ) ]

exp (mi) - ( tp - t ) / ( t - t )1 Tr r r -2 - m=V2h7kbsinh (m£)

See Pi g . 5 .17

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Section 5.1. Extended Su rf ac es —N o Internal Heating-


5.1.32 1 0 ,

p . 394

Single T, straight,

rectangular fin.

t = t Q , x = b.

A l l s u r f a c e s c o n v e c t i v e l y

cooled by h , t f .

%LI

n 1

b

id

It

• ^

Vertical section:

t - t^

t> = —x m xb

— = FJ co sh (m x) + "V2u/v tanh (m a) s in h (mxx)]

|s in h {m b) + V2u /v ta nh (m a) £cosh (m b) - lj )

Horizontal section:

t - t , cos h (m-Ty)= F £'t Q - t f " cosh (m a)

F tan h (m a)m a

y

, See F ig s . 5 .18 a, 5 .18b

co sh (m b) + V2u/v tan h (m a) si nh (m^b)

m = VhTiw, m = Yh/kux y

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5.1.33 10,

p. 413

Double T, straight,

rectangular fin.

t ** V xi = b i -

All surfaces convectivelycooled by h, t f .

'/AAX 2 b 2

H

m\ u i, b, U,

EI

Ve r t i c a l s e c t i o n s :

t - t f

- — ^ - j - = F^T cosh (nijjjXj ) + ^ si n h (n > x l x 1 )] , 0 < x1 + K l ( c o s h ( m x l b l ) " D ] . 0 < xx x2 " ^ [ S l n h ( m x 2 b 2 ' * V 2 V V 2 t a n h ( r ay 2 a2 >

x (cosh (mx 2 b 2 ) - 1 )] , 0 < x2 < b 2 .

Hor izon ta l s e c t i o n :

t - t f co sh (m YX)

t Q - t f

= F l cosh 0 " ^ ) ' ° < Y l < a l

t - t. cosh (m„2y 2)=— = F F y , 0 < v < a

t Q - t f 1 2 c o sh ( my 2 a 2 i ' *2 2

V = i v ^ t a n h W ' 0 < y i < a i

f y 2 = ^ a 7 t a n h ' m y 2 a

2 ' ' 0 < y 2

< a 2

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5.1.33 continuedF =

1 cos h (in _b ) + B sinh (m b )

2 cosh (ra jc,t)2) + 1 ' 2 u 2 ' / v 2 t a n h ^ mv 2 a 2 ' S i n h { m x 2 b 2 '

B = F

2 " Vv

2/ v

i [s i n n ( m

x 2b

2) +

'V 2 u

2/ v

2 t a n h ( m

y 2a

2 '

x cosh (m 2 b_ )l + 2Vu /v tanh (ra b )

Oli

mxl = W** ' \2 = V 5 E 7 1 ^ 2

m x = V 2 h / k u 1 , m 2 = V 2 h / k u 2

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5.1.34 2, Insulated wires adjoined on t - t m 1

p . 179 a co nv ec t iv el y coo led p la te t - t^ 1 +(fe Kj( r H)/K ( r Hj]( e . g . , t h e r m o c o u p l e w i r e s ) ,t = t , r - co.

Wire ther m, con d. = k .

In su la t i on the rm , cond . = k .P l a t e the rm, cond . = k .

P

tS U T2 , l f 2 .

B =Wire r a d iu s = r . ~ VET + Y k Tw wl w2Wi re i n s u l a t i o n t h i c k n e s s = 6 .

kpWd^ + h2 ) ( l / h i + S / k ^ l1 7 2

rw

w ' H = "V2(h + h2 ) / k w

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Section 5.1. Extended Su rf ac es —No Internal Heating.


5.1.35 9,

p . 157

ito

Composite finite rod.

t = t Q , x = 0.

t = t 2 , x = x 2 .

A 1 , A_ - areas.

C. , C_ = cir cumfe re nces.

Lh, , t f ph 2 ,t f

cosh (A) si nh (B) + H si nh (A) co sh (B) + (T /T ) si nh (Dx/x )

T ~ si nh (D) cosh (A) + H cosh (D) si nh (A)

0 < x < x.

A = m 2 (x 2 - X l ) , B = m 1 (x 1 - x) , D = m ^ , T = t - t f

E " V W l V W z V »1 = V h l C l / k l A l ' m 2 = V h 2 C 2 A 2 A 2

h— xi—Û x.F — * x

2 J l cosh (A) sinh (B) + H sinh (A) cosh (B) + (T /T ) sinh (m b - rax)

sinh (D) cosh (A) + H cosh (D) sinh (A)

\ < X < X 2

A = m 1 x 1 , B = m 2 (x 2 - x) , D = m 2 (x 2 - x ), T = t - t f

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Section 5.1. Extended Surf ace s—N o Internal Heating.


5.1.36 9, Case 5.1.35 with: t - t_ cosh (m l x. - IILX)

p . 157 q = 0, x = x . t - t cosh (m x ) cosh (m, x 2 - "• x ) + H si nh (m x ) sinh (m x - m x )

0 < x < x .

(t - t f ) cosh (m 2 x 2 - ra 2x)

t n - t- H cosh (m.x ) cosh (m x - m x ) + sinh (m x ) sinh (m_x - H L X )

X . < X < X . .

H = V 5 ^ V W A ' m i = V h l C l / k l A l ' m2 = Vh 2 C2 / k2 A 2 - >"2"2 2""2' "1 1 "1 1 ' 1 ^ - j r i ' l ~ r 2 y ~2 2' "2 2

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Section 5.1. Extended Su rfa ces —No Internal Heating.


5.1.37 31 Concave parabolic fin See Fig. 5.19 for valu es at <b(N ,N ) .c s

radiating to non-free space.

T = T , x = 0.

K_ = incident absorbed

N c = (2eoF . 2T^)/kb 0, N S = K 2 / 2 e a F T * .adiating to non-free space.

T = T , x = 0.

K_ = incident absorbedF = view factor.

radiation. .

Profile: f (x) = - | (1 - x/l) .

5.1.38 31 Convex parabolic fin

radiating to non-free space.

Conditions same as for case

5.1.37 except profile is:

f (x ) = - | V l - x/l.

See Fig . 5.20 for valu es of <j>(N , N ) ,

N and N given in case 5.1.37.c s

5.1.39 81 Straight rectangular fin of

variable conductivity.

k = k 0[l 3 ( t - t 0 , ] .

See Ref. 81 for approximate temperature and efficiency

solutions.

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Section 5.1. Extended Surfaces—No Internal Heating.

Case No. References Descr iption Solution

5.1.40 83 Sheet fin for square

array tubes with

convection boundary,

h f t f .

*

f

o

oooo

- t — 1 - —

I J I

oooo

J J I

+x> +x>

fc- fcf

fco- fc f

2 , Z K

0 lV 5 IOx - mA - 2 + <Y - ^ )

n=-»» m=-°°jca jut

K ^RUST) + i n {R>ET) ^ ^ K Q [AVBT(m 2 + n 2 ) J

n=- c ° m=-°°

n, m 7 s 0 .

21T

K > § l [ ( A / R )2 - IT]

K (KVST) - I ^ R Y B I ) V V K [AVIi(ni 2 + n 2 j], 1 /2

= - m m = - 0 3

K 0 ( K V 5 T ) + I 0 (HVBT) ^ ^ K 0 [AVBl(ni 2 + n 2 ) ]1/2

n=-°° 111=-"

n , m ? 0 .

X = x/ b, Y = y/ b, R = r /b , Bi = h b/k, A = a/b

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mi

to

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Section 5.1. Extended Su rfa ce s—N o Internal Heating.


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Section 5 . 2 . Extended Sur fac es—W ith In te rnal Hea t ing .

Case N o . References Descr ip t ion Solution

5.2.1 1 0 , Straight, rectangular f i n . (t - t )

p . 19 4 q = q Q , x = 0.

q x = 0 , x = b .

1,-h.t,

iUl

I

f . i_ cosh (mx) . . , . , g' ' 'b— mk b = -—- • I „ . - s inh (mx) + a

tanh (m&) q 0m

Optimum profi le:

sinh (2mA) i - g 0 / h b ( t 0 - t f )

2mil 1/3 - q 0/hb(t 0 - t f ) ' ° i V h b ( t 0 " V < 1 / 3

m = -V2h/kb, t = t Q , x = 0

5 .2 .2 1 , p . 246 S t ra igh t in f i n i t e rod .t = t Q , x = 0.

h.t,kq'"

F 7

t - 1 ,fco- fcf

= 1 + 2 1 -km (t f l - t f ) Miti

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Section 5.2. Extended Surfac es—W ith Internal Heating.


U1i

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Section 5.2. Extended Surface s—Wi th Internal Heating.


inI

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f\

I1

IREPRODUCED FROM

BEST AVAILABLE COPY

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Section 6.1. Infinite Solia s—N o Inte \i Heating.


6.1,1 9, p. 54 Infinite plate source.

t = t Q , - b < x < b , x = 0.

t = t . b < x < - b , T = 0.i-rr - [«* K - K)+ e r fK + F0 ] *°° < X < <

See F i g . 6 .1

x = "b x = b

x

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Section 6.1. Infinite So li ds —N o Internal Heating.


6 .1 .1 .1 74, Case 6 . 1 . 1 wi th a p l a t e of

p . 424 d i f f e r e n t p r o p e r t i e s .

© 0 ©

l^k^-TTil (-H rfcn=l

(2 n - 1) - X

2VFO.b l

+ e r fc <2n - 1) + XZW% 1

J , - 1 < X <+1

t - t.— = K e r £ c / X " 1 \ - K ( 1 + H ' V d H ) 1 1 - 1

\ D2/ n= l

/ X - 1 + 2nVa^/ot \x e r f c [ -—=• , 1 < X < - 1

V " ^ /

P2C

2k

2 1 + K

Interface temp

t - t7 - = V . K ? > (-H) e r f c / F o ) , X = 1fc » X + K (1 + K ) 2 £ V b 1 '

Por Fo .„ s m a l l :D Z

fc 2 ~ fc » _K . / X - 1e r f co °° ziss-y a + K )-

2K , / X - 1 + 2 Vl\erfc l—2 V P o ^

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Section 6.1. Infinite Sol ids —No Internal Heating.


6.1.1.2 74, Case 6.1.1 with a pulse source(t

p . 427 of heat of strength Q J/m" at

X = 0, T = 0.

t = t Q, -«> < x < °°, T = 0.

V P l C l b 1 / ( X 2 \ V n-1-^H^/\? ( _ H )

{ (2n - X) '+ e x p

n = l

(2n + X)'4 Fo, - 1 < X < 1

« t ^ VP^c^b

IX (-H) n - l

x exp[ fx. - 1 + (2n - D V ^ T T ]2 Ij_L_l _ _ 2_ J_f, K X < - 1

K = Wi n l - KP 2 c 2 k 2 ' 1 + K

6 . 1 . 2 9 , p . 5 4 C a s e 6 . 1 . 1 w i t h : °?_ _ 1 -_ Xt = t Q ( b - x ) / b , -b < x < b, t - t a

T = 0 .

t = t m , b < x < - b , T = o.

* K - K)+ H^ «*K + K) - * ««j . ; ' a T ,

b TT ( L a

- (x + b) 'exp

• (x - b) '

a 2 J2 exp

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Case Ho. References Description S o l u t i o n

6 . 1 . 3 9 , p . 56 I n f i n i t e r o d s o u r c e .

t = t Q , |x | < d , |y | d ,

| y | > b , T = o.

t - t

I - T T ; = i [e r f K - F ox)+ e r f K + K)]x [e r f ( F O - F o * ) + er f ( jFo , + Fob y)]

• + •

6 . 1 . 4 7 4 , p . 4 32 I n f i n i t e c y l i n d e r s o u r c e ,

t = t Q r r < r 0 , T = 0.

fc = fc= o r > V T = °"

exp(-X Fo)C 0 " ^ " Jo

\ \ = K f e x p ( - X2 Fo)c 0 » IT SO

j 0 a R ) j 2 ( X )

J 1 ( X ) YQ(X) - JQ( X ) Y1(X)

J 0 ( X R ) J1 ( X )

f " j1 ( X ) Y0 ( X ) - J0 ( X ) Y1 ( X ) ' j2

a , » > •

dX , R < 1

See Fi g . 6 .2

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6. 1. 5 9, p. 55 Sphe rical sour ce,

t = t 0 , r b , T = 0

t - t

t = t 0 , r ; • K 0 )

2 ] )2 Fo

See Fi g. 6.3

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Section 6.1. Infinite Soli ds— No Internal Heating.


6.1.5.1 74, Case 6.1.5 with a sphere of

p. 429 different thermal properties.

x erfcA-t-^ _ p V F O 7 \ - e c f\ 2 V F O ; 1 X 2 V F 0 7 / I

fc 0 ' R ( ^ z Vi o ^ J K 2 " ^ V F O ^

- V l j - exp(pjto 2 - P 2 R - l)erfc/ j-~^ - P. V F O ^

t - t

^ 2 _ + K ^(K 2 - 1) (K n

• ^ - e r f c / - 1 + 2 ^ " A + — K ^ ^" 2 - 1 \ 2 V ^ ; ( K 2 - l ) ( K i + l )

x e x p f p2 F o 2 - P 2 R - 1 + 2'Va 2 / a 1 êrfc/ R - 1 + 2y^/\

V ^

P 2 V F O ^ \ \ , R > 1

P l ° l k l k l K 2 " l K 2 ~ X

K l = V ^ ' * 2 = V ? 1 = K2 + V K 1 ' ? 2 = ^ ^f (+R) = f (-R) - f(+R)

http://2vfo/

http://2vfo/

http://2vfo/

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Section 6.1. Infinite Soli ds— No Internal Heating.


6.1.6 9, p. 6x Adjoined semi-infinite regions,

t = t , x > 0, T = 0.

t = t 2 , x < 0, T = 0.

P V I K O - ] , - - 1 ,OQ < j; < 00

6 .1 .7 7 , p . 89 Spe c i f i ed t emp era tu r e

d i s t r i b u t i o n ,

t = f ( x ) , - » < x < °°, T

CO

vl / > + X0)exp(-X )dX

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S e c t i o n 6 . 1 . I n f i n i t e S o l i d s — N o I n t e r n a l H e a t i n g .

C a s e N o . R e f e r e n c e s D e s c r i p t i o n So lu t ion

6.1.8 9, p. 88 Adjoined semi-infinite

regions with contact

resistance,

t = t , x > 0, T = 0 .

t = t_, x < 0, T = 0.

«2

t - t

t - t,

t^ \ m r h I1 + A [ e r f K )+ e x p( v+ Hf2/4)x e r f c (Fo + HO /2Yj 1 , x > 0

x e r f c ( | F o * | + H2a 2/2YJ , x < 0

= hyP H = h ( 1 + A > H = - & < 1 + A )k ^ C t ' H l k , A ' H 2 k ( 1 A )

1 2

6.1.9 9, p. 88 Adjoined semi-infinite

regions—no contact

resistance.

tt

t , X > 0 , T = 0 .

t 2 , X < 0 , T = 0 .

__JL . _ i_ [x + A e r f ( Pc 0j ,

t~rr 2 - TÂ e r f (K 0' x < 0

A = ^k l "2

x > 0

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6.1.10 9, p. 89 Plane source of heating

between adjoined semi-

infinite regions.

t = t . , < X < <=, T = 0.

\ = q 0r x = o, x > o= heat source strength.

X »i

( t - t >k , , .J- 1 = T - ^ T ierfclFo ),

nO. 1 + A \ x/'x > 0

"01

( t - V k 2q 0 a 2

= r T T i e r f c ( l F o x l ) ' x < 0

k * a1 2

6 . 1 . 11 I F p . 3 4 1 L i n e s o u r c e o f h e a t .

9 , p. 2 61 t = t , 0 < r < rQ, T = 0.

i l l

t - t

t „ - t It Fo0 °°

e x p (- K 2 ) • > V r0 * °

I f l i n e i s a s t e a d y s o u r c e o f s t r e n g t h q :

(t - t )k

"i/Vf

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6.1.12 1 , p . 34 2 C y l i n d r i c a l s h e l l s o u r c e .t " V r i K r < V T =t = V r > rQ , T = 0.

t - tz~ = (Fo * - Fo 2 ] e x p [ - ( F o *2 + F o *2 \ J„ /2 i Fo Fo \

t o "f c- V r o V L V r r

i / J ° V r r J

-v=i

6.1.13 9 , p . 247 S p h e r i c a l s u r f a c e a t s t e a d yt e m p e r a t u r e ,

t = t . f r > t , T = 0.

t = t 0 , r = rf l , T > 0.

t ' fci 1 / *- — - = - e r f c ( F or - Fo

0 l \ ; )

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6 . 1 . 1 4 9, p. 247 Spher i ca l su r face wi th

convection Ipoundary.

t = t . , r > r Q , T = 0.

B i- t.i _ _

t t - t . (Bi + 1)Rf l

e r f c fPo - Fo | e x p [ ( B i + 1){R - 1)

+ (Bi + l ) F o ] e r f c Fo* - Fo* + (Bi + 1 ) Y F O I .r o J |

IP 6 . 1 . 1 5 9, p . 248 Spher i ca l su r face wi th

s teady hea t f lux ,t = t i f r = r Q , T = 0.q r = V r - V

( t - t )k x r , *~ = =r e r f c (Fo - Fo

% r0 ; )

- exp(R - 1 + Fo)e rfc( Fo^ - Fo^ + V F O | |\ r n

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S e c t i o n 6 . 1 . I n f i n i t e S o li d s— N o I n t e r n a l H e a t i n g .

C a s e H o . R e f e r e n c e s D e s c r i p t i o n S o l u t i o n

6 .1 .1 6 1 , p . 336 P o in t so u rc e .t - t." °° C 0 / „ *2 \— = - exp - Fo ,i " fc°° A C U V r /= t . , o < r < v x = o. TT^X, = 7 T ^ e x p l" F°r"i ' r > V ro * °

t = t m , r > r , T = 0. X °° 6Vira T

If point is a steady source of strength Q fl :

( t S t ) k rf l

V r 0 " 47r3 / 2V a 1 ? V T

— / e x p l - X F o JdXff A/= V r '

F o r T •+ <*>:

( t - t j k r 1Q 0

= 41T

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ction 6.1. Infinite Solids—No Internal Heating.

se No. References Description Solution

17.1 1, Spherical shell source.

p . 340 t = t . , r. < r < r1 1

T = 0 .

0 '

t - fc ~ R 0 - 1 [ ( R - l ) :

t. - t_ 6VTTRVFO e x p L 4Fo exp(R + 1)

4Fo ' R > V R 0 * °

R = r/r.

I f the shell i s a steady source of strength Q :

(t - O k VoTV r i 47t 3 / 2

For T •+ °°:

(t - t J kr

OO

exp

, 2,2(r - r ^ X

4a - exp(r + r ^ 2 * 2

4ad)

_14ir

1 . 1 7 . 2 1 , p . 342 Plane sou rce.

t = t t , -a <

t - o |x| > i, x = o.9, p. 263 t = t., -A < x < +J-, T = 0.

/MAA

rtA( y

I I

t - t_

1

If the plane is a steady source of strength q

r r ^ f - = ik exp (~ F o x 2 ) ' i x i * l>*+ °

(t - t j k

1/Vx A

2 2exp(- x X /4a)dX

= VFO"o exp I (-«0-Jf ««.(»;„ i)

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Section 6.1. Infinite Solids—No Internal Heating.

Case N o . References Description S o l u t i o n

6.1 .18 9 , p . 335 Cyl inder wi th s teady

surface temp,

t = t i f r > r Q , t = 0.t

= v r

- vT y

°-

CO

o i J n

JQ( X R ) Y0(X) - YQ(XR)J0(X)Fo) —

See Fi g . 6 .4

<Jr

n

Jg{A) + Yg(X)

1/2k ( t „ - t . )

0 l

For Fo < 0 .02 :

1t - tt)

( t 0 - t . ) " V5

ax_x

. „ « , - ^ . i - l ( P ) - , if c . . . . . -

32R

6 .1 .19 9 , p . 337 C yl ind r i ca l su r face wi th

convec t ion

t = t i f r > r Q r T = 0.

' ""*: ^ f , B i fboundary. - — = -2 — It . - t_ if I exp(- X'Fo)

J 0 ( X R ) [ X Y1 ( X ) + B i YQ(X)] - YQ(XR)[XJ1(X) » Bi J 0(X>] ^

[ X ^ t X ) + B i J

0 ( ^ ) ] 2 + [X ^ lX ) f B i YQ{X)J

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Section 6.1. Infinite Solids—No Internal Heating.


6.1.20 9, p. 338 Cylindrical surface with

steady heat flux,

t = t i f r > r Q , T = 0.qr = V r

"" V T > °-

I

( t ~ V k 2 f r 2 1 J

0 < X R ) Y1 ( X ) " Tf 0(XR)J (X)dX

— — — = - - / L 1 - e i cP ( ^ F

°>J r , 5 — T^ 0 * JQ ^ jJ (X) + Y * ( X ) J X

2

For Fo < 0.02:

(t - t.)k

V o

For Fo » 1

(t - t.)k

i'K <lFo . , /R - 1\ 3R + 1 ^ = - .2 . /R - 1\ i - = 2 V - i er fc (- ^ =) - - j j - VPS" I e r f c ^ - ^ )

= An /^ f^ - 0.57722v o ~*"U 2>

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Section 6.1. Infinite S^ Ld s—N o Internal Heating.


6 . 1 . 2 0 . 1 7 4 , C a s e 6 . 1 . 2 0 w i t h tw o d i f -

p . 434 e r e n t t h e r m a l m a t e r i a l s ,

t = t . , r > 0, T = 0.

g c = q n , r = r0 , T > 0.

n

( t. - t . ) k J„ (A R)J . t t )dA

Vo / J L v ^ X4< < > +

(t - t4 ) k

X R i * • v o \ ^ a X R ' *

^h =^ / " [ l - expô j ~^f-

* K J i ( X , 7o W ?x ) - > § • Vx > J i W r x

(<j) + il l' )

J , (A)

, R < 1

(* + r )

dX , R > 1

"42

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Section 6.1. Inf ini te So l id s—N o Internal Heat ing.


i

6 .1 .21 9 , p . 342 I nf ini te conducti vi ty

cylinder in an in fi ni te

medium.

t = t Q , r > r Q , T = 0.

t = t , r < r Q , T = 0.

CO

fc " t 0 40 f , , 2 _ %dAR < 1

B = 2 p 0 c 0 / p l C l

A = [XJ„(X) - B J Â ) ] 2 + [XY 0(X) - BY^X)] 1

See Fig . 6.5

6.1.22 9, p. 342 Infinite conductivity

cylinder with steady heat

source in an infinite medium.

t = t , r > 0 , T = 0 .Heat rate = Q per unit length.

(t - t Q )kCO

1 - e x p ( - X 2 F o ) ] ^ - , R < 1X 3 A

B and A defined in case 6.1.21.

See Fig. 6.6.

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Section 6.1. Infinite Solids— No Internal Heating.


6.1.23 9, p. 346 Cylinder with properties fc » j(XR)J<X)dXdifferent from surrounding — = —fr I exp(-A Fo , ) — _ r _ r — , R < 1

medium.

t = t L , r < r Q , T = 0.

t - t r , N J ( X R ) J (X)d)rounding = -£ / exp(-A Fo ) — - p - ^ T

fc - * 0 2Y f ( , 2 , \ J l ( X ) [ J 0 ( e X R ) " ' - Y 0 ( 6 X R ) ' t ' ] d X

= —*- I exp -X Fo, I = -

0

i|) * Y ^ c X j j j j j B X ) - B J0 ( X ) J1 ( B A )

<> = y J ^ X j y ^ B A ) - B Jo ( A ) Y1 ( 0 X )

R > 1 .

A X y B - V 5 ^ , Y = k A00

S e e c a s e 6 . 1 . 4 f o r e q u a l m a t e r i a l p r o p e r t i e s .

6 . 1 . 2 4 9 , P o i n t s o u r c e o f r e c t a n g u l a r ( t - too)4Trrk

p . 4 0 2 p e r i o d i c w a ve h e a t i n g . Q~ " e r c * "'**' T ^ + 1

CO

( S T ) - T i

e x p ( - T X " " ) j e x p f - X2 + T X 2 j - e x p ( - X2 ) J s i n ( F o * X ) d X

x[l - exp(-X2)]

Q(T) = 0 , x < 0. „ 2 j

Q ( T ) = Q Q , n T Q < T < n TQ

+ T.f n = 0 , 1, . . .

Q(T) = 0 , n tf l + T < T < (n + 1 ) T0 . 0 < T < T .

n = N umb er o f c y c l e s .

t = t ^ , r + co. T x = T 1 / T 0 , T = T / T Q , Fo* = r/V5f^

>* r

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Se ct io n 8 . 2 . So l id s Bounded by P la ri e Sur faces— With In t e rn a l He a t in g .

Case No . Refe rences D esc r ip t io n S o l u t i o n

8 . 2 . 3 9 , p . 1 3 1 I n f i n i t e p l a t e w i t ht ime dependenti n t e r n a l h e a t i n g ,t = t . , -Jt. < x < ±SL,

T = 0.t = t , X = ±1, T > 0.

q " ' = f ( T ) .

q"'

h-HU*|

fu i. \ *» V ( - 1 ) " r (2n + l ) 1( fc " t i ) = iiT Z Un + 1) C O S L 2 — "X J

n=0

x J f ( T ) exp -ir (2nJ o

+ l )2 a (T - x ' ) / i2 J d t

8. 2.4 9 , p . 132 Case 8 . 2 . 3 wi thf t t ) = q £ " e "b T .

( t - t j k

I n * ' *

i 1 i fc os (yy^d) .-] m , . ,

2 ^ (- 1 )" co s (X^X) exp (-X* Fo)

n=0

X = (2n + 1 ) TI /2n

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m

O iB

•rtJJ

100>£HflcM0)+Jc c-H 0

-w0 4J

f u& O• 0 m•A 41M a00 )

4»t >•rf <aM 4 s-l cC 01W u0 )

<M

, Sr-( PiM>

»0o z•"* 1 ..-1 1 »o 2d > 1 wto 1 O

6-2Q

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i

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Section 6.2. Infinite Sol ids —Wi th internal Heating.


6 . 2 . 1 9, p . 265 Cy l in d r i ca l s o u r c e of h e a t i n g ,

q ' " = q j " , 0 < r < r1 #

-SL < z < +1 , T > 0 .

q " ' = 0 , r > i t , | z | > J .,

T > 0.

t = t, . , r > 0,

| z | > 0, T = 0.

q"' = 0

•1

— 8 ,

5— = ^ r / 1 - e x p ( - r / 4 a u ) e r f ( X0 *' r l r i J 0 V™

( t - t . ) kdu , r = 0, Z = 0

For S, + » :

( t - t . ) (c

7^T- = * £ ' e xp(~ ^ ) ] « E " ™ ) ' r = °' z =o rq 0 1

Po = OT/r

6 . 2 . 2 9 , p . 347 Case 6 .1 .23 wi th a cy l ind er

and i n t e rna l h e a t i n g .t = t Q f r > 0, T = 0.— i i i — n t i i

to - —( t

~ fc

0)k

0 _4_ f 11 - expi-\2

F o1)\j Q(\R)Jia)a\

q " 1 , 0 < r < rn .T > 0. (t " t n ) k- Vk Q 2 f [l - exp(-X FOjfl [ J0(BXR)(|) - Y0(SXR)4)ldX

x 3 [ *2 • *2 ]R > 1,

1, <|)(and ii def in ed in case 6 .1 .23 .

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Section 6.2. Infinite Sol id s—W ith Internal Heating.


6 . 2 . 3 9 , p . 348 Sphe re w i th i n t e r n a l h e a t i n g ( t - tn ) k _ *2

i n an i n f i n i t e m e d iu m ,

t = t Q , r > 0 , T = 0.

- V " FcflrL"rl ' 4 L

1 - 2R i e r f c

*0 0

' " = q j " , 0 < r < rQ ,

T > o

0 , r > r .

T > 0.

2 Fo .3 ,•=—-— i e r f c

0 < R < 1 .

( t - t . ) k * 2 ,

*0 0

0 ' FO f. 2 . /R - 1 \ x .2 /R + 1 \ * .3 c /R - 1\

+ F o i e r f c , R > 1 ,

F o = 2-vSr / r

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Section 6.2. Infinite Sol ids —Wi th Internal Heating.


6.2.4 9, p. 349 Infinite conductivity sphere r• * • -u *• ' ^ u ( t " 0* 1 1 + Bi 6 2 _ .2i n a n i n f i n i t e m ed iu m w i t h - — = — ——. — K Bi, , . 2 3 B l TT

c o n t a c t r e s i s t a n c e . ^ 0 0 L

t = t , r > 0, T = 0. » 2

q ' " - q j " , 0 < r < rQ , T > 0 . X ^ ^ ^ 2 " f V„ » . - 0 ° r > 0. r > 0° K [ u 2 d + B i , - K B i ] 2

+ [ u 3 - K B i „ ] '

0 < R < 1 .

f \ o- .c- / For sm all va lue s of T:

< V ' > r c r k l ( t " V " _ K F o- 3 [1 - (K Bi Fo ) /2 + . . . Ja ' ' ' r0 0

F o r l a rg e v a l u e s o f T:

a ' " r 2

q 0 0

lpL + Bi 1_ 2 + B i(2 - K) 13L B i " * F o "2 B . K 7 i A / ^ J - J

3P cK = " T T ^

P 0 C 0

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Section 6.2. Infinite Sol id s—W it h Internal Heating.


lto

6 . 2 . 5 7 4 , p . 426 I n f i n i t e p l a t e w i t h i n t e r n a l

h e a t i n g i n a n i n f i n i t e

medium,

t = t_ , -I < x < I, T = 0.t = tm , «. < x < -£ , T = o.

i U "

fc J . i .

= 1 + PO Fo , er fc (2n - 1J_ + X2VFo7

+ 4 Po Fo,

t - t„

tfi " ^

1 1 + K L, * 'n=l

- r i r [-* (1*4) * * - 2 <2«- (§ * $

l < x < l .

n=l

'2riVa / a + X - ie r f c i W S T -

2 / 2nV5q7c^ + X - 1+ 4 P o F o2 i e r fc ^== , 1 < X < - 1

K = 1

V , H = f — f , f (+X) = f (-X) - f(+X)P 2 C 2 k 2 1 + K

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Section 6.2. Infinite Sol ids —Wi th internal heating.


ito01

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Section 6.2. Infinite Sol ids —Wi th Internal Heating.


I

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7. Semi-Infinite Solids — Tran sient

o oD111OIDDOo:i l

ea<

<><UJ W

CQ

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Sect ion 7 .1 . Sem i-Inf ini te Sol ids—Ho Inte rna l Heat ing.

Case Ho. References Descript ion Solution

7.1.1 9, p. 61 Constant surface temperature, t - t / *\bx

t = t Q + bx, x > 0, T = 0.

t = t f x = 0 , T > 0 .s

0 s

For b = 0:

V _2k ( t s - t Q ) - V i r

0 s

See Fig. 7.1

i

7.1.2 7, p. 89 Var iable initial temperature,

t = f (x), x > 0 , T = 0.

t = t , x = 0 , T > 0 .s

t - t s ~ cryir , / «,L[-(H*)}--.[- (^)1B

f (B) = £ W

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Section 7.1. Semi-Infinite Sol ids —No Internal Heating.


7.1.3 5, p. 80 Convection boundary

74,

p. 206

t = t i f x > 0, x = 0.

t , .h

t - t .1 _

fcfi-fcl

h ( t f - 1 . )

1 - erf (FO^)

~ expK + Blx ^ i 1 ' e r f K + B i x * ^ l l

= e x p ( c r r h2 / k 2 ) e r f c ( Vo x h A ) S e e F i g . 7 . 2

T 7.1.4to

2, p. 275 Ramp surface temperature. t - t.

t = t., x > 0, T = 0.

t = t. + bT, x = 0, T >_ 0.

t = t . + b r .

bT( l + 2 Fo^ 2 ) erfc ( Fo ; ) - § o ; exp(-Fo; 2)

7.1.5 1, p. 258 Steady surface heat flux,

t = t., x _> 0, T = 0.

g^ = q Q , x = 0, T > 0.

(t - t.)ki ^ 5 f - " "* F o x «fc(Fo*) exp(Fo^)

(t - t )k

- " d - x = o0V5r" VW See Fig. 7.7

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S e c t i o n 7 . 1 . S e m i - I n f i n i t e S o l id s — N o I n t e r n a l H e a t i n g .

Cases No. Ref eren ces D e s c r i p t i o n Solution

7.1.6 9, p. 62 Time dependent surface

temperature.

t = t Q , x = 0, 0 < T < T 0 .

t = t^, x = 0, T > T Q .

t = t , x > 0, T = 0.

t - t.erfc K).

t - t

0 < T < T„

t. " t,±- = erfc(F 0; ) ^ - ^ erf c(^-2—-y ) T > T„


temperature,

t = t., x > 0, x = 0.

t = b T, x = 0 , T > 0.

bT-

^ i = (l 2 F o ^ e r f c ^ ) - | = Fo* exp (-Fo 2)

7.1.8- 9, p. 63 Time dependent surface

temperature.t = t . , x > 0, T = 0.

l

t = bVx, x = 0, T > 0.

t - t .

b t- = ex pl -2 Fo "J - Vix Fo erf cfF o J

by/r-

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Section 7.1. Semi-Infinite Sol id s—No Internal Heating.


/.1.9 9, p. 63 Time dependent surface

temperature.

t - t.

^53* - A (f • i)i"-*.K)t = t., X > 0, T = 0-it = t t + br"^ 2 , x = 0, T > 0. See Fig. 7.6

br'n/2.

I7.1.10 9, p. 64 Time dependent surface

temperature,

t = t., x > 0, T = 0.

t = e , x = 0, T > 0.

t - t^ - i = | e x p ( - x V b/ ty e r f c / F O * - Vbr\

+ exp (xY b/ct) erfc( Fo + VbTJ

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7.1.11 9, p. 74 Convection boundary with t - t.

step fluid temperature

t = t , x > 0, T = 0.

t f = V 0 < T < Tr t _ t

t f = t 2 , T > V

t f . h -

Iui

;— = e rf c (FO*) - e x p [ B ix + ( H2 /4}Jer fc |Fo* + (H/2)J

i - = e r f c ^ F o * ) - e x p [ B ix + ( H2 /4)jerfc |Fo* + (H/2)J

F h a ( T - T0 ) - |e x p [ B ix + jj j

, 0 < T < T,

+ •— ^ I e r fc^ " ' i 2Vct(T - TQ)

x e r fc2Va(T - xQ )

+ k | ' T > T o

H =2hV5r

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S e c t i o n 7 . 1 . S e m i - I n f i n i t e So l id s — N o I n t e r n a l H e a t i n g .

Ca se N o . Re fe r ences De sc r i p t i on So l u t i o n

7 .1 .1 2 9 , p . 76 S t ep su r f ac e hea t f l u x . ( t - t . ) k

t = t . , x > 0 , T = 0. IQ*^*= 2 i e r f c

v x / T \ « & < T - x0)J^ = q Q , X = 0, 0 < T < TQ .

^ - 0, X - 0 , T > V T > T

0

See ca se 7. 1. 5 for 0 < T < T. .

_j ' 7 . 1. 13 9, p . 76 Time dep ende nt su rfa ce t - t . = 1, x = 0, 0 < T < T0* heat f l ux .

t = t l f x > 0 , T « 0. t - t i = | s i n_ 1 ( V T0 / T ) r x = 0, T > T0

q = k/Vcrrir, x = 0 ,

° * T * V 2 / V ^ V xpf- Fo* 2,l + u 2 , lq x = 0 , x - 0 , t > TQ . t - t i = ? y L

x ' u 2 ^ d u , T > T 0

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Section 7.1. Semi-Infinite So li ds —N o Internal Heating.



heat flux.

t = t . , x > 0 , T = 0.

q^ = l ^ 2 - x = 0, T > 0,

n = - 1 , 0 , 1 , 2 . . .

( t - t ^ H I + n /2 )_ / o - /•> _ \ ' X U f I ' U

^ ^ 2 " r( | • f)

( t " X l , - 2 ( 4 )n / 2 r ( l + n / 2 ) in + 1 e r f c

q 0VOT Tn/2 ;

See F ig . 7 .8 for n = 2K)

V 7 . 1 . 1 5 3 ,p . 402

p l a t e a nd s e m i - i n f i n i t e s o l i dc o m p o s i t e w i t h c o n s t a n t

s u r f a c e t e m p e r a t u r e .t = t . , - b < x < <*> x = 0 .t = t , x = - b , T > 0.

n=0 '

e r f c (2n + l ) b - x , - b < x < 0,

fc2 " fci _ _ 2 V n _ f u n + l ) b +6x'

n=0

6 - V a i / a 2 , X = k 2 6 A r x B = p ^ y

, x > 0

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7.1.15.1 74, Case 7.1.15 with:

p. 409 t = t ,0 < x < b,

T = 0.

t = t_, x > b, T = 0.

t = t , x = 0, T > 0.o

t - tt~=\ = r f ° ( F°x) * 2 H n _ 1 e t f c ( 2 n o b * F ° x )

n=l

t, - t

(t \ ) (1 K) S ^^ r f 0 [n=l

(2n - 1) Fo. + Fo, :]•O < X < 1 .

t - t1 | _ = J U L y H"" 1 erfc TFO* - Fo* + Fo,t - t. 1 + 1 L L x x I

° n=l

.. + (2n - 1)B Fo.b b.a

( fcl - V K

(t Q - t x ) (1 + K) erfc Fo - Fo. +x b2 K

(1 + K)'{t Q - tx)

y H n - 1 erfc |Fo* - Fo* + 2nB Fo* , X > 1 .

n=l

P l c l k l _ 1 - K _ W 2

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S e c t i o n 7 . 1 . S e m i - I n f i n i t e S ol id s — N o I n t e r n a l H e a t i n g .

Case No. R efere nces D e s c r i p t i o n Solution

7.1.16 7 , p . 99 Periodic surfacetemperature,t = (t„- t J cos I — — \

x = 0, T > 0.t - t , x > 0 , t = 0 .

0

l T o / '

i ^ = e xp ( - V J x ) c os (u,x - V J x ) , S ee F ig 7 .3- t„

V" "oT = maximum su r fa ce tem per a tu remT 0 = p e r i o d of p e r i o d i c t e m p e r a t u r ea) = 2irn/Tn

Maximum temp:

fcmax ~ fc0 _ o m ( yljl x \

Time for t to rea ch xsmax

St _ m Xfc0 n 2 "Vamrx

, m = Q, 1, 2,

H e a t t r a n s f e r r e d i n t o s o l i d , x < x < x :

q V a A 0 1 T I % \ i i r \1-— r- r- = cos l u x . - - I - cos (a)X_ - 7 )

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Section 7.1. Semi-InLinite Solids—No Internal Heating.


7.1.17 7, p. 106 Periodic fluid temperature,

t.£ =(tf M t 0 ) cos &2L\

T > 0. \ ° /

t = t . JC > 0, T = 0.0

t - to

t C w - tfM o

exp (-yiT* U)T 25 - a n

Cl + (2/H) + ( 2 / H 2 ) ] 1 / 2

Mi)]

t f .h -

H = — 2 = -, ID = 2mr/ximlc °

t = max fluid temperature

T . = period of periodic fluid temperature

Maximum temp:

fc M- fc oexp ( - # * )

( t f M V [l + (2/H) + ( 2 / H 2 ) ] 1 / 2

7.1.18 9, p. 68 Square wave surfacetemperature.

0.*• V X

2mT. < T < (2m + 1)T ,0 0

m = 0, 1, 2, . . . .

t = t - t , x =ra D

(2m + 1)T < T <0

(2m + 2)T .

0,

" T> nTo X

° ' L

°_ /t2n + 1 ) 1V 2 ar 0 J

irr

T = period of periodic temp0

t = mean surface tempm r

L = deviation of surface temp from t

7.1.19 2, p. 248 Quarter-infinite solid.

7.1.20 2, p. 248 Eighth-infinite solid.

7.1.21 2, p. 248 Semi-infinite plate.

Dimensionless temperatures equal product of solution for

semi-infinite solid (case 7.1.1 or 7.1.3) and solution for

.infinite plate (case 8.1.6 or B.1 .7) . See Figs . 9.4a and 9.4b.

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7.1.22 4, p. 84 Infinite cylinder in asemi-infinite solid.Cylinder assumed raassless.

Time for soil to reach steady stat e:

B | = C(d/D + 1/Bi I 1 " 4 4 ,D

I

2 d

C = 4.6 for const, cylinder temperature.

C = 6.0 for const, heat flux from cylinder.

The heat transferred during the time given abo ve:

-2SLKD (t c - t f )

12(d/D + l/Bi D).25

Temperature of cylinder = t .

Initial temperature of semi-infinite solid = t f .

t > t,.c r

The time for th e cylinde r to ret urn t o temp t aft ersteady state is achieved and heating is stopped:

^ = (d/D + VBi/" f (^f-D \ c f

See Fi g. 7.4 for va lue s of f,

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7.1.23 9, p. 264 Semi-infinite surfaceheating,t = t., 0 < x < <*>,

- C D < r < +ot>, T = 0 .

% - v x =

° --oo < y < 0 .

q = 0 , x = 0 ,

0 < y < +°°.

( t - t . , f c 1

q (T/ax Vn

Poer fc K)+S*Ei W2) . , x = 0.

7.1.24 9, p. 264 Infinite strip heatedsurface.t = t., 0 < x < o°,

-=o < y < +=o 1 = 0 .

q x = q Q , x = 0,

-a < y < +a.g^ = 0, x = 0,

y > a, y < -a.

: = -n=- 1 erf / 1 + erf /

V -« {* l ^ a+ e r f( 2^ J

(zVKîr)Ei 1 + Y.

2VFo\i - y

\2VFO " \ 0

Ei 1 - Y

2Vfo"

x = 0 .

See Fig. 7.5

http://2vfo/

http://2vfo/

http://2vfo/

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7.1.25 9, p. 264 Heating through acircular surface.

% ' V ° < r " VX = 0, T > 0.c r = 0 , r > r , x = 0 , T > 0 .

t = t . , 0 < r < ° ° ,

0 < x < °°, T = 0 .

( t - t . )k

V c ~ = 2 > f \ r e r f c K )

^ x V1 + ( r c / x ) 2]ierfc , r = 0

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Case Ho . References Description S o l u t i o n

7.1 .26 9, p. 265 Heating through ar ec t angu l a r su r f ace .t = t . , x > 0 , y > 0 ,Z >. 0, T = 0 .

% - V-w < y < +w,-% < z < +&, x = 0 ,

T > 0.

g^ = 0, x = 0

|y | > w, |z j > I,

T > 0.

Maximum temp:

('max ~ fc i)k

q 0w

Average temp:

m l )kq 0w l

| s i n h - 1 (L ) + L s i n h- 1 QS\, -W < y < +w,

-£ < z < +S,, x = 0.

2 - 1— jsi nh (L) + L sinh•HiHH + i

- ^ ^ ] | . - » < y < +w, -S. < z + £ , x = 0 .

L = l /w

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ection 7.1. Semi-Infinite So li ds —N o Internal Heating.

ase No. References Description S o l u t i o n

.1 .27 9, p . 401 Recta ngu lar wave

su r f ace hea t f l ux .

t = t , x •*•<*>, i > 0.CO

q = q n ( T ) , x = 0 , T > 0 .q n (x) = or x > o.

9 0 ( T ) = q „ ,n x 0 < x < nxQ + V

n = 0 r 1, . . .

V T ) " °' n T 0 +

A T, < T < ( n + l ) Tn .ui x u

n = No. of cy c l es .

At x = 0:

( t - t_) k

V a T o

( t " t I k

2 T_ ^ Vr + - — [(1 - Tx) V ^ - I [T v T)/ViF],

0 < T < T,

< * > 2 ~- 2 T r 1/2 ~\_ _ _ = __ ^ vr + - — ^ _ T i , V5-_ ( T . T i , _ x { T ] / T ) / V n J ,

T < T < 1 ,

CO

I ( T 1 # T) = /

q„(r)-

exp (- T X2) 1(1 - T ) exp (-X2) - exp (-X2 + X 2 ^ ) + T 1

X Z [l - exp (-X 2)]

dX

T = T/TQ , Tx = T l A 0

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S e c t i o n 7 . 1 . S e m i - I n f i n i t e S o l id s — N o I n t e r n a l H e a t i n g .


7 . 1 . 2 8 9 , p . 419 S e m i - i n f i n i t e c y l i n d e r S e e c a s e 9 . 1 . 2 0w i t h c o n v e c t i o n b o u n d a r y.

t = t Q , 0 < cQ< c . . , z > 0 , t = 0 .

t = t , , 0 < r < r Q , z = Q, T > 0 .

Convection boundary

of h , t . a t r « t . i z > 0 , T > 0.

7 .1 .29 9 , p . 419 Case 7 .1 .28 wi th : See c a s e 9 . 1 . 2 1t - t x , r = r 0 , z > 0,

T > 0 .t = t n , 0 < r < r , z = 0 , T > 0 .

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7.1.30 19 , Plate and semi-infi-p. 3-83 nite solid composite

with constant surfaceheat flux,t = t Q, 0 < x 1 < 6, 0

< X < <*> r T = 0 .

k •* • » .

q = q", x = 0, T > 0.

k 2 ( t r V _ i2 q" Vo^r 2 m V F o 2 L

exp (m Fo J erfc (m VF o ) -1

+ — , 0 < x < 6

VF 1

Heat flux at interface:

q /q" = 1 - exp (m Fo j erfc (m VF o ) , x = 6 ,

• = (P 2 c 2 / p x cx)

See Fig. 7.9

7.1.31 74 , Semi-infinite rod withp . 211 convection boundary.

t = t . , x > 0 , x = 0 .

t = t_ , x = 0 , T > 0.

— - ± - = 1 exp (VB Td X ) e r f c ( j ^ - B i d F o d

+ e x p ( B i , X) e r f cd n "— — + V B T T F O T2 V F o , d d

T"t , . n

. cross section aread = : r , X = x/d

perimeter

Heat flux at x = 0:

Vk ( t f - t.

= = exp (-Bi F o J + V B T , erfc (VBi Fo,)IT VF oT d d d d d

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Case No . References Description

7.1.32 74 , Semi-infinite platep. 46 5 with fixed surface

temperatures.

i

v

1

Solution

r } - = + I 2 , *=i*~ sir> (A > exp ( -X2 Fc>)fc o " fc i n n=l n n \ n /

CO

+ ^ 2, "^n 1" s i n < x X) [exp (A n Y) erfc

x rPo* + X n VfoJ + exp {-X nTC) erfc ( F O * - X Vfo )

( Xn F O) r f c ( F Oy ) ] '

A n = nir.

- 2 exp

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Section 7.1. Semi-Infinite Solids—N o Internal Beating.


7.1.33 74,

p . 466

Semi-infinite cylinderwith fixed surfacetemperatures.t = t i f 0 < r < r Q ,

o < y < °°, T = o .

t = t ± , 0 < r < r Q ,

y = 0, T > 0.

fc • V r = r 0 ' ° < y

t - t.- t . ,— J . (A R) r / o \ / * \

n=l

+ exp (X Z) er f c (Fo + X V F 5 )

+ ex p (-X Z) er f c Fo - X V F O \ " | .

< =», T > 0 . w = o

K7H

7.1.34 74,

p . 466

Case 7.1.33 with:convection boundaryh , t . at r = r , and

t = t Q , 0 < r < r Q ,

Z = 0 , T > 0.

t - t. 2 Bi J 0 (X n R) exp (-X nZ)

£L J 0 ^ ( B i 2 + X n )

Bi J n (X„R)^ - Bz J [A R] r / * \+ / 7—= =\ exp (X Z) e r f c (X VFO + Fo }

~ J„(X ) f B i 2 + X 2 U n V n * 'n=l 0 n \ n /

exp {-X Z) er f c f\ VFo - Fo*)

W " B i

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Section 7.1. Semi-infinite So li ds —N o Internal Heating.


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Sect ion 7.1. Semi-inf ini te So l i ds —N o Internal Heat ing.

Case No . Reference s Description ' Solution

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Section 7.2. Semi-infinite Sol ids— With Internal Heating.


7.2.1 9, p. 79 Steady surfacetemperature andinitial temperaturedistribution,t « T. + bx, x >. 0,

T - 0.

t « t Q, x * 0, T > 0.

t - t, 0 * * * 2 / * \ 2 * *— = (1 + Po + Po Fo ) erf (Fo I + jpr Po Fo exp (" »D

bx _ * _ *2+ Po Fo .T. x

POa' • ''" at

7.2.2 9, p. 79 Case 7.2.1—variableheating,t • T, + bx, x >_ 0,

T = 0.

t

uq ' " - q,

tQ, x = o, T > o.

- q ^ " e_ B x

.

(t - t o)k0" bx

T, Po- b " 1 ) * -Sxer f (Fo ) + 1 - e p

i • X - B * e r fc (VX - Fo*) - \ e

A + p x e r f c (V* + Fo * , ,

q ' 1 •Po*= — ~ , X = <* 32T

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Section 7.2. Semi-infinite Sol ids —Wi th Internal Heating.


7.2.3 9 , p . 79 Case 7.2.1 withband heating,t = T., + bx, x >_ 0,

T = 0 .

t •> t , X • 0 , T > 0 .

g ' " - q j " ,0 < x < d.

q ' " = 0 , x > d .

t - t,— - = Po* l l - 4 i e r fc / F O j+ 2 i2 e r f c IFo* (1 + X) 1

i 2 e r f c [ F O * ( 1 - X ) ] j + § * + e r f ( F G * ) ., 0 < X < l

t - t— - = 2 Po* J i2 e r fc fo*(X - 1)1 + i2 e r f c |Po*(X + 1)1

i 2 e r f c ( F o * ) ) + ^ + e r f ( F O * ) ,i

* q • • > axPo _ JO

kT,

I , X > 1 .

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7.2.4 9, p . 80 Insulated boundaryand band heating.t » t . , i > 0 , T = 0 .

X = 0 , T > 0.

(t - t .) k

g^ = 0,

q ' " = q ^" , 0 < x < d,

T > 0.

q ' " = 0, x > d.

, . , . * T = 1 - 2 i 2 e r f c [ F O * { 1 - X)l - 2 i 2 e r f c [ F O * ( 1 + X)l ,

0 < X < 1 .

. . .aT 2 j i 2 e r f c [F o

d

( X 1 }] " i 2 e r f° [F

° d( X + 1 }] ) '

X > 1

IHJ

7. 2.5 9, p. 80 Insul ated boundaryand variable heating,t = t Q + bx, x > 0,

T = 0 .

a = 0, x = 0, T > 0.q . . . = q j . , e - 6 x t

x > 0 , T > 0 .

r r n = — * " exp (-Bx) + — ^M 0 P o \F o Po

X \ X X

+ 2X1 erfc K)+ j exp (X

2

- Bx) erfc f\ - Fo*} + j exp (X2

+ Bx) er fc (x + Fo*)

P o* l i " «

kbx, X = B Var

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ection 7.2. Semi-infinite Sol ids —Wi th Intecnal Heating.

ase Ho. References Description S o l u t i o n

. 2 . 6 9 , p . 323 Pl a te and se m i- in f i - . . .n i t e so l i d com posi te . ' 1~ (T 1t = t 0 , -b < x < °°fT = 0.

t = t , x = - b , T > 0.

*0 '-b < x < 0, x > 0 .

g-'-ar = 1 - 4

e i e r fc

0 , X '< 0, T > 0 .X .2 „

i e r fc+ X

n=0

f(2 n + 1) - X~| X .2 . / 2 n + X \L -2Vfo- J + 1 + X * e r f c \Z^ }

( Yv f e * ' ) ] - * < - < ° •( V t « ) k" V k l 4 r . n f . 2 _ / 2 n + j x \

^ — = (in) z B { 1 e r f c V T v f s - Jn=0

. f(2 n + 2) + 6x1 , • 2 „ f( 2n + 1) + 6x1 I v -, nf c

L 2 v i s — J -2 1 e r f c

[ 2 V P 5 — J J *x >

° -

+ i 2 e r f

6 = VSp5T2, x = k26 /klf 3 = f = - i FO = ¥-, x = *b

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Case Ho . References Description So lu t ion

7 .2 .7 9, p. 307 Time dependent hea tingand convection boundary.t = t , x > 0 , T = 0.

q " 1 - q ^ " T S / 2 ,s = - 1 , 0, 1, . . .

V h -

( t - t | J ) k- i ., r m + (s/2 )], , , „ T l + ( s / 2 ) [ 1 + ( a / 2 fl F o ( s / 2 ) + l , _n 4 , s+2q^ 'OT — J Fo^~' (-Bi x )

e x p Fo + Bi VP o

X X I X ,(s i + B i2 Fo \ e r f c (\

s+2 / n . in

<* 7.2.8 9, p. 308 Con stant tempera turesurface and bandbea t ing ,t = t - , x > 0, x >» 0 .t = t Q , x = 0, x > 0.

q " ' = 0, 0 < x < x ,

x > x„ .

q " " = q , ; " .X]_ < x < x2 .

Surface hea t f lux:

%'• a t = 2 ie r f c (Fo \ - ie r f c |F o 1 , x =

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Section 7.2. Semi-infinite Solids —With Internal Heating.

Case No.. References Description Solution

7.2.9 5419,p. 384

Exponential heating,t = t Q / x > 0, T = 0.

q " ' • = q ' " e x p ( -u x ) .

k (

2~q,

fc" V / *\ e x p ( _ 2 M F o x ) r 2i

^k = l e r f c K ) — i i —- L - «* »n_ HEjaa [ e x p (.2 .„ F c g e r f c (F o* _ H )

- ex p [ 2 H F o j e r f c ( P o + M^ .

Sur face temp:

M t w " V = 0 5 6 4 2 _ 1 - e x p (M2 ) e r f c (M)2 q ( J

1 , Va r r 2M , x 0

M = U Vo f

See F i g . 7 .10

7.2.10 74 , p . 383 P la nar hea t pu lse andc o n v e c t i o n b o u n d a r y.I n s t a n t a n e o u s h e a tpu ls e occur s a t T = 0 ,x = x , w i t h s t r e n g t h

• Vm

t = t , X > 0 , T = 0 .

( t - t £)kx 1

Qa = 2 V ^ ° | e x p

- Bi ex p Bi (X -

(X - 1 ) '4 Fo + exp

(X -

1) + Bi Fo e r f cX + 1

2 Fo + Bi VFn]l

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Section 7.2. Semi-infinite Solids—Wi th Internal Heating.

ito

C a s e N o . R e f e r e n c e s

7 . 2 . 9 541 9 ,

p . 384

D e s c r i p t i o n S o l u t i o n

E x p o n e n t i a l h e a t i n g .t = t Q , X > 0, T = 0.

q i i i = g ^ i i x p J.^J

7.2.10 74, p. 383 Planar heat pulse andconvection boundary.Instantaneous heat

pulse occurs at T = 0,x = x, with strengthQ J

m

t = t

2

r X > 0 , T = 0 .

k ( t - t >o- /_ *\ e x p ( "2 M F o

. . . ,— = i e r f c ( F o

2 qAM

VccF V x) -

*

X

M^ -

exp (M)

iexp (M)

4M exp ( - 2 M F O x )o ) erfc (FOXJ \ X - )

- exp( 2 » * > » ) ('

erfc [Fo + -)iS u r f a c e t e m p :

k ( V t o )

2 q J ' W= 0 . 5642 1 - ex p (M ) er f c (M)

2M x = 0

M = UYOfT

S e e F i g . 7 . 1 0

(t - t f ) k x x

Qot= 2-VTTFO exp (x - i r

4 Fo + exp (X - 1)4 Fo

- Bi exp.2

(X - 1) + B i Fo er fc X + 12 Fo

+ Bi YFn

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Sect ion 7.2 . Sem i- inf ini te Sol ids—With Inte rna l Heat ing.

Case No. References Des cription Solution

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Section 7.2. Semi-infinite Soli ds—W ith Internal Heating.


i• o

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OB

SB9«

H

s


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Case No. References Description So lu t ion

8 . 1 .1 9 , p . 94 In f in i t e p l a t e w i th s t eadysurface tempera ture .t » t ( x ) , 0 < x < & ,T = 0.

t = t , x = 0, I, x £ 0.K>V1

t - t Q = 2 > A s in (rnrx) exp (-n V Fo)

n = l

*„ = /= / f (X) sin (mrx) dx

See case 8 .1 .25 for con vect io n coo l in g a t x = i ..

8 . 1 .2 ° , p . 96 In f in i t e p l a t e w i th s t eadysurface temperature andl i n e a r i n i t i a l t e m p e ra t ur e .

t = t Q + b x , 0 < x < J L ,T = 0.

t = t , x = o, a, T >_ o.

t - tb S L

L = 2 V Szlln - 1 2 2sin (mrX) exp {-n IT FO )

n=l

7

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Section 8.1. Solids Bounded by Plane Sur fa ces —N o Internal Heating.

Cas e (to. Referenc es Descript ion Solut ion

8.1.3 9, p . 97 In f in i te pla te withsteady surfa ce temperature and bi l ineari n i t i a l t e m p e r a t u r e .t = ( tc - t0){ l - | x | ) /0, + t Q ) , -i. < x < + S L ,

T •» 0.t = t , X = tl , T > 0.

D

CO

fc ~ t Q = -i V i c o s [ ( 2 n + 1) 1 ["_ ( 2 n + 1 ) 2 ^ F o ]' e ^ O i r 2 \ <2n + l , 2 |_ * " M ^ L 4 J

n= 0 l

= 1 - |X | - 2 V F O " J ( - l )n f i e r f c (2 n + l X ' )

["(2n + 2) - |x|~ |lL 2VIS Jl

N A ^ A ^ I

i e r fc

— *0

>w\MSee Fig . 8.1

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Section 8.1. Solids Bounded by Plane Su rf ac es —N o Internal Heating.


8 .1 .4 9 , p . 98 In f in i t e p l a t e w i ths t eady su r f acetemperature and bi-p a r a b o l i c i n i t i a ltempera ture .fc - < V V <*2" * 2 ) /

z2

+ t , -a < x < +1,c

T = 0 .

t = t Q , x = ±1,

T > 0 ( s ee ca se 8 .1 .3 ) .

t ~ fc0 32 XTt - t n - 2 2

±UJ (2n + 1) TTX expn O ( 2 n + 1 ) '

£_ i l n^ l i i p o

See Fi g . 8 .2

8 .1 .5 9 , p . 100 In f in i t e p l a t e w i ths teady surfacetemperature and cosinein i t i a l t empera tu re . / vt ~ fc0 - ( tc " V <* ( i f )+ t , -I < x < +&cT = 0 .

t = t n , x = ±1,T >_ 0 .

t„

V fco( f x ) e x p ( - ^ F o )

UsA/^/V/

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Section 8.1. Solids Bounded by Plane Sur fac es —No Internal Heating.


8.1.6 9, p. 97 Infinite plate with74, p. 113 steady temperature and

constant initialtemperature,t = t±, -I < x < I,

T

t 0' ±1.T > 0.

N V » ' V * >

^

+^/\s*

— * < >

oo

T=T: =2 E ^ « V> *** {< FO)i 0 . n x 'n=0

Mean temp:

£ £ - » £ * - ( • * »X

n=0 n

*« " fc 0 /Fo^ — ^ = 1 - 2 ^ / - , FO < 0.1

Beat loss:

Q , V S i n ^ X n i1 * e x P (- Xn F o) ]2£pc (t.- t.) Z Z , X |"X + sin (X ) cos (X )'

* ^ i 0 , n I n n nn= l •- J

X = (2n + l)n/2, See Fig. 8.3

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Section 8.1. Solids Bounded by Plane Sur faces—No Internal Heating.


8.1.7.1 74, Infinite plate withp. 220 convection boundary

(general).t = f (x), -SL < x < +1 ,T = 0.

-h. t.

t - t f = 2X cos (X X) exp (-X Fo 0 ) f 1

— " n . ",, ' I [f W - t-1 cos (X X) dXsin (X ) cos (X ) I L fj n

n n Jn

„ A cos

L x +n=l

cot (X ) = X /Bi

X given in Table 14.1n

8.1.7.2 3, p. 294 Infinite plate with74, p. 223 convection bound ary—

constant initial temp.t = t . , -I < x < +£,

lT = 0.( s e e c a s e 8 . 1 . 7 . 1 )

t - t„ ~ s in (X ) co s (X X) ex p l-\ F o )

t . - t - Z , X + s in {X ) co s (\ )n=l

co t (X ) = X /B in n

t . - t f

t - t

1 - cos ( V B I X) exp ( - Bi Fo) , Bi < 0 .1

r~ = e r f Z 1 " X ) - e x p | B i ( l - X) + B i2 F o ] e r f c f1 ~ X + Bi "VFo)i f \ 2VFO/ L J \2 VFO /

fc l1 + X 1 - exp U3 i( l + X) + B i2 F o | e r f c ( •*• * X + B i VFo") ,\ 2 V F O " / L J \2 VF^ /

+ erf + B i VFO"1VFO

Fo < 0 .1 .S e e F i g s . 8 . 4 a , b , c , d a n d e a n d 9 . I d .

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8.1.8 1, p. 268 Infinite plate withtime varying surfacetemperature.t = t . , -I < x < S.,

l

T = 0. t = t . + b t ,

x - ± I, T > 0.

I

t ( T ) -

' I I J

- t ( T )

( t - t±) a= Fo +

[(2n + 1) | xj

» * " » ' n r 0 (2n + 1)

x c o s I(2n + 1)

See Table 8 .1 , F i g . 8 .16 and F i g . 8 . 2 5 .

8 . 1 . 9 9 , p . 104 I n f i n i t e p l a t e w i t ht ime varying sur facetemperature (general) ,t = t . , -i. < x < l.

T = 0.

t =

X = ± I, T > 0.

t i + f {T ) ,

, 2 2- V (2n + 1) (- 1 )" _ f (2n + 1) "Tf „- t . = Tt y J tJ — * — F o exp — j—* Fo

n=0

x cos [ • t t t 7 J a ']/"»[" , ^ V "1 [' ""•>] dT'

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Section 8.1. Solids Bounded by Plane Surfaces—No Internal Heating.


8.1.10 9, p. 104 Infinite plate withexponentially varying i _ -bx cos (X"^9) _ 4_ S T (-1)surface temperature. T - t . e „/Trr, it ,n ,•., "-, ,-> . i > 2 „ 2 / ; l r ,,2nt = t. , -I < x < I, * cos (VP3) n = Q (2n + 1 ) |_1 - (2n + 1) u /4Pd J

I

T = 0.

t = t. + T (1 - el

X = ±1, T > 0.

_ bX)i x e x p [_ iaaî iV F oj c o s [ i 2 n _ ^ ii J L XJ (

b ft (2n + l ) 2 i r 2 a / 4 A2

^ 8 .1 .11 9 , p . 105 In f in i t e p l a t e w i th °°i. ex po ne nt ia ll y var yin g i _ ^bx cosh (XT^d) _ 4 X" (-1)

IT ^->

t = t . , -i, < X < I,sur fac e temp erature . I - t . ^ ^ TT ^ ( 2 n + 1 } r + 4 M / „ 2 ( 2 n + 1 ) 2 - j

T (2n +. l )2 i i 2 „ 1 f~(2n + l)ir „1exp - J j - 1 Fo co s P -—<— XJ= 0.bT

t = t. + Tei

x = ±JL, T > 0.

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Case No. References Descr iption Solution

8.1.12 1, p. 299 Infinite plate withperiodic surfacetemperature.fc - "ta - W

( ? ) • " *±1

rax mn5 cos [ ( 2 T T TA0 ) + *]

t = max su rf ace tempmx

t = mean sur fa ce tempmn

cycle time

t W -

M ^ « V b « l

t

-tlr)

5 =

r 2 2f-^Xa) + f 2(XO)

f^(o) + t\{a)

1/2

4> = t an

See Table 8.2

£ xia) f 2 (xa ) - f 2 (a) f^xa)

f x (a ) ^( xa ) + f 2 (a) f 2 (xa)

f x (a ) cos (a) cosh (a ), f»(a) = si n (a) sinh (a)

a = VirPd = JSVir/aT.

Heat stored during half cycle:

2X.pc (tmx

t ) F(a)/0mi l /

F(a) =df ^ (a) + df 2 (a)

1/2

See Fig. 8.5

df x (a ) co s (a) sinh (a) - sin (a) cosh (a)

df _ (0) = sin (0) cosh (o) + cos (0) sinh (0)

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Section 8.1. Solids Bounded by Plane Surfac es—N o Internal Heating.


8.1.13 9, p. 105 Infinite plate with t - t.periodic surface — — _ = A sin (w + 9 + $)temperature. m it - t t , -» < x < I,

T = 0 .

CDI

ID

t<T) -

t = t, + (t - t . )i m i

s i n (on: + 9 ) , x = ±Sb,T > 0.t = maximum sur fa cemtemperature.

^ \ » » * ^ . ^ *^

\~H

• t (T)

H

+ 4ir Y ( - l ) n ( 2 n + 1) [4 Pd cos (9) - (2n + 1 ) 2 TT2sin (9)}/ , 2 4 4~ 16 Pd + {2n + 1) ITn=u

[ - ( 2n + I )2 * 2 Fo /4 ] co s [ i 2E_±_ l )JL x ] .

co sh (X V2 Pd + co s (X -V2 Pd)

x exp

A =cosh { V 2 Pd) + cos (Y 2 Pd)

c os h ( x ^ + X ^ f i ) '

1/2See F ig . 8 .6a .

a rg

. c o s h W ^ + V 2 l)S ee F i g . 8 . 6b .

Pd = 2. to/a .

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8.1.14 9 , p . 1 05 I n f i n i t e p l a t e w i t hp e r i o d i c s u r f a c et empera tu re on ones i d e / c o n s t a n tt empera tu re on t heo t h e r ,t = t . , 0 < x < I,

T = 0 .

* = fci + <f c . Vs i n (u t+ 9) , x = % ,T > 0.t = t . , x = 0 , T > 0.t = maximum surfacem

temperature.

-t(r)

t - t .— = A s i n (wr + 6 + <>)

m I

CO

V n ( - l ) n [n 2TT2 s in (9) - cos (9) PdJ£•> A A 7 A ?

n TI + urtr/ar= l

x exp ( - n V F o ) sin (n IT X)

co sh (XV2 Pd) - co s (xVz~Pd)cosh (V2~Pd) - cos (V2~"Pd)

1/2

$ = argsinh (x>pf + X-y/^l i)

, Pd = I a/a

hH

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Sec t ion 8 . 1 . So l id s Bounded by P l ane Su r f aces—No In t e r na l He a t ing .

Case No . Re fe rences De sc r ip t i on So lu t i on

8.1.15 9 , p . 108 I n f i n i t e p l a t e w i t h « X (T - T " ) X { T- T ' )s t e a d y p e r i o d i c fc 0 2 V ( - 1 )n . e n - e n

su rfa ce tem per a ture t , - t„ " rr A , n s l n ( m r X } X T. , . 1 0 n= l , n

on one s i de , co ns t an t 1 - et empera tu re on t heo t h e r . T* = x - nff, m » 1 .t = t Q , X = 0, T > 0. fc - X n ( T - T 1 - T ' - ) Xn ( T- T " )

t - t 2 ^ ( H n

fc = V* =*- ir^r = * 2. -H— in { m T X ) s fi

m T < T < ID T + T , X ° n = 1 1 - e n

m = 0, 1, 2 , . . . .» t - t 0 , x - i . mr T " = T - (mT + T l ) f m » 1 .p :

T < T

- i ,(

v1 )

1

T

- * = a n

2

, vT = pe r iod o f cy c l e . n

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8.1.16 9, p. 109 Infinite plate withsteady surfacetemperature on oneside, insulated on theother,q = 0, x = 0, T > 0.

t = t 1 # x = I,mT < T < mT + T ,

m = 0 , 1 , 2 , . . . .

00

t o

t = t , x = SL , mT

+ T x < x < (m + 1 )T .

T = p e r i o d o f c y c l e .

fcl " fc0 * n 4 * > + 1 ° ° S L 2 X J

X (T.. --TM X T -T ' )n v 1 ' n v

e - e

1 - eX Tn, T ' = T - mT, m » 1

00

_ _ ? _ . i S (-1 )" cos f(2n * l) i t 1- t,t

X (T-T - T" ) X (T -T " )n 1 ne - e

1 - eX T

n

, T " • = t - (mT + T ) , m » 1

X = o (J2n + 1) 2 T T 2 / 4 A 2

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Section 8.1. Solids Bounded by Plane Surfac es—No Internal Heating.


00

I

8.1.17 4, p. 82 Infinite plate with48 steady surface heat

flux and convectionboundaries.t = t , 0 < x < I, T = 0.

%f q 0 r X = «. , T > 0.

tf, h -

( t - V h Bi fio" 1 xp (X - l ) 2

4 Foq 0 " 2 V , |-xp (X - l ) 2

4 Fo

- Bi Po exp (Bi + Bi + 4 Bi Fo

See Fig. 8.18.

For Bi + » :

<fc ' V k VF5 f r (x - i ) 2 i

v - 2vr b 4 Fo J ' tt|

+ exp(X + 1) '

4 Fo

> — ( J v K * *5) ] '

(X + 1 ) '4 Fo

See case 8.1.20 for special solutions.

h-'-H

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8.1.18 144774 , p. 174 flux and variable

Infinite plate with „ r „ °°steady surface heat q 0 1 ,_ , \ ,„ i, . * . ,, 2 V 1 r „ , v , Cl

— £ - . J (X - 1) + (X - 1) + y + Fo - — £ — cos \m ( X - 1)Jt h e r m a l c o n d u c t i v i t y,t = t Q , -I < x < +1,

0 0 L IT n= l n

2_2X = 0. k = k ( t ) . a is x exp (-n ir Fo)c o n s t a n t .q x = V X = l' " lm

Wok (T) dT

k(t)

- » - x

T = ( t - t0 ) / t 0 , kQ= k ( tQ )

For k( t) = kQ [ l + 3 (t - tQ ) ]

tf k(T) dT = t - fcn S t n / t - t

'0 0

For f$ = 0, se e Fi g . 8. 7.

£(^)For 0 = 0 and q - t e rm ina t ed a t T = D , s ee F ig . 8 .1 " .

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Section 8.1. Solids Bounded by Plane Surfaces—No Internal Beating.


8.1.18.1 3, p. 279 Infinite plate withsteady surface heatflux.t = t t , -I < x < +SL,

T = 0.

q ^ q„ , X = ±1, T > 0.

( t " V k „ X-TJL- = F ° 2 6

X = nnn

n=l X v 'cos (X X)

n

M- -H

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8.1.19 3, p. 350 Infinite compositeplates with convectionboundary and infinitethermal conductivity.

t = t 2 ,0 < x < i.2, T > 0.

fc V l2 < * A2 + VT > 0.

t = t,

T - 0.

< ^ - 0, x - 0,

h_ is contact coefficient

between 1 and 2.

. Q, 0 < x < *j+ i . r

-X„T -\,Tt - t X,e _ X.e

fc o- fc f x x - x 2

/ - X „ T - x n-r4 - " l ^ f . f e 2 -e 1

t„ - t c " t„ - t* 3 \ X, - X,

X l + X 2 - b l + b 2 + b 3

X l X 2 * b l b 3

bl - 2 A C 2 V b 2 - V P 1 Cl V b 3 - V P 1 Cl S.

V*\'// V.-+<~\ k-»~

1• h r t f

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8.1.20 3, p. 476 Infinite plate withsteady surface heatflux and convectionboundary,t = t f , 0 < x < %, T = 0.

c^ = q Q , x = 0, T > 0.

(t - t f )k

(t - t fn

z \ V5Fo7, 0 £ T < T r

4 0 - i p - t f • & • * « - - A ]

-h.t.

K-«H

x \ 1 - exp[ l + (Bi/3 )]Bi Fo [1 - (T 0 A)]

1 + 2(Bi /3)+ 2(B i 2 /15)T > T„

T Q = penetration time to X = 1

= 1 2/5CL

See case 8.1.17 for general solution.

8.1.21 3, p. 349 Infinite plate withunsteady surface heatflux; convectionboundary and infinitethermal conductivity,

t = t , 0 < x < £,, T = 0.a = q.cos (cot) , x

(t - t f )hCOS (WT - b) -

V m + to2 ^ 2

ex p (-mx)

m = h/pct, b = tan (to/ra)

k->«>|-h.t,

k - f — I

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Section 8.1. Solids Bounded by Plane Su rfac es—N o Internal Heating.


8.1.22 2, p. 248 Semi-infinite plate,quartet-infinite plate ,infinite rectangular bar,semi- infiniterectangular bar,rectangularparallelepiped.

Dimensionless temperatures equal the product of solutionfor semi-infinite solid (case 7.1.1 or 7.1.3) andsolution for infinite plate (case B.1.6 or 8 .1 .7) .Se e Figs. 9.4a and 9.4b.

8.1.23 9, p. 113 Infinite plate withunsteady surfaceheat flux.

( t - t ,) k

V T4 r - 2m + 1 r (m/2 +1 , VFO- y i -i. r t c r <2 n + i> -x i* 2 n=0 L 2 VFO" J

. .m+1 ,+ I e r f c (2n + 1) + X"|

2 V?o J

k-i-l

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Sec t ion 8 . 1 . So l id s Bounded by Plane Surfaces—No Inte rna l Heat ing .

Case No. References De scr ip t io n

8.1 .24 9 , p . 114 Case 8 .1 .23 wi tht = t . , x » 0, T > 0.

S o l u t i o n

(t - t . )k

q 0 *T

1 ' " ' = 2 n H " 1 T (m/2 + 1) VF 5 J i1m+1m/2 n=0

fe f( 2 n -f- 1) - X le t t c L — n S — J

.m+1 .I e r f c

f (2n + 1 ) + x ]

L 2 V F O " J

8 .1 .25 .1 9 , p . 120 In f in i t e p l a t e w i ths t eady su r f acetempera ture , convec-ti o n boundary andv a r i a b l e i n i t i a ltempera ture ,t = f (x ), 0 < x < I, xt = t Q , x - 0, x < 0.

• " 1 * 0

( t

co / 2 2\V (Bi + \ ) s i n i\ X)

- V = 2 y exp (-X F o V °> "n=l x ' Bi + X + Bi

n,-1

* / [ t 0 - f (X)] si n (Xn X) dx- J o

Bi = hVk f * co t (A } + Bi = 0n n

See case 8. 1 .1 for t = t , x = I, x > 0

•h-«H

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Section 8.1. Solids Bounded by Plane Su rf ac es —No Internal Heating.

Case No . References Description So lu t ion

8 .1 .25 .2 74 , p . 239 In f in i t e p l a t es teady surfaceand convect ionboundar ies .

" i t h t - t n v fai2 + *

2

) ["i - « * <* >1 / , \

t = t , 0 < x < SL,

T - 0 .

t - t f l , X = C, T > 0 .

I—I—I

X cotn

See Fig . 8 .24

n=l [ Bi + Bi + \

+ Bi = 0

* 0 " - M ,

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8.1.26 7 4 , p. 236 Infinite plate withparabolic initialtemperature andconvection boundary.

o < x < a, T = o.t = t , x = 0 , T = 0 .ct = t , x = ±1, T = 0,

* - fcf 1 + 4t s _ t f n = 1 K X n /

s i n (X ) cos (X X)n nX + s i n (X ) cos (X )n n n

exp «»oX n tan (X n) = *i t

H" -HCDI

-h . t .

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Section 8.1, Solids Bounded by Plane Sur fa ce s— No in. .. ,u. Heating.

Case No . References Des cript ion Solut ion

8.1.27 9, p . 125 Infinite plate withsteady surface tempand convectionboundary,t = t , 0 < x < H, T = 0.

t • t 0 , x = 0 , t > 0.

-h.t,

B i X1 + B i

- 2 B^ s in (X X) exp f -X Fo 1

n=l (Bi + B i 2 + X 2 J si n (X )

V cot (X \n V V Bi = 0

h-*—I8.1.28 9, p. 126 In fi ni te pl at e with a

steady surface temp.Convection boundaryand linear in it ia ltemperature,t = ( t j- t Q)x/l + t Q ,

0 < x < l, T » 0.t = t Q , x = 0, T > 0.

•h,t.

fc - fc 0 X +[Bi ( t f - trf/^ - t Q ) - Bi - 1

Bi + 1

2 JBi [(t f - t 0 ) / ( t l - t 0 )]

^ sin (X X) exp ( - X 2 Pol

n=l ( Bi 2 + Bi + X 2 } si n (X n )- Bi - 1

X co t (X ) + Bi = 0n n

h-i-H

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Section 8.1. Solids Bounded by Plane Su rfa ce s—N o Internal Beating.


8.1.29.1 9, p. 126 Infinite plate withtwo convectiontemperatures,t = t , -I < x < +1, T = 0.

fc" fc o _ B J X _ yt - t - 2 (Bi + 1) B 1 _4-

s i n ({5 X) exp « » )n=l ( B I 2 + Bi + e 2 ) s i n (B )

t n , h-

osIo

f-i-^[-i-*-l

- h . t f

i »•' f cos (X n X) exp (-X^ Fo)2 ~ 4- 7 2~\

n=l (Bi" + Bi + X 1 cos (X )

X tan (X ) = Bi, B co t (6 ) + Bi = 0n n n n

8.1.29.2 74, p. 239 Infinite plate withtwo convect ioncoefficients,t = t Q , 0 < x < +1,

T = 0 .

W - h 2 , t,

t - fco l + B i £ l X

fcf - to~ 1 + B i s , i + ( B W B i i 2 )

oo

-Xvn=l

B i J>l 1 / 2cos (X XJ+ - r s i n (X X) e xp [-X Fcn X n \ n

( r . .-, s in (X ) co s (X ) + XA n = | [ 1 + K l / B i 4 " z s i n C X ^ "

+ " X — s i n ( X n>n

HM Bi i l l = h^/ k, Bi ^ = h 2 Vk

co t (X )n B i U X n / V BilZI

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8.1.30 9, p. 127 Infinite plate withconvection boundariesand time dependentfluid temperature.t = t . , -I < x < +8,, x = 0 .

t f = 4> (T) .

-h . t ,

asl

°° , 2

t - t . = 2 Bi IA co s (X X) ex p i-\ Fo)n n r \ n /

I2 n= l (&i + X 2 + B i 2 J cos (Xn)

<.( exp ( a x V A2

) * ( T1

) dT'Ja

X ta n (X ) = B i.n n

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S e c t i o n 8 . 1 . S o l i d s B o u n d e d by P l a ne S u r f a c e s — N o I n t e r n a l H e a t i n g .


8.1.30.1 7 4 , p . 316 C a s e 8.1.30 w i t h t - t± cos (V P S X ) e xp (-Pd F o)t, = t_ - ( t „ - t . ) - D T . — ;— = 1 - : — —

f f m »» * • t f m - c. c o s ( V ? 3 ) _ J , ^ s i n { V M )

A cos (X X ) e x p (-X 2 F o )n n \ n /

n=l 1 - X 2/Pdn

(•

,,n+l _ „. A .2 ,2\l/2-1) 2 Bi 1 B l + X 1 '

n X n ^ B i 2 + Bi + X 2 )

Mean Temp:

t - t,m i exp (-Pd Fo)

^ m " fc i V p d " f c o t { V P d ) - ; rr V P d " l

» 2 I e x p2 B i "

« » )»-l i* (Bi 2 * Bi * 1*) [l -(X=/ M)]

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Section 8.1. Solids Bounded by Plane Surf ace s—No Internal Heating.


8.1.31 9, p. 127 Case 8.1.30 withase 8.1.30 with ^ _ °° / , 2 „ \

t _ f v " » . i ik . .x , ,M i - v > • * KF° ) .• <T< x ,t " ~ -T r 0 i n=l(Bi + X + Bi ) — -'

[ t l r T > T 0 . V n )

cos (Xn)

fc- fcl , , " f [T + ( 1 - T> e x p (

X

n F oo ) ]- _ . - l - 2 Bi ^ T - r -c l c i n= l ( Bi + X + Bi ) co s {X )

x co s (X X) exp (-X Pol

< j> T - (t - t ) / ( t - t . ) , Fo. = OCT / i 2 , X tan (X ) = Bii O i l i O O n n

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Ref-Case No. erences Description S o l u t i o n

8 . 1 . 3 2 9 , C a s e 8 . 1 . 3 0 w i t hp . 127 t f = ( t Q - t j ) s i n

I * 3 2 5 0.

ooI

t - t . Bi M

~ = ~M ~2 S i n ( W T + E + B 0 " B l 'co- fci

+ 2 Bi

iS„

2 Fo*2c o s ( e ) - s i n ( e )

v- X'co s (X X) ex p (-X FoA

n ^ n /

„ = ( * + 4 F ° M A n ) ( x * + B i 2

+ B i ) c o s < Xn)

~n * * * * *M e = co sh (X Fo ) cos (X Fo ) + i s in h (X Fo ) (X Fo ) s i n (X Fo )

1 it it * * * * 1tM e = (Fo ) sinh (Fo ) cos (Fo ) - Fo cosh (Fo ) sin (Fo ) + Bi cosh (Fo )

* r * * * * * *x cos (Fo ) + i I Fo sinh (Fo ) cos (Fo ) + Fo cosh (Fo ) sin (Fo )

+ Bi s inh (L) si n (L)J

F o = I V(»)/2a , X tan (X ) = B in n

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8.1.33 9, p. 127 Case 8.1.30 with89 t = t A+ bT, x > 0.

( t - V ° „ B i X 2 - B i - 2= F o + — •

t>r2 Bi

„ . V cos (X X) exp (-X Fo 1+ 2 Bi \ n \ n I

n=l X 2 (Bi 2 + X 2 + Bi\ cos (X )n \ n / n

X tan (X ) = Bi , See Fig. 8.26.n n

(See Fig. 8.1.25 for Bi = ™.)

perfect contact withan infinite plate ofinfinite conductivity,t = t., 0 < x < Jl + SL ,

T = 0.

q = 0 , x = 0 , T > 0.

- q x = q Q , X = l 1 + l 2 .

T > 0.

2

I—/,-*K~l

( t - t. )k' V _M_ L + i- 3 + M 1~£~ ' 1 + M L 1 2 " 6{1 + M)J

V c o s ^ x j e x p ^ F o J

n= l X 2 (x 2 + M + M) cos (X )n \ n / n

X co t (X ) = -Mn n

« - q . E o i = V / S v x = x / *i

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Section 8.1. Solids Bounded by Plane Su rfa ce s— No Internal Heating.


8.1.35 9, p. 129 Case 8.1.34 withq = 0, x = I + I , T > 0.

t = t 2 , i.x< x < «, 2, T = 0.

t = t., 0 < x 0. (t - t.)k

= X - 2MY sin (X n X) exp (-X 2 FOjJ

n=l X |X 2 + M 2 + M ) cos (X )n \ n / n

CDi

X tan (X \= Mn \ n/

8.1.37 9, p. 128 Cas e 8.1.34 witht = t., x = 0, T > 0.

t = t 2 , & 1 < x < % 2, T = 0.

t = t. , 0 < x 0.

( t - v( t 2 - t . ) = 2M 2 sin (X X) exp « * • , )

n=l ( X 2 + M 2 + M) sin (X )\ n / n

X tan i\ ) = Mn \ nl

8.1.38 9, p. 129 Case 8.1.34 withb Q , x = 0, X

0, x = I, + &,, T > 0.

t = t Q , x = 0, X > 0.( t ~ V . , f ( X n + 2 ) s i n ( v ) e x p ( - X n F o l )(t„ - t.) " _4, ^ ^ 2 + M 2 ^

n=l

X tan (X ) = Mn n

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8.1.41 9, p. 129 Ca se 8.1.40 with <* .2 ,, „, / ,2 „ \— X cos X X) exp l-X Fo, I

n n V n 1/q = f ( T ) , x = 0 , (t - t ) _ 2 a _ Y

T > 0. kJT n=l I^Bi - X 2 / M V + (1 + 1 /M)' X 2 + Bil cos (X )

x f f(T') exp (\âz'/l Z) dt ' .

Jo

X tan (X ) = Bi , - X 2 / Mn n 1 n

OD 8 . 1 . 4 2 9 , p . 1 2 9 C a s e 8 . 1 . 3 4 w i t h a ( t - t . ) (M + l ) k 3 B i + 6 + M B iii > c o n t a c t r e s i s t a n c e h r - r = F on + — (1 - X) - ——— — — — —

q -J l. M 1 2 6 B i ( 1 + M)e t w e e n t h e t w o c

p l a t e s .

+ 2 3 i „ . „ V - X2 - M B ic

( X 2 - M B i ) ta n (X } = B i X .V n c / n e n

P = X + X ^Bi + B i - 2 M B i ) + M B i2 (1 + M) X2 .n n n \ c c c / c n

B i c - h c V k l •

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Section 8.1. Solids Bounded by Plane Sur faces—No internal Heating.


8.1 .43. 19, p. 324 Composite plates of Region 1, -& 1 < x < 0:different materials.fc V "V x < V (t - t„)x = 0.

t - v

x = -X^ , T < 0.

t = t n , x. = l 2 ,

T > 0.

( t " t p ) = < •-»> - 2 y exp - X 2 F O ,

cos (A X) sin (B U ) - a sin (A X) cos (3LA )ri li n n

A Td + K3 2 L) sin (A ) sin (PlA ) - B (K + L) cos (A ) cos (BLA )ln L n n n n J

Region 2 , 0 < x < l^'

_ _ r ^ - (t i ~ V ~ + K £1exp -A Fo 1

sin (0LA - 0A X)n n

A [(X + K3 2L) sin (A sin ( S L A - 3 (K + L) cos ( A ) cos (SLA )]

cot (A ) + a cot (3 U ) = 0n n

B = V V ^ ' K = k 2 A l ' F ° l = V/ll' L = W X = X / £ l

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8.1.43.2 7 4, p. 441 Composite p la te s ofd i f f e r e n t m a t e r i a l sw i th i n su l a t edsurface and convection boundary,

t = t . , 0 < x < I. +

-h , t ,

rrf - x ~ 2 1 Hp c o s (x n x / v s) e x p ( -x n L 2 p oi / A ) ' ° < x K

f l n=l ir v ' v '

L .

t - t.T^T = x " 2 A xlJ (c o s K (x " L> 1 c o s ^ n1- / ^ )

f i n=l n ' L J

- K s i n l"x (X - L)l s in (XnL/VE) } exp (-xh^Fo^h) , L < X < 1 .

• [(X * *>** S P ) S i n ( Xn > + X„ (TP)( l + KL/VA ) ra ( Xj] cos ^X n L/VS^ + [ ^ i + K L / VA + ^ g p ) c o s (X )

- \ (SP) (* +

i & )s i n

<V] K s i n

<x

,y*> •LX X—"- (1 + L ) t a n (X L / V A ) = 1 - - T T ( 1 + L ) t a n (X )

- K ta n (X ) ta n (X L/ih) .n n

Bi = oyK.,, L = V V F O l = *f/ . A - a ^ , K = ^ J

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i

8.1.44.1 9, p. 416 Rectangularparallelpiped.t = t . , 0 < x < w ,

0 < y 0 .

t = t . , a l l

o t h e r f a c e s , T > 0 .

t - t . , , v V" s i n (imrY) s i n (mrZ) s in h | ( 1 - X)-v/X Ii _ _ X6_ y y L \ mf n,o Jt n - t . ~ 2 ^ , ^-t . . , , ,

0 i IT — " — " - - . 1 . M i

as co

m n si n h (X )m , n , o

O O 0 0

m=0 n=0

i. s i n (nnrY) si n (mrZ) s in (JITTX) exp (-X . Fo )ID f n t Xf w

0 m n X_m,n,Jl

X „ = A2TT2 + (ntirw/b2) + (mrw/d)2I Q , n , x <

m = 2m + 1, n = 2n + 1, X = x /w, Y = y /b , Z = z /d

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S e c t i o n 8 . 1 . S o l i d s B o u n d e d by P l a n e S u r f a c e s — N o I n te r n a l H e a t i n g .

C a s e N o . R e f e r e n c e s D e s c r i p t io n S o l u t i o n

8.1.44.2 7 4 , p . 287 R e c t a n g u l a r p a r a l l e l e - » » mp i p e d w i t h c o n v e c t i o n 0 _ . V y *\boundary.t = t±, -i.x < x < + £ r

- l2 < y < +A 2 ,- J l 3 < z < +A , T = o.

n = l m=l s=lAn l A m 2 s 3 o s < * „ 1X ) « » < \ n 2Y>

B

• t l . t .

x c o s ( x ^ z ) e x p (\ 2

nl ? ox + X 2 P o 2 + x j 3 F o 3 )

( n , . , s )+ l / 2 + x2 \l/2A = J . l i i n ,m ,s , i j

n ' m ' s ' i X . ( W + B i . + X 2 . )n , m , s f i \ I i n , m , s , i /

co t (X . ) = ( r T - J X . , Fo . = ax/I 2.v n , m , s , i ' \ B i . / n , m , s , i l l

Bi i = WI../IC, i = 1, 2, 3 , X = x Ar Y = y / £2 , Z = z A 3

<1 A

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8.1.45 9, p. 417 Case 8.1.44.1 with

t = t Q sin (UK + e) ,

x = 0, 0 < y 0.

00 00t - t . V ^ " s i l > ivm?) si n (mrz) M s i n (urr + c + d> )M^n m,n- fc i _ i6 yt - t 2 " —^0 i n m=0 n=

QO 00 00 ^ — O

„- V"* V* V" & s i n (nffiY) si n (mrz) s i n (JITTX) (W COS E -X n s i n E) exp(-Xar /w )

- f Z 2 2 7-i— - 2 —x-- mf f*m=0 n=0 H=Q ( W + \ _ 1 m n

\ m,n,x./

m,n »P »m,n s i n h { [ ( S r w / b )

2

+ ( ^ w / d ) 2 + i w ] 1 / 2 }

2 *~ "~W = w 0)/a, See ca se 8. 1. 44. 1 f or X ,.,, X, Y, Z, m, nm n . < £

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8.1.46 9, p. 419 Wedge with steadysurface temperature. i 6 L 2 V , ,.n r . , „ n ,

t - t . . o < e < e n , t n - t. - 57 + e" L <-« [ S l n < 2n + D /BJi 0 0 l 0 0 n= l

T = 0.

t = t , e = o, T > o.t = t . , e = e 0 , T > o. / exp (-< rruVr

2

> a ( u ) d u

J n u s

09I

s = (2n + 1) TT/8.

Fo r t = t Q , 9 = 0, 8 0 , T > 0:

t _ t = 1 " | ~ ^ s i n < s 9 >J e x P ( -o ru 2 / r 2 ) —J (u)du

0 n=0

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8.1.47 33 Composite slabs withramp surfacetemperature.t = t , 0 x < 5L X+ %,

T = 0.

t = fc

i + ( t

o " Vfcrc

'x = 0, 0 < t < 1 /b.

t = t , x = 0 f T>" l /b .

t. - t. = 2

S

k-»oo

i

Y Pd (x^ + M) exp f-A^ Fo ) [l - exp / x 2 / Pd ) j si n (X X)

n=l \^ (\^ + M 2 + K)

\ ta n {A n) = M, M = A

1 P 1

C

1 / J ' 2 P 2 C 2 ' X = x / X l

See cases 8.1.34 to 8.1.42 for o the r con dit io ns.

h-it-H*-je2-H

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Section 11.1. Change of Phase—Plane Interface.


1 1 . 1 . 9 9, p . 292 C o n v e c t i o n b o u n da r y a t

o r i g i n a l s o l i d - l i q u i d

b o u n d a r y.

t = t 1 # x > 0 , T = 0 , t x > t m .

t = m e l t i n g t e m p .

t - t F B i 2

_ § _S = B i T-^{1 + 2 Po )t - t , x 2 x'

m £

P Bi

IT' 1 + 3 1 F

° x >+ . . . , 0 < x < w .

V \ . ,

F 2 B i 3

F B i x F O x . _ _ x ( 1 + P ) F o 2 + . .

F = k s ( t m - V / a s P Y ' F o x = V / x ' Bi x = h x A s

(solid) (liquid)

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Section B.l. Solids Bounded b y Plane Surfaces—No Internal Heating.


8.1.49 19 , p. 3-30 Infinite plate ofinfinite conductivitywith convectionboundary and harmonicfluid temperature.fc f - W (t mx - • t )mnC O S (U K ) .

t = mean temp,mn

t = max temp,mx

I

V"—i • h . t ,

t - tmx mn Ve*COS (TU) - \p)

+ 1

~ exp ( -' + 1 V pe l J

9 = u p e V h , \|i = tan (9)

I n s t a n t a n e o u s s u r f a c e h e a t f l u x :

J I _ . eh {t - t )mx mn

See Fi g . 8 .9

mx mn" |ZT?~s i n (Tto - i|))

(« -2 i -* |

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Section 8.1. . Solids Bounded by Plane Surfaces—Ho Internal Beating.

C, .a No. References Description Solution

8.1.50 40 Infinite plate ofinfinite conductivitywith surface reradia-tion a nd circularpulse heating,t = t Q, T = 0,

c r ^ T 4 .

See Pi g . 8 .12 for maximum temp era ture va lu es :

/3naj£

D =' he a t i n g du ra t i on

Sc = 'W111 l r r / D >

Ik->»

|— 28—|

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Section 8.1. Solids Bounded by Plane Sur fac es—N o Internal Heating.


8.1.51 19 , p. 3-44 Infinite composite424366

plates in perfectcontact, convectionboundary on one side,insulated on other side,t = t Q , 0 < x < & 1 +

S2, T = 0.

o = 0, x = 6 + 6 , T > 0.

See Fig. 8.14 P l 6 l= f

V P i k i D / ° iD = heat ing durat ion

fc 2inax " fc 0t„ - t„

|k,^H

t f . h * \ JHH^-H

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8.1.52 19, p. 3-49 Infinite plate with44 radiation heating68 or cooling.

t • t Q , 0 < x < S,

9 r - ^ ( T s - T x = o ) -

See Figs. 8.15a-e for heating.

See Figs. 8.15f-l for cooling,

t - t„

t •* t= f (Fo)

source temp.

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Section 8.1. Solids Bounded by Plane Sur fa ces —No Internal Heating.


8.1.53 19 , Infinite plate withp . 3-68 time-linear surface49 heat flux.

t = t 0 , 0 < x < 6,

T = 0.

gx " W ^ ' > °-

k ( t -

«0«Jfii Fo |

" F oD ^ k 2x 3

1 2

% = c t D / 62 , X - x/6

¥ - - f e ) .*< *><F ° D 'F ° D > 1 .

For Sc = q n , a x ( 1 - T / D ) ;

MH

k (t - t n ) x 2 ±

= Fo + —- - X + — + (solution forV 3 Vq x = q n u

T/D) •

D «• duration ofheating.

TT < Fo < Fo , Fo„ > 12 D D —

8 . 1 . 5 4 1 9 , C a s e 8 . 1 . 5 3 w i t hp . 3 -6 8 a = a s in wT / t> .150 ^ ™ *X

S e e F i g s . 8 . 1 9 a & b

&S* ( t - t Q ) = f ( 2 i r t / D ) , QQ= ^ D

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Section 8.1. Solids Bounded by Plane Sur fac es—No Internal Heating.


8.1.55 1 9 , p . 3-73 Porous infinite5-l plate with steady

surface heat fluxand fluid flow,t = t Q , 0 < x < S,

x = o.t fl = fluid source

09I - V k f c ,

k-«H

See F i g s . 8.20a & b

y i t - tn)

F = ,G f f 5 /k e

k e = Pk f + (1 - P) k s

G = fluid mass flux

P = plate porosity

Steady state solution:

^T7l = p exp (-Fx/6)

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8 . 1 . 5 6 1 9 , Two i n f i n i t e p l a t e sp . 3 -7 5 i n p e r f e c t c o n t a c t52 w i t h a s t e a d y su r f a c e

h e a t f l u x .t = t 0 , 0 < x x < fiir

0 < x 2 < 6 2 , t = 0.

^ = q 0 , x x - 0 , T > 0 .

See Pi gs . 8 .21 a-d

( t ~ V k iV i

f (Fo >

COI I1

H i * K H

8 .1 . 57 19,p . 3-7548

Case 9 .1 .5 6 w i thk 2 + » .

See Fi g . 8 .22

q 6 - = f < F ° l >V i

8 . 1 . 58 19 ,p . 3-7553

Case 8 .1 .5 6 w i thk_ + ™ and

% * Snax S l n 2 ffT/D-

See F ig s . 8 .2 3 a , b , and ct ~ fc0

t h - t„ = £ U 1 T T / D )

eqb 0

D = du ra t i o n o f p u l s e

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8.1.59 74 , p. 245 Thin pl a te with twoc o n v e c t i o n c o n d i t i o n s ,t = t , -A < x < +1,

T = 0.

CO

I

V h i

t , . h . V h - i —

h,.t«fcr^"\ H

t - t_ cosh K z 0tvE

fcf - *o cos h (B i ^ 2 L ) • ^ s in h ( B i ^ ) L

- 2X n s i n X n C O S ( X n X ) e x p [" ( X n * B i l w L 2 ) F o w J

(Xn + B il wL 2) [X „ + s i n( MC O S( X

n ) ]

X n t an (X n ) = B i2 w , B i l w = h l W A, B i 2 w = h ^ / k

X = x /w, L = V w

Mean temp:

t_ - t _ t anh (B i * '2 L )B i ," " t ° " » ^ - ^ - 2 " - K 2 -)

X ' °n «» f- ('n * " l . L ' ) ">.]n=l

X 2 + B i . L2

n l w

B iB_ = lw

\2 fa i2 +B i , + X2 )n \ lw lw n /

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8.1.60.1 74 , p. 320 Infinite plate withevaporation and convection boundary.

t = t . , -S , < x < +S, ,

T = 0 .

-bxm = m . e , x = ±SL.

m = e v a p o r a t i o n r a t e .

H- -H

-h. t«

t - t .M co s (VPdY) ex p (-Pd Fo)

tf ~ fci co s (VPd) - ^ r VPd s in (Ypd)= 1

-2n=l

1 - M

1 - Pd/X

B i

A c os (X X) exp -X Fon n n

2 s in ( X n ). M =

YÔ fcf - fcwbn X + s i n / X \ c o s (X \' h ( t . - t . ) t c - t .

n ( n | I n | f i ' f 1

co t / X ) = X / B i , t . = we t b u lb t emp .

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8.1.60.2 74, p . 321 Case 8.1.60.1 with t - tasymmetrical boundary i

s

t f ,h

h X + s in / X )cos /X \' "m X_ - c o s /X _ \ s i n fX

co t /X ^= X /B i , t an /X \= -X /Bi

Mean temp:

" l- t.m l

i -« - )

- 2 — - 2 l { X - 2 ) X 2 / . 2 + B . + x 2 )n=l n \ n /

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Section 8.1. Solids Bounded by Plane Surfaces —No Internal Heating.


i

o

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Section 8.1. Solids Bounded by Plane Surfac es—No Internal Heating.

Case No . References Description Solutioni _ _ _ _ _ — _ —

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Section 8.2. Solids Bounded by Plane Surfa ces—Wit h Internal Heating.


8.2.1 3, p. 276 Infinite plate withuniform internalheating.t = t . , -S L < x < S L ,

iT = 0.

t = t ± , X = ±SL. T > 0,

CDI

KJ

( t - V k 1 2~ = f (1 - X Z) - 2

X = (2n + 1J1T/2n

See Fig. 8.8

y 1=11:n ^

n=0 n

exp ^-X 2 Fo\ cos (X X)

k*. i , - » j

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8.2.2.1 74 , p . 363 In f in i t e p l a t e w i ths teady surface hea tf lux and var iablein t e rna l hea t i ng .t = t . , -a < x < +i,

x = o .q " ' = q , J " ( 1 - * / * ) •\ = q„» x = ±a, x > o.

CDI

i nU)

%" - < 1 o

i i " V k Fo* i *2

™ cos f(2n - l)7rxl cos (X X)

_ ^ ( 2 n . } 4 ~n=l

°.n

1 - exp f-xjj Fo ) + *

« n=l \ n i

t - W - f )cos ,X nX)n=l W

x expX = mrn

8.2.2.2 74 , p. 363 Case 8.2.2.1 with (t - t,)k. 1 I I — — • • •= q > " (1 - X - ) . — i - - f Fo - J ("I)" cos (X nX, [l - exp (-X* Fo)] + *

0 n=l n

X and * given in case 8.2.2.1n r

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Section 8.2. Solids Boi id by Plane Surfaces— With Internal Heating.


8 . 2 . 2 . 3 74, p. 364 Case 8 . 2 . 2 . 1 w i th

q > " = g j " exp (-bX). irjr • H t" - '-b] * I t 1" -1' - H 7W77)0 n=l n \ n/

(X X) j l - exp (- A 2 Fo) + $COS

X and $ given in case 8.2.2.1

8.2.2.4 74, p. 364 Case 8.2.2.1 with (t - t )k± —= Fo (1 + Pd Fo) + $

T 1 ' l A *1 + * > • q- " 9?

4 given in case 8.2.2.1

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Section 6.1. Infinite Solids— No Internal Heating.


6.1.23 9, p. 346 Cylinder with properties fc » j(XR)J<X)dXdifferent from surrounding — = —fr I exp(-A Fo , ) — _ r _ r — , R < 1

medium.

t = t L , r < r Q , T = 0.

t - t r , N J ( X R ) J (X)d)rounding = -£ / exp(-A Fo ) — - p - ^ T

fc - * 0 2Y f ( , 2 , \ J l ( X ) [ J 0 ( e X R ) " ' - Y 0 ( 6 X R ) ' t ' ] d X

= —*- I exp -X Fo, I = -

0

i|) * Y ^ c X j j j j j B X ) - B J0 ( X ) J1 ( B A )

<> = y J ^ X j y ^ B A ) - B Jo ( A ) Y1 ( 0 X )

R > 1 .

A X y B - V 5 ^ , Y = k A00

S e e c a s e 6 . 1 . 4 f o r e q u a l m a t e r i a l p r o p e r t i e s .

6 . 1 . 2 4 9 , P o i n t s o u r c e o f r e c t a n g u l a r ( t - too)4Trrk

p . 4 0 2 p e r i o d i c w a ve h e a t i n g . Q~ " e r c * "'**' T ^ + 1

CO

( S T ) - T i

e x p ( - T X " " ) j e x p f - X2 + T X 2 j - e x p ( - X2 ) J s i n ( F o * X ) d X

x[l - exp(-X2)]

Q(T) = 0 , x < 0. „ 2 j

Q ( T ) = Q Q , n T Q < T < n TQ

+ T.f n = 0 , 1, . . .

Q(T) = 0 , n tf l + T < T < (n + 1 ) T0 . 0 < T < T .

n = N umb er o f c y c l e s .

t = t ^ , r + co. T x = T 1 / T 0 , T = T / T Q , Fo* = r/V5f^

>* r

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Sec t ion 8 .2 . So l id s Bounded by P le^ e Sur faces— With In t e rn a l He a t in g .

Case No . Refe rences D esc r ip t io n Solution

8.2.3 9, p. 131 Infinite plate withtime dependentinternal heating.t = t . , -a < x < ±SL,

T = 0 .t = t , X = ±1, T > 0.

q ' " * f ( T ) .

c -« J - s i •&$» - P ^ -]n=0

x J f fr) exp -IT (2n

J o+ l )2 a (x - T ' ) / X2 J * : '

8. 2 . 4 9 , p . 132 Case 8 . 2 . 3 wi thf ( r ) = q j " e " b T .

( t - t j kV * l fcos (xVgd) ."1 o % m , . .

2 V t - l>" cos U n X ) exp ( -X2 Fo)

+ ™ \ X (l - X 2 /Pd )n=0 n \ n /

X = (2n + 1 ) IT /2n

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Section 8.2. Solids Bounded by Plane Surface s—With Internal Heating.


8.2.5 9, p. 132 Case 8.2.1 with

q' «• • f ix ) . fr"J-5:2H«-Pi}r k , nn=l

1 - exp (-=¥-)]

x I £ (X) cos ( ~ x )J-X

dX

CDI

m

8/10/2019 6224569


Section 8.2. Solids Bounded by Plane Sur fac es— Wit h Internal Heating.


8 . 2 . 6 9 , p . 1 3 2 I n f i n i t e p l a t e w i t h u n i fo r m4 1 i n t e r n a l h e a t i n g a nd

c o n v e c t i o n b o u n d a r y,t = t f , -l < x < +1, T = 0 .

t „ h -

(*- (,—|*-X,-»|

-h, t ,

( t - t j k B i „ . . 2f T . B l B i X

^ = 1 + T'~2~- ' T

co s (X X)2 B i '

~ X2 f \ 2 + B i 2 + B in = l n V n J cos (Xn)exp K -)

X ta n (X ) » B i , S e e P i g . 8 . 1 3n n

8 . 2 . 7 9 , p . 1 3 2 I n f i n i t e p l a t e w i t h u n i fo r mi n t e r n a l h e a t i n g a nd c o nv e c t i o n b o u n d a ry o n o n e s i d e ,t = t , 0 < x < «., T = 0 .

t = t - , x = 0 , T > 0 .

- M f

{ t " t f ' k ( 1 + B i / 2 ? X _ x f

l ' " i 2 1 + Bi

+ 4 Bi^ s i n {X X ) f l - c o s (X )~1

, X 2 (\ 2 + B i 2 + Bi \ s i n (2X )n = l n \ n / n

e x p « *>)

X c o t (X ) + B i = 0n n

H-<M

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8.2.8 9, p. 311 Infinite plate withtime-dependent heating,t - t , 0 < x < I, T = 0.

t = t n , x = 0, I, x > 0.

q - = l * 8 / 2 ,

S - - 1 , 0, 1, 2, . . . .

1

I — A — I

(t - t 0 ) k

Bar +s/2 (1 + s/ 2)1 - T(2 + s /2) 2 +2

^•['"-(^•'"-(l^)]}

8.2.9 9, p. 404 Infinite plate with variable internal heating. ( t - t„)kt = t n , 0 < x < l, x = 0. „ i . ip2 B ] cos (Vl )

u 9 0t = t Q , x = «., T > 0 .

0 ' _ 1 cos(Vlx) _

. t i t _ - H Iq j " + B t t - t Q ) .a = 0 , x = 0, T >. 0.

16 V 1-1)" exp {L-t2n -v \)2

T\2

+ tej go/4J cos L(2n + l)nx/2j* ~ f4B - (2n + l ) 2 i i 2 ] (2n + 1)

x

:

n=0

B = &8T A

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Section 8.2. Solids Bounded by Plane Surf ace s—Wi th Internal Heating.


8.2.10 9, p. 405 Case 8.2.9 withconvection boundaryh, t at x = I.

( t - t f l ) kBi cos (V EX ) 1_

, , , .2 ~ B [ B i cos (VTf) - V T sin (V"B) B90*"*

+ 2 Bi^ cos {X X) exp (B - X 2 ) Fo

\ = o (B- Xn ) [ Xn B i ] O S < V

D

X tan (X ) = Bi, B = 04 /kn n

8.2.11 19 ,p . 3-29

Infinite plate of infinite G H n (l - (Q/G)\ + 9 = LH/Cconductivity, variablespecific heat, convection :boundaries and steady heating.

(1 + q^ " £)/hL(t - t f J, C = (1 - c f ) / c Q , H = hT/pc£

t = t , 0 < x < 28,, T = 0.

c = c Q + S(t - t Q ) .

a • * > = a l ' ' .

%

- * • x

H evaluated at fc„ , c, evaluated at t_.0 E f

L = H x (surface area/volume) = 1

9 = (t - t 0 ) / ( t f - t Q )

- h , t f

I — 2 S - H

8/10/2019 6224569


Section 8.2. Solids Bounded by Plane Sur fa ces —Wi th Internal Heating.


8.2.12 19 , Case 8.2.11 for infinitep . 3-29 square cod and cub e.

See case 8.2.11, set: L = 2 for squareL = 3 for cube

rod

8.2.13 19 ,p. 3-32

Infinite plate of infiniteconductivity with surfaceradiation and steady heating,t = t , 0 < x < 2£, T = 0.

q t = adr ( i* - T 4 ) .

T = source temp.

4=% -

In + 2 t a n -1 ( 8 / N ) = 2 t a n _1 ( l / N ) + 4 M N 36 + N) (1 - H ) " |(6 + N) (1 + N)J

/ 4 4 \ 1 / 4 3

6 = T/TQ , N = [q^'SL/a&T0 + 8*J , M = a^T^/pcl

<*r - ^ r

H-2JH

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Section 8.2. Solids Bounded by Plane Surfac es—W ith Internal Heating.


8.2.14 19 ,p . 3-33

09I

Case 8.2.13 w i t h s i m u l t aneous su r f ace conve c t iona n d r a d i a t i o n .^ = h ( tf - t ) + a-jr^T4 - T 4 ) ,

x = 0, 211, x > 0.

M = A i *n ( F T ^ ) + A Jin9 - R \ A

wnrj + r^(9 - n ) 2 + d >2l

2 2( i - n.) + <t> :

A

4 r - i /e - n\ .. - i A - n\"l

^ = JBVU - n2 [n Vv - n2 + (v/2 + n2)]J

A2 = jsVv - n2 [nVv - n2 - (v/z + n2 ) ] |

A 3 = n / (v2 + 8n4)

A, = (V/2 - V2 ) / ( V 2 + 8I14)4

•\f 2 \ ( 2

R1 = - n - > v - r i , R2 = - n + ' y - n* = Vn + B/4, n = ^672

\ V 3

*-(££+&^F + (£-JF ^rv = - J N4 + 32/4

s = h/a#-T„

N = I q " ^ / ^ ^ + 9* + s9fl1/4

8/10/2019 6224569


Section 8.2. Solids Bounded by Plane Su rfaces —With Internal Heating.


8.2.15 19, Infinite plate of infinitep . 3-34 conductivity with prescribed

heating rates.t = t f l , T = 0.

a. st ep input; _TTT = 1 ' q ?"p = 1Tnax Tnax

g i l l Qf I Ib . li near input; ^nr = T , «,,-, ' = 0.5

Tnax TnaxQ I I • Q I I I

c. li near input; _rrr =» (1 - T) , * , , , p =0 . 5Tnax Tna~ax

r l l l

sax) • 11n ' 2 — 0 ' "

d. circ ula r puls e; g , , , = si n irr , •, , , D

Toax Tnax

0.5

b. ^ e = 7 2

PC«5Q">

peg

= T( 2 - T)

sin27TT

e. power pu lse; JL1L = ( 3 T ) - 3 e ~ ( l A - 3 ) f aLlL- = 0 .5473 e. &*.Tnax Tnax

0.06767 f -( I *1

)"(l/T-3)

f. power pul se, J l l = ( 5 T ) - 5

e - ( V c - 5 ) f QLLL_ m 0 < 2 7 9 5

Tnax Tnax

pC6

Q "- 6 = 0.006869/'- r- + ~Z7 + Z + x J

\ 6 T 2T T /

• ( l /T-5)

exponential pulse; _rrr = < 1 0 T ) e 1 _ 1 ° T , £7775 = 0.2717 g. &~ 9 = 1-001 [ l - (10T + 1) e 1 0 T JTnax <Tnax D

9 = t - t_, D = heating duration

T = x/D, Q''' = total heat input per unit volume during time D

See Pig. 8.10

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Section 8.2. Solids Bounded by Plane Surfa ces— With Internal Heating.


8.2.16 19 , Infinite plate of infinite See Fig . 8.11p . 3-35 conductivity with surface

reradiation, steady surfaceheating, and steady internal

heating.t = t Q / T = 0.

^ = q " + O&^l - T 4 ) , x = 0 , 26 .

T = source temp.

k-

q"

I — - ' 5 — |

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8.2.17 7 4 , p . 3 8 6 I n f i n i t e p l a t e w i t h t wos y m m e t r i c a l p l a n a r h e a tp u l s e s a n d c o n v e c t i o nb o u n d a r i e s ,t = t f , -ft < x < +&, T = 0 .

I n s t a n t a n e o u s p u l s e o c c u r sa t T = 0 , x = ± x , w i t h

2s t r e n g t h Q (J /m ) .

t f , h -

/

• x 1

X

- h , t f

(t - t f ) U= 2 I

n = l

, ; : T;—; 7,—r co s (X X, ) c o s (X X)X + s i n (X ) c o s (X ) n 1 ' n

x exp {< )•X ta n (X ) = B i .n n

For t = f (x) , -H < x < +«,, T = 0:m

Xt - t f ) k &

03 = 2 / T : : 7\— ; 7\—r co s (X X) ex p (-X Fo)<L >. + s i n (X ) c o s (X ) n r \ n /

n = l

1

x / [ f (X ) - tf ] c o s (X X)dX .

Mean temp:

( t m - t f) k X , ^ . B i2

c o s (Xn\) e x p ( - X2

F o j& ~ s i n (X ) X ( B i2 + B i + X 2 V

n = l n n \ n /Fo r B i = °°:

( t - t f) k X .

Q5= 2 Y ( - l )n + 1 c o s J"(2n - 1)TTX "I c o s [( 2n - l ) i r x ]

n = l

e x p [ - ( 2 n - 1 ) IT 2 F o / « ] .

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8.2.18 74, p. 387 Infinite plate witha planar heat pulse,t = t , 0 < x < I, T = 0.

Instantaneous pulse occursat x = x. ,

T = 0 w i th2

st rength Q(J/m ) .

rri

(t - tQ ) k £ V 2 2= 2 > s i n (mrx ) sin (mrX) exp (-n v Fo)

£j in=l

LO

h-M

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Section 8.2. Solids Bounded by Plane Surf aces— With Internal Heating.


8/10/2019 6224569


Section 8.2. Solids Bounded by Plane Surfac es—Wi th Internal Heating.


8/10/2019 6224569


1°

n

I

i

REPRODUCED FROM

BEST AVAILABLE CO PY

8/10/2019 6224569




9.1.1 2, p. 240 Infinite cylinder with74 , p. 131 steady surface temp.

t = t i , 0 < r < r Q , T = 0.

t = t 0 , r = r Q , T > 0.

i

t ~ t,~ 0 „ V 1 ( ,2 _ \ V n

n=l

J (A ) = 0, s ee Fig . 9.2

Cumulative heating:

J, (X R)s--»sM--k-)â=l n

e 0

= 7 I r u p o ( t o - t i )

0 x n

9.1.2 X, p. 269 Infinite cylinder with time9, p. 328 dependent surface temp.

t. f 0 < r < r Q , T 0.

t. + cr, r = r Q , T > 0.

t - t

cr 2 /ai = Po + J (R 2 - 1) + 2 V exp (-A Fo) J ^ J L L

n 1 nn=l

W = °See Table 9.1 and F ig . 9.8

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9.1.3.1 74 , p. 270 Infinite cylinder wit hconvection boundary andvariable initial temp.

n=l

W > eX P l' Xn Fol

fn= l J 0 < V + J l ( X n >

B i

w - V i ( V

R)dR

9.1 .3. 2 3, p. 298 In fi ni te cyli nder withconvection cooli ng.t = t t , 0 < r < r Q , T =

h,t.

fc- fc f . V B i V \ R ) °*P ( - X 2 Fo)

fci - fc f " Z , (X 2

+ Bi 2 ) J n ( X )n=l \ n / 0 V n

X J,(X ) + Bi J.(X ) = 0n 1 n u n

See Fig s. 9.1a -c

Cumulative heating:

J . Bi 2 | l - exp (-X 2 Fo) | ,

n=l n \ n /

For Fo < 0.02, R » 0:

( t " t i ) , n . /Fo" . . A - R\ __ , „. Fo

(t„ -1.) • 2 B i v r i e r f c vr^re)+ 4 B l vs

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9.1.4 9, p. 199 Infinite cylinder with initialtemp varying with radius.

t = t . - b r 2 ,

0 < r < r , T = 0 .

t = t 0 . r = r 0 , x > 0 .

t - t„

b r Iexp (-X 2

n F o) J Q ( R V

n = l

J 0 ( X n ) = 0

n 1 n

l j J (X ) + 2 J (X )I 1 n 2 n

9 . 1 . 5 9 , p . 2 01 I n f i n i t e c y l i n d e r w i t h p er i o d i c s u r f a c e t e m p e r a t u r e ,t =» t . , 0 < r < r_ , x = 0.

t = ( t - t . ) s i n (wr + e ) ,ni i

r n , T > 0 .

t = m a x f l u i d t e m p ,m

'( t - t.)sin(wT + e)m i

t - t .

t - t .m I

r e a lI n ( R Vi Pd) 1

. . . . . . . . ,i i n ( v* pa) e x p C i ( w r + e ) l

+ 2 2n = l

e x p (-Xn F ° ) Xn [ P d

2co s (e ) - X s i n ( e )n

( x 4 + P d 2 ) J (X )\ n / J- n

J 0 ( R V

J o ( V = ° ' p d = r o t J / a

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Section 9,1. Solids Bounded by Cylindrical Sur fac es —No Internal Heating.


9 . 1 . 6 . 1 9 , p . 20 2 I n f i n i t e c y l i n d e r w i t h89 conv ec t ion coo l ing and

l i n e a r t i m e - d e p e n d e n t

f lu id t emp ,t = t t , 0 < r < rQ , t = 0 .

Ib

( t - t . ) a exp

b r

o' - 4 V - V

n l n ' B i J 0 ( X n >

Mean temp:

t - t .m i • i / w L l

Kii J»( *_ R)

~ X2 ( B i2 + \ 2 ) j n ( \ )n=l n \ n) 0 n

«•>)bx

n=l n \ n /

See F i g . 9 .9

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9.1.6.2 74 , p. 317 Case 9.1.6.1 with fc yjg F ( j )

-o r x _ , 0t_ = t , - (t.. - t. )e T r~ = l ,f f m £ m l ' fcfm

_ti V p a j n { V p d ) - ^ P d J , ( Vfd)0 B l X

2Bi V W * e X P (' X n F °)

n=l W ( Xn + B i 2 ) [ l - ( X n / M ) ]Mean temp :t - t . 2J (VPd) exp (-Pd Fo)

£ fcfm " fci VPd J Q ( VPd) - i j - Pd jj_ VPd

exp- 4 Bi '

2 2S »5 W * « * ) L1 - ( » M J

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9.1.6.3 74 , p. 320 Infinite cylinder with fc _ M J n ( V P T R ) exp ,-Pd Fo)convection and evaporation i_ _ 0boundary. t f - t. J Q ( VPd) - ( VPd/Bi) ( VP<3)

1 - T T rr=r r A cos (A R) exp (-A Fo ]

t = tj, 0 < r < r Q, T = 0.

m = m n e = evaporation rate.0 - 2

h-h n=l

~{\j S " < W ( 4 • • ' * ) ' = h <=

h( t c - t . ) '

0 n l n n

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9 . 1 . 7 9 , p . 20 2 I n f i n i t e c y l i n d e r w i t h p er i o d i c c o n v e c t i o n b o u n d a r y,t = t . , 0 < r < rQ , T = 0.

t . = ( t - t . ) s i n (<DT + e)1 m it = wax f lu id tem p,m

Jh,t .

t - t . f Bi I (R Vi pd) exp (ion + ie)= r e a l TT

t - t.m I

i[VT"Pd I1 ( YlHPd) + Bi IQ ( VFPd)] + 2 Bi

V / 2 \ X n C P d o o s ( e l " s i n ( e ) l J o ( X n R )

A ex p ^ P o ) (V + r d

2Wx2

+ B r W >n=l V n / V n / 0l n

V i { V = B i Jo ( V ' P d = r o u / a

9 . 1 . 8 9 , p . 203 In f i n i t e cy l ind e r wi ths teady su r face hea t f lu x ,t = t . , 0 < t < rQ, T = 0.

q r = V r = r 0 ' T > °-

( t _ V k 2 V / 2 \— *— = 2 Fo + (R /2 ) - (1/ 4) - 2 y exp / -A* FojV o n = l

\£J , (X ) = 0 , See F i g . 9 .3I n ' '

0For Fo < 0 . 02 , R » 0:

, /Fo" . , / l - R\ ^ F o (l + 3R) . 2 . (\ - R\t - t t ) k

V o 2R

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Section 9.1. Solids Bounded by Cylindrical S urf ace s— No Internal Heating.


t - t„9.1. 9 9, p. 207 In fi ni te hollow cylinder

with steady surface temp. c ~ 0 V / -,2 „ \t - t „ r / < r < r n , x = 0. 1 " ^ = * 2 «-P K F ° )

t » t Q , r = c i f r Q , t > 0.

IDi

n=l

J0 (Vn> [ J Q ( X n R ) W ~ W W * ]V W + W

VWVV " V WW = °

R = r/r Q , Fo = oer/r 0

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9.1.10.1 9, p. 207 Infinite hollow cylinder withdifferent surface temp,t = t. f r. < r < r Q r T = 0.

t = t Q , r = V T > 0.t = t , r = r i , x > 0.

f- = TT £ exp (-X* Fo)

n=l

W V R VJo ( a„ > + W T .

1

C W - < W W ] J0(an)UQ(RXn)

n = lJ^(a ) - J?(b )O n 0 l n

exp « »•)

T 0 [ ( T 1 / T 0 ) l n l 1 / R ) + &n(R/Ri)]T J lnd /Rj )

a n = V n ' b n = V °0 (R X n' = W ' W ~ W V X n R >

W i ' W ~ W V V W i ' " °

R = r/r 0 , Fo = a x / r Q

See F ig . 9.7a for t, = t., and Fig. 9.7b for t„ = t.,1 l 0 l

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ction 9.1. Solids Bounded by Cylindrical Sur fac es— No Internal Heating.

se No. References Description Solution

1.10.2 74, p. 282 Infinite hollow cylinder with °°two convection boundaries and _ y / 2 \variable initial temperature- f Z* n ^ \ n ^ l n' ' *general case. n=lt = f ( R ) , r.< r < r , T = 0 .

V ^

W(XnR) = - [ B i . Y 0 ( X n ) + X n Y l ( X n ) ] a Q ( X n R )+ [ W V + B i i W ] W > .

2T 2 , 2En " T K K V W " W W ] 'R 0

x [ R [£(R, - tf ] W(Xn,R)dR j(xj + BiJ) [Bi. J ^ ,

+ W V ] 2 - (x n + Bil) CBii V W " W W ]2 ) " . 1

CBii W + V l M CBi0 V W " W W ]" LBi0 J 0 ( W " W W ] P1 V V " W V ] = ° '

B ii - W * Bi 0 " V i / k ' R = r / r 0

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9.1.11 3, p. 409 Infinite hollow cylinderwith steady insidesurface temperature.

i , _ x i \2 _ \ 1 n 0 n 1 n o n

v ^ T = " 4 e x p ( n Fo ) Vo ' v - W i 'B i '= l

W = W V W - V W W

Al(R.) = VX

n) J

l( X

n V - V V V WW W i > - J i f t „ | Y o ( V i ' - °Po = ocr/r 0 , R = r /r Q

9.1 .12 74, p. 200 In f in it e hollow cyli nder with (t - t .) ksteady surf ace heat f lux.t = t i ? c ± < r < r Q , x = 0.

- q r - q 0 / * = r 0 , x > o.

q r = 0, r = r i f T > 0.

V o 1 - R' h - i ' r - " "» fe * r h »x An (R 1 ) + -

j . a R . ) j , a >+ i | U Y ir u l l A n * i , u l * V4JJ nTl^l 'W W

[ J 0 < * „R > V W - ^ V l ^ f t ^ ) ] exp (-X* Fo)

Po = Qtr/r 0 , R = r /r Q

j i u . ] . a ) = Y a p..?J. (Xjl n x l n l n i i n

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9 . 1 . 1 3 9 , p . 21 0 I n f i n i t e c y l i n d e r w i t h v a r ia b l e i n i t i a l t e m p e r a t u r e a n dc o n s t a n t s u r f a c e t e m p ,t = f ( r , 9 ) , 0 < r < rQ , x = 0.

t = t Q f r = rQ , X > 0.

CO 0 0

t - fc0 = 2 I [V» c o s <n 9 ) + Bn ,m s i n ( n 6 )]m=l n=0

x J (XmR) exp K-

)-1 - I f

\ = r / / R| f(K ,9) - tn" | Jn (X R)dR dB

°'m * [ w ]2 I 4 [ o J ° m

1 IT

\ = - / | R p E ^ e ) - t . l cos (n9) J (X r )dR dBn'm * [ W ] I L ° J' 0 - I T

1 IT

n,m * O W ] * Jo 4,I / R|"f(R,9) - t J s i n (n9) Jn ( X r aR ) dR

n •i.tr

de

J (X ) = 0n 4 m'

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n 9.1. Solids Bounded by Cylindrical Surfaces—No Internal Heating.

No. References Description Solution

9, . 211 Infinite cylinder with vari- °° °°

able initial temp and convec- . . V V r» / , , ,-. , >n T , I T,, / I 2 r,

. . , t - t = ) > A cos (n9) + C sin (n9) J (XR) exp (-X Fo)

tion boundary-general case . f Z_, £-, \_ n,m n,ra 'J nl

n V n /= f ( r , 9 ) , 0 <r < r , x = 0. ra=l n=0

x 2 - 1 -*A = —j-z 5? I I Rff (R,9) - t,~| J.(X R)dR d9

° ' m n(x 2 + Bi 2 ^ f j i j t l 2 l J L fj 0 m

2X 2

m

""•( " - )[W]

1 IT

x / / R[~f{R,9) - t 1 cos (n9)J (XR)dR d9

2X 2

mn ' m " TT(X2 + Bi 2 - n 2 ) [ J (X )1 :i - n ; ivvj 2

1 .It

x / / R[f(R,9) - t j sinf (R,9) - t , 1 sin (n9)J (X R) dR d9* • n m

' 0 -i r

K? J X J + B i J n < X J = °n m n m

See case 9.1.3.2 for f(r,9) = .

or

case 9.1.3.1 for f(r,9) = f(r)

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Section 9.1. Solids Bounded by Cylindrical Surf ac es —N o Internal Heating.


9.1.15 9, p. 212 Infinite cylindrical sectorwith steady surface temperature. 0 8 V sin(j8) \* / , 2 _ \ j m f _ T ,_, .._

t = t i f 0 < r < r Q, 0 <P9 < 9 Q, T — r - 1 2 ^ Y 1 «* K ^ T ^ T P j j ' n=0 m=l D m "'O

= 0

t = t Q , r = r Q , 9 = 0, x > 0. j = (2n + l)ir/8 , J.(X ) = 0

— 0

.1.16 2 , p . 248 Semi-infinite cylinder. Dimensionless temperature equals product of solutionfor semi-infinite solid (case 7.1.1 or 7.1.3) andsolution for infinite cylinder (case 9.1.1 or 9.1.3).Se e Figs. 9.4a and 9.4b.

1.17 2, p. 248 Finite cylinder. Dimensionless temperature equals product of solutionfor infinite plate (case 8.1.6 or 8.1.7) and solutionfor infinite cylinder (case 9.1.1 or 9.1.3).Se e Figs. 9.4a and 9.4b.

9.1.18 Inf in i te cy l inder in asemi - in f in i t e so l id .

See case 7.1.22

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Section 9.1. Solids Bounded by Cylindrical Su rf ac es —N o internal Heating.


9 . 1 . 1 9 9 , p . 4 18 F i n i t e c y l i n d e r .t = t„, 0 < r < V

o < z < a , T = o .t = t 1 # 0 < r < r Q ,

z = 0, T > 0 .t =

v0 < r < r , z = I, T > 0.

C o n v e c t i o n b o u n d a r y h , tQ

at r = r Q , x > 0.

t - t" ~ "0 V" B i J p t ^ n ) s i n h [ ( 1 - Z ) Xn L ]

fcl " fc0 ~ ( B i2 + X 2 ^ J„(X ) s i n h (X L)n= l \ n / 0 n n

"I I

- 4ir Bi

mJ (RX ) s i n (nmZ) e xp j - [ x2 + (imr/L) 2 j F o J0 " w V

n = l m = l K L ) 2 + lm)2] [Bi* + Xn ] V VVi'V = B i WB i = h r Q A , R = r / r Q , Z = z /X , , F o = a r / r2 , L = V r 0

9 . 1 . 2 0 9 , p . 4 18 C a s e 9 . 1 . 1 8 w i t h c o n v e ct i o n b o u n d a r y h , t

a t z = I, T > 0 . h~ ko= 2 Bi

~ J,,(RX ) I X c o s h ["X L ( l - Z)"| + B i s i n h1V(1 - 4

- 4 B

n-1 (B i 2 + Xn) W [Xn c o s h <LV + B i s i n h <LV ]

" y » . ( B i2 L 2 4 6; ) J o ( R X n ) s i n ( z em ) e x p [ - f a + ^ / L

2 ) F Q ]

Z

n

Z , fBi 2

+ X 2 ) ( B i2 L 2

+ B 2 + B i L ) ( L 2 X 2

+ B 2 ) J n ( Xn = l m= l V n / V ra / \ n m / 0 n

B co tB = -B i L , L = H/ t„m m u

Se e cas e 9 .1 .1 9 fo r X , B i , R , Z , and Fon i i i

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Section 9.1. Solids Bounded by Cylindrical Surf ac es —N o Internal Heating.


9.1.21 9, p. 419 Semi-inf inite cylinder withconvection boundary.t = t Q , 0 < r < r Q , z > 0,

T = 0 .

t = t ± , 0 < r < rQ ,z = 0, T > 0.

Convec t ion boundary h , t .

a t r = rQ , z > 0, T > 0.

t - t, J 0 ( R Vh-k^" ^ ?Bi2 + X

2

) J n ( X )n=l \ n / 0 n

2 exp (-X Z) + exp (X Z)n n

x e r f c\2 Po*

+ Z Fo 1 - exp (-X Z) e r f cn 2 FoZ FO

X J (X ) = B i J (X )n 1 n O n

Bi = h rQ / k , R = r / rQ , Z = z / rQ , Fo = rQ / 2 yjaJF

9 . 1 . 2 2 9 , p . 419 S e m i - i n f i n i t e c y l i n d e r w i t hs t e a d y s u r f a c e t e m p e r a t u r e ,t = t Q , 0 < r < rQ , z > 0,T = 0

t fcl' r = r 0 ' Z > ° 'T > 0.

t = t ( J , 0 < r < rQ ,

Z = 0 , T > 0 .

fc- fco J « < » * >

n=l

V O n ' I 2 \ *Z X J (X ) 2 e x p \ TXn F o j e c f ( Z F o

+ e x p ( X

nZ )

(z Fo* + — M + exp (-ZX ) er fc (z Fo* " - _ )V 2Fo / n ^ 2 Fo 'x e r f c i z Fo +

See case 9 . 1 .2 0 f or X , Fo , R,and z

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9.1.23 9, p. 420 Cone with steady surfacetemperature.

t = t . , o < e < e Q , T = o.t = t Q f e = e 0 , T > o .

= I +~ {2n + 1)T [<n + l ) / 2 ] P n (X)

^ 2 . nf (n/2) [dPn(X)/dn] x = x

n=0 0

•f x p ( - o r u / r * ) J n + 1 / 2 (u )du

..3/2

P (X„) = 0 (Legendre polynomial)n 0

24 33 Infinite composite cylinder,t = t., r 3 < r < r , T = 0.

t = t Q , r = r 3 , T > 0.

q r = 0, r = r ± .

k = °°

1 - ^ 2 - = TT J exp (-X 2 Fo) FG [x;J 0 ( R 2 X n ) - NX n J l (R 2 X n ) ]

n=l

F =v w

[Wall "['n - N ( N " 2 / R2>Xn] " K J 0 <R

2V " " V l ' V n ^G ° J0 (R V KWn' " "Wn'] " V n'

^KvVi,1

- H V I ( R 2V]{R - D 2 P, C

R = r / r i ' N = ^ — ^ T ~ ' J ° ( R 3 X n > = °(R 3 - 1) P 2 c 2

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Section 9.1. Solids Bounded by Cylindrical Surf aces —No Internal Heating.


9.1.25 33

oI

boundary and insulated end. -~ fc i V B i J 0 ( X n R ) c o a h [ X n L ( 1 ~ X>]t = t,, 0 < r < r rt , 0 < x < I, t n - t. - Z A 2 + B i 2 \ c M h

n=l \ n / O n n

Finite cylinder with convectional,r 0

T = 0. t = t Q, 0 < r < r Q ,

X = 0 , T > 0. q = 0 ,

0 < r < r Q, x = Si , T > 0.

Convection boundary h,h 2 , t Q : at r = r 0 .

h,t.

I

V —| x = A

- 271

2 1(2m + l)BiJ0 (X n R ) co s [(2m-^~ -"J (1 - X)nJ

«=0 „ - l L ' K + ( 2 m + X > V / 4 L 2] ("n + B i 2 ) WL [~,2 A (2m + 1 )2 T T 2 "11x exp | -Fo |_Xn + J — - - J J |

X J . (X J = Bi J (A,,)n l n O n

Bi = h rQ / k , L = J /r 0 , R = r / rQ , X = x / £ , Fo = or r / r2

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Case No . References Descr iption Solution

t - t. *- Bi, Bi„ J n (X R) cosh fX L (l - X)lX £. U n i- n £_

9.1.26 33 Case 9.1.25 with convection

V t. at r = r Q. ^ L . = , - =Convection boundary 0 i , (X + Bi. ) J. (X ) X sinh (X L) + Bi_ cosh (X L)

n= l \ n 1 / O n L n n 2 n J2 , t 0 : a t x = 0.

4in

" " , X m B i l B i 2 ^ + B i2 J 0 (X n R ) COS [ Xm L(l - X)]

Z , ^ (X2 L 2 + X 2 ) (X 2

+ B i 2 ) J n (X ) [(X2

+ B i 2 ) + B i , l11=1 n=l V n m/ \ n 1 / (T n |_\ m 2 / 2J

* e x p [-FO ( x v + xy]V i ( V = B i i J o ( V ' x » t a n ( V = B i 2B i = h r A , B i = h l A , L = V r . , H = r / r , X = x / i

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Section 9.1. Solids Bounded by Cylindrical Sur fac es— No Internal Heat ing.


9.1.27 33

I

Finite cylinder with alinear time dependenttemperature on one end.t = t.,, 0 < r <c r Q ,

0 < x < %, T = 0.t = t t + b(t Q - tjjx,

x = 0, 0 < T < b.t = t Q , x = 0, T > b.

c ^ = 0, 0 < r < r Q , x = l ,

T > 0. Conv ect io n boundaryh/ t . : at r = r Q .

t = f{r)-

^ x

h,t r

IX =

Lt 1

Bi J Q (3 n R) cosh [e n L(l - x)]

o " t i £, (ef + B i2)j . (3„) cosh te Dn= i \ n / o n n

B Bi j n ( 6 R) cosh f3 L( l - xjlin 0 n L m J

t „ -4_Pd

CO CO

m=0 n=l . f r 2 + *m) ( B n + B i 2 ) W S i n «*»>

1 « * [ f a (6n + B>2) ] « • [ "F o (Bn + e > 2 )

0 m = (2m + UTT, U ( B ) = Bi j n (3„)in n 1 n O n

Fo = oct/r jj, Bi = h r Q /k , L = l/a, R = r /r Q , X = x/B.,

Pd = r Qb/a

9.1.28 33 Case 9.1.27 with convectionboundary h_, t- at x = 0.

t £ = t. + b ( t 0

0 < T < 1/b. t = t Q , T > 1/b.

Solution identical to that given in case 4.1.25 exceptmultiply the second s ummation series by

Pd {exp l^rr (X: + X~/L") | - 1}[ k « • > >2) ]Convection boundary h . , t . :

at r = [..

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9.1.29 6

UPl

Infinite rod attachedto an infinite plate,t = t Q , 0 £ x <.<D ,

0 < r < » , T = 0.g^ = q Q , x = 0, X > 0.

©b — ^

tf?mmzz;

te5S5S555 ?£

3 • s »exp(-n 2 TT 2 Fo) ,

X = 1 , R + » •

(fc - V k l H* - V k l l / k 2 P 2 C 2 \ 1 / 2

q 0b Lg 0

b J^ 1 V k l P l c l /R+oo

x J Fo j 4 i 2 erfc ( =) +4 i 2 erf c (=) 4 i'erfcV « 4) 1 / 21

2 VFo J

W 2 / T ( 2 n + 2 ) 2 + R oim l( „ /n \ , f /n + l\

- erfc p * 4) 1/21 , (nfîiliir). 2~?Fo J e t f C \ L i~F5 J /

, X = 1, R =

F o = o ^ T / b " , X = x/b , R Q = r Q / b

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Section 9.1. Solids Bounded by Cylindrical Surfa ces— No Internal Heating.


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Section 9.1. Solids Bounded by Cylindrical Surfa ces —No Internal Heating.


Ito

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9.2.1 9, p. 204 Infinite cylinder with steady »internal heating and constant 0' 1 ... _2 - V

4 (1 - R ) - 2 2 ,

J 0 ( R X n ) exp (-X 2 F O )

surface temperature. r 2 a " *0*0 n=l i n n

J„(X ) = 0

See Fig. 9.5

9.2.2 9, p. 205 Infinite cylinder with90 steady internal heating

and convection boundary,t = t £ , 0 < r < r Q , T = 0.

<t t f ) k

r„q„• * »

" exp (-A Po) J n ( R A n )R 2 + 2/Bi) - 2 Bi J •" - "

*i x 2 ( B I 2 + x 2 ) j n(x )n=l n \ n/ 0 n

Bi j (AJ = X J (X ),O n n l n

See Fig. 9.10.

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Sec t i o n 9 .2 . So l i d s Bounded by Cy l ind r i ca l Su r f ace s —Wi th I n t e rn a l He a t i ng .

Case No. References De scr ip t ion S o l u t i o n

9 . 2 . 2 . 1 74 , p . 372 Case 9 . 2 .2 w i th q' ' • = q j - ' e " 1 * 1 t ~ t

i

t = t . r 0 < r < r , T = 0.t f _ t i P d L J0 ( V P d , - 4 p Jl ( v P d , J

CO

1 ( 1 - ^ ) * n W » » (- Xn P0)

exp (-Pd Fo)

A =2 Bi J„(A ) A0 n n

^"v V (x; + B»)' W

9 .2 .3 9 , p . 204 In f i n i t e cy l i nde r w i thexponen t i a l t i ne -dependen ti n t e r n a l h e a t i n g .t = t Q , 0 < r < rQ , T = 0.fc - V r - V T * °"

<* - V exp t-bt) ( VR > y >Pd J n (VT3)

W

. I I I _ — • I I

- « « " •-br

2_ V exP ("^K^n 1

P d Z , X f x 2 / P d - l ) J ,( X )n = l n V n / I n

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Section 9.2. Solids Bounded by Cylindrical Su rfaces —With Internal Heating.


9.2.4 3, p. 363 Infinite pipe with steady Pipe temp:fluid flow and sudden heatgeneration in the wall. (t - t.)h P. / I \ 1 T + F r *F l u i d : t = t . , T = 0. E l ^- = A T+ + ( r ^ T ) d " e > " e ( ^ " D

a ' " R (X - l l / i 2 \A - 1 / u.P i p e : t * t . , T = 0 . q 0 V 1 ) M

F l u i d v e l o c i t y = V.

\o

x * ( E , T * ) - X<f>(£,T*> + ^ - i - j . ) e _ X T A ( ? , T \ A ) ] , T + > (X - 1 )C.

No ax i a l cond uc t i o n . F lu id t emp :

Wa l l h e a t i n g : g• " = q' " ,T > 0 . u

( t . - t . ) h P. __ . + r

0 Insulated £ 1 i - = X T + - (1 - e ~* T ) - e"^J, outer wall q ( J " A p ( X - 1 ) / X 2

JM(M(/«l<lti«/<(tt(tllltM<«q _ „ . ** ~ X ^ * ~ x [ < V r + X " 1 ) , I ' ^ ' T ) _ A * ( u . T * 5 - e A ( C , T , X ) ] , T + > (X - l) Cv-<^

6 e ~6 d6h / M J J M M / M M / M M M d ± £ r n f T

I — ' * *(E,T ) = > - S — r /n fo <"'> 2 J

F, » b - x / v , X = 1 + b f / b . T* = b„ (x - x / v ) ,1 r

P P a> T

b f - h P i / P j C j A j . b p = h P i /PpCpAp , , T

+ - b p x + ( 5 f T * , . £ j £ _ J fin+le-fl d f i

A , = f l ow a r e a , A = p i p e s e c t i o n a r e a , n = 0 °

P. = p i p e i n s i d e p e r i m e t e r . °° r /. -, n .-(X-l)T1 A«,T*,») = y M * - i ) ] f « n - 6 *

S e e F i g s . 9 . 6a, 9. 6b, and 9 .6c . *-* fn l) Jn= 0 * ' 0

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Section 9.2. Solids Bounded by Cylindrical Su rfac es—W ith Internal Heating.


9.2.5 9, p. 405 Infinite cylinder withlinear temperature-dependent internal heating,t = t 0 , 0 < r < r Q , T = 0.

Vr = r Q , T > 0.

V'T > 0.

(t - t 0 )k x J Q ( R V P 5 ) X ^ exp [( -X n + Po)Fo] J Q( R X n )4 2 z-'-r*. ~ B J

Q ( V P o ) P O

'0 0~ X (Po - X 2 )j 1 (X )n=l nV n' 1 n

W = °' P ° B Vk ' / r

02

If 6 > 0, steady state solution exists for Po < A..'.

9.2.6 19, Infinite cylinder ofp. 3-29 infinite conductivity,

variable specific heat,convection boundary, anduniform heating.

See case 8.2.11, set h = 2, radius

9.2.7 19, Case 9.2.6 with a shortp. 3-29 cylinder of length 2&.

See case 8.2.11, set L = 3, radius = i.

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9 . 2 . 8 63 H o ll ow i n f i n i t e c y l i n d e rw i t h e x p o n e n t i a l l y s p a c ev a r y i n g h e a t g e n e r a t i o n ,t = t . , r . < r < rQ . x = 0.

t = t^ Cr ) , r = r1 # T > 0.

t = t 2 ( T ) , r - r 2 , T > 0.

q " 1 • im e x p [ - B ( r - c , ) ] .

I

CD

{t - t . ) kB a • 2 J ?

n =l "O^n* " 0 " " n " 2 '2 = * I 2 ? W e XP (~Xn F°)

l/tJ n < x J

FO

T ^ T ) exp W-) d(Fo)

^ f (Xn) ( S ^ ) 2 / mx ex p ( \2 Foj d(Fo)J

( t - t . ) kS aR = r / r i r F o = t r r / rl f T ( T ) =

f*2:( Xn) = / exp [-[Jr^R - 1)] RB0(X nR)dR

B 0(X n R, = J0 ( X n R ) Y 0 ( X n , - V X n R , J 0 ( V

WVVV = W V W

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Sect ion 9.2. Sol ids Bounded by Cylindrical Surface s—Wi th Internal Heat ing.


9 .2 .9 74 , p . 375 In f i n i t e cy l i nde r w i th a ( t - t . ) ks tead y sur f ace hea t f lux —and s t e ady i n t e r na l he a t i n g . 1 n "r ot = t . , 0 < r < r„ , T = 0.l 0q

Po +

• " = *o" 0 < r < rQ , T > 0 . ^ _ g L FQ _ i ( 1 _ 2R2 ) _2 y

z " V C = r 0 ' T " °" L nt l

J0(X nR) exp ( A ) -

ICO

a' txj - o

9.2 .10 74 , p . 375 Case 9 . 2 . 9 wi th• I I I = C T * e •q ' " ( l - R ) . " - y - i „

J - = J Fo + *' r"0 0

[ 2 J W - W ]X n J 0<V

Ll2J Q (X n R) [ l - exp (-X' Fo) ] + *

$ g iven in case 9 .2 . 9

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Section 9.2. Solids Bounded by Cylindrical Sur face s—Wi th Internal Heat ing.

Case No. References Descript ion Solution

9.2.11 7 4 , p. 375 Case 9.2.9 with

q' = q£"(l + bT>.- - Fo ( 1 + ~ Pd Fo l + <j>

a ' r 2

q 0 r 0

given in case 9.2.9

9.2.X2 7 4 , p. 376 Case 9.2.9 with

.•it . „ I M' " eap (-br).

(t - t.)k -

...,2 Pd

given in case 9.2.9

V r o

<=> 9.2.13 74, p. 376 Case 9.2.9 with

qu i * q' cos (br).

(t - t )k5 — = £r sin (Pd Fo) +

q r*

*> given in case 9.2.9

9.2.14 74, p. 376 Case 9.2.9 withq . . . = q . . . b r n - <* - v * (P d Fo>"

q . , .2 » + l

given in case 9.2.9

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Section 9.2. Solids Bounded by Cylindrical Surfaces—Wit h Internal Heating.

i


9.2.15 74 , p. 393 Infinite cylinder with ._ . 2 » 2 „ „ , , , , „ . _ / ,2 _\, . .7 ( t - t , )p c r„ , _~ A J„ (A R , ) J . (X R) ex p (-A Fo)pu lse he a t in g on a J f H 0 _ 1. V n 0* n V 0V n ' ^ V n /c y l i n d r i c a l s u r f a c e Q1 ~ TT Z , L/ L + ^ 2 \ j 2 . ând con vec t ion boundary n=l \ n / 0 nh , t f , 0 < r < r0 , T = 0 . „ 0 / R =

Instantaneous pulse heatingof strength Q. occurs at For Bi •+ »:

1

" * 'T

" "• » - v °« o i £ ' . " . " ' . ' W -g (-'S*>)

Mean temp:

( t - t j p c r2 , Z X J (X E, )B exp (- \2 Fo)m f 'K 0 l V n O n l ' n r V n /JSl * ^ J2 rA )1 n -1 J l ( V

4 B i 2

1 1 A 2 ( A 2

+ B i 2 )

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Sect ion 9.2. Sol ids Bounded by Cylindrical Surfa ces— With Internal Heating.


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Section 9.2. Solids Bounded by Cylindrical Sur face s—Wi th Internal Heating.


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Til•9

Hi

IREPRODUCED FROM

BEST AVAILABLE COPY

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Section 10.1. Solids Bounded by Spherical Sur fac es—No Internal Heating.


10.1.1 2, p. 239 Sphere with steady •=74, p. 126 surface temperature. 0 2 V (-1) n , 2 2 „ , . , „„.

t = t A, 0 < r < r fl , T = 0. t^ - t Q TT £, nn=l

' - V r • V x > °

- * - i £ [— ( 3a n^ s ) - — ( " S T ^ ] - •» «i-n=l

Mean temp:00

fcm " fc0 6 V 1 , K2 « ,

? I ^ y * ° * n= in

Cumula t ive hea t ra te :

0 * n=l n

See F i g . 10 .1

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10.1.2.1 74, Sphere with convectionp . 254 boundary-general case

ol

so

t = f ( r ) , 0 < r < rQ , t ' fcf = 2

T = 0.

h.t.

V Xn S l n ( XnR ) ex p K F o ) f „ r F , . ,. "1Z T x - si n (X ) co s (X )1 R / R L f ( r ) " fc f J

n=l L n n n J "0

x s i n (X R) dRn

t a n ( A n > - (1 - Bi)See Table 14.2 of roots

10.1.2.2 3,p . 298

Sphere with convectionboundary.t = t £ , 0 < r < r Q , T = 0 . t, - t, 2 Z

B i sin (X ) ex p (-X Pol si n (Xn \ n / nX - s i n (X ) co s (X )X Rn n n n

R)

n=lX co s (X ) = (1 - Bi ) s i n (X )n n n

Cumulative heat rate:

" [ s i n (Xn) - Xn co s tX nQ [l - exp {-\\ Fo)]

Q " 6 Z 7in=l

X 3 fX - sin {X ) cos (X )1n •- n n n ~>

Se e F igs . 10. 2a, 10.2b, and 10.2c

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Section 10.1. Solids Bounded by Spherical Sur fac es—N o Internal Heating.


10.1.3 9, p. 233 Sphere with initial tempf(r) and surface temp4 > ( T ) J general case. - 1 2

n=l

•If

2 2exp{-n TT Fo) s in (rnrR)

Rf (R) sin (nirR)dR - nir( •l)nFoj exp(n 2 ir 2 Fo X)* (X)dX

10.1 .4 9, p . 235 Sphere wit h sur fa ce temppropo rt i onal to t ime.t = t . , . 0 < r < r Q , T = 0.

t Q = br + t i # r = r Q , T > 0.

fc " fc i /. , R 2 - l \ 2 V _tlll , K0 i \ / i F o R , n

n = l

Fo) s in (nlTR)

See Table 10. 1 and Fi g. 10.8

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10.1.5 9, p. 235 Sphere with variableinitial temp. t - t„ 8fc = ( t c " to* <r0 " r , / ( r 0 + fc0> fcc " fc0 " n3R % (2 n + 10 < r < rQ , T = 0 . n ~fc V r B V T > °-

Y ~ j exp[-T l2{ 2 n + l )2 Fo ] s in [( 2n + 1) TTR]

?

10.1.6 9, p. 236 Sphere with variable initialtemp (parabolic).

fc = t c V ( I ^ H o2 c o)'0 < r < r , T = 0 .

t = t Q , r = r Q , T > 0.

t - t„

' c - fc 0

12

7 T 3 R I ^-1 2 2

sin (ffitR) ex p( -n li Fo)

n=l

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Section 10.1. Solids Bounded b y Spherical Surfa ces —No Internal Heating.


10.1.7 9 , p . 237 Sphere with variable initialtemp (trigonometric). ~ _0 _ 1 ._„. , Jl „ .._ .. . , , _ , , i , i . 7 I"~ = n Sin (TTR) exp(-TT FO )t = t Q + ( t ^ r ) s in (irr/r 0), t c - t Q R r i

oU1

10.1.8 9 , p . 237 Sphere with variable initialtemp (exponential). .. 0 2TTfc = fc0 + ( t c " V e x pL b ( r " r oU' fcc ~ fc0 " R

0 .< r < rQ , xt = t ( ) , r = r Q , T > 0.

n=l (nV + B 2 ) 2

2 2ex p( -n IT Fo)

x si n (nTTR) [ ( - l )n + 1 ( B 2 - 2B + n2TT 2) - 2B exp (-B)]

B = br„

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Section 10.1. Solids Bounded by Spherical Su rf ace s— No Internal Heating.


10.1.9 9 , p . 236 Sphere with two initialtemperatures.

J

t - 1 0 2 Y" r s i n '• ri'nR

1'> i

t r^x = m Z L—m— - R i s i n <n i t Ri > J1 0 n=l2_2x sin (mrR) exp (-n n Fo )

10.1.10 9 , p . 237 Sphere with polynominalinitial temperature. t - t,

t - t x - t Q + b x r + b 2 r 2 + b 3 r 3 h - 0

+ . . . . , 0 < r < r , T = 0 .t = t Q , c = r Q t T > 0.

S - = 2 | « i . ( «m ( i H )I H 1 * - ^ [ ( A2 - 2,n=l

x ( - l )n + 1 - 2 ] + - ( n V _ enTT) ( - l ) n + 1

+4„4n it 5„5n TT

x [24 - < n V - 1 2 n V + 2 4 > ( - l )n + . . . . ] e x p ( - n2 n 2

Fo)

bX " V o / R ' b2 = b 2 r 0 / R ' b 3 = V o ' * '

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Section 10.1. Solids Bounded by Spherical Surface s—N o Internal Heating.


10.1.11 9, p. 238 Sphere with convection boundary89 and changing fluid temperature.

t = t., 0 < r < r , T = 0.

t f = bT + tA , T > 0 .

oI

( t - V " „ R2 Bi - 2 + Bi 2 B i

—i— =Fo + —m + i r -b r :

s in (RX )e x p ( -F o X )

, X 2 |~X2 + B i (B i - 1 )1 s i n (X )n = l n L n -I nI

X cot{\ ) +n n Bi = 1

Mean temp:

( t m -t n. = Fo - 1

15( ' •

See Fig. 10, 9

V 6 X P (~Xn F ° )^ X 4 ( X 2

+ B i 2 - B i )n= l n V n '

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S e c t i o n 1 0 . 1 . S o l i d s B ou nd ed by S p h e r i c a l S u r f a c es — N o I n t e r n a l H e a t i n g .


1 0 . 1 . 1 1 . 1 7 4 , C a s e 1 0 . 1 . 1 1 w i t h t - t . „ „ . . , v ^ T ^ > , „ J r, >, , . , _ K T i _ _ , R Bi s i n ( VPd R) ex p ( -Pd Fo)

- 1) s i n ( V P H ) + V"PcT co s

A R s in (A R) ex p ( -A2

Fo j

P " 3 1 7 t , = t „ - ( t „ - t . ) e "b T . t . - t . "" x ( B i - 1 ) s i n (V Ta ") + V M c o s ( V P 3 )£ £m rm l £m I

oI

y~ (l - A 2 / W ) Ai= l \ n / n

( - D n + 1 [^ + (Bi - I )2 ]

- 2 Bi

n= l

,21 1/2A

( A 2 + B i 2 - B i ]

Mean temp:

fcm " fci _ 3 Bi Ctan ( V?d ) - YPTLI exp (-Pd Fo)t f - t ± P d [ ( B i " l ) t a n ( - V P d ) + V P d J

00 / 2 \

_ ^ ex p (~A Fo)- 6 B i2 ^ V n '

, A 2 ( A 2 + B i 2 - B i) ( l - A2 / P d )n = l n \ n / \ ti '

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Section 10.1. Solids Bounded by Spherical Surfa ce s—N o Internal Heating.


10.1.12 9, p. 238 Sphere with convection boundaryand changing fluid temp.t = t., 0 < r < r„ , T = 0.

i 0t = (t - t.) sin (OTT + G ) , T > 0.r m it - max fluid temp.

m r

o

t " t i 2 B it - t . Pd Rm I

X rX

P d L p d s i n ( e ) " c o s ( el l ( B i " 1 ) s i n ^V

\£ x C M H * X

l (X

n + B i 2

B i

) « » < Vx ex p f -Fo \ J + — - s i n (urr + e + $ + <t>2)>

Pd = <»)rn/a

X c o t (X ) + B i = 1n n

A e = s in h (u ) co s (u ) + I co sh (u> ) s i n (u> )

i < l >2A e = a) c o sh (to ) + (B i - 1) si n h (ui )

co = R "Vpd/2, ii) = (1 + i) V Pd /2

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10.1.13 9, p. 240 Sphere enclosed in afinite medium of infiniteconductivity.t = t . , 0 < r < r , x = 0.

t = t Q , r x < r < r 2 , x = 0.

q r = 0 , r = r2 , T > 0 .

a

o

Tem p o f t h e s p h e r e :

fc- fco 0 _ 1 2M V" nt . - t „ M + 1 " 3R Z w 2

M 2 X 4 + 3 (2M + 3 )X2 + 9- n

M X + 9 |M + l ) rn= l n n

s i n (RX )

x s i n (X ) ex p ( -X Fo J , 0 < R < 1 .Temp o f t he su r ro un d in g med ium:

oot - t

3 1 C 1

S e e F i g . 1 0 . 3

t j - t „ M + 1 M 2 A 2 + V n 1/n= l n '

H 2 U 2 / 3 \ " 'M = Kc7 (R 2- V ' R = r / r l ' t a n <V

1 < R < R_

3X

3 + MXn

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10.1.14 9, p. 241 Case 10.1.13, except the sur- Temp of surrounding mediumrounding medium transfers ^heat by convection to afluid at temp t .

t = t., 0 < r < r , T = 0.

fc - V r > V T °-Exterior surface area = A.

t - ti

- i L 0, - t„ Z ,

(3 Bi R2MX=) exp (-A* P o J

. M2X 4 + 3X^{3 + 3M - 2M Bi R_) - 9 Bi R , ( l - B i R_)n= l n n 2 Z 2

P,c_2 2 / 3 \B i = h r2 A . , H = — ( R2 - l ) , R = t / rL

tan(X )3X

3 (1 - B i R_) + MX2 n

\

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Section 10.1. Solids Bounded by Spherical Surfaces—No Internal Heating.


10.1.15 9, p. 242 Sphere with steadysurface heat flux. < fc - V k

3 p o + 5 R 2 - 3 2 V sin (R 2 Xn ) exp (-X R F O )

t = t 0 < r < r , T = 0. q n r 0 " 10 " R L f s i n ( X )

n=l n n^ r V r V T °'

i

tan(X ) = Xn n

See Pig. 10.4

10.1.16 9, p. 246. Hollow sphere with two R. (T- /T. - R )R T /T . cos (mi) - R.sur fa ce tempera tures . T_ = _i t 0 i i 2_ V 0 i it = f ( r ) , c . < r < r „ , T = 0 . T. ° R R Rir Z . nt = t . , r = r , , T > 0 .

t = t 0 , r = r 0 , x > 0 .

n=l

2( 1 - R .)* 2 2 "v " i Vx s i n (nnR ) e x p ( - n n Fo) + ) s i n

n=l

(nTiR )

2 2x ex p( -n ir Fo)/ \ , + R) f (R ) s i n (nlTR ) dR

R* = (r - r , ) / ( r n - r , ) , R = r / r . , Fo = a T / ( r . - r . )2

• i " v 0 i' "0 i'

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ection 10.1. Solids Bounded by Spherical Surfaces—No Internal Heating.

ase No. References Description Solution

0.1 .1 7 9, p. 247 Hollow sphere withsteady surface flux,t = t t , r A < r < r Q , T = 0.

t = t i t r = r i r x > 0.

^ r = V r = V T > °'

?

(t - t ^ k (R Q - R)

V i R QR+ 1 V ( 1+X D 1/2 S i n ( X n R Q -X n R )

\=1 K K + <

R

0 1 ) X

n J,2

x exp (-X Fo)

tan (X R„ - X ) + X = 0n 0 n n

R = r/r . , Fo = orr/r .I I

0. 1.1 8 9, p. 350 Sphere of i nf in i teconductivity enclosedby she ll of fi ni teconductivity.

t - t,

fc i- fc 0

0 _ iJKR

'I

sin [X (1 - R)] exp (-X n Fo)

^ 2K(1 - R. )X n + 4R t X n s i n 2 fr^l - R^] - K sin [ 2\ n ( l - R.)]

R. < R < 11

R,K X cos fX (1 - R.)l = (x 2 R? - K) sin [\ (1 - R.)li n L n i ' J v n x ' u n v i ' J

2K = 3 P 2

C 2 / p i c l ' R = r / , , : 0 ' F o = c c r / r Q

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Section 10. 1. Solids Bounded by Spherical Sur fac es— No Internal Heating.


10.1.19 9, p. 351 Composite sphere.

oi

fc V r = V T > °t - V 0 < r < V -r - 0. __±_ = _ 0 ^ ^ s i n ( R V s i n ( V s i n [ A < v 1 } X J

l u , nn=l

x ex p (- X 2 Fo n ) , 0 < R < 1 .

CO

fc - fc Q 2 R Q

t- ~ t . ~ R , . .1 ° n=l n

x exp (-X n Fo 2 ) , 1 < R < R Q .

(|)(X ) = K X s i n 2 TA X (R„ - 1)1+ A X (R„ - 1) s i n 2 (X )n n ' - n u - ' n O n

+ [(1 - KA)/A X n ] s in 2 (X n ) s in 2 [A X Q (R Q - 1)] .

X cotfA X (R„ - 1)"| + 1/A + K [X co t (X ) - l ] = 0 .n L n O ' J ' L n n J

Fo= CTC/r*, A = V\te 7. K = \ / * 2 , R = r / r . , A(R Q - 1) i rrational

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Section 10.1. Solids Bounded by Spherical Sur fac es— Wo Internal Heating.

Case No. References Descriptio n Solution

10.1.20 4519,P. 3-57

Sphere with radiation coolingt = t Q, 0 < r < t x , T = 0.

q = a #"(T 4 - T 4 ) , r = r,.^ r \ r= r x s / 1

T = sink temp.

See Fig. 10.6

oI

10.1.21 72 Prolate spheriod withsteady surface tempt = t., throughout solid, T = 0.

t

T > 0

t , on surface of solid.

See Figs. 10.7a & b fort - t .

1t -w

t .x

vs Fo

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Section 10.1. Solids Bounded by Spherical Surfa ces— No Internal Heating.


10.1.22 74, p. 320 A sphere with convectionand evaporation boundary.t = t., 0 < r < r „ r T = 0.

I 0

m = m e = evaporation rate.

t - t.= 1 - R M Bi s in ( VPd" R) ex p (-Pd Fo)

(Bi - 1) si n (VP d) + VPa" cos (VP d)

H L i - ( p d A2 y n v n ;

n=l

A =n

( - l ) n + 1 B i [ A2

+ (Bi - I , 2 ] 1 / 2

2 2X + Bi - Bin, K = Y mQ / h ( t f - tt)

A cot(A ) = 1 - B in n

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Section 10,1. Solids Bounded by Spherical Sur fac es— No Internal Heating.


o

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ol

} •CD

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Section 10.2. Solids Bounded by Spherical Sur fac es— Wit h Internal Heating.


10.2.1 1, p. 307 Sphere with steady surfacetemp and internal heating,t = t. f 0 < r < r Q , x = 0.t = v r - vT" ° -q " ' = q i " , 0 > r > r n , T > 0.

(t - t Q) k

*0 '0i'-«2'*±Ife-^)-""T R ~;; "n

n=l

2_2x sin (nirR) exp (-n ir F o ) .

See Fig. 10.5 for condition t. = t ,

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Section 10. 2. Solids Bounded by Spherical Surfaces —Wit h Internal Heat ing.


10.2.2 9, p. 243 Sphere with linear internalheating distribution,t = t Q , 0 < r < r n , T = 0.

t - t Q , r = r Q , T > 0.

q - - q J " ( E 0 - r > / r 0 ,

0 < r < r Q , T > 0.

(t - t Q ) k x

J 0 0

f j U - 2R Z + R 3 )_ J3__ V* _ _ _ 1

^ n=0 ( 2 n + 1 }

x exp [- (2n + 1) TT Po] si n [(2n + 1) TTR]

>fOa

10.2.2.1 74,p . 366

Case 10.2.2 with

«'" = %"^° n>b •» 1, 0, 1, 2, . . .t - t, , , 0 < r < r 0 , T = 0

t = t Q , r = r Q , T ? 0 ,

t^rr ' 2 R [ e r f c PYvfe" V ~ e r £ c ( 2 Yv * 5+ R )J

U - r ( 2 + | )

n=l

Po Po1 + (b/2)

h+2

V 1 T.b+2 . /2 n - 1 - R2 . R L 1 " ^ V 2VF5 j

b+2 c /2 n - 1 - R't ,b+2 . (^f^)])n=l

P o = q - ' T b / Z r J / b ( t 0 - t l

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Section 10.2. Solids Bounded by Spherical Surfaces—With Internal Heating.


10.2.3 9, p. 243 Case 10.2.2, with

q - - < J " ( r ; - r 2 ) / r ; f ^ ^ - fe (1 - R 2 , (V - 3R 2 ,

0 < r < r n , T > 0. H 0 0 n=l

(t - t n)k , 9 •> 12 V (-l)"" 1

"0'2 2x exp(-n li Fo) sin (nltR)

10.2.4 9, p. 244 Case 10.2.2, with( t -

Vk

1 r 2 - ,q'-'r. 5 — = - 5 - [l - exp{-TT Fo)J sin (TTR)- J — a s l n d r r A 0 ) . q ' " r 2 ^ R

0 < r < r , T > 0 .

10 .2 . 5 9 , p . 244 Case 10 .2 . 2 wi th ( t - t . )k

q ' - ' r 2 B~

uas e io . - i . i wi tn i t - t JK , 1q » . = q ' - - e x p [ b ( r - r0 ) ] . — _ 2 _ . i j 11 - f - ( l - ^ ) exp(BR - B) - | ( | - l ) eB )

s i n (mrR) , 2 2 „ %

j j 2 e xp (- n TT Fo)2_ V s i n (mrR) Q n m , ^ 2TTR Z ,

n = 1 n (n TI + B )

[ ( - l )n ( 2 B - n 2Tf 2 - B2) - 2Be"B ]B = rQ b .

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Section 10.2. Solids Bounded by Spherical Surfac es—With Internal Heating.

Case M o. References Description Solution

10.2.6 9, p. 245 Case 10.2.2 with

q , ., . q - ' e ^ .i_ "br [sin (RVTdL _ I _2 _Pd LR sin( Vpd) "J 7 r 3 R

CD

V (-1) s in (mm)Z . _ , 2_2 „ ,

2 2ex p( -n fl Fo) ,

n=l

Pd = rjjb/ct

n (n TT - Pd)

ito

10 . 2 .7 9 , p . 245 Case 10 .2 .2 w i th

q " ' = q j - ' e " *1 * ,r x<c< r 0 .q " > = 0, 0 < r < r,

2a" ' ' r0 0

_ e j f s i n (RV"Pd) _~| s in (V Pd - RV pd) I" R Pd l [ s i n (V~Pd~) J s in (V Pd ) J

x [V co s (R jVPd) - ^ L g . s i n ( R jVPd j J

0 0

+

h I ,z \ M 1 [- '-1

'"+

T S F

s i n

"""Vn h I t — Prl l l -(n TT - Pd)

R cos (mtR,) sin 2 2(nuR) exp(-n TI Po) .

Pd = rjj b/a .

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10.2.8 9, p. 245 Case 10.2.2 with ,,. . ,,,

. i n - * . - . n - - - _ * c ~ zn> K

T > 0

q ' " - f(r), 0 < r < r Q , ^ _ = _^_ y ±_ exp(-n'ir' Fo) sin (nirR) / R'f(R") sin (mrR')dR'fc 0 2 V 1 2 2 /

-= = - f " > ^ exp(-n IT Fo) sin (nTfR) / R'f(R')E* TT ZR ~ n 2 J

n=l 0

jj I R , 2 f ( R ' ) dR' + I R'f(R') dR" - I R' 2f(R') dR'

10.2.9 9, p. 245 Sphere with steady internal ,. . .. •» • ,i „ , / , 2 „ \90 heating and convection ( t ~ t f ) l { 1 2 „ . „ . , „ B i V s l n ( V > e x P H n

F ° )boundary. l t l 2 = ? ( 1 ~ + 2 / B x ] ~ 2 T 2

- = t f , 0 < r < rQ

b ou nd ar y. „ m , 2 6 R ^- ^ 2 ( \ 2 ^ r>- 2 „- \ , i >t = t *0 < r < rn , T = 0. *0 r 0 „ - l X J Xn + B l " B l J s l n <V

q . . . = q ^ . . # o < r < rQ , X n c o t (Xn> = 1 - B i

T > 0.

S e e F i g . 1 0 . 1 0

h .t f

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Sec t i on 1 0 .2 . So l i d s Bounded by Sp he r i ca l Su r f a c e s— Wi th In t e r na l Hea t i n g .

Case N o . Re fe r ences De sc r i p t i on S o l u t i o n

10 .2 .9 .1 7 4 , p . 366 Case 8 .2 .9 w i th

q " ' = qJ"e"" b T -Pd L

x exp(-Pd Fo)

B i s i n ( V P C T R ) "1R E(Bi - 1) s i n fy Pd)+ V Pd cos

.A s in (X R) exp (-X Fo ).Po \ n n \ n /

n=l nRX

2 T si n (X ) - X co s (X )~|L n n n J

X - s i n (X ) co s (X ) ' t a n ( Xn } ~ 1 - Bi

1 0 . 2 . 1 0 9, p. 246 Sphere wi th p a r a b o li c i n i t i a l ( t - t , ) ktemp and c o n v e c t i o n boundary. 1* 1 2 |" l_ 1 1 I" 2 Bi

0 < r < r , T = 0 ( t e m p d i s t .for steady state with const,surface temp = t 1 ) .

. i n _ — 111q ' " , 0 < r < r n , T > 0.

^ . s in (X n R ) e x p ( - X n Fo j -i

*£*_ (x* + B i 2 - Bi) sin (X ) -In= l v n / n

2

X n cot (X n, = 1 - Bi, Po = ° J° h)

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Section 10.2 . Solids Bounded by Spherical Surface s—With Internal Heating.


10 2.11 9, p. 350 Sphere of infinite con- (t - t )kductivity enclosed by a 0__ _ 1 /i 2^.\ _ Mshell of finite a l " r 2 3 ^ R V R

" sin [ X n ( R Q - 1)] sin [X JR p - R)] exp (-X* Fo)

^ X n J2K(R 0 - l )X n + 4 X n s in 2 [ X ^ - 1) ] - K sin [ 2X n (R 0 - 1)])

K Xn cos [X n ( R c - 1)] = (x* - K) sin [X n (R Q - 1)] , 1 < R < % .

= 3 p 2 c 2 / P 1 c 1 , R = r/r., Fo = a 2T/r.

snexi or rinite q ' " ^conductivity. ° 0t = t Q , 0 < r < r Q , T = 0.

q . . . = g^.., o < r < r Q f

T > 0.

q'" = 0, t . < r < r Q ,

T > 0.

otoOl

0.2.12 19 , pp. 3-29 Sphere of infinite con- See case 8.2.11, set L = 3, radius = &ductivity, variablespecific heat, convectiveboundary, and steadyheating.

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Section 10.2. Solids Bounded by Spherical Surfac es—With Internal Heating.


10.2.13 74 , p. 370 Sphere with a steadyinternal heating andsteady surface flux,t = t., 0 < r < r Q, T = 0.

- q r - q 0 , r = r Q, x > o .

*o"> 0 < C < r 0 >T > 0.

Ito

(t - t i )kFo + $

= Po 3 Fo - j£ (3 - 5R 2 ) - 2sin (X R) exp (-X Poj

t ana ) = X nn nn=l

X R sin (X )n n

10.2.13.1 74 , Case 10.2.13 withp. 370 q" ' = q '" ( l - r /r n).

(t - t.)k

• t i r

2 4 ^

2 - (2 + \ 2 ) cos (X ) sin (X R)V n/ n n

V r o n=l5 2X R sin (X )n n

x [l - exp(-X 2 Fo)] + $

$ given in case 10.2.13.

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10.2.13.2 74, Case 10.2.13 withP-370 q l I I = q , . . a _ R 2 h

( t - V 2 Foa' "t0 r 0

- 2^ sin (X R) [l - exp (-X 2 Fo)]

z —~ — = ~—+ *n=l

X R sin (X )m m


10.2.13.3 7 4,

p. 370

oI

Case 10.2.13 with. 1 1 1 — n i i l

= q A " exp(-bR).

( t " fc i )k 3 Fo [2 - exp(-b) (b 2 + 2b + 2)

q ' • 'r^0 r 0

2X+ 2b

n=l

- exp(-b) (2 + b + 2b + X ) sin (X ) sin (X R)n \ v\j_ n n

X 2 s in 2 (X ) (X 2 + b 2 ) 2 Rn n \ n '

x J"l - ex p( -X 2 Fo ) 1 + *


10.2.13.4 74,p. 370

Case 10.2.13 with. 1 1 1 — n 1 1 1q'"( l + br),

( t ~ V k / 1 x5 — = Fo (l + ± Pd Fo) + $q 0 r 0


10.2.13.5 74,p. 371

Case 10.2.13 withq " ' = q^-'expf-br),

<t - t.)kT~ = ^ L 1 " exp(-Pd Fo)J + 0

q ' " r 2H 0 0

Pd


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Section 10.2 . Solids Bounded by Spherical Surf ace s—W ith Internal Heating.


10.2.13.6 74 ,p. 371

Case 10.2.13 withq ' " = q '"cos(bT).

(t - t >k x

,,,2 " id i n ( P d P o ) + *

9 given in case 10.2.13.

10.2.13.7 74, Case 10.2.13 withp. 371

T " ' = < - . • '

v*( t - V k (Pd F o ) n

r .,,, 2 " n + 1 F o + *

o* given in case 10.2.13.

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Section 10.2. Solids Bounded by Spherical Surfac es—W ith Internal Heating.


10.2.14 74 , p. 390 Sphere with pulse heatingon a spherical surface anda convection boundary,t = t f , 0 < r < r Q , T = 0.

Instantaneous pulse ofstrength Q occurs at

r = r x , T = 0 .

, 1 / 2( t - t f ) p c r ^ x ^ . [ ( B i - l ) 2 + X 2 ] A n s i n ( X ^ ) s i n (XnR )

{£ ~ 4TF Z , Bi R. Rl

A = ( -1 )nn+1

n = 1 x ex p ( -X^ Fo ) .

2 B i [ V * ( B i - l ) 2 ] 1 / 2

X 2 + B i 2 - Bi

oI

t a n « x

n > - r ^ - R = r / r oFor B i •* »s

( t - t f ) p c r g^ - / r^- s i n (mrR .) s i n (nTO) ex p (-n TT Fo)21T l_ j *^ i

For r. = 0 :n= l

3 °° , 2(t - t f ) p ^ _ j ^ V X n S i n ( X n R ) e x p ( - X2 ^ o )

21TR Z , X - s i n (X^Jcos fX^)n = l

Mean temp:

( t - t . ) p c r ^ , ^ - r , o - r / 2• Z 0 . I \ [ ( B i - l )2 + X 2 1 X B s i n (X R )

Q, 4irR, B i A J L nJ n n n 1n= l

x ex p (-X Fo)

6 Bin , 2 / , 2 , „. .2X 2 ( X 2 + B i 2 - B i )

n v n '

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Section 10.2. Solids Bounded by Spherical SurCaces—With Internal Heating.


iuo

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Section 10.2. Solids Bounded by Spherical Surfac es—W ith Internal Heating.


ou

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osr

R E P R O D U C E D F RO M j §

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Section 11.1. Change of Phas e—Pl ane Interface.


11.1.1 9, p. 285 Initial surface maintained at Solid temp:constant temp: freezing liquid.t = t n , J t = u , T > 0 .

t = t , x > 0, T = 0.

t = me l t i ng t emp ,m

°Y•w i «

(solid)

V V1 "

(liquid)

' f c s - fc 0fcm" fc0

er f K ) e r f ( X ) , 0 < x < w.

Liquid temp;

t„ - t.a I / * \ i— ——t _ t = e r f c l F o ^ ) / e r t c ( X V c r / aA ) , x > w.m ~ 1 ^ '

XVVTTC

S

( t m " Ve x p ( - X2 ) , ( t m - V h e X P fo V * & )

er f (X) ( t m - t Q ) k s

2 a

erfc~(X)

w = 2X y a xs

11 . 1 . 2 9, p . 287 S o l i d r e g i o n m a i n t a i n e d a tm e l t i n g te m p : f r e e z i n g l i q u i d .t = t , x < w, T > 0 .mt = t „ , x > 0, x = 0.

t , < t , t = me l t i ng t emp .1 m m

Liquid temp:

= X V Te r f c / p o ^ Ve x p C - X ) , x > w .

w = 2X-ya.T

X exp(X 2 ) erfc(X) = ( t m - t ) C J / Y V T

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Section 11.1. Change of Phase —Plan e Interface.

_ase No. References Description Solution

11.1.3 9, p. 287 Melting of a solid in contact Solid temp:

with another solid.

t = t Q , x > 0, T = 0.

t = t, > t m, x = 0, T > 0.

t = me l t in g temp.m

'NU,

ito

v vk .(solid)

I W k I (liquid) (solid)

fc

S" fc0 = e r f c ( F ox s ) / e r f c ( X V c ^ / c T ) , x > w .

Liquid temp

= er f (F o ^ l / e r f (X) , 0 < x < w .

e x p ( - X 2 ) + " s^ Ô - V " P ^ V v _ XYV

e r f { X ) k . V 5 ~( t , - t ) e r f c (X Va . / a ) c X , ( t l " V£ s 1 m SL s

w = 2X Va - t

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Section 11.1. Change of Pha se— Pla ne Interface.


11.1.4 9, p. 288 Liquid freezing on solid-

properties of solidified

liquid different from

original solid.

t = t „ , x < 0 , T = 0 .

t

t 0 , X < 0, T

t , , x > 0, X = 0 .t = me l t in g temp,m

O r i g i n a l s o l i d :

VA"csi - fcot - t„ VS" + K er f (X)m 0

[ l + e r f (F o*s l ) ] , x < 0 .

S o l i d i f i e d l i q u i d :

fcs2 - fc0 V * + K e r £ ( F° x s 2 >

m 0 V A + K e rf (X) , 0 < x < w

L i q u i d :

>1 ' i l 1 \T ai2- kn2(solid) I (solid) (liquid)

t„ - t ,t _ = e r f c ( F o ^ J / e r f c (XV a ^ / c ^ ) , x > wm i.

X Y V T K e x p ( - X2 ) ] " f c ^ ^ m - V 6 X P ( - X aB 2 / af t>c s 2 ( t m " t 0 ) VA"+ K e r f (X) k , V X ( t - t . ) e r f c (XVa , / a . )

s2 X, m 0 s2 X,

w = 2X V aT r , K = k , / k . , A = a , / a0s2 si s2 s i s2

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Section 11.1. Change of Pha se— Pla ne Interface.

Case No. References Description Solut ion

11.1 .5 9, p . 289 Melting of a so lid in con tactwith another so l id— prope rt iesof l iquid different thanor ig ina l so l id .t = t, > t , x < 0, T = 0.

1 mt = tQ , x > 0, T = 0.

w = 2 X V ^ t

X Y V T Tc A ( t l V

k^Vaêxp ( -Xz)

H^7l + *s l V ^ e r f ( X )

+ k * ^ < t x - y erf c (XVapoT^)

See cas e.1 1.1 .4 for temperaturest = meltin g temperature ,m

11. 1.6 9, p . 289 S olid me lts over temp range For cas e 11 . 1 . 1 , change equation for X to :

>2,f tm l to Wt - t ^ , x > 0, T = 0.

H * °H + Y / ( t m 2 " W '

e x p [ t a s - H)V VaJ e rf c ( X V a ^ ) ( tm 2 - t m l ) k £ V ^

erf (X) (t.ml V s * 5 *

For case 11 .1. 4, change equation for X to :

K e x p [ ( a5 2 - y X 2 / ^ ] e rfc a ^ ^ \ ) _ ( tm 2 - t ^ ^ V T JV3T+ K er f (X) (t.ml t„> V^7

Use c." for c« in the above e quatio ns and solutions given in

cases 11.1.1 and 11.1.4.

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Section 11.1 . Cha nge of Phas e—P lane Interface.


11.1.7 9, p. 290 Change of volume during t. - t ( t X ( p s - P„ ) /a

— -= = e r f c I Fo „ + n

s o l i d i f i c a t i o n . t_ - t , I xl Pt = t Q , X = 0 r T > 0.

t = t 1 # x > 0, T = 0.

P s > P r

t = m e l t i n g t e m p e r a t u r e .

I 2 ) ( t l - V h ^ e * p ( - * 2 P s V P & a l ) X Y V ?e r f (X) ~ (t_ - t n ) k „ Vo 7 e r f c ( X p „ V c T / pDVa; ) c^ ( t m - t n)exp(-X )

*m " O ' ^ s7 ^

W = 2 X V CX Ts

J s T BW'

II " x

w

P. - 'WM(solid)

t a . a s , k e , p s

(liquid)

1 1 . 1 . 8 9, p. 291 Cons t an t hea t f l ux a t


bounda ry.

t - t , x > 0, T = 0 .m

q j£ = q Q , x = 0, T > 0.

t s o l i d t e m p , t < t .s r s m

( t s - V k sa s V

Qx2x

_ - 2 ^ ( 1 + 2 Fo J + - ^ - T d + 1 2 F o - + 1 2 P o » >x- 4X 12x

. . . , 0 < x < w .

w = QT - \ Q 3 T 2 + | Q 5 T 3

Q = q 0 / a s P s Y • F ° x = < V / X * ' T = V

t = me l t i ng t emp ,m

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Section 11.1. Change of Phase—Plane Interface.


1 1 . 1 . 9 9, p . 292 C o n v e c t i o n b o u n da r y a t


b o u n d a r y.

t = t 1 # x > 0 , T = 0 , t x > t m .

t = m e l t i n g t e m p .

t - t F B i 2

_ § _S = B i T-^{1 + 2 Po )t - t , x 2 x'

m £

P Bi

IT' 1 + 3 1 F

° x >+ . . . , 0 < x < w .

V \ . ,

F 2 B i 3

F B i x F O x . _ _ x ( 1 + P ) F o 2 + . .

F = k s ( t m - V / a s P Y ' F o x = V / x ' Bi x = h x A s

(solid) (liquid)

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Section 11.1,, Change of Phase —Plan e Interface.


11.1.10 55

19rp. 3-87

Solidification on plane semi-

infinite solid with convection

coefficient between solid-solid

and liquid-solid,

t = t , x < 0, T = 0.

c = 0.

t = melting temperature.

h 2 ( ts m V

pyk

N =

= i - o r N - 1 I BiM 2 LN(Bi + 1 ) - lJ " N

V fc 0 " Vh (t

s m V, Bi =

h ws

See Fig . 1 1. 1

(Solid) (Solid)

P.c

(Liquid)

-•Vô

11.1.11 5619 ,p . 3-88

Case 11.1.10 with so li di fie d

layer of finite heat capacity

and heat flow at x = 0 given

as q = bt , where t i s tempw w

at x = 0.

See Fig s. 11.2a and b

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Section 11.1. Change o f Phase—Pla ne Interface.


11, . 1 . 1 2 1 9 ,77p . 3. 88

11 . 1 . ,13 1 9 ,p . 3 --88

Case 11.1.10 with heat flow4

at x = 0 given as q = btM .See Figs . 11 .3a and b

iCD

Decom posit ion of a sem i- Char la ye r:

i n f i n i t e s o l i d .

t = t Q , x >_ 0, T = 0.

t = t , x = 0 , T > 0 . .

t . = decom posit ion temp.

£ - bVT.

Char Virgin

©

I — i - 1

(D

t - 1 ,

w d

erf KJerf {b/2Va^> , 0 < x <_ I

Virgin layer:

t - t , e r f c KJt w - t d " e r fc" (b /2 V a p , x > I

e x p ( - b 2 / 4 a 1 ) / P ^ 7 / t d - t A exp f b 2 / 4a 2 ) _ i r p / c ^ yb

e r f ( b /2Va £) \ P-f^ \* v " t j « f c ( b / 2 V ^ ) 2 { tw - t f l )

See Figs . 11 .4a and b

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Section 11.1. Change of Phase—Pl ane Interface.

Case N o. References Description Solution

11.1.14 571 9 ,p . 3-93

i

Ablation of a semi-infinite

solid by convection heating,

t = t , x >_ 0 , T = 0.

t, = decomposition temp,d

See Figs. 11.5a, b and c

2cd" fc 0fc _ ° = 1 - exp [ ( h A ) O T d ] erfc [(h/k) Va Tj ]

T . = Time for free surface to reach temp t,d a

Steady state ablation velocity and temp distribution:

v =h < f c a " V

"ss p ( t d - t Q) + If]

^ = e x p [ - > ( * - * , ]

H = heat of ablation

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Section 11.1. Change of Phas e—Pl ane Interface.


11.1.15 58 Case 11.1.15 with heating by See Fig. 11.6

19, a constant surface flux q " . Preablation time:p. 3-95

T * r ^ - ^yTd= 4 p c k \ T 5 r r ~ 7

A b l a t i o n v e l o c i t y :

' s s p [ c ( t d - t Q ) + H]

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Section 11.1. Change of Phas e-Pl ane interface.

SolutionCase No. References Description

V

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Section 11.1. Change of Pha se—P lane intecface.

Case No. References Description

i

t o

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Section 11.2. Change of Phase—Nonplanar Interface.


11.2.1 9, p. 29 Liquid freezing ona cylinder.t = t , 0 < r < R .mt = t , r -»• «, t < tx i nt = me lt ing temp.

.£) ,

Liquid temp:

hî = E i (_ F O* 0 m (-x2)' r > r

s-X 2 expCX2) Ei (-X2) + ^ ( ^ - t x ) / Y = 0 .

r = 2X VaTrs I

(Solid) (Liquid)

11 .2 .2 9 , p . 295 L iqu id f r eez ingon a sphere.t = t , 0 < r < R .t = t", r * » , t x < t m .

(Solid) (Liquid)

Liquid temp:

H ~ h = 2X^n ~ fcl exp ( -X2) X V 7 e rf c (X) (2 Fo*

exp(-3)rS,

- - er fc r > r_K)) •X 2 exp(X2) j exp ( -X2 ) - XTT e r fc (X)J = ^ ( tm - tj

r s = 2X V v

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Section 11.2. Change of phase—Nonplanar Interface.


11.2.3 9,p. 296

Steady line heat sinkt = t , r > 0, x = 0,

t > t . sink rate = q.

t = me l t i ng temp .

So l i d t e mp :

( t s " V k s 4ir Ei (-2) Eif.-X" ) , d < r < r s .

r = 2X VC TTs s

Liquid temp:

( t , - Mr > r

m 1

xp( -X2) /4 i r + - f t - t ) e x p ( - X2 a s / a t ) / E i ( -\\zaj = X 2 a s p Y / q0 .

11 .2 . 4 9 , p . 296 F reez ing on s teadytemp cyl inder.t V c V T > 0 r

fc0 < V

* = V r ' r

o ' T = °*

t = me lt ing temp,m

Sol id temp:

^ 4 " T - * n( r /r ) / t a ( V r ) , r Q < r < r scm c 0

2 r s * » « Vr 0 » " r s + ^ = 4 k s ( t m - t 0 , T / ^

(Solid) (Liquid)

http://-//zaj

http://-//zaj

http://-//zaj

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Sec t ion 11 .2 . Change o ii Phase—Nonplanar In t er fac e .

Case No. Refe renc es 1 . - jr ipt ion So lu tio n

11 .2 .5 19 , F reez ing i n s id e a t ube , 2p . 3 -87 t ube r e s i s t an ce neg l ec t ed , ( m " a ' _ Bi T(X_ l \ 2 _2 „ / l \" l

conv ect ion co ol i ng a t r = rQ . pyk " 2 L U i + 2 j ( 1 ~ R ' " R . n UAJ ' R - 1 -t = t m , r = r 1 . Bi = h6 /k, R = c i/r Q

t = melting temp.

Liquid

Solid

11.2.6 19 , Case 11.2.5 with freezing on 2p. 3-87 outside of tube and convec- ' m a' _ Bi*" |/ 1 _ 1 \ ,Jl .. , _2

tion cooling on inside. pyk ^ lv ° 'i n ' v " """'Bi 171 1 \ ,Jl .. s „2

, R > 1

11 .2 .7 19 , Case 11 .2 .5 w i th f r eez in g jin I s - ~ tJp . 3 -88 in s id e a sphere . - 1 f e r a i = ^ f [ ( i i " 0 <* " H3) + f d - R2>] . R < 1

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Section 11.2. Change of Phase—Nonplanar Interface.


11.2.8 19,p. 3- 88

Case 11.2.6 with freezingon outside of sphere.

n ^n, " Vo a B i 2

[&-) 3 3 2(R J - 1) - f (R - 1) R > 1

I

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Se ct io n 11 .2 . Change of Phase—Nonplanar in te r f ac e .


i

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Sect ion 11.2. Change of Phase— Nonpla nar Interface.


4.CD

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oBaeW" lM a

on


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Section 12.1. Traveling Boundaries.


1 2 . 1 . 3 9 , p . 389 Sem i - in f in i t e s o l i d wi ths t e a d y p e r i o d i c temp,t = ( t f l - t ^ c o s t o r r + 9 ) ,x = 0 , X > 0 .So l id i nc rea s ing a tv e l o c i t y u .

t - t_ "* = e x p [ u x / 2 a - x V b c o s (<j>/2)] cos [UT - x V b s i n (<j>/2) + e ]

^ + f = b exp(i<t»4a

12.1.4 9, p. 389 Cylindrical boundarytraveling in aninfinite solid.t « t j, r > t fl , t > 0.fc = V r r o - °Cylinder traveling atvelocity u.

J3-

t - 1 „- ..„ _ e I (O)K (UR) cos (n6)0_ X n n nt 0 - t . " 2 K_,

n=0

,(U)

+ I y exp (-a 2 F o ) e cos (n6) i (U)IT {_ , n n

n=0

C° exp(-Fo X 2 ) f j (XR)Y (X) - Y (XR)J (X)~l XdXI L n n ii n J

10 ( X 2

+ u 2 ) [ a 2(X ) + Y2 ( X ) ]

e Q = 1, e n = 2 if n > 1, Fo = a r / r 2 , R = r / r Q

U = u r Q/2a

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12.1.5 9, p. 391 Thin rod with fixed endtemps and convectioncooling,t = t , 0 < x < %, T = 0.

t = t , x - I, T > 0.

t = t Q , x = 0, T > 0.

The end boundaries traveling

at velocity u.p = rod p e r ime te r.A = cod s ec t i o n a r ea .

inh L\U2X 2 + Bi X 2 Jinht , - t„

X + Bi X J exp(DX - D)

sinh [V o + B i J

+ 2ir exp(DX - 0) £ ( - ^ " y i n im x)

n=lB i + 0" + n it

["exp l-Fo (Bi + U + n n )JB i = h p i 2 / k A , Fo = ore A 2 , U = uA/2a , X = x/SL

Iu

•4T

•-T

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i

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toim

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5"aeratosa.Hs:

REPRODUCED FROMBEST AVAILABLE COPY

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x/6

IG . 1 .1 . Tempera tu re d i s t r i b u t i o n and mean t emp era tu re i n a po ro us p l a t ec a s e 1.1.4, so u r ce : Re f . 2 , p . 221 , F ig . 9 .2 ) .

0 0 2 0 4 0 6 0 8 10

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F I G . 1 . 3 . Te m p e ra tu r e d i s t r i b u t i o n i n a h e a t g e n e r a t i n g p o r o u s p l a t e ( c a se1.2.6 , so ur ce : Ref. 2 , p . 223 , F i g . 9 .3 ) .

0 1 2 3 4 5 6 7 8 9 10 11 12-+ b/a (or a/b)

FIG . 1.4. Ratio of mean and maximum temperature excesses in an electricalcoi l of rectangular cro ss section (case 1.2.8, source: Ref. ?., p . 180,

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FIG. 1.5. First-term approximation to the maximum temperature in a solidrectangular rod with internal heating, ^ = -V q 0 " >/f$k 2b, \ | )2 = "V<3o'''/3k 2a(case 1.2.10, source: Ref. 2, p. 19 8, Fig. 8 . 1 6 ) .

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•±" 1 0 " 3

FIG. 1 .6a . Maximum tem pe ra tu re v a r i a t i o n s on th e co o le d su r fa ce of a f l a tp l a t e h a v i n g e q u a l l y s p a c ed a d i a b a t i c an d c o n s t a n t t e m p e r a t u r e s t r i p s on t h

o p p o s i t e s u r f a c e ( ca se 1.1.30, s o u rc e : Re f. 2 7 ) .

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0.8 1.0

IG. 1.6b. Heat flux through slabs held at a uniform temperature on oneurface and having equally spaced constant temperature strips on the othercase 1.1.30, source: Ref. 88 ) .

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ora

ha/k

FIG. : . .7 . Maximum tem pera tu re v a r i a t i o n of the coo led su r fa ce of a f l a t havi-(> a l t e r n a t i n g a d i a b a t i c an d c o n s t a n t h e a t - f l u x s t r i p s on t h e o p p o s i tsu r f ace ( ca se 1 . 1 .3 1 , sou rce : Re f . 28 ) .

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10 J

10^

o

10 1

10 L

"I 1 I I II I I

Insulatin

10 -2

hw/k

PIG. 1.8. Te m p e r a t u r e s at the s p o t c e n t e r of a s p o t - i n s u l a t e d p l a t e h av in g aun ifo rm in t e rn a l hea t sou rce ( ca se 1 . 2 . 13 , sou r ce : Ref. 29 ).

• ) 0 J c 1—i—r i M I I I 1—i—i i i i i 11 1 — i — I - T i i i i I 1—i—i i I I 11_

Insulating spot-

h, t „

oi n

' i i 1 1 i i i i i i i 1 1 i i i i i 1 1 1

10'

hw/k

10 J

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0.2 0.4 0.6 0.8

(y/B)/(1/Bi + 1)

1.2 1.4

PIG. 1.10. Hotspot temperatures along plate/ rib centerline (Bi = hS,/k)(case 1.1.39, source: Ref . 19, p . 3-126, Fig. 77 ).

0.5 1.0 1.5 2.0 2.5 3.,Ratio b/a

0.5 1.0 1.5 2.0 2.5 3.0Ratio b/a

PIG. 1 .11 . Con duc t ive shape f a c to r s fo r a r ec t an gu la r s e c t i o n con ta in i ng a

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H" 1.0

rcA

FIG. 2 . 1 . Heat l o s s from Ins u l a t ed t ub es ( ca se 2 . 1 . 3 , so u rc e : Re f. 3 ,p . 1 2 3 , F i g . 3 . 1 8 ).

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b

4

I 1 ' 1 1 T12.251 \

1

1

t m ^ l \ -

^ 3 1 \+ 1 \

£ Vb- r0 = 2.00 1 \1 \ -

+ 2 \ 1 \

§

1

" '1.75^ 1 \

\ —~ l.bU.

^ 1 \\ —

CSvV0 i — 1 — o . o o — 1

+o

*E

0 0.2 0.4 0.6 0.8R

1.0

FIG. 2 .2 . Temperature d is t r ib u t i o n and mean temperature in a cy l ind er w i thtempera ture dependent hea t source (c a se ? 2 .2 .4 , sourc e : Ref . 2 , p . 189 ,F i g . 8 . 11 ) .

J

F IG . 2 . 3 . Tem p er atu re d i s t r i b u t i o n i n an i n f i n i t e p l a t e w i t h a c y l i n d r i c a lhea t Bource , B = V ( hx + h 2 ) / kw r g , t 0 = t at r 0 , t«, = t a t r = °° (ca se 2 . 2 . 1

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0.1

0.01

0.001

= o 0 . 0 00 (1

0.001

0.01

0.0010.05

K = 10, R = 100K= 10 , R = 1K = 10, R = 1 0 -

K = 1, R = 100 -

0.4 0.5

FIG. 2 .4 . The i n s i d e s u r f a c e t e m p e r a t u r e o f an i n f i n i t e t u b e w i t h t e m p e r a t udependent co nd uc t iv i t y and hea t ing ( ca se 2 .2 .1 7 , sou rce : Re f . 16 , F ig . 1 ) .Equation (5) g iven in c a s e s o l u t i o n .

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s/r 0

FIG. 2. 5 . Maximum (hot sp ot) tem pera tures in the cr os s s e c t i o n of ahea t -gene ra t ing s o l i d (cases 2 .2 .1 8 and 2 . 2 .1 9 , source : Ref . 64 , F ig . 3 ) ,

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1/P = b/r0

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0.01 0.1 1 10 100

G. 3.1. Steady temperature of thin, nonrotat ing spherical shell in uniformdiation field (case 3.1.10, source: Ref. 19 , pp. 3-112, Fig. 6 8 ) .

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PIG. 4 .1 . Surface temperature of a se ra i- in fi ni te sol id with heat ing on thesu rf ace over width 2b, which moves at ve loc i ty u (case 4 .1 . 8 , s ource : Ref. 9,p . 270, Fig . 34) .

3 -

2 -

1 -

• Heat sourcewidth - I

' I 1 1 1 1 1 '

A< °~ , \\ s H = 0 L = 1

/ / H = 0 . l S \

y^ H=I.O ^-

1 1

L e a d i n g e d g e — ^

» < • 1 •~~ " v | ^ ^-

x/8

f f l l d f l d

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o • ' ' ' • • — •

-0.10 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

0 - radFIG. 4.3a. S u r f ac e " t em p e r a t u re of an i n f i n i t e c y l i n d e r w i th a r o t a t i n gsur fac e band sou rce , <> =» 0 .01 (c a se 4 . 1 .1 0 , so u r ce : Ref . 30 ) .

n i T-0.1 0-0 .08 -0.0 6-0.0 4-0.0 2 0 0.02 0.04 0.06 0.08

f l - rad

FIG. 4 .3 b . (Same co nd iti on s a s fo r F ig . 4 .3a ) ,

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-1.5

(a)

f l ( R . r )(4/Pe) 2 0 _|_

Nu = 2.0

Pe = 20.0

Pe = 2.0

-1.0 -0.5

Nu = 0.2

Pe = 20.0

-1.5 -1.0 -0.5 0 0.5 1.0 1.5

(b)

FIG. 4 .4 . Temperature dis t r ibut ion i n a cylinder wi th a ring heat source (case 4.1.11, sou rce: R e f . 3 4 ) ,9 - (t - tf)2nrQk/Q0.

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e ( R . f ) Nu = 0.2R0 =1 .2BPe = 20.0

- Pe = 2.0

SIR,?)

Nu = 2.0R 0 = 1.25Pe = 20.0Pe = 2.0

PIG. 4.5. Tempecature distrib utions in a hollow cylinder with an inside ring heat source (case 4.1.12,source: Ref. 34 ). 9 defined in Pig. 4.4.

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n — i — i — i — i — i — r " i — i — i — r " 1 — I — i — i — r - 1 ' '2 'n = c o ; y = yb ( x / C )2 ;0 =

Vb

U-

n = - 1 ;y = Vb ( x / B | ; 0 =

2s/T .u b = — - - C V h T k y ;

9 b

u b l , ( ub )

n = O;y = yb ( x / C )1 / 2 ; 0 =

V 2 " ." b= — 7 — - K V h / k yb

2 l("b)u b l 0 (u b )

^?*&

tanh u h

n = 1/2;y = yb;tf>=-

u b = v ^ - f i V h 7 k yb "u b

J I I I I I I I L J I I L1 J I I L J I I L1.0 2.0 3.0

xVhTkVb

4.0 5.0

FIG. 5 .1 . Per formance of p in f i n s (cases 5 .1 .2 , 5 .1 .1 4 , and 5 .1 .16 through5 .1 .1 8 , so urc e : Ref. 8 and Ref . 7 , p . 55 , F i g . 3 .1 4 .

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J 1_

i= 1 /2 ;v = ;uh = » V h / k vb

tanh u b

= 1/3; y = V b ( x / C )1 " ; ub = — | ^Jhjk^

' 2/3 3

3 u b L, /3 <'Jb)= 0 ; y = yb ( x / B ) ; ub = 2 8 V ' h / k yb

2 ' i K '

i = - 1 ; V = V b ( x / B )3 / 2 ; u b =4C-. /h/kyb

"b Mub) _2, = ± oo; y = yb ( x , ' C )2 ; u b =Cx/h7kyb

2

1.0 2 . 0 3 .0 4 . 0 5 . 0

ev^RVb"

IG . 5 .2 . Perfo rmance o f s t r a ig h t f i n s ( ca ses 5 .1 .4 , 5 .1 .5 , 5 .1 .7 , 5 .1 .8 ,our ce: Ref. 8 and Ref. 7, p. 56, F ig . 3 .1 5 ).

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0.5 -

0.4

1.0 2.0 3.0 4.0

B V h 7 k V b

5.0

FIG. 5.3. Performance of circumferential fins of rectangular cross section(case 5.1.10, source: Ref. 8 and Ref. 7, p. 57, Fig . 3 . 1 6 ) .

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. 5.4. Curve for calculating dimensions of circular fin of rectangular

ofile requiring least material (case 5.1.10, source: Ref. 1, p . 234,

g. 11.11). a - Q/nhx£(t 0 - t f ) .

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1.0 2.0 3.0

FIG. 5 . 5 . Per formance of cy l in d r i ca l f i n s of t r i angu lar p r o f i le (case 5 .1 .1 1 ,sou rce: Ref. 8 and Ref . 7 , p . 58, F ig . 3 .1 7 ) .

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( r e * - r o) , / 2 h 7 k r0

FIG. 5,6. Efficiency of an infinite fin heated by square arrayed round rods(case 5.1.19, source: Ref . 11 and Ref. 10, p. 135, Pig. 2.22). t% = (2/W)s.

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(r e--r 0)V5R7E5;

FIG. 5 .7 . E ff i c i en cy of an in f in i t e f in hea ted by eq u i l a te r a l t r i angulararrayed round rods (case 5 .1 .2 0 , so u rc e: Ref. 11 and Ref. 10 , p. 136,F i g . 2 . 2 3 ) . r* * ( 2 V 3 7 n )1 / 2 s .

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Z = T0 A " e

FIG. 5. 8 . Fin parameter as afunc t ion o f Z fo r a s t r a ig h t f i nr ad i a t i ng t o f r ee space ( ca se 5 .1 .2 4 ,sour ce : Ref . 10 , p . 208, F ig . 4 . 3 ) .

Z = V T e

FIG. 5.9. Fin efficiency for astraight fin radiation to free space(case 5.1.24, source: Ref. 1 0, p. 209,

Fig. 4 .4 ) .

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0 0.2 0.4 0.6 0 .8 1.0 1.2 1.4 1.6 1.8

Profile number C(K1 T ^ / 2 k b )1 / 2

FIG. 5.10. Heat flow relationship for a straight fin of rectangular profileradiating to nonfree space (case 5.1.25, source : Ref . 10, p . 216, F ig. 4 . 8 ) .

0.4 0.8 1.2

Profile number « ( K ^ T j^ k b ) 1 7 2

F IG . 5 . 1 1 . E f f i c i e n c y o f a s t r a i g h t f i n o f r e c t a n g u l a r p r o f i l e r a d i a t i n g tf ( 5 1 2 5 R f 10 216 F i 4 9 )

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X = — = 0 .75

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Profile number K.,Tê2/kbn

IG. 5 .1 2a . Fin e f f i c i e n c y f or th e l o n g i t u d i n a l r a d i a t i n g f i n o f t r a p e z o i d a l

r o f i l e w i t h a t ape r r a t i o o f 0 .75 ( ca se 5 .1 .2 6 , source : Ref. 10 , p . 223,i g . 4 . 1 2 ) .

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Profile number K ^T j^ /k b rj

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0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Profile number K ,T 3(2/kb0

F I G . 5 . 1 2 c . F i n e f f i c i e n c y f or t h e l o n g i t u d i n a l r a d i a t i n g f i n o f t r a p e z o i dp r o f i l e w i th a t ape r r a t i o o f 0 .25 ( ca se 5 .1 .2 6 , sou rce : Re f. 10 , p . 224 ,F i g . 4 . 1 4 ) .

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

\ Profile number K^rg ^/kbn

FIG. 5 .12d . F in e f f i c i en cy for t he l on g i t ud ina l r ad i a t ing f in o f t r i a n g u l a rp r o f i l e ( ca s e 5 . 1 . 2 6 , so ur ce: Ref. 10, p . 225 , F i g . 4 . 1 5 ) .

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1.2 1.3 1.4 1.5 1.6

Temperature ratio Z = TQ/T

1.7

PIG. 5 .1 3 . P ro f i l e number of cons tan t - tem pera ture -grad ien t lo ng i tu d i na lr ad i a t in g f in as a fun c t ion o f ba se - to - t i p t empera tu re r a t i o ( ca se 5 . 1 . 28 ,so ur ce : Ref. 10 , p . 233 , F i g . 4 .1 8 ) .

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K 2 / K 1 T 0

0.4 0.8 1.2

Profile number £2

l<1T^/kb

1.6

FIG. 5.14. Efficiency of constant-temperature-gradient lon gitudina l radiatingfin as a function of profile number (case 5.1.28 , source: Ref . 1 0 , p. 232 ,Fig. 4.17).

0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80Radius ratio p = r Q /r e

FIG 5 15a Radiation c i n efficiency of radia l fin of rectangular profile

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-©.> •

uca>\uM-4 -01

c

0.25 0.35 0.45 0.55 0.65

Radius ratio p = r Q /r e

0.75

FIG. 5,15b. Radiation fin efficiency of radial fin of rectangular profile.

Taper ratio, X = 1.00; environmental factor, K /K T = 0.20 (case 5.1.29,

source: Ref. 10, p. 251 , Fig. 4.25).

0.25 0.35 0.45 0.55

Radius ratio p = r n / r e

0.65 0.75

FIG., 5.15c. Radiat ion fin efficiency of radial fin of rectangular profi le.

Taper ratio, X = 1.00; environmental factor, K / K T n = 0.40 (case 5.1.29,

source: Ref. 10, p. 251, Fig. 4.26).

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0.25 0.35 0.45 0.55 0.65 0.75


FIG. 5 .15d . R adi a t io n f i n e f f i c i e n c y of r a d i a l f in of t r a p e z o id a l p r o f i l e .

Ta p e r r a t i o , X = 0. 75 ; e n v i r o n m e n t a l f a c t o r , K2 / K , T = 0 . 0 0 ( c as e 5 . 1 . 2 9 ,

s o u r c e : Ref. 10, p . 252, Fi g. 4 . 2 7 ) .


FIG. 5.15e. Radiation fin efficiency of radial fin of trapezoidal prof ile.

Taper ratio, X = 0.75; environmental factor, K./K.T. = 0 . 2 0 (case 5.1.29,

source: Ref. 10,p. 252, Fig. 4.28).

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0.25 0.35 0.45 0.55 0.65 0.75

Radius ratio, p = r Q /r e

FIG. 5.15h. Radiation fin efficiency of radial fin of trapezoidal profil e.4

Taper ratio, X = 0.50; environmental factor, K

2 / K i T n = ° - 2 0 < c a s e 5.1.29,

source: Ref . 10 , p. 254, Fig. 4.31).

0.25 0.35 0.45 0.55 0.65 0.75

Radius ratio, p = r n /r e

FIG. 5.15i. Radiation fin efficiency of radial fin of trapezoidal prof ile.

Taper ratio, X = 0.50; environmental factor, K./ K. K = 0.40 (case 5.1.29,

source: Ref . 10, p. 254, Fig. 4.32).

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0.25 0.35 0.45 0.55 0.65 0.75

Radius ratio, p = r Q /r e

PIG. 5.15J. Radiation fin efficiency of radial fin of triangular profile.

Taper ratio, X •» 0.00; environmental fac tor , K / K T « = 0.00 (case 5.1.29,

source: Ref . 10, p. 255, Fig. 4.33).

0.25 0.35 0.45 0.55 0.65 0.75

Radius ratio, p = r n /r e

PIG. 5.15k. Radiation fin efficiency of radial fin of triangular profile.

Taper ratio, X = 0.00; environmental factor, K /K T = 0.20 (case 5.1.29,

source: Ref . 10 , p. 255, Fig. 4.34).

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-Q->uc

0.6

0.5

0.4

0.3

0.2

0.1

Z^^^^ —

—

^5^$^^r?^ Profile number

I I I I I I I0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80

Radius rato, p = r Q /r e

P IG . 5 .1 58 ,. R a d i a t i o n f i n e f f i c i e n c y o f r a d i a l f i n of t r i a n g u l a r p r o f i l e .Tap er r a t i o , X = 0 . 0 0 ; e n v i r o n m e n t al f a c t o r, K j A iT n = 0 . 4 0 ( c a se 5 . 1 . 2 9 ,s o u r c e : R ef. 1 0 , p . . 2 5 6 , P i g . 4 . 3 5 ) .

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Base paramater mR,

FIG. 5 .16a . E ff i c i en cy of the s i de ofa capped cy l i nd er f i n (case 5 1 3 0

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0.4 0.8 1.2

Base parameter mB

1.6

FIG. 5.16b. Efficiency of the top of a capped cylinder fin (case 5.1.30,source: Ref. 10, p. 277, Fig. 5.7).

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0.4 1.6 2.0 2.4

Parameter ml

2.8 3.2 3.4 3.6

FIG. 5 .1 7 . Heat flow ra t io q^/qg as a fun ctio n of mb and temp eraturee x c e s s r a t i o ( t^ - t f ) / ( t g - ff ) for the doubly he ate d recta ng ular f in(case 5 .1 .3 2 , source : Ref. 10 , p . 410 , F ig . 8 . 1 0 ) .

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0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6m x 1 b l

I G . 5 . 1 8 a . E f f i c i e n c y of t h e v e r t i c a l s e c t i o n of a s t r a i g h t , s i n g l e Tee f i nor u » v (case 5 .1 .3 2 , sou rc e : Ref. 10 , p . 398 , F i g . 8 .4 ) .

1.0

0.9

0.8

0.7•e?

| 0.6

% 0.5i

0.4

0.3

0.20.1

IK 'i i i i i i

-

\\ / S . \ / S . y— Values of m .a.

-

- -

i 1 1 1 1 1 1

» « • « . -

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6Parameter r n x 1 b 1

FIG. 5 .18b. E f f i c i e n c y o f t he h o r i z o n t a l s e c t i o n of a s t r a i g h t , s i n g l e Teef i n fo r u • v (case 5 . 1 . 3 2 , sou rce : Re f. 10 , p . 399 , F i g . 8 .5 ) .

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as

100

908070

6050403020100

V I I l I l I

~ \ -- TO^ -

_ y\ —Y*. A/2R -

\

^—1.5^ - 3 . 0^ - 5 . 0^—10.0

-

i i i3?===—_ -

i i i I I— 1 ' i

FIG. 5.21. Effectiveness of sheet f in with square array tubes (case 5.1.40,source: Ref. 83, p. 29 4, F ig . 2 ) .

J 0.5

< J \0 .05M

1

. 0 2 \ N AVi -

-

^î^fc 1,

1 ^ O C s ^

^î^fc 1,

FIG. 6 . 1 . Temperatures i n an i n f in i t e r eg ion o f which t he r eg ion | x | sfl b:\ 0.01 = F.u '1

oTN -

O^**-

0.5-

~1 "~

2 ^ = >.. ,., I ^ S2 ^ = >R

FIG. 6.2 . Temperatures in an i n f i n it e re gi on of which the re gio n r < rQ i si n i t i a l l y a t t e m p e r a t u r e t 0 (case 6 .1 .4 , sou rce : Ref. 9 , p . 55, F ig . 4b ) .

i i

1.00 0 5 ^

O.ION

W>2\\0 .0 l ' = Fo ' '1

0.50 . 2 0 s

0.50

1.00

1 I ^ ^ ^ S :

FIG. 6.3. Temperatures in an infinite region of which the region r < r n isinitially at temperature t 0 (case 6.1.5, source: Ref. 9 , p . 55 , F ig. 4c ) .

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— 0.4

PIG. 6 .4 . Temperatures in an in f i n i t e reg ion with s teady temperature t0 onthe su rfa ce r = r 0 ( case 6 .1 .1 8 , source : Ref. 9 , p . 337 , F ig . 4 1 ) .

log 10(Fo)

FIG. 6.5. Temperature in a cylinder of infinite conductivity, initially attemperature t l r in an Infinite medium inditially at t 0 . Numbers on thecurves are values of 2PQCQ/P1C (case 6.1.21, source: Ref. 9, p. 34 2,

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o

J, 0.2

F IG . 6 . 6 . Te m p e r a tu r e i n a c y l i n d e r o f i n f i n i t e c o n d u c t i v i t y , i n i t i a l l y tem pe ratu re tg , i n an i n f i n i t e medium. Numbers on the cu rv es are va lue s 2Pn co /P ic i ( ca se 6 . 1 .2 2 , sou rce : r e f . 9 , p . 343 , F i g . 46 ) .

0.2 1 2 10 20 100 200 1000 2000 10,000Fo-»

PIG. 6. 7. Temperature response of s o l i d sphere 0 < r < r 1 ( w i th kj_ + M andi n i t i a l l y a t t 1 g , embedded in an i n f i n i t e s o l i d ( r > r ^) i n i t i a l l y a tt 2 g (case 6 .1 .2 6 , source : Ref. 19 , pp . 3 -64 , F i g . 37 ) .

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0.001 0.00 5 0.010 0.050 0.100 0.500 1.000 5.000 10.00 0

F o -

PIG. 6.7. Temperature response of an infinite solid with a constan t sphericalsurface temperature of t Q (case 6.1.13, source: Ref 74, p. 430, Fig. 10.9).

0.6

(III)0.5

0.4t

~° 0.3

0.2

0.1

0 0.02 0.04 0.06 0.08(1)0.1I - *J I I

- • I _

l l l \ - II

III > II

I I I I S N

1.0

(II)

0.9

0.8

0.7

0.6

10.0

(I)0.99

0.98

0.97

0.50 0.2 0.4 0.6 0.8(11)1.00 40 80 120 160(111)200

20 40 60 80 10010 1 1 1 1 0.2b

0.08 - 0.20

0.06 0.15

0 .04 — - 0.10

0.02 —

I 1 1 1

0.05

00 200 400 600 800 1000

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. 3 - «3"I I

11 0.1

~ I J -

0.01

FIG. 7 .2 a . Temperature d i s t r i bu t io n in the se m i- in f i n i t e so l i d wi th con vec t io noundary co nd i t io n (case 7 .1 .3 , sou rce : Ref. 5 , p . 82 , F ig . 4 .6 ) .

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0.01 . 0.05 0.10 0.50 1.00 5.00 10.00 50.00

G. 7.2b. The dimensionless excess temperature vs the number BijjfFOjj) 1/ 2 andious Fourier numbers for semi-infinite solid with a convection boundary (case

1.3, source: Ref. 74 , p. 207, Fig. 6.2 ).

1.0'

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1.00 0.2 0.4 0.6 0.8 1.0

x

F IG 7 3 Te m p e r at u re d i s t r i b u t i o n i n a s e m i - i n f i n i t e r e g i o n when t h e s u r f a c

V-» 1 i i 1\*\° —

_ - v \

* \* VO — • ^

— C P V

7^° 2^s».^° 2^s». —JO

~h¥ '<? —/ * / / &4 y/.& _7 A /A /fi i 1 1 1

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1008060

40

o 20o

* „ 10

6

4

2

0 0.2 0.4 0.6 0.8 1.0 1.2

* & )

FIG. 7.4. Function f T(t- tf) /(t c - t f )] (case 7.1.22, sour ce: Ref. 4, p . 84,Fig. 4.15).

- 2 - 1 0 1 2

x/a

FIG. 7.5. Tempera ture distribution acr oss a heated strip of width 2a on thesurface of a semi-in finite solid (case 7.1.24, source: Re f. 9, p. 265,Fig . 3 3) .

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•i- 0.6 -

"2- 0 .4-

1.4 1.6

FIG. 7.6. Temperature response of semi-infinite solid (x >_ 0) with surfacetemperature t g suddenly increasing as power function of time, t s = t: + b r n ,(case 7.1.9, source: Ref. 19, pp. 3-66, Fig. 39 ).

1.0

1/ /— 0.61/V7T —

0.4

0.2

0

V \ I I I I ISemi-infinite solid

\ \ q 0 = const —

\ \i k i 3 t ~~J TIT a —

39

k 3t" " " q0 3x

' kl t- t, ) N ^

I t -q Q \ ^ r I t -0.4 0.8 1.2 1.6 2.0 2.4

FIG. 7 .7 . Temperature response1, temperature gradient, and heating rate in ase m i - i n f in i t e s o l i d a f t e r exposure t o a cons t an t su r f ace hea t f lux q0

(case 7 . 15 , sou rce : Ref. 19 , pp . 3 -8 1 , F i g . 49 ) .

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IG . 7 . 8 . Tem pera tu re r e sponse of a s e m i - in f i n i t e so l i d a f t e r sudden exposu ro a l i n e a r l y i n c r ea s i n g su r f ace hea t f l u x fo r a du ra t i o n D , Q = D qm a x / 2 ,

0 = q m a x T /D (case 7 .1 .1 4 , so urc e : Ref . 19 , pp . 3 -82 , F ig . 50 ) .

rri\/Fch

8 10 12 14 16

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My/it-0.6

Fo* -+x

FIG. 7.10. Temperature d i s t r i b u t io n s ina s e m i - i n f i n i t e s o l i d w i t h e x p o n e n t ia lh e a t i n g ( ca s e 7 . 2 . 9 , s o u r c e : Ref. 19,pp. 3-85, Fig. 52).

1.0

0 . 8 -

i 0.6U

J2 0.4

0 . 2 -

\ 0=Fo1

_ o . o rH ^

6 O J * ^ « S S N \ ^

— o T i o ,. ^ ^ -

0.4iQ

V

1.00

T r - - ^1 T r - - ^0.2 0.4 0.6

X0.8 1.0

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1 .Uo . o p " ÔuT"" <

I I^ ^ 0 = F o

I

0.8ouT""*^.

-

'„ 0.6 — —

4-<

^ 0.4 ^0 .40 \ \

0.21.0

N \ -

I I

N \ -

0.2 0.4 0.6 0.8

X

1.0

FIG. 8.2. Temperatures in a slab with a parabolic initial temperaturedistribution (case 8.1.4, source: Ref. 9, p. 98, Fig. 1 0 ) .

FIG. 8.3. Temperatures in the infinite plate with a constant intialtemperature and steady surface temperature (case 8.1.6, source: Ref. 9,

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1.000

0.100 -

0.010 -

0.0010 1 2 3 4 10 30 50 70 9 0 110 130 150 300 500 700

Fo

FIG. 8.4a. Hidplane temperature of an infinite plate of thickness 21 and convectively cooled.(case 8.1.7, source: Ref. 12, p. 227 and source: Ref. 5, p. 83, Fig. 4-7).

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0.0001 0.0005 0.0010 0.0050 0.0100 0.0500 0.1000 0.5000 1.000Fo

FIG. 8.4b. Surface temperature of an Infinite plate of thickness 21 andcorrection boundary (case 8.1.7.2, source: Ref. 74 p. 226, Fig. 6.6c).

i

0.0001 0.0005 0.0010 0.0050 0.0100 0.0500 0.1000 0.5000 1.0000

Fo

FIG. 8.4c. Surface temperature of an infinite plate of thickness 21 andconvection boundary (case 8 1 7 2 source: Ref 74 p 226 Fig 6 6d)

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1.0

0.9

0.8

0.7>' 0.6J?

> 0-5

i 0.4

0.3

0.2

0.1

00.

' ' I ' ' I 'j ^ 5 ? ^- 0.2 ^ ^ y j ^ 5 ? ^

— —D-4"""""^ / /A

/ /1/ -

— — °-6-*^ /// -

JU& /-

/ -

- ^ / -

A.Oi l l . i 1 i i . i

01 0.1 1.0

1/Bi

10.0 100.0

PIG. 8 . 4 d . Tem perature as a fu nc t io n of midpiane temperature for an i n f i n i t ep l a t e o f t h i c k n e s s 2i and co nv ec t ive ly coo led ( ca se 8 . 1 . 7 , so ur ce : Ref . 5 ,p . 8 6 , F i g . 4 - 1 0 ) .

1.0

0.8

a0.6 9 6" 6' •=" fi, <o ST , _

O O' o' O' O' O' O' O' ©' •- ' <V *l>" £" S '

B i2 F o

FIG. 8.4e, Relative heat loss from an infini te plate of thickness 2% andconvectively cooled (case 8.1. 7, source: Ref. 5 , p . 90, Fig. 4-14 andRef. 1 3 ) .

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G. 8.5. The function F(O) (case 8.1.12, source: Ref. 1. p. 301 , Fig. 14-3),

FIG. 8.6a. Variation of amplitude of the steady oscillation of temperature inan infinite plate caused by harmonic surface temperature (case 8.1.13,source: Ref. 9, p. 106, Fig. 13 ).

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-600

-500 -

- 4 0 0 -

-300

- 2 0 0 -

FIG . 8.6b. Variation in pha se of the steady oscillati on of temperature in aninfinit e place caused by harmon ic surface temperature (case 8.1.13 , source :Ref . 9 , p . 107, F i j . 14 ) .

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Iinm

Fo

FIG. 8.7. Temperature distribution in an infinite plate with no heat flow at x = 0 and constant heat fluxq 0 at x = I (case 8.1.18, source: Ref. 9, p. 113, Fig. 15 ).

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1.0

0.8

so 0.6

S °- 4

0.2

1.0

I I I

- 0.8

o l T ~

-

=— ~0.4

0.3

.

-

0.2

0.1

—0.06

—0.02 -Fo

I I I 1 — *

0.2 0.4 0.6X

0.8 1.0

F IG . 8 . 8 . Te m p e ra tu re s i n an i n f i n i t e p l a t e w i t h c o n s t a n t i n t e r n a l h e a t i n g and surface t empera tu re t ^ ( ca se 8 . 2 . 1 , sou rce : Re f . 9 , p . 131 , F ig . 20 ) .

F IG . 8 . 9 . Q u a s i - s t e a d y s t a t e te m p e r a tu r e o f an i n f i n i t e c o n d u c t i v i t y p l a t

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FIG. 8.10. Plate temperature responsefor prescribed heat inputs (case 8.2.15,source: Ref. 19, p. 3-35, Fig. 18 ) .

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iooo.o~l " ~800.0 finnn (q" + q " ' S + x 7 t * ) / o 7 t 2600.0

400.0 \

200.0

1.00.0001

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

1.0000

o.ftg/pCS)

-0 .90 i 1 r i r-0 .80

-0 . 70-0 . 60

0.50 jr- ( q " + q '"« + a.?\^lo&\\

-0 .40-0.34

-0 .30-0.24

•0 .20

— 0 . 1 4

0.10

-0.07

-0 .05

Thin platecooling

-0.04-0.02-0.02

-0.01

J L10 14 18 22 16 30

(b) (o.ftppC5)

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Thin plate

20 4010

q m a x

D / P C 6 t 0

(a)

FIG. 8 .1 2 . Maximum tem per a tur e of t h in in su la te d p la te sudde nly exposed to

c i r c u l a r h e a t p u l s e w i t h s u r f a c e r e r a d i a t i o n : ( ca se 8 . 1 . 5 0 , s o u r c e : R ef. 1 9p . 3 -38 , F i g . 2 0 ) .

Fo

0.20 0.16 0.12 0.08 0.04 00.20

0.16•*I

~ 0.12

? 0.08

0 . 0 4 -

l\ I I 1

A/ 0 . 5 // / 1.0X.

1

A/ 0 . 5 // / 1.0

Thick plate ^ / 2 /

^ ^ V / ^ S - ' 3 0 ^

&^\ i

^ ^ ^ 3 ^ 0 . 0 -

I I \0 0.04 0.08 0.12 0.16 0.20

Fo(a)

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0.2 J _ -i.0.1 0.2 0.3

1*2. max " W V

0.4 0.5

FIG. 8 .14a. I n s u l a t i o n w e ig h t f or s u b s t r u c t u r e p r o t e c t i o n t o t2 m a x i nh e a t i n g du ra t i on D , Wx = P i^ i ( ca se 8 .1 .5 1 , sou rce : Re f. 19 p . 5 -48 ,F i g . 2 7 ) .

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(a)

/i = n + (n + D /B i , - , n = / 32 C 2 f i 2 / p 1 C l 5 l

1.0

5 10 20

Bi , =h6,/k 1

F o:

( b )

F I G . 8.14b. T e m p e r a t u r e r e s p o n s e o f t h i c k p l a t e (0 < _ x <_ 6 ) c o n v e c t i v e l y

h e a t e d a t x . » 0 and in p e r f e c t c o n t a c t a t i t s r e a r f a c e x = 6 w i t h a

t h i n p l a t e (0 < x 2 < 5 2 ) i n s u l a t e d a t x 2 = 6 2 : ( a ) x . , / 6 , = 0.

( b ) x . / ^ " 1 ( c a s e 8 . 1 . 5 1 , s o u r c e : R e f . 1 9 , p . 3 - 4 6 , F i g . 2 6 ) .

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y = t s1.0

0.8

0.6

0.4

0.2

o tiS/k-^ 1

i wn/ T T ^••

0.5 / / // 0.33 / / / / /— / ' / 0.2 / iff/ / a i /' / / 0.05

/ / ^ - Th n plate

/ / / 0 - 0 3 3 /

P^ZZ- | |

^ • 0 2 t s = const

Heating: tg/t,. = 0 —

x/ 5 = 0

10.1 0.5 1

(a)5 10 50 100 50 100

50 100

FIG. 8.15. Temperature response o f thick plate (0 <_ >: < <5) wi th insulated rear face x = 6 after suddenexposure to uniform radiative environment t s at x = ("•: ""(a) hea tin g, tn / t s = 0, x/6 = 0, (b) heating,to / t s

= ° ' x/& = 1 ' < c> heating, t 0 / t s = 1/4, x/6 = ;, (d) heating, t 0 / t s = 1/4, x/5 = 1 (case8.1.52, source: Ref . 19, pp. 3-50, Fig . 29 ) .

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0.1 0.5 1 5 10 50 100 0.1 6.5 1 5 10 50 100 500

(f ) Fo (h)FIG. 8.15 (Cont.) . Temperature response of thick plate (0 <_ x <_ &) with insulated rear face x = 6 aftersudden exposure to uniform radiative environment t s at x = 0: (e) heating, t Q / t s = 1/2, x/6 = 0, (f)heating, t 0 / t s = 1/2, x/6 = 1 , (g) cooling, t 0 / t s = 6, x/6 = 0 , (h) cooling, t 0 / t s = 6, x/ = 1 .

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500

500

IG. 8.15 (Cont.). Temperature response of thick plate (0 <_ x £ 6) withnsulated rear face x = 6 after sudden exposure to uniform radiativenvironment t 8 at x = 0: (k) cooling, t 0 / t a = 2, x/6 = 0, (1) cooling,

0 / t s = 2, x/S = 1.

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500

Thin plate.

Cooling: tn /ts = 2x/8 = 1

50 100 500

FIG. 8.15 (Cont.). Temperature response of thick plate (0 <_ x <_ 6) withinsulated rear face x = 6 after sudden exposure to uniform radiat iveenvironment t_ at x = 0: (i) cooling, t g/ t a • 4, x/6 = 0, (j) cooling,t 0 / t s - 4, x/5 - 1.

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PIG. 8.16. Temperature response of thick plate (0 £ x £ 6) with surfacetemperature t„ at x = 0, increasing linearly with time and rear face x = 6insulated or exposed to uniform convective environment t Q (case 8.1.8,source: Ref. 19 , pp. 3-60, Fig. 33 ).

FIG. 8.17. Surface temperature response of a plate of thickness 21 exposed toa steady heat flux <jn on both sides for a time duration D (case 8.1.18,source: Ref. 19 , pp. 3-67, Fig. 40 ).

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•P 0,

J?

5 10 50 100

(a)

50 100

Fo

(b)

FIG. 8. 18 . Temperature response of a p la te with steady heat lnq qg at x = and convection boundary at x - 0 to tg (a) x /£ - 1, (b) x/fc = 0 (case 8 .1 .17,s o u r c e : Ref. 19, pp. 3-6 9, Fig. 41) .

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18

16

14

12

10

efo

Q D / 2 J I 52

Thick plate

= q m a x s i n2

nfl/Dx/5 =0

3 -

2 -

IThick plate

1 1aD/27tfi2 -

1 1x/6 = 1 0.1—v _ — a — —

0 . 6 - \ \

°-4- \ \ 400% <<S§ô.i°-3 —\ \4 tf/y/\/ 0.08

~~ 0 . 2 - A *0.16-M2

0.14-2cW0 12^225cOt£

• J X / / 0 . 0 6 -— Thin plate-

/ \ S ^S 0.03' ^ V ^ - O ^ 0.025i i ^ i - r - o . 0 2

,**« S = ^ q _ - n n i d0 1 2 3 4 5 6 7

2TTT/D

FIG. 8 .15 . Temperature resp onse of an i n f i n i t e pl at e exposed to a ci rc ul a rheat pulse q^ a j t s i n 2 (TTT/D) at x = 0 and insu la ted a t x = 6 (a) x/5 = 0,(b) x/6 = 1 (case 8. 1. 54 , so ur ce : Ref. 19, pp. 3-70, Fig . 42 ).

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0.05 0.1 0.5 1 5 10

OLeTlf>1

(b)

FIG. 8.20. Temperature response of an infinite porous plate after suddenexposure to a constant heat flux at x = 0 and cooled by steady flow throughplate from x = 6 of a fluid initially at t 0 (a) x/6 = 0 , (b) x/5 = 1 (case8.1.55, source: Ref. 19 , pp. 3-74, Fig. 4 4 ) .

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1 I '•Two-layer plateq constlk 2/k,)â x/a 2 = 1/2

FIG. 8.21. Temperature response of an infinite plate with a constant heat fluxq« a t X i = 0 , in contact with a plate at x-± = 6^ and insulated o n exposedsurface o f second plate x 2 = 5 2 (case 8.1.56, source: Ref. 19, pp . 3-76,Fig. 4 5 ) .

8/10/2019 6224569


Two-layer platey i ' i T=^~I—r

(k,/k 1 )VVs = 2 >°2 ' * 1 ' V " , ' " 2x , = S r

l\.1/4

( S ^ l V S ^.1 /2

4 °°<"4 oo^

£J I L J . I ' l1 0 1

i (0

0.2

0.4

0.6

0.8

1.02 3 4 5 6 7 E S

Fo,1

v -i i i i i- Two-layer plate

fc 2/k,)^/u 2

1 —

= 4.A ' i ' i '

- \ vs 1 / 4 ^-—'

\

: \ "~ 1/2—1

" - 1 / 4 - ~ _ _ _ _ _ ^

r;, 1 , 1 J i ,

n _r;, 1 , 1 J i , 1 . 1 , 1 ,

0

- 0.2

- 0.4

- 0.6

0.8

0 1 2 3 4 5 6 7 8

(d> Fo,

1.0

FIG. 8 .21. ( c o n t i n u e d ) .

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FIG. 8.22. Temperature response of an infinite plate exposed to a constant heatflux q " at xi = 0 and in perfect contact at xi = 61 with a plate of thickness62 insulated at x 2 = S 2 (case 8.1.57, source: Ref. 19, pp. 3-78 , Fig. 46) .

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Two-layer plate

n = p 2 C 2 6 2 / p 1 C l S l = 0 . 1W t 0 - S i « W W n + 1 > k i

FO;/2TT = 3 , 0 / 2 ^ 2

G. 8.23. Temperature response of an infinite plate exposed to a circular heatse at ip = 0 and in perfect contact wi th an infinite conductivity plate ofckness &2 (case 8.1.58, source: Ref . 19 , pp . 3-78, P ig . 4 7 ) .

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1000I I

Two-layer plate

_ (b)

0.01

1000Two-layer plate

p 2 C 2 S 2 / p 1 C , 6 1 = 10

1 0 0 -

- (0

0.01

F IG . 8 . 2 3 . ( c o n t i n u e d ) .

8/10/2019 6224569


I

FIG . 8.24. Temperature response of an infinite plate surface having steadytemperature and convection boundaries (case 8.1.25.2, source: Ref. 74 , p. 240,Fig. 6.12).

FIG. 8.25. Temperature of a flat plate having a ramp surface temperatureequal to t 8 = t + br (case 8.1.33, source: Ref. 74, p. 305 , Fig. 7. 1) .

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2Fo

1 + 2/Bi

2Fo

1 + 2/Bi

G. 8.26. Transient surface temperature of a slab convectlvely coupled to anearly changing environment temperature equal to tg - t + bt (case 8.1.33,ource: Ref. 89 ).

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i

0.00110 16 24 30 60 90 120 150 350

Fo

FIG. 9.1a. Axis temperature for an infinite cylinder of radius r Q (case 9.1.3, source: Ref. 5, p. 84,Fig. 4-8 and Ref. 12 ) .

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1.0

0.8

0.6

1 0.4

0.2

1 I 1 1 L _-| •

- r / r 0 = 0 . 2 - ^ ^ ^ >

— °ji_—-"^'^ / //t ' -

/ III -— Q&y^ /ll -

/1 1 -— / /1

0.8 . / / /

^***^ / I -

— Q-\S / -1 . 0 , /

r ^ 1 i i 1 , I , ,

0.01 0.1 1.0

1/Bi

10 100

IG. 9 .1b . Tempera tu re a s a func t ion o f ax i s temp era tu re i n an i n f i n i t eyl inder of rad ius r Q ( c a s e 9 .1 .3 , sou rce : Re f . 5 , p . 87 and Ref . 12 ) .

1.00.90.8

0 .i 0.

°|cP o.0 .0.0.0 .

765.4,3.2.101 0 "5

^ . f ^ i i j * . . t i t § / # #

a S e o o S ^ fy <o ' ,/ < o < a f i ,oo o o o o o <a o o ^ V* y <v <o

IG. 9 .1c . D i m e n s i o n l e s s h e a t IOSB Q /Q0 o f an i n f i n i t e c y l i n d e r o f r a d i u s0 w i t h t im e ( c as e 9 . 1 . 3 , sou rc e : Ref. 5 , p . 90 , F i g . 4-15 and Ref . 1 3 ) .

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I .U0.90.80.70.6

V ' ~ r I I I I . I . _Plate: m = 1Cy linder: m = 2Sphere: m = 3 —

i Range of these lines:\ F o B > 0 . 2

0.5 \ B ic < 0 . 0 1 -•3-

1 0.4 \*-*"3-

i0.3 \ _

0.25 - \

0.2 - \

0.15

0.1 I , I i I , I , I , \

U. IU

0.08 \ 'I I

0.060.050.04

-

0.03 —

0.02 -

0.010.008

v -

0.0060.0050.004

\ -

0.003—

\ -0.002 - \ -

0.001 I I I I \

mBifFog

FIG. 9.Id. Center temperatures for plates , cylinders, and spheres for smallvalues of h (case 9 .1.3, 8.1.7 and 10.1., source: Ref. 5, p . 89 , Fig. 4-13and Ref. 12 ), C is half thickness of radius.

FIG 9 2 Temperature distribution in an infinite cylinder with initial

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PIG. 9 .3 . Temperature in an in f i n i t e cy l inde r w ith constant heat f l u x at thesur face (case 9 .1 .8 , sou rce : Ref. 9 , p . 203 , F ig . 25) .

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T = S(X)

Semi - in f i n i t e so l i d

T = S ( X , ) S < X2 )

Qua r t e r - i n f i n i t e so l i d

T = S(X1 ) S ( X2 ) S ( X3 )

E ig h th - i n f i n i t e s o l i d

T = P(Fo) T = P(Fo)S(X) T = P(Fo)S(X1 )S (X2 )

I n f i n i t e p l a t e

* — 2 8

Semi - i n f i n i t e p l a t e Q ua r t e r - i n f i n i t e p l a t e

T = P(Fo , )P(Fo2)

V

u~| — Z * 1

In f in i t e r e c t angu l a r ba r

.28,

T = P(Fo,)P(Fo2)S(X) T = P(Fo , )P(Fo2) P ( Fo 3 )

r28,

K - 2 8 ,

Semi - in f i n i t e r e c t a ngu l a r b a r

A 28,

h+«3

±\.- \Rectangula r para l le lep iped

• 26,

, T = C(F o,)

i

A.

I n f i n i t e c y l i n d e r

T = C(Fo1)S(X)

\S

T = C ( F0 l ) P ( F o 2 )

Y-2r,c b

-2r,

S e m i - i n f i n i t e c y l i n d e r S h o r t c y l i n d e r

S = s emi - i n f i n i t e so l i dP = plateC = cy l i nde r

FO T = G T / 62 = a r / r2

X , = x1 / 2 v / 5 f = F o J

T = ( t - tw ) / ( t 0 - t w )

FIG. 9.4a. Product solutions for internal and central temperatures in solidswith step change in surface temperature S(X) given in Fig 7 2 P(Fo) given

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0.01 -

0.001

Fo, or/5?

G. 9.4.b. Central temperatures in an infinite plate, an infinite rod, anfinite cylinder, a cube, a sphere, and a finite cylinder of length equal to

s d iameter, with all surfaces at temperature tg (case 7.1.29, 7.1.20,1.16, 9.1.17, and 8.1.22, source: Ref. 2, p. 248, Fig. 1 0 - 8 ) .

V°

1.0

0.8

0.6

i 0.4 -

0.2

^ Q T 6 ^ - 4 = O I I I- 0 4 ^ ^ v

-o • 3 - _ ^ S s \ N s v -

- 0 - 2 — . — . ^ \ \ ^ v

— 0 - 1 5 . N . N ,

= 0 • 1 — _ _ _ _ _ _ _ ^ ^ * s N^vk -- 0 • 08 _ _ ^ ~ " - - « ^ _

— n • n f i - — ^ " ~ " " "

0 • 0 1— 0 • 0 ° ~ Fo

I I l ^p-^0.2 0.4 0.6 0.8 1.0

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100 .0

s 10.0 h

Juy>

• 9 -

AA?

•3-

0.4 0.8 2.0 3.0 4.0fo re?

F I G . 9 . 6 a . ( c a s e 9 . 2 . 4 , s o u r c e : R e f . 3 ,p . 3 5 6 , F i g . 7 - 5 ) .

F IG . 9 . 6 b . ( c a s e 9 . 2 . 4 , s o u r c e : R e f . 3 ,p . 3 6 4 , F i g . 7 - 8 ) .

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0.1 1 10 50

S/ IX-1)PIG. S .6 c. (case 9 .2 .4 , sou rce : Ref. 3 , p . 365 , F ig . 7-9)

F-87

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0.001 0.005 0.010 0 .050 0.100 0 .500 1.000 5 .000 10.00 0

Fo, OT/TQ

FIG. 9. 7a. Temperature re sponse of a hollow cy li nde r with a ste ady in si desurface temperature equal to the in i t i a l temperature , t^ (case 9. 1. 10,so urce : Ref. 74, p. 156, Fig . 4. 24 ) .

0.001 0.005 0.010 0.050 0.100 0.500 1.000 5.000 10 .000

Fo, ar/rg

FIG. 9.7b. Temperature response of a hollow cylinder with a steady outsidesurface equal to the initial temperature, t (case 9.1.10, source: Ref. 74,p. 157, Fig. 4.25).

F-88

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T — i - r - i i i i

0 . 8 0 " " " " " " "~ _ ^ - —

I I I

- /

-r i

0 . 6 6 " " " ^ ^ - - " " ^

^ 0.50 ^ \ ^ ^ ^ ^

I I I I

t 5 = t, + bT

I I I

I

I

0.8

0.6

a0.4

0.2

0 0.1 0.2 0 .3 0 4 0.5 0.6 0.7 0.8 0.9 1.0

Fo

FIG. 9 . 8 . Tem pera ture of an i n t i n i t * cy l in der having a ramp sur fac etemperature equa l to ts = t± + bx (c ase 9 .1 .2 , so ur ce : Ref . 74, p . 312,F i g . 7 . 3 ) .

4Fo

1 + 2/Bi

4Fo

1 + 2/Bi

FIG. 9 .9 . Tr an sie nt surfa ce temperature of a c y l i n d e r c o n v e c t i v e l y c o u p le d t oa li n e a r ly chan ging environment temperature equal to tf • t j + br ( ca se 9 . 1 . 6 ,sour ce : Ref . 89 ) .

F-89

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100 i i i i i 1 1 i i i i t i i i

J i I • ' • • • I I I I i i • • '10Fo

100 1000

FIG. 9. 10 . C en tral temperature of an i n f i n i t e cyl inde r w ith uniform int er n alheat in g and co nv ec t ion boundary (cas e 9 . 2 . 2 , source : Ref . 9 0 ) .

F-90

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IID

0.400

0.8 -

0.6

0.4 -

0.2 -

1 R -1

^ 0 . 9 5 ^

o.9o /

y 0.85 /

/ O.BoX/ ' /

—/' / 0.75 /

' / 0.66

— / ' / 0.50 •^-0.25

/ >^0.33X 'o

10 0.2 0.4 0.6 0.8 1.0

(1-R)

0.004 0.010 0.4 0.100

FIG. 10.1. Temperature distribution in a sphere of radius r Q with initial temperature t± and surfacetemperature t 0 (case 10.1. 1, source: Ref. 74, p. 127, Fi gs. 4.15 and 4 . 1 6 ) .

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1.000

I

-i . 0.050+•«

'._ 0.03 0

0.001

3.0 5.0 7.0 9.0 15.0 25.0 35.0 45.0 70.0110.0 170.0 250.0Fo

PIG. 10.2a. Center temperature of a convectively cooled sphere (case 10.1.2, source: Ref. 5, p . 85,Fig . 4-9 and source : Ref . 12 ) .

8/10/2019 6224569


1 c V ^ n

0.02 0.1 0.5 2.00 5.00 20.00 100.00

1/Bi

FIG. 10 .2b . Tem pera ture as a func t ion of ce nt er tem pera tu re of a co nv ec t lv e lycoo led sphe re ( ca se 1 0 .1 .2 , sou r ce : Re f. 5 , p . 88 , F ig . 4-12, and Ref . 12 ) .

a id 0.5 -

B i2Fo

FIG. 1 0 . 2 c . D l m e n s l o n l e s s h e a t l o s s Q/QQ o f a con vec t ive ly coo led sphe re( c a s e 1 0 . 1 . 2 , s o u r c e : Ref . 5 , p . 9 1 , F ig . 4-16 , and Ref . 1 3 ) .

F-93

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FIG. 10.3. Temperature of an infinite conductivity medium surrounding aspherical solid (case 10.1.13, source: Ref . 9, p. 241, Fig. 30 ).

0.2

o

0 -

- -0.1-

-0.2

-0.3

-

I I I

Ai

±1f

2.8

2.4

2.0

1.6

1.2

0.8

0.4

-

I I I

Ai

±1f

2.8

2.4

2.0

1.6

1.2

0.8

0.4 I

I I I

f A

I I I

\

i

0

i±1

f

2.8

2.4

2.0

1.6

1.2

0.8

0.4 I

I I I

f A

I I I

\

i

0

i±1

f

2.8

2.4

2.0

1.6

1.2

0.8

0.4 I

I I I

f A

I I I

\

„«I ^~*-~**5w

i

0

i±1

f

2.8

2.4

2.0

1.6

1.2

0.8

0.4 I

I I I

f A

I I I

\

—

i

0

i±1

f

2.8

2.4

2.0

1.6

1.2

0.8

0.4 I

I I I

f A

I I I

\

—

__ 0. 0 8 ( Xy/

i

0

i±1

f

2.8

2.4

2.0

1.6

1.2

0.8

0.4 I

I I I

f A

I I I

\

0.2 0.4 0.6 0.8

R

1.0 0 0.2 0.4 0.6 0.8

Fo

FIG. 10.4. Temperature in a sphere caused by a steady surface heat flux (case10.1.15, source: Ref. 9, p. 242, Fig. 31).

F-94

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1 -0

1 -0

FIG. 10 .5 . Temperature d i s t r ib ut io n in a sphere wi th s teady in te rn a l hea t ing(case 1 0 . 2 . 1 , source : Ref . 9 , p . 244 , F ig . 3 2) .

F-95

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0.010 0.100 1.000 10.000

Fo

10.00

10.6. Temperature response of solid sphere (0 <_ r £ r-^) cooling by radiation to a sink temperature of(a) r/ r x = 0, (b) r/c 1 = 1/2, (c) r/ r x = 1 (case 10.1.20, source: Ref. 19, pp. 3-57, Fig. 30 ) .

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T- T, 0 . 6

T w -T

i

0.02 0.06 0.10 0.40 0.80

F o , —

FIG. 10.7a. Tr a n s i e n t t e m p e r a t u r e d i s t r i b u t i o n a t t h e c e n t e r p o i n t (x = y= z = 0 ) fo r v a r io us p ro l a t e sph e ro id s ( ca se 1 0 . 1 , 2 1 , sou rce : Raf. 72 , F ig . 1 ) ,

1.0

0.8

T - T i 0.6

T w - T i

0.01 0.04 0.08 0.20 0.601.00_ a r

FIG. 10.7b. Transient temperature distribution at the focal point (x = y = 0,z - L2) fo r various prolate sol ids (case 10. 1.21, source: Ref„ 7 2 , F i g . 2 ) .

F-97

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FIG. 10 .8. Temperature of a sphere having a ramp surface tep.perature equal tot s = t£ + br (case 10.1.4, source: Ref. 74 , p. 3 10, Fig. 7.P 1 .

0.5

6Fo1 + 2/Bi

FIG. 10 .9 . Tr an s i e n t su r f ac e t empera tu re o f a sphe re con vec t ive ly coup led to al i n e a r ly chang ing env i ronmen t t empera tu re eq ua l t o t f = tj_ + b t ( ca s e 1 0 . 1 . 11 ,so u r ce : Re f. 89 ) .

F-98

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100.0

10.0

I

CNO

r-[—i i i 11 I I 1—I I 111 1 1—i i i i i I J 1—i i i i I I

R = 0 80.0 —

J I I I I I I100.0 1000.0

FIG. 10.10. Central temperature of a sphere with uniform internal heating andconvection boundary (case 10. 2.9 , source: Ref. 9 0 ) .

(h s Vk)(t m-t a )r

FIG. 11 .1 . Solidification depth in semi-infinite liquid for which solidified. phase has negligible thermal capacity and is exposed to convective environmentst a at free surface x = 0 and t 0 at interface x = w (case 11.1.10, sou rce :Ref. 1 9 , pp. 3-88, Fig. 54 ).

F-99

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0.01 0.05 0.1 0.50 1

(b 2/pck)T-

50 100 500 1000

10

5

0.5

0.1

0.05 -

0.010.001 0.005 0.01

(b 2/pck)T

100

5 10

50 100

50 100

500 1000

G . 11.2. Tempe ratur e response and solidification depth in semi-infinite

quid initially at t^ (a) surface temperature for convective cooling q = b t w

x = 0, (b) solidification depth (case 11.1.11, source: Ref . 19, pp. 3-89,g. 5 5 ) .

F-100

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0.010.001

0.05 0.1

(b2

tJL/pck)r

0.006 0 .01

50 100 500 1000

5 10

50 100

50 100

500 1000

( b 2 t 6 m / p c k )

F IG . 11 . 3 . Te m p e r a tu r e r e s p o n s e a nd s o l i d i f i c a t i o n d e p t h i n s e m i - i n f i n i t el i q u i d i n i t i a l l y a t t ^ w i th r a d i a t i v e c o ol in g q = b tw a t x = 0 (a) su rf ac et em pera tu re a t x = 0 , (b) s o l id i f i c a t i o n dep th ( case 1 1 . 1 . 12 , sou rce : Re f . 19 ,p p . 3 -90 , F ig . 55 ) .

F-101

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- 0.4 -

Semi-infinite solidt = constw0 < x = £ l

00.0001 0.0005 0.001

b/2V 7

0.005 0.01FO:

0.05 0.1 0.5 1

x l

0.01Fo'x2

PIG. 11 .4 . Temperature response of decomposing s e m i- in f i n i t e s o l i d af t er suddenchange in su rf ac e tem perature from tg t o t,,: (a) 0 £ x £ SL , (b) x £ SL(case 11 .1 .1 3 , s ource : Ref. 19 , pp . 3- 91 , F ig . 56 ) .

F-102

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10

5

0.01.

(a)

0.1

• W ^ d - V

Semi-infinite solid

t = const( h / p H ) t ta - td h / ^ 7 J = 1 / 2

0.5 1

r / r d - 1

10

5

i i i /

1

0.51 0 - 5 0 -

1 . 0 -0 . 5 -0.1 -

dv

0.1

\ / ,y\ ^ - V ^ ' V o 1

0.05

n n iV i

Semi-infinite solid( h / p H M ^ - t ^ ^ / T ^ I

1 1

(b)

0.1 0.5 1

r /Td - 1

10

0.05

0.010.1

vwv

Semi-infinite solid

(h /pH)( ta- td) v/5 d7d"=2

(c)

0.5 1

T/r, - 1

10

FIG. 11 .5 . Ab la t ion dep th o f s e m i - i n f in i t e so l id (x >_ 0) a ft e r suddenexposu re to co n ve ct iv e environment t- (> t ^ ) : (a) (h/pH) ( ta - t d )^ td /a = 1/ 2, (b) (h/pH) ( ta - t d ) STga = l , (c) (h/pfl) ( ta - t d )/T a/ a •» 2 (case 1 1 .1 .1 4 , source : Ref . 1 9 , pp . 3-94 , F i g . 5 8 ) .

F-103

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10

5

1

0.5

0.1

0 .05

0.011—<•

i—i—n— i | i i i I I 1—i—i—i—i I i i 11

Semi-infiniteq " = const

0.01

- V ? C ( td - t 0) /2Hv

0.05 0.1 5 1050 100

< r / Td ) - 1

FIG. 11 .6 . Ab lat io n depth of s e m i- in f i n i t e s o l i d (x >_ 0) af te r suddenexposure to con sta nt surface heat inpu t q" (case 1 1 .1 .1 5 , sou rce : Ref . 19 ,pp. 3 -96 , F ig . 59 ) .

F-104

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TABLE 1.1. Mean and maximum temperature excesses and their ratio for

electrical coils of rectangular and circular cross sections, 9 = t - t_,m mean 0

6„ = t - t„ (case 1.2.8, source: Ref. 3, p. 220, Table 10-3) .0 max u

Cross Section, Rectangular Circular,

b/ a » 10 5 2.5 1.5 1 r = 2a//i

0.66 0.625 0.58 0.28 0.32

1.00 1.00 1.00 0.59 0.64

0.6 6 0.625 0.58 0.5 2 0.485 0.475 0.50

TABLE 1.2a. Values of ( t - t ) / ( t f - t ±) on the

surface y = 0 of a se mi- in f in it e st r ip

(case 1.-1.29, sou rce : Ref. 15, Table 2) .

x/w hw/k hw/k hw/k hw/k hw/k

= 100 = 10 = 1 » 0.10 = 0.01

0.05 0.889 0.448 0.094 0.011 0.001

0.10 0.931 0.594 0.146 0.018 0.002

0.20 0.964 0.725 0.212 0.027 0.003

0.30 0.973 0.781 0.250 0.032 0.003

0.40 0.977 0.807 0.270 0.035 0.004

0.50 0.978 0.814 0.277 0.036 0.004

TABLE 1.2b. Heat transfer ra tes through the surface y = 0 of a semi -infi ni te

s t r i p (case 1. 1. 29, source: Ref. 15, Table 1) 9 = (t - t ) .

hw/k

1000 200 100 20 10 1.0 0.1 0.01 0.001

Q/k6 8.603 6.827 5.984 3.971 3.107 0.792 0.097 0.01 0.001

% Error 1.8 0.5 0.3 0.08 0.05 0.02 0.02 0.02 0.02

U k / q ' - ' a ^ e

; 2 k / q " ' a 2 ) 9 n

F-105

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TABLE 1. 3. Conductance dat a for he at flow normal to wal l cu t s in an i n f i n i t e

plate (case 1 .1 .37, so ur ce : Ref. 19, p . 3- 124). These groups of data are for

the four kinds of wall plates shown in case 1.1.37.

(a) K/Kuncut

(b) K/Kuncut

c/a b/a = 1/2 1/4 1/8 c/a b/a = 1/2 1/4 1/8

4 0.902 0.760 0.646 3/2 0.7 47 0.520 0. 381

2 0.818 0.618 0.480 3/4 0.575 0.315 0.217

1 0.704 0.465 0.339 1/2 0.430 0.209 0.146

1/2 0.610 0.363 0.235 1/3 0.296 0.113 0.080

1/4 0.564 0.303 0.182

(c) K/Kuncut

(d)

c/a d/a b/a = 1/2 1/4 1/8 c/a d/a

1/8 0.822 0.634 0.478 0.402

3 3/8 0.767 0.546 0.374 3/ 2 0.469

5/8 0.721 0.480 0.311 0.502

12/8 0.599 0.339 0.190 0.866

1/16 0. 740 0.486 0.342 1.010

3/ 2 3/16 0.698 0.430 0.284 1.083

5/16 0. 661 0.390 0.2421/2 1/2

12/16 0.570 0.299 0.164

0 0.627 0.403 0.270

1/2 1/16 0. 561 0.312 0.186

3/16 0. 527 0.276 0.152

K/Kuncut

0.507

0.479

0.329

0.362

0.338

1/16

0.310

0.254

0.213 0.141

P-106

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I

TABLE 2 . 1 . Va l u es of A. for a t r i a n g u l a r c o o l i n g a r r a y o f c y l i n d e r s ( c a se 2 . 2 . 1 8 , s o u r c e :

Re— 64 , Table 2) .

8 / r o A l A 2 *3 A 4 A5 A6 *7

4.0 - 5 . 0 5 0 7 2 ( 1 0 -2 ) - 8 . 0 8 9 1 ( 1 0 ~ 4) - 1 . 2 6 2 ( 1 0 ~ 5) - 1 . 9 7 ( 1 0 ~ 7 ) - 3 . 0 8 ( 1 0 ~ 9) - 4 . 8 ( 1 0 - 1 1 ) - 7 ( 1 0 - 1 3 ,

2.0 - 5 . 0 4 9 8 8 ( 1 0 -2 ) -8 .0542(10~ 4) -1 .217(10~ 5 ) - 1 . 6 4 ( 1 0 - 7 ) -1 .33(10~ 9) 2.8(10 -1 1 ) 2 ( 1 0 ~ 1 2 )

1.5 -5 .02447(10~ 2) - 6 . 9 9 2 0 ( 1 0 ~ 4) - 1 . 3 7 4 ( 1 0 -6 ) 8.34(10~ 7 ) 5.23(10~ 7 ) 2.4(10 -9 ) 8 ( 1 0 - 1 1 )

1.3 - 4 . 9 0 7 9 2 ( 1 0 ~ 2) -2 .1203(10~ 4) 6,354(10~ 5 ) 5.42(10~ 6 ) 2 . 9 9 ( 1 0 ~ 7) 1.3(10~ 8 ) 4 ( 1 0 - 1 0 )

1.2 - 4 .6 9 3 9 8 ( 1 0 - 2 ) 6.7812(10~ 4 ) 1.749(10~ 4 ) 1.31(10~ 5 ) 6.27(10~ 7 ) 1.7(10~ 8 ) - K 1 0 - 1 0 )

1.175 -4.60073 (1 0 - 2 ) 1.0619(10~ 3 ) 2.205(10~ 4) +1 .56(10~ 5) 6.32(10~ 7 ) 9.8(10~ 10 ) - 2 ( 1 0 ~ 9 )

1.15 - 4 . 4 8 3 4 5 ( 1 0 -2 ) 1.5384 (10~ 3) 2 . 7 3 6 ( 1 0 -4 ) 1.76(10~ 5 ) 4.21(10~ 7 ) -5 (10~ 8) - 7 ( 1 0 " 9 )

1 . 1 0 - 4 . 1 5 6 9 4 ( 1 0 -2 ) 2 .8029(10~ 3) 3 .782(10~ 4) 1.08(10~ 5 ) - 2 . 4 0 ( 1 0 -6 ) -5 (10~ 7) - 5 ( 1 0 " 9 )

1 .05 -3 .68059(10" 2) 4 .3358(10~ 3) 3 . 1 6 9 ( 1 0 ~ 4) 5.67(10~ 5 ) - 1 . 5 1 ( 1 0 ~ 5) - 1 . 7 ( 1 0 - 6 ) - 9 ( 1 0 " 8 )

TABLE 2 .2 . Va lue s o f 6 f o r a squa re coo l i n g a r r a y o f cy l i n d e r s ( c a se 2 . 2 . 19 , so u r c e : R e f . 64 , Tab l e 1 )

3 / r o 6 1 S 2 6 3 6 4 5 5 6 6 6 7

4.0 -1 .25382 (1 0 _ 1 ) - 1 . 0 5 8 3 ( 1 0 -2 ) -6 .120(10~ 4) - 3 . 8 9 8 ( 1 0 ~ 5) - 2 . 4 2 ( 1 0 ~ 6) - 1 . 5 ( 1 0 ~ 7 ) -8 do" 9 )2.0 -1 .25098 ( 10 - 1 ) -1 .0428(10" 2) - 5 . 7 1 3 ( 1 0 -4 ) - 3 . 1 7 7 ( 1 0 -5 ) - 1 . 3 9 ( 1 0 ~ 6) -2 (10~ 8) 3 (10~ 9)

1.5 -1 .22597 ( 10 _ 1 ) -9 .060 (10~ 3) -2.08 (10~ 4) 3.33 (10~ 5) 8.1 (10~ 6) 1 d O " 6 ) 1 ( 1 0 - 7 )

1.3 -1 .17022 (1 0 _ 1 ) -5 .995 (10~ 3) 6.13 (10 ~ 4) 1.83 (10~ 4) 3.1 (10~ 5) 4 (10~ 6) 4 ( 1 0 - 7 )

1.2 -1 .10421 (1 0 - 1 ) -2.387 (10~ 3) 1.566(10~ 3 ) 3.50 (10~ 4) 5.3 (10~ 5) 6 ( 1 0 - 6 ) 5 (10" 7)

1.15 -1.05352 ( 10 - 1 ) 3.172 (10 - 4 ) 2 .233(10~ 3) 4.44 (10~ 4) 5.8 (10 ~ 5) 4.3(10" 6 ) 1 ( 1 0 _8 >

1.1 -9.8721 (10~ 2) 3 . 6 5 9 6 ( 1 0 -3 ) 2 . 9 1 K 1 0 " 3 ) 4.655(10~ 4 ) 2.44(10~ 5) - 1 . K 1 0 " 5 ) -5. ( 1 0 - 6 )

1.05 -9.0358 (10~ 2) 7.310 (10~ 3) 3.210U0 - 3 ) 2.06 (10~ 4) -1 .20(10~ 4) - 5 . B ( 1 0 ~ 5) -1. ( 1 0 " 5 )

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TABLE 8.1. Values of (t - t,)/(t - t.) for an infinitely wide plate1 s 1

whose surface temperature increases proportionally to t i m e , t

(case 8. 1. 8, sour ce : Ref. 1, p. 268, Table 1 3 . 5 ) .

t . + bxi

Fo

X 0.08 0.16 0.20 0.32 0.40 0.80 1.60 2.00 4.00

0 0.004 0.045 0.074 0.170 0.231 0.464 0.694 0.752 0.875

0.33 0.030 0.104 0.142 0.245 0.305 0.522 0.728 0.779 0.889

0.5 0.082 0.194 0.238 0.346 0.402 0.594 0.770 0.814 0.906

0.66 0.214 0.353 0.398 0.498 0.545 0.698 0.830 0.862 0.930

0.8 0.418 0.548 0.586 0.664 0.699 0.802 0.889 0.911 0.955r

1 1 1 1 1 1 1 1 •1

TABLE 8.2. Values of the function X, (case 8.1 .12, so urce : Ref. 1,

p . 299).

X

a 0 1/8 2/8 3/8 4/8 5/8 6/8 7/8 1

0 1 1 1 1 1 1 1 1 1

0.5 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.99 1

1.0 0.77 0.77 0.77 0.78 0.79 0.81 0.85 0.91 1

1.5 0.47 0.47 0.47 0.48 0.52 0.58 0.68 0.83 1

2.0 0.27 0.27 0.28 0.30 0.36 0.45 0.58 0.77 1

4.0 0.04 0.04 o.or. 0.08 0.13 0.22 0.37 0.64 1

8.0 0.00 0.00 0.01 0.01 0.02 0.05 0.14 0.36 1

00 0 0 0 0 0 0 0 0

F-108

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TABLE 9 . 1 . Values of ( t - t . ) / ( t - t . )1 s 1

f o r a n i n f i n i t e l y l on g c y l i n d e r whose

s u r f a c e i n c r e a s e s p r o p o r t i o n a l l y t o t i m e

( t - t . = cT) (case 9 .1 .2 , s ou rc e :Ref . 1 , p . 269 ) .

R F o

0 . 0 8 0 . 1 6 0 . 3 2 0 . 8 0

0 0 . 0 1 6 0 . 1 2 3 0.354 0 . 6 9 1

0 . 3 3 0.054 0 . 1 9 1 0.420 0.725

0 . 5 0 . 1 2 2 0.287 0.505 0.768

0 . 6 6 0. ' , 68 0.443 0. 628 0.828

0 . 8 0.470 0 . 6 2 1 0.755 0.8881 1 1 1 1

TABLE 1 0 . 1 . Tem peratu res for a sp he re whose

s u r f a c e t e m p e r a t u r e i n c r e a s e s p r o p o r t i o n a l l y t ot ime ( tQ = t . + b t ) ( ca se 1 0 .1 .4 , sou rce :

Ref. 1 , p . 269 , Table 13 -7 ) .

R P o

0 . 0 1 6 0 . 0 8 0 . 1 6 0 . 3 2 0 . 8 0

0 0 . 0 0 0.054 0 . 2 1 9 0.506 0.792

0 . 3 3 0 . 0 0 0.090 0.290 0.560 0.725

0 . 5 0 . 0 0 0 . 1 6 2 0.385 0.626 0.844

0 . 6 6 0 . 0 2 0 . 3 1 2 0.529 0.722 0.884

0 . 8 0 . 1 4 0 . 5 1 6 0.686 0 . 8 1 9 0.925

1 1 1 1 1 1

F-109

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235n

ato

REPRODUCED FROMBEST AVAILABLE COPY

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SECTION 13. MATHEMATICAL FUNCTIONS

1 3 . 1 . The Error Function and Related Functions.

2 f 2erf(x) = y== I exp(-z ) dz

H w n-0 (2n+l)nl

- 1 - exp£x )

VX

2 x3

erf(0) = 0 , erf(~) = 1

er f (-x) • - erf(x)

erfc(x) • 1 - er f (x)

i n erfc(x) = r • _ 1 e r f c ( z ) d

2n+l, for small values of x

i erfc(x) « erfc(x)

1 2ierfc(x) » -& exp(-x ) - x erfc(x)

i 2erfc{x) = j [erfc(x) - 2x ierfc(x)]

2n in

erfc(x) » in _ 2

erfc(x) - 2x in _ 1

erfc(x)

l ^ e r f t x ) - ^ e x p ( - x 2 )

d 2 .. . 4x , 2 .— _ erf (x) - - T E exp(-x )d x z ^

forlarge

* valuesof x

13-1

8/10/2019 6224569


Z-ZT

•L zz •; •;•:•: z z s E •; •; s ; s ; : z z z z?

s s g s c s g s g s •;: 11z

i l i i l l l i i l i i l i i i l i i l i l l i l l l i l l l l i l lo o o o o o O a a o O a a o o o

o © o oS S 3 , emu V-i o O W

o u o» w IIM> IB V) 09

f . 3 M a 2en ui A. u M

hi a at o en

O O O O O O O O O O O O O O O H

I 1 1 iO O O

o o a o M M Ul Jh *• in Ut 91 91

U» P P *• J IS -J si W ~J N O IO K) W en m - J to - j u»' - • • oi o u> o ui m ui ui o

o o o o o o c - o o O O O O O Q O O

10 <A o « _C U3 10 10 O lO 10

10 ID U? U3 10 U) 40 10 U) to 10-- IS >0 10 10 CO C3 - - - - - - - » 5 » R

j o> oi wi m» O O* WU N M

0 10 10 " N » « « ? > F ' t = ° R ° S o i 3 K - « 5f o w w a j i o f c o m o f ? ? '

W O J-" O Ul CD *O Ul O N Ul h) Q CO 10H « r > * - * U J W i W O N > K l M S 5 S S

o o o o o o o o o o a a o O O O O O O O ) - '

i i S So o o o a H> H M w m - j o w ui CD w

IS Ol M ID c n u u i s j u i i o i a - j w u i o« a ?• s «q - j g o w y . » > i y > = » * o » = S * »

10 UJ C f v u i U l < * e n W C " > M

Ul U 10 03 U l U l M C F i U I I A t O M

O O O O O O O O O O O O O O O O O O O O *-• -•

g g g g g g g p o S H " H at- >o en in M ui

o o c a o o o o o

h - M M M W U l U l W *_ t n C D O * - J t - ' U I U ) *

«J » CO K> U) O * H O* 10 V>

C\ CT1 *-l 00 "O

U -J K> M IO

O O O O

u n am -a ui w ui

© o o o o

l l i l l l

o o o o

o o a a o o ow u m en* B B . i E S m W O N *

G O O D O O © O O O O O O

I I I I I I I I I Io o o o o o a o

o o o r~ o o o o o a o o o o o o a o oo o o o o o—

—

o o o oo o o o o o

o o o o o i- 1 \0 o\ * • pCk - J

o o a o o O P O O O O o o D o a o

o o o o o O O O O O O O h-n) o ro uien ui CD en

o o o o o o o o

a o o o o

o o o o oD O O O O O o o o o

o a o o o oO O Q O O D O O O Oo o o **m en m o

w w w

H" Ul W

»

8/10/2019 6224569


oo

1

t13-3

13.2. Exponenti al and Hypet bolic Functions.

e = 2.71828 . . .u

exp(u) = e

u -usinh (u) = - — i ~ ^ — , sinh (0) = 0 , sin h (=>) =2

ue

+ - u6

2

2e

u - 1tanh (u) = — , tanh (0) = 0 , tanh («)

2u ,e + 1

8/10/2019 6224569


TABLE 13 . 2. Exponential funct ions (source: Ref. 2, pp. 3 6 7 - 9 ) .

0.00 1.000 1.000

0.02 1.020 0.980

0.04 1.041 0.9610.06 1.062 0.942

0.08 1.083 0.923

0.10 1.105 0.905

0.12 1.128 0.887

0.14 1.150 0.869

0.16 1.174 0.852

0.18 1.197 0.835

0.20 1.221 0.819

0.22 1.246 0.802

0.24 1.271 0.787

0.26 1.297 0.771

0.28 1.323 0.756

0.30 1.350 0.741

0.32 1.377 0.726

0.34 1.405 0.712

0.36 1.433 0.698

0.38 1.462 0.684

0.40 1.492 0.670

0.42 1.522 0.657

0.44 1.553 0.644

0.46 1.584 0.6310.48 1.616 0.619

0.50 1.649 0.606

0.52 1.682 0.594

0.54 1.716 0.583

0.56 1.751 0.571

0.58 1.786 0.560

e " u

0.60 1.822 0.549

0.62 1.859 0.538

0.64 1.896 0.5270.66 1.935 0.517

0.68 1.974 0.507

0.70 2.014 0.497

0.72 2.054 0.487

0.74 2.096 0.477

0.76 2.138 0.468

0.78 2.182 0.458

0.80 2.226 0.449

0.82 2.270 0.440

0.84 2.316 0.432

0.86 2.363 0.423

0.88 2.411 0.415

0.90 2.460 0.407

0.92 2.509 0.3980.94 2.560 0.391

0.96 2.612 0.383

0.98 2.664 0.375

1.00 2.718 0.368

1.02 2.773 0.361

1.04 2.829 9.353

1.06 2.886 0.3461.08 2.945 0.340

1.10 3.004 0.333

1.12 3.065 0.326

1.14 3.127 0.320

1.16 3.190 0.313

1.18 3.254 0.307

e u

1.20 3.320 0.301

1.22 3.387 0.295

1.24 3.456 0.2891.26 3.525 0.284

1.28 3.597 0.278

1.30 3.669 0.272

1.32 3.743 0.267

1.34 3.819 0.262

1.36 3.896 0.257

1.38 3.975 0.252

1.40 4.055 0.247

1.42 4.137 0.242

1.44 4.221 0.237

1.46 4.306 2.232

1.48 4.393 0.228

1.50 4.482 0.223

1.52 4.572 0.2191.54 4.665 0.214

1.56 4.759 0.210

1.58 4.855 0.206

1.60 4.953 0.202

1.62 5.053 0.198

1.64 5.155 0.194

1.66 5.259 0.1901.68 5.366 0.186

1.70 5.474 0.183

1.72 5.584 0.179

1.74 5.697 0.176

1.76 5.812 0.172

1.78 5.930 0.169

13-4

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TABLE 13.2. ( C o n t i n u e d ) .

u e u e" u u e u e" u u e u

e" u

1.80 6.050 0.165 2.40 11 . 0 2 3 0.091 3.00 2 0 . 0 8 0.050

1.82 6.172 0.162 2.42 11 . 2 4 6 0.089 3.10 2 2 . 2 0 0.045

1.84 6.296 0.159 2.44 1 1 . 4 7 3 0.087 3.20 2 4 . 5 3 0.041

1.86 6.424 0.156 2.46 11 . 7 0 5 0.085 3.30 2 7 . 11 0.037

1.88 6.554 0.153 2.48 1 1 . 9 4 1 0.084 3.40 2 9 . 9 6 0.033

1.90 6.686 0.150 2.50 1 2 . 1 8 0.082 3.50 3 3 . 11 0.030

1.92 6.821 0.147 2.52 1 2 . 4 3 0.080 3.60 3 6 . 6 0 0.027

1.94 6.959 0.144 2.54 1 2 . 6 8 0.079 3.70 4 0 . 4 5 0.025

1.96 7.U?9 0.141 2.56 1 2 . 9 4 0.077 3.80 4 4 . 7 0 0.022

1.98 7.243 0. 138 2.58 1 3 . 2 0 0.076 3.90 4 9 . 4 0 0.020

2.00 7.389 0.135 2.60 1 3 . 4 6 0.074 4.00 5 4 . 6 0 0.018

2.02 7.538 0.133 2.62 1 3 . 7 4 0.073 4.20 6 6 . 6 9 0.015

2.04 7.691 0.130 2.64 1 4 . 0 1 0.071 4.40 8 1 . 4 5 0.012

2.06 7.846 0.127 2.66 1 4 . 3 0 0.070 4.60 9 9 . 4 8 0.010

2.08 8.004 0.125 2.68 1 4 . 5 8 0.069 4.80 1 2 1 . 5 1 0.008

2,10 8.166 0.122 2.70 1 4 . 8 8 0.067 5.00 1 4 8 . 4 0.007

2.12 8.331 0.120 2.72 1 5 . 1 8 0.066 5.20 1 8 1 . 3 0.006

2.14 8.499 0.118 2.74 1 5 . 4 9 0.065 5.40 2 2 1 . 4 0.004

2.16 8.671 0.115 2.76 1 5 . 8 0 0.063 5.60 2 7 0 . 4 0.004

2.18 8.846 0.113 2.78 1 6 . 1 2 0.062 5.80 3 3 0 . 3 0.003

2.20 9.025 0.111 2.80 1 6 . 4 4 0.061 6.00 4 0 3 . 4 0.002

2.22 9.207 0.109 2.82 1 6 . 7 8 0.060 6.50 6 6 5 . 1 0.002

2.24 9.393 0.106 2.84 1 7 . 1 2 0.058 7.00 1 0 9 6 . 6 0.001

2.26 9.583 0.104 2.86 1 7 . 4 6 0.057 7.50 1 8 0 8 . 0 0.001

2.28 9.777 0.102 2.88 1 7 . 8 1 0.056 8.00 2 9 8 1 . 0 0.000

2.30 9.974 0.100 2.90 1 8 . 1 7 0.055 8.50 4 9 1 4 . 8 0.000

2.32 1 0 . 1 7 6 0.098 2.92 1 8 . 5 4 0.054 9.00 8 1 0 3 . 1 0.000

2.34 1 0 . 3 8 1 0.096 2.94 1 8 . 9 2 0.053 9.50 1 3 3 6 0 0.000

2.36 1 0 . 5 9 1 0.094 2.96 1 9 . 3 0 0.052 1 0 . 0 0 2 2 0 2 6 0.000

2.38 1 0 . 8 0 5 0.093 2.98 1 9 . 6 9 0.051

1 3 - 5

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13.3. The Gamma Funct ion.

co

I ' M = / u*"" 1 e " u du, x > 0

Hx) = x x e - x ^ | L12x „„„ 2

139 571

288x" 5184x 3 2 4 8 8 3 2 0 X4 + ...

T(x + 1) = r<x)for large positive values of x

TABI£ 13.3. The gamma function (source: Ref. 17 , p. 136)

X T(x) X T(x> X T(x) X T(x) X T(x)

1.01 0.99 433 1.21 0.91 558 1.41 0.88 676 1.61 0.89 468 1.81 0.93 408

1.02 0.98 884 1.22 0.91 311 1.42 0.88 636 1.62 0.89 592 1.82 0.93 685

1.03 0.98 355 1.23 0.91 075 1.43 0.88 604 1.63 0.89 724 1.83 0.93 969

1.04 0.97 844 1.24 0.90 852 1.44 0.88 581 1.64 0.89 864 1.84 0.94 261

1.05 0.97 350 1.25 0.90 640 1.45 0.88 566 1.65 0.90 012 1.85 0.94 561

1.06 0.97 874 1.26 0.90 440 1.46 0.88 560 1.66 0.90 167 1.86 0.94 869

1.07 0.96 415 1.27 0.90 250 1.47 0.88 563 1.67 0.90 330 1.87 0.95 184

1.08 0.95 973 1.28 0.90 072 1.48 0.88 575 1.68 0.90 500 1.88 0.95 507

1.09 0.95 546 1.29 0.89 904 1.49 0.88 595 1.69 0.90 678 1.89 0.95 838

1.10 0.95 135 1.30 0.89 747 1.50 0.88 623 1.70 0.90 864 1.90 0.96 177

1.11 0.94 740 1.31 0.89 600 1.51 0.88 659 1.71 0.91 057 1.91 0.96 523

1.12 0.94 359 1.32 0.89 464 1.52 0.88 704 1.72 0.91 258 1.92 0.96 877

1.13 0.93 993 1.33 0.89 338 1.53 0.88 757 1.73 0.91 467 1.93 0.97 240

1.14 0.93 642 1.34 0.89 222 1.54 0.88 818 1.74 0.91 683 1.94 0.97 610

1.15 0.93 304 1.35 0.89 115 1.55 0.88 887 1.75 0.91 906 1.95 0.97 988

1.16 0.92 980 1.36 0.89 018 1.56 0.88 964 1.76 0.92 137 1.96 0.98 374

1.17 0.92 670 1.37 0.88 931 1.57 0.89 049 1.77 0.92 376 1.97 0.98 768

1.18 0.92 373 1.38 0.88 854 1.58 0.89 142 1.78 0.92 623 1.98 0.99 171

1.19 0.92 089 1.39 0.88 785 1.59 0.89 243 1.79 0.92 877 1.99 0.99 581

1.20 0.91 817 1.40 0.88 726 1.60 0.89 352 1.80 0.93 138 2.00 1.00 000

13-6

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th e p o s i t i v e s i gn used i f -n /2 < a rg u < 31T/2 and the n eg a t iv e s i gn used i f

-3TT/2 < ar g u < n / 2 .

I (u) - I (u)_ , . _ 7J_ —n n

nw

" 2 s i n (mr)

For n any positive integer:

n+2jKn( u , = < - i >

n + 1

| m ( u /2 ) + Y | in , „ , + ^- I iag__j=0

n+32 m 1 + S ra_1

.m=l m=l j=0

V - 1where f o r ]= 0r y m =

m=l

For large va lues of u :

v«->/H»^»'^ffi-••'*••»•'»4 - J (u) = J . (u) - s J (u) = - J (u ) - J _,, (u)du n n-1 u n u n n+1

| - y (U ) = y (U ) - £ Y (u) = T; Y (u) - Y «u)au n n— 1 u n u n n+1

4 - I (u) = I . (u) - - I (u) = I _,, (u) + - I (u)du n ' ' n - lx ' u nv ' n+1 u n '

jr K (u) = -K (u) - £ K (u) = £ K (u) - K (u)au n n-1 u n u n n+1

13-7

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1 3 . 4 . B e s s el F u n c t i o n s .

3 , „ / ^ + 2 ja ( u ) V ( - l ) J ( u / 2 ) " , n iJ

n

( u ) L j i r { n + j + i)i s r e a l and u may be comp lex

j=0

J (u) = (-1) J (u ) , i f n i s an i n t eg ern -n

J_ (u) co s (nir) - J__(u)V (u) = — : —

n s i n (nit)

J ,„/•,* n + 2 3TIY

B ( u ) = 2 ( i n ( u /2 ) + Y | J n ( u , - I ^ i ^ f ^ '

j=0

j+ n j n-1

^ m "1 + m "1 - S ( u / 2 ) " n + 2 j (n -_ J - 1)1

_m=l m=l J j=0

w he re y = 0 . 5 7 7 2 . . . i s E u l e r ' s c o n s t a n t , a n d n i s an y p o s i t i v e i n t e g e r

j YQ(u) = {In (u/2 ) + y}3Q{M ) + ( u / 2 )2 - ( l + j •&Z1L(21)

• ••WW--V d i ' ( 3 1 T

Vu )

Z , j i r t n + j +J-0

For la rge va lues o£ u :

I

n

( u ) " ^ r o r a

^.i* ^ ))

± = = e x p [ - u ± (n + | ) * i ] | l + 0 ( u ' 1 ) ) ,V2TTU

13-8

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TABLE 13.4.1. Zero and first-order Bessel functions of the first kind

(source: Ref 2, pp. 369-71) .

u J 0 (u) Jj_(U) u J 0 (u) •^(u)

0.0 1.0000 0.0000 3.0 -0.2600 0.3391

0.1 0.9975 0.0499 3.1 -0.2921 0.3009

0.2 0.9900 0.0995 3.2 -0.3202 0.2613

0.3 0.9776 0.1483 3.3 -0.3443 0.2207

0.4 0.9604 0.1960 3.4 -0.3643 0.1792

0.5 0.9385 0.2423 3.5 -0.3801 0.1374

0.6 0.9120 0.2867 3.6 -0.3918 0.0955

0.7 0.8812 0.3290 3.7 -0.3992 0.0538

0.8 0.8463 0.3688 3.8 -0.4026 0.0128

0.9 0.8075 0.4059 3.9 -0.4018 -0.0272

1.0 0.7652 0.4400 4.0 -0.3971 -0.0660

1.1 0.7196 0.4709 4.1 -0.3887 -0.1033

1.2 0.6711 0.4983 4.2 -0.3766 -0.1385

1.3 0.6201 0.5220 4.3 -0.3610 -0.1719

1.4 0.56690.5419 4.4 -0.3423 -0.2028

1.5 0.5118 0.5579 4.5 -0.3205 -0.2311

1.6 0.4554 0.5699 4.6 -0.2961 -0.2566

1.7 0.3980 0.5778 4.7 -0.2693 -0.2791

1.8 0.3400 0.5815 4.8 -0.2404 -0.2985

1.9 0.2818 0.5812 4.9 -0.2097 -0.3147

2.0 0.2239 0,5767 5.0 -0.1776 -0.3276

2.1 0.1666 0.5683 5.1 -0.1443 -0.33712.2 0.1104 0.5560 5.2 -0.1103 -0.3432

2.3 0.0555 0.5399 5.3 -0.0758 -0.3460

2.4 0.0025 0.5202 5.4 -0.0412 -0.3453

2.5 -0.0484 0.4971 5.5 -0.0068 -0.3414

2.6 -0.0968 0.4708 5.6 0.0270 -0.3343

2.7 -0.1424 0.4415 5.7 0.0599 -0.3241

2.8 -0.1850 0.4097 5.8 0.0917 -0.31102.9 -0.2243 0.3754 5.9 0.1220 -0.2951

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TABLE 1 3 . 4 . 1 . (Continued.)

u J 0 (u) Jû) u J 0 (u) ^ ( u )

6.0 0.1506 -0.2767 9.0 -0.0903 0.2453

6.1 0.1773 -0.2559 9.1 -0.1142 0.2324

6.2 0.2017 -0.2329 9.2 -0.1368 0.2174

6.3 0.2238 -0.2081 9.3 -0.1577 0.2004

6.4 0.2433 -0.1816 9.4 -0.1768 0.1816

6.5 0.2601 -0.1538 9.5 -0.1939 0.1613

6.6 0.2740 -0.1250 9.6 -0.2090 0.1395

6.7 0.2851 -0.0953 9.7 -0.2218 0.1166

6.8 0.2931 -0.0652 9.8 -0.2323 0.0928

6.9 0.2981 -0.0349 9.9 -0.2403 0.0634

7.0 0.3001 -0.0047 10.0 -0.2459 0.0435

7.1 0.2991 0.0252 10.1 -0.2490 0.0184

7.2 0.2951 0.0543 10.2 -0.2496 -0.0066

7.3 0.2882 0.0826 10.3 -0.2477 -0.0313

7.4 0.2786 0.1096 10.4 -0.2434 -0.0555

7.5 0.2663 0.1352 10.5 -0.2366 -0.0788

7.6 0.2516 0.1592 10.6 -0.2276 -0.1012

7.7 0.2346 0.1813 10.7 -0.2164 -0.1224

7.8 0.2154 0.2014 10.8 -0.2032 -0.1422

7.9 0.1944 0.2192 10.9 -0.1881 -0.1604

8.0 0.1716 0.2346 11.0 -0.1712 -0.1768

8.1 0.1475 0.2476 11.1 -0.1528 -0.1913

8.2 0.1222 0.2580 11.2 -0.1330 -0.2028

8.3 0.0960 0.2657 11.3 -0.1121 -0.2143

8.4 0.0692 0.2708 11.4 -0.0902 -0.2224

8.5 0.0419 0.2731 11.5 -0.0677 -0.2284

8.6 0.0146 0.2728 11.6 -0.0446 -0.2320

8.7 -0.0125 0.2697 11.7 -0.0213 -0.2333

8.8 -0.0392 0.2641 11.8 0.0020 -0.2323

8.9 -0.0652 0.2559 11.9 0.0250 -0.2290

13-10

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u J 0 (u) Jû) u J 0 (u) J x (u)

12.0 0.0477 -0.2234 13.6 0.2101 0.0590

12.1 0.0697 -0.2158 13.7 0.2032 0.0791

12.2 0.0908 -0.2060 13.8 0.1943 0.0984

12.3 0.1108 -0.1943 13.9 0.1836 0.1165

12.4 0.1296 -0.1807 14.0 0.1711 0.1334

12.5 0.1469 -0.1655 14.1 0.1570 0.1488

12.6 0.1626 -0.1487 14.2 0.1414 0.1626

12.7 0.1766 -0.1307 14.3 0.1245 0.1747

12.8 0.1887 -0.1114 14.4 0.1065 0.1850

12.9 0.1988 -0.0912 14.5 0.0875 0.1934

13.0 0.2069 -0.0703 14.6 0.0679 0.1998

13.1 0.2129 -0.0488 14.7 0.0476 0.2043

13.2 0.2167 -0.0271 14.8 0.0271 0.2066

13.3 0.2183 -0.0052 14.9 0.0064 0.2069

13.4 0.2177 0.0166 15.0 0.0142 0.2051

13.5 0.2150 0.0380

13-11

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TABLE 13.4.2. Zero and first-order Bessel functions of the second kind (source:

Ref. 2 p . 371-3).

u Y 0 (u> Yx (u) u V u ) ^ ( u )

0.0 -co -CO 3.0 0.3768 0.3247

0.1 -1.5342 -6.4590 3.1 0.3431 0.3496

0.2 -1.0811 -3.3238 3.2 0.3070 0.3707

0.3 0.8073 -2.2931 3.3 0.2691 0.3878

0.4 -0.6060 -1.7809 3.4 0.2296 0.4010

0.5 -0.4445 -1.4715 3.5 0.1890 0.4102

0.6 -0.3085 -1.2604 3.6 0.1477 0.4154

0.7 -0.1907 -0.1032 3.7 0.1061 0.4167

0.8 -0.0868 -0.9781 3.8 0.0645 0.4141

0.9 0.0056 -0.8731 3.9 0.0234 0.4078

1.0 0.0883 -0.7812 4.0 -0.0169 0.3979

1,1 0.1622 -0.6981 4.1 -0.0561 0.3846

1.2 0.2281 -0.6211 4.2 -0.0938 0.3680

1.3 0.2865 -0.5485 4.3 -0.1296 0.3484

1.4 0.3379 -0.4791 4.4 -0.1633 0.3260

1.5 0.3824 -0.4123 4.5 -0.1947 0.3010

1.6 0.4204 -0.3476 4.6 -0.2235 0.2737

1.7 0.4520 -0.2847 4.7 -0.2494 0.2445

1.8 0.4774 -0.2237 4.8 -0.2723 0.2136

1.9 0.4968 -0.1644 4.9 -0.2921 0.1812

2.0 0.5104 -0.1070 5.0 -0.3085 0.1479

2.1 0.5183 -0.0517 5.1 -0.3216 0.1137

2.2 0.5208 0.0015 5.2 -0.3312 0.0792

2.3 0.5181 0.0523 5.3 -0.3374 0.0445

2.4 0.5104 0.1005 5.4 -0.3402 0.0101

2.5 0.4981 0.1459 5.5 -0.3395 -0.0238

2.6 0.4813 0.1884 5.6 -0.3354 -0.0568

2.7 0.4605 0.2276 5.7 -0.3282 -0.0887

2.8 0.4359 0.2635 5.8 -0.3177 -0.1192

2.9 0.4079 0.2959 5.9 -0.3044 -0.1481

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u VQ( u ) V x ( u ) u Y 0 ( u ) ^ ( u )

6.0 -0.2882 -0.1750 9.0 0.2499 0.10436.1 -0.2694 -0.1998 9.1 0.2383 0.1275

6.2 -0.2483 -0.2223 9.2 0.2245 0.1491

6.3 -0.2251 -0.2422 9.3 0.2086 0.1691

6.4 -0.2000 -0.2596 9.4 0.1907 0.1871

6.5 -0.1732 -0.2741 9.5 0.1712 0.2032

6.6 -0.1452 -0.2857 9.6 0.1502 0.2171

6.7 -0.1162 -0.29459.7 0.1279 0.2287

6.8 -0.0864 -0.3002 9.8 0.1045 0.2379

6.9 -0.0562 -0.3029 9.9 0.0804 0.2447

7.0 -0.0260 -0.3027 10.0 0.0557 0.2490

7.1 0.0042 -0.2995 10.1 0.0307 0.2508

7.2 0.0338 -0.2934 10.2 0.0056 0.2502

7.3 0.0628 -0.2846 10.3 -0.0193 0.2471

7.4 0.0907 -0.2731 10.4 -0.0437 0.2416

7.5 0.1173 -0.2591 10.5 -0.0675 0.2337

7.6 0.1424 -0.2428 10.6 -0.0904 0.2236

7.7 0.1658 -0.2243 10.7 -0.1122 0.2114

7.8 0.1872 -0.2039 10.8 -0.1326 0.1973

7.9 0.2065 -0.1817 10.9 -0.1516 0.1813

8.0 0.2235 -0.1581 11.0 -0.1688 0.1637

8.1 0.2381 -0.1332 11.1 -0.1843 0.14468.2 0.2501 -0.1072 11.2 -0.1977 0.1243

8.3 0.2595 -0.0806 11.3 -0.2091 0.1029

8.4 0.2662 -0.0535 11.4 -0.2183 0.0807

8.5 0.2702 -0.0262 11.5 -0.2252 0.0579

8.6 0.2715 0.0011 11.6 -0.2299 0.0348

8.7 0.2700 0.0280 11.7 -0.2322 0.0114

8.8 0.26590.0544

11.8 -0.2322 -0.01188.9 0.2592 0.0799 11.9 -0.2298 -0.0347

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TABLE 13.4.2. (Continued.)

_u * 0 ( u ) Y x ( u )

12.0 -0.2252 -0.0571

12.1 -0.2184 -0.0787

12.2 -0.2095 -0.0994

12.3 -0.1986 -0.1190

12.4 -0.1858 -0.1371

12.5 -0.1712 -0.1538

12.6 -0.1551 -0.1689

12.7 -0.1375 -0.1821

12.8 -0.1187 -0.1935

12.9 -0.0989 -0.2028

13.0 -0.0782 -0.2101

13.1 -0.0569 -0.2152

13.2 -0.0352 -0.2182

13.3 -0.0134 -0.2190

13. 4 0.0085 -0.2176

u Y 0 ( u ) Y x ( u )

13.5 0.0301 -0.2140

13.6 0.0512 -0.2084

13.7 0.0717 -0.2007

13.8 0.0913 -0.1912

13.9 0.1099 -0.1798

14.0 0.1272 -0.1666

14.1 0.1431 -0.1520

14.2 0.1575 -0.1359

14.3 0.1703 -0.1186

14.4 0.1812 -0.1003

14.5 0.1903 -0.0810

14.6 0.1974 -0.0612

14.7 0.2025 -0.0408

14.8 0.2056 -0.0202

14.9 0.2066 0.0005

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TABLE 1 3 . 4 . 3 . Zero and f i r s t order modified Bes sel func tion s of the f i r s t kind

( s o u r c e : Ref. 2, p . 373-4) .

u I 0 (u ) I ^ u ) u I 0 (u ) 3^(11)

0.0 l.ÔOO 0.0000 3.0 4.881 3.953

0.1 1.0025 0.0501 3.1 5.294 4.326

0.2 1.0100 0.1005 3.2 5.747 4.734

0.3 1.0226 0.1517 3.3 6.243 5.181

0.4 1.0404 0.2040 3.4 6.785 5.670

0.5 1.0635 0.2579 3.5 7.378 6.206

0.6 1.0920 0.3137 3.6 8.028 6.793

0.7 1.1263 0.3719 3.7 8.739 7.436

0.8 1.1665 0.4329 3.8 9.517 8.140

0.9 1.2130 0.4971 3.9 10.369 8.913

1.0 1.2661 0.5652 4.0 11.30 9.76

1.1 1.3262 0.6375 4.1 12.32 10.69

1.2 1.3937 0.7147 4.2 13.44 11.71

1.3 1.4693 0.7973 4.3 14.67 12.82

1.4 1.5534 0.8861 4.4 16.01 14.05

1.5 1.6467 0.9817 4.5 17.48 15.39

1.6 1.7500 1.0848 4.6 19.09 16.86

1.7 0.8640 1.1963 4.7 20.86 18 .48

1.8 1.9806 1.3172 4.8 22.79 20.25

1.9 2.1277 1.4482 4.9 24.91 22.20

2.0 2.280 1.591 5.0 27.24 24.34

2.1 2.446 1.746 5. 1 29.79 26. 68

2.2 2.629 1.914 5.2 32.58 29.25

2.3 2.830 2.098 5.3 35.65 32.08

2.4 3.049 2.298 5.4 39.01 35.18

2.5 3.290 2.517 5.5 42.70 38.59

2.6 3.553 2.755 5.6 46.74 42.33

2.7 3.842 3.016 5.7 51.17 46.44

2. 8 4.157 3. 30 1 5.8 56.04 50.95

2.9 4.503 3.613 5. 9 61.3 8 55. 90

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TABLE 1 3 . 4 . 4 . Zero and f i r s t order modified Besse l funct ions of th e second kind

( s o u r c e : Ref. 2, p. 3 7 4 - 5 ) .

u iV u > K( u

> u ¥oM K ( u )

0.0 CO CO 2.0 0.072 0.089

0.1 1.545 6.270 2.1 0.064 0.078

0.2 1.116 3.040 2. 2 0.057 0.069

0.3 0.874 1.946 2.3 0.050 0.060

0.4 0.710 1.391 2. 4 0.045 0.053

0.5 0.588 1.054 2.5 0.040 0.047

0.6 0.495 0.829 2. 6 0.035 0.042

0.7 0.420- 0.669 2.7 0.031 0.037

0.8 0.360 0.549 2.8 0.028 0.032

0.9 0.310 0.456 2. 9 0.025 0.029

1.0 0.268 0.383 3.0 0.022 0.026

1.1 0.233 0.324 3.1 0.020 0.023

1.2 0.203 0.277 3. 2 0.018 0.020

1.3 0.177 0.237 3. 3 0.016 0.018

1.4 0.155 0.204 3. 4 0.014 0.016

1.5 0.136 0.177 3. 5 0.012 0.014

1.6 0.120 0.153 3. 6 0.011 0.013

1.7 0.105 0.133 3.7 0.010 0.011

1.8 0.093 0.116 3.8 0.009 0.010

1.9 0.082 0.102 3. 9 0.008 0.009

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1 3 . 5 . Legendre Polynomials

The Legendre polynomial of degree n, of the first kind:

P ( u ) = ( - i )n / 2 l • ' — < n - X ) Ti - H U U - 1 L 2n* ' l ; 2 • 4 • 6 ••• n I 21

n(n - 2)(n + 1 } (n + 3) 4 , "l „ „ ,+ — i '•' ,. , ' ' - u + , , " , n = 2, 4, 6, ....

41 -1

n , , , t (n - l ) /2 1 » 3 • 5 • n r (n - 1) (n + 2) 3V U ) = » - X ) 2 • 4 • 6 — (n - 1) L u " 31 u

+ {n-l) {n-3nn + 2 ){n + A) 5 . . . . J

# n „ ^ ^ ? | u ] < 1

P Q( u ) = 1, P x (u) = u, P 2 (u ) = (3u 2 - l)/2 ,

P (u) = (5u 3 - 3u)/2 r P (u) = P 5 u 4 - 30u 2 + 3)/8 ,

P 5 / 2 ^ . v . 7 . i ? -n ^ - [ i - a f c j i- y - u2

+ n(n - 2) ( n ^ 1) ( , + 3) fl4 + .„.-] , n = 3 , 5 , 7 , . . . ,

O ( u , = ( _ i ) n / 2 ^ • i • 6 — a r _ <n - l)<n + 2) 3U n vuj \ i.) l • 3 • 5 • •• (n - 1) L u 31 u

a. (n - 1) (n - 3) (n + 2) (n + 4) 5 . 1 . , ,

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0 0 W ) - iQ ±W = QQ(u) Pû) - 1,

Q 2(u) = QQ(u) P 2 (u) - 3u/2

Q 3(u) » QQ(u) P 3 (u) - 5u2 /2 + §,

Q 4('J) = Q Q(u) P 4 (u) - 35u3/8 + 55 u2 /24 ,

Qn(u) = Q0(U) Pn(u , - - f - r - i l p n _ i ( u ) _ 1 2 2 ^ | L P n _ 3 ( u )

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TABLE 1J . 5. The f i r s t fiv e Legendre polynomials of the f i r s t kind ( sou rce :

Ref. 2, p. 3 7 5 - 7 ) .

u ^ ( u ) P 2 (u) P 3 (u) P 4 (u) P 5 (u)

0.00 0.0000 -0.5000 0.0000 0.3750 0.0000

0.01 0.0100 -0.4998 -0.0150 0.3746 0.0187

0.02 0.0200 -0.4994 -0.0300 0.3735 0,0374

0.03 0.0300 -0.4986 -0.0449 0.3716 0.0560

0.04 0.0400 -0.4976 -0.0598 0.3690 0.0744

0.05 0.0500 -0.4962 -0.0747 0.3657 0.0927

0.06 0.0600 -0.4946 -0.0895 0.3616 0.1106

0.07 0.0700 -0.4926 -0.1041 0.3567 0.1283

0.08 0.0800 -0.4904 -0.1187 0.3512 0.1455

0.09 0.0900 -0.4878 -0.1332 0.3449 0.1624

0.10 0.1000 -0.4850 -0.1475 0.3379 0.1788

0.11 0.1100 -0.4818 -0.1617 0.3303 0.1947

0.12 0.1200 -0.4784 -0.1757 0.3219 0.2101

0.13 0.1300 -0.4746 -0.1895 0.3129 0.2248

0.14 0.1400 -0.4906 -0.2031 0.3032 0.2389

0.15 0.1500 -0.4662 -0.2166 0.2928 0.2523

0.16 0.1600 -0.4616 -0.2:.98 0.2819 0.2650

0.17 0.1700 -0.4566 -0.2427 0.2703 0.2769

0.18 0.1800 -0.4514 -0.2554 0.2581 0.2880

0.19 0.1900 -0.4458 -0.2679 0.2453 0.2982

0.20 0.2000 -0.4400 -0.2800 0.2320 0.3075

0.21 0.2100 -0.4338 -0.2918 0.2181 0.3159

0.22 0.2300 -0.4274 -0.3034 0.2037 0.3234

0.23 0.2300 -0.4206 -0.3146 0.1889 0.3299

0.24 0.2400 -0.4136 -0.3254 0.1735 0.3353

0.25 0.2500 -0.4062 -0.3359 0.1577 0.3397

0.26 0.2600 -0.3986 -0.3461 0.1415 0.3431

0.27 0.2700 -0.3906 -0.3558 0.1249 0.3453

0.28 0.2800 -0.3824 -0.3651 0.1079 0.3465

0.29 0.2900 -0.3738 -0.3740 0.0906 0.3465

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TABLE 13.5. (Continued.)

u ^ ( u ) P 2 (u) P 3 (u) P 4 ( ) P 5 (u)

0.30 0.3000 -0.3650 -0.3825 0.0729 0.3454

0.31 0.3100 -0.3558 -0.3905 0.0550 0.3431

0.32 0.3200 -0.3464 -0.3981 0.0369 0.3397

0.33 0.3300 -0.3366 -0.4052 0.0185 0.3351

0.34 0.3400 -0.3266 -0.4117 0.0000 0.3294

0.35 0.3500 -0.3162 -0.4178 -0.0187 0.3225

0.36 0.3600 -0.3056 -0.4234 -0.0375 0.3144

0.37 0.3700 -0.2946 -0.4284 -0.0564 0.3051

0.38 0.3800 -0.2834 -0.4328 -0.075? 0.2948

0.39 0.3900 -0.2718 -0.4367 -0.0942 0.2833

0.40 0.4000 -0.2600 -0.4400 -0.1130 0.2706

0.41 0.4100 -0.2478 -0.4427 -0.1317 0.2569

0.42 0.4200 -0.2354 -0.4448 -0.1504 0.2421

0.43 0.4300 -0.2226 -0.4462 -0.1688 0.2263

0.44 0.4400 -0.2096 -0.4470 -0.1870 0.2095

0.45 0.4500 -0.1962 -0.4472 -0.2050 0.1917

0.46 0.4600 -0.1826 -0.4467 -0.2226 0.1730

0.47 0.4700 -0.1686 -0.4454 -0.2399 0.1534

0.48 0.4800 -0.1544 -0.4435 -0.2568 0.1330

0.49 0.4900 -0.1398 -0.4409 -0.2732 0.1118

0.50 0.5000 -0.1250 -0.4375 -0.2891 0.0898

0.51 0.5100 -0.1098 -0.4334 -0.3044 0.0673

0.52 0.5200 -0.0944 -0.4285 -0.3191 0.0441

0.53 0.5300 -0.0786 -0.4228 -0.3332 0.0204

0.54 0.5400 -0.0626 -0.4163 -0.3465 -0.0037

0.55 0.5500 -0.0462 -0.4091 -0.3590 -0.0282

0.56 0.5600 -0.0296 -0.4010 -0.3707 -0.0529

0.57 0.5700 -0.0126 -0.3920 -0.3815 -0.0779

0.58 0.5800 0.0046 -0.3822 -0.3914 -0.1028

0.59 0.5900 0.0222 -0.3716 -0.4002 -0.1278

13-20

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TABLE 13. 5. (Continued.)

u P-û) P 2 (u )

0.60 0.6000 0.0400

0.61 0.6100 0.0582

0.62 0.6200 0.0766

0.63 0.6300 0.0954

0.64 0.6400 0.1144

0.65 0.6500 0.1338

0.66 0.6600 0.1534

0.670.6700 0.1734

0.6B 0.6800 0.1936

0.69 0.6900 0.2142

0.70 0.7000 0.2350

0.71 0.7100 0.2562

0.72 0.7200 0.2776

0.73 0.7300 0.2994

0.74 0.7400 0.3214

0.75 0.7500 0.3438

0.76 0.7600 0.3664

0.77 0.7700 0.3894

0.78 0.7800 0.4126

0.79 0.7900 0.4362

0.80 0.8000 0.4600

0.81 0.8100 0.48420.82 0.8200 0.5086

0.83 0.8300 0.5334

0.84 0.8400 0.5584

0.85 0.8500 0.5838

0.86 0.8600 0.6094

0.87 0.8700 0.6354

0.88 0.8800 0.6616

0.89 0.8900 0.6882

P 3 ( u ) P 4 ( u ) P 5 (u )

-0.3600 -0.4080 -0.1526-0.3475 -0.4146 -0.1772

-0.3342 -0.4200 -0.2014

-0.3199 -0.4242 -0.2251

-0.3046 -0.4270 -0.2482

-0.2884 -0.4284 -0.2705

-0.2713 -0.4284 -0.2919

-0.2531 -0.4268 -0.3122

-0.2339 -0.4236 -0.3313

-0.2137 -0.4187 -0.3490

-0.1925 -0.4121 -0.3652

-0.1702 -0.4036 -0.3796

-0.1469 -0.3933 -0.3922

-0.1225 -0.3810 -0.4026

-0.0969 -0.3666 -0.4107

-0.0703 -0.3501 -0.4164

-0.0426 -0.3314 -0.4193

-0.0137 -0.3104 -0.4193

0.0164 -0.2871 -0.4162

0.0476 -0.2613 -0.4097

0.0800 -0.2330 -0.3995

0.1136 -0.2021 -0.38550.1484 -0.1685 -0.3674

0.1845 -0.1321 -0.3449

0.2218 -0.0928 -0.3177

0.2603 -0.0506 -0.2857

0.3001 -0.0053 -0.2484

0.3413 0.0431 -0.2056

0.3837 0.0947 -0.1570

0.4274 0.1496 -0.1023

13-21

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u P x (u) P 2 (u)

0.90 0.9000 0.7150

0.91 0.9100 0.7422

0.92 0.9200 0.7696

0.93 0.9300 0.7974

0.94 0.9400 0.8254

0.95 0.9500 0.8538

0.96 0.9600 0.8B24

0.97 0.9700 0.9114

0.98 0.9800 0.9406

0.99 0.9900 0.9702

1.00 1.0000 1.0000

P 3 (u) P 4 (u) P 5 (u)

0.4725 0.2079 -0.0411

0.5189 0.2698 0.0268

0.5667 0.3352 0.1017

0.6159 0.4044 0.1842

0.6665 0.4773 0.2744

0.7184 0.5541 0.3727

0.7718 0.6349 0.4796

0.8267 0.7198 0.5954

0.8830 0.8089 0.7204

0.9407 0.9022 0.8552

1.0000 1.0000 1.0000

13-22

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13.6. Sine, Cosine, and Exponential Integra ls.

- X

Sine in t e g ra l : S i (x)

Si(<=°) = IT/2

/ "¥**••

Cosine integral: Ci(x) = - / c o s ^ du =i n(yx) - j 1

0

An Y = 0.577215... (Euler's constant)

Ex pon ent i a l i n t e g r a l : E i (x) = - / —— du, <0 , °° > x > 0

x

CO

/ ^ du, <0,

*i(x) = Joga r i th mic i n t e g r a l : S i (x) = / . " •• = E i ( i i x)

0

«,i(e x ) = Ei(x )

FOE smal l va lues of x :

S i (x ) = x ,

JU(x ) = -x /£ n ( l / x )

C i(x ) = Ei (-x ) = Ei (x) = J ln d/ pc ) = Y + An(x) - x + ( x2 / 4 ) +

For la rge va lues of x :

Si (x ) = TT/2 - cos (x )/ x

Ci (x ) = s i n (x ) / x

I i ( x ) = ( eX /x ) (1 + 11 /x + 2 i / x + 3 /x + '•'), | x | » 1

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TABLE 1 2 . 6 . Values o f t h e s i n e , c o s i n e , l o g a r i t h m i c , and e x p o n e n t i a l i n t e g r a l s .

( s o u r c e : R e f . 1 8 , p . 6 - 9 . )

S i ( x ) Ci(x) E i ( x ) E i ( - x )

0.00 +0.000000 -CO —CO

0.01 +0.010000 -4.0280 -4.0179

0.02 +0.019999 -3.3349 -3.3147

0.03 +0.029998 -2.9296 -2.8991

0.04 +0.039996 -2.6421 -2.6013

0.05 +0.04999 -2.4191 -2.3679

0.06 +0.05999 -2.2371 -2.1753

0.07 +0.06998 -2.0833 -2.0108

0.08 +0.07997 -1.9501 -1.8669

0.09 +0.08996 -1.8328 -1.7387

0.10 +0.09994 -1.7279 -1.6228

0.11 +0.10993 -1.6331 -1.5170

0.12 +0.11990 -1.5466 -1.4193

0.13 +0.12988 -1.4672 -1.3287

0.14 +0.13985 -1.3938 -1.2438

0.15 +0.14981 -1.3255 -1.1641

0.16 +0.15977 -1.2618 -1.0887

0.17 +0.16973 -1.2020 -1.0172

0.18 +0.1797 -1.1457 -0.9491

0.19 +0.1896 -1.0925 -0.8841

0.20 +0.1996 -1.0422 -0.8218

0.21 +0.2095 -0.9944 -0.7619

0.22 +0.2194 -0.9490 -0.7042

0.23 +0.2293 -0.9057 -0.6485

0.24 +0.2392 -0.8643 -0.5947

0.25 +0.2491 -0.8247 -0.5425

0.26 +0.2590 -0.7867 -0.4919

0.27 +0.2689 -0.7503 -0.4427

0.28 +0.2788 0.7153 -0.3949

0.29 +0.2886 -0.6816 -0.3482

- C O

-4 .0379

-3.3547

- 2 . 9 5 9 1

-2 .6813

-2 .4679

-2 .2953

- 2 . 1 5 0 8

-2.0269

-1.9187

-1 .8229

-1 .7371

-1 .6595

-1 .5889

- 1 . 5 2 4 1

-1 .4645

-1 .4092

-1 .3578

-1 .3098

-1 .2649

-1 .2227

-1 .1829

-1 .1454

-1 .1099

-1 .0762

-1 .0443

-1 .0139

-0.9849

-0 .9573

-0 .9309

1 3 - 2 4

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TABLE 1 3 . 6 . (Continued.)

X Si(x) Ci(x) Ei(x) Ei(-x)

0.30 +0.2985 -0.6492 -0.3027 -0.9057

0.31 +0.3083 -0.6179 -0.2582 -0.88150.32 +0.3182 -0.5877 -0.2147 -0.8583

0.33 +0.3280 -0.5585 -0.17210 -0.8361

0.34 +0.3378 -0.5304 -0.13036 -0.8147

0.35 +0.3476 -0.5031 -0.08943 -0.7942

0.3S +0.3574 -0.4767 -0.04926 -0.7745

0.37 +0.3672 -0.4511 -0.00979 -0.7554

0.33 +0.3770 -Q.4263 +0.02901 -0.73710.39 +0.3867 -0.4022 +0.06718 -0.7194

0.40 +0.3965 -0.3788 +0.10477 -0.7024

0.41 +0.4062 -0.3561 +0.14179 -0.6859

0.42 +0.4159 -0.3341 +0.17828 -0.6700

0.43 +0.4256 -0.3126 +0.2143 -0.6546

0.44 +0.4353 -0.2918 +0.2498 -0.6397

0.45 +0.4450 -0.2715 +0.2849 -0.6253

0.46 +0.4546 -0.2517 +0.3195 -0.6114

0.47 +0.4643 -0.2325 +0.3537 -0.5979

0.48 +0.4739 -0.2138 +0.3876 -0.5848

0.49 +0.4835 -0.1956 +0.4211 -0.5721

0.50 +0.4931 -0.17778 +0.4542 -0.5598

0.51 +0.5027 -0.16045 +0.4870 -0.5478

0.52 +0.5123 -0.14355 +0.5195 -0.5362

0.53 +0.5218 -0.12707 +0.5517 -0.5250

0.54 +0.5313 -0.11099 +0.5836 -0.5140

0.55 +0.5408 -0.09530 +0.6153 -0.5034

0.56 +0.5503 -0.07999 +0.6467 -0.4930

0.57 +0.5598 -0.06504 +0.6778 -0.4830

0.58 +0.5693 -0.05044 +0.7087 -0.4732

0.59 +0.5787 -0.03619 +0.7394 -0.4636

13-25

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Si(x) Ci(x) Ei(x) Ei(-x)

0.60 +0.5881 -0.02227 +0.7699

0.61 +0.5975 -0.008675 +0.8002

0.62 +0.6069 +0.004606 +0.8302

0.63 +0.6163 +0.01758 +0.8601

0.64 +0.6256 r0.03026 +0.8898

0.65 +0.6349 +0.04265 +0.9194

0.66 +0.6442 +0.05476 +0.9488

0.67 +0.6535 +0.06659 +0.9780

0.68 +0.6628 +0.07816 +1.0071

0.69 +0.6720 +0.08946 +1.0361

0.70 +0.6812 +0.10051 +1.0649

0.71 +0.6904 +0.11132 +1.0936

0.72 +0.6996 +0.12188 +1.1222

0.73 +0.7087 +0.13220 +1.1507

0.74 +0.7179 +0.14230 +1.1791

0.75 +0.7270 +0.15216 +1.2073

0.76 +0.7360 +0.16181 +1.2355

0.77 +0.7451 +0.17124 +1.2636

0.78 +0.7541 +0.1805 +1.2916

0.79 +0.7631 +0.1895 +1.3195

0.80 +0.7721 +0.1983 +1.3474

0.81 +0.7811 +0.2069 +1.3752

0.82 +0.7900 +0.2153 +1.4029

0.83 +0.7989 +0.2235 +1.4306

0.84 +0.8078 +0.2316 +1.4582

0.85 +0.8166 +0.2394 +1.4857

0.86 +0.8254 +0.2471 +1.5132

0.87 +0.8342 +0.2546 +1.5407

0.88 +0.8430 +0.2619 +1.5681

0.89 +0.8518 +0.2691 +1.5955

-0.4544

-0.4454

-0.4366

-0.4280

-0.4197

-0.4115

-0.4036

-0.3959

-0.3883

-0.3810

-0.3738

-0.3668

-0.3599

-0.3532

-0.3467

-0.3403

-0.3341

-0.3280

-0.3221

-0.3163

-0.3106

-0.3050

-0.2996

-0.2943

-0.2891

-0.2840

-0.2790

-0.2742

-0.2694

-0.2647

13-26

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X Si(x) Ci(Jt) Ei(x) Ei(-x)

0.90 +0.8605 +0.2761 +1.6228 -0.2602

0.91 +0.8692 +0.2829 +1.6501 -0.2557

0.92 +0.8778 +0.2896 +1.6774 -0.2513

0.93 +0.8865 +0.2961 +1.7047 -0.2470

0.94 +0.8951 +0.3024 +1.7319 -0.2429

0.95 +0.9036 +0.3086 +1.7591 -0.2387

0.96 +0.9122 +0.3147 +1.7864 -0.2347

0.97 +0.9207 +0.3205 +1.8136 -0.2308

0.98 +0.9292 +0.3263 +1.8407 -0.2269

0.99 +0.9377 +0.3319 +1.8679 -0.2231

1.00 +0.9461 +0.3374 +1.8951 -0.2194

1.0 +0.9461 +0.3374 +1.8951 -0.2194

1.1 +1.0287 +0.3849 +2.1674 -0.1860

1.2 +1.1080 +0.4205 +2.4421 -0.1584

1.3 +1.1840 +0.4457 +2.7214 -0.1355

1.4 +1.2562 +0.4620 +3.0072 -0.1162

1.5 +1.3247 +0.4704 +3.3013 -0.1000

1.6 +1.3892 +0.4717 +3.6053 -0.08631

1.7 +1.4496 +0.4670 +3.9210 -0.07465

1.8 +1.5058 +0.4568 +4.2499 -0.06471

1.9 +1.5578 +0.4419 +4.5937 -0.05620

2.0 +1.6054 +0.4230 +4.9542 -0.04890

2.1 +1.6487 +0.4005 +5.3332 -0.04261

2.2 +1.6876 +0.3751 +5.7326 -0.03719

2.3 +1.7222 +0.3472 +6.1544 -0.03250

2.4 +1.7525 +0.3173 +6.6007 -0.02844

2.5 +1.7785 +0.2859 +7.0738 -0.02491

2.6 +1.8004 +0.2533 +7.5761 -0.02185

2.7 +1.8182 +0.2201 +8.1103 -0.01918

2.8 +1.8321 +0.1865 +8.6793 -0.01686

2.9 +1.8422 +0.1529 +9.2860 -0.01482

3.0 +1.8487 +0.1196 +9.9338 -0.01304

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x Si(x ) Ci(x) IT (x) E i ( -x )

3.1 +1.8517 +0.08699 +10.6263 -0.01149

3.2 +1.8514 +0.05526 +11.3673 -0.01013

3.3+1.8481 +0.02468

+12.1610 -0.02

89393.4 +1.8419 +0.004518 +13.0121 -0.0 2 7890

3.5 +1.8331 -0.03213 +13.9254 -0.0 26970

3,6 +1.8219 -0.05797 +14.9063 -0.0 2 6160

3.7 +1.8086 -0.08190 +15.9606 -0.0 25448

3.8 +1.7934 -0.1038 +17.0948 -0.0 24820

3.9 +1.7765 -0.1235 +18.3157 -0.0 2 4267

4.0 +1.7582 -0.1410 +19.6309 -0.0 23779

4.1 +1.738T -0.1562 +21.0485 -0.0 2 3349

4.2 +1.7184 -0.1690 +22.5774 -0.0 22969

4.3 +1.6973 -0.1795 +24.2274 -0.0 2 2633

4.4 +1.6758 -0.1877 +26.0090 -0.0 22336

4.5 +1.6541 -0.1935 +27.9337 -0.0 2 2073

4.6 +1.6325 -0.1970 +30.0141 -0.0 2 1841

4.7 +1.6110 -0.1984 +32.2639 -0.02

16354.8 +1.5900 -0.1976 +34.6979 -0.0 21453

4.9 +1.5696 -0.1948 +37.3325 -0.0 2 1291

5.0 +1.5499 -0.1900 +40.1853 -0.0 2 1148

6 +1.4247 -0.06806 +85.9898 -0.0 3 3601

7 +1.4546 +0.07670 +191.505 -0.0 3 1155

8 +1.5742 +0.1224 +440.380 -0.0 4 3767

9 +1.6650 +0.05535 +1037.88 -0.04

124510 +1.6583 -0.04546 +2492.23 -0.0 5 4157

11 +1.5783 -0.08956 +6071.41 -0.0 5 1400

12 +1.5050 -0.04978 +14959.5 -0.0 6 4751

13 +1.4994 +0.02676 +37197.7 -0.0 6 1622

14 +1.5562 +0.06940 +93192.5 -0.0 7 5566

15 +1.6182 +0.0462B +234 956 -0.0 7 1918

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X Si(x) Ci(x) X Si(x) Ci(x)

20 +1.5482 +0.04442 140 +1.5722 +0.007011

25 +1.5315 -0.00685 150 +1.5662 -0.004800

30 +1.5668 -0.03303 160 +1.5769 +0.001409

35 + 1.5969 -0.01148 170 +1.5653 +0.002010

40 +1.5870 +0.01902 180 +1.5741 -0.004432

45 +1.5587 +0.01863 190 +1.5704 +0.005250

50 +1.5516 -0.00563 200 +1.5684 -0.004378

55 +1.5707 -0.01817 300 +1.5709 -0.003332

60 +1.5867 -0.00481 400 +1.5721 -0.002124

65 +1.5792 +0.01285 500 +1.5726 -0.0009320

70 +1.5616 +0.01092 600 +1.5725 +0.0000764

75 +1.5586 -0,00533 700 +1.5720 +0.0007788

80 +1.5723 -0.01240 800 +1.5714 +0.001118

85 +1.5824 -0.001935 900 +1.5707 +0.001109

90 +1.5757 +0.009986 1 0 2

+1.5702 +0.000826

95 +1.5630 +0.007110 1 0 4

+1.5709 -0.0000306

100 +1.5622 -0.005149 1 0 5

+1.5708 +0.0000004

110 +1.5799 -0.000320 10 6 +1.5708 -0.0000004

120 +1.5640 +0.004781 1 0 7

+1.5708 +0.0

130 +1.5737 -0.007132 00 1/2TT 0.0

1 3 - 2 9

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SECTION 14. ROOTS OF SOME CHARACTERISTIC EQUATIOHS

TABLE 14.1. First six roots of X tan X = C. (Source: Ref. 74, p. 217.)

c X l X 2 X 3 X 4 X 5 X 6

0 0.0000 3.1416 6.2832 9.4248 12.5664 15.7080

0.001 0.0316 3.1419 6.2833 9.4249 12.5665 15.7080

0.002 0.0447 3.1422 6.2835 9.4250 12.5665 15.7081

0.004 0.0632 3.1429 6.2838 9.4252 12.5667 15.7082

0.006 0.0774 3.1435 6.2841 9.4254 12.5668 15.7083

0.008 0.0893 3.1441 6.2845 9.4256 12.5670 15.7085

0.01 0.0998 3.1448 6.2848 9.4258 12.5672 15.7096

0.02 0.1410 3.1479 6.2864 9.4269 12.5680 15.7092

0.04 0.1987 3.1543 6.2895 9.4290 12.5696 15.7105

0.06 0.2425 3.1606 6.2927 9.4311 12.5711 15.7118

0.08 0.2791 3.1668 6.2959 9.4333 12.5727 15.7131

0.1 0.0311 3.1731 6.2991 9.4354 12.5743 15.7143

0.2 0.4328 3.2039 6.3148 9.4459 12.5823 15.7207

0,3 0.5218 3.2341 6.3305 9.4565 12.5902 15.7270

0.4 0.5932 3.2636 6.3461 9.4670 12.5981 15.7334

0.5 0.6533 3.2923 6.3616 9.4775 12.6060 15.7397

0.6 0.7051 3.3204 6.3770 9.4879 12.6139 15.7460

0.7 0.7506 3.3477 6.3923 9.4983 12.6218 15.7524

0.8 0.7910 3.3744 6.4074 9.5087 12.6296 15.7587

0.9 0.8274 3.4003 6.4224 9.5190 12.6375 15.7650

1.0 0.8603 3.4256 6.4373 9.5293 12.6453 15.7713

1.5 0.9882 3.5422 6.5097 9.5801 12.6841 15.8026

2.0 1.0769 3.6436 6.5783 9.6296 12.7223 15.8336

3.0 1.1925 3.8088 6.7040 9.7240 12.7966 15.8945

4.0 1.2646 3.9352 6.8140 9.8119 12.8678 15.9536

5.0 1.3138 4.0336 6.9096 9.8928 12.9352 16.0107

6.0 1.3496 4.1116 6.9924 9.9667 12.9988 16.0654

7.0 1.3766 4.1746 7.0640 10.0339 13.0584 16.1177

8.0 1.3978 4.2264 7.1263 10.0949 13.1141 16.16759.0 1.4149 4.2694 7.1806 10.1502 13.1660 16.2147

10.0 1.4289 4.3058 7.2281

14-1

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TABLE 14.1. (Contii)ued.)

X 2

15.0 1.4729 4.4255 7.3959 10.3898 13.4078 16.4474

20.0 1.4961 4.4915 7.4954 10.5117 13.5420 16.5864

30.0 1.5202 4.5615 7.6057 10.6543 13.7085 16.7691

40.0 1.5325 4.5979 7.6647 10.7334 13.8048 16.8794

50.0 1.5400 4.6202 7.7012 1C.7832 13.8666 16.9519

60.0 1.5451 4.6353 7.7259 10.8172 13.9094 17.0026

80.0 1.5514 4.6543 7.7573 10.8606 13.9644 17.0686

00.0 1.5552 4.6658 7.7764 10.8871 13.9981 17.1093

CO 1.5708 4.7124 7.8540 10.9956 14.1372 17.2788

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TABLE 14.2. F i rs t f ive root s of 1 -• X cot X = C. ( s o u r c e : Ref. 20 p. 442.)

c X l\

2X 3 X 4 X 5

0.000 0.0000 4.4934 7.7253 10.9041 14.0662

0.005 0.1224 4.4945 7.7259 10.9046 14.0666

0.010 0.1730 4.4956 7.7265 10.9050 14.0669

0.020 0.2445 4.4979 7.7278 10.9060 14.0676

0.030 0.2991 4.5001 7.7291 10.9069 14.G683

0.040 0.3450 4.5023 7.7304 10.9078 14.0690

0.050 0.3854 4.5045 7.7317 10.9087 14.0697

0.060 0.4217 4.5068 7.7330 10.9096 14.0705

0.070 0.4551 4.5090 7.7343 10.9105 14.07120.080 0.4860 4.5112 7.7356 10.9115 14.0719

0.090 0.5150 4.5134 7.7369 10.9124 14.0726

0.100 0.5423 4.5157 7.7382 10.9133 14.0733

0.200 0.7593 4.5379 7.7511 10.9225 14.0804

0.300 0.9208 4.5601 7.7641 10.9316 14.0875

0.400 1.0528 4.5822 7.7770 10.9408 14.0946

0.500 1.1656 4.6042 7.7899 10.9499 14.1017

0.600 1.2644 4.6261 7.8028 10.9591 14.1088

0.700 1.3525 4.6479 7.8156 10.9682 14.1159

0.800 1.4320 4.6696 7.8284 10.9774 14.1230

0.900 1.5044 4.6911 7.8412 10.9865 14.1301

1.000 1.5708 4.7124 7.8540 10.9956 14.1372

1.500 1.8366 4.8158 7.9171 11.0409 14.1724

2.000 2.0288 4.9132 7.9787 11.0856 14.2075

3.000 2.2889 5.0870 8.0962 11.1727 14.27644.000 2.4557 5.2329 8.2045 11.2560 14.3434

5.000 2.5704 5.3540 8.3029 11.3349 14.4080

6.000 2.6537 5.4544 8.3914 11.4086 14.4699

7.000 2.7165 5.5378 8.4703 11.4773 14.5288

8.000 2.7654 5.6078 8.5406 11.5408 14.5847

9.000 2.8044 5.6669 8.6031 11.5994 14.6374

10.000 2.8363 5.7172 8.6587 11.6532 14.6870

11.000 2.8628 5.7606 8.7083 11.7027 14.7335

16.000 2.9476 5.9080 8.8898 11.8959 14.9251

21.000 2.9930 5.9921 9.0019 12.0250 15.0625

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c X l X 2 X 3 X 4 X 5

31.000 3.0406 6.0831 9.1294 12.1807 15.2380

41.000 3.0651 6.1311 9.1987 12.2688 15.3417

51.000 3.0801 . 6.1606 9.2420 12.3247 15.4090101.000 3.1105 6.2211 9.3317 12.4426 15.5537

TABLE 14.3. F i r s t f ive roots of J (X )m n

(Source: Ref. 20, p. 443.)

2.4048

3.8317

5.1356

6.3802

7.5883

5.5201

7.0156

8.4172

9.7610

11.0647

8.6537

10.1735

11.6198

13.0152

14.3725

11.7915

13.3237

14.7960

16.2235

17.6160

14.9309

16.4706

17.9598

19.4094

20.8269

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TABLE 14.4. First six roots of X J 1 ( X n ) - CJ Q( X ) = 0. (Source: Ref. 74.

p. 217)

X6

0 0 3.8317 7.0156 10.1735 13.3237 16.4706

0.01 0.1412 3.8343 7.0170 10.1745 13.3244 16.4712

0.02 0.1995 3.8369 7.0184 10.1754 13.3252 16.4718

0.04 0.2814 3.8421 7.0213 10.1774 13.3267 16.4731n 06 0.3438 3.8473 7.0241 10.1794 13.3282 16.4743

0.08 0.3960 3.8525 7.0270 10.1813 13.3297 16.4755

0.1 0.4417 3.8577 7.0298 10.1833 13.3312 16.4767

0.15 0.5376 3.8706 7.0369 10.1882 13.3349 16.4797

0.2 0.6170 3.8835 7.0440 10.1931 13.3387 16.4828

0.3 0.7465 3 .9091 7.0582 10.2029 13.3462 16.4888

0.4 0.8516 3.9344 7.0723 10.2127 13.3537 16.4949

0.5 0.9408 3.9594 7.0864 10.2225 13.3611 16.5010

0.6 1.0184 3.9841 7.1004 10.2322 13.3686 16.5070

0.7 1.0873 4.0085 7.1143 10.2419 13.3761 16.5131

0.8 1.1490 4.0325 7.1282 10.2516 13.3835 16.5191

0.9 1.2048 4.0562 7.1421 10.2613 13.3910 16.5251

1.0 1.2558 4.0795 7.1558 10.2710 13.3984 16.5312

1.5 1.4569 4.1902 7.2233 10.3188 13.4353 16.5612

2.0 1.5994 4.2910 7.2884 10.3658 13.4719 16.5910

3.0 1.7887 4.4634 7.4103 10.4566 13.5434 16.6499

4.0 1.9081 4.6018 7.5201 10.5423 13.6125 16.7073

5.0 1.9898 4.7131 7.6177 10.6223 13.6786 16.7630

6.0 2.0490 4.8033 7.7039 10.6964 13.7414 16.8168

7.0 2.0937 4.8772 7.779 7 10.7646 13.8008 16.8684

8.0 2.1286 4.9384 7.8464 10.8271 13.8566 16.9179

9.0 2.1566 4.9897 7.9051 10.8842 13.9090 16.9650

10.0 2.1795 5.0332 7.9569 10.9363 13.9580 17.0099

15.0 2.2509 5.1773 8.1422 11.1367 14.1576 17.2008

20.0 2.2880 5.2568 8.2534 11.2677 14.2983 17.3442

30.0 2.3261 5.3410 8.3771 11.4221 14.4748 17.5348

40.0 2.3455 5.3846 8.4432 11.5081 14.5774 17.6508

50.0 2.3572 5.4112 8.4840 11.5621 14.6433 17.7272

60.0 2.3651 5.4291 8.5116 11.5990 14.6889 17.7807

80.0 2.3750 5.4516 8.5466 11.6461 14.7475 17.8502

100.0 2.3809 5.4652 8.5678 11.6747 14.7834 17.8931

00 2.4048 5.5201 8.6537 11.7915 14.9309 18.0711

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TABLE 14.5. First five roots of J 0 (X n )Y 0 (C X n ) = Y Q (X n ) J 0 ( C X n ) . (Source:

Ref . 9, p. 493.)

c X l X 2 X 3 X 4 X 5

1.2 15.7014 31.4126 47.1217 62.8302 78.5385

1.5 6.2702 12.5598 18.8451 25.1294 31.41332.0 3.1230 6.2734 9.4182 12.5614 15.7040

2.5 2.0732 4.1773 6.2754 8.3717 10.4672

3.0 1.5485 3.1291 4.7038 6.2767 7.8487

3.5 1.2339 2.5002 3.7608 5.0196 6.2776

4.0 1.0244 2.0809 3.1322 4.1816 5.2301

i14-6

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SECTION 1 5 . CONSTANTS AND CONVERSION FACTORS.

1 5 . 1 . M a t h e m a t i c a l C o n s t a n t s .

e

Hn 10

T I

2 . 7 1 8 2 8 1 8 . . . .

2 . 3 0 2 5 8 5 1 . . .

3 . 1 4 1 5 9 2 6 . . .

Y = 0 . 5 7 7 2 1 5 6 . . . ( E u l e r ' s c o n s t a n t )

1 5 . 2 . P h y s i c a l C o n s t a n t s .

S t a n d a r d a c c e l e r a t i o n

of g r a v i t y

J o u l e ' s c o n s t a n t

9 . 8 0 6 6 5 m / s

3 2 .1 7 42 f t / s2

J = 1 . 0 N 'm /Jc= 7 7 8 .1 6 f f l b f / B t u

S t e £ a n - B o l t z m a n n

c o n s t a n t a = 1 .35 5 x 1 0 ~ 1 2 c a l / s ' c m 2 - K 4

= 1 7 12 x 1 0 ~1 2 B t u / h r « f t 2 - R 4

= 5 . 6 7 3 x 1 0 "8 W /m 2 'K 4

U n i v e r s a l g a s

c o n s t a n t R = 8 .3 14 3 J /mo l«K

» 0 .0 82 05 i l ' a t i v ' i ii o l 'K» 1.9859 B t u / l b m * m o l ' R

« 1 5 4 5 .3 3 f f l b f / l b m ' m o l ' R

" 6 . 4 1 x 1 0 j / k g * m o l ' K

1 5 - 1

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1 5 .3 . Conve r s ion Fac to r s .

TABLE 1 5 . 1 . Con vers ion factors for l eng th .

Vim in. ft yd

1 m

1 cm

1 lint

1 A1 in.

1 ft

1 yd

¥o

= 1 100 10" 1 0 i u 39.37 3.2B0 1.0936

= 0.01 1 1 0 4 1 0 8 0.3937 0.0328 0.0109

= 10" 6 l O " 4 1 1 0 4 0.3937 x 10 ~ 4 0.0328 x 1 0 - 4 0.0109 x 10

- I D "1 0

l O "8

io-4

1 0.3937 x 1 0- 8

0.0328 x 10 ~4

0.0109 x 10- 0.0254 2.540 25.4 x 1 0 4 2.540 x 1 0 8 1 0.0833 0.0277

- 0.3048 30.48 30.48 x 1 0 4 30.43 x 1 0 8 12 1 0.3333

= 0.9144 91.440 91.440 x 1 0 4 91.440 x 1 0 8 36 3 1

r 4

-8

TABLE 1 5 . 2 . Conve r s ion f ac to r s for a rea .

cm in. ft yd

1 cm

l m 2

1 in . 2 =

1 f t 2 =

1 y d 2 =

10 '

6.4516

929.034

8361.307

10

1

6.4516 x 10 4

929.034 x io

8361.307 x 10 4

0.1550 1 .07639 1.1960 x 10

1550 10.7639 1.1960

1 6.9444 7.7160 x 10

144 1 0.11111

1296 9 1

-4

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TABLE 15.3. Conversion factors for volume.

Uli

™ 3

cm . 3in. f t 3 ml liter gal

31 cm « 1 610.23 x 10 ~ 4 35.3145 x 1 0 - 4 9 9 9 . 9 7 2 x 1 0 - 3 9 9 9 . 9 7 2 x 10" 6 264.170 x lO " 6

1 i n . 3 = 16.3872 1 5.7870 x 10" 4 16.3867 16.3867 x 10" 3 432.900 x lO " 5

1 f t 3 = 283.170 x 10 2 1728 1 28.3162 x 10 3 28.3162 7.4805

1 ml = 1.000028 610.251 x 1 0 " 4 353.154 x 10 ~ 7 1 0.001 264.178 x 1 0 " b

1 liter = 1000.028 61.0251 353.154 x 1 0- 4

1000 . 1 264.178 x 10 "J

1 gal => 37 85.4 34 231 133.680 x 10 " 3 3785.329 3.785329 1

TABLE 15.4. Conversion factors f or mass.

lb slugs 9 kg ton

1 l b = 1 0.03108 453.59 0.45359 0.00051 slug = 32.174 1 1.4594 x 1 0 4 14.594 0.016087

1 g = 2.2046 x 10' 3 6.8521 x 10" 5 1 ID " 3 1.1023 x 1 0 - 6

1 kg = 2.2046 6.8521 x 10' 3 10 3 1 1.1023 x l o - 3

1 ton = 2000 62.162 9.0718 x 1 0 5 907.18 1

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TABLE I S .5 . Conversion fa ct or s for d en si ty .

l b m / f t 3 slug/in. lbm/in. 3 lbm/gal . 3g/cm

1 lbm/ f t 3 = 1 0.03108 5.787 x 1 0 - 4 0.13386 0.01602

1 s lug / f t 3 = 32.174 1 0.01862 4.3010 0.515433

1 lbm/in. = 1728 53.706 1 231 27.680

1 lbm/gal = 7.4805 0.2325 4.329 x 1 0 "3

1 0.119831 g/cm 3 62.428 1.9403 0.03613 8.345

• I

I

15-4

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TABI£ 1 5 . 6 . C o n v e r s i o n f a c t o r s f o r p r e s s u r e .

l b f / i n / d yn e/ cm k g f / c m i n . H g mm Hg i n . H O a tm b a r

1 l b f / i n . 2 = 1

1 d y n e / e m2 = 1 4 5 . 0 3 8 3

x 1 0 ~7

1 k g f / c m2 = 1 4 . 2 2 3 4

1 i n . H g = 0 . 4 9 1 2

1 ram H y = 0 . 0 1 9 3 4

1 i n . H O = 0 . 0 3 6 0 9

1 a tm

1 bar

1 4 . 6 9 6 0

= 1 4 . 5 0 3 8

6 8 9 . 4 7 3

x 1 0 2

1

9 8 0 . 6 6 53

x 1 0 J

3 3 8 . 6 4

x 1 0 2

1 3 3 3 . 2 2 3

24 .883

x 1 0 2

1 0 1 . 3 2 54x 10

1 0 6

0 . 0 7 0 3 1

1 0 1 . 9 7 2

x l O - 8

1

0 .03453

1 . 3 5 9 5

x 1 0 ~3

2 . 5 3 7

x 1 0 ~3

1 . 0 3 3 2 3

2.0360

295.299

x 1 0 ~ 7

28.959

1

0.03937

0.0735

29.9212

1.01972 29.5299

51.715

750.062

x 1 0 " 6

735.559

25.40

1

1.8665

760

750.0617

27.71

4.0188-4

x 10394.0918

13.608

0.5358

460.80

401.969

0.06805

986.923

x 1 0 " 9

967.841

x 1 0 3

0.03342

1.315

x 1 0 - 3

2.458

x 1 0 ~ 3

1

986.923

x 1 0 " 3

0.06895

i o - 6

980.665

x 1 0 ~ 3

0.03386

1.333

x 1 0 - 3

2.488

x 10

1.01325

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TABLE 15 .7 . Con version fa ct or s for en ergy .

f f l b f abs j ou l e i n t j ou l e c a l I .T. c a l

1 f f l b f 1 1.35582 1.355597 0.32405 0.32384

1 abs joule = 0.73756 1 0.999835 0.23885 0.238849

1 in t j ou l e = 0.737682 1.000165 1 0.239045 0.238889

1 ca l 3.08596 4.18401 4.1833 1 0.99934

1 I .T . ca l = 3.08799 4.18676 4.18605 1.000657 1

1 Btu 778.16 1055.045 1054.366 252 .161 251.9961 i n t kW'hr = 265 .567 x 104 360.0612 x 1 0 4 360 .000 x 104 860.565 x 1 0 3 860.000 x 103

1 hp'hr = 198.0000 x 10* 268.4525 x 1 0 4 268 .082 x 104 641.615 x 1 0 3 641.194 x 103

1 l i t e r - a t m = 74.7354 101.3278 101 .3111 24.2179 24.2020

B t u i n t kW»]hr hp»hr l i t e r ' a t m

1 f f l b f

1 abs joule

1 i n t j ou l e

1 ca l

1 I .T . c a l

1 Btu1 int kWhr

1 hp'hr

1 l i t e r ' a t m

128.5083 x 10

947.827 x 10"6

0.947988 x 10

396.572 x 10 '

396.832 x 10"

13412.76

2544.46

0.09604

-3

376.553 x 10

277.731 x 1 0 ~ 9

2.777778 x 1 0 ~ 7

116.2028 x 10 ~ 8

116.2791 x io ~ fi

293.018 x 10 ~6

1

0.74558

281.718 x 10 ~ 7

505.051 x 10 '

372.505 x i o - S

3.7256 x l o " 7

155.8566 x io"

155.9590 x io"

293.010 x 10 ~6

1.3412

1

377.452 x 1 0 ~ 7

133.8054 x 10

986.896 x 10

9.87058 x 10

412.918 x lo"

413.189 x lo"

10.4122255.343 x 1 0 2

264.935 x 1 0 2

1

-3

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Table 15.8 . Conversion factors for specific energy.

abs jou l e /h ca l /g I .T. cal /gm Btu / lb f t - lb£ / lbm in t . kw hr/g hp h r / lb , . 2 . 2t t / s

abs jou le /g - 1 0.2390 0.2388 0.4299 334.53 2.777 x 10" 7 1.690 x 10" 4 10763

c a l / g - 4.184 1 0.9993 1.7988 1399.75 1.162 x 1Q ~6

7.069 x 10~ 4 4 .504 x 10 4

I .T. ca l /g - 4.186 1.0007 1 1.8 1400.69 1.163 x 10" 6

7.074 x 10" 4 4 .506 x 10 4

Btu/lb . 2.326 0.5559 0.5556 1 778.16 6.460 x l O - 7 3.930 x in " 4 25,037f f l b f / l b m - 2.989 x 10~ 3 7.144 x 10"4 7.139 x 10~ 4 1 .285 x lo"" 1 8.302 x 1 0~ 1 0 5.051 x l o" 7 32.174

int . RWhr/g - 3.610 x 106

860,565 86C,000 \ 54 8 x 106

1.2046 x 109

1 608.4 3, 876 x ID1 0

hp-hr / lb • 5919 1414.5 1413.6 2545 1.980 x 10 6 0.001644 1 6. 370 x 10 7

f t W - 9.291 X 10" 5 2.220 x 10" 5 2.219 x 10" 53, 994 x 1 0 " 5 0.03108 2.580 x l o " 1 1 1.567 x 1 0 - 8

1

Table 15.9. Conversion factors for specific energy per degree.

abs j o u l e / g • >K Cal/g-K I .T . c a l / g ' K B t u / l b ' R Ws/kg-K

abs joule/g*K = 1 0.2390 0.2388 0.2388 1 0 3

ca l /g -K = 4.184 1 0.9993 0.9993 4184

I . T. c a l / g - K = 4.186 1.0007 1 1 4186

b tu / lb -R

Ws/kg-K

4.186

l O" 3

1.0007

2.390 x 10"-4

2 .

1

,388 x 1Q~•4

1

2.388 x I d- 4

4186

1

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Table 15.10. Conversion fac to rs for thermal conduct ivi ty .

cal/s«cm-°C Bt u/hr «f f° F Btu/hr -ft '" F/i n. W/ m-°C

1 cal/s'cm* C = 1 241.9 2903 418.6

1 Btu/hr'ft- F = 4.13 x 10~ 3 1 12 1.73

1 Btu/hr'ft 2- F/in. = 3.45 x 10~ 4 0.0833 1 1.44 x 10 1

1 W/m C = 23.89 5780 69350 1

Table 15.11. Conversion fa for thermal ''and momentum diffusivities

ft 2/hr stokes m2 /hr 2/m /s

ft 2/h r = 1 0.25806 0.092903 2.58 x 10~ 5

stokes =

m2 /hr =

3.885

10.764

1

2.778

0.36

1

io- 4

1.778 x 10~ 4

m2 / s = 38,750 1 0 4 3600 1

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TABLE 15.12. Conv ersi on factors for heat flux.

Btu W k c a l c a l

f t 2 « h r2

m . 2hr«m2

S"cm

B t u / f t 2 « h r = 1 3 . 1 5 4 2 . 7 1 3 7.536 x 1 0 ~5

W /m2 = 3 . 1 7 0 x 1 07 1 8.600 x 1 07 23892

k c a l / h r - m = 0.3687 1 .163 1 360002c a l / s ' c m = 1 32 77 4 1 8 6 8 . 2.778 x 1 0 ~5 1

TABLE 15.13. Conv ersi on factors for heat transfer coefficient.

Btu W c a l k c a l

n r . f t

2 . ° F m 2 ' ° C2 o„

s»cm • Ch r - m 2 ' ° C

B t u / h r - f t2 « ° F = 1 5.678 1 . 3 5 6 x 1 0 "4 4.883

W/m 2 '°C 1 . 7 6 1 x 1 07 1 2 3 9 1 8.600 x 1 0 7

c a l / s e c - c m - ° C = 7 37 6 4 . 1 8 6 x1 0 4 1 36000

k c a l / h r - m2 - ° C = 0.2049 1 .163 2.778 x 1 0 "5 1

15-9

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SECTION 16. CONVECTION COEFFICIENTS

16.1. Forced Flow in Smooth Tubes.

16.1.1 Fully Developed Laminar Flaw (Source: Ref. 21)

— = 4.364 (constant heat rate)k

— = 3.658 (constant surface temp)

16.1.2 Fully Dev elo ped Turbul ent Flo w (Soui'ce: Ref . 21)

£ £ = 6.3 + 0.003 (Re Pr ), Pr < 0.1 (constant heat rate)

— = 4.8 + 0.003 (Re Pr ), Pr < 0.1 (constant surface temp)

j p = 0.022 Pr * Re " , 0.5 < Pr < 1.0 (constant heat rate)

— = 0.021 Pr " Re , 0.5 < Pr < 1.0 (constant surface temp)

jp- = 0.0155 P r ° ' 5 R e ° * 9 , 1.0 < Pr < 20

f- = 0.011 P r ° ' 3 R e ° - 9 , Pr > 20

k

Pr = Prandtl number, Re = Reynolds number

16.2 . Forced Flow Between Smooth Infinite Parallel Plates.

16.2.1. Fully developed laminar flow (Source: Ref. 2 1 ) :he— = 4.118 (constant heat rate on both sides)

hs— = 2.69 3 (constant heat rate on one side, other side insulated)

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hs— = 3.77 (constant surface temperature on both sides)

h sr— = 2 .43 ( co ns t an t su r f ace t em per a tu re on one s i de , o th e r s id e in s u la t ed )

s = spa c ing o f p l a t e s

1 6 . 3 . Fo rced F low P a ra l l e l t o Smooth Se m i - I n f in i t e F l a t P l a t e s Laminar F low.1 6 . 3 . 1 . Laminar f low :hxk

temperature)(Source: Ref.7)

= 0.332 P r 1 / 3 R e 1 / 2 [ l - (X Q/X ) 3 / / 4 ] ~ 1 / 3 , (constant surface

X = unheated starting length

X = distance from leading edge

J P = 0.453 P r 1 / 3 R e x

1 / 2 [l - (X Q/X) 3 / 4 ] 1 / 3 , (constant heat rate)

(Source: Ref . 21)

1 6 . 3 . 2 . Tu rb ule nt f low (Source : Ref. 2 1) :

f = 0 .0 2 95 P r ° -6 R e x

0 ' 8 [ l - ( X 0 / X ) 9 / 1 0 ] " 1 / 9 , (cons tant su r fa ce

temperature)

hx 0.6 0.8 T 9/101-1/9— = 0.0307 Pr R e x |_1 - (X Q/X) ' J ' , (constant heat rate)

1 6 . 4 . Fully Developed Flow in Smooth Tube Annul .

16.4.1. Laminar flow (Source: R ef . 2 1 ) :

h

i <a

Q -d

i > N U

i ik " 1 " <q o/qX

h (d - d .) Nuo o i oo

i - VqX(Subsc r ip t s i and o r e f e r t o i nne r and ou te r su r f a ce s , q i s su r f a ce hea t f l ux ,

N u. . and Hu ar e in ne r and oute r su r fa ce N us se l t numbers when only one

su r fa ce i s hea t ed and 0 i s an in f lu enc e co e f f i c i en t g iven in Ta b le 16 .1 . )

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TABL E 16.1 . Tube -annu lus solutions for constant heat rate in fully developed

laminar flow and temperature profiles.

r.r Nu .. Nu 0. 0l o 11 oo l o

0 °° 4.364 °° 0

0.2 8.499 4.883 0.905 0.1041

0.4 6.583 4.979 0.603 0.1823

0.6 5.912 5.099 0.473 0.2455

0.8 5.580 5.240 0.401 0.299

1.0 5.385 5.385 0.346 0.346

16.4.2. Turbulent flow (Source: Ref. 2 2 ) :

7 (d - d.) - 0.023 R e ° ' 8 P r ° " 4 (d / d . ) ° ' 4 5 . R e . , 1 0 4 , A d = d - d.k o I . . Ad o i Ad o I

16.5. Forced Flow Norm al to Circular Cylin ders.

16.5.1. Local Coefficients (Source: Ref. 2 2 ):

& = 1.14 P r ° - 4 R e ° ' 5 [l - (G / 90 ) 3 ] , 0 < 6 < 80°

0 = cylinder angle from stagnation point

16.5.2. Average Coefficients (Source: Ref. 7 ) :

M = 0.43 + C Re"? P r ° ' 3 1

k d

TABLE 16. 2 .

R S d

1-4,000

4,000-40,000

40,000-400,000

0.533 0.500

0.193 0.618

0.0265 0.805

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1 6 . 6 . Forced Flow Normal to Spheres.

1 6 . 6 . 1 . Average Coefficients (Source: Ref. 7 ) :

^ = 0.37 R e , - 6 P r 0 " 3 3 , 20 < Re < 150,000

ifl = 2 + 0.37 R e ° -6 P r 0 - 3 3 , Re., < 20k d a

1 6 . 7 . Free Convection on Vertical Plates and Cylinders.

1 6 . 7 . 1 . Local Coefficients (Source: Ref. 7 ) :

jp = 0.508(Pr)1/ 2

(0.952 + P r ) ~1 /

'4

( G r x )1 / 4

f GrPr < 1 04

|* = 0.0295 ( P r ) 7 / 5 [ l + 0.494 ( P r ) 2 / 3 ] ~ 2 / 5 ( G r )2 / 5 , GrPr > 1 0 4

1 6 . 8 . Free Convection on Horizontal Cylinders. >

1 6 . 8 . 1 . Average Coefficients (Source: Ref. 23):

f = C (Gr d Pr) 1"

TABLE 16 .3.

Gr d Pr C m

0 - 1 0 " 5 0. 0 0

1 0 " 5 - 1 0 - 1 0. 7 1/16

1 0 - 1 - 1 0 ~ 4 1. 4 1/7

1 0 4 - 1 0 9 0. 3 1/4

1 0 9 - 1 0 1 2 0 13 1/3

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16.9. Free Convection From Horizontal Square Pla tes.

16.9.1. Average Coeffici ents (Source: Ref . 23) ;

•- 1 = C (Gr T P r ) m , L = plate dimensionk L

TABLE 16.4.

Condition Gr PrL

Upper surface heated or

lower surface cooledLower surface heated orupper surface cooled

Upper surface heated orlower surface cooled

1 05

- 2 x 1 07

0.54 1/4

3 x 1 0 5 - 3 x 1 0 1 0 0.27 1/4

2 x 10 7 - 3 x 1 0 1 0 0.14 1.3

16.10. Free Convection from Spheres.

16.10.1. Average Coef ficients (Source: (Ref. 5 ) :

^ = ?. + 0.43 (G r d P r ) 1 / 4 , 10° < G r d P r < 1 0 5

16.11. Free Convection in Enclosed Spaces.

16.11.1. Enclosed Verti cal Air Spaces (Source: Ref. 1 ) ;

G r B < 2,000

0 .1 8 {Gr )1 / 4 ( s / L ) 1 / 9 , 2,00 0 < Gr < 20,00 0s s

I 0.0 65 (Gr )1 / 3 ( s / L ) 1 / ' 9 , 20,0 00 < Gr < 1 078 S

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k e = e f f e c t i v e t h e r m a l c o n d u c t i v i t ys = width of a i r sp ac eL = le ng th of a i r spa ce

1 6 . 1 1 . 2 . Enc losed H or iz on ta l Ai r Spaces (So urce : Ref. 1 ) i

0.195 (Gr )1 / 4 , 1 04 < Gr < 4 x 1 05

s s

0.068 (Gr )1 / 3 , 4 x 1 05 < Grs s

1 6 . 1 2 . F i lm Con dens a t ion .

1 6 . 1 2 . 1 . V e r t i c a l P l a t e s — A v e r a g e C o e f f i c i e n t s ( S o ur ce : R ef. 5 ) :

P A ( P a - P v ) g

1/3

= 1.47 Re„-1/3 Re < 1800

&

H â " p

v) g

1/3

= 0 .007 R e j j0 , 4 , *•% > 1800

R% ' < 4 h A t v „ > / W *At = temp d i f f e ren ce be tween sa tu ra t e d vapor and w al li j = he a t o f ev ap ora t io nUg,ko ,p , , = v i s c o s i t y , th e rm a l c o n d u c t i v i t y and d e n s i t y o f l i q u i d

a t s a tu ra t ion vapor t emp

1 6 . 1 2 . 2 . H or i zo n ta l Tubes—Average C oe ff ic ie n ts (Source: Ref. 5) :

j £ = 0 .725 i(H ~ P I ) 9 H vd

W * A t v w

1/4

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16.13. Pool Boiling,

r -| 1/2 r CjAt -i

At = temp difference between wall and saturated vaporwv

c„,u«,Pj,Pro = specif ic heat, viscosity, density and Prandtl

number at saturated liquid temperature

a = surface tension at liquid vapor interface

i« = heat of evaporation

C , = surface coefficient (see Tabl e 16.5)

TABLE 16.5. Values of C and n (source: Ref. 19 ).

Surface-fluid combination C 8 f n

Water-nickel 0.006 1.0Water-platinum 0.013 1.0

Water-copper 0.013 1.0

Water-brass 0.006 1.0

CCl.-copper 0.013 1.7

Benzene-chromium 0.101 1.7

ri-Pentane-chromium 0.015 1.7

Ethyl alcohol-chromium 0.0027 1.7Isopropyl alcohol-copper 0.0025 1.7

35% K 2 C 0 3 -copper 0.0054 1.7

50% K.CO.-copper 0.0027 1.7

r»-Butyl alcohol-copper 0.0030 1.7

16-7

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SECTION 17. CONTACT COEFFICIENTS

An empirical correlation developed by Shevts and Dyban (Ref. 24) gives

estimated thermal contact coefficients for many common ferrous and non-ferrous

metals in contact, including dissimilar metals .

hr/k = ( T T / 4 ) [l + 8 5 ( P / S ) 0 - 8 ]

k = thermal conductivity of the gas phase

P = contact pressure

r = height of a micro-element of roughness plus the '.leight of the wave

for one surface

S = permissible rupture stress

A theoretical approximation of contact coefficients developed by French

and Rohsenow (Ref. 25) can be found by using Fig. 17.1 and the following

properties:

C = constriction number, P/M

P = contact pressure

M => Meyer hardness of the softer contact material

B = gap number = 0.335 C™1

m = 0 . 3 1 5 ( V " A 7 r ) 0 " 1 3 7

A = interface area (one side)

I = effective gap thickness, 3.56 (JL + I ) if (i, + I ) < 280

uin. (smooth contacts), or 0.46 (&, + I ) > 280 uin . (rough

contacts)

H.,%. • mean (or rm s) depths of surface roughnessk, = equivalent conductivity of interstitial fluid, for liquids use

k. = k evaluated at t = (t., + t,)/2, for gases:

k ôK f 1 + 8 Y (v/v) + a 2 - a x - a 2 ) /Pr (y + U J U ^ + a 2

iole e t3

+ £ 1 + £ 2 " e i £ 2

17-1

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k = fluid conductivi ty at zero contact pressure

Pr = Prandtl number

'v = mean molecular velocity

Y = ratio of specific heats

v = kinematic viscosity evaluated at t

a = accommodation coefficient

e = surface emissivitya = Stefan-Boltzmann constant

K = conductivity number , k (k + k )/2k k,

k. = conductivity of first solid evaluated at

t =[\ + ( k ^ + ^ t j j / tk j + k 2 ) J /2

k = con du ctivity of second so lid evalu ated att = [ f c 2 + ( k ^ + k j V / t k j , * ^ ) ] ^

Some thermal contact' data assembled by P. J. Schneider from various sources

(Ref. 19) is given in Fig. 17.2. Also, some additional data from Re f. 26

showing the effects of machining processes and surface matching is given in

Fig. 17.3.

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TABLE 17.1. Interface conditions for contact data given in Fig. 17.2.

RMS surface Mean contact

Curv e Mate rial pair finish (Uin.) Gap material temp (°F)

1 aluminu m (2024-T3) 48-65 -4vacuum (10 mm Hg) 110

2 aluminum (2024-T3) 8-18 —4vacuum (10 ram Hg) 110

3 aluminum (2024-T36-0 (not flat) —4vacuum (10 mm Hg)

1104 aluminum (75S-T6) 120 . air 200

5 aluminum (75S-T6) 65 air 200

6 aluminum (75S-T6) 10 air 200

7 aluminum (2024-T3) 6-8 (not flat) lead foil (0.008 in.) 110

8 aluminum (75S-T6) 120 brass foil (0.001 in.) 200

9 stainless (304) 42-60 —4vacuum (10 mm Hg) 85

10 stainless (304) 10-15 —4vacuum (10 mm Hg) 85

11 stainless (416) 100 air 200

12 stainless (416) 100 brass foil (0.001 in.) 200

13 magne sium (AZ-31B) 50-60 (oxidized) —4vacuum (10 mm Hg) 85

14 magne sium (AZ-31B) 8-16 (oxidized) —4vacuum (10 mm Hg) 85

15 copper (OFHC) 7-9 —4vacuum (10 mm Hg) 115

16 stainless/aluminum 30-65 air 200

17 iron/aluminum - air 80

IB tungsten/graphite — air 270

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TABLE 17. 2. Interface conditions for contact data given in Fig . 17.3 .

Roughness

RMS (Uin.)

Block

Fluid

in Temp

Curve Material Finish 1 2 Gap (°F) Condition

a Cold rolled steel Shaped 1000-1000 Air 200 Parallel cuts , rusted

b Cold rolled steel Shaped 1000-1000 Air 200 Parallel cuts , clean

c Cold rolled steel Shaped 1000-1000 Air 200 Perpendicular cuts , clean

d Cold rolled steel Hilled 125-125 Air 200 Parallel cuts , rusted

e Cold rolled steel Milled 125-125 Air 200 Parallel cu t s , clean

f Cold rolled steel Shaped 63-63 Air 200 Perpendicular cuts , clean

g Cold rolled steel Shaped 63-63 Air 200 Parallel cuts , clean

h Cold rolled steel Lapped 4-4 Air 200 Clean

i 416 Stainless Ground 100-100 Air 200

J 416 Stainless Ground 100-100 Air 400

k 416 Stainless Ground 30-30 Air 200

X 416 Stainless Ground 30-30 Air 400

m Stainless Milled 195-195 Air Clean

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* * ^ 6 X 1 0 " 'i-i1-1

5 X 1 0 "4 X 1 0 '3 X 1 0 "1

2 X 1 0 "1

10" 1 ,8 X 10- 2

6 X 10" 2

4 X 10-22X10-2' o -2 ,8X10-36X10-34X10-32 X 10" 3

1 0 ~3 .

8 X 1 0 "4

6 X 1 0 i4 X 10"42 X 10 -4

8 X 10-f6 X 10"?4 X 1 0 "5

2 X 10"s

1 0 ~5

FIG. 1 7 . 1 . Thermal contact coefficient from theory (source: R e f . 2 5 ) ,

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J — I I I . . . I

5 10 50 100 200 500 1000

p - lb/in. 2

FIG. 17.2. Thermal contact coefficient data (source: Ref. 19 ), 1

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0.016

u

coo

100 200 300 400

Contact pressure — lb / in .2500

Parallelcuts

Perpendicularcuts

P IG . 1 7 . 3 . T h er m al c o n t a c t c o e f f i c i e n t fo r b a r e s t e e l s u r f a c e s , ( s o u r c e :

Re f. 26 , s e c t i o n G502 .5 , p . 5 ) .

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Table 18.1 Thermal properties of selected metals (source: Refs. 35 to 3 9 ) .

Properties at 30 0 K Thermal conductivity (k) , (W/m-K) x 10' T

(kg/m 3) x 10 * 3 (J/kg-K) x 1 0 " 3 (W/m-K) x 1 0 ~ 2 (m ? /s ) x 10* 50 100 200 400 600 flOO 1000 1200

O

Aluminum: Pure 2.70 0.88 2.4 1.01 235. 117. 12.3 3.0 2.4 2.4 2.3 2.2

2028 (4.5* Cu, 1.5* Kg) 2.77 0.84 1.3 0.56 0.08 0.17 0.40 0.65 1.0 1.4

5086 (4.1* rig, 0.5* Mn) 2.65 0.90 1.3 0.55 0.08 0.17 0.40 0.63 0.85

6063 (0.7* Hg , 0.1* Si) 2.70 0.92 2.1 0.85 0.86 1.7 2.8 2.1 2.0

Beryllium: Pure 1.85 1.84 2.0 0.59 i a . o ' 34.8 40.0 9.9 3.01 1.61 1.26 1.07 0.89 0.73

Chromium: Pure 7.19 4.56 0.90 0.03 3.9 6.0 3.2 1.6 1.1 0.B7 0.B1 0.71 0.65 0.62

Copper: Pure 8.96 0.38 4.0 1.17 196. 105. 12.2 4.8 4.1 3.9 3.8 3.7 3.6 3.4

Commercial bronze (10* Zn) 8.80 0.41 1.8 0.50 0.20 0.42 0.81 1.2

Yellow brass (35* Zn) 8.47 0.41 1.2 0.35 0.59 0.94 1.3 1.5

Naval brass (38* Zn, O.B* 5n) 8.41 0.38 0.75 0.23 0.85 1.01

German silver (22 * Zn, 15* Hi) 8.55 0.40 0.25 0.07 0.028 0.07 0.15 0.17 0.20

Cupronickel (20* Ni, 2* Zn) 8.94 0.38 1.5 0.44 0.11 0.20 0.38 0.92

Constantan (40* Ni) B.90 0.42 0.23 0.06 0.08 0.15 0;17 0.19

Manganin (13* Hn, 4* Ni) 8.19 0.40 0.22 0.07 0.015 0.032 0.082 0.13 0.17 0.28

Gold: Pure 19.32 0.13 3.2 1.27 28.2 15.0 4.2 3.5 3.3 3.1 3.0 2.9 2.B 2.6

Iron: Pure 7.87 0.45 0.80 0.23 7.1 10.0 3.7 1.3 0.94 0.69 0.55 0.43 0.33 0.2H

Wrought iron (<0.S* C) 7.87 0.46 0.60 0.17 0.5B 0.49 0.39

Gray cast iron (3.0* C, 0.6* Si) 7.15 0.42 0.29 0.10 0.31 0.34 0.29 0.22 0.20

5AE 1095 steel (0.9* C, 0.3* Hn) 7.83 0.47 0.44 0.12 0.08 0.22 0.34 0.43

AISI 4340 steel (1.9* Ni, 0.7* Cr) 7.84 0.46 0.36 0.10 0.26 0.31 0.38 0.38 0.34

Nickel steel (9* Ni, 0.8* Hn) 7.96 0.46 0.29 0.08 0.18 0.24 0.33 0.34 0.33

invar (31* Ni, 5* Co) 8.00 0.46 0.14 0.08 0.12 0.16 0.19 0.22

SAE 4130 steel (1* Cr, 0.5* Mn) 7.86 0.46 0.41 0.11 0.06 0.16 0.17 0.37

AISI 304 stainless (19* Cr, 10* N1) 7.90 0.38 0.15 0.05 0.008 0.021 0.06 0.09 0.13 D.17 0.20 0.22 0.25 0.28

AISI 316 stainless (1 7* Cr, 12 * Ni) 8.00 0.46 0.14 0.40 0.008 0.021 0.06 0.09 0.12 0.15 0.18 0.21 0.23 0.25

Lead: Pure 11.34 0.13 0.35 0.24 1.78 0.59 0.44 0.40 0.37 0.34 0.31

Solder ( 80* Pb, 20* Sn) 10.20 0.37

Solder (50* Pb, 50* Sn) 8.B9 0.18 0.46 0.29

Lithium: Pure 0.53 3.97 0.77 0.37 6.1 7.2 2.3 1.1 0.88 0.72

Magnesium: Pure 1.74 1.00 1.56 0.90 11.7 13.9 3.8 1.7 1.6 1.5 1.5 1.5

(6* Al, 2* Si) 1.84 1.00 0.70 0.38 0.50 0.62 9.BO

(12* Al, 2* Si)

Molybednum: Pure1.81

10.220.990.25

0.561.4

0.31

0.55 1.45 2.77 3.0

0.30

1.8

0 4 41.4

0.68

1 j 1.3 1.2 1.1 1.0

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Table 1B.1 (Continued).

Properties at 300 K Thermal conductivity (It), (W/m-K) x 1 0 ~ ?

P, c, k, a,

fetal (kg/m 3 ) x 10~ 3 (J/kg-K) x 1 0 " 3 (W/m-K) x 1 0 " 2 (m ?/s) x 10* 10 K 20 50 100 ?0D 400 600 800 1000 12O0

Ni cke l : Pure

Duranickel (4.5* Al)

8.90

8.26

0.45

0.54

0.91

0.19

0.23

0.04

6.0 8.6 3.4 1.6 1.1 0.B0 0.66 0.67 0.72 0.76

Monel (30* Cu, 1.4* Fe) 8.84 0.43 0.23 0.05 0.08 0.14 0.17 0.20

Inconel X-750 (16% Cr, 8* Fe) 8.51 0.46 0.12 0.03 0.010 0.025 0.06 0.08 0.10 0.13 0.17 0.21 0.25 0.29

Nichrome (20* Cr) B.40 0.45 0.13 0.03 0.14 0.17 0.21 0.25 0.28

Nichrome V (24* Fe, 16* Cr) 8.25 0.44 0.10 0.03 0.12 0.15 0.19 0.22 0.26

Niobium: Pure 8.57 0.27 0.54 0.23 2.2 2.3 0.76 0.55 0.53 0.55 0.5B 0.61 0.64 0.68

Platinum: Pure 21.45 0.13 0.71 0.25 12.3 4.9 1.1 0.78 0.72 0.72 0.73 0.76 0.79 0.83Plutonium: Pure 19.B4 0.15 0.67 0.23 0.03 0.48 0.90

Rhenium: Pure 21.04 0.14 0.48 0.16 14.0 8.4 0.96 0.60 0.51 0.46 0.44 0.44 0.45 0.46

Silver: Pure

Sterling (7.5* Cu)

Eutectic (28* Cu)

10.49

10.38

10 . 6

0.24 4.3

3.5

• 3.3

1.71 168 . 51. 7.0 4.5 4.3 4.2 4.1 3.9 3.7 3.6

00Tin: Pure 7.30 0.22 0.67 0.42 19.3 3.2 1.2 0.85 0.73 ' 0.62

1S3

Solder (40* Pb) 8.44 0.19 0.51 0.32 0.43 0.56 0.52 0.54 0.52

Tantalum: Pure 16.6 0.14 0.58 0.25 1.08 1.5 0.72 0.59 0.58 0.58 0.59 0.59 0.60 0.61

Tungsten: Pure 19.3 0.13 1.8 0.72 82. 32. 4.7 2.4 2.0 1.6 1.4 1.3 l.Z 1.1

Titanium: Pure 4.51 0.50 0.22 0.10 0.14 0.28 0.40 0.31 0.25 0.20 0.10 0.20 0.71 0.2?

. A-110 AT (5* Al, 2.5* Sn) 4.46 0.53 0.075 0.03 0.088 0.11 0.14

Uranium: Pure 19.07 0.12 0.28 0.12 0.098 0.16 0.19 0.22 0.25 0.30 0.34 0.39 0.44 0.40

Zinc: Pure 7.13 0 .39 1.2 0.43 43. 10.7 2.1 1.2 1.2 1.2 1.1

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T M L E I B , 2 . T h e r n a l p r o p e r t i e s o f m i s c e l l a n e o u s a o l l d s { s o u r c e : R e f s . 3 5 t o 3 8 ] .

P r o p e r t l e g a t 3 Q 0 K T h e r m a l c o n d u c t i v i t y ( k ) , ( W /m 'K ) > 10

(kg/aT) x 10 "J ( J /kg- l t )k.

( m / a ) * 1 0 1 0 K 1D0 200 400 600 1000 1200

C e r a m ic s * ( F o l y c r y a t a l l i n e , 9 9 . 5 'p u r i t y , 9 8 % solid}

A 1 2 0 3

B«0

MgOS i 0 2 [ h i g h p u r i t y f u s e d )

T W 2

TiO

s c o 2

G l a s s e s t

B o r o s i l i c a t e ( P y r e x )

S o d a l i n e ( 7 5 1 S i O )

Vi t r e o u s s i l i c a ( 1 0 01 S i O j ]

Zi nc crown (65% SiO )

I n s u l a t i o n s ; ( P e r h i g h t e m p s . )

A l u q t i n a , f u s e d ( 9 0 1 A 1 , 03 )

A s b B S t o s p a p e r [ l a m i n a t e d )

D i a t o n a c e o u s e a r t h .s i l i c a ( p o w d e r )Pirebrick (58% SiO ,

37% A 1 2 0 3 )

Magnesite (65% MgO)

Micro quart* fiber (blanket)Rockwool (loose)'

Zirconia (grain)

Insulations: (For low temps.)

Expanded glass (Foanglas)

Fiberglass (board}

Fiberglass (blanket)

Silica aerogel (powdei)

polystyrene foaa (latmabs)

polystyrene foam (10~ atmab s)

3 . 8 4 0 . 7 9 3 6 . 0 0 . 1 2

2 . 9 7 1 . 0 0 2 7 2 . 0 0 . 9 2

3 . 2 1 0 . 9 2 4 8 . 0 0 . 1 62 . 2 1 0 . 7 5 1.4 0.008

9 . 5 8 0 . 2 3 1 3 . 0 0 . 0 6

3 . 9 1 0 . 7 1 8 . 0 0 . 0 3

5 . 2 B 0 . 4 6 1 .6 0.007

2 . 2 1 0 . 7 1 1 .1 0.007

2 . 5 2 0 . 6 6 0 . 9 0.005

2 . 2 1 1 . 0 0 1.4 0.006

2 . 6 0 0 . 6 7 1 . 1 0.006

2 . 7 0 0 . 8 3 5 . 2 0 . 0 2

0 . 3 5 0 . 8 4 0.048 0.002

0 . 8 1 0 . 9 2

0 . 1 9 1 .13

0 . 0 5 0 . 8 4

0 . 1 6

1 . 8 1 D.50

0 . 1 7

0 . 1 8

0 . 0 5

0 . 0 8

0 . 0 8

0 . 0 8

0 . 3 4 0.005

0.050 0.002

0.036 0.009

0.046

0 . 1 8 0.002

0.062

0.040

0.038

0.022

0.036

0 . 0 1 7

10. 550 . 2 6 0 . 160 . 100 . BO.

4 2 4 0 . 1 9 6 0 . 111 0 . 7 0 0 . 4 7 0 .

7 5 0 . 3 5 0 . 2 2 0 . 1 4 0 . 9 0 .6 . 9 11 . 15 . 18 . 22 . 29 .

ISO. 100 . 70 . 50 . 40 .

1 D 0 . 7 0 . 5 0 . 4 0 . 3 0 .

1 7 . 18 . 19 . 19 .

6. 9 .

a.12 .

11 .

15 . 19 .

0.31

0.12

0.07 0.12

0.51

0.24

0.17

0.24

0.12

0.73

0.79

0.61

0.48

0.66

1.9

1.0

5.2

0.89

1.0

2.1

1.3

6.6

1.3

1.3

2.2

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TABU: IB. 2 . (Cont in ued . I

COI

P r o p e r t i e s a t 3 0 0 K T h e r m al c o n d u c t i v i t y ( k ) , Wmfl) x 10~2

P, c , k . a .M a t e r i a l < k g / m3) x 1 0 - 3 ( J A g - K ) x 10 ~3 W/m-K (r a 2 / s ) * 10* 10 K 20 50 100 200 400 600 800 1000 1200

P l a s t i c s :

A c r y l i c , PHKA ( P l e x i g l a s ) l . i a 1 . 4 6 0 . 1 6 0 . 0 0 1 0 . 6 0 . 7 1.2

N y l o n 6 1 . 1 6 1 . 5 9 0 . 2 5 0 . 0 0 1 0 . 3 0 . 9

P o l y v i n y l c h l o r i d e ( r i g i d ) 1 . 4 0 1 .00 0 . 1 5 0 . 0 0 1 1.5 1.6 1.5

Te f l o n , P T F E 2 . 1 8 1 . 0 5 0 . 4 0 0.002 1.0 1 .5 2 . 0 2 . 5 4 . 5 5 . 5

P o l y e t h y l e n e , h i g h d e n s i t y 0 . 9 5 2 . 3 0 0 . 5 0 0.002

R o c k s :

G r a n i t e 2 . 6 0 0 . 7 9 3 . 4 0 . 0 1 7 30. 2 " .

M a r b l e 2 . 5 0 0 . 8 8 l .B 0.00B 1 7 . 11 .S a n d s t o n e 2 . 2 0 0 . 9 2 S . 3 0.026 4 4 . 30 .

s h a i a 2 . 6 0 0 . 7 1 1 . 8 0 . 0 1 0 15 . 14.

H o o d s : ( a c r o s s g r a i n , o v e n d r y )

B a l s a 0 . 1 6 0.059

D o u g l a s f i r 0 . 4 6 2 . 7 2 a . i i 0 . 0 0 1

O a k , r e d 0 . 6 ? 2 . 3 8 0 . 1 7 0 . 0 0 1

P i n e , w h i t e 0 . 4 0 2 . 8 0 0 . 1 0 0 . 0 0 1

Redwood 0 . 4 2 2 . 9 0 0 . 11 0 . 0 0 1

M i s c e l l a n e o u s :C a r b o n b l a c k ( p o w d e r ] 0 . 1 9 0 . 8 4 0 . 0 2 1 0 . 0 0 1

C a r b o n ( p e t r o l e u m c o k e ) 2 . 1 0 0.B4 1 . 9 0 . 0 11 21 . 25 . 26 . 3 0 .

G r a p h i t e ( AT J )

( U t o g r a i n s ) ' 1 .73 0 .B4 1 2 5 . 0 0 .B9 S . 25 . 1 7 0 . 5 B 0 . 1 2 0 0 . 11 8 0 . 9 5 0 . 7 7 0 . 640 .

( i t o g r a i n s ) 1 . 7 3 0 . 8 4 9 8 . 0 0 . 6 7 4 . 19 . 1 3 0 . 4 2 0 . B 6 0 . 9 0 0 . 7 3 0 . 5 9 0 . 4 9 0 .

G r a p h i t e ( p y r o l y t i c )

( | t o g r a i n s ] 2 . 2 0 0 . 8 4 2 0 0 0 . 0 1 0 . B BID. 4 2 0 0 . 2 3 0 0 0 . 5 0 0 0 0 . 3 2 0 0 0 . 1 4 6 0 0 . 9 3 0 0 . 6 8 0 0 . 5 3 0 0 .

( 1 t o g r a i n s ) 2 . 2 0 0 . 8 4 9 . 0 0 . 0 5 2 7 0 . 11 0 0 . 1 0 0 0 . 3 9 0 . 1 5 0 . 7 0 . 40 . 30 . 20 .

C o n c r e t e 2 . 2 8 0 . 6 7 l . B 0 . 0 1 2

Mica 1 .96 0 . 8 8 0 . 4 3 0.002

R u b b e r ( h a r d ) 1 .19 1 .88 0 .16 0 . 0 0 1

G y p s u n b o a r d 0 . S 2 0 . 1 1

Sand (dry) 1 .52 0 . 8 0 0 . 3 3 0.003

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TABLE 1 8 . 3 . Thermal p r o p e r t i e s of some me ta ls a t 10 K above t h e i r me lt in g

po int (so ur ce: Refs. 35 and 38 ).

mp, k, P. c, YMetal K W/m'k k g / m 3 kJ/kg 'k kJ/kg

Aluminum 933.2 91 2390 1.09 395.4

Copper 1356. 167 7940 0.49 205.

Iron 1810. 41 7020 0.82 281.6

Lead 600.6 16 10700 0.21 24.7

Lithium 453.7 43 520 4.25 663.2

Mercury 234.3 7 13650 1.34 11.3

Potassium 336.8 54 820 0.84 61.5

Sodium 371.0 87 930 1.34 114.6

NaK (eutectic) 262. 13 850 1.00 ~

Tin 505.1 31 6980 0.23 60.2

Zinc 692.7 50 6640 0.50 102.1

18-5

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TABLE 18.4. Room temperature total emissivities (source: Ref . 3 7 ) .

Silver (highly polished ) 0.02 Brass (polished) 0.60

Platin um (highly polished) 0.05 Oxidized copper 0.60

Zinc (highly polished) 0.05 Oxidized steel 0.70

Alumi num (highly polished) 0.08 Bronze paint 0.80

Monei metal (polished) 0.09 Black gloss paint 0.90

Nickel (polished) 0.12 White lacquer 0.95

Copper (polj :..)ied) 0.15 Whit e vitreous enamel 0.95

Stellite (polisned) o.ia Asbestos paper 0.95

Cast iron (polished) 0.25 Green paint 0.95

Monel me tal (oxidized) 0.43 Gray paint 0.95Aluminum paint 0.55 Lamp black 0.95

18-6

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TABLE 1 8 . 5 . To t a l e m i s a i v i t i e s o f m i s c e l l a n e o u s m a t e r i a l s ( s o u r c e : R ef . 3 7 ) .

Temp, Emis- Temp, Emis -

H a t e r i a l °c s i v i t y M a t e r i a l °c s i v i t y

A l l o y s I r o n , r u s t e d 25 0 .65

2 0 N i - 2 5 C r- 5 5 F e , o x i d i z e d 2 00 0 .90 w r o u g h t , d u l l 2 5 0.94

500 0.97 o x i d i z e d 350 0.94

6 0 N i - 1 2 C r- 2 B F e , o x i d i z e d 2 7 0 0 . 8 9 L e a d , u n o x i d i z e d 100 0.05

560 0 .62 o x i d i z e d 200 0.63

BONi-20Cr, ox id ize d 100 0.87 M e r c u r y, u n o x i d i z e d 2 5 0 .10

600 0.87 100 0.12

1300 0.89 Molybdenum, un ox idi ze d 1000 0 .13

Aluminum, unox id ized 25 0.022 1500 0 .19

100 0.028 2000 0.24

500 0 . 6 0 M o ne l m e t a l , o x i d i z e d 200 0 .43

o x i d i z e d 200 0.11 600 0.43

600 0.19 N i c k e l , u n o x i d i z e d 25 0.045

B i s m u t h , u n o x i d i z e d 25 0.048 100 0.06

100 0.06 1 500 0.1 2

B r a s s , o x i d i z e d 200 0.61 laao 0 .19

600 0 . 5 9 o x i d i z e d 200 0.37

u n o x i d i z e d 25 0.035 1200 0.85

100 0.035 P l a t i n u m , u n o x i d i z e d 25 0.037

C a r b o n , u n o x i d i z e d 25 0 .81 100 0.047

100 0 . 8 1 500 0.096

500 0 .81 1000 0 .152

Chromium, unox id ized 100 0.08 1500 0 . 1 9 1

C o b a l t , u n o x i d i z e d 5 00 0 . 1 3 S i l i c a b r i c k 1 00 0O.80

1000 0.23 1100 0 .85

Columbiura , unox id ized 1 5 00 0 . 1 9 S i l v e r, u n o x i d i z e d 1 00 0 .02

2000 0.24 500 0.035

C o p p e r, u n o x i d i z e d 1 00 0.02 S t e e l , u n o x i d i z e d 100 o.oel i q u i d 0 .15 l i q u i d 0 . 2 8

o x i d i z e d 2 0 0 0 . 6 o x i d i z e d 25 0.80

1000 0.6 200 0 .79

c a l o r i z e d 1 0 0 0 .26 600 0.79

5 00 0 . 2 6 S t e e l p l a t e , c o u gh 4 0 0 . 9 4c a l o r i z e d , o x i d i z e d 200 0.18 400 0.97

600 0.19 c a l o r i z e d , o x i d i z e d 200 0.52

F i r e b r * c * 1000 0 .75 600 0.57

G o l d , u n o x i d i z e d 100 0.02 Ta n t a l u m , u n o x i d i z e d 1500 0.21

500 0.03 2000 0.26

Gold enamel 100 0 . 3 7 Ti n , u n o x i d i z e d 25 0.043

I c o n , u n o x i d i z e d 100 0 .05 100 0.05

o x i d i z e d 1 0 0 0.74 Tu n g s t e n , u n a x i d i z e d 25 0.024

500 0.84 100 0.032

1200 0.89 500 0 . 0 7 1

c a s t , u n o x i d i z e d 100 0 .21 1000 0 .15l i q u i d 0 .29 1500 0 . 2 3

c a s t , o x i d i z e d 200 0.64 2000 0.2S600 0 .78

Z i n c , u n o x i d i z e d

300 0 .05c a s t , s t r o n g l y o x i d i z e d 40

250

0 .95

0 .95

18-7

8/10/2019 6224569


TABLE IS.6. Electrical resistivity of some common metal s (source: R e f. 3 7 ) .

R e s i s t i v i t y , Temp. S p e c i f i c Mel t ing

llfi*cm c o e f f i c i e n t g r a v i t y . p o i n t .

Metal 20°C 20°C g/cra °cA d v a n c e . S e e c o n s t a n t a n

Aluminum 2.824 0.0039 2 . 7 0 659

Antimony 4 1 . 7 .0036 6 . 6 6 3 0Arsen ic 3 3 . 3 .0042 5 . 7 3 —Bismuth 120 . 0 0 4 9 . 8 2 7 1

Brass 7 . 0 0 2 8 . 6 900

Cadmium 7 . 6 .0038 8 .6 321

C a l i d o . S e e nichrome

Climax 87 .0007 e . i 1 2 5 0

Cobal t 9 . 8 .0033 B . 7 1 1480

Constantan 49 . 00001 8 .9 1190

Copperi annealed 1.7241 .00393 8 . 8 9 1 0 8 3

hard-drawn 1 . 7 7 1 .00382 8 . 8 9 —E u r e k a . S e e cons t an tan

E x c e l l o 92 .00016 8 . 9 1 5 0 0

G as carbon 5000 - . 0 0 0 5 — 3500

German s i l v e r , 18% H i 3 3 .0004 B.4 1100

Gold 2 . 4 4 .0034 1 9 . 3 1 0 6 3

I d e a l . S e e cons t an tan

i r o n , 99.984 pure 10 . 0 0 5 7 . 8 1 5 3 0

Lead 22 .0039 11 . 4 3 2 7

Magnesium 4 . 6 . 0 0 4 1 . 7 4 6 5 1

Manganin 44 . 00001 8 . 4 910

Mercury 95.783 .00089 13.546 - 3 8 . 9

M o l y b d e n u m , drawn 5 . 7 . 0 0 4 9 . 0 2500

Monel metal 42 .0020 8 . 9 1 3 0 0

N i c h r o m e3 100 .0004 8 . 2 1 5 0 0

Nicke l 7 . 8 . 0 0 6 8 . 9 1 4 5 2

Pal ladium 11 .0033 1 2 . 2 1 5 5 0

Phosphor bronze 7 . 8 . 0 0 1 8 8 . 9 750

Plat inum 1 0 . 0 0 3 2 1 . 4 1 7 5 5

S i l v e r 1 . 5 9 .0038 1 0 . 5 9 6 0

S t e e l , E . B . B . 1 0 . 4 . 0 0 5 7 . 7 1 5 1 0

S t e e l , B . B . 1 1 . 9 . 0 0 4 7 . 7 1 5 1 0

S t e e l , S iemens-Mart in 1 8 . 0 0 3 7 . 7 1510

S t e e l , manganese 70 . 0 0 1 7 . 5 1 2 6 0

Tantalum 1 5 . 5 . 0 0 3 1 1 6 . 6 2850

T h e r l o a 47 . 00001 8 . 2 —T i n 1 1 . E . 0042 7 . 3 2 3 2

T u n g s t e n , drawn 5 . 6 .0045 1 9 3400

Zinc 5 . 8 .0037 7 . 1 419

"Trade

18-8

8/10/2019 6224569


TABIB I B .7 . Thermal conduc t iv i ty in tegr a l s of misce l l aneou s ma te r i a l s {source : Ref . 76] .

Q - 7 *_) , k_ - J t dt , W/n x 10~ 2

L i X - ir

C o p p e r s S t e e l M i • c. A l l o y s

Temp, H i-p ur ity A l u m i n u m S ta in - In co ne l G l a s s a n d p l a s t i c s SoftK O.F.H.C. anneale d Brass (Pb) 2024-T4 6063-T5 SAE 1020 les s (annealed) glas s nylon persp ex Constantan Ag sold er sold er

6 6. 1 166 0.053 0.080 0. 850 0.088 0.0063 0.0133 2. 11 X 10 J 0. 32 x 10 ' 1.18 x 10"' 0.024 0.059 0.425

8 14. 5 382 0.1290.197

2. 05 0.231 0.0159 0.0348 4 ..43 0. 60 2.38 0.066 0.148 1.05

10 25. 636 0.229 0.347 3. 60 0.431 0.0293 0.0653 t. .81 1. 48 3.59 0.128 0.268 1.B3

15 61. 4 1270 0.594 0.872 9. 00 1.17 0.0816 0.182 13. 1 4. 10 G.69 0.375 0.688 4.16

20 110 1790 1.12 1.60 16. 5 2.22 0.163 0.356 20, 0 8. 23 10.1 0.753 1.25 6.86

25 169 2160 1.81 2.51 25. B 3.52 0.277 0.592 27. 9 13. 9 14.4 1.24 1.92 9.66

30 228 2410 2.65 3.61 36. 5 5.02 0.424 0.882 36. 8 20. B 19.6 1.B1 2.67 12.5

35 285 2580 3.63 4.91 48. 8 6.74 0.607 1.22 47. 1 29. 0 25.9 2.44 3.52 15.3

40 338 2700 4.76 6.41 62. 0 8.67 0.824 1.60 58. 6 3B. 33.0 3.12 4.47 1B.1

50 426 2860 7.36 9.91 89. 13.1 1.35 2.47 84. 6 60. 49.5 4.57 6.62 23.4

60 496 2960 10.4 14.1 117 18.1 1.98 3.45 115 B5. 6B.3 6.12 9.12 2B.5

70 554 3030 13.9 18.9 143 23.6 2.70 4.52 151 113 B8.5 7.75 12.0 33.6

76 586 3070 16.2 22.0 158 27.1 3.17 5.19 175 131 101. 8.75 13.9 36.7

SO 606 3090 17.7 24.2 167 29.5 3.49 5.66 194 142 110. 9.43 15.2 15.B

90 654 "140 22.0 30.1 190 35.5 4.36 5.85 240 173 132 . 11.1 IB.7 44.1

100 700 3180 26.5 36.3 211 41.7 5.28 8.06 292 204 155. 12.8 22.6 49.4

120 788 3270 36.5 50.1 253 54.5 7.26 10.6 408 269 200. 16.2 31.1 60.3

140 874 3360 47.8 65.4 293 67.5 5.39 13.1 542 336 247. 19.7 40.6 71.4

160 956 3440 60.3 83.1 333 80.5 11.7 15.7 694 405 294. 23.2 51.0 82.6

ISO 1040 3520 ' 73.8 100 373 93.5 14.1 IB.3 85B 475 342. 26.9 62.2 93.8

200 1120 3600 B8.3 119 413 107 16.6 21.0 1030 545 390. 30.6 74.0 105

250 1320 3800 128 171 513 139 23.4 26.0 1500 720 510. 40.6 105 133

300 1520 4000 172 229 613 172 30.6 35.4 1990 895 630 . 51.6 138 162

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Table 18.8 Specific heat of miscellaneous solids (source: Rets. 35 to 38) .

Specific heat (c) , J/ g-K

Material 10 20 50 100 200 400 600 800 1000 1200

Aluminum; Pure 1.4 8.9 142 481 790 161 1026 1092 1054 1096

2024 418 515 737 920 1029 1318

Beryl H u m 0.389 1.61 19.2 199 1113 2134 1046 2804 3022 3223

Copper: Pure 0.86 7.7 99 254 360 423 427 434 439 443

Brass 30 % Zn) 0.65 10.5 120 255 360 401 42B 448 469

Gold 2.2 15.9 72 108 122 134 135 138 140 142

Iron: Pure 1.24 4.5 55 216 389 485 582 677 774 665

AISI 316 stainless 0.80 13.1 110 265 420 510 560 580 590 600

: SAE 1010 steel 1.88 6.69 75 251 418 628 669 690 711 724

Lead 13.7 53 103 118 121 141 121 10B 103 96

Nickel 1.62 5.8 68 232 385 502 580 539 540 540

Platinum 1.12 7.4 55 100 121 133 142 146 150 155

Silver 1.8 15.5 108 187 226 243 251 259 267 280

Tin 4.2 4 0 130 189 200 242

Tungsten 0.234 1.89 33 88 122 133 138 142 146 130

Zinc 2.5 26 171 293 352 401 472 489 460 434

Carbon: Graphite 1.38 71 41.8 138 418 1004 1339 1590 1716 1799

Ceramic: A 1 2°30.08 0.84 14.2 125 502 920 1088 1172 1213 1255

Glasses; Borosi Licate

(pyrexi 4.2 25.9 125 272 544 837 1088

Quartz 16.7 16.7 96 255 502 879 1046 1172 1213 1255

Plastics Acryl c, PMMA

(plexiglas) 22 79 250 540 360 1640

NEMA G10 18 89 240 360 630 1110 1600

Teflon PTFE 20 75 200 380 700 1300

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REFERENCES

1 . M. Ja ko b, Heat T ra ns fe r, V ol . I (Wiley, New York, 19 49 ).

2. P. J . Sc hn ei de r, Condu ct ion Heat Trans fer (Addison-Wesley, Reading , 19 55 ) .

3 . V. S . A rp ac i , Conduct ion Hea t Tra nsfe r (Addison-Wesley, Reading , 19 66 ) .4. S . S. Ku ta t e l a dz e and V. M. B or i s ha ns k i i , A Con c i se Encyc loped ia o f Hea t

T ra ns fe r (Pergaiaon, New York , 19 66 ).

5 . J . P. Holman, Heat T ra ns fe r (McG raw-Hill , New Yo rk, 19 63 ).

6. R. B. B ir d , W. E. S te w ar t , and E. M. L ig ht fo ot , T ra ns po rt Phenomena

(W iley, New York, 19 60 ).

7 . E. R. G. E ck er t and R. M. Drake J r . , Heat and Mass T ra ns fe r (Mc Graw -Hill ,

New Y or k, 1959) .

8 . K. A. G ard ne r, " Ef f ic i an cy of Extended S ur fa ce s , " T ra ns . ASHE 67, 621-631

(1945) .

9. H. S. Ca rsla w and J . C. J a e g e r , Condu ct ion of He at In S o l i d s , 2nd Ed.

(Oxford Press , London, 1959).

10. D. Q. Kern and A. D. K ra us , Extended Surfac e Heat Tr an sfe r (McG raw-Hil l ,

New Y ork , 1972) .

1 1 . E. M. Spa rrow and S. H. L in , I n t . J . Heat Mass T ra n sf er 7_, 951-952

(1964) .12. M. P. H e i s l e r, "Tempera tu re Ch a r t s fo r Ind uc t ion and Cons tan t Tempera tu re

H e a t in g ," T ra n s . ASMS 69 , 2.T7-236 C.947).

1 3 . H. G rob er, S . Erk , and U. G r i g u l l , Fundamenta ls of Heat Tran sfer

(Mc Graw -Hill , New York, 19 6 1) .

14. R. E . D. S te w ar t and F. C. W ess l in g J r . , On Heat Conduct ion in S ol id s

Hav ing Tempera tu re Dependen t P rope r t i e s , " Sand ia Labs , A lbuque rque ,

SC-RR-70-334 (1970).

15 . J . H. VanSant and M. B. Lar so n , "Heat Tr an sfe r From a Se : i \ i - Inf i n i te

Rectangular S t r i p , " J . Hea t Tr an s fe r 85 , 191-192 (1 963) .

1 6 . J. H. VanSant and J . H. P i t t s , " V a r i a b l e P r o p e r t y E f f e c t s on t h e I n s i d e

Su rface Temperature of a Heated Tub e ," J . Heat Tr an sf er 88 , 340-341

(1966) .

17 . B. 0 . P e ir c e and R. M. F o s t e r , A Shor t Table of I n te g r a l s (Ginn , Naw York ,

1 9 5 6 ).

18 . E. J a h n k e and F. Emde, T ab les of Fu nc t io ns , 4t h Ed. (Dover, New Yo rk,1945) .

R-l

8/10/2019 6224569


19 . W. M . Rohsenow and J. P. Hart nett, Eds ., Handbook of Heat Transfer

(McGraw-Hill, New Yo rk, 1973) .

">.. G. E . My er s, Analytical Meth ods in Conduction Heat Transfer (McGraw-Hill,

Hew York , 1971) .

21 . W. M. Kays , Convective He at and Mass Transfer (McGraw-Hill, New Yo rk ,

1966) .

22 . J. G. Knudsen and D. L. K a t z , Fluid Dynamics and Heat Transfer

(McGraw-Hill, New York , 1958) .

2 3 . W. H . McAdams, Hea t T ra n sm is s i o n , 3 rd Ed . (M cGr aw-H i l l , New Yor k , 1 9 5 4 ) .

2 4 . I . I . S h e v t s an d E . P. D y b a n , " C o n t a c t H e a t T r a n s f e r B e tw e e n P l a n e M e t s l

S u r f a c e s , " I n t . C hem . E n g . 4_, 6 2 1 - 2 4 1 ( 1 9 6 4 ) .

<J5. H . Fe ne ch and W. H . Rohsenow " P re d i c t i o n o f The rma l Conduc t ane o f M e t a l l i c

S u r f a c e s i n C o n t a c t , " J . H e a t Tr a n s f e r 8 5 , 1 5 - 2 4 ( 1 9 6 3 ) .

2 6 . N . D . F i t z r o y , . E d . , H e a t Tr a n s f e r D a t a B ook ( G e n e r a l E l e c t r i c C o . ,

S c h e n e c t a d y, 1 9 7 0 ) .

2 7 . J . H . Va n S a n t , "Te m p e r a tu r e V a r i a t i o n s o n t h e S u r f a c e o f a S t r i p - H e a t e d

F l a t P l a t e , " J . H e a t T r a n s f e r 8 9 . 3 7 2 - 3 7 3 ( 1 9 6 7 ) .

2 8 . J . H. Va n S a n t , Te m p e r a t u r e o f a F l a t P l a t e H a v i n g A l t e r n a t i n g A d i a b a t i c

a nd C o n s t a n t H e a t F l u x S t r i p s o n O ne S i d e , L a w r e n c e L i v e rm o r e L a b o r a t o r y,

L iv e r m or e , CA, OCRL-7990 (1 9 6 4 ) .

2 9 . J . H . Va n S a n t , "T he S p o t I n s u l a t e d P l a t e , " N u c . E n g . D e s i g n 8_, 2 4 7 - 2 5 0( 1 9 6 8 ) .

3 0 . N . R . D e s R u i s s e a u a n d R . D . Z e r k l e , " Te m p e r a t u r e i n S e m i - I n f i n i t e a n d

C y l i n d r i c a l B o d i e s S u b j e c t e d t o M o vin g H e a t S o u r c e s a nd S u r f a c e C o o l i n g , "

J . Hea t Tr an s f e r 9 j2 , 45 6 - 46 4 (1970) .

3 1 . K . K o t a n a nd O . A . A r n a r, "On t h e O p t i m i z a t i o n o f t h e D e s i g n P a r a m e t e r s o f

P a r a b o l i c R a d i a t i n g F i n s , " ASME P ap er 6 5- H T- 4 2 ( 1 9 6 5 ) .

3 2 . M. J . P a s c u a l , S . D . B e c k , a n d J . E . Z w e i g , Te m p e r a t u r e F i e l d I n a S l a b

C o n t a i n i n g An I n s u l a t i n g F i l a m e n t , W a t e r v l i e t A r s e n a l , W a t e r v l i e t , N . Y. ,

WVT-6511 (1965) .

3 3 . K . O . B e a t t y , J r. and A . A . Arm s t rong , The Measu rem en t o f The rma l

P r o p e r t i e s o f N o n m e t a l l i o M a t e r i a l s a t E l e v a t e d Te m p e r a t u r e s , U . S . A t o m i c

En erg y Comm., Oak R id g e , TN, ORO-170 ( 1 9 5 6 ) .

3 4 . R . G . W a t t s , " Tem p e ra tu r e D i s t r i b u t i o n s i n S o l i d an d H o l l ow C y l i n d e r s Due

t o a M o v in g C i r c u m f e r e n t i a l R i n g H e a t S o u r c e , '1 J . H e a t Tr a n s f e r 9 1 ,

4 6 5 - 4 7 0 ( 1 9 6 9 ) .

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8/10/2019 6224569


35 . Y. S . Tou louk ian , e t a l . , The rmophysica l P ro p e r t i e s of M a t t e r, V o l s . 1 ,

2, 4 & 5 (IF I/P len um , New Y ork, 197Q).

36 . A. P a r r i s h , Ed . , M echanica l En gi ne er ' s Referenc e Book, 11 th Ed. (CRC

P r e s s , C leve land , 1973) .

37 . R. C. W eas t , Ed . , Handbook of Chem istry and P h y si c s , 5 5th Ed. (CRC P r e s s ,C l e v e l a n d , 1 9 7 4 ) .

3 8 . T. Ba um eis te r, E d . , S tand ard Handbook for M echan ica l En gin eer s , 7 th Ed.

(M cG raw -H ill, Mew Yo rk, 1967) .

39. T. Lyman, Ed . , M eta ls Handbook 8 th Ed. , Vol . 1 , "P ro pe r t ie s and S e l e c t i o n

of M et a ls " (Amer ican S oc ie ty fo r M eta ls , M eta l s Park , OH, 196 1) .

40. D. M. T e l le p , Te mpe rature Respons e of Thin Sk ins Under Low Heat F lux

C o nd i t io ns , Lockheed M is s i l es and Space C o. , Sun nyva le , CA, IAD 349,(1958) .

41 . M, P. H e i s l e r, "Tempera tu re Ch a r t s fo r In t e rn a l Hea t Ge ne ra t ion , " T ra ns .

ASME 78_, 1 18 7-11 92 (1956) .

42 . W. H. Ho l t e r and J . H . G lov e r, " I ns u l a t io n Tem pera tu re fo r t he Tr an s i en t

Hea t ing of an In su la t e d In f i n i t e Me ta l S l a b , " J . ARS 30 , 907-908 (19 60) .

4 3 . J . H. Grover and W. H. H o l t e r, "S olu t ion of th e T ra ns ie n t Heat Con duct ion

E q u a t i o n f o r an I n s u l a t e d I n f i n i t e M e t al S l a b , " J e t P r o p u l s i o n 2 7 ,

1249-1252 (1957).44. R. D . Ze rk le and .7. E . Su nde r l a nd , "The T ra ns i e n t Tempera tu re D i s t r ib u t i o n

in a S l ab Subje c t t o Thermal R ad ia t i on ," J . H eat Tr an sfe r 87 , 117-133

(1965) .

4 5 . D. L . A ye rs , "Tr an s ie n t Cool in g of a Sphere in Sp ac e ," J . Heat Tra ns fe r

£ 2 , 180-182 (1970).

46 . N. Ko nopl iv and E. M. Spar row , "T ran s ie n t Heat Condu ct ion i n

NonH mogeneous Sp he ri ca l S ys te m s," Warme-und Sl of fu be rtr ag un g 3» 197-2100 ) .

47 . h. B. Newman, "The Tem perature -Tim e R e la t i on s R e su l t in g From the

E l e c t r i c a l He at ing of t h e Face of a S la b ," Tr a n s. AICHE 3_, 598-613 (1934)

4 8 . M. D. Mikhaylow, "T ra ns ie n t Tem pera ture F ie ld s i n S h e l l s , " NASA,

Wash ing ton D .C . , TT F- 55 2 , ( 19 69 ) .

49 . C . D . C o u l b e r t e t a l . , Te m p e r a t u r e R e sp on s e o f I n f i n i t e P l a t e an d S l a b s

to He a t Inpu t s o f Sh or t D ura t ion a t One Su r fa ce , D ep t . Eng . , Un iv. Ca l i f . ,

Los A nge les (195 1).

R-3

8/10/2019 6224569


0 . J. A. Kuhn, Temperature Distribution in a Finite Slab Caused by a Heat

Rate of the Form q(9)=( q /2) (1-Cos w6 ) , Lockheed Miss iles andmax

Space Co. , Sunnyval e, CA, TXN/548 (1961).

51. A. R. Mend elso hn, "Transient Temperature of a Porous-Cooled Wall ,"

J. AIAA 1, 1449-1451 (1963).

52. H. Halle and D.E. Taylo r, Transient Heating of Two-Element Slabs Exposed

to a Plane Heat Sourc e, Chicago Midway Labs , Chicago, CML-59-M-1

(1959).

53. P. J. Schneider and C.F. Wilson Jr ., "Pulse Heating and Equilibration of

an Insulated Substructure ," Spacecraft and Rockets 7, 1457-1460 (1970).

54. H. C. Hottel and C. C. Williams II I, "Transient Heat Flow in Organic

Materials Exposed to High-Intensity Thermal Radiation," ASME Heat Transfer

and Fluid Mechanics Institute (1954) .

55. A. L. London and R. A. Seban, "Rate of Ice Formation," Tran s. ASME 65 ,

771-778 (1943).

56. R. R. Cullom and W . H. Robins, One-Dimensional Heat Transfer Analysis of

Thermal-Energy Storag e for Solar Direct-Energy-Conv ersion Systems, NASA,

Washington D.C., TN D-2110 (1964).57. J. E. Sunderland and R. J. Gros h, "Transient Temperatures in a Melting

solid," J. Heat Transfer 83 , 409-414 (1961).

58. H. G. Landau, "Heat Conduction in a Melting Solid," Qua rt. Appl . Ma th . 8_,

8194 (1950).

59. J. W. Tatom, "Shell Radiation," ASM E Paper 60-WA-234 (1960).

60. S. Katsoff, Similitude in Thermal Model s of Spacecraft, NASA ,

Washington D.C., TN D-1631 (1963).

6 1 . E. L. Knuth, "An Engineering Approximat ion for Resistance of Certain Two-

Dimensional Con ductor s," Int. J. H ea t Mass Transfer T_, 270-272 (1964).

62 . J. G. Bar tas , "Gas Side-Wall Temper ature in Rib-Backed Liquid-Cooled

Combustion C hamber s," Jet Propulsion 27 , 784-786 (1957).

63 . J. E. Schmidt and G. Sonnemann, "Transient Temperatures and Thermal

Stresses in Hol lo w Cylinders Due to Hea t Generation," J. He at Transfer 82 ,

273-279 (1960).

64 . E. M. Sparrow, "Temperature Distrib ution and Heat-Transfer Results for an

Internally Coo led , Heat-Generating Solid," J. Heat Transfer j}2, 389-392

(1960).

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. D. R. Bur net t, "Transient Temperature Measurement Errors in Heated Slabs

for Thermoco uples Located at the Insulated Su rfaces," J. Heat Transfer 83 ,

505-506 (1961).

6. J. J. Brogan and P. J. Sc heid er, "Heat Conduction in a Series CompositeWall " J Heat Transfer 83 506-507 (1961)

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