DOE Heat trasnfer equations

/f^ UCRL-52863

Conduction heat transfer solutions

James H. VanSant

u

March 1980

.:'•• •:•.•! i s UHUMITEE

CONTENTS

Preface v Nomenclature vii Introduction . 1 Steady-State Solutions

1. Plane Surface - Steady State 1.1 Solids Bounded by Plane Surfaces 1-1 1.2 Solids Bounded by Plane Surfaces

—With Internal Heating 1-27 2. Cylindrical Surface - Steady State

2.1 Solids Bounded by Cylindrical Surfaces —No Internal Heating 2-1

2.2 Solids Bounded by Cylindrical Surfaces —With Internal Heating 2-33

3. Spherical Surface - Steady State 3.1 Solids Bounded by Spherical Surfaces

—No Internal Heating 3-1 3.2 Solids Bounded by Spherical Surfaces

—With Internal Heating 3-10 4. Traveling Heat Sources

4.1 Traveling Heat Sources 4-1 5. Extended Surface - Steady State

5.1 Extended Surfaces—No Internal Heating . • 5-1 5.2 Extended Surfaces—With Internal Heating 5-31

Transient Solutions 6. Infinite Solids - Transient

6.1 Infinite Solids—No Internal Heating 6-1 6.2 Infinite Solids—With Internal Heating 6-22

7. Semi-Infinite Solids - Transient 7.1 Semi-Infinite Solids—No Internal Heating 7-1 7.2 Semi-Infinite Solids—With Internal Heating . . . . 7-22

8. Plane Surface - Transient 8.1 Solids Bounded by Plane Surfaces

—No Internal Heating 8-1 8.2 Solids Bounded by Plane Surfaces

—With Internal Heating 8-52

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9. Cylindrical Surface - Transient 9.1 Sol ids Bounded by Cylindrical Surfaces

—No Internal Heating 9-1 9.2 So l ids Bounded by Cyl indrical Surfaces

—With Internal Heating 9-24 10. Spherical Surface - Transient

10.1 Solids Bounded by Spherical Surfaces —No Internal Heating 10-1

10.2 So l ids Bounded by Spherical Surfaces —With Internal Heating 10-19

11. Change of Phase 11.1 Change of Phase—Plane Interface . . , ' . . . . . n - i

11.2 Change of Phase—Nonplanar Interface 11-13 12. Traveling Boundaries

12.1 Traveling Boundaries 12-1 Figures and Tables for Solutions F-i Miscellaneous Data

13. Mathematical Functions 13-1 14. Roots of Some Characteristic Equations 14-1 15. Constants and Conversion Factors 15-1 16. Convection Coefficients 16-1 17. Contact Coeff ic ients 17-1 18. Thermal Properties . 16-1

References R-l

i v

PREFACE

This text is a collection of solutions to a variety of heat conduction problems found in numerous publications, such as textbooks, handbooks, 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, scientists, technologists, and designers of all disciplines should find this material useful, particularly those who design thermal systems or estimate temperatures and heat transfer rates in structures. 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 categories are used to classify the problems. A case number is assigned to aach 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 equations, 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 equations, and boundary conditions for conduction heat transfer. Some discussion is given on the use and interpretation of solutions. Also, some example problem solutions are included. This material may give the user a review, or ever some insight, on the phenomenology of heat conduction and its applicability tc specific problems.

Supplementary data such as mathematical functions, convection correlations, 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 throughout, making this text more readable than a collection of different references. 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 functions, this should not pose a formidable task for most users. Modern computers can make complicated calculations easy to perform. Even many electronic calculators can be used to compute complex functions. If, however, these tools are not available, one can resort to hand computing methods. The table of mathematical functions and constants would be useful in this case.

Heat conduction has been studied extensively, and the number of published solutions is large. In fact, there are many solutions that are not included in this text. For example, some solutions are found by a specific computational process that cannot be described briefly. Moreover, new solutions are constantly appearing in technical journals and reports. Nevertheless, this collection contains most of the published solutions.

The differential equations and boundary-condition equations 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. This practice is prescribed in many texts on applied mathematics, electrical theory, heat transfer, and mass transfer.

A search for particular solutions has frequently been a tedious and difficult task. Too often, countless hours have been spent in searching for a problem solution. Locating and obtaining a proper reference can require considerable effort. Also, it is frequently necessary to study a theoretical development in order to find the applicable solution. In so doing, there are sometimes misinterpretations which lead to erroneous results. This text should help alleviate some of these problems.

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

VI

NOMENCLATURE

2 A = A r e a , ro b = Time constant, s c = Spec i f ic heat, J/kg* C C = Circumference, m d, D = Diameter, depth, m h = Heat transfer c o e f f i c i e n t , W/m • C k = Thermal conductivity, W/m* C m = "V hC/kA, m""1

d, L = Length, m 2

q = Heat flux rate , W/m 2

%,' %.> <3_ = Heat flux in x , y , z d irect ions , W/m q " 1 = Volumetric heating r a t e , W/m Q = Heat transfer rate , W r, R = Radius, m t , T = Temperature, C, K Y = Ve loc i ty , m/s w = Width, m x, y , z , = Cartesian coordinates , m

2 a = Thermal d i f f u s i v i t y , k/pc, m / s B = Temperature c o e f f i c i e n t , C Y = Heat of evaporation, J/kg Y = Latent heat of fus ion, J/kg A = Difference e = Emissivity for thermal radiation

„. ... ,_. (actual heat transferred) n = Fin e f f ec t iveness , ) h e a t transferred without f ins) . _. „ . (actual heat transferred <p = f i n errect iveness , ( h e a f c t T a R S t s c t ^ f r o m i n f i n i t e conductivity f ins) p = Density , kg/m a = 2 "V ax, m

-2 -4 a = Stefan-Boltzmann constant , W/m *K T = Time, s Jf = Radiation configuration—emissivity factor

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DIMENSIONLESS GROUPS

Biv Bi = Biot modulus = hi/k Bh' Bf = Bi Fo = hx/pcJ, P o r 2 Fo = Fourier modulus = at/X Fo*v

* , Fo = Modified Fourier modulus = l/(: Fo*v Gr = Grashof number = gSAtH /v

Kiv Ki = Kirpichev number = q£./kAt Nu d, Nu = Nusselt number = hd/k *\, 2 Pd = Predvoditelev modulus = bS, /a

*°v 2 Po = Pomerantsev modulus = q'"H /kAt Pr = = Prandtl number = v/a R = Radius ratio = r/r Re d, Re = Reynolds number = vd/V X = Length ratio = x/SL Y = Width ratio = y/w Z = Length ratio = z/5.

MATHEMATICAL FUNCTIONS

exp = Exponential function Ei = Exponential integral erf = Error function erfc = Complementary error function inerfc = Complementary error function integral I = Modified Bessel function of the first kind n J = Bessel function of the first kind n K = Modified Bessel function of the second kind n &n = Natural log Y = Bessel function of the second kind n P = Legendre polynomial of the first kind r = Gamma function

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INTRODUCTION

1. HEAT CONDUCTION

Energy i n the form of h e a t has been used by man ever s i n c e he began

walking on t h i s earth. Moreover, the transfer of heat i s e s sent ia l to our very ex i s tence . Not only do our own physiological functions require some form of heat transfer , but so do most l i f e - sus ta in ing processes of nature and many man-control led a c t i v i t i e s . The importance of the thermal sciences in the t o t a l sphere of science can, thus , hardly be disputed.

Conduction i s one of the three principal heat transfer modes, the others being convection and radiat ion. I t i s customarily distinguished as being an energy dif fusion process i n materials which do not contain molecular convection. Kinetic energy i s exchanged between molecules resul t ing in a net transfer between regions of di f ferent energy l e v e l s , these energy l e v e l s are commonly cal led temperature. Part icu lar ly , heat conduction in metals i s mainly a t t r i buted to the motion of free e lectrons and in s o l i d e l e c t r i c a l insulators to the longitudinal o s c i l l a t i on s of atoms. In f l u i d s , the e l a s t i c impact of molecules i s considered as the heat conduction process.

The process of heat transfer in materials has been studied for many centuries . Even early Greek philosophers, such as Lucretius (c. 98-55 B.C. ) , meditated on the subject and recorded their conclusions. Much l a t e r , the famous mathematical p h y s i c i s t , Joseph B. J. Fourier (1768-1830), developed a mathematical expression that became the basis of pract ica l ly a l l heat conduction s o l u t i o n s . He postulated that a loca l heat f lux rate in a material i s proportional to the l oca l temperature gradient in the direction of heat flow:

where g^ i s the heat flow i n the x-direct ion per unit area as i l l u s t r a t e d in Fig. l a . Material properties are accounted for by including a proportional i ty constant:

* X = - K ! X - , (2,

1

• i .

I - « » * at - « » *

t - « » *

— • dx •* X

/{///ft

(a) (b)

FIG. 1. I l l u s t r a t i o n of heat flow and temperature gradient.

there the constant k i s ca l l ed thermal conductivity. (The minus sign must be .ncluded to s a t i s f y the second law of thermodynamics.) This equation i s a i l e d Fourier's law for heterogeneous i so trop ic continua.

A simpler form of Fourier's law i s for homogeneous i sotropic continua. For example, consider a p la te of this of type material having isothermal surfaces md insulated edges as shown in Fig. 1. The plate has a width Ax, surface area A, and thermal conductivity k. The heat flow in the p la te i s expressed as

Qx = - ] = -kA ( t 2 - V

Ax (3)

Chis expression becomes Eg. (2) when Ax diminishes to zero.

fx lim At . 3t A = %. = ~ K Ax+O Ax = " K 3x (4)

The heat f lux q i s presumed to have both magnitude and d irec t ion . Thus, Lt can be given as a vector , g which i s normal to an isothermal surface. For example, in Cartesian coordinates

q = q x i + j + * z k (5)

2

where i , j , k are unit vectors in the x-, y-, and z-directions, respectively. Since Eq. (4) defines q^ = -k3t/3x, and similarly qy = -k3t/3y, q z = -k3t/3z, we can state

q = -k (i3t/3x + j3t/3y + k3t/3z) (6)

i = -kVt . (7)

In anisotropic continua the direction of the heat flux vector i s not necessarily normal to an isothermal surface. Example materials are crystals, laminates, and oriented fiber composites. In such materials we may assume each component of the heat flux vector to be linearly dependent on a l l components of the temperature gradient at a point. The vector form of Fourier's law for heterogeneous anisotropic continua becomes

q = -K • Vt , (8)

where K is the conductivity tensor; the components of this tensor are called the conductivity coefficients.

In Cartesian form, Eq. (8) i s

*x - -("ll £ + k12 £ + *13 H)

S ' -(k21 fe + *22 ft + k23 l l ) < 9 >

/ 3t 3t 3t\ % = ' ^ 3 1 3x + k32 3y + k33 Szj *

To compute heat flow by Fourier's law, a thermal conductivity value i s needed. I t can be estimated fran theoretical predictions for some ideal materials, but mostly, i t i s determined by measurement and Fourier's law, Eq. (2* -. As illustrated in Fig. 2, thermal conductivity can have a large range, which depends on materials and temperature. For example, copper at 20 K has a thermal conductivity of approximately 1000 W/m*K and diatomaceous earth at 200 K has a conductivity of 0.05 W/m'K. Consequently, heat flow in materials can have a very large range, depending on a combined effect of temperature gradient and material property.

Thermal properties of some selected materials are given in Section IS.

3

1000 1 r 1 r

100

10

•a c o u

I -

0.1

0.01 j L

-Teflon

Copper

Lead

. Fused quartz

' Diatomaceous earth

_L J L 200 400 600 800

Temperature—K 1000 1200 1400

FIG. 2. Thermal conductivity of some selected solids.

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2. DIFFERENTIAL EQUATIONS OF HEAT CONDUCTION

Solutions to heat conduction problems are usually found by some mathematical technique which begins with a differential equation of the temperature field. Theâppropriate 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 spherical. Then analytical or differencing methods 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 energy) 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 through six faces is

"net x+Ax *y ~ ^+Ay z ~ °z+Az ' (10)

FIG. 3. Coordinate systems for heat conduction equations: (a) rectangular, (b) cylindrical, (c) spherical.

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where typ ica l l y

Q = -AyAz

Q = -AxAz y Ml'

Q = -AxAy

and k , k , and k are d irec t iona l conduct iv i t ies . x ' y z An increase in internal energy of the element i s represented by

AI = AxAyAzpc | ^ , (11)

where t is the mean temperature of the element, p is the material density, and c is its specific heat.

Internal energy sources can be expressed as

QUI = qtii A xAyA Z f (12)

where q" ' is the unit volume source rate. Examples of internal heating in materials are joule, nuclear, or radiation heating. Summing these energies in accordance with the energy conservation law yields

For the limits Ax, Ay, Az -»• 0, we obtain

3 t BCL. 3qy 3q z

p c 3T q 3x 3y 3z » t i 4 >

6

where

. 3t q x = x 3x '

q y " " y 3y '

q = - k -5— . ^z z oz

Using Eg. (14) as a general differential equation, we can derive the following specific equations.

2.1 Rectangular Coordinate System

For isotropic heterogeneous media

D c 3 t = 3_/ 3t\ 3_/ i t \^ i_/ 3t\ +

p c 3x 3X \ k 3 X / / + 3y \ K Syj 3z \ K 3 Z > / q " ( 1 5 )

For isotropic homogeneous media this becomes

3t _ k_ 3 T " pc

3 2t , 3 2t 3 2 t l 3x2 3y 2 3z 2J

+ — = art + a . (16) pc pc

When q'" = 0 , Eq. (16) becomes Fourier's equation. In steady-state conditions, 3t/3x = 0 and Eg. (16) becomes the Poisson

equation. When q'" = 3t/3x = 0, Eg. (16) reduces to the Laplace equation. Nonisotropic materials, such as laminates, can have directionally

sensitive properties. For such materials the conduction differential equation in two dimensions is expressed in the following form:

2 2 pc || = (k ? cos 2B + k n sin 2 B ) -| + lk ? sin 2 B + 1^ cos 2 B ) ~

2 + ( k c - k T 1 ) ( s i n 2 B ) y y - + q'" , (17)

7

•-X

FIG. 4. Coordinate system for a nonisotropic medium.

where k_ and k are directional thermal conductivities, and 3 is the angle of laminations as indicated in Fig. 4. When the geometrical axes of the nonisotropic material are oriented with the principal axes of the thermal conductivities, then Eg. (17) simplifies to the form of Eq. (14)

n 3 t p c a ? X 4 + * 4+*-'

3x 2 * 3 y 2

(18)

2.2 Cylindrical Coordinate System

Rectangular coordinates can be transformed into cylindrical coordinates by the relations x = r cos 9, y = r sin 9, and z = z. The partial differential equations (15) and (16) transformed to cylindrical coordinates are thus

P c 37 = 7 37 ( r k 3?) + 7 39 \ k 39 ) + 3l (k 37) + q (19)

t _ » (& + i & i_ i*t . aft\ 3^1 T U 2 r 8 r r a a e 2 a .V p c

(20)

For nonisotropic materials with the conductivity and geometry axes aligned as in Eq. (18) the differential equation is

P c 3 ? " r 3 r l r 3 r ) +

r 2 3 e 2 + k

Z 3 z 2 + * (21)

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2.3 Spherical Coordinate System

A transformation from rectangular to spherical coordinates can be accomplished by subst i tut ing the relat ions x = r s i n i|i cos ij>, y = r s i n ty s in § and z = cos \|i into Eqs. (15) and (16), which y i e l d the partial d i f f e r e n t i a l equations for i so tropic heterogeneous and homogeneous materials , respect ive ly .

3t 1 3 / 2 . 3 t \ A 1 3 / . 3 t \ P° 37 - ~2 3T( r k 37) * 2 s . n 2 ^ 36 ( k 39 j

+ —JL— IV ( k sin I f ) + q ' " (22)

r sin ty x '

a t m / a 2 t 2 a t 1 3 2 t 1 3 t \ <?•" * " "W E 3 t r 2 s i n 2 * 3 6 2 r 2 tan * 8 * ) • * " ^

The d i f ferent ia l equation for nonisotropic materials with al igned conductivity and geometric axes i s

n „ 3t . k r 3 2 t kd) 3 2 t , *Sl» 3 . ,., 3t PC j r = — — 7 + -5—*-= 5- + - ^ JT. s in \p •55- + q' . (24)

3 T r 3 r 2 r 2 s i n 2 i|» 3<|)2 r 2 s in * 8 * **

3 . SPECIAL DIFFERENTIAL EQUATIONS

Some defining equations can have implied assumptions and boundary condit ions . They are usual ly employed in spec ia l cases to s implify the method of so lu t ion . However, Fourier's law i s the bas i s for deriving these specia l equations.

3 .1 Combined Conduction-Convection

Thin materials having r e l a t i v e l y high thermal conductivity have very small l a t e r a l temperature gradients . If the surfaces are convect ively heated or cooled, the convection condition becomes part of a heat accounting on a d i f f e r e n t i a l element. Referring to Fig. 3 , l e t Az be the thickness b of a th in s o l i d . On the surface z and z + Az, the s o l i d has a convection boundary described by q = h (t - t ) , where t . i s the convection f lu id temperature.

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Applying the same principles used to develop the general equations for rectangular coordinate systems should result in

/ c £ + 2 S<*-v-« , i , + fc( k l f ) + fc(kij)- (25)

If the geometry is a thin rod of circumference C, the appropriate equation is

p°{f + r < * - t f > - « , M + f c ( k i i ) - < 2 6 > 3.2 Moving heat sources

The general heat conduction Eq. (15) can also be used for moving heat sources, but a simpler quasi-steady-state equation can be derived by coordinate transformation. If the coordinates are relative to the traveling source, the temperature distributions appear to be stationary. For example, if a point source of strength Q is moving at a velocity U parallel to the x-axis, the transformation would be x = x' + UT, where x' is the x-direction distance from the source. By substitution in Eq. (16) we can obtain

I F ('fc-Vfc(»|*)*fc ( * £ ) • » . & •« • • • - . • The applicable equation for a moving source in a thin rod that is

convectively cooled is

^ I X T V ^ I X T ^ " ^ 1 ! i t - v • < 2 8 > The moving source strength is accounted for in the boundary conditions

for a particular problem solution.

4. BOUNDARY CONDITIONS

Solutions to heat conduction problems require statements of conditions. For general solutions there must be given at least a definition of the solution region, such as infinite, semi-infinite, quarter-infinite, finite, etc. Additionally, limits can be specified for any of these regions.

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If s p e c i f i c solutions are needed, then complete conditions must be defined. They could include, for example, i n i t i a l , internal , and surface condit ions . Other conditions might include property def in i t ions . Whether deriving a solution or searching for ex is t ing s o l u t i o n s , one must decide which conditions are applicable to the problem and how they can be sui tably expressed.

4 .1 I n i t i a l Condition

Unsteady-state problems must have an initial condition defined. Mostly, this implies a temperature distribution at x = 0 but, also, internal or surface conditions could have initial values. A problem solution for x = 0 depends, of course, on whatever is specified at x = 0.

4.2 Surface Conditions

The most commonly employed surface conditions in heat conduction problems are prescribed surface convection, temperature, heat f lux, or radiat ion. I t i s even acceptable to prescribe two of *.hes3 for the same surface, such as combined radiation and convection. Other surface conditions could include phase change, ablat ion, chemical react ions , or mass transfer fror a porous s o l i d .

4 .2 .1 Convection boundary

Conduction and convection heat transfer ra tes on a surface are equated to s a t i s f y continuity of heat f lux according to Newton's law:

a t ( x h T )

-k 3 x ' = h [ t ( x b , T) - t f ] . (29)

where t_ is the temperature of the convection fluid, and x. is the boundary location. The convection coefficient h must be determined from suitable sources that give predicted values satisfying the conditions of the fluid. (Some correlations of convection coefficients 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 values for the coefficient. However, h is usually defined as

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TABLE 1. Sample convection coefficient values.

Fluid Condition h, W/nT-K

Air Free convection on v e r t i c a l plates 10 Air Forced convection on p la te s 100 Air Forced flow in tubes 200 Steam Forced flow in tubes 300 Oil Forced flow in tubes 500 Water Forced flow in tubes 2,000 Water Nucleate boil ing 5,000 Liquid helium Nucleate boil ing 8,000 Steam Film condensation 10,000 Liquid metal Forced flow in tubes 20,000

Steam Dropwise condensation 50,000 Water Forced convection boi l ing 100,000

h • t(K b? T ) - t £ ' ( 3 0>

where t can be given as

t f = B [ t <x b, x, - t j + tm ^ (31)

and where 3 £ 1, and t^ is the temperature outside the thermal boundary layer of the fluid. In this respect, one must 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 sense. 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 rate.

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4.2.3 Heat flux

Fourier's law defines the flux on a boundary by

3t(x b ,T) * = " k ~ 3 x • < 3 2 )

An adiabatic surface can be defined by either setting q in Eq. (32) or h in Eq. (29) to zero. Inversely, if the solid's temperature di.stribution has been solved, then Eq. (32) can be used to determine the surface heat flux. Understandably, heat flux can be, in particular cases, time and position dependent.

4.2.4 Thermal radiation

Heat transfer from an opaque surface by radiation can be expressed as

T 4 41 3 t ( V T )

a*\v (xb, T) - T*J = -k — s , (33)

where a is the Stefan-Boltzmann radiation constant, J*" is the combined configuration-emissivity factor for inuiltiple-surface radiation exchange, and T is the sink or source temperature for radiation. Because Eq. (33) is a nonlinear expression, it is frequently difficult to find exact solutions to problems with this condition.

A common method for dealing with radiation problems is to treat the radiation boundary as a convection boundary. According to Eq. (29) we can write

3t(x , T) - k — ^ h r[t(x b, x) - t j , (34)

where h r = oy[T(x b, x) + T ] [T 2(x b, T) + T* J .

Using this method means that T(x. , T) must first be estimated in order to compute a value of h . After a value for T(x. , T) has been computed from the problem solution, then the estimated value for h can be improved. This, of course, becomes an iterative process.

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4.3 Interface Conditions

4 . 3 . 1 Contact

Two contacting s o l i d s , either similar of dissimilar, w i l l almost always have some interface thermal resistance to heat flow between them. The magnitude of th i s res is tance can depend great ly on the condition of the two contacting surfaces. Properties that can e f f e c t the surface condition include c l ean l ines s , roughness, waviness, y i e l d strength, contact pressure, and the thermal conduct ivi t ies of the so l ids and in ter s t i tua i f lu id . Since there are so many influences on the contact thermal resistance, i t i s d i f f i c u l t to theoret ica l ly predict i t s value. Consequently, experimental resu l t s are frequently used. Some representative values of the inverse thermal contact res i s tance , commonly referred to as the thermal contact c o e f f i c i e n t , are given i n Section 17.

The generally accepted def ini t ion of the contact coe f f i c i ent i s

h o - A ^ ' ( 3 5 )

where q i s the steady heat flux corresponding to a f i c t i t i o u s interface temperature drop of At. defined by extrapolating the virtual l inear temperature gradient in each so l id to the contact centerl ine . This temperature drop, which i s i l l u s t r a t e d in Fig. 5, would diminish to zero i f the interfaces were in perfect contact .

4 . 3 .2 Phase change

Other interface conditions include those caused by endothermic reactions such as melting, s o l i d i f i c a t i o n , sublimation, vaporization, and chemical d i s s o c i a t i o n .

A statement of an interface reaction condition defines the difference of heat flux across the interface . If the interface which separates two phases of a material i s located at x - x . , the heat balance for a phase change react ion i s given in the form

3t (x ) a t (x ) dx

h -hr~ - h ~^r- =Yp2 * r • <36> 14

Length -*

FIG. 5. Illustration of interface contact between solids.

whece y is the latent heat or chemical heat capacity, and subscripts 1 and 2 refer to the two phases.

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

15

distribution by using Fourier's law, i.e. Eq. (2). If cumulative heat flow is required, a time and surface integration of local heat flux is necessary.

where n is the direction normal to the surface s. Steady-state solutions can be considered as the infinite-time condition for

unsteady-state solutions. That is, problems which have a time-asymptotic solution exhibit steady-state solutions for x •*• <*>. Thus, steady-state solutions can be derived from transient solutions.

A steady surface temperature condition can be implied from a convection boundary condition. For h •* °°, the surface temperature approaches the fluid temperature. Therefore, a solution which includes a convection boundary can be transformed into a constant temperature boundary solution by solving for the implied limiting case.

5.2 Dimen&'ionless parameters

Grouping particular variables yields dimensionless numbers that can be useful. Symbolically, they can shorten an eguational expression. But, they can also give insight to the behavior of heat transfer in a particular problem.

One very useful parameter is the Biot number, Bi = ht/k. which results from convection boundary conditions. This parameter is proportional to the ratio of the conduction resistance to the convection resistance. Thus, we could say that for

Bi > 1, conduction is highest resistance to heat transfer, Bi < 1, convection is highest resistance to heat transfer, Bi « 1, the solid behaves like k = ».

2 Another dimensionless parameter is the Fourier number, Fo = crr/S. , which is found in transient solutions. This number is a dimensionless time value, but it is also considered an indicator of the degree of thermal penetration

2 into a solid. Since crt/S. = (kTAt/£)/(pc8.Ai.;. it is proportional to the ratio of conduction heat transferred to thermal capacity. Thus, an increasing Fo value implies approaching thermal equilibrium.

16

The product of Bi and Fo numbers yields the parameter Bf = ht/pc& 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 as 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"'Jl2/kAt. This number is a ratio of internal heating to heat conduction rates. Large values of Po imply large temperature differences will occur in the solid.

The parameter Fo = 1/2S&Z 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 = bfl, /d, where b is the inverse time constant. Small values of Pd imply a slow changing condition. 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 insulation. 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

2ir ( t x - t 2 ) q = .

1 . r 2 ^ 1 . r 3 __ 1 A 1 -.— In — + ;— Jin — + —— + — r -h r l k2 r 2 r l h l r 3 h 3

17

From the problem descript ion t x = 65°C t . = 20°C 4 r. = 1.25 cm r = 1.33 cm r , = 2.33 cm k x = 400 W/m-°C (from Table 18.1) k 2 = 0.038 W/m«°C (from Table 18.2) hd/k = 0.0155 P r ° ' 5 R e 0 , 9 (from Sect . 16.1) h l = ^ w a t e r / 2 1 ^ ( 0 J 1 5 5 P r ° ' 5 R e 0 - 9 ) kwater = ° ' 6 5 9 w / m * ° c l a t 6 5 ° c > Pr = 2.73 Re = 2pvr]/n p = 980 kg/mJ

v = 0.5 m/s p = 4.3 x 10" 4 kg/m's

Re = 2(980)(0.5)(0.0125) = M 4 8 8

4.3 x 10

h x = (0 .659/0 .025)(0 .0155)(2 .73)°" 5 (28 4 8 8 ) 0 , 9 = 6895 W/(m2-°C) h 3 = ( k a i r / 2 r 3 ) C lâ*1)1* (from Sect . 16.8) k a i r = ° - 0 2 5 w/m*0c

Pr = 0.71

Gr = gB( t 4 - t 3 ) ( 2 r 3 ) W g = 9.8 m/s B = 1/T 4 = 1/293 K - 1

V =16 .55 x 10" 6 m 2 / s

G . (9.8) (0 .0466) 3 (0.71) = ^ _ = ^ a (293) (16.55 x l O ' V

C = 1.14, m = 1/7 (from Table 16.3)

h 3 = (0.025/0.0466) (1.14) ( 8 7 7 4 ) 1 / / 7 = 2.24 W/m2«°C

q = 2n(65-20) 1 „ / 1 . 3 3 \ . 1 » / 2 . 3 3 V

400 £ n (1^5 ) + 0 38 la(l^2) (0.0125) (6895) (0.233)(2.24)

2ir(45) 1.55 x 10" 4 + 14.76 + 0.012 + 19.16

= 8.33 W/m

18

^-u 8.33 2nr3h 271(0.0233) (2.24) = 25.4"C

Using this new estimate of (t. - t.), we can recalculate h,.

h 3 = 2.24 (25.4) 1 / 7 = 3.56 W/m2*°C, q = 10.54 W/m.

Additional iterations on h, would little improve this result. Note that the copper tube and water film have a small effect on the

results because they present little resistance to heat transfer by comparison to the insulation and air film.

5.3.2 Transient heat conduction in a slab

A billet of 304 stainless steel measuring 2 x 2 x 0.1 m thick and having a uniform temperature of 30 C is heated by sudden immersion into a 450°C molten salt bath. The mean convection coefficient is 350 W/m • C. Determine the time required for the center temperature of the billet to reach 400°C. The solution is found in the solution table (case 8.1.8 and Fig 8.4a):

t - t. r_ 400 - 450 30 - 450 = 0.119

From Table 18.1 k = 21 W/m«°C#

-6 2, a = 7 x 10 m /s k 21 hi (350)(0.05) = 1.2 .

— h . t f

From Fig. 8.4a

O T A 2 =3.4,

_ = 3.4& 2 _ (3.4H0.05) 2 =

a 7 x 10" 6

19

5.3.3 Transient heat conduction in a semi-infinite plate

For the conditions given in 5.3.2, find the temperature at 0.05 m from the end and sides of the billet.

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

hVax = 35oV(7x 10~ 6) (1214) = ^ M

1.2

0.05

lV 2^/ca 2V(7 x 1 0 - 6 ) (1214)

= 0.27

S(X) = 1 - 0.45 = 0.55 (from Fig. 7.-) W i V r

I

t ~ t t = P(Fo) S(X) = (0.119) (0.55) = 0 . 0 6 6

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 yellow brass extending 4 cm from the flat surface and 2 cm wide. 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.

v \ l *^\

T b

I : t (

20

_ f » Bi cos (X n X)ix n c o s h [ X n ( B - Y)j + Bi s i n h [ X n ( B - Y)l}

1 " t t ^ cos (X ) (B i 2 + X 2 + Bi) fx cosh (X B) + Bi sinh (X B)"l n=l n n L n n n J

t-J£ t

X tan (X ) = Bi (characteristic equation) B = b/a = 4/1 = 4 X = x/a, Y = y/a B i = I r = t 5 3 i 3 o * 0 1 > = 0 , ° 0 4 ( k v a l u e f r o m T a b l e 1 8 - 1 )

From Table 14.1 X x = 0.0632, X 2 = 3.1429, X 3 = 6.2838, X 4 = 9.4252, Xg = 12.5667,

Heat loss without the rib attached would be q = hA (tx - 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 Rectangular 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.

= \ k a " V(130M0.01> = 6 ' 3 8 5 m

l = 0.04 + 0.01 = 0.05 m c mfc = 0.3193 c

tanh {mSLj_ = t a n h ( 0 . 3 1 9 3 ) _ Q ^ ml 0.3193

c q = 2h(a + b) ( ^ - t f)(|) = 2(53) (0.05) (130) (0.968) = 667 W/m

21

5.3.6 Semi-infinite plate heat transfer

Find an equation for the heat transfer rate through the edge of the semi-infinite plate described in case 1.1.5 with f(x) = t .

Using the given temperature solution and Eqs. (2) and (37) we can find the heat transfer in the following manner:

. 3t k .. . . 3T . m

fc " fcl S = _ k 37 = " I ( t 2 " V 3? ' W h e r e T = t~^t

9Y|Y = Q = ~2 Z S i n ^ " ^ C 1 ~ c o s ( n 7 r G n=l

00

Q Y | Y = o = £ / ^XIY = o ^ = 2 k ( t 2 " V i It, s i n ^ ^ C 1 - c o s t ™ ) ] ^

oo go

= 2k(t 2 - t ) 2 , El - costfflT)]2 = J k ( t 2 - t L ) 2 , J ' n - 1» 3 ' 5 ' n=l an n=l

22

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

Case No. References Description

1.1.1 19, Convectively heated and p. 3-103 cooled plate.

w -h,,t. 2' '2

Solution

h i ( t i - V B i x + 1 + (h j /h 2 )

t - t . B ^ + 1 fc

2 - h Bi

x

+ 1 + W

(-• 1 .1 .2 19, The composite p l a t e . H p. 3-103

{ to - V

k, h 2

V ^ V i^^w«A/

V h o-

h n - 2 h n - 1

* * « ^ M ^ ^

Vi -1

\ i - 1

Mi'S.

]£ (wi/k1 + 1/IK) + l/h 0

i - 1

Temp in the j t h layer:

j - l ^ Hi/î + l / h i ) + {x^/kj + ( l / h 0 )

fcj ~ fc0 _ i f l t - t„ " n n 0

J > 1

J ai/ki + l/h£) + (l/h0) i=l

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

Case No. References Description Solution

1.1.3 1, p. 138 Plate with temperature dependent conductivity, k = k + B (t - t x).

<ww^

fcl-vfc2 rv A m £

\—*~~\

t - h - i H+ 2 f v f c

2 - v f -L

k = <k. + k, ) /2

,J, 1 .1 .4 2 , p . 221 Porous p la te with internal f l u i d flow.

0.

t - t,

t x i x t 2 , x = 6.

P = Porosity .

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

P = porosity

fc2 "0

- f c o = expT- v»-

t - fc0 exp (£fx)

v p

fc2 - fcl " " » ~ f ? " X ] , _» < x < 0

M* Mean temp:

V ^ = | ^ [1 - «*p«p«>] , 0 < X < 5

S ~ V M _ TJX" » 5 * = * p k p (1 - P) ' ""f ~ k f (1 - P)

(See F ig . 1.1)



1.1.5 2, p. 122 Semi-infinite plate. 9, p. 164 t = t:, x a 0, H, y > 0.

t = f(x), 0 > x > A, y = 0.

V

I UJ

t - t x = 2 2 _ exp(-rarY) sin (mrX) n=l

1 x f ff (X) - t i"| sin (mrX)dX

For £(x) = t 2s

-t = f(x> t - t

n=l

1.1.6 3 r P- 250 Rectangular semi-infinite rod. «° <= , , ™ + n

t = t x , x = 0. t " t 5 V - ^ l ~ X J e x p

t = t_ on other surfaces. ^ - 2 2 L (X n x/w) 2 + (Xmx/Jl) 2

1 2 n=0 m=0 X X n m

x cos (X_) cos (\ ) n ID

Xn = (2n + 1)£ , \ = (2m + 1)^-n 2 m 2

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

Case Ho. References Description Solution

1 .1 .7 2, p . 130 Rectangulac i n f i n i t e rod. t « P j U ) , 0 < i < I , y = 0.

i

t a t i + tu* 'in + 4v

t = F - (x ) , 0 < x < &, y = w t = G x(x) t * G, (x ) , x = A, 0 < y < w.

. o s x * * . y = w. Y sinh (airc/L) . . _ , , , x = 0, 0 < y < w. *l - 2 Z sinh (iw/i.) S l n ( n l t X L )

n=l

w t = G,(v)-»

t = F1(x)

i

t = F2(x)

II - 2 sinh [fo - ')] n=l

A i

(X) s in (mtx)dx

sinh (niF/L) S i n ( r t I t X L ) I V X > S i n < n 1 t X ) f l X

I— t = G2(y) *1II " 2 £ f ^ u i l S f f i s in (iffy) | G,(X) s in (nTTY)dY sinh (nir/L)

n=l i n=l

L = l /w

• ' f t

s in (niTX) , 2 1 " ~ ncosh n=l

F^x) = F 2 (x) = t 2 , G 2(y) = G 1(y) = t



1.1.8 2, p . 147 Thin rectangular p l a t e , t = t Q , x = 0, 0 < y < x.

0.

w.

in

t = t Q , 0 < x < l, y t - t 0 = 2

t = t Q , 0 < x < «,, y t = G(y), x = Si, 0 < y < w.

6. 0.

h r t 0 at z h 2 , t Q a t •

V sinh [(Bi + n 2 i r 2 L 2 ) \ ] . . „„. y &— 2 2 2 1.1 s i n ( n i r Y )

~L sinh [(Bi + n V l . V J

x j [G(Y) - t Q ] s i n (fflTV) dY

Bi <- (h x + h2)i./k6, L - Jl/w

Z.I-' j t = G<y)

t Q hjj.tnJ

1 .1 .9 4 , p. 41 I n f i n i t e rectangular rod in a semi - in f in i t e s o l i d .

/7777&777777777777777777777777

d

JL

fciô

R ~ 1 - An f 3 . 5 ( d + k / h ) ]



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

k < tl " V d ° ~ .. , ,0.24 - °'5 < w < "

n , t 0 0.42/ 1 i^k) d k J Horizontal plate:

i 1 K ii 1

k ( t i " fcn> fl Q s : i 2 _ _ , o . 5 < £ < 1 2 /A 1 \ 0 ' 3

J VW B l w /

1.1.11 4, p. 43 Thin rectangular plate on the kwir(t - t ) surface of a semi-infinite Q = / / M \ — solid. A n ( b )



1.1.12 4, p. 44 Thin rectangular plate in an 2irwk(t1 - t ) infinite solid. Q =- — —

in m

T 1.1.13 5, p. 54 Rectangular parallelpiped "J with Wall thickness of &. Q t = P£(dw + db + wb) + 2.16(d + w + b) + 1.2<5J(t2 - tj_

= total heat flow through six walls

i Ll _', _ i T a 1



1.1.14 4, p. 37 Infinite hollow square rod.

„ h 2 , t 2

2irk(t 2 - t x ) 9 = It o l.OBw Tlk

h l r 0 2 r 0 2 h 2 W

1.1.15 9, p. 166 Rectangular i n f i n i t e rod. t = f ( x ) , 0 < x < a, y = 0.

T t » t , 0 < x < a, y = b. t = t , , x • 0 , 0 < y < b. t = t , , x = a , 0 < y < b .

t - t = Y A n s in (mix) sinh j (1 - Y) ( ^ Y l coaech 0&\ n=l

1 A » 2 I [f (X) - O s i n (nirX)dX , L = l/v

w—» V

For: f(X) = t .

-» x

f(x) I l ^ i - - 1 ^ } s in (nnx, sinh [(1 - V, £ * ) ] cosecn ( * ) ,

n=l n = 1, 3, 5, 7



1.1.16 9, p. 167 Rectangular infinite rod. t = f (x), 0 < x < I, y = 0. a » 0, 0 < x < I, y = w. a = 0, x = 0, 0 < y < w.

i

V w< y

• / y y y ^ 1 !

t = f (X)

t - t f = 2 ^ ( B i 2 + X 2 ) cos (X X) cosh [*(1 - Y)WX ]

nTl [ ( B i 2

+ X * ) + B i ] c o s h <XnW,

/ x I j"f (X) - t / 1 cos (X X)dX

X tan (X ) = Bi , Bi = hfc/k , W = w/Jt n n

For: f (x) = t ,

fc- fcf fcl" fcf

= 2 B i ~ cos (XflX) cosh [(1 - *)WAn]

, 1 B i 2 + P 2 J + Bi cos (X ) cosh n=l \ n / n



1.1.17 9, p. 168 Case 1.1.16 with q = 0, y = w , Q-«x<J!.is replaced by convection boundary h,t f.

t - t f = 2 ~ (Bi 2 + X*J cos (AnX){Xn cosh [Xn(W - Y)] + Bi sinh [\n(W - Y}]}

^ [(Bi2 + X 2 ) + Bil{X n cosh (XRW) + Bi sinh <\,W)}

X [f(X) - t f"| cos (XnX)dX 1 i x

'0

X tan (X ) = Bi , Bi = hJL/k , W = w/Jl n n

For: f(x) = t ^

t - t f ^ . Bi cos ( x

n

x ) U n cosh [Xn(W - Y)] + Bi sinh [Xn<W - Y)]} fcl " fcf ~^ cos (Xn>r^Bi2 + XM + Bi"J[Xn cosh (XnW) + Bi sinh ( nW)]



1.1.18 9 , p . 166 Case 1.1.16 with t = t £ , y a w, 0 < x < £ .

t - t . « 2 " i - B ( B i 2 + X 2 ) cos (X X) sinh fX (W - Y)"| r1

* *" — "*• x I [ f (X) - t f ] cos (XnX)dX " 2 * » : j + Bij sinh (XnW)

X tan (X ) = Bi , Bi = hH/k# W = w/S, n n

For: f(x) = t , :

i P

fcl" fcf 2Bi \ r( B i

2 • x2) n

+

n=l L\ n /

cos (X X) sinh TX (W - Yfl n L n J Bi cos (X ) s inh (X W) n n

1.1 .19 9, p. 169 Case 1.1.16 with t • t x , y = 0, 0 < x < I. t = t , y = w, 0 < x < S,.

t - t t x - t f

.. ^ cos (XX) {sinh f X W - Jf)"| - sinh (X Y) (t - t )/<t_ - t . ) } •-— = 2 B i > — = - ^ -

n=l [^Bi 2 + X 2 ) + Bi l cos (Xn) sinh <XnW)

X tan (X ) = Bi , W «= w/X, n n



1.1.20 9, p. 178 Rectangular parallelpiped. t » t, , x = 0 , 0 <"y < w , z " c0 1 6 V V fsinh (L - LX) + T sinh (LX)] sin (mtY) sin (nmZ)

1 t, - t„ = _2Z Z <nm) sinh (L) ' 0 < z < d . t = t 2 , x = & , 0 < y < w , 0 < z < d . Remaining surfaces at t Q.

"0 TT n=l m=l n = 1, 3, 5, m = 1, 3/ 5,

L 2 = (nirVw)2 + (Jim£/d)2 , z = z/d T = (t2 - t 0 ) / ( t l - t Q)



1.1.21 9, p. 179 Rectangular parallelpiped. t = t t , x = 0 , -w < y < +w f -d < z < +d . t = t 2 , x = i. , -w < y < +w , -d < z < +d . Remaining surfaces convect ion boundary with h , t , .

t - t . Jl> ~ Tsinh (L - X) + T sinh (LX)3 cos (X Y) cos (B Z) t _ . _, z _ x x ^ n TO fcl " ** ~ _ , cos (X ) cos <B ) f x 2 + B i 2 + Bil^B2 + B i 2 D 2 + Bi D^sinh (h)

-—, r-i i sinn ^L, - a.) v T sinn tidw J COS JA XJ cos IP zj x x ^ n TO

~ „ , cos (X ) cos <B )Tx 2 + B i 2 + BilZe2 + B i 2 D 2 + Bi D^sinh n=l m=i n m |_ n J\ m /

X tan (X ) = Bi , B tan (B ) = Bi D, Bi = hw/k n n m m

L 2 = X 2 & 2 /w 2 + B / V d 2 , D = d/w, X = x/i., Y = y/w

Z = z/d , T = ( t 2 - ^ / ( ^ ~ t f )

Section 1.1. Solids Bounded by Plane Surfaces—Mo 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 are correction boundaries with h,t .

oa oo fc ~ fcf . _ . 2

[A Bi sinh (L - LX) + L cosh (L - LX)] cos (X Y) cos (B Z) _ _ L * « sin 2* 2, [A Bi t, - t. 2. Z. [A Bi sinh (L) + L cosh (L)] NM cos (X ) cos <B)

1 * n=l jn=l

A = £/w , N = X 2 + Bi 2 + Bi , M = &2 + Bi 2 C 2 + BiC. n in X , 3 , L , D , X , Y , Z, and Bi are defined in n m case 1.1.21 •

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



m

1.1.24 9, p. 428 I n f i n i t e s t r i p with stepped temp boundary, t - t f l . y - 0 . - » < x < +°» . t = t 1 , y = w , x > 0 . t = t 2 , y = w , x < 0 . k = k , , 0 < y < w , x < 0 • k = k , 0 < y < v f x > 0 •

h t „ *, . fc " fco = y i * . t , - t n

= y

w > k i k 2 { , f - *• - *•

1 " „ U = y { fc2 - fcl

= y

2 [(Vfcl> - f (-1)" * ( k l + k2> n=l n

sin (mtY)exp(-mtx) , X > 0

sin (nirY)exp(nTTX) , X < 0

X = x/w , Y = y/w



1.1.25 9, p. 452 Heated planes on a semi-infinite medium. t = t x , x < -I , y = 0 t = t 2 , x > . + J , , y - 0 a => 0 , -I < x < +8, , y = 0 .

Q = J C O S h-l (^lti _ V , _Xi < x < -a

"1 • V ~*

+£

1.1.26 9, p. 453 Heated parallel planes in Heat flow from bottom side of semi-infinite plane: an infinite medium. t = t , , x > . O f y = s .

Heat flow from top side of semi-infinite plane: t l Q = £ ^[(irxj/s) + l] (^ - t 2) , 0 < x < X j L .

C = k [(Xj/s) + (1/TI)J (tj - t 2) , 0 < x < x x .

t = t 2 , -» < x + «

Yt +oo

i 1 r I—x J X1

+oo

Section 1.1. Solids Bounded by Plane Surfaces—Ho Internal Heating.


1.1.27 9, p. 454 Infinite right-angle cocnec. t » t l f x > 0 , y » 0 . t = t l f x = 0 , y > 0 . t » t j . , x > Wj , y = w 2 . t » t j , i • ^ , y > » 2 .

2 . 2\

w 2 W ; L nw 2 ^ w J it y 4 W l w 2

| — tan" 1 ( — ) ( t , - t_) , 0 < x < x, , 0 < y < y, w l \ w 2 / 1

X I *1 i ,

w0 I * y . w 1 , -K » -K

For w. = w_ = w F x. = y = x:

1.1.28 9, p. 462 A wedge with stepped surface temp, t = t x , 0 < r < r Q , e = ±e 0 . t = t 2 , r < r Q . 9 = ±e Q

r0 *?

t - t L L l /*° s in X p l n { r 0 / r ) ] cosh(A6)d\ t 2 - t x " 2 + ? j j , \ cosh ( \ e Q )



1.1.29 15 Semi-infinite strip with convection boundary.

x = 0 y = 0 fcf-fcl \ exp(-miY) s i n (mix) C* 2 2 . ' n=l mr + (n IT / B I )

1. 3 , 5,

Convection boundary at y = 0 . Heat transfer into strip at y = 0:

-s^.fi—»—...i..... k(t 1 "£' " n=l n + (n TT/Bi) See Tables 1.2a and 1.2b.

h, tf * = w

1.1.30 88 27

Periodic strip heated plate q^Wjy) = 0, on 2b wide strips. t(w,y) • t , on (2a - b) wide strips spaced 2a on centers.

-y = 2a- b

See Fig. 1.6a and b for values of maximum differences on y = 0 surface, i.e. Ct(0,a) - t(0,0)J = At . and heat transfer.

tf, h —

£-y = b r *•'

-w-

-y = -b

•y= -2a+ b

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


1.1.31 28 Case 1.1.30 except Q = surface heat flux on (2a - b) wide strips spaced 2a on centers.

t(x, y) - t f k . £ L - §j. (1 + BiX) + 2j Qa

°° r Bi i ^ sin (rotB) cos (raiY) cosh (nirX) + — sinh (fflrX)J /. 2~T id T n=l n sinh (nnw) + —• cosh (miW) '] B = b/a , Bi = ha/k » W « w/a , X = x/a , Y = y/a . See Pig. 1.7 for values of T = t(0 , a) - t(0 , 0),

1.1.32 29

i 10

Spot insulated infinite plate with constant temp on one face and convection boundary with an insulating spot on the other.

h,t f

T777?77m \ w

? r w

See Fig. 1.9 for values of 0(r/w,z/w) = [t(r , z) - tf]/Tt<<» , w) - t f 1 at r = 0 , z = w

Section 1.1. Solids Bounded by Plane Surfaces—Ho Internal Heating.


1.1.33 32 Infinite plate containing an insulating strip. t = t 1 , - 0 0 < x < < * > , y = w . t = t Q , - » < x < » , y = -w q t » 0 , 0 < x < A , y = 0 .

i

t<x , y) - t, t

' *' U 0 _ 1 . i _ n „ - l f VI F 1

f y •

1 2i"l + cosh CTT(2X - L)1 cos T2TTY)]} p = 1 + c o s h (TTL)

T TH 2 sinh Cu(2X - L)3sin (21TY) 1 + cosh (TTL) G =

For L •*• <° :

F = 2 cos (2irY)exp(-2TTX) - 1 , 6 = 2 sin <2TTY)exp(-2TrX)-l ,

X = x/w , Y = y/w , L = V w 1.1.34 19, Infinite thin plate with

p. 3-110 heated circular hole. t ~ «*. K 0(Br/6)

t = t x , r = r x . fc3 " fc» " V B V 5 ) , r > r ,

if h 1 ' t 1

h 2 , t 2

B = -VBix + B i 2 , t^ = {tx + H t 2 ) / ( 1 + H), H = B i x / B i 2



1.1.35 19, p. 3-110

Case 1.1.34 with t. replaced by a heat source of strength g.

k5(t - t j KQ(Br/5) q = 2v{Bi1/S) K^Brj/S) ' r > r l B and t_ defined in case 1.1.34 ,

1.1.36 19, Case 1.1.34 with h„ = 0. p. 3-111 t - tx i Q (Br/6)

= In(Br,/6) ' c > r 1 t 3 ~ fcl

B given in case 1.1.34 .

i

1.1.37 60 Infinite plate with wall 19, cuts as shown. Heat flow p. 3-123 normal to cuts,

q

See Table 1.3 for conductance data K/K K = ka/fi, uncut Q = K t x - t 2

uncut"



1.1.37.1 84 Infinite medium with single and multiple insulating cuts.

See Ref. 84 for temperature and heat flow solutions.

1.1.38 61 Case 1.1.37, cut (c), 19, with only one cut. p. 3-124

Conductance: it

USL - d) /a] + (d/b) + (4/ir) In tsec (TT/2)[1 - (b/a)]J Q = K t x - t 2

I ro to

1.1.39 62 Infinite rib on an 19, infinite plate with p. 3-126 convection.

t = t w , x =• ±«/2 , y > A . t = t w , x > |fi/2| , y = I . Convection boundary at y = 0

5

tfr

See Pig. 1.10 for temp at x = 0 , y >_ 0

t f;n



1.1.40 79 Finite plate with centered hole. k(t, V ¥+ ^(f)

_2w \TTr/_ 2TT

/X 1*1 U

I iFH 1.1 .41 79

i

Tube centered in a finite plate.

_L

T I T - ® -

K w H T

2irk(t 1 -V r < d

-eft-w/d C 1.00 0.1658 1.25 0.0793 1.50 0.0356 2.00 0.0075 2.50 0.0016 3.00 4.00

0.0003 , 1.4 x 10

00 0

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


1.1.42 79 Tube in a multi-sided infinite solid.

2nMt, V £n ®- C{n)

n = No. sides r l * E 2 / ' 1 0 £ o E n = 3

n C<n) 3 0.5696 4 0.270B 5 0.1606 6 0.1067 7 0.0761 8 0.0570 9 0.0442

10 0.0354 00 0

1.1.43 79

l

Infinite square pipe.

T w 1

2 u k ( t 1 - t 2 ) q = 0.93 In (w/d) - 0.0502 ' w / d * 1 ' 4

2 n k ( t 1 - t 2 ) 0.785 Jin (w/d) , w/d < 1.4

1.1.44 87 Partially adiabatic rectangular rod with an isothermal hole.

q - Sk(tx - t Q) See Fig. 1.11 for values of S.

.r/vwyyyy.

W/W//7?



in

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


i M

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


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

t = t 2 , x = «, .

fc-fcl = X + PO X(l - X)

h-H i

to 1.2.2 4, p. 50 I n f i n i t e p la te with

convection boundaries.

V h i —

t - t_ 1 - PO (1/Bi + 1) ?_ _ 2 Po_ + Po 2

t - t 1 + Bi + H + Bi + 2

Bi.^1 + Po(l/Bi2 + 1/2)] (1 -X)

_h 2.t, 1 + Bi + H

H = h 2/h L



1.2.3 3, p. 130

i 00

Infinite plate with temperature dependent conductivity. k = k f + 0<t - t f ) .

(t - t j k 4^ = jr-l+Vl + 2B( l -X 2 , ] q'- 'L

V - 7 -k(t)

J-iÛ-U

B = Bq>"L 2/2k^ See Fig. 1.2.

_ h , t f

1.2.4 Z, Infinite plate with p. 215 temperature dependent

internal heating, t = t Q , x = ±H , q'" = q £ " + P(t - t Q)

q"'(t)

k

(t - t 0 )k L cos (VPOg X)

- l q . . . , t 2 * ° 3 cos CVPOg)

- l

Po& = 6Jl 2/k

Mean temp: ( t m - V k 1 ,i t a n ****$ q'"l2 Pol ,i

t a n ****$



1.2.5 2, Infinite plate with p. 217 radiation heating,

t = t , x = 0 ,

q ' • • = q Y e * Y X . q = radiation energy flux. Y = mean radiation absorption coeff.

(t - tQ)lCY l - yxe "Y

A _„-YX

i to

V-_^



1.2.6 2, p. 223

i o

Porous plate with internal fluid flow. P = Porosity.

u,Vk f ,p f , c q"\k p . P

H & H

— r ^ = jr- L1 - e J + x , o < x < (See Fig 1.3)

—^ - i - U - « V« , — < X < ^ P

Mean temp:

V:_h._|(. . . S i 5 r 2 \ P ^ ( v ^ M

p.ucA P f

u c & _ q'"Jj.(l - P) *P ~ k (1 - P) ' h = k f ( 1 - P) ' C = P f uc



1.2.7 2, p. 227 Porous plate with internal fluid flow and temperature dependent internal heating, t = t 2, x = l,

q ' " = q j " d + B t ) , P = Porosity .

i to

" - V ' W 0

q'"(t)

Bt + 1 e P l 7 ^ 1 * 1 D2 \ l^(*Jl±± D l ,\ 6 t 2 + 1 - E ^ t 2 + 1 e - 1 ) - E \ f i t 2 + 1 " V

« ° 2 D l

E = e - e D I = v 2 + [ ( v 2 > 2 - P O B P °2 = v 2 - [ v 2 , a - p oe? B t j _ - <D1 - D 2 ) ( 6 t 2 + 1 } ' * f c o S p ~ E

E* - ECp

p.ucS, = V

2 - D 2e \ S p = k ( ; . p ) , Po = q - - - B i 2 / fcj

1.2.8 3 , p . 220 Rectangular rod. t = t Q , x = ±b, - a < y < + a.

t = t Q , -b < x < +b, y = ±a .

) - 2 q - - a

°^ft VA b

^ ( - 1 ) " cosh [_(2n + D^XJ cos I (2n + 1 ) | Y |

£(2n + D j l cosh [{2n + 1 ) | B 1

^ , „

n=0

See F ig . 1 .4 , Table 1 .1 . Y = y / a , X = x /a , B = b/a

Section 1. 2. Solids Bounded by Plane Surfaces—With Internal Heating.


1.2.9 3, p. 469 Infinite triangular bar. t = t Q , x = ±y. t = t Q , x = I.

(t - t0)k

X = y/8,

= §- (X2 - *2> In (J)

i



1.2.10 2, p. 197 Infinite rectangular rod with temperature dependent internal heating.

<* - y v e + t n

4Po„ cos (XX) cos (X Y) n m

[•" = q ,J"( i + 0 t ) .

CO GO

S I J L n=lm=l V m [ X n A + ( X m / A ) - P O B ]

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

^

A q"'<t)

X = 2ir{2n - 1) , X = 2ir(2m - 1 ) n m

X = x/b , Y = y/a , POg = qj'-abS/k , A = a/b

See Fig 1.5 ,

^

1.2 .11 9, p . 171 I n f i n i t e rectangular rod (t - t )'k 1 , with convection boundary. - — = 7— + ^-(l - X) - 4

2 Bi k = conductivity in x -d ir . q " ' a 1 k = conductivity in y - d i r . s i n (X ) cos (X X) cosh (X AY) I n ' * n n

V h i

X n [ 2 X n + s i n ( 2 X n > ] [ ( J 5 B ^ ) s i n h < A V + c o s h ( A X n > ]

X n tan (x n ) = B i r B ^ = tâ /k^ B i 2 = h 2 a / k x

A = b ^W X = x / b ' Y = Y / a ' B = b / a



1.2.12 9, p. 423 Infinite plate with point source, t = t Q , z - 0 , l. Source of strength Q is located at r. , 9 , b.

(t - t p ) U

n=l in (fflTZ) s i n (mtB)K0(ffliR)

R = J2 [** + r l " 2 r r l C O S ( 9 " G l } ]

1 u

1 M'P>

B = b/l , Z = z/H

1.2.13 29 at r = 0 , z = 0 , w are given in P ig . 1 .8 .

Spot insulated i n f i n i t e Solutions of f"t{r , z) - t ~|/ft(o° , w) - t "J = 6(r/w , z/w) plate. L f J L f J

Insulated on one side and an insulating spot with convection boundary on the other.

h,t

w r J

77777777777777777777777777,

4z



at



Section 2.1. Solids Bounded by Cylindrical Surfaces—No Internal Heating.


2 .1 .1 4, p. 37 In f in i t e hollow cyl inder. 2Trk(t0 - tL)

fcn U , / ^ ) + (V&i.) + (VBi0>

-"oô t - t„ In ( r / r Q ) t . - t Q ~ Jin ( r . / r 0 , + ( 1 /B i 0 + (1/Bo^

Bi . = h . r . / k , Bi = h n r„/k l l l 0 0 0

2 .1 .2 19, p.3-107

The composite cyl inder. q =

* ( t n - V

t„h

n-1 / r \ n

i=l " " ' i=l •

Temp in the j t h layer:

t - 1 , n 1

j - l

h [ ki \ r i / r i h i J kj Vi) r j h j n-1 . n j L.ln (H±JS + y _i_

r j > 1

Section 2.1. Solids Bounded by Cylindrical Surfaces—No Internal Heating


2.1.3 3, p. 121 Insulated tubes. 9 =

2Trk(t i - t f )

An (*H See Fig. 2 .1

Max heat l o s s occurs when t = k/h.

V to 2.1.4 1, p. 138 Infinite cylinder with

temperature dependent thermal conductivity, k k Q + 3 ( t k = k Q at r Q. k = ki at t^

V-q = S-n ( r Q /

- v 0

*0 =

V

(t

S-n ( r Q /

- v 0

*0 = Vi +

2^K fcn(r0/r)

J lnUg/r^ ( t 4 V

km " ( k 0 + k i ) / 2



2.1.5 2, p. 148 Cylindrical surface in an infinite medium, t = f ($) , r = r r

0 0 , .n t - t. 1 2# \~) [An c o s ^^ + B n s i n { n * ) ] ' r > r

, 27T

, n=ijf [f<4»-tj

n=? f [^+>-^]sin

cos {nif) d(|)

(n<|>)d$>

2.1.6 3, p. 226 Infinite cylinder with specified surface temperature, t - f ( $ ) , r = r n .

t=f(#)

uu

t = aQ + ^ R [a cos (n((i) + b s in (n^H n=l

21T /

• -if n 7T I

f(<l»d<t>

0 2TT

a_ = r I f (<!>) c o s (n<j>)d<t>

'0

t= fW),r = r„

b_ = $• I £M>) s i n (n<t>)di|>

' 0

For f(<))) = t Q , 0 < <j> < TT

[0 , Tt < <)> < 21T

n=l

Section 2.1. Solids Bounded by Cylindcical Surfaces—No Internal Heating.


2.1.7 3, p. 228 Infinite cylinder with specified surface heat flux, q = f (*) , r = r n.

t - t,

<3 CO

= a + 2* R \ a c o s W + b

n

s i n WH

2ir :

V> m aih / f ( * ) d * ' a n = irh(l i n/Bi) /

n irh(l + n/Bi) I

21 1 J f(d>; s i n (w|>)d<t>

f((j)) cos (ni)))d(|)

I q Q s i n (()>) , 0 < <)> < IT F o r « = 0 , TT < 4 > < 2ir

(t-tyh i J. R s in ((})) 2_ V , B , 2 n cos (2mr) = „ 2C1 + (1/Bi)] " * 2 , « ( 4 n 2 _ ^ + ( 2 n / B i ) - ]


Case Mo. References Description Solution

2.1.8 2, p. 133 Infinite half-cylinder with specified surface temperature. t = t , $ - 0 and IT.

t = f ($) , r - r n .

t = f (#

«s; fit t ~ fc

0 = f Z R" sin (n<{»J [f «(>) - t Q ] sin (n<|>)d<t>

For f (<p) = t :

I

t - t "1 u0 " n=l For f ($) = t 1 # sin (4))

r- = R sin (1) - ; ^ ; R" sin (ntf>) fcl " fc0 n n=l n

2.1.9 3, p. 230 Cylindrical shell section with specified surface heat flux and temperature, t = t. , r = r.. q = f (<|0 , r = r Q .

t - t . + a 0 £n (R) + £ *Jn\*f - ^ " J cos (X^) n=l

r n A

u ^ 0

/ X -X \ 2r A n X n l R 0 n + R 0 n ) = W Q J [ f <•> " fci] C O S < V ) d *

Xn = mt/(t>0 , R = r / r i



2.1.10 2, p. 148 Infinite quarter cylinder, t = t Q , <|> = 0 and TT/2. t = f (*) , r = r Q.

t = f(0)-j<

«> u/2 ' • V f l R 2 " s i n ( 2 n ) < t > [ l t m _ t o]

n=l Ja

s in (2n<j>)d<)>

n=l For f(<|>) = t . :

fc- fc0 2 „ - 1 tT^T: = ¥ t a r i

2R2 s i n (2d»l

X - K 4 J

V 2 .1 .11 2, p. 133 Fini te cylinder with two

surface temperatures, t - t L , z = 0. t = t x , r = r Q . t = t 2 , z = I.

t - t sinh (X Z)J„(X R) u u l , V ""•" ^ " ' " O 1 n ' t , - t " " Z X sinh (XL) J (X 2 1 _. n n i. n

n=l J 0 ( V = ° ' Z = z / r 0 ' R = r / r 0 ' L = A / r 0

For t 2 = f (r)

t - t , , r - sinh [\ Z)J (A R) /•

r Q n = 1 s i n h ( X ^ J ^ X ^ JQ



2.1.12 3, p. 253 Case 2.1.11 with t = t , r = 0 , r = t. t = t 1 , r = r Q , 0 < ( | ) x - t Q

= T T Z nI 0(X nR 0) n=l

t - t, t

„ ~ I (X R) sin (X Z) sin (m<|» I I ' * n=l m=l

X = rnr, R = r/A , z = z/Jl n

nml (X R„) n n 0

KJ i

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

fc " fc0 " 2 A n W e X p ( ' X n Z )

=1 1

2 J R[f(R) - t0]JQ(XnR)dR

A = n J i<V

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

If £ (r) = t ,

t~ fco . V e x P ( - X n 2 > V X n B

t l " t o " -A X «»W



2.1.14 3, p. 234 Case 2.1.13 with t = t. , z = 0, and convection boundary h , t at r = r„.

t - t V B i J 0 ( X n R ) e x p ( - X n Z )

~ a* +Bi 2} J.tt ) n=l n o n t x - t f

= 2

n 1 n O n

i

2.1.15 3, p. 238 Infinite rod with a traveling boundary between two temperature zones, t = tlf- r = rQ, z < 0. t = t2, r = r Q, z > 0. Velocity of boundary (z = 0) = v.

•bi

V^r—I n=l [-<v»T /2. exp [J] + [l + < V P , 2 ] 1 / 2 } P Z ] ' Z < °'

t - t fcl - fc2 £ n=l V i ( V [x • c v w ^

exp [ | l - [ l + (X n /P) 2 ] 1 / 2 )p z] , Z > 0 .

Z = Z / r 0 , P = vr Q/a, J (A ) = 0


Case No, References Description Solution

2.1.16 3, p. 238 Finite rod with band heating, q = 0, z = + b. TS —

ro I

2b U-2I8-*nz

^ Heating band

I

t - t

Convection boundary at r = r Q, ^s -b < z < +b. Surface heating (q ) at r = r., -8. < z < +8, .

~ "f L V s i n ( X n L ) I 0 ( X n P R ) c o s ( X n Z )

Vk = H + 2 1 AjBi I„(XnP) + V > W > ] n=l

A = MI, Bi = hr„/Jc, L = V b , z = z/b n o R = r/rQ, p = r Q/b

2.1.17 4, p. 37 Eccentric hollow cylinder, t = t 2, r = r 2. t = t^, r = r .

q = In V(R + 1)

2irk(t2 - t x)

S 2 • V(R - l ) 2 - S 2

V(R + D 2 - s 2 - V(R - D :

R = r 2/ r l ' S = s/ r i

See Ref. 80 for other solutions to eccentric cylinder.



2 .1 .18 4 , p . 39 Pipe in semi- inf in i te s o l i d .

d > 4r,

2 l T k 1 ^ 2 " fcl>

dr-teH-i;-^ h 2 , t 2

to I

h i r i * S d k 9 A 1 kx 2 k 2 ' k x r 2

2.1 .19 4, p . 40 Row of pipes in semi-i n f i n i t e s o l i d .

h 2 . t 2

27ik(t 2 - t x )

5 ^ + A n ^tFîl , for one pipe .

Wu

?3p d

1 w>

k d

s » I« s

„. V l n . h 2 d

n d B l i - — • B l 2 " ~ F ' D - 1

*Lv\



2.1.20 4, p. 40 Row of pipes in a wall.

I T77777777777777777777777777777?

to I

4vrk(t 2 - t x )

BV* n [«'D S l n h ( 2 t^])] , for each pipe .

h r h d B il=-P' B i2=^ D=I

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

k

I*

q i = 2 i rk( t 1 - t Q ) z1

1 " ^ U / fcl - fc0 ^ S 2

+ M -z, =

Z 2 = iln

d2>

1 <V

(-:) - f ^S?!1^)2

For q,, interchange indices 1 and 2



2.1.22 4, p. 43 Circular disk-on the q = 4rflk(t2 - t^ 9, p. 215 surface of a semi-infinite

solid. . . t = v z * -. i i i i _ = 2 s i n - i t , - t , IT t_ 2 1 [ ( R - l ^ + I ^ + t c H + l ) 2 * ! 8 ] 1 ^

j^fe k

i "I z = z A 0

T

2.1.23 4, p. 44 Circular disk in an infinite solid. t = t - f z •+ ± «.

q = 8r 0 k(t 2 - t x )

<2 f. ^ .|

1 1 k

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

M 0

*H ? k 1 •2r, -n

4 ^ r 2 ( t l - t Q )

An ft) + Jin 4r„

'(^J , r1 « d « r 2



2.1.25 4, p. 42 Vertical cylinder in a semi-infinite solid. 2D

//////S/S/V/A

d

1

M 0

l&n H^)] B i , \

+ — i r r 0 k ( t l - t Q )

k

to I

D = d/r r

w 2.1.26 9, p. 216 Two semi-infinite regions of different conductivities

t - t„ 21c ,

connected by a c ircular disk. fcl " fc0 ' T ( k l + k 2 )

t = t Q , z + +». t = tjy z -»• - » . q z = 0, r > r Q , z = 0.

x s in - 1 [(R - I , 2

+ Z2f/2 * [(R + I , 2

+ Z 2 ] 1 / 2 \. z < o .

Z 4

M

w/w/mss. maw///,:

fci~ fco

x s in

2k, = 1 - l K ^ + k 2 )

[(R - I ) 2

+ Z 2 ] l / 2

+ [<R + I ) 2

+ Z 2 ] 1 / 2

g = [ 4 r ( J k 1 k 2 / ( k 1 + k 2 , ] ( t 0 - t l ) , Z = Z / r Q

. z < o



2.1 .27 9, p. 218, F in i t e cylinder with: 69 t = f ( r ) , z = 0 , 0 < r < r

t = t Q , z = I, 0 < r < r Q . Convection boundary at r = r f l , 0 < z < A,

with h, t 0

0" t - t, J0V&n) sinh (CI - Z ] U n )

0 - 2u n s i ' n=l

sinh (LA )

2X 2 n " n (B i 2 + \ \ ) J 2 (A

- J^ »pW " t 0] J 0 ( R X n , d R

L = £/r0. Bi = hr0/k, XnJ' (XJ + Bi J Q(X n) = 0, X R > 0

For f(r) = t x:

t - t, fci " fco " ^ ( B i 2

+ x 2 ) J n ( X n=l \ n/ 0 n

Jn(EXn) sinh (CI - Z]U n)

) sinh (LX ) n

2.1.28 9, p. 219 Case 2.1.27 with t = f(r), z = 0, 0 < r < r . fc" t 0 = 2 C n J 0 ( R X n > Remaining surfaces convection boundaries h, t_. n=l

X cosh (CI - Z]LX ) + Bi sinh (CI - Z]LX ) X cosh (LX ) + Bi sinh (LX ) n n n

C , X , L, and Bi are defined in case 2 .1 .27 . n n

Section 2.1. Solids Bounded by Cylindrical Surfaces—No internal Heating.


2.1.29 9, p. 220 Case 2.1.27 with t = f(z). => , . _, In(mrR) r = r , 0 < z < &. t - t = 2

t = t Q, z = 0, %, 0 < r < r Q. n=l ^, lQ(rarR) r

1 i^v s i n ( m r z ) J [ f <z> ~ fco] s i n < n l T Z ) d z

R = c/l

2.1.30 9, p. 221 Case 2.1.27 with t » £ (z), » r, ,, _, . _. . n „. _ ,, _.T „ * < . * * , V [Xn C O S ( X n 2 ) » B l H s i n (XnZ> W * ]

r = r Q, 0 < z < H. t - t Q = 2 £ 7 ~ ^ 2 TT ' Remaining surfaces convection n=l l B lS, + X n + 2 B lS,) I0 ( n R0 )

boundaries h, t.. 1

x \ f(Z)lX cos (X Z) + Bi„ s i n (X ZlldZ /f{Z)[X n cos (XnZ) + B i & s i n (X nZ)]!

"TO

t an (X ) = (2X B i 0 ) / ( X 2 + Bii? ) , X > 0, R = r/8. n n x> n t. n

Section 2.1. Solids Bounded by Cylindrical Surfaces—No Internal Heating-


2.1.31 9, p. 220 Finite cylinder with strip heating and cooling, q r = -q^ r = r Q, S. > z > & - b. q = 0, r = r Q, % - b > z > -A + b. q r - -tq^ r - r Q, -* + b > • > -A. q = 0, z = ±1, t = t Q . t = t Q, z = 0, 0 < r < r Q.

q o i

(t - t Q)k 8 <-, (-l)ni0(mnR)

T2 £ o • 2 l1<™ f Ro )

s i n (nnrb/S,) s i n (nnrZ)

m = 2n + 1, R = r/2Jl, Z = z/2A, B = b/2&

jUIUHUltA • / / / / / / / / / / / /

•7T7777777777frr7777frrrm. \

l.\.yi 9, p. 220 Finite hollow cylinder with: t = f {z) r r = r i r 0 < z < %. t = t Q, r = r Q / 0 < z < «,. t = t Q, z = 0 and i, r. < r < r Q.

t - t Q = 2 00 1

y |T sin (raiz) / [f (Z) - t QJ sin (mrZ)dZ n=l * "0

F Q = I0(mrR) K0(niTR0) - K Q (nnR) I 0(mrR Q)

F = I (HITR^ ^(mrRg) - K 1(mrR i) I^rniR^

R = r/A



2.1.33 9, p. 221 Case 2.1.32 with q = f(z), CD 1

t Q = - H ^ •£- sin (mrz) / f (Z) sin (mrz)dz r = ct, 0 < z < H. i

Remaining surfaces at t . n=l

F defined in case 2.1.32

Fx = Ij^mm^ K0(niTR0) + K^mrR^ I 0 ( M T R 0 )

For f (z) = q . , w < z < (SL - w)

i = 0 , 0 < z < w a n d ( J l ~ w ) < z < 2 , :

(t - t )k p —s = > —— cos (nirw) sin (rnrz), n = 1,3,5, V *• n 2 F,

n=l 1

W = w/£



2.1.34 9, p. 222 Case 2.1.32 with t = f (r), 2 » j2 T2 z = 0, r,. < r < r n. fc " fcn = T 2-i - - - *o- " ~ "o

1 Remaining surfaces at t Q . n=l [J0<V - J0<XnV] S i n h ( L X n >

x / [Rf<R) - t Q ] 0Q(RXn)dR

u Q {X n ) = 0, XR > o, R = r / r i f z = z / £ , L = A / ^

V U 0 ( R A n > = V ^ n * Y 0 ( V o ' " V * n V Y 0 ( R X n >

2.1.35 9 , p . 222 Case 2 .1 .32 with t = f ( z ) , r = r . , 0 < z < & . t - t < : = 2 2 [X f l cos (XnZ) + Bi s i n (XnZ)]G(R,n)

i t u - a - «,. t _ u f " , ' Z J r'~2 .2 7 1 Remaining surfaces convection n=l [}n + ^ + 2 B l J l J G ( V n )

boundary with h, t , . f 1

x / [f(Z) - tf"|["X cos (X Z) + B i & s i n (X Z)] dz •'o

ta., (Xj = 2XnBV(xJ - BijJ), Xn > 0

6(R.n) = W C W W -BlaK0(Xna0)]

+ K 0 ( R X n>[ X nWV + B i I 0 ( X n V ]



2.1.36 9, p . 223 Semi- inf ini te cylinder with: t = f ( z ) , r = r Q , z > 0. t = t Q / z = 0, 0 < r < r Q .

Jn(*& ) t - t„ 1 J7(TT / [ f ( Z ) -tol"* ( -V + X n Z >

n=l * n J0

exp (-X Z + X Z)dZ n n

W '0.\n>0.z =z /r Q

2.1.37 9 , p . 223 Case 2 .1 .36 with t = f ( z ) , r = r Q , z > 0. Convection cooling at z = 0 with h, t_ .

JnO&J t - ^ • 1 w - / [ f ( z )-^

n=l X n J0

exp (-X Z + X Z) n n

exp (-X Z - X Z)jdZ

J 0 ( X n ) = 0 , X n > 0



2.1.38 9, p. 223 Finite cylinder with heated disks on ends. q z = r z = 0, i, 0 < r < r L

Remaining surfaces insulated.

(t „ - t „))c l n {mrR,)

'A

<ii

'//,

YW////////A T r i lro

Ys

V.

to o J

^^~ =1 - 5 I ^hdr [v-v w > ir" Aî n V n 7 T V - K^nirRg) I^mrR^] , n = 1, 3 , 5,

R = r /% t „ = m m,0 t . = mean temp, over region 0 < r < r i r z = A t 0 = mean temp, over region 0 < r < r. r 3 = 0

2.1.39 9 , p . 224 Case 2 .1 .32 with t = t ^ z = 0 and JL, r. < r < r . t = f ( z ) , r = r Q , 0 < z < l. q = 0, r = r . , 0 < z < I.

/-t = f(r) z = S

+ \/////7///////// V/Aj I'O f ///////////////, ,<*

t - t ( J = 2 V 1 > — s in (mtZ)

n=l 2 •'0

1

f [ f <Z> " fco] s in (nirZ)dZ

F = Ij^mm.) K (mtR) + K (nirR.) I (rorR)

F 2 = \xKm*.£ K0(raiR0) + ^(niTRj) I 0<mtR 0)

R = t/l, z = z/l



2.1.40 9, p. 426 I n f i n i t e cylinder in an i n f i n i t e medium having a l inear temp gradient. At large distances from cyl inder , t = t f l z .

i to

2.1.41 9, p . 434 Sector of a cyl inder, t e = o, 9„ t 0 < 6 < 9„

V 0 < r < rQ,

V r = V

Z, 0 < r < r.

Cylinder, conductivity = k . — Medium conductivity = k .

t n z

fc

0 W + v

< k i + k o ^ r ' J r > r„

Z = z / r r

t - t 1 2 . - 1

r = ? t a n

s in (u8/9 0 )

Isinh ITT- J,n(l K*-©]



2.1.42 9, p . 451 Two infinite cylinders in an infinite medium. q =

2irk(t 1 - t 2)

cosh (*hiH R = r ] / r 2 , S = s/r 2

to 2.1.43 9, p. 462 Spot heated semi-infinite solid, to q z = q Q , 0 < r < r Q , z = 0 .

q z = 0, r > r 0 , z = 0 . t = t , r > 0 f z - * - « > .

00

o qo = l (t - t„)k

r exp (-XZ)J Q(XR)J 1(X)^|-

q 0

1 1



2.1.44 73 Infinite cylinder with two internal holes. t = t Q, r = r Q . t = t., hole surface . hole radius = X .

q = Sk(t„ V S = shape factor

See Fig. 2.6

i IO

2.1.45 79 Infinite cylinder with multiple internal holes. q =

Znkf^ - t 2 )

®-M^ , r 2 < r l f n > 1 Jin

n = No. holes



2.1.46 79 Ellipsoidal pipe,

t, 9 =

T.wtv^ - t 2)

In /a2 * b2\

2.1.47 79 Ellipsoid with a constant temperature slot.

2nk(t x - t2) «,n ( * * * )



2.1.48 79

to

Off-center tube row in an infinite | plate. q =

4T & © © © Y

2TTk(tx - t 2 ) / 6 ( a / 2 d ) \ , v

uu

9 2 (a /d ) = £ exp itrs(n + | V / 2 d cos [(2n f l ) f fa/2d] n=l

e l (0) ire a

00

3 e 3 0

9 2 (0) = 2 exp

n=l

0 = 1 + V exp (iirsn 2/2d)

n=l

e 0 = i + 2 £ ( ~ l ) n exp (iTrsn2/2d) n=l



2.1.49 79 Single off-center tube in an infinite plate. g

1 -Hfrh-

2TTk(t1 - t 2 )

X2J XyJ

T~f

to 2.1.50 79 i

CO

Tube centered in a f inite plate. ;

W-«M

i = 2irk(tj_ - t 2 )

Jin \irr ^ 2w

Section 2.1. Solids Boundedjby Cylindrical Surfaces—No Internal Heating.


2.1.51 79 Heat flow between two pipes in an insulated infinite plate.

^^mmmmm^ q =

2-nMt! - t 2) ira + Jin (•i)

M 2.1.52 79 i

Tube in a semi-infinite plate.

I—)2rU-

f^^^^W

2Trk(tx - t 2 )

2w Virr/

2.1.53 79 Ellipsoid in an infinite medium. 4TT

Q = rVl - a 2 / b 2 M ^ - t 2 )

arctanh (Vl - a 2 /b 2 )



2.1.54 79

l M

Ellipsoidal shell. 4irk(t, Q = V arctanh (c/b.,) - arctanh (c/b )

c = \ b x - a x = \ b 2 - a.

2.1.55 79 Rod in an infinite medium. t 2 ; x, y , z

4-irkt^ - t 2 ) An (2£/r) ' l

,A \k 2i- 37

T < 0-1

2.1.56 79 Short cylinder in an i n f i n i t e Q = c k ( t x - t 2 ) medium. t = t 2 ; x , y , z + » .

Zl 2r

ML 0 8 0.25 10.42 0 . 5 12.11 1.0 14.97 2 .0 19.87 4 . 0 27.84

• 2 « -


Case No. References Descc iption Solution

2.1.5? 79 Toroidal ring in an infinite medium. t = t 2 ; K t y t z - K » .

4TT k ( t x - t 2 ) 2 = in Gtjzj ' r l / E 2 > 2 0

2.1 .58 79 I

Thin flat ring in an infinite medium. t = t 2 ; x,y,z->=».

4ifk(t - t )

2.1.59 79 Two parallel rods in an infinite medium.

S i r M ^ - t 2 ) s > 5r

5 k 4 ^ i_

28

t , _ /

3 * For p a r a l l e l s t r i p s of width 2w, use r = w/2 .

Section 2.1. Solids Bounded by Cylindrical Surfaces—Ho Internal Heating.


2.1.60 79 Two aligned rods in an infinite medium.

22-

*7 ] t C Y„

4TTk(t1 - t 2) , s - 2% > 5r

For aligned strips of width 2w, use r = w/2

2.1.61 79

M i

Parallel disks in an infinite medium.

H taM^ - t2)

2[I - arctan (r/s) | , s > 5r

2.1.62 86 Infinite cylinder with symmetric isothermal caps. q -

kiKtj - t 2 ) 2 In C2{1 + cos 9) /s in 6]



to i UJ



Section 2.2. Solids Bounded by Cylindrical Surfaces—With Internal Heating.


2.2.1 4, 51 Infinite cylinder with convection boundary. 7 ^ - * " " 1 q'" E

i

2.2.2 4, 51 Hollow infinite cylinder with (t - t )k with convection boundary on — outside surface. q t = 0, r = r i

q'-rj 1 (lit 1 ~ R l ] + l ~ R 2 + 2 R i l n < R ) |

r/rif Bi = hrQ/k



2.2.3 4, 51 Hollow infinite cylinder with (t - tf)k ±\2 f2 convection cooled inside "

i

surface. q r = 0, r = t..

I M r * q- " C

l l i i K " 1 ] + i - R 2 + 2«o A n <R>

R = r/r., Bi = hr./k

2.2.4 ?., p. 189 Infinite cylinder with temperature dependent heat source. fc = V r = r 0 • q... = q^"»(i + gt) .

Bt + 1 J 0 ^ R ) „ . ,,, 2.„ e t o + l J„(VPO-) e . ° °

Mean temp:

Max temp:

g t m + 1 2 J l ( V g 5 i ^

et + i K max 5 t o + ' * 0(VK:>

See Fig. 2.2 .



2.2.5 2, p. 189 Case 2.2.4 with:

in in

t = t Q , r = r Q l 0 < <|> < TT.

t = t . f r = r Q f ir £ <|> £ 2TT.

q . . . = q ^ « ( i + gt) .

t _ _ l V 5 ! ' + 2 V sin n6 J n ^ R> fco + t i 2 J 0 < V P 5 ; > n = 1 Jn(VPo^)

n = 1, 3, 5,

p ° f $ = & q r r o A

2.2.6 2, pg 191 Case 2.2.4 with convection boundary.

•O Bt + 1 0 t f + l

J Q " ^ * )

VPOT ' P ofJ = ^ " r u A

Mean temp: 3 t Q + l 2^0/Po^)

VP^|_J0(VP5;) _-!! i J i ( vp^)J

See Ref. 85 for nonuniform convection boundary solution.



2.2.7 2, p. 193 Infinite hollow cylinder with

to I CJ at

temperature dependent heat 3 t source.

^ H [ r ° <VP5~) (0t n + 1)

Yn (VPc"R.) 3' (&tL + l) - o " " 0 " i '

T(3t + i ) [(St. + l) J o ( V g 5 g R i } " J 0 ( V P 5 ; )

J 0(VPS;R)

Y 0(VPO;R)

B = J o ( R i V 5 5 3 ) Y o ( V 1 V - V " ^ V V ^ >

p°3 = ^ o " r o A ' R = r / r o

2.2.8 2, p. 194 Infinite hollow cylinder .. ... (t - t )K . 1 ±

with temperature dependent 5— = -\—^ + 2P0" thermal conductivity, t = t„, r = r.. fc " V r " V k = k0[l + B(t = t0>].

q ' " r r 3 (1 - R 2)

(l - R*) In (1/R)

(k) An(5-

1/2 _1 Po,

Po p = q ' " r 0 3 A 0 , R = r/r Q



2.2.9 6, p. 272 Electrically heated wire with temperature dependent thermal and electrical conductivities.

* = V r = r0 ' kt / k t o = 1 + e t ( t - t o ) • k e / k e O = 1 + 3 e(t - t ( )) .

B - R f + B [ 8 - 6 t 16 J) k e O r u E \ R = 1

kto L

k = thermal conductivity It = electrical conductivity E = voltage drop over length L

2.2.10 2, p. 175 Cylindrical heat source in an infinite plate. t - t, r + - .

h r0 \

\ - t . 2 , h 2

t - %> KQCRBJ

q' /kW " 2IT Bkj^CB]

See Fig .

B =

2 .3 Fig .

B = r ^ l ^ + h^)/kw

** V l * h2 f c2

h 1 + h 2

q' = heating rate in cylindrical source



2.2.11 9, p. 224 Finite cylinder with steady surface temperature.

CD

r -3a.

I ^ V " Z^ Z2 4_ V V n T r R )

2 t = t , z = 0 and A, 0 < r < r Q . q ' " 4 t = t Q , r = r Q , 0 < z < SL.

bl 2 " 3 ti "3lo<™»V sin (mr2) ,

n = 1, 3, 5, . . .

Z = z/A, R = t/Z

2.2.12 9, p. 224 Finite hollow cylinder. t = t 0 , r =• r Q , 0 < z < & . t = t n , z = 0 and %, t i < r < tQ . q = 0 , r = r . , 0 < z < & .

( t ^ V * _ z^ z 2

q«"a 2

4_ V sin (rnrZ) . _1 ,3 L _3 F„'

n=l

n = 1, 3» 5,

F = ij tmrRJK.CmrR) + I (nnR)K (mrR )

F 2 = ij^tn^JKjjtmiRjj) + ^(miR^K^mtR^

z = z/l, R = t/%



2.2.13 9, p. 224 Case 2.2.11 with q'" "%"[} + 3(t- tQ)]

( t " V a COB [(VPQ>2 - VPQZJ] I_ q i t i j j 4 po cos (VPo/2) Po

A ^ ^ ^ ^ S l n < n 1 l Z )

4. v 0 n _ _ l i c

nti V W n X n X2 = (2n + 1 ) 2 T T 2 - Po, Po = q : " e A 2 / k , Z = z/%, R = r / £ n u

V 2 .2 .14 9, p . 423 F i n i t e cylinder with l i n e heat

source. t = t Q , r = r 0 , 0 < ss < H. t " t , 0 < r < r 0 , z = 0 , I. Line source of strength q' per

unit length i s located at r , 9 . ,

b < z — cos (imrB) s i n (imrz) - 2 ^ S m-1

i _ (imrR)

2 TirTFnÔ^n^11^ - V^V V^V] n=0 *0

x E n cosfntf - 6^ . 0 < R < \

E = 1 i f n = 0, E = 2 i f n > 0, m = 2m - 1 n n B = b/Jl, R = r / £ , Z = z /£

For R. < R < R interchange R for R t .



2.2.15 9, p. 423 Case 2.2.14 with a point ,. . .. 0 « source of strength Q located - = - > sin (nmZ) sin (muZ^ at r.f 8^, z^ . m=l

" I (irniR) x 2 îjsagrfn^^n'"^ - ô"*^^] *n 4 0' n=0

y. cos|"n(9 - 9 )"J , 0 < R < R x

f For R < R < R Q, interchange R for R . ° R o E/A, Z = z/£, E = 1 if n = 0, E = 2 if n > 0

n n 2.2.16 9, p. 423 Infinite cylinder with (t - t )kr "

point source of strength Q. 0 = 2? £, c o s C n < 9 " 91 J1 t - t 0 , r - r 0 . n=^" Source located at t v \ . x y e*P<-Xl z l > J

n

( X n R ) V W z = 0- \ > 0 * [K»>- J f i w ] 2

h-z ? W = °' Z B Z A 0 ' R = r A 0

Section 2,2. Solids Bounded by Cylindrical Surfaces—With Internal Heating.


2.2.17 16

to I

Hollow cylinder with temperature dependent heating and thermal conductivity. fc B V r " V q r = 0, r - r Q. k * V r a V «•" -<»i"r t - 0. k = k Q + a(tQ - t). q... = q ^ M + b(t 0 - t),

Solution by numerical method . See Fig. 2.4 for values of t at r = r. .

I

Pi " X " <Vr0>' K " V a t 0 » R = a V b r 0 ' 6 = %"/btQ For a = b = 0:

c 0 c i '" t ) l n n .2 , 1 . / 1 \ 1

'0 0

(Eq. (5) in F ig . 2.4)



2.2.18 64

V

Triangular arrayed cylindrical cooling surfaces in an infinite solid. t ' V r = r o •

0\QJZ o

I q"'

(t - t 0 )k ^

q'"s 2 to (R)

See Table 2.1 for values of A. . See Fig. 2.5 for t values , max S = s/r0, R = r/rQ

Equilateral triangular array

^>z?mfz i . T

. c^WlB " c P i « • VI # $ •

h • ^•; • i -JA I \ 'A m •

Section 2.2. Solids Bounded by Cylindrical Surfaces—With Internal Heating. • :o


2.2.18 64 Triangular arrayed cylindrical cooling surfaces in an infinite solid.

(t - t0)k yj

q'"s 2 IT Sn (R) - \W- ©*] v F

0 *

i £ «f1 - («H - ««• j-x

Table 2.1 for values of A. Pig. 2.5 for t m a x values . sA 0r R - r/rQ

Equilateral triangular array

1 , , . ._, _ • • - * '



2.2.19 64 Case 2.2.18 with a square array of holes.

o efe - | 2 r 0 | ~ - | » - 2 s - - |

Square array

q'"s „, 1 > D2 = 7 an <*> - life) " U ) J

j = i

See Table 2.2 for values of 6. See Fig. 2.5 for t values, max



2.2.20 67 Eccentric, hollow, infinite See Ref. 67 for eguational solution. cylinder.

to i 2.2.21 73 Cylinder cooled by ring of

internal holes. See Ref. 73 for max temperatures.

Insulated surface



I



10 i

Section 3.1. Solids Bounded by Spherical Surfaces—No Internal Heating.


3 .1 .1 . 4, p. 37 Spherical shel l . Q -

4nr 0 k(t i - t Q ) ( r ^ ) - 1 + (tp/rj) ( k / h ^ ) + k/h 0 r Q

t - t„ <r 0/r) - 1 *i " fcn <*,/'«> - 1 + (r n / r , ) (k /r ,h . ) + k/r„h( - i -0 ^ O ' l 0 " i / w v / i i " i ' 0 0

u l

3.1.2 92 Composite sphere. 4»< f ci - V n-1 y i_/i L _ \ + v _J^ £*A"i *i*J £rjh

_ _ ^ _ - , D > 1 Si"*!

t. = local temperature in j layer



3.1.3 If p. 139 Sphere with temperature-dependent thermal conductivity. t » t 1 ( t » t t . fc = V r = r0 ' k - k Q + B(t - t Q) .

u I to

4Trrnk (t. 0 m l V <Vri>

km = < k0 + k i > / 2

V t m % V t = h

k~ fc0 k m ( r 0 / r ~ 1 }

3.1.4 2, p. 137 Sphere with variable surface temperature, t = f(9), r = r Q .

f(fl)

t a V a r nP (cos 9) ^^ n n n=0

2n n 2r

IT — / f (9) P (cos 9) sin 9 d9 ro 1 n



3.1.5 2, p. 137 Case 3.1.4 with: t * t 0, 0 < 9 < TT/2 t = o, ir/2 < e < Ti . + 32 R 5 V C O S D ) • '

3.1.6 2, p. 137 Spherical shell with specified surface temperatures. t fc " V r = r o • t a f (8 ) , r = r . .

ui i

U>

R 2 n + l _ R 2 n + l fc0 = 1 ( n + I) (^H-l-!) (") P » l C O S 9 )

TV

x I P^(cos 9)[f (cos fl) - 1] sin 6 d6

R = r / r .

t = f(fi)



3.1.7 4, p. 41 Sphere in a semi-infinite 4nr k{t - t 0) solid. Q = i 1 +

y//,y//////////////,

MS 2[(d / r Q ) + (1/Bi)]

h ; t i Bi = hr„/k 0'

I

3.1.8 4, p . 43 Hemisphere in the surface Q = 2irr Q k(t 0 - t . ) of a semi-infinite sol id.



3.1.9 9, p. 423 Sphere in an infinite medium having linear temp gradient, nor large distances from sphere t = t.z. Sphere conductivity

A,

t 3 k o TQ " 2kQ + kj a» 0 < R < 1, 8 - z/r0

is k, and medium k o .

conductivity is ^ = [ R + ( R ) (2kQ + k^J cos 9, R > l

3.1.10 59 19,

Irradiated spherical thin shell. See Fig. 3.1 for T/T' . _ \ in <I_ = source heat flux p. 3-111 s

T B sink temp . s a = absorptivity .

S.

a Gl \ V e 2 + X /.



3.1.11 79 Sphere in an infinite medium. Q = 4TT ' Q M ^ " *2* t » t,f r * - .

3.1.12 79 Two spheres separated a large distance in an infinite medium.

0 QE

4nk (^ - t2) 2(1 - r/s) > 5r

for s > 2r, error « 1%



3.1.13 79 Two spheres separated a small Q = ^ ^ _ + r / s + ( r / s ) 2 + ( r / s ) 3 + 2 ( r / g ) < distance (see cane 3.1.12) in an infinite medium. + 3( r / s ) 5 + J, 2r < s < 5s

3.1.14 79 Two spheres of different radii •irMtj - t 2 )

i

in an infinite medium.

3© ^ r—s—H

( r 2 / 9 ) T - 2r_/s , s > 5r n

1 - ( t 2 / s ^ '

2r,

»wsejM';''-3&!*W

Section 3 ,!. Solids Bounded by Spherical Surfaces.

-No intecnal Heating.



W i

Section 3.2. Solids Bounded by Spherical Surfaces—With Internal Heating.


3 . 2 . 1 Spherical s h e l l with spec i f i ed ins ide surface heat f lux. fc a V r = r0 '

q r = q i ' r = r i '

«t - t Q ) k q ' " r .

<J<r i ' i 6<3, RR„ 0 R RR„

R = r / r i

o

3 . 2 . 2 1, p . 190 Sphere with temperature dependent heating,

r. t - t Q , r L 0 * + 3 ( t - t 0 )

<t - t n ) 3

*0

t - t,

0 , p 1 s i n (RVPO) . _ . 2 . . 1 L _ j_ f p o „ & r y k

sin (V5o)

'max ~ fc0 — = 1 - R 2 , Po = 0

t 0 s i n (TTR) , / = -

(tmax - V B ypo s in (VPO)



3.2.3 6, p. 274 Sphere with nuclear type heating, (t - t . )k fc " V r =

rr 0 ' ,- ^f .1 [„ - A - ffu - .«,]

*0 0

I

3.2.4 9, p. 232 Solid sphere with convection boundary. f~ = £&. - R 2 + (2/Bi)]

q'-'r? 6

See Ref. 85 for nonuniform convection boundary solution.



3.2.5 9, p. 232 Solid sphere in an infinite medium. k = k Q, 0 < r < tQ . k = kj_f r > c Q . q... = q ( J " , 0 < r < r Q . g"' = 0, r > r Q .

<* ~ *J kn 1 2 — = id - R + 2/Bi + 2kn/k.), 0 < R < 1 „i..r2 6 0 1 q0 0

(t - t )k

<3n"r 2 = 1/3R, r > r Q

0 0 h = contact coefficient at r = r t = t„, r > » . Bi = hr Q/k 0

Section 3.2. Solids Bounded by Spherical SucCaces—With Internal Heating.


u



w i

Section 4.1. Traveling Heat Sources.


4.1.1 2, p. 189 Traveling line heat source in 9, p. 267 .-..,_ ,.j r an infinite solid.

t « t Q f x - oo. v •*• ±°°.

*• » v

fc - (-S>o(s^^) v = velocity of line source q' = heating rate in line per unit length

4.1.2 2, p. 291 Traveling line heat source in an infinite plate, t = t Q , x •*• <*>, y -»• ±oo.

T w

f—I tn.h-,—^

( t - V w 1 / vxx I > 1 + h 2 , /v \ V T 7 - 1 I

v = velocity of line source q' = heating rate of line

'o-"a



4.1.3 2, p. 286 Traveling plane heat source in an infinite medium, t = tfl, x •*• ».

(t - t„)kv . v

(t - t0)kv 1, x « 0

q" = heating rate of the plane

i 4.1.4 1, p. 352 Traveling point heat source

' p" in an infinite solid, t * t Q f r -*• ».

fc" fc°. k - e x p L u < * + Y > 1 g'/kr 4n e x p L 2a J

g' = heating rate of the point

î



4.1.5 9, p. 268 Traveling plane source in an infinite rod. t = t0, x * ±».

t0.h X x v

AU e x p \ 2a ) q"

U = (v2 + 4a 2 hC/kA) 1 / 2

v = velocity of source q' = heating rate of source C = circumference of rod A = cross-sectional area

i u> 4.1,6 9, p. 268 Traveling point source on the surface of an infinite plate and no surface loss, t = t f l, r •*• ».

''''''jdr. ,%• H

'^^ - k «.© •* i ^[ -ar /D n=l

cos (2|p)exp{vx/2a) 2 2 ^ 2 r = x + y v = source velocity in x-direction q' = heating rate of point source

Section 4.1. Traveling Heat Soucces.


4.1.7 9, p. 268 Traveling line source on the surface of an infinite plate with no surface losses, t = t Q, x •* ±«.

(t - t 0)vwk g'a

1 + N — cos(mrz/w) (1 - N) exp(vx/2a)

n=l

/,

TV ^<m

N = [ l + (2afflr/vw) 2] 1 / 2

v = source velocity in the x-direction g' = heating rate of source

4.1.8 9, p. 269 Traveling infinite strip source on the surface of semi-infinite solid with no surface losses, t = t Q, x + ±».

<«= - v f c v 2 r* + B

a q'a " TT I no "'x-B

exp (X)KQ[(Z2 + X 2 ) 1 / 2 ]dX

thW/A

X = vx/2a, z = vz/2a, B = bv/2a

See Fig. 4.1 v = strip velocity in the x-direction q' = heating rate of strip



4 . 1 . 9 30

i m

Traveling i n f i n i t e s t r i p source on a s e m i - i n f i n i t e s o l i d with convection boundary. t = tf, x •*• ±°°, and z .

Source = q

( t ( x , z ) - t f )kirv X+L

2aqD

TTH exp (HZ)

X-L Vz^T™2

00

/ T exp (H: 2 T 2 ) er fc (Z/2T + Ht)

x {erf [ (X + L)/ar + T ] - erf[(X - L)/2T + t3}dt

v = v e l o c i t y of s t r i p H = 2ah/kv, L = vZ/2a, x = vx/2a , z = vz/2a

See F ig . 4 . 2 .



4.1.10 30 Traveling band source on an infinite cylinder with convection cooling.

(t - t f)irk £T + — V sin (n6) exp [-W(l - R) / l£]

y R n=l

i en

/w cos {w 2Fo - r>4> - [W(l - R)/V2] - IT/4 } + Bi cos Cw2Fo - n6 - W(l - R ) / V 2 ] \ V nfW2 + B i 2 + V2 W Bi) /

Source = q,

W = r0"Vno)/a

a) = rotation speed of source 26 = angular width of source See Fig. 4.3 a and b.



4.1.11 34 Infinite cylinder with moving ring heat source on i t s surface.

(t - t f )2nr n k - 2 X n W>

n - (Nu2/4) + X2 W

-J

.h.t, ,-t(z,r,t) e x p ( f ± V ^ e 2 / 1 6 ) + x ; ) g

V(Pe 2 /16) + \2 + for £ < 0 - for C > 0

Nu = 2hr /k, Pe - 2vr /a, X, = Z * ** , R = r/r 0

NU J n ( X j = 2X a.(X n) O n n 1 n

See Figs. 4.4 a and b.



4.1.12 34

i OB

Infinite cylinder with a moving ring heat source on its inner surface.

t(z.r.t)

{t - t f )2Trr .k

£—fiinB~neat source"

• I \ n=l

J n ( ^ R) J i«V

* n ( * -») V n ' ^ ( X J J ) ' 0 1 V

Xn[Jl(XnV W - W W V ]

^ [ W V W " W V W ] . C / 2 r , -•

A n • . " _ exp (Pe/4)±V(Pe 2 /16) +X 2 , n V(Pe 2 /16) +X 2 L n J

+ for? < 0 - for£ > 0

[ww - ww] w

\ - "I {[WV W - WVW]* + [WWV " WWo^n)

-f[WW " WW] R = r / r . , Nu = 2 h r . / k , Pe = 2 v r . / a , C = Z + OT

i ' i i ' ' * r. l See F i g s . 4 .5 a and b .



4.1.13 70 Infinite cylinder traveling through temperature zones.

r Z. h,.t. z. h 2 , t 2

* — , ' — / >

Zone boundary

Temperature solution given in source Ref. 70.

4.1.14 7i

i Traveling plane source in a thin rod with change of phase.

Temperature solution given in source Ref. 71.

Melt zone Source

4.1.15 71 Traveling plane source in an Temperature solution given in source Ref. 71. infinite medium with change of phase.



.»

Section 5.1. Extended Surfaces—No Internal Heating.


5.1.1 1, p. 209 Infinite rod. t - t Q, x - 0. t - t.f x •*• ».

in I

I S«

V 3

Solution

- r— » exp (-mx) C 0 Zt

5.1.2 7, p. 43 Finite rod, insulated end. t • t . , x = 0. a " 0, x • a.

. ; z: h,tf

E i

Total heat loss:

Q = VhCicA tan (ml) <tQ - t f )

t ' fcf = cosh Qn(& - x)] S - ^ e cosh (m£)

See Fig. 5.1.

Section 5.1. Extended Surfaces—Ho Internal Heating.


5.1.3 7, p. 43 Finite rod with end loss, t = t Q, x = 0.

mkA £ h 2 / m k ) + tan (ml) (to - V 1 + (h /mk) tan (mJl)

(J L -h 2.t.

P t - t^ cosh rm(8. - x)] + (h,/mk) sinh [>(«, - x)] r _ *•

V^" 2' 1' fco" cosh (mil) + (h /mk) sinh (mS,)

m = kA

•J 5.1.4 7, p. 44 Straight rectangular fin.

h.tf

< • *

Use m =-Jjfo i n c a s e 5 > 1' 3 •

tanh (m& ) . See Fig. 5.2

Recommended design criteria: 2k/hb > 5 , Minimum weight criteria: 2S,/b = 1.419Y2k/hb .



5.1.5 7, p. 52 Straight triangular fin. , ^(zÎ) 2h. q = V5hkb — — — , S - — i0(2-VHa)

t - t f l0(2-VgTc) t„ - t„

kb

1 i^ziU) c o " c f i0(2-ca*) >OTi0(2Ve«;)

, See Fig. 5.2.

Optimum l/b r a t i o : V b = 0 . 6 5 5 j | | ,

5.1.6 1, p. 228 Straight fin of minimum mass, t - t £ h(t - t )x fc0 " fcf x - q'

q" = total heat loss Profile:

2k(*-q . \ 2

Y - 2k(*- h ( t 0 - v ) I = Q' I = h(t - V


Case No. References Description Solut ion

5 . 1 . 7

l

10, p . 95 Straight f i n of convex parabolic p r o f i l e .

f w - f V F •fix)

t - t

m = -V2h/kb

q = kbm(tQ - t f ) 2/3

-1/3V

(H (H

* = S T i ^/ad"*)

/ 3 ( f raa)

-1/3

, See F ig . 5.2

5 .1 .8 10, p . 175 Straight f i n of trapezoidal t - t f . JJ parabolic p r o f i l e . t - t \Jl/

f(x)=|(f)2 3 =- i + | V l + 4 m V

- S & (t - t j

1

q " - l ^ 0 -f

0 + 1 ' See Fig . 5.2



5.1.9 10, p. 175 Straight fin of trapezoidal t - t f ^(B )*0(8) + 1 (3 )K (3) profile.

i

fco - fc

f" VWV + WW h3

q = 3

(t0 - V f1!^1!",' " W W "H2COS e [ W W + W V B c >

3 c V e e , I l ( e c ) " V ^ e ' V ^ o * = IHV W W + W W

3 = 2H \ x + - ^ - tan 9)

2 tan , B = 2H e

- tan 9) 2 tan 9

3 = 2H ... c c tan 9)

2 tan 6 , H = W k s i n 6,

*0 -* + -i



5 .1 .10 7, p . 54 Cyl indrical fin—rectangular p r o f i l e .

I (x m)K (x.m) - K ^ x ^ ) I ^ x ^ ) Q = 2Trxbbkm(t0 - t f ) ( x ^ a ) R Q {x^m) + ^ ( X ^ B ) I Q {x^m)

m = V2h/kb

t - t f Kx (xem) I (rm) + ^ ( x ^ ^ f r m ) fc0 " fcf ~ V ' e ^ V V 0 + ^'V^flV

v t 1 ~ ( x e / ! £ b ) 2 ] I 0 (x b m) + 0K o(x bm) , See F ig . 5.3

Minimum material: use F ig . 5 . 4 .

5 .1 .11 7, p. 58 Cyl indrical f in—triangular 9 1 profile.

v [ i - ( V V 2 ] L I V 3 ( * , + B l - v * l _ 2 / 3 ( 5 ) ^ B I 2 / 3 ( g )

m = V2h/kb, 6= | x bm, -2/3

"/ \ 3 / 2 l

See Pig . 5 .5 . L 2/3 fe)]

Section 5.1. Extended Surfaces—No internal Heating.


5.1.12 10, p. 106 Cylindrical fin— hyperbolic profile.

b x b

n

o - 2lrkMbx3/2 ,t t . fo/aty '-a/s 'V - ^ ^ V W V ]

<fc - fcf> /» \ I / 2 C r2/3<V I1^ - ^ / a ^ e ' 1 - ! ^ 8 ' ] ( t 0 " V ~ W P ^ V W V ~ I-2/3 ( Be ) I-l/3<Bb )]

, 1 / 4o (1 - pi \ 1 / 2 Pa /3 <Be> l-2/l ( V " '-2/3 <V J 2/3 ( Bbfl

T * [ M = V2hAbx-b. B e = §Mxf 2 , B b - f M»f 2 - » - 3 / 2 , B = § M X 3 / 2

>3/2 n - (Ke - x b ) J " M/MKSITP, P = V x e

Section 5.1. Extended Surfaces—Mo Internal Heating.


5.1.13 1, p. 234 Cylindrical fin of minimum material.

m I

CD

hlTX.

Q = " T ( X e / x b + 2 ) ( x e / x b " 1 } ( t 0 " V

Profile:

, , h r e [ l /x \ 2 1 x 1 x e ' *™ = — |3\r) "2x; + 6x-

5.1.14 10, p. 114 Pin fins—cylindrical type.

.1 *CK* Q = j kd2m tanh (mfc) ( t Q - t f )

+ = t a n h

n j m i > ' ) , m = V3h7kd , See Pig . 5 .1

J ^ f = r



5.1.15 10, p. 115 Pin fins—rectangular type. Depth of fin = b.

Same as for case 5.1.14 except (7 d J is replaced by (bd),

.1 C ..-/F7

5.1.16 10, p. 117 Pin fins—conical type. m 1

Q = E ^ I . ^ . ( t _ t ) U Bl I , (M) { 0 V

t - t t . nj- ^(MVx/JO

t 0 - t f \ x I l ( M ) , See Fig . 5 .1

<t> = 41, (M)

, M = 2V2 iti&, m = "V2h/kb MI^M)



5 .1 .17 10, p . 119 Pin fins—concave parabolic

type. irkb

81 [V9 + 4M2 - 3j ( t n - t f ) 0 - f

I

1 + W g W 2 2

• 1 + M"2) 1/2 , M = V-ln/kb S. , See Fig. 5.1

5 .1 .18 10 , p. 121 Pin fins—convex parabolic type. Q 164 IQ(M) l C 0 V

3/4

t Q - t f IQ(M) ' " 3 \ k b

2 *1<M> M IQ(M) , See Fig . 5.1



5.1.19 10, p. 134 Infinite fin heated by square See Fig. 5.6 arrayed round rods.

m i

- *-«o

5.1.20 10, p. 135 Infinite fin heated by equilateral-triangular arrayed round rods.

•h,t f

u. r®\ w rz

V) © °© E — 8n

See Fig . 5.7



5.1.21 10, p. 185 Straight rectangular fin with nonuniform cooling. h(x) = (Y * U h a ( £ ) Y . h = av coeff. a

h(x),tf

in I

I

t - t. -Ml fc0 ~ fcf " 2 1 / ( Y + 2 >

. l / (Y + 2) I 1±1 ( V

\ -L. w + * "" (°t) U_ Y+2 "H? Y+2

(0)

A « |"lY + 2)^fr, + 1 ) ' ^ L <mJl) 2 ^ + 1 > .

l/(Y+2)

Y+2

r (Y+l)/(Y+2) ( V l-(y+i)/(y+2)laa]

rri±i" j t+2

Y+2.

Y + *

_ 2-VY + 1 "I Y + 2

5 .1 .22 10, p . 191 Same condit ions as case

5 .1 .21 with:

h(x) = h. (1 + a) Ix/SL + c) ' „ . ,a+l a+1 (1 + c) - c

*- fc

f /u^rv^wv -^""Î'V fc0 - fcf " VV [ V V W V " ^n 'VVl 'V

-.0 2(1 - n)

i-cw^1""!!

u = 2n 1 - n \ (4 1/2 (x/a + o + c ) 1 ' " ( 1 + c ) d - n ) / n . c ( l - n ) / n mil

u Q = u, y. = 0; Uj « u, x » i , n •> — , m = V2h g/kb



2m8, ./- / xc\ n = — , u = nVi exp(- ^ ) 5.1.23 10, p. 187 Same conditions as case 5.1.22 with: h ( x ) = h r l - a e ^ - c x / H ) I • " "VS exp(c), B - nVi

°° "aLl- (a/cf-exp'ÔjJ- „ n ^ a n .,

I

integer: Q = 2iaZ[l - (a/c)Cl - exp(-c)3j * A3 - A4

A l " [ V l * " V l f * 0 [ J n - l < W " J n + l ( 3 ) ] A2 = [j^rt.) - J n + 1(*)] [v n. l (B) - Yn+1(B)]

&3 = J n ( B ) [ V l ( * » - V l ( * » ] A 4 - V w [ J n - i « » - J » n W ]

If n is not an integer:

. A l - *2 2(mH)2{l - (a/c) CI - exp(-c)]) A3 " A4

Al = ["".nm ~ n l f - " ' / ) + » Jn-l < S )]

A 2 = [-Jn(*) + iKJ^(*)][-nJ.n(B) - SJi^tP)]

A 3 = Jn<B)[-nJ_n(#) - *J1_n«l»]

A4 = J - n( B > [ " n J n ( * ) + Hi-l** 1]



5.1.24 10, p. 205 Straight fin of rectangular profile radiating to free space. t = t , x = 0. T = T , x = SL. e Space temp = 0 K.

in i

OK U / - i H c —10 W, h— i -

J20OGT3

= B(0.3, 0.5) - Bu(0.3, 0.5)

B = complete beta function B u incomplete beta function

see Figs. 5.8 and 5.9

'V'3 - V ' , z . T.A. 20oeir/kb

2 3 Optimum length: b/Z = 2.486aeT0/k



5.1.25 10, p. 211 Straight fin of rectangular profile radiating to nonfree space. T = T Q, x = 0. T = T , x = S,.

e K = incident radiat ion absorbed from surroundings.

i >JuLl

k b T 0 \hkbZ 5

4 3 2 5 Z V Z + Z + Z + Z + 1 ~ K 1 T 0

QVkbTQ

, Kx = 20E, Z = T Q / T e

See F i g s . 5.10 and 5.11

Optimum length: b/SL2 = 2.365aeT^/k

1/2



5.1.26 10, p. 218 Straight trapezoidal fin radiating to nonrree space. T = T , x = 0. K,= incident radiation 2

I

absorbed from surroundings

ToJ / / 2

bo J Y<± ± r

V'T 1—»-x

See Pigs. 5.12a, 5.12b, 5.12c, and 5.12d. X = b /b„ = 0.75, 0.50, 0.25, and 0 (triangular fin), e 0 K = 2 ae.

5.1.27 10, p. 228 Straight fin of least material Profile: radiating to free space. T = 0, x = 0.

f(x) = J2-2kaeT: or. • J2_

2aeT„

1/2



5.1.28 10, p. 230 Straight fin with constant Profile: temperature gradient

K S, T radiating to nonfree space. . . 1 0 5 K 2 Z

Tn-T„ ui I

dT 'o dx I

T - T dT J) e dx ~ I '

10kZ3(Z - l ) 2 |[x * fc - «]» - i - =$-ftc. - «])

K 1 T S K x J lTg (Z 5 - 1}

Q = 5(Z - 1) K 2 £

Z = T Q / T e

K = 2 ae

z 5 - L 5 Z 5 ( Z - 1) KjT*

K_ = incident radiation absorbed

e,mm 1/4

A2'"l' *



5.1.29 10, p. 247

Radial fin of trapezoidal profile radiating to nonfree space. T = T 0, r = r 0.

Pin efficiency: (See Pigs. 5.15a-5.15Jt) A = b /b„ = 0, 0.5, 0.75, 1.0 e u K 2 / K 1 T 0 = °' ° ' 2 ' °* 4

Profile No. = K^rjj/kbg

r 0 ' • " ' "* | r e

b 0 -\

T ° A l~— J K,

K x = oe, K_ = incident radiation absorbed



5 .1 .30 10, The capped cy l inder , Side:

p . 275 convection boundary. t - t cosh (mx) + sinh (mx) p. , (mr ) / I (mr )1 t = t 0 , x = «,. Insulated ins ide surfaces .

t„ - t ~ cosh (mJl) + sinh {mi,) Tl, (mr )/I„(mr )1 O r L l e u e J

sinh (mS.) + [cosh [ml) - l ] [ l (mr ) / I (mr )1

* ~ mJl{cosh (mil) + sinh [ml) [i-finr ) / I Q ( m r e ) ] }

Top: t - t f I 0 ( m r ) / I 0 ( m r e )

(7) t„ - t ~ cosh (mJl) + sinh {mi,) Tl, (mr )/I„(mr )1 O r L l e u e J

sinh (mS.) + [cosh [ml) - l ] [ l (mr ) / I (mr )1

* ~ mJl{cosh (mil) + sinh [ml) [i-finr ) / I Q ( m r e ) ] }

Top: t - t f I 0 ( m r ) / I 0 ( m r e )

U1 I-1

to

-Hh- i

t Q - t cosh [mi.) + sinh (mfc) p ^ r a r ^ / I g t m r n

21. (mr )/I_(mr ) 1 e u e mr {cosh (mil) + sinh (nfl) p^Cmr ) / I 0 ( m r n |

m = "Vh/k6, See Figs. 5.16a, and 5.16b

8 t Q



5.1.31 10, Doubly heated straight, t ~ t t p. 407 rectangular fin.

t = t 0 , x = 0. fco-fcf

t = t%, x = l. q Q = kbil

in

I /-M|

^ 1

exp(mx) - 2B sinh (mx)

q_ = kbm |exp (mil) - 2B cosh (mS,) ( t - t )]]

B = -«_-<H^V/_< tcJiL,..« r o 2 s inh (mi,)

See Fig . 5.17



5.1.32 10, p. 394

Single T, straight, rectangular fin. t = t Q, x = b. All surfaces convectively cooled by h, t_.

Vertical section:

t - t„ — = Ffcosh (m x) + "V2u/v tanh (m a) sinh (rnx)l t Q - t £ L x y x J

<h = -5- jsinh (m b) + V2u/v tanh (ma) [cosh (m b) - l]} vx ra..b * x y u x

Horizontal section:

in N

'VL r r t ^

• ^

t - 1 , fc0" fcf

cosh (m„y) <y> cosh (m a)

y

F tanh (ma) h = *•— , See Figs. 5.18a, 5.18b "y m„a

F = cosh (m b) + yivfv tanh (m a) sinh (m b)

m = "Vh/kv, m = Vh/ku x y



5.1.33 10, p. 413

Double T, straight, rectangular fin. fc = V x i = b r All surfaces convect ively cooled by h, t f .

1 * a %

• J f

x 2 b 2

1 1-

« — a 2 —

—«2

•a %

• J f

*1 b 1 u 1

— - a 1 — •

«- v 1

Vert ical s e c t i o n s :

t - t f

T^rq = ^ [ c o s h ( m ^ ) + Kj_ sinh ( m ^ ) ] , 0 < X j < ^

t - t =

y ^ " V 2 [ O M h 2«2> + V 2 V V 2 t a n h l m y2 a 2 )

x sinh ( m x 2

X 2 ) ] ' ° < X 2 

x (cosh (n>x2b2) - 1)] , 0 < x 2 , = —=— tanh (m . a . ) , 0 < y. < a. ¥yl m ^ yl 1 1 1

p i? 1 2 1> „ = tanh (m _a_) , 0 < y_ < a_ *y2 m y 2 a 2 y2 2 ' 2 2



5.1.33 continued 1 cosh (m , b,) + B sinh (m , b,) xl 1 xl 1

i to Ul

2 cosh («nx2b2) + V2u^7v^ tanh (m 2

az^ s i n h ( mx2 b2'

B = P 2Vv 2/v 1 [sinh (i\2b2) + V2u 2/v 2 tanh (m 2a 2 }

x cosh (m,c2b2)] + 2 V u]/ vi t a n n ^yl 1 3! 1

m x l = V2Vkv l f m x 2 = "V2h/kv2

m y l = V2h7K^, m y 2 = V2h/ku2



5.1.34 2, p. 179

m l to lb

Insulated wires adjoined on a convectively cooled plate (e.g., thermocouple wires). t = «:„, r - «. Wire radius = r . w Wire insu la t ion thickness = <S. Wire therm, cond. = It . w Insulat ion therm, cond. = k . P la te therm, cond. = k .

t - t m ,

nMi

B = Vk~T + VFT wl w2

kpWthj + h 2 ) ( l / h . + 6 /k 1 ) 1/2

H = "V2(h + h )/kw

"2' lf2

Section 5 . 1 . Extended Surfaces—No Internal Heating.

Case No. References Descript ion Solution

5 .1 .35 9 , Composite f i n i t e rod. cosh (A) sinh (B) + H sinh (A) cosh (B) + (T /T ) s inh (Dx/x ) p . 157 t = t , x = 0. T^ = sinh (D) cosh (A) + H cosh (D) sinh (A)

t = t 2 , x = x 2 . 0 < x < x x

Aj , A_ = areas .

CV C 2 = circumferences. ft _ ^ - ^ j , B = m ^ - x ) , D = m ^ , T = t - t f

I | H = V h l C l k l V h 2 C 2 k 2 A 2 ' m l = V h l C l / k l A l ' m 2 = V h 2 C 2 / k 2 A 2 f-h,,tt _h 2 , t f

*°\~ 1 P 2 T*1 „ cosh (A) sinh (B) + H sinh (A) cosh (B) + (Tg/TJ sinh (m b - m x)

h—xi-*i F^r-X* Ul I to T ~ sinh (D) cosh (A) + H cosh (D) sinh (A)

X . < X < X

A = IILX^ B = m 2 ( x 2 - x), D = m 2 ( x 2 - x x ) , T = t - t f

H = V h 2 C 2 k 2 A 2 / ' h l C l k l A l ' m i a n d m 2 a S a b o v e '



5.1.36 9, Case 5.1.35 with: t - t f cosh ( m ^ - nyc) p. 157 q = 0, x = x_. t„ - t. cosh (m x ) cosh (ra2x2 - m ^ ) + H sinh (m^) sinh (m2x2 - ny^)'

0 < x < x • (t - tf) cosh (m2x2 - m2x) t - t = H cosh (ra1x1) cosh (m2x2 - m ^ ) + sinh (m.^) sinh ( m ^ - m ^ ) '

X., < X < X j ,

in i w

H = V h 2 C 2

k 2 A 2 / h l C l k l A l ' "l = V h l C l / k l A l ' m 2 = V h 2 C 2 A 2 A 2



5.1.37 31

i ro

Concave parabolic fin radiating to non-free space. T = T. , x = 0.

D K_ = incident absorbed radiation. Profile: f (x) = -|(1 - x/l)\

See Fig. 5.19 for values at (J>(N ,N ).

N c = (2eo-FA2T^)/kb0, N s = K2/2ECTFT* .

P = view factor.

5.1.38 31 Convex parabolic fin radiating to non-free space. Conditions same as for case 5.1.37 except profile is:

See Fig. 5.20 for values of <j>(N , NJ. N and N given in case 5.1.37. c s

f (x) = -~ "VI - x/JL

5.1.39 81 Straight rectangular fin of variable conductivity, k - k |l + B(t- t 0)].

See Kef. 81 for approximate temperature and efficiency solutions.



5.1.40 83

01 I to

Sheet fin for square array tubes with convection boundary, h ft f .

o o o

*

o o o

4 4

(. (.

o o I I

•loo -loo

t ' t t J 2 K 0IVBT[<X - mA)2 + <Y - nA) 2 ] )

n=-°° m=-°a jaa ^£0 J~ n 1/2

K0<RVBT) + I0(RVBT) ^T ^ K0[&VBT(m2 + n 2)J n=-co m=-°°

n, m / 0 .

2n HVBT[(A/R) 2 - IT]

•400 4CO

K^KV&T) - I^KVBI) Y ^T K 0[Aî(m 2 + n 2 3 ,1/2

n=-°° m= •fCO ^£0 "^ ^ 1/2

K0(RVBT) + I 0(R^T) £ J K0[AYBT(III2 + n 2 0

n=-°° m=-°°

n, m ^ 0 .

X = x/b, Y = y/b, R = r/b, Bi = hb/k, A = a/b

Section S.l. Extended Surfaces—No Internal Heating.


i to

Section 5,1. Extended Surfaces—No Internal Heating.


to o

Section 5.2. Extended Surfaces—With Internal Heating.


5.2.1 10, Straight, rectangular fin. p. 194 q = q Q. x = 0.

g = 0, x = b.

I I u>

( t ~ t f ) .. cosh (mx) . . , , g'"b rakb = -—, >•„. - sinh (rax) + J

q Q tanh (mi!,) q 0m

Optimum profile: sinh (2m&) 1 - q 0 / h b ( t - t f )

2m£ - 1/3 - q0/hb(t0 - t f) ' ° ± V ^ O ~ fcf) < 1 / 3

m = Y2h/kb, t = t , x = 0

5.2.2 1, p. 246 Straight infinite rod. t = t Q, x = 0. 1 + 2 1 -

km (tQ - t f) MSB to|/- h' tf

F EZ3

Section 5.2. Extended Surfaces—With Internal Heating.


i u M

Section 6.1. Infinite Solids—No internal Heating.

Case No. References Descr iption Solution

6 . 1 . 1 9 , p . 54 I n f i n i t e p la te source.

t = t , - b < x. < b , T = 0. t = t ^ b < x < - b , T = 0.

— ^ = f [erf (FO* - Po^) + erf (PO* + Fo*)] , - » < x < 00

See Fig . 6 .1

x = ~b x •* b

l

Section 6.1. Infinite Solids—No Internal Heating.


6.1.1.1 74, Case 6.1.1 with a plate of p. 424 different properties.

I to

© 0 ©

t-^- = 1 - r ^ T i ( - H j H e r f c p 2 " - 1 * - ^

* e r f c R 2 " - X> + X 1 ) , - 1 < X < . 2i¥%l J)

CO

l"*£/ 1 + " n=l

+1

fcn - fc«, K + X

/ x - 1 + 2nvsr^rr\ x erfc f • 1 , 1 < X < - 1

" P 2 C 2 k 2

Interface temp:

t - t_

, H 1 - K ;. + K

2K fc0 " ^ 1 + K ( 1 + K) 2

For Po. - small:

y < - H ) n _ 1 er fc^Fo*^, X = 1

n=l

fc2 - *» K fc0 " fc- " X + K

, / X - 1 \ 21 er fc / |

\ 2 V F O - ; (i +

2K f / X - 1 + 2 l V ^ — erfcl K) 2 V f ^



Case 6.1.1 with a pulse source 2

6 . 1 . 1 . 2 74, p. 427 of heat of strength Q J/m* at

x = 0, x = 0 2 TTVFOTV \ F o l / f.

( t - t i ) p , c 1 b

Q

t = t , -» < X < <*>, T = 0.

x {exp (2n - X)'

4 Fo, + exp

n=l

(2n + X)' 4 Fo, - 1 < X < 1

(t - t ^ p ^ j b

I u

^ y ^ { l + KJVTMPO^ ^ (-H) n-1

x exp I ^ ^ V F : ; 1 ^ 1 2 ) . i<*<-i K = P r l ^ f H = l - K

P 2 C 2 k 2 1 + K

6 .1 .2 9 , p. 54 Case 6 .1 .1 with: t - t. co 1 - X t = t Q ( b - x ) / b , -b < x < b, t Q - ta

T = 0. t = t m , b < x < - b , T = 0.

rf K _ F o x ) + Hêr f K + F ° x ) - x e r f K) /at , + J - 2 ~ i e x p (x + b)' + exp • ( x - b ) ' 2 exp m



6 . 1 . 3 9 , p . 56 I n f i n i t e rod source.

t = t Q , | x | < d r | y | < b, T = 0 .

t = t a , | x | > d, | y | > b , T = 0.

t - t 1 1 5 = - = J [ e r f ( F o ; - Fo*) + .rf (*£ + F o * ) ]

x [erf (FO* - Fo*^ + erf M * * )]

6 .1 .4 3 , p . 407 i n f i n i t e cylinder source. fc " V r < r 0 ' T = °' t = ^ r > r Q , T = 0.

t = t_ 1 +

0 0

exp(-X^Fo) J 0(XR)Y 0(X) - 3 0 ( \ )? 0 (XRT1 ^

J*(X) + YJ(X)

See Fig . 6.2



6 . 1 . 5 9 , p . 55 Spherical source. t - t ^ ^ t * \ i / * * \ ,_ . . . r ~ = r erffFo - Fo ] - — erf(Fo + Fo ) t = t 0 , r < b, T = 0. t„ - t^ 2 \ r cQ) 2 \ r tQJ t = t^, r > b, x = 0.

* \ 2

V - j e x p ^ - FO^) ] exp[(Fo r* + F o ^ ) 2 ] 2 Fo VT r

See F ig . 6.3



6.1.5.1 74, Case 6.1.5 with a sphere of p. 429 different thermal properties.

I I A . . i (^\hi?± exprP2P . P ( 1 T R ) i fc«- fc0 R l K 2 - V l X + K 1 L l 1 1 J

AI_S . P VFOT\ - « f A-iSM , R < i .

W x V l 2 V ^ f " Merfc/^^\ + F^ M W 2~

x erfc/

t - t _ , , ,„ „ , / R _ X

erfc

1 ~ (K, - 1) (K

fco " ^ R i laVioTi K2 " x \vnq.

K 2 + K V l f e X P ( P 2 F ° 2 " P 2 R " *)*rtC(^= ~ P 2 V B 2 |

1- , £ / R - 1 + 2 ^ 7 5 l \ , K2 + K l K 2 " X \ 2 ^ j ( K 2 - D ( K 1 + D

/ 2 . x /R - 1 + 2 0 : x exp(p^Po2 - P2R - 1 + 2>5 2 7^)e r fc / ^ _

~ V 5 5 ! )) ' R * X

K - ^ 1 K - ^ P " 2 - 1

P - Vll 1 " P 2 C 2 k 2 ' 2 " V X " K 2 + V K 1 ' 2 X + K l

f(+R) = f(-R) - f(+R)



6.1.6 9, p. 61 Adjoined semi-infinite regions. t = t , x > 0, T = 0. t = t_, x < 0, T = 0.

t - t, i - r f - = | [ e r f ( p o ; ) + l ] / - » < x

I -J 6 .1 .7 7 , p. 89 Specif ied temperature

d i s t r i b u t i o n . t = f ( x ) , -co < x < <*>, t

00

w J f (x + \a)exp(' -X2)dX



6.1.8 9, p. 88 Adjoined semi-infinite regions with contact resistance, t = t , x > 0, T =0. t = t2, x < 0, T = 0.

t - t,

x erfc I

t - t,

00 •c k,

t~rt2 ' TTJ I1 + A [ e r f K ) + e 3 * ( H i x + H f i / 4 )

(POx + H f l / 2 ) ] ) ' X > °

sp^q = r r ^ [ e r f I Poxl - e 3 * ( v + »la» x erfc(| Fo*| + H2o2/2^1 , x < 0

0 h ^ , H . i lLpL , H - J&(1 + A, k, ot- 1 k,A 2 k *

1 2

6.1.9 9, p. 88 Adjoined semi-infinite regions—no contact resistance.

0 .

0 .

t - t, —-f- - ^ [ l • A erf (*£)] f x > 0

t = t , X > 0 , T

1 2

t - t t = t , X < 0 , T t-r^ = rir e r f ( l p o x l) ' x <

k, > a „ 1 2



6.1.10 9, p. 89 Plane source of heating between adjoined semi-infinite regions, t = t., -°° < x < °°, T = 0. ^ = q„, X = 0, T > 0

= heat source strength.

k 2 k,

at I is

-x t.

( t " V k i i / *\ a<ri

{ t " V k 2 I /, *,\ q 0 C T 2

k * a 1 2

6 .1 .11 1, p . 341 Line source of heat.

9 , p . 261 t = t , 0 < r < r , X = 0 . fc = *=«' E > r

0' T = °-t T T T " F15 e x p ( " F o r j ' r > r Q , r Q + 0

If line is a steady source of strength g :

0 VVT



6.1.12 1, p. 342 Cylindrical shell source. t - t t = t Q , t i < r < r Q , T = 0. t 0 - t a

(F\ ' F % ) e x P [ " ( F o r * 2 + F °r ' ) ] J o( 2 i F o r * \ )

*»' r > r 0 , T = 0. i =V=T

I H

6.1.13 9, p . 247 Spherical surface a t steady t - t temperature. t - t t = t ± , r > r Q f T = 0. t = t 0 , r = r 0 , T > 0.

i 1 / * * \ — = — erfcfPo - Po I fci R V r V

l 0



6.1.14 9, p. 247 Spherical surface with t - t. . i B I , . _ , . _ , „ . „_ , expQBi + 1)(R - 1) convection boundary. t , - t . (Bi + 1)R

t =» t t , r > r Q , T = 0.

j er fc jPo - Fo J

(Bi + l )Fo]er fc |Po* - Fo* + (Bi + 1 ) V F O | |

£ 6 .1 .15 9 , p . 248 Spherical surface with ( t - t . ) k - — = ^ lecfclFo* - Fo* ) 0 R L I r V steady heat f l u x . q Q r 0

t - t . , r = r Q , x = 0. / * * , - Y l q = q , r = r . - exp(R - 1 + Fo)erfcfFo t - F o r + V f o J

Sec t ion 6 . 1 . I n f i n i t e Solids—No I n t e r n a l Hea t ing .

Case No. Ref _r •.ices Desc r ip t i on So lu t ion

6 .1 .16 1 , p . 336 Point sou rce . t - t r"* t = t . . , 0 < r < r „ , T = 0. „ _ ,. — e x p / - P o * 2 j , r > r „ , r n - 0 0_

i' u " L " V L ~ "* t. - t " ^r^n^y *"r / ' * ' *0' '0 fc = fcco' c > V T = °"

0 I

t i _ t o ° 6 > ^ V If point is a steady source of strength Q Q:

; = — z - z : — I expl- X Po IdX V r 0 4TT3/2Va fe \ "'

For T ->• <*>:

( t - t j k r 1 Q 0 4TT



6.1.17.1 1, p. 340

i

Spherical shell source. t t., r. < r < r I I

0. 0'

too' r > r Q, T = 0.

fc- fc» Ro - 1 r ( R - i ) 2 i r ( R + i ) 2 i tpr-^ • S ^ m e X PL" 4FO J " e X PL" 4Fo J ' R > V R0 * °

R = r/ri

If the shell is a steady source of strength Q.: ( t - *»>* V5- f i_

*0 ' I 4ir r - / ^ X exp

( 2 , 2 (r - r ^ X 4a exp to dX

For T •+• °°:

( t - t )kr _1 4ir

6 . 1 . 1 7 . 2 1, p . 342 Plane s o u r c e .

9 , p . 263 t = t . , -9. < x < +£, T

t = t^, |x| > a, x = o.

I

t - t co 1 / *2\ , , 7 — = —=- expf- FO ) , x\ > a, a + o t . - t irFo c \ x / • '

If the plane i s a steady source of strength q n

g 0A w l / 72 e x p ( ~ x^2/4a)dX *VVr

a T A. * J x2

= VFO" exp(- Fo* 2) - -L| - e r fc /FoÂ



6.1.18 9, p. 335 Cylinder with steady surface temp. t = t^, r > r Q f T = 0. t = v r - v T > °-

I

00

0 1 J.

See F ig . 6.4

2 J 0(XR)Y Q(X) - YQ(XR)J 0(X) e x p ( - X Fo)-

jJ(X) + Y*,(X)

q r 0 , _ - 1 / 2 1 1 / F o V / 2 _,_ 1 _ ( t n - 1 . ) • ( 7 T F O ) + 2 - i ( r j + 8 F o -0 i

For Fo < 0.02

( t - V 1 f /R - 1 \ +

( t n - t | ) = VI e r f c ( l ? f 5 ) +

(R - 1 ) V F O

4R 3/2 i e r f c (~2Vfo)

_,_ (9 - 2R - 7R 2) .2 _ / R - l \ ^ + „ D 5 /2 l e r f c ( ^ V f o ) +

32R'

dX X

R = 1

6.1 .19 9, p . 337 C y l i n d r i c a l sur face with _

convect ion boundary, t - t ^ r > r Q , T = 0.

• ^ - - 2 Si f i " fcf M

e x p ( - X Fo)

JQ(XR)[XY1(X) + Bi * 0 (X)] - YgtXR^XJ^X) + Bi JQ(X)] ^

[XJ^X) + Bi J Q ( M ] 2 + [XYX(X) + Bi Y Q (X)] 2 X



6 .1 .20 9, p . 338 Cyl indrical surface with

steady heat f lux ,

t = t r r > r Q , T = 0. q r = V r = V T > °-

( t - t , ) k - t..)k 2 /" T 2 1 V X R ) Y 1 ( X ) ~ V X R ) J l ( X ) d X

— j Li - • * » *-°>J — — [jjtt) + Y*<A)]A3

For Po < 0.02:

0\ H Ul

(t - t . ) k

V o

For Po » 1:

i , A / F O . . /R - 1 \ 3R + 1 ^=r- .2 . /R - 1 \ ^ — = 2 \ - x e r f c ( - ^ W = ) - - ^ - VFO I e r f c ( - ^ = ) +

( t _ V k /4Po\ -*— = Anp^f ] - 0.57722 V o \ R 2 '



6.1.20.1 74, Case 6.1.20 with two dif-p. 434 ; erent thermal materials,

t = t., r > 0, T = 0. q r = q0r r = r Q, T > 0.

en I

( t - V i 4 f r / 2 \i J o ( X R > v A ) d A

VO IT2 1 L *\ VJ X 3 ,2

ô^f X R )* - Y o|/ f x^

J , ( X )

(• + *")

dX , R > 1

K Ji< x' yo(vf xj - ^ J o ^ Y i ^ x )

* - K J i ( X , J o ( > / 5 A ) " ^ V A ) J i ( ^ T A



6.1.21 9, p. 342 Infinite conductivity cylinder in an infinite medium. t - ' t y r > r Q , T =0. t = tjl, r < r Q , T = 0.

I k,c 0 , /J 0

fc - fc0 4B / " fcl " fc0 " n 2 i

exp(-A 2 Fo) f | , R < 1

B = 2 P 0C 0 / P 1 C 1

A.':=!.[AJ0(\) - BJX(X)]"2 +|^Y0(^) - BYÂ)] 1

See Pig. 6.5

6.1.22 9, p. 342 Infinite conductivity cylinder with steady heat source in an infinite medium, t = tQ; r > 0, T = 0. Heat rate = Q per unit length.

(t - tQ)k 00

exp(-X2 Fo)] 3^, M < IK

B and A defined in case 6.1.21.

See Pig. 6.6.

k,c 0,P Q



6.1.23 9, p. 346 Cylinder with properties different from surrounding medium. t = t x , r < r Q, T = 0. t = t Q, r > r Q, x = 0.

I

6.1.24 9, Point source of rectangular p. 402 periodic wave heating.

Q(T) = 0, T < 0. Q(T) = Q„, nt n < T < nt + T l f n = 0 , 1 ,

t _ fc0 4Y f /,2 \ V X R ) J l ( A ) d X

i .

J^Xjfj^XRJlf) - Y0(BXR) T[dA

x (4>2 + i|>2) R > 1 .

i|i = Y^CXJJJJCSX) - BJ^XJJ^BX)

<j> = yJ1(X)Y0(eX) - 6JO(X)Y1(0X)

e = v s ^ , Y = v k o See case 6.1.4 for equxl material properties.

(t - t„)4irrk = erfc (f??) - Tl + 1

0' '"0 0 1,

Q(T) = 0 , n T 0 + T < T < (n + 1 ) T 0

n = Number of cycles, t = t m , r -»• =°.

00 exp(-TX2) exp(-X 2 + TjX 2) - exp(-X 2)| sin(Po*X)dX

x[l - t X p(-X 2 )]

0 < T < T,

r i = T i / T o ' T = T / T o ' F o * = trtv^l

Q(r)


Ref-Case No. erences Description Solution

6.1.25 9, Line source of rectangular (t - t )4irk CO *

p. 402 periodic wave heating. Q~Q = E i < " F o / 4 T> <Ti _ 2>

Same conditions as for case 6.1.24.

- 2 /

°° exp(-TA 2)J 0(Fo*A)[(l - Têxpl-X2) - exp(-A 2 + OÂ2) + r JdA

0 x[l - exp(-A2)]

T l = T l / T 0 ' T = T / T 0 ' P ° = t / ^ d f o i

H

6.1.26 46 Infinite conductivity sphere See Fig. 6.7 19, in an infinite medium, p. 3-63 t = t x Q, r < r x, T = 0.

0 < T < T .

Section 6.1. Infinite Solids—No internal heating.


Section 6.1. Infinite Solids—No Internal Heating. , . • ' ; ' . ; . . • • • • • ' ' >


to

Section 6.2. Infinite Solids—With Internal Heating.


6.2.1 9, p. 265 Cylindrical source of heating. ( t - t . ) k

q u i _ q w i f o < r < r l f

-a < z < +z, T > o. q ' " = 0 , r > r £ , | z | > A, T > 0 .

t = t ± , r > 0, | z | > 0, X = 0.

_^ . - / t -HHHi4) du, r = 0 , Z = 0 M 0 ' 1 -i J0

F o r SL •*• ">i

( t - t , ) k ^ • " t — ( - ^ - i - C - A . ) , r = 0, Z = 0

to to

q'" = 0

a 0 *1

Fo = orr/r ,

1

— %

6 .2 .2 9 , p . 347 Case 6 .1 .23 with a cy l inde r and i n t e r n a l hea t ing , t = t Q , r > 0, T = 0. — I l l — — I I I

CO - - j

fc " V ' O _ 4 f ll - exp(-X2Po1)Jj0{\R)J1(X)dX q A " r 2 IT2 i , 4 ^ 2 , ,,,2T

q l " , 0 < r < r n . T > 0 .

*0 "0

( t - t n ) k

[*2 + * 2 ] , R < 1 .

- V k Q 2 f [ l - exp(-X F o ^ ] [J Q (BXR)I)) - Y0(BXR)ip]dX

S, <f>,and I)J defined in case 6 . 1 . 2 3 .

R > 1 .



6.2.3 9, p. 348 Sphere with internal heating (t - tQ)k „„*2

j ' " r 2 3 0 0

Sphere with internal heating ( t - t»)k *2 r , / i _ R\ Q / l + R \ . . . . . „ 5 I ° _ _ ! _ 2 R i 2

e r t c / ^ - ^ \ - 2R i 2 e r f c ( ^ ^ r ) in an i n f i n i t e medium. q« '<r 2 4 L V Fo / \ Fo / t = t . , r > 0, T = 0.

q . . . = q . . » , 0 < r < r , * 0 2 Fo 4-^{"try-^^m q " 1 = 0 , r > r Q ,

T > 0 . 0 < R < 1 .

T ( y \ i f c _: t o , k FO* 2

B V - 7 F ' ô"r2o _ 2 R

+ Fo* i 3 e r f c ( 5 - | - L ) l / R > 1 .

Fo = 2"tf«T/r0



6.2.4 9, p . 349 Inf in i te conductivity sphere in an infinite medium with contact resis tance, t = t , r > 0, T = 0. q ' " = q'Q", o o. q " ' = 0, r > 0, x > 0.

(t - t Q )k x

i n 2 *o r o

l_jh_Bi 6 K 2 B . 2 Bi it

oo

exp(Fo u )du lQ [ u 2 ( l + Bi) - K B i ] 2 + [ u 3 - K Bi u]

I to

a 1 . P l ' C V k 1

For small values of T:

<fc - V k K Fo

^0 0

[1 - (K Bi Po)/2 + . . . J

0 < R < 1 .

For large values of T:

( t - t 0 ) k

• ^

+ Bi Bi

K = 3 P 1 C 1

p o c o

Section 6.2. Infinite Solids—With Intt. . Heating.


6.2.5 74, p. 426 Infinite plate with internal heating in an infinite medium, t = t Q f -Z < x < I, T = 0. t = t m , a < x < -a, T = o.

1 l q ' "

JULA*

1 + Po Fo,

+ 4 Po Fo, \, - 1 < X < 1

1 V , ^ n - l j . r <2n - 1) + x"j i - r n c 2 (_H) e 2^sr n=l * u

1 i2erfc [ ^ - ^ ] J .

- Tti [«*» (!*£)+ 4 p ° F ° 2

i 2 e r f c (f*J)]

K(l + H) V , m n - l L f / 2 n V V 5 I ^ X - 1 \

V 2 ^ ) \ + 4 Po Fo i ' e r f c , 1 < X < - 1

K" £hr 'H = i ~ r l ' f ( + x > = f < ~ x ) - f ( + x )

P 2 ° 2 k 2 1 * K

Section 6.2. Infinite Solids—With internal heating.


i

Section 7.1. Semi-Infinite Solids—No Internal Heating.


7.1.1 9, p. 61 Constant surface temperature, t - t / *\ h ~- = erf(Fo ] + ^T-t = t. + bx, x > 0, x = 0.

t = t , x = 0 , T > 0 . s 0 S

For b = 0:

V _2 k(ts - tQ) " W

0 s

See Fig. 7.1

i

7.1.2 7, p. 89 Variable initial temperature, t = f (x), x > 0, T = 0. t = t , x = 0, T > 0.

' -<* • * I «>{-[- (^) 2] - -[- H*fy f (B) = f(x)



7.1.3 5, p. 80 Convection boundary 74, p. 206

t = tj_, x > 0, T = 0.

t f ,h *, )

t - 1 . i _

fcf-fci 1 - erf K)

- e x p ( B i x + B i 2

F o x ) [ l - e r f ( F o * + B i ^ ) ]

,. *° „ » = exp(aTh 2 /tt 2 )erfc(V5f h/k) See Fig. 7.2 h ( t f - t . )

"i1 7 .1.4 2 , p . 275 Ramp surface temperature. t = t . , x > 0 , T = 0 . t = t . + bT, X = 0, T _> 0.

t = t, + br.

t - t. bT

i - ( l + 2 Fo* 2) erfc ( F O * ) - £ Fo*. exp(-Fo* 2 )

7 .1 .5 1, p. 258 Steady surface heat f lux, t = t . , x £ 0, T = 0. <3X = qQf x = 0, T > 0.

(t - t , )k *2 i ^ - - ^ p°r e r f ° K ) + e x p K ) <* - V 2

q0Var" "Vf , x = 0 See Fig . 7.7

Section 7 .1 . Semi-Infinite Solids—No Internal Heating.


7.1.6 9, p. 62 Time dependent surface temperature, t » tQ, x = 0, 0 < T < t £

t = t x, X « 0, X > X 0. t = t., X > 0, T = 0.

fc " fci / *\ — T 7 - erfc(Fo x > 0 < T < x 0

t - t. t, - t, i TTT - » f o K ) + v ^ T T e r f c( 2V a (T X- v ) ' T > To

7.1.7 9, p. 63 Time dependent surface temperature. t = t., x > 0, x = 0. t = bT, x = 0, T > 0.

br-

1 ^ - (l + 2 Fo* )erfc(Fo*) - |= Fo* exp (-Fo^2)

7.1.8 9, p. 63 Time dependent surface temperature. t = t,, x > 0, x = 0. t = bVr, x = 0, x > 0.

bV7-

t - t. bx - = expf-2 Fo*

2j - Vir Fo* erfc^Fo'j



7.1.9 9, p. 63 Time dependent surface temperature.

t - t.

bf n/2 i = A(f + 1 ) i W ( P O ; )

t = t., x > 0, T = 0. t = tj + bx n' , x = 0, T > 0. See Pig. 7.6

br' .n/2.

7.1.10 9, p. 64 Time dependent surface temperature, t = t., x > 0, T = 0.

t = e ^ , x = 0, x > 0.

t - t i 1 bx

= i exp {- xVETS) erfc /FO* - Vbr\

+ exp(xVbAx)erfc(Fo + Vbrl



7.1.11 9, p. 74 Convection boundary with t - t step fluid temperature t = t . , x > 0, T = 0.

t f = t x , 0 < T _< T t £ - V T > T.

t..h-

Ol

-—•^-—- = erfc(Fo*^ - expJBi + (H 2 /4 ) | er fc Fo* + (H/2)| , 0 < T < T,

v t - t = e r f c ( F O x ) " e x p [ B i x + ( H 2 / 4 j | e r f c [ P o * + ( H / 2 ) ]

+ i — ) erfc t . - t . 1 i 2"Va(T - TQ) - exp

x erfc hVa(T - Tn)

2Va(T - x 0 ) ,T > T

2KV5T

Section 7.1. Serai-Infinite Solids—No Internal Heating.


7.1.12 9, p. 76 Step surface heat flux, t = t., x > 0, T = 0. 3 = - = 2 ierf c( Fo ) - 2 V - — — ierfc/ * \ ,

" o 1 ^ v « ' T \2va(T - t o ) y g^ = q Q, x = 0, 0 < X < T Q. c^ = 0, x * 0, x > x Q. T „

0 See case 7.1.5 for 0 < T < x .

7.1.13 9, p. 76 Time dependent surface t - t. = 1, x = 0, 0 < x < x m heat flux.

t = tL, x > 0, x = 0. fc " fci = f sin-1(VxJJ7x) , x = 0, x > x Q

x = k/vSnT, x = 0, 0 < x < T 0 . A T „ / ( T - T „ ) q x = 0, X = 0, X > X Q.



7 .1 .14 9, p. 77 Time dependent surface heat f lux. t = t . , x > 0, T = 0. q x = q n T n / 2 , x =* 0, T > 0,

n = - 1 , 0, 1, 2 . . .

( t " V k _ H I + n/2) q0V5? t n / 2 n

< * • ! )

, x = 0, x > 0

< t _ t i n72 = 2 l 4 ) n / 2 r ( 1 + n / 2 ) i n + l e r f c K ) q0Var T

See Pig. 7.8 for n = 2,

V 7.1.15 3, p. 402

Plate and semi- inf in i te so l id composite with constant surface temperature, t = t . , -b < x < <=, T = 0. t = t , x = - b , T > 0.

t o " ^ „T 0 | L a i J

- B erfc r<2n + l )b - x~|j I °1 J)' -b < x < 0 ,

t„ - t , tt • r f r I «" " '°[ ' 2" * l{> * -] • • » •



7.1.15.1 74, Case 7.1.15 with: t - t / *\ V" n i / * * \ p. 409 t = t„0 < x < b, t-ri- - e c f c K) ? H 2 H e r f ° I2" F°b * F V

T = 0. t = t , x > b, x = 0. t = t , X = 0, T > 0. O t , - t

(tQ -\[) (I • I) 1 H n _ 1 e r f c [ ( 2 n " X ) F°b * F o x ] ' n=l

o < X < 1

i

t - ^ - l - = f i T [ Z H n erfc [Po x - Fo x + Fo b + (2n - 1)B F o J n=l

( t l - V K

f + ( t Q - t x ) ( i + x) e " c

* * Fo - Fo. +

x b

2 K

(1 + K) ( t 0 - tx)

Y H n - 1 erfc | F O * - Fo* + 2nB Fo*J, X > 1

n=l

. P l " l k l n 1 - K . _JJ ~ P,c„k„ ' H - 1 + K ' B "" V 2 2 2

Section 7*1. Semi-Infinite Solids—No Internal Heating.


7.1.16 7, p. 99 Periodic surface temperature. . . t - (t„-v c o a ( — ) ' X = 0 , T > 0.

t = t , x > 0 , t = 0 . 0

t - t ^ = exp ( - V J x) cos (u>x - V ^ x) , See Fig 7.3

T = maximum surface temperature TQ = period of periodic temperature a) = 2Ttn/Tn

J,

Maximum temp: fcmax " fc0

exp ( - « • )

Time for t to reach x: max

Si m — +

fc0 n 2 Yawnr" , m = 0, 1, 2,

Heat transferred in to s o l i d , T. < T < T_: q V c t / T 0 1 r / TI\ / Tf\T

-r-rr r - r = ° °S ( U)T, - -r I - COS (<0T_ - -)\



7.1.17 7, p. 106 Periodic fluid temperature. fc _ e x p (,^) C Q S \ w - ^ . - (-LJj| t f > =( t f M - m i — y — — t + ( 2 / H J + ( 2 / H 2 ) ] 1 / 2

t = t , x > 0, X = 0.

<h-SJ ax h

H = — S r - , u = 2mt/x mrlt °

t -„ = "iax f lu id temperature fM T- = period of periodic f lu id temperature

I

Maximum temp:

fcM- fcO exp ( - ^ • ) ( t fM " V [ l + (2/H) + ( 2 / H 2 ) ] 1 / 2

7.1.18 9, p . 68 Square wave surface temperature. t - t a + t,,, « - 0, 2raT0 < x < (2m + l)x , m = 0, 1, 2, . „ . . t = t - t , x = 0, (2m + 1)T < x < (2m + 2)X 0'

T = period of periodic temp t = mean surface temp m t_ = 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 8.1.7). See Figs. 9.4a and 9.4b.



7.1.22 4, p. 84 Infinite cylinder in a semi-infinite solid. Cylinder assumed mass less.

Time for soil to reach steady states

«*•» C(d/D + l / B i D ) 1 , 4 4 ,

i

2, WAMMWAWM.

C = 4.6 for const, cylinder temperature. C = 6.0 for const, heat flux from cylinder.

The heat transferred during the time given above: _2SL

SQ 2(t c- t f) = 12(d/D + V B i D ) 1.25

Temperature of cylinder = t . Initial temperature of semi-infinite solid = t. t > t e. c f

The time for the cylinder to return to temp t after st jady state is achieved and heating is stopped:

3 | = (d/D + l/Bi D) 2' 3 f D m See Fig. 7.4 for values of f,



7.1.23 9 , p . 264 Semi- in f in i t e surface hea t ing , t = t . , 0 < x < « , -oo < y < +oo, x = 0 .

q x = q„, x = 0, -oo < y < 0. q = 0 , x = 0 , 0 < y < +<=°.

i

( t - t A ) k ±

qoVoa ~ Vn frfc K ) + # Ei ( " F o

y

2 ) J ' * = °

7.1.24 9, p . 264 I n f i n i t e s t r i p heated su r face . t = t i r 0 < x < ">, -oo < y < +00 , = 0 .

q = q , x = 0, ^x ô - a < y < +a. q x = 0, x = 0, y > a, y < - a .

(t - t ^ k ^ F o

%*

ii*_\ E i f. /ujrfl _ /^_*f E i L (L^L\2'

x = 0 ,

See F ig . 7.5



7.1.25 9, p. 264 Heating through a circular surface. "3 = <3n» 0 < r < r , x = 0, T > 0. ^ = 0, r > r o, x = 0, T > 0. t = t., 0 < r < », 0 < x < °°, T = 0 .

(t - t.)k '• ^ - = 2 / P o — ierfc

ierfc [F°*-y/l + ( V M f ' r = 0



i

7.1.26 9, p. 265 Heating through a rectangular surface, t = t i f x >. 0, y >. 0, z >_ 0, T = 0 . % m V -w < y < +w, - i < z < +a, x = o, T > 0. g^ = 0, x = 0 IYI > w» h i > *» T > 0.

Maximum temp

Cinax - tj)k o T _i _i /i\" -—= - sinh (L) + L sinh x (±) , -w + L sinh "" frll» -w < y < +w,

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

Average temp:

2 . 3/2 ] | *-—-^J J I, -w < y < +w, -A < z + A, x = 0.

L = S,/w

Section 7.1. Semi-Infinite Solids—No Intecnal Heating.


7.1.27 9, p. 401 Rectangular wave surface heat flux, t = t. , x •*• », T > 0. q = q0(T)f x = o, T > o. q 0(T) = 0, T > 0. q 0(x) = q 0, n t 0 < T < nT Q + V

n = 0, 1, . . . q„(T) = 0 , n t 0 +

X% < T < (n + 1 ) T Q .

n - No. c i cyc les .

i

At x = 0:

(t - t_)k 2 T ^ ^ = - = y f T 1 V T + - - [ d - T ^ V T - ! ( T l f T)/V5f],

0 < T < T,

(t - O k ^ r - = T T i w + W ^ [ ^ - V ^ - ( T - V 1 / 2 - I ( V w/vr].

T < T < 1 .

q 0lr)- x O

:(T r T> = j f exp (- T X 2 ) U l - T ) exp ( -X 2 ) - exp ( -X 2 + X 2

X 2 [ l - exp (-X2)] V + T J dX

T = T/T 0 , T x = T l A 0


•j


7.1.28 9, p. 419 Semi-infinite cylinder See case 9.1.20 with convection boundary. t = t Q, 0 < r Q< r , z > 0, T = 0. t = t x, 0 < r < r Q, z = 0, t > 0. Convection boundary of h, t Q at r = r Q, z > 0, T > 0.

7.1.29 9, p. 419 Case 7.1.28 with: See case 9.1.21 £ t = t i r r = r Q , z > 0,

T > 0 .

t = t 0 , 0 < r < r 0 , z = 0 , T > 0 .



7.1.30 19, Plate and semi-infi-p. 3-83 nite solid composite

with constant surface heat flux, t = t Q, 0 < x x< <S, 0 < X < <*>, T = 0 .

q = q", x = 0 , T > 0.

I

vvy. 2 g" VcQr " 2 m i^== [exp (m 2 P o 2 ) erfc (m V F O ^ -lj

+ — , 0 < x. < 6

Heat flux at interface: q /q" = 1 - exp fm Fo 2 J erfc (m V?o~) , x = 6 ,

m = ( P 2 V P 1 °1> SeL Pig. 7.9

7.1.31 74, p. 211

Semi-infinite rod with convection boundary. t = t A, x > 0, T =s 0.

t - t

t = t f, x 0, T > 0.

T t „h

tprt - I e xP ^"d x ) e r f c ( iTBJ ' B i d F o d)

+ exp ( Bid X) erfc ( T ^ ^ ^ T ^ I )

. cross section area .. d perimeter , X = x/d

Heat flux at x = 0:

k( t ! ° - t . ) = ?V?oT e x p ( " B i d F°d> + V i r d e r f c ( V 5 I d ^ a '



7.1.32 74, Semi-infinite plate p. 465 with fixed surface

temperatures.

t = t., 0 < x < SL,

i

0 < y < ° ° , T - 0 . t = t., x = 0, i.,

0 < y < =>, T > 0. t = t Q, 0 < x < K y = 0, X > 0.

1

Solution

H±i_ . x + 1 Y - t ^ sin (XX) exp (-X* To) t - t. it iS n n \ n / o l n-l

00

1 y S=lil s i n (x x) [exp (XV) erfc * ATi n n L n

x /FO* + X n VFO") + exp (-XnY) erfc M?o* - X R VFoj

(~Xn P o ) e r f c ( F°y)] • - 2 exp

X = mt. n



7.1.33 74, p. 466

I

Semi-infinite cylinder with fixed surface temperatures. t = t i r 0 < r < r„ , 0 < y < » , T = C.

t = t ± t 0 < r < cQ, y = 0, x > 0. t " V r = V ° < y

n=l

+ exp {X Z) erfc [Fo + X V F O ] n \ z n /

+ exp (-X Z) erfc / F O * - XnVFo\"j .

< <*>, T > 0 . W " ° r. 7.1.34 74,

p . 466 Case 7.1.33 with: convection boundary h, t at r = r , and t = t Q , 0 < r < r Q , Z = 0 , T > 0 .

2 Bi J 0(X nR) exp (-XflZ)

~ J„ (X ) (si2 +\2) n=l 0 1 n \ n /

Bi J . (X R) r - B i J . (A R) r / * \ + Z T ,. t / B . 2 , , 2 \ H? (V> e r f c ( X n + F o z

n=l W ( B l + X n ) L

- exp (-XnZ) erfc ^ VFO" - Fo*)j .

W " B i

Section 7.1. Semi-infinite Solids—No Internal Heating.

Case No. References Descciption

Section 7.1. Semi-infinite Solids—No Internal Heating.

Case No. References Description ' Solution

10

Section 7.2. Semi-infinite Solids—With Internal Heating.


7.2.1 9, p. 79 Steady surface temperature and initial temperature distribution, t = T, + bx, x 0, T ••- 0.

t = t , x = 0, t > 0.

t - t, 0 * - = (1 + Po + * * 2 / * \ 2 * * / *2\ Po Fo *) erf {Fax) + j= Po Po x exp f- F 0 j £ J

^ bx _ * „ *2 T PO FO . T- x

Po q " ' OT kT,

7.2.2 9, p. 79 Case 7.2.1—variable heating, t = T., + bx, x 21 0, T = 0 .

t " t , x = 0, x > 0.

(t - t o )kB*

r x PO \ P O / * -8x

erf (Fo ) + 1 - e R

+ \ e X - e x erfc (VX - Fo*) - \ e X + 0 X erfc (Vx + Fo*) .

Po = a" " * q 0

k?xe> , X = aB"T



7.2.3 9, p. 79 Case 7.2.1 with band heating, t = T, + bx, x >. 0, T = 0.

t " t, —- = Po* jl - 4 i 2 erfc ( F ° * ) + 2 i 2 e r f c |Po* (1 + X)l

2 i 2 e r f c [FO*(1 - X)lj + ~ + erf (FO*), 0 < X < 1

= 2 Po* ( i 2 erfc [FO*(X - 1)1 + i 2 er fc [Vo*(X + 1)1

2 i 2 « f c K)) + i 7 + e r f K )

t - 1

| , X > 1 .

* q i " OT Po _ ^0


Case Nu. References Description S o l u t i o n

7 . 2 . 4 9 , p . 80 I n s u l a t e d boundary and band h e a t i n g . t = t . , X > 0 , T = 0 .

a = 0 , x = 0 , T > 0 .

q " ' = q A " r 0 < x < d,

T > 0 .

q ' " = 0 , x > d.

( t - t , ) k , - = 1 - 2 i e r f c

q^"aT

( t - t . ) k

[FO*{1 - X)] 2 i ' e r f c [FO*(1 + X)l ,

0 < X < 1 .

- L ^ = 2 i 2 e r f c [ F O ^ X - 1 ) ] - i 2 e r f c [pdjjx + 1)] ) .

X > 1

-J I to

7 . 2 . 5 9 , p . 80 I n s u l a t e d boundary and v a r i a b l e h e a t i n g , t = t Q + b x , x > 0 ,

T = 0 .

o = 0 , x = 0 , T > 0.

q ' " = q ( J " e * & X , X > 0 , T > 0.

(t - tQW - ^ - exp (-SX) + Po_.

* * Po Po

\ X X

+ 2X\ e r f c K) + | exp (X 2 - 3x) e r f c ( x - F o * ) + | exp (X 2 + 0x) e r f c ( x + F o * )

Po q^'OT

kbx , X = B Vor



7.2.6 9, p . 323 Plate and semi-infi ni te solid composite. ^ 1 V 1 . . V Q n ,2 , / (2n + 1) + X\

t - t f t . -b < x < - f

4 0 n=0 L

n .2 , f(2n + 1) - x | X .2 ^ /2n + X\

- e x e r f c L 2-vfe J + i + x x e r f c \2rm')

O'Yvfe * ")]- -

0 T = 0 .

t = t , x = - b , T > 0.

J 0 ' -b < x < 0, T > 0

q " 1 = 0 , x < 0 , T > 0 . X .2 .

x e r fc 1 + X -1 < X < 0

i'o" E? ( t 2 l V ' h , _ __4 V 0 n f ,2._^_ / 2 n + J x \

\ 2Vf5 / "0*"l 4 V a n f , 2 .

n=0 l

i 2 e r f 0 [ ^ » «»] - 2 i 2 e r f c [ ( * > , + # + " ] } , x > 0.

, _ • X - 1 V x

6 = V^7o 2, X = k 2«A x, 3 = YTT' P o = JT"' x * b



7.2.7 9, p. 307 Time dependent heating and convection boundary. t = t . , x > 0, T = 0. q ' " = q ^ ' T S / 2 , s = - 1 , 0, 1, . . .

(t - t 0 ) k , , , l+(s/2) D. + (s/2)] T (s/2)+l

r p . + (s/2)] qA/'OTC (-Bix)

s+2

x exp (Bi + Bi Fo \ erfc (Fo + Bi -v/Fo \

« 7.2.8 9, p . 308 Constant temperature surface and band heating, t = t Q , x > 0, T = 0. t = tQ, x = 0, T > 0. q'" =0, 0 <x < x , X > Xj.

q... = q<" , X < X < X .

Surface heat flux:

qA" at = 2 ierfc [Fo* \ - ierfc |Po* ] , x = 0



7.2.9 54 19. p. 384

Exponential heating. t = t , x > 0 , t = 0 .

2 qj "V5t ierfc K) exp (-2 MFo ) [l - exp (M)2]

l to

" e X p4M M > [exP ("2 M F°x) e r f c (Fox - M ) - exp (2 M Fo*\ erfc (FO + JM 1 .

Surface temp:

* ( y t p J = 0 5642 - 1 - exp (M 2) erfc (M) =

M = ji Vaf See Fig. 7.10

7.2.10 74, p. 383 Planar heat pulse and convection boundary. Instantaneous heat pulse occurs at x = 0, x = x, with strength

• V m

t = t f , x > 0, T = 0.

tf,h-

(t - t f) k x x

Qa = 2VnFo (X - 1) 4 Fo

- Bi exp (X - 1) + B i 2 Fo erfc X2+p' 0 ^ Bi Vfo I



i 00

Section 8.1. Solids Bounded by Plane S u r f a c e s — N o Internal Heating.

Case N o . References Description Solution

oo l

8.1.1 9, p. 94 Infinite plate with steady surface temperature. t = t(x), 0 < x < %,

T = 0. t = t Q, x = 0, H, X >, 0.

t - t Q = 2 I*. sin (rmX) exp I-n Ti Pol n=l

w1

-»n

A = / f(X) sin (mrx) dX

See case 8.1.25 for convection cooling at x = 8,.

8.1.2 9, p. 96 Infinite plate with steady surface temperature and linear initial temperature. t = t _ + bx, 0 < x < J l f

T = 0. t = t Q, x = 0, l , T >_ 0.

t - t, • 0 . 2 V (-D 8, - t ~ ir Z n

n-1 2 2 sin (mrx) exp {-n ir Po)

n=l

K M

Section B . l . Sol ids Bounded by Plane Surfaces—No Internal Heating.


09 l to

t - t„ 8 .1 .3 9, p . 97 In f in i t e p late with steady surface temper- *"0 8 V 1 ature and bi l inear t_ - t n " „2 Za / 0 _ A

i n i t i a l temperature. 0 w " n~0 < 2 n + 1 }

(2n f 1"] exp 12U&L. u 2

F o " 4

t = ( t c - tQ> («- - | x | ) / a + t 0 >, -s. < x < +£, T = 0. t = t 0 , x = ±1, T > 0.

IsA"/^

W W

= 1 n=0

| > n 4- 2) - l x | l \ L 2VfS J)

i e r f c

See Fig. 8.1



8.1.4 9, p. 98 Infinite plate with steady surface temperature and bi-parabolic initial temperature. t = (te- t Q) (a2- x 2 ) /

I2 + t , -I < x < +«., c T = 0. t T > 0 (see case 8.1.3),

t ~ fc0 32 V-t " tn " 2 Z ±lll cos

n=0 See Fig. 8.2

(2n + I)" (2n + 1)

2 TTX exp [.ia-iid^^j

t 0, x = ±a,

T 8.1.5 9, p. 100 Infinite plate with steady surface temperature and cosine initial temperature

Vo (¥) -(-4-) t - t„ <v to ) TOS

+ t , -2. < x < +«. c

T " 0. t = t Q, x = ±£, T >_ 0.

(ft)



8.1.6 9, p. 97 Infinite plate with 74, p. 113 steady temperature and

constant initial temperature. t t., -i. < x < I,

T t T > 0.

V x = ±1,

I W A ' V A '

e Irt-V^v/*

I

CTJ: - 2 E "4^ C° S <V> 6XP fXn F 0) 1 ° n=0 n

Mean temp:

1 n=0 n

^ . 1 - 2 ^ " . P O < 0 . 1

l 0 Heat loss:

2£pc J 2 V S l n!l Xn L 1 - *** (~Xn F°)] (t.- t.) ~ Z X TX^ + sin (X,,) cos (AJ" -. w„, Z X [X ..... ... . i 0 _ t n I n n

X = (2n + l)u/2, See Fig. 8.3


Case Do. References Description Solution

8.1.7.1 74, Infinite plate with p. 220 convection boundary

(general). t = f (x), -«, < x < +1, T = 0.

t - t f = 2 n=l

X cos (X x) exp (-X Fo. ) C _n n ** V n l) j rf ( x ) _ fcj c o s ^ ^

n JO X + sin (X ) cos (X

t.,h-

CO J.

cot (Xn) = X ^

X given in Table 14.1 -h,t,.

8.1.7.2 3, p. 294 Infinite plate with 74, p. 223 convection boundary—

constant initial temp. t = t . , -«, < x < +fl,

i T = 0. (see case 8 .1 .7 .1)

t - t „ ~ s in (X ) cos (XX) exp (-X^Fo) t . - t . Z / X + s i n (X ) cos (X )

n=l cot (X ) = X /T8i n n

fc- fcf t . - t f

t - t

1 - cos (V§i X) exp (-Bi Fo) , Bi < 0 .1

TT = e r f / * " X- ) - exp m i d - X) + Bi 2 Fo] e r f c f1 " X + Bi VFO) V C £ \ 2 V F 3 / L J \ 2 VFO" /

fc Z 1 + X \ - exp feid + X) + Bi 2 Fo] e r f c f 1 + X + Bi V F O ) , \ 2 V F O / L J \2 VFo '

+ erf

See F igs . 8 .4a, b , c , d and e and 9 . Id .

Section 8.1. Solicis Bounded by Plane Surfaces—No Internal Heating.


8.1.8 1, p. 268 Infinite plate with time varying surface temperature, t = t.f -I < x < H, T = 0. t = t. + bT, X = + «,, T > 0.

os I en t(r)-

s\/ss/\ -t(T)

(t - tj) a = Fo +

n=0 x cos |{2n + 1) | X1

See Table 8.1, Fig. 8.16 and Fig. 8.25.

8.1.9 9, p. 104 Infinite plate with time varying surface temperature (general), t = t., -«. < x < %, x = 0. t = t ± + f(T), X = ± H, T > 0.

. = l t J^ <*> * VW" wo *„[ ( 2 n V ' V F o ] n=0

at'

Section 8.1. Solids Bounded by Plane Surfaces—Mo Intecnal Heating.

Case No. Sefecences Description Solution

8.1.10 9, p. 104 Infinite plate with J.I1A.4.AIX L 6 J / 4 0 L C H 4 U I "~

exponentially varying i _ -OT cos (X^3) 4_ ^ (-1)" surface temperature. T - t . e ,-ssr, ir ^ ,., . , , fi #•> ^ , , 2 „ 2 . . „ , 2 l t - t 1 # -£ < x < * f

x O O B ( V 5 3 ) n=0 ( 2 n + 1 J L 1 " ( 2 n + X) * / 4 p d J T = 0. t = t . + T (1 - e " b T ) , x = ±1, x > 0.

T (2n + 1) 2TT 2 „ 1 l"(2n + 1)TT v"j exp I- J j - ^ PoJ cos j - 1 j - 1 — Xj ,

b ? (2n + l ) 2 ir 2a/4A 2

8.1 .11 9, p. 105 In f in i t e p la te with ment ia l ly varyi lace temperature t . , -2, < x < H,

In f in i t e p la te with «° exponentially varying i DT cosh (xVPd) £ V (-1)" surface temperature. T - t . = e ^ ^ " 7T ^ ( 2 n + X ) N + 4 P d / 7 r 2 ( 2 n + rf]

T (2n + l ) 2 n 2 _ 1 f(2n + 1)TI „"1 x exp - J r-* Fo cos -1 —<— X

, ... . - , 2 _ 2 T = 0 .

t = t . + T e b T ,

x = ±1, T > 0.



8.1.12 1, p. 299 Infinite plate with periodic surface temperature.

2

t =

cos

(t - t ) 1 mx mn

±1

t(T)—H

M

- t (T)

"rax mn S— = X, cos [(2TTT/T 0)+ *]

t = max surface temp mx t = mean surface temp mn t„ = cycle time

£ = f (XCT) + fjtXCT)

L

f i«» + f2<°>

1/2

= tan

tâ)

-1

See Table 8.2

tx{a) f 2(xa) - f 2 (a) f^xa)

fâ ) f^Xa) + f 2 (a) f 2(xa)

cos (a) cosh (a), f (a) = sin (a) sinh (a)

a = VriPd = fcVn/aT,,

Heat stored during half cycle: 2llpc (t

mx x) F(a)/a

P(CJ) df* (a) + df* (cr) fj (a) + t\ (a)

1/2 See Fig. 8.5

df (a) = cos (a) sinh (a) df2(o)

sin (a) cosh (a) sin (0) cosh (0) + cos (a) sinh (a)



8.1.13 9, p. 105 Infinite plate with t - t.

00 I IP

tM-

periodic surface temperature. t = t t, -a < x < a, T = 0. t = t . + (t - t )

i m i sin (urc + 9), x = ±1, T > 0. t = maximum surface m temperature.

m 1 = A s i n (our + 9 + <t>)

K-i-

<^^* ^ V « M

•t(r)

-H

+ 4 ' I ( - l ) n ( 2 n + 1) [4 Pd cos (9) - (2n + l ) 2 n 2 ain (6)]

n=0 16 Pd 2 + (2n + l ) 4 II4

x exp [- (2n + l ) 2 IT 2 Po/4] cos [ ( 2 n \ l ) l t x ]

1/2 cosh (X -VTPd + cos (X V2 Pd) cosh (V2 Pd) + cos ("V 2 Pd)

= arg

, See Fig. 8.6a

See Fig. 8.6b .

Pd = Hio/ti •

Section 8.1. SolidB Bounded by Plane Surfaces—No Internal Heating.


09 i

8.1.14 9, p. 105 Infinite plate with periodic surface temperature on one side, constant temperature on the other, t = t i # 0 < x < Jl,

t = t. + (t - t.) I ' m i

sin (tur+ 9), x = I, T > 0. t = t., x = 0, T > 0. t = maximum surface m temperature.

-t(r)

t - t . t m - t x

— = A s in (wr + S + if)

on

A n „ V n ( - l ) n [ n V sin (9) - cos (9) PdJ ^ - 4 4 2.4 2 n=l n V + urV/aT

x exp (-n IT Fo ) s in (n ir X)

A = cosh (xV2~Pd") - cos (xVTTd)

<|> = arg

cosh (VFi'd) - cos (V2~Pd)

sinhfxVH,^,)

1/2

sinh f ^ SO , Pd = K. w/a

K-'H



8.1.15 9, p. 108 Infinite plate with » X (T.-T*) X (T-x") steady periodic fc " *Q = 2. Y (-1)" s i n - n - - n

surface temperature t, - t„ IT £-•, n (rntX) e ^ ~ surface temperature t , - t n ~ A n *-. n B i " v " , l i W X T . . . 1 0 n=l . n on one s ide , constant 1 - e temperature on the other. T ' = T - nff, m » 1

t - t 0 . K - 0. T > 0. - X n ( T - V T - M X n ( T - T " ) t - v * • l' tr^-f- = f Z • f c J j r - s i n (nirx) * ri aff < T < *r + T x , X ° n = 1 1 - e n

T • • = T - (mT + T . ) , m » 1 , <f t » t Q , X = %, iflj L V U i "1

T B period of cyc le . n '



8.1.16 9, p. 109 Infinite plate with steady surface temperature on one side, insulated on the other. q = 0, x = 0, T > 0. t = t l f x = a, mT < T < mT + T , m = 0, 1, 2, . . . . t = t_, x = I, mT + T x < x < (m + 1)T. T = per iod of cyc le .

0 4 V ( -1 )" m a f(2n + 1)TT "I i - r ^ = 7 n Z 2 n + i c ° s L 2 XJ

1 -

A ( T , - T ' ) A ( T - T 1 ) n ( 1 ' n x

e - e

1 - e X T n

, x ' = t - mT, m » 1

00 fc " fc0 4 V (-1)" f"(2n + l)ir J

X (T-T - T " ) X ( T - T " ) n 1 n e - e

1 - e X T n

, T " = T - (mT + T^, m » 1 ,

X = a (2n + l ) 2 i r 2 / 4 £ 2

n



8.1.17 4, p. 82 Infinite plate with 48 steady surface heat

flux and convection boundaries, t = t Q, 6 < x < %, x = 0.

GO I

V So; * = l' T > °" t f = t 0 , T > o.

t f , h -

(t - t„)h „. /=-0' Bl /Fo *0 2 \ TT

r-„-2 „ - Bl Po exp (

See Fig. 8.18

For Bi •*• » ;

< f c - V k V F 5 (

exp

Bi + Bi + 4 Bi Fo x

See case 8.1.20 for special solutions.

hiH

Section 8 .1 . Sol ids Bounded by Plane Surfaces—No Internal Heating.


8.1.18 14 47

In f in i t e p late with steady surface heat

74, p . 174 f lux and variable thermal conductivity» t = tQ, ~Sb < x < +«,, T = 0. k = k ( t ) . a i s constant. q x = V X = "" "A-

2 (X - U * - (X - 1) + j + Fo - ^" Z , \ cos [rnr (x - 1)] TT n=l n

x exp (-rnr Fo) J - / 1

k(T)dT

T = (t - t 0 ) / t ( ) f kQ= k ( t 0 )

V k(t)

For k(t) = kQ [ l + 3 (t - t Q ) ]

f f k ( I ) dT = fc - fc0 , B t 0 ( f c - M ' k 0 j n t„ + 2 \ t„ /

k(T) dT • '0 "0

For 3 = 0 , see Fig. 8.7.

For 3 = 0 and q Q terminated at T = D, see Fig. 8.17.



8.1.18.1 3, p. 279 Infinite plate with steady surface heat

OB i

1*4— —i-*|

(t - t.)k flux. t = t . , -1

1 T = 0 .

< x < +H, V

V V x

J

= ±%, T > 0 .

1

x = n

n=i A

rnt



CO i

8.1.19 3, p. 350 Infinite composite plates with convection boundary and infinite thermal conductivity. t = t2,0 < x < & 2, x > 0. t = t 1 ? l 2 < x < %2 + i.x,

X > 0. t = t Q, 0 < x < «,x+ l 2 ,

T = 0. q = 0, x = 0,

h„ is contact coefficient between 1 and 2.

fcl " fcf X l e - X 2 e

fc0 - fcf " X l " X 2

fco - fcf " fco ~ H 3 \ xi " x2 X l + X 2 = b l + b 2 + b 3

X l X 2 = bl b3

bl = h2/P2 C2 V b2 - V P 1 Cl V b3 = V P 1 Cl \

% 2

k-> • h , , t f

-*,-K*H



8.1.20 3, p. 476 Infinite plate with steady surface heat flux and convection boundary, t = t-, 0 < x < %, x = 0. qx = g0 ' X = °' T > °*

-h,t,

00 I

K-H

'^V'l-WPo ^

(t - tf)k \ VSPO/ . ° < T i T

0

* 0 * i<l-tf +[fe +j(l-*]

, r [1 +(Bl/3)] Bi FO [1 - (T0/T)]li x {1 - exp r f \ L 1 + 2(Bi/3)+ 2(Bi /15) JJ

T. = penetration time to X = 1

= l2/5a See case 8.1.17 for general solution.

T > T„

8.1.21 3, p. 349 Infinite plate with unsteady surface heat flux; convection boundary and infinite thermal conductivity, t = t f, 0 < x < %, T = q = q.cos (urr), x = 0.

***•**"+ ». — * x

k-»«> % ' k-»«> fc

*•

-h,t.

(t - t f)h \f~2 2 cos (arr - b) 2 2

m + u exp (-nrr)

m = h/pcS,, b = tan""1" (ui/m)

hH



8.1.22 2, p. 248 Semi-infinite plate, quarter-infinite plate, infinite rectangular bar, semi- infinite rectangular bar, rectangular parallelepiped.

Dimensionless temperatures equal the 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 8.1.7). See Figs. 9.4a and 9.4b.

OS i

8.1.23 9, p . 113 In f in i t e plate with unsteady surface heat f lux , t = t . , 0 < x < I, T = 0. q^= 0 , x = 0, T > 0.

m/2 „ \ = V ' x = *" T > 0 , m «= - 1 , 0 , 1 .

m/2

DO

n , /-> -,. vs— V -m+1 c l"(2n + 1) - ~A\ T (m/2 + 1) VPo ? i erfc J • n=0 L 2VFO" J

f(;2n + 1) + X~| L 2VFO J

. .m+1 -+ l erfc

V-i-A

Section 8 . 1 . Sol ids Bounded by Plane Surfaces—No Internal Heating.


8.1.24 9, p. 114 Case 8.1.23 with t = t , r x = 0, T > 0. (t - t . )k

q / T '

i- = 2 m + l r ( m / 2 + X) y ^ ^ i i m+1 V 2 riÔ

erfc [ < 2 n ^ > - X ] - i ^ e r f c [ < t o * ^ + X ]

8 .1 .25 .1 9 r p. 120 Inf in i te plate with steady surface temperature, convec-

i t ion boundary and to variable i n i t i a l

temperature. t = f (x) , 0 < x < a, T = 0. t = t Q , x = 0, T < 0.

-h, t n

co / 2 2 \ ^ . (Bi + X ) s i n a X)

(t - V - 2 > exp U* ro)\ g - S ^ ± -n=l V n / Bi + X + Bi

n r-1

x / [ t Q - f (X)] s in (XR X) dX -'0

« Bi = hVk, X cot (X ) + Bi = 0 n n

See case 8 .1 .1 for t = t , x = I, T > 0

j—«-H



8.1.25.2 74, p. 239 Infinite plate with _ steady surface temp 0 and convection boundaries. *£-*«>

= 2 ( Bi 2 + Xp [l - cos (Xn)-| _}

n= + A I i - cos |AJ / 9 \

t = t Q, 0 < x < A, T = 0. t = t Q, x = 0, T > 0.

X cot (\ \ + Bi = 0

See Fig. 8.24.

f*-*-00 I o

-h.t.



8.1.26 74, p. 236 Infinite plate with parabolic initial temperature and convection boundary, t - tc- (tc- ts)(x/A)2, o < x < a, T = o. t = t , x = 0, T = 0 . c t = t , x = ±SL, T = 0.

s

* - f c f . , t B ~ ^ S /_L_ _i\ sin (X ) cos (X X) X n + sin (Xn) cos (Xn) exp « » * )

X tan (XJ = Bi»

j- i-J -i- l 00

t f,h- •h,t.



8.1.27 9, p. 125 Infinite plate with steady surface temp and convection boundary, t = t Q, 0 < x < I, x t = t , x = 0, T > 0.

-h.t,

t f - t o Bi X 1 + Bi - 2 Bi

Y sin (XnX) exp (-X2 Foj

n=l (Bi + Bi 2 + X 2 J sin (X ) \ n / n

X cot n w + Bi = 0

K-iH 8.1.28 9, p. 126 Infinite plate with a

steady surface temp. Convection boundary and linear initial temperature, t = (t^- tQ)x/S, + tfl, 0 < X < Sir T = 0. t = t Q f X = 0, T > 0.

-h,t.

= X + [Bi (t f - t n)/(t 1 - t p) - Bi - 1

Bi + 1

2 {Bi [(tf - t 0)/( t l - t0)]

- Bi , ^7 sin (X X) exp f-X2 Po) n=l fBi" + Bi + X M sin (Xn)

X cot (X ) + Bi = 0 n n

h-'H



8.1.29.1 9, p. 126 Infinite plate with two convection temperatures. t = t Q, -fc < x < +i, T = 0.

(< ") t " t 0 B i X „. V s in (gnX) exp \-&n Po; t f - t Q " 2 (Bi + 1, " B l ^ ^ . 2 + B . + 0 2 J s . n ( ^

tn.h"

I to [••i-^l*-*—|

+ 2 " B i

V cos (XnX) exp ( -X 2 Fo) L-i /_ .2 . _ . . , 2 \

-h,t« n=l ( B i 2 + Bi + X 2 J cos (X )

X tan (X ) = Bi , 0 cot (8 ) + Bi = 0 n n n n

8 .1 .29.2 74, p. 239 In f in i t e p late with two convection c o e f f i c i e n t s , t = t Q , 0 < x < +1, T = 0.

t - t„ 1 + B i £ l X

t - - t n 1 + B i , , + ( B i . . / BI , , ) S.1 '•%\/ aJ-i.v

t 0 . h r

00

k n Bi

cos (X XJ+ , n X U s i n (X X) n exp (< F°a) •

A = B l l l / B i U I -h 2 .t,

s in (X ) cos (X ) + X n n n 2 s i n (X )

+ - j - s in (Xn)

h-«-H B i u = h l V k , B i A 2 = h 2 V k

,. ,. , / xn B i n \ L +

B i n V 1



8.1.30 9, p. 127 Infinite plate with convection boundaries and time dependent fluid temperature, t = t , -S, < x < +S,, T = 0. t f= <t>(T).

t f.h-

| * - { - *

-h , t f

L-«-J p-*-»i

. , , n- ci f *• c o s <V> e x p K F o) t - t. = 2 Bi ~r £ —. - rr 1 r n=l f Bi + X„+Bi ) cos (Xn)

xf exp ( a X ^ x ' / A 2 ) * ( T ' ) dT'

X tan (X ) = Bi . n n

Section 8.1. Solids Bounded by Plane surfaces—No Internal Heating.


8.1.30.1 74, p. 316 Case 8.1.30 with t - t i cos (VPH X) exp (-pa Fo) f f m £ m i e fcfm " fci " * cos CVPI) - J j V 5 3 sin (V5d)

00 I

in

A = n

Mean Temp:

Y An cos (XnX) exp (-X2 Foj

1=1 1 - X2/Pd n

(-1)"* 1 2 Bi ( B i 2

+ X 2 ) 1 ' 2

Xn ( B i 2

+ Bi + X 2 )

fcni~ fci exp (-Pd Fo) fci Vpa (cot (VfoT) - g j VPS"!

V exp (-X 2 Fo) - 2 Bi2 2 ^ ^ ~

n=l X2 (Bi 2 + Bi + X 2 ) [l -(X 2/Pd)]



8 .1 .31 9, p . 127 Case 8.1.30 with

" 1*1. T > V

fc- fci . , B . f cos g n X) exp ( - A ; F O ) , O < T < T 0

c " - n=l fei" + X^ + BiJ cos (Xn) 0 i

t - t. Y

tT^t * x " 2 B 1 _4, & T + (1 - T) exp

n=l n M %)]

B i 2 + X 2 + Bi J cos (Xn)

x cos (X X) exp (-X Poj

CO I to an

T = ( t n - t . ) / ( t - t ) , Po. = ( H / r , X tan (X ) = Bi o i l i u u n n


Ref-Case Ho. erences Description Solution

8.1.32 9, Case 8.1.30 with p. 127 t.= (tn- t.); sin 74 r u i ' ' (HIT + £) + t.ri p. 325 1

T > 0.

00 I ro

t - t. Bi M — = —~ sin (COT + e + e Q - &x) fco-fci

+ 2 Bi

iB„

2 Po *2 cos (E) - sin (E) cos (X X) exp (-X Foj

nfx (l + « Po*4/xJ) (*n + B i ' + B i ) c o s < V

M e ° = cosh (X Fo ) cos (X Fo ) + i sinh (X Po ) (X Fo ) sin (X Fo )

^1 A * it it it it . . it M e = (Fo ) sinh (Fo ) cos (Fo ) - Fo cosh (Fo ) s in (Fo ) + Bi cosh (Fo )

x r * * * * * , * x cos (Po ) + i I Fo sinh (Fo ) cos (Fo ) + Fo cosh (Fo ) s in (Fo )

+ Bi sinhs (L) s i n (L)j

Fo = S, V372a , X tan (X ) = Bi n n

Section 8.1. Solids Bounded by Plane Surfaces—No internal Heating.


8.1.33 9, p. 127 Case 8.1.30 with 89 t = t.+ bt, T > 0. <fc - V a Bi X 2 - Bi - 2

— = F o + Til WT

+ 2 Bi cos (XnX) exp (-X2 Fo j n^l X 2 (ai2 + X 2 + Bi) cos (Xn)

X tan (X ) = Bi, See Fig. 8.26. n n (See Fig. 8.1.25 for Bi = ».)

perfect contact with an inf ini te plate of inf ini te conductivity. t - t . , o < x < a 1 + i 2 , t = o. q = 0 , x = 0 , T > 0. ^x - q x - q Q , X = t 1 + l 2 , T > 0.

(t - t , )k

i V M L .X 2 , 3 + M "I

~S^~~ = 1 + M LF°1 + 2 " 6(1 + M)J

y cos ( y Q exp (-XJ; F o J

i£ l X2 (\2 + M2 + u) cos (A)

2 -"o

X cot (X ) = -M n n

p,c JL ,

I—/,—(•«,*(



8.1.35 9, p. 129 Case 8.1.34 with g = 0, x = S, + J, r T > 0. t = t2, a 1< K < %2, x = o. t = t., 0 < x < SL , T = 0.

t - 1.1 . v c o s ft x ) e x P I -* 2 p o - , i i = i + 2M / \ n 1/

fc2 - V * + H n=l (V + M 2 + M ) cos (A ) \ n / n X n cot (XR) = -M

8.1.36 9, p. 128 Case 8.1.34 with t = t., x = 0, T > 0. (t - tt)k

Vl X - 2M Y sin (XrX) exp (-Xn FOjJ

n=l X (x 2 + M 2 + M^ cos (X ) n \ n / n

CD i

X tan |X )= M n \ n/

8.1.37 9, p. 128 Case 8.1.34 with t = t., x = 0, T > 0. i t = t 2, Jl1< x < SL2, T = 0. t = t. , 0 < x < JL, T = 0. q x = 0, x = J^ + %2, T > 0.

CD <fc - V v t t , - t j } = ™ A

sin (X X) exp «-0 (t - t ± ) ( t 2 " t i ) n=l (\2 + M 2 + M ) sin (X ) \ n / n

X tan i\ \ = M n \ n/

8.1.38 9, p. 129 Case 8.1.34 with t = t Q, x = 0, T > 0. q = 0 , x = « . 1 + i l 2 , T > 0 .

(t - t t)

(fc0 " V = 1 - 2 | ^ 2 ^ M 2 ) s i n ( x n x ) e x p ( - X ; F 0 l ) n=l X (x 2 + M 2 + M )

n V n / X n tan (Xn) = M



8.1.39 9, p. 129 Case 8.1.34 with (t - t.)k(l + M) V °' X = ll* V T > °- ZJT = M Fo + 1 - X + ? (1 - X) 6 ( 1 + M )

V s qo' x = °' T > °" ! ( 1 + M ) f ( « 2 + X n ) cos (XnX) exp (-X2

n F o J 4 1 X2

n (X2

n + M 2

+ M ) n=l

A cot (A ) + M = 0 n n

m o

8.1.40 9, p. 129 Case 8 1.34 with _ - / 2 \ c Q s ft / ^2 \ convection boundary 2 _ \ V I n / n \ n 1 / h . t f at x = i x + £ 2 . ( t t - t 2 ) = ^ r / B . _ x 2 \ 2 + A 2 + X 2 / M + B . -] c o s }

t = t 2 , Ax< x < 4 2 , L\ 1 n ) n n l j n

T = 0. X n t a n ^ n 1 = B i l " X n / M ' B i = ^ i / * !

(See Ref. 82 for solution of material 1 of low conductivity and material 2 of high conductivity)



8.1.41 9 r p. 129 Case 8.1.40 with j ^ ^2 "1„

x

.1 .40 with °° ,2 ,, „. / .2 „ \ f ( T ) , x = 0, ( t - t . ) _ 2 a _ y A n cos (XnX) exp (-X n Fc> )

T > 0. kA3 n=l ["(si, - X 2 / M ) 2 + (1 + 1/M)' A 2 + Bi

x f f<T') exp (&n'/l2) * : ' . Jo

cos (X ) n

X tan (X ) = Bi , - X2/M . n n I n

os 8 .1 .42 9, p. 129 Case 8.1 .34 with a (t - t . ) (M + l)k , 3 Bi + 6 + M Bi i contact res istance h . *—-£ F ^ + j (1 - X) - 6 Bi (i + M) °

VI 1 C between the two ° p l a t e s .

00 2 — X_ - M Bi .

+ 2 Bi c (1 + M) J K ~ M B l c / 2 v n 4l * n « » <Xn) n n ^ V n /

( x J - M B i J t a n (Xn) = B i c X n .

P n = X n + Xn ( B i c + B l c " 2 M B i c ) + M B i c ( 1 + M> Xn •

B i c " h c V k l '



8 . 1 . 4 3 . 1 9 , p . 324 Composite p l a t e s of Region 1 , - a i < x < 0: d i f f e r e n t m a t e r i a l s , t = t n , -& n< x < «,,,

00 I Ul M

X = 0 .

t = v

x =» -A.,, X < 0.

t - t Q , x = l 2 , X > 0.

< f c l - V ( L + K ) n^l n X

cos (A X) s i n (BLA ) - a s i n (A X) cos (BLA ) n n n n X Hi + KS2L) s i n (A ) s i n (BLA ) - B (K + L) cos (A ) cos {BLA )1 n •- n n n n J

Region 2 , 0 < x 2 '•

CO

<fc - V L - X _ V , 2 „

s i n (BLA - SA X) n n A [(1 + KB2D s i n (An) s i n (BLAJ - B (K + L) cos (An) cos (BLAn)]

co t (A ) + a co t (BLA ) = 0 n n

B =ycT[7cy K = k 2 A l f F o 1 = a.£/l\, L = l 2 / l v X = x /4 j



8.1.43.2 74, p. 441 Composite plates of different materials with insulated surface and convection boundary, t = t i f 0 < x < SLX +

00

-h.t,

t - t t ~ t ""l

i - r f - - 1 - 2 2 n r " a ( X n X / V S ) « * K ^ V * ) ' ° r i n=l n * ' < X < L

d ~ T - 1 - 2 5 XT |C O S [ \ <X " L)] C O S (V*> f l n=l n ' L J

- K sin |~X (X - L)l s in (X LA'S) ) exp £-X*L2Fo /A") , L < X < 1 .

x( l + KL/VS) cos (Xj] cos ( \ y V K \ + [ ( l + KLA5 + i g r - ^ ) cos (X )

" Xn & P ) (l + m) Sin <V] K sin W*> • LX X —? • (1 + L) tan (X L/VA) = 1 - - T (1 + L) tan (X ) Bi n Bi n

K tan (X ) tan (X L/^) . n n

Bi = hSLx/K2, L = l^/i.2, P o l = O j T / ^ , A = <Xx/0. , P l C l k l 2 ' K - P 2 c 2 k 2



8 .1 .44 .1 9, p. 416 Rectangular parallelpiped. t = t . , 0 < x < wt

0 0 CD

CD I

t ~ t i _ i6 y t - t ~* 2 " — 0 i n m=0 n=

s i n (imrY) s i n (mrz) s inh | ( 1 - X) \/X

0 < y < b,. 0 < 2 < d. t = 0. t = t 0 . x = C ), 0 < y 0.

t = t i # a l l

othe

i

:r f aces , T > 0 othe

i f /

t b

JJ /

0 0 CO 0 0

m n s inh (X ) ra,n,o

a s i n (mitY) s i n (mtZ) s i n (fcirX) exp (-X_ „ Fo ) m,n tx, w

m=0 liFO ISO m n X. 'm,n,£

X „ = J l V + (mirw/b2) + (mrw/d) ?' m,n,)t

m = 2m + 1, n = 2n + 1, X = x/w, X = y / b , Z = z/d



8 .1 .44 .2 74, p. 287 Rectangular p a r a l l e l e piped with convection boundary, t = t ± , -lx < x < +SLX,

- * 2 < y < + a 2 , -& 3 < z < +1 3 , T = o.

V ' fco

0 0 CD

= 1 2 2 2 n=l m=l s=l

A n l Am2 A s 3 ° ° s <*nlX ) O O S <\n2 Y )

X COS (hi*) «P (*nl *°! + *£ 2 * ° 2 + X^3 P O 3 )

( n , m , s ) + l / 2 2 \ l / 2 A , = J _ l 1 \ * n f m . s , i j

X n , m , s , i ( B i i + B i i + X n , m , s , i )

cot (X .) = ( r f - } \ „ n . , Po. = or/A? n ,m,s , i \ B I . / n ,m, s r i i i

B i £ = hlt^/k, i = 1, 2, 3 , X = x/J^, Y = y / £ 2 , Z = z/& 3



8.1.45 9, p. 417 Case 8.1.44.1 with t = tQ sin (ur + e}, x = o, o < y < b, 0 < z < d, T > 0.

, , v 1 V s i n (imtY) s i n (mrZ) M s i n (urc + e + 4 ) -J* 5 " x _ 16 \ \ ^ _ _ " _ ' m,n T m f n «

OS I

fc0 " fci IT 2 ra=0 i£u m n

00 00 00 , „ ^ ^ ^ Jl s i n (imrY) s i n (mtz) s i n (JluX) (W cos e -X o s i n e ) exp(-Xar/w )

~ ~~v ^ *-* £-> / 2 2 \ 1 1 m=0 n=0 A=0 r + X , n n

V m,n,J6/

= s i n h { ( l - X ) [ ( ^ w / b ) 2 -t- (nirw/d) 2 + i v f l 1 / 2 }

Vn«* c^ r n) s i n h ( [ ( - ; ; / b ) 2 ; {-W)2; i w]v 2j W = w w/a, See case 8 ,1 .44 ,1 for \ . , X , y , Z, m, n

Section 8.1. Solids Bounded by Plane Surfaces—Mo Internal Heating.


8.1.46 9, p. 419 Wedge with steady » surface t - P e r a t u r e . _ ^ = | 2 £ ^ n ^ { 2 n + ^ ^

0 i 0 0 n=l T = 0. t = t , 9 = 0, T > 0. t = t . , e = e 0 , T > o. r exp (-<rcu 2/r 2) d u

s

s = (2n + 1) Tt/Sp

For t = t Q f B = 0, 8 0 , T > 0:

t - t .

h-H = i

'0 n=0 i | - 2, s i n < s 6>( e x P (~«u 2 A 2 ) " s J (u)du



8.1.47 33

00

09

Composite slabs with ramp surface temperature. t = t ± , 0 x < &J+ Z2 , x = 0.

x = 0 f 0 < X < 1. " • t = t , x = 0 , T > " l / o . ^ - 0 , x = «,, + K,2 •

4 - ^ = 2

nTL

Pd (* n + M2) e x P (~ X

n

F o ! ) L1 - e x P ( X

n / p d ) j s i n ftR

x>

X n (*n + « 2 + M )

X n t a n ( V = " ' M = S l P i c i ' a 2 p 2 0 2 ' X = x / X , l

See cases 8.1.34 to 8.1.42 for other conditions.

\*-li-*\*-i3-*\



8.1.48 19, p. 3-29 In f in i t e plate of h ( t " V 3/2 i n f i n i t e conductivity, For n = 1/2: J - ^ r a~^ = 2 (Bi F o ) J / exp (-Bi Fo) transient f lu id temp and convection boundary. t » t Q , T = 0 . t f - t„ + Srn.

pcS, 3

(Bi Fo) j - 1

j ~ l (2j + 1) (j - 1) 1

t..h-w MO

k->-~ For n = 1: h < f c - V

pel &

v2 (t - t„)

= exp {-Bi Fo) + Bi Fo - 1

~ ) g-^- = 2 |_1 - Bi Fo - exp (-Bi Fo)J + (Bi Fo)'

h-2*H

Section 8.1. SolidB Bounded by Plane Surfaces—Ho Internal Heating.


8.1.49 19, p. 3-30 Infinite plate of infinite conductivity with convection boundary and harmonic fluid temperature. fcf " W (tmx ~ W COS (UfT ) . t = mean temp, mn t = max temp, mx r

t - t mn t - t mx mn

cos (TIJ) - \))) Ve 2 + i

9 = pcX,/h, i|> = tan" 1 (6)

Instantaneous surface heat flux:

q A 9

r exp (-" + 1 v pel)

h (t - t ) - , =-mx mn >/l + 9 2

See Fig. 8.9

sin (TO) - i|i)

k->°° tf,h- — h,tf

| — 2f—*|



8.1.50 40 Infinite plate of infinite conductivity with surfabe reradia-tion and circular pulse heating, t = t Q, T = 0,

g =aj?T 4.

See Fig. 8.12 for maximum temperature values:

max (^ a x \ 'pott J

D = heating duration

CD i

*x = V 1 " m/D'

— <*r k -*•«>

H- 2?H



8.1.51 19, p. 3-44 Infinite composite 42 43 66

plates in perfect contact, convection boundary on one side, insulated on other side, t

See Fig

t Q , 0 < x < S± +

<S2, T = 0.

c^ = 0 , x = S^ + 6 2 , T

t f , h -

> 0 .

. 8.14 P 1 5 1 = f Aamax ~ fco\

D = heating duration

K*HH



8.1.52 19, p. 3-49 Infinite plate with radiation heating 44

68 or cooling. t = t Q, 0 < x < 6,

% -**(*!-*I-o)-

See Figs. 8.15a-e for heating.

See Figs. 8.15f-l for cooling,

t - t„

H f(Fo)

T = source temp.

a i



8.1.53 19, Infinite plate with p. 3-68 time-linear surface 49 heat flux.

t = t Q, 0 < x < 6, x = o.

h-«H

k ( t ' V PO /PO X 3 5X2

Y 5 ) 1 „ ~i^6 = io U - 12 + — " X + 16/ ' 2 < F ° < Fo D ,

^ D ^ 1

Fo = aD/S"", X - x/6

For ^ ^max T/D)

k (t -

*o 6

X 2

2

D = duration of heating.

± < Fo < FO D , Fo D > 1 .

8.1.54 19, Case 8.1.53 with p . 3-68 q = <i s in TTT/D. 150

See Figs . 8.19a 6 b

2fi£$. (t - t Q ) = f(2irr/D), QQ= q xD C 0



8.1.55 19,p. 3-73 Porous infinite 51 plate with steady

surface heat flux and fluid flow. t = t Q, 0 < x < 5,

t = fluid source

CD i *» in

-t 0,k,,c f

k-M

See Figs. 8.20a & b

f(ax/6 2)

P = G fc f6/k e

k e = Pk f + U - P ) k s

G = fluid mass flux p = plate porosity

Steady state solution:

^Tfi = p exp (-Fx/6)



8.1.56 19, Two infinite plates p. 3-75 in perfect contact 52 with a steady surface

heat flux. t = t Q, 0 < * 1 < 6 Z, 0 < x 2< 6 2, T = 0.

\ = V xi = °' T * °"

co I J

See Figs. 8.21 a-d

<fc - V k i *0 61

ewo^)

8.1.57 19, Case 8.1.56 with p . 3-75 k •*• oo. 48 i

See Fig. 8.22

= f (FOj)

8.1.58 19, p. 3-75 53

Case 8.1.56 with k, •* >» and a = a sin TTT/D.

See Figs. 8.23 a, b, and c t - t„

eqb 0 f (2TTT/D)

-ur.ition of pulse



8.1.59 74, p. 245 Thin plate with two convection condit ions, t = t Q , -& < x < +£,

T = 0 .

l - j

V h i 2w]

f ' " 2

- h 2 , t (

-i\H "V,

t - 1 „ cosh ( B i ^ 2 X ) £VBT

fcf " fc0 cosh (BiJ/2 L ) * Sinh (Bi^jL

2w

2 y X2 s in X n cos (XnX) exp [- (x^ + B i ^ L 2 ) F o J

" 2 n=l ( X 2 + B i l w L 2 ) [ X n + s i n ( X n ) C O S ( X n ) ]

K tan(X n)= B i 2 w , B i l w = h l w / k , B i 2 w = h2w/k

X = x/w, L = Jt/w

Mean temp:

' . " ' o Bi , . . .,

tanh

t f ' t ( ) B i ^ 2 L + P * L2 t a n h l w B l 2 w

V X n B n — [- fa + B i l w * ' ) F ° w l

-4, X? + Bi,.. L2

n=l "n lw

. " i »

'" >'.$ 2 'Bi? + Bi . + X

lw lw I)



8.1.60.1 74, p. 320 Infinite plate with evaporation and convection boundary. t = t., -S, < x < +8,, T = 0 . m = n»0 e , x = ±&.

m = evaporat ion r a t e .

M -I

t f .h -

-H

h.t,

m

t - t

t f - t . i . _ M cos CVPoX) exp (-Pd Fo)

cos (VP3) - —• VPd s in (VPd)

n=l 1 - Pd/X"

A cos (A X) exp -X Fo n n ' r n

2 s i n ( X n ) . , M = •ymo f wb

"n ~ X + sin/X \cos/X ) ' " ~ h ( t . - t . ) ~ t , - t . n \ n) \ n) f l f i

cot/X )= X / B i , t . = wet bulb temp.

Section 8 . 1 . Sol ids Bounded by Plane Surfaces—No Internal Heating.


8 .1 .60 .2 74, p . 321 Case 8 .1 .60 .1 with asymmetrical boundary conditions.

oa i

t - t . M l + "2 V / M l + M 2 \ / s

1 X n=l

mx - M2)Bi X 2(1

" V " " V ( M 1 " M 2 } / \

TBIT" ' 1 2 \ s i n <V e*e K F o ) m=l

2 s in A =

(*n) 2 cos , » * -

( \n ) n ~ X + sin/X )cos/X ) ' m ~ X - cos/X \sin/X

cot(X n )= X n /Bi , tan(Xm)= -X^/Bi

Mean temp:

k -) fc.- fci . M l + » 2 , V A M 1 ^ « 2 \ B i e x p ^ F o ;

* f - f c i 2 ^ . V 1 " 2 / X 2 ( B i 2

+ B i + X 2 ) n=l n \ n /



00 in o



oo i



8.2.1 3, p. 276 Infinite plate with uniform internal heating, t = t i # -J, < x < &, T = 0. t = t i t x * tl, x > 0,

00 I to

(t - t.)k T^h" = I (1 - x2) - 2 2 6XP (< F°) COS «V° V * n=0 n X n = (2n + 1)71/2

See Pig. 8.8

I — J t - -l~\



8.2.2.1 74, p. 363 Infinite plate with steady surface heat flux and variable internal heating, t = t i f -fl, < x < +£/ T = 0. q ' " = %" (1 - «/«•). < ^ = q 0« x = ±SL, x > 0.

k-&-H I Ul

<*<T q"'

-k»]

-"0

(t - t^k P o 4 ^ cos [(2n - 1)TTX] cos (X RX) q u " *' 2 1,a i-1 ( 2 n " 1 ) 4

x 1 - exp C-X^ PoJ + * «. 00

a " 'ii,

x exp X = mr n

B.2.2.2 74, p. 363 Case 8.2.2.1 with , I I I _ „i 11 (1 - X"),

(t - t.)k J _ i _ - f to - 2 <-l) n -J cos (XnX) [l - exp (-*J Fo)] + * 0 n»l n

X and $ given in case 8.2.2.1 n

Section 8.2. Solids Bounded by Plane Sucfaoes—With Internal Heating.


8.2.2.3 74, p. 364 Case 8.2.?.l with _ q--i = q£" exp (-bX). - 1 - 4 - = 2g [l - exp (-„,] + V [i - (-!,» exp ,-b>] f

nX) [l - exp (-X* *>)] + $ x cos (X X) 1 - exp

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) + $

J q".= q."(l + bT). ,2 it*

$ given in case 8-2.2,1



8.2.2.5 74, p. 364 Case 8.2.2.1 with q'" = <3o" exp (-be),

(t - t.)k _4 r:.._[i-««p (-MFo>] + * g0 *

$ given in case 8.2.2.1

8.2.2.6 74, p. 364 Case 8.2.2.1 with .in - » i n q'" cos (DT)

<fc - V k i q,,. ,2 Pd sin (Pd Po) + *

00 J.


8.2.2.7 74, p. 365 Case 8.2.2.1 with q"\ = q'" bx".

( t ~ fci)k (Pd Fo) n _ -a'" I2 n + 1




8 .2 .3 9 , p . 131 In f in i t e p late with time dependent internal heating. t = t . , -i, < x < ±a, T = o. ; t = t.,'"xi= ±a, T > o. q ' " = f (T) .

CO

I

\-*-l-*\:*-l-*\

e-"d-»5 (£&r-[•****«] n=0

xj f(x) exp T-Tr2(2n + l ) 2 a(T - T ' J A j a r " Jo

8.2 .4 9, p. 132 Case 8 .2 .3 with

ffr) = q ' " e •bt ( t " V k

o 1_ [cos (xVPd q > " I

[s^BM-™ 2 ^ ( - l ) n cos (AnX) exp (-A^ Fo)

X = (2n + 1 ) 7T/2 n



8.2.5 9, p. 132 Case 8.2.1 with q"' = f(x).

Solution

n=l

x / f (X) cos 0^ xj dX

CD I Ul •J


Case No. References Description Solu t ion

8.2.6

03 I

U1 03

9, p . 132 I n f i n i t e p l a t e with uniform 41 i n t e r n a l heat ing and

convection boundary. t = t , - 1 < x < H i T = 0 .

- h . t f

MH H W

( t - V* B i , , Bi Bi X 2

g- ' - r

2 Bi '2 cos (X X) n 4, X 2fx 2

+ B i 2

+ B l ) n=l n \ n / exp

+ Bi" + B I ) cos (X ) ' n

K ») X tan (X ) = Bi , See P ig . 8.13 n n

8.2.7 9, p . 132 I n f i n i t e p l a t e with uniform .. _ . . . _ i n t e r n a l heat ing and con- ' ~ f _ (1 + Bi/2)X x

1 + Bi " 2 vect ion boundary on one s ide . a,l,S? t = t , 0 < x < X., T = 0. 9

t = t f , x = 0 , T > 0. . i n _ „ i 11 1 y 0

+ 4 Bi sin (X X)fl - cos (X l"

, X 2 (X 2 + B i 2 + Bi ) s in (2X ) n=l n \ n / n '2 exp « «>)

• h , t f

X cot (X ) + Bi = 0 n n'

K*H



8 .2 .8 9, p. 311 In f in i t e p late with time-dependent heating, t = t , 0 < x < H, T = 0.

t = t^, x = 0, A, T > 0. q . . . = fr8/2, s - - 1 , 0, 1, 2 , . . . .

00 1 to

\*-l^\

(t - t Q )k

Bar l+s /2 " (1 + s/2) ! l - r/(2 + s /2)2 s+2

n=0 L

8.2 .9 9, p. 404 In f in i t e p late with variable internal heating, t = t Q , 0 < x < SL, T = 0. t = t , x = S., T > 0. q'" = q^" + 3 ( t - t „ ) . q = 0, X = 0, X >_ 0.

• x

:

<*- y k if. «^2 " B l

c o s ( V l x ) 1

cos (VI )

16 V (-1)" exp {[-(2n + 1) V + 4 B ] Fo/4f cos L(2n + DTTX/2. + * ^ f4B - (2n + 1) V ] (2n + 1)

n=0 L

B = eavk



8.2.10 9, p. 405 Case 8.2.9 with convection boundary h, t Q at x = I.

(t- tQ)k Bi cos (Vlx) l l l n 2 ~ BTBi cos (V"B) -iTS sin (V~B) ~ B «0 * L

+ 2 Bi ™ cos (XnX) exp [(B - Xfl) Fo ]

„ (B - \ 2 ) \\2 + Bi(Bi + 1) cos (XJ n=0 V n/ L n J n

X tan (X ) = Bi, B = 3A A n n

8.2.11 19, p. 3-29

Infinite plate of infinite G Jin [l - (9/Gf) + 6 = LH/C conductivity, variable specific heat, convection boundaries and steady heating.

0. t = t , 0 < x < 21, T c = c Q + B ( t - t Q ) .

n o %

k-t-oo

-h,tf

G = (1 + q £ " X.)/hL(t - t f ) , C = .1 - c f)/c Q, H = hc/pcA

H evaluated at t Q , c f evaluated at t f

I. = R. x (surface area/volume) = 1 9 = (t - t Q)/(t f - t Q)

\^2i.*\



8.2.12 19, Case 8.2.11 for infinite p. 3-29 square rod and cube.

See case 8.2.11, set: L = 2 for square rod L = 3 for cube

8.2.13 19, p. 3-32

00 I en

Infinite plate of infinite conductivity with surface radiation and steady heating, t = tQ, 0 < x < 2%, T = 0. <Jr = °& 1 = source temp. rt< • > = « « • •

in R| + B (1 + S i + 2 t a n " 1 < 6 / N > • 2 t an^d /N) + 4 MN3

« - < ) • 9 = T / T Q , N = (ci'Q"l./a&'ri

0 + 9*J , M = a^r^t/pcS, 1/4

q'"

h~2AH

"Ir



8.2.14 19, p. 3-33

OS I to

Case 8.2.13 with simultaneous surface convection and radiation.

g x = h ( t f " t } + t S r |

x = 0, 2%, T > 0. «- *}

/9 - R \ /e - R 2 \ A 3

M = A i *n [wnrj+ A2 *n \e^R-j + rJ

^ = jsVv - n 2 [n Vv - n 2 + (v/2 + n2)]}

A 2 = fsVv - n 2 [nVv - n 2 - (v/2 + n2)]J

(9 - n ) 2 + 4>2~] ( l - n ) 2 + 4>2_

- l

- l

a 3 = n / ( y 2 + 8n 4 )

A. = (v/2 - v 2 ) / (v 2 + an4) 4

R 1 = - n - V v - n r R 2 = - r i +"Vv - r\

<t> = V n + B / 4 , n = V 3 7 2 ,1 /3

V = ^ J + 3 2/4

s = h/aJTT„

N = Ic q'"i./a&-T* + 9 4 + s 9 f

,1/4



8.2.15 19, Infinite plate of infinite p. 3-34 conductivity with prescribed

heating rates, t = t Q, T = 0.

a. step input; *rrr = 1 , ?,,D = 1 Tnax ^max

„ I I I _ Q' ' ' b. l inear input; jjhvr = T r 5 i n p = 0 . 5

Tnax Tnax _ n u t

Tnax

q>"D Tna~

c . l inear input; )*,,-,' = (1 - T) r ^. n D = 0 . 5 Tnax

en a ' ' * 2 — Q ' ' ' <*» d. circular pulse; g , , , = s i n TIT , " i n D = 0 . 5

Tnax Tnax

e . power pulse; f ^ l = (37)" 3 e " ( 1 A ~ 3 ) . S ^ _ = 0.5473

o ' " 5 - ( 1 / T - 5 ) Q'' ' f. power pulse, g , , , = (5T) e , . , , D

"rnax ^max

0.2795

a £ £ « - 9 = 7 = • Q l I « " l

h £ C S _ fl _ r2 b- QTTT O - T

c p r 8 = ? ( 2 - 7 ,

sin2irx

PCS ^ - 9 = 0 .

pC6 Q " ' 9 = 0.

0 6 7 6 7 A + l ) e - ( l A - 3 >

006869/-47 + ~Z7 + Z + x ) \ 6 T 3 2T T /

- ( l / T - 5 )

g. exponential pulse; ^LLL . ( l o 7 ) . X - " t ? fi^_ . 0 . 2 7 1 7 g . flSL. 9 . ^ [ l - (10T + 1) e " 1 0 * ] % ax Tna

9 = t - t 0, D = heating duration x = TA>/ Q''' = total heat input per unit volume durin;j time D See Fig. 8.10



8.2.16 19, Infinite plate of infinite p. 3-35 conductivity with surface

reradiation, steady surface heating, and steady internal heating.

See Fig. 8.11

t = tQ, x = 0. g x = q" + o?^ - T 4 ) , x = o, 26. T = source temp.

00 i on

• V k-q" -i*

hr-25-H

Section 8.2. Solids Bounded by Plane Surfaces—With internal Heating.


8.2.17 74, p. 386 Infinite plate with two symmetrical planar heat pulses and convection boundaries, t = t f, -ft < x < +1, T = 0. Instantaneous pulse occurs at t = 0 , x = ±x, with

2 strength Q(J/m ).

oo l

en ui X , X ,

h-H

- h , t f

*-M

( t - t c )k& Qa

= 2 V !1 A> X + s in (X

,i . 7T—- cos (X X,) cos (X X) (X ) cos (X ) * n 1 n n=l

x exp « " ) • X tan (X ) = Bi . n ri

For t = f (x) , -H < x < +SL, T = 0:

( t - t f ) k £ = 2 ) ^ — - — : rr^- -r—- cos (X X) exp (-X 2 Po)

Zs X + s in (X ) cos (X ) n ' r \ n / n=l

1 cos (X X)dX . n '

Mean temp:

( t - t ) U m f Qa

For Bi = ai.

( t - tf)kSi

^ , B i 2 cos ( X ^ ) exp (-X 2 FoJ

, s i n (X )X (B i 2 + Bi + X 2 ) n=l n n \ n /

Qa = 2 Y ( - l ) n + 1 cos [(2n - DTTXJJ cos [(2n - l ) n x ]

n=l

x exp [-(2n - l ) 2 i r Fo /4] .



8.2.18 74, p . 387 Inf in i te plate with a planar heat pulse, t = t Q , 0 < x < Z, x «= 0. Instantaneous pulse occurs at K = x . , T = 0 with strength Q(J/m } .

-Q nf\

(t - t )k«, ^ , 2

= 2 \ s in {mx^ s i n (n7rX) exp (-n ir Po) Qa

n=l

00 I a\

JLJ M H

Section 8.2. Solids Gounded by Plane Surfaces—With Internal Heating.


CD i en

Section 8.2. Solids Gounded by Plane Surfaces—With Internal Heating.


CO l 03



9 .1 .1 2 , p. 240 In f in i t e cylinder with 74, p. 131 steady surface temp.

t = t . , 0 < r < r„, T = 0. i 0 fc = V r • V T * °-

IS I

t ~ t 0 , V 1 / , 2 „ \ 3 0 ( X n

n=l

J.(A ) = 0, see Fig . 9.2 u n Cumulative heating:

n=l n

Q0 " V ^ ~ t i )

9 .1 .2 1, p . 269 I n f i n i t e cylinder with time 9, p. 328 dependent surface temp.

t = t . 0 < r < r 0 , x 0.

t B = t t + « , r r Q , T > 0. 2,} ' F° + 4 ( R 2 " X) + 2 2 6 X P ("An F o ) , 3 W>

c r 0 / a n=l K W W See Table 9 .1 and Fig. 9.8



9.1.3.1 74, p. 270 Infinite cylinder with convection boundary and variable initial temp, t = f (r), 0 < r < r , T = 0.

t - t 2 V W > exp(-X2

nFo) >

" " n=l J 0<V + J 1<V -4

1

/Rp(r) - t f jJ 0 (X n S)dR

B i J 0 ( X n ) = X n J l ( X m )

9.1.3.2 3, p. 298 Infinite cylinder with convection cooling, t = t i r 0 < r < r Q, 1 = 0.

h,t.

t - t f , V Bi J 0 ( X n R ) exp (-X; Fo)

**-*«• k « + B i 2 ) w x J . a > + Bi j n {X > = o

n 1 n O n See Figs. 9.1a-c

Cumulative heating:

1 - exp (< ")] Qo Z , X 2 ( x 2

+ B i 2 ) n=l n \ n /

For Fo < 0.02, R » 0: (t - t . )

' Q 0

= l r V c ( t i " V

1 1 " V „ _ . /Fo" . . A - R\ . . „ . Fo

t e - 1 . ) - 2 B I V R ~ i e r f c \2^ws)+ 4 B l vi x ( iR + f - B i ) i 2 e r f c (ivfl) + ••••



9.1.4 9, p. 199 Infinite cylinder with initial temp varying with radius. t = t t - br 2, 0 < r < cQ, T = 0 . t = t Q, r = r Q f T > 0.

to I

t - t„

br! exp

= 2 (< ") V RV n=l

\ b c o a 0(X n) = o

X 2 J 2 ( X ) n 1 n

1 J l (X n, + 2 J 2(A n)

9.1.5 9, p. 201 Infinite cylinder with periodic surface temperature, t = t., 0 < r < r„, T = 0. t t , 0 < r < r Q, T t r

(t t.) sin (wr + e). r 0, T

t = max fluid temp, m

'(t m - t)sin(cOT + e)

t - t. l_ t - t. m I

real I Q (R -Vi Pu, iiTRTpdY e x p L 1 ^ + e)]

J!, exp (-X Fo) X Pd cos (e L, A 4 , „ J 2 \ ,

) - Xn sin (e)

n=l (xW) w

W

W = °' p d = r o a ) / a

M&SKt'



9.1.6.1 9, p. 202 Infinite cylinder with . 89 convection cooling and i _

linear time-dependent .2 fluid temp. 0

, / , , \ ^ exp (-\l Fo) J (A R) K1 " R + Si) + 2 B i 1 2 / 2 2\

t = t i # 0 < r < r Q, T = 0.

Mean temp: (if) fcm- fcj „ 1 L , 4 \ , , „.2 y e*P r X n F o ;

n=l n V n )

i \ / see Fig. 9.9

Section 9 . 1 . Solids Bounded by Cylindrical Surfaces—No internal Heating-


9 .1 .6 .2 74, p. 317 Case 9 .1 .6 .1 with t - = t_ - (t_ - t . ) e f fro ' fm i '

-br t - t . J ( Vpd R) exp (-Pd Fo) _ _ _ _ _ _ !

fm i Vpd J Q ( Vpd) - ~ Pd ^ t Vpd)

- 2 Bi J 0(X nR) exp i V "0'"n"' " * (~Xn F ° )

n=l W (>n + B i 2 ) t " (XnH]

_ i in

Mean temp fcn,- fci = 1

2J. ( V"Pd) exp (-Pd Fo)

fcfm " fci Vfd J Q ( Ypd) - ~ P6 J1 Vpd

exp 4 Bi



9.1.6.3 74, p. 320 Infinite cylinder with convection and evaporation boundary. t = t i f 0 < r < r Q, T = 0. in = m.e = evaporation rate.

t - t . MJ ( VPd~R) exp (-Pd Po) t f - t . = 1 ~ J Q ( V P 3 ) - (-VPd/Bi)J 1( VPa)

vo

/h.t, IrFfeil. -A cos (A R) exp n n K ») A = Bi

n ~ v v K + B i 2 ) J„(A ) / J , (X ) = \ /Bi 0 n" l v n n

, M Y<m 0>

h ( t f - t . ) '



9.1.7 9, p. 202 Infinite cylinder with periodic convection boundary, t = t i f 0 < r < r Q, T = 0.

l

( t t . ) s in (turr + e ) , t = max f luid temp. m e

t - ti j Bi I 0 (R"ViPd) exp (iurr + ie) V^7 = c e a l (i[vrra i l ( VTM, + B i v vrwgj+ 2 B i

" / , 2 \ X n [ M COS ( £ ) ~ S i n [ e ) 3 J 0 a n R )

n=l Vn / \ n / O n

X „ J i f x J = B i J n < X J ' P d = r n w / a

n 1 n u m o

9.1 .8 9, p. 203 In f in i t e cylinder with steady surface heat f lux. t = t i # 0 < r < r Q , T = 0.

q r = V r = r 0 ' T y °"

\£

(t - t . )k , „ / 3 \ J 0 < X n R )

—-~— = 2 Fo + (RV2) - (1/4) - 2 > exp (-X„ Fo) - ~ 1

q ° ° „=1 ^ / X n W J (X ) = 0, See F ig . 9.3

0 For Fo < 0 .02, R » 0 (t - t . ) k ~ t i ) k _ IFO" . , (X - R \ . Fo(l + 3R) .2 , / l - R\ . J-J— = 2 ^ ierfc ( p ^ ) + ^ R 3 / 2 i erfc ( y ^ ) + .



9.1.9 9, p. 207 Infinite hollow cylinder with steady surface temp.

to i oo

tTTt: = " 2 6 X P R F o) "i L 0 n=l

V»iV [ W W ~ V W W "VW + Jo<V

V W V V " V W W - ° R = r / r Q , Fo " a r / r 0



9 .1 .10 .1 9, p . 207 In f in i t e hollow cylinder with dif ferent surface temp. T „ V _ / , 2 _ \ Y V V ° V „ A0 1 " V ri < r < V T = ° fc = V r = c 0 ' T > °' t = t , r •> r . , T > 0.

to i vo

n=l

x > "- = =-* exp f-A Pol n=l J 0 < V " J 0 < V V ' T0 [ ( W J l n ( 1 / R ) + *n(R/R i )]

an " RiV bn " V V*V ' W ' W " ' W W * Wi> W " W W Wi> - ° R = r / r Q , Po = aT/rp

See Fig. 9.7a for t = t . , and Fig. 9.7b for t = t . .



9.1.10.2 74, p. 282 Infinite hollow cylinder with

to I

two convection boundaries and t _ t V E e x p ^ 2 FO) W(X n,R). a- £ iî n \ n / n variable i n i t i a l temperature-

general case, t a f(R), r.< r < r , T = 0 .

n=l W(XnR) - - p i i V V + X n V X n > ] J 0 ( X n R )

+ [ x n J l ( x n , + B i . a 0 (X n )] y 0 ( X n R ) !

2 H 2 , 2 E n = r X n [ B i 0 J 0 ( X n V " V l < W ]

x /" R [f (R) - t f ] W(Xn,R,dR {(X2

n + BiJ) [Bi. J 0 (X n , •a

+ V l M * ~ (An + O EBii VXnV " WW]*) ' . 1

E B i i V X n > + X n W ] [ B i 0 V X n V " X n W V ]

" [ B i 0 V X n V " V l < X n V ] C 8 1 ! W " X n Y l ( X n>] " °-B i i = h i r i / k ' B i o = h 0 r i / k ' R = r / c 0



9.1.11 3, p. 409 Infinite hollow cylinder with steady inside surface! temperature, t = t.y r.. < r < r Q, T = 0. t = t ^ J r = r i # T > 0.

g r = 0, jr = r Q , t > 0.

t - t . tprr, =1 - 2 2. - KFo) w » M ¥

n=l A„{R.) = Y„(X )J„(X R.) - J„(X )Y„(X R.) 0* i' 0 ' n 0 n l 0 n 0 n l A (R ) = Y (X ) J (X R ) - J (X )Y (X R ) l i 1 n J. n i. l n l n i Y, (X )J . (X R.) - J , (X )Y.(X R.) = 0 1* n 0 n i ' 1 ' n ' 0 V n i '

Po = o t r / r 0 , R = r / r Q

N»

9.1.12 74, p. 200 Infini te hollow cylinder with (t - t . )k steady surface heat flux. t = V r £ < r < r Q , T = 0. - q r = q 0 > r = r Q f T > o. q = 0 , r = r . , T > 0.

<*n r

O'O 1 - R7 l

i i 2 F ° - i + r - R i *n

S i /

x «,n {R^ + j + f TT J 1 < W J 1 < V

[ j n (X R)Y,(X R.) - Yn(X R)J.(X R.)"| exp (-X 2 Fo^ L0 n l l n i 0 n 1 n I ' J \ n /

Po = a r / r Q , R = r / r Q

J . rtR.JY. (X ) = Y (X R.) J . (X ) x n i l n x n i 1 n



oo co 9.1.13 9, p. 210 Infinite cylinder with variable initial temperature and _ y y r A 9 + B s i n 9)-| constant surface temp. 0 Z* ^ L n » m n' m J t = f(r,0), 0 < r < r Q, t = 0. t = t Q, r = r Q, T > 0.

m=l n=0 , J n(X mR) exp (-X* Fo 2)

= " =• / / R[f ( R r 9 ) " tnl J

n(X™R>«lR d 9

5 ' m ir [JA<Ol2 I L l ° J ° m

. 1 IT

L 0 v m'J 0 -IT

. 1 - T T

= - / / RTf(R,9) - t„"l cos (n6)J (X r)dR d9 n'm * ft VI 2 - i i

1 TT

„ ™ = ~ 5 I / R l " f < R ' 9 > " fcnl s i n (n9)J (X R)dR d9 n ' m TT |J' ( M 2 I 1 L ° J • " m [W] 0 -7T

J „ < X J = o n m

Section 9 . 1 . Sol ids Bounded by Cylindrical Surfaces—No Internal Heating.


9.1.14 9, p. 211 In f in i t e cylinder with vari - « °° able i n i t i a l temp and convec- _ V V r f t C Q g Q) s i n (Q)-, j ( X R ) / _ £ p \ t ion boundary-general case. f £,£-•]_ n,m n,m J n n \ n / t = f ( r , 9 ) , 0 < r < r , T = 0. m=l n=0

,2 _1 _TT

IdR d6 A „ r a = - 7 - 5 =r 2 I f Rff(R,B) - tJl JQ(XmR)<

2X2

m

10

x I I Rff(R,B) - t f"j cos (n9)J (XR)dRd9

'0 -TT

m

- \ ( x W - n 2 ) [ W ] 2

1 TT

* \ \ R[f(R,9) - t f"| s in (n9)Jn(XmR)dR d9 0 -TT

X J' (X ) + Bi J (XJ = 0 in n m' n v m' See case 9 . 1 . 3 . 2 for f ( r , 9 ) = t i

or case 9 . 1 . 3 . 1 for f (r,8) = f (r)

<$ffi§®? e«Sav;'r£Ks!Es



9.1.15 9, p. 212 Infinite cylindrical sector with steady surface temperature. 0 8 V sin(j9) V „,_ / , 2 „.\ j m f D T .„, . t = t . , 0 < r < t n , 0 < 0 < e n , t T ^ T " = ? L 2n + 1 L e X P Tm F o j f -.2 / ^ j ' ^ 1

T = „* n=0 m=l L j • J 0

dR

t = t Q , r = r Q, 6 = 0, x > 0.

10 I — 0

j = (2n + l)ir/60, Jâj = 0

9.1.16 2, p. 248 Semi-infinite cylinder. Dimensionless temperature equals product of solution for semi-infinite solid (case 7.1.1 or 7.1.3) and solution for infinite cylinder (case 9.1.1 or 9.1.3). See Figs. 9.4a and 9.4b.

9.1.17 2, p. 248 Finite cylinder. Dimensionless temperature equals product of solution for infinite plate (case 8.1.6 or 8.1.7) and solution for infinite cylinder (case 9.1.1 or 9.1.3). See Figs. 9.4a and 9.4b.

9.1.18 Infinite cylinder in a See case 7.1.22 semi-infinite solid.



9.1.19 9, p . 41S F i n i t e cy l i nde r . t = t Q , 0 < r < r Q , 0 < Z < St,, T = 0 . t = t 1 # 0 < r < r Q , Z = 0, T > 0 .

0 < r < r 0 , z = I, T > 0. Convection boundary h, t_ a t r = r , T > 0.

to I

t - t„ t, - = 2 Bi J 0 (RX n ) sinh [(1 - Z)XnLJ

4it Bi 1 " fc° n _ , (Bi2 + A2,) J . U J sinh | H | 11=1 \ n/ o n n

CD co ( r 2 „ ^-, mJ-(RX ) s in (imrZ) exp I - U + (nrn/L)

~ ~ |"(X L ) 2 + (mir) 2 1 | B i 2 + A 2 1 J„ (X , n-X m=l L n J L n j 0 V n

X J , (X ) = Bi J . (X ) n l l n 0* n' Bi = h r Q / k , R = r / r Q , Z = z/l, Fo = onr/r 2 , L = V r Q

9.1.20 9, p . 418 Case 9.1.18 with convect ion boundary h , t a t z = $., x > 0. fcl" ^

2 Bi ^. J 0 ( R ^ \ , c ° s h r*n LJ1 - Z J] + B I s i n h [ v ^ 1 ~ z3i n-l ( B i 2 + Xl) W [Xn c o s h ^ n ' + B i s i n h ( L V ]

i Y V Pm ( B i 2 L" * e m) J 0 ( R V 5 i n ( 2 V «"> RXl + g > 2 ) F o " 1 nfl m=l ( "* + *n) ( B i 2 ^ + ^ + B i L) (^n + <) W

3 cotg = -Bi L, L = Vr„ m m 0 See case 9.1.19 for X , Bi, R, Z, and Fo n

ft^w-'"



9.1.21 9, p. 419 Semi-infinite cylinder with _ « j (RX ) convection boundary. 0_ . V" 0 n t = V 0 < r < r„ , « > 0, 4 - t Q - £ ( B . 2 + X^) J ^ )

2 exp (~X Z) + exp (X Z)

T = 0 .

t - t x , 0 < r < r 0 , x e r f c

2 = 0 , T > 0 .

Convection boundary h, tQ \ i J l ' ^ n ^ = B * Jo'^n^ at r = r , z > 0, T > 0.

f — 2 _ + Z Fo*] - exp (-X Z) erfc I — ^ - z P o *) \2 Fo / n \2 Fo /.

li = hr Q /k , R = r / r 0 , Z = z/tQ, Fo = r 0 /2-ôT

A 9.1.22 9, p . 419 Semi-infinite cylinder with steady surface temperature. Z I9_ = , _ V "Q'""n' Jo » ^ L\2 _ \ „ „ . , * t •t = 0. t = t x, r = r Q, z > 0

erf (Z Fo ) + exp (X Z) -inrinite cylinder witn i /pi t f dy surface temperature. z ~ z0 . V J 0 ( B V I. / ,2 „ \ y o < r < V z > o , v ^ - 1 - £ x^TT) [ 2 . « • K P o )

erfcfz Fo* + — ^ A + exp (-ZX ) erfc |Z Fo* ~) \ 2Fo / n \ 2 Fo*/

X r T > 0. \ 2Fo"/ " \ 2 Fo" t = tQ, 0 < r < r Q, S e e o a g e 9 > 1 - 2 0 f o r x , Fo*, R,and Z z = 0, T > 0.



9.1.23 9, p. 420 Cone with steady surface temperature. t = t,, o < e < e Q, T = o. t = t Q, e = e 0, T > o.

t - t. i 0 l

= 1 + ^ (2n + l )r [(n + l ) /2] PJX)

^ 2 . nT(n/2) [dPn(X)/dn] x = x

• /

exp (-orru/r )J n + 1 / 2(u)du 3/2

P (X„) = 0 (Legendre polynomial) n u

^ 9.1.24 33 Infinite composite cylinder, t = t., r 3 < r < r., T = 0. t = t Q l r = r 3, T > 0. q = 0, r = r,.

k = °°

1 "0 ) r - r r = IT J « P ("*n *>) FG [x2j0(R2Xn) - NX n J l(R 2X n)]

n=l

F = vw [VXnR3>] 2 [ ^ n " «(« " V*2»2J ~ [ X„V R2 Xn> " N V l <R2Xn>] '

G - J0{RXn> [ ^ D ( R 2 V " «VVn>] ~ V » V

*[Xn J0(R2Xn> - V l ' V n ' ]

(R - 1) ' p c

( R 3 " X ) P 2°2



9.1.25 33

vo I

Finite cylinder with convection boundary and insulated end. t = t., 0 < r < r Q, 0 < x < T = 0. t = tQ, 0 < r < r Q f

x = 0, x > 0. q x = 0, 0 < r < r Q, x = %, T > 0. Convection boundary h, h

2 ' V a t r = V h.t.

I

*' fc0 " fci = 2 \ (X2 + Bi 2 ) n=l V n /

Bi J0(^nR) cosh [XnL(l - X)J

|x = £,

J„(A ) cosh (X L) D n n - 2TT

™ ^ . (2m + l)BiJQ(XnR) cos [ ( 2 m

2 ~ *) (1 - X)7ij

~ , L2 [x 2 + (2at + 1>V/4L 2 1 ( \ 2 + B i 2 ) J„(X ) m=0 n=l L" J \ n / O n '

WV = B i W Bi = hrQ/k, L = «/r Q , R = r /r Q , X = x/l, Fo = orr/r2



9.1.26 33 Case 9.1.25 with convection " . x r x _ n h , t-.j at r = r Q . i_ _ 2 V 1 2 0' n L n -I Convection boundary fc0 " fci , (x2 + B i 2 ) J„(X ) Tx sinh (X L) + B i , cosh (X L)l

n=l \ n J. / Q n l_ n n £ n J h 2 , t Q : at x = 0.

to

" • Xm B ^ B i ^ • B i ; J 0(X nR) cos [X m L(l - X)] Z

n 4; (xV + x 2 ) (x2

+ Bi 2 ) J 0 ( X ) [(x2

+ Bi?) + Bi,l m=l n=l \ n m/ V, n 1 / 0 n ]\, m 2 / 2J

x exp [-FO ( x 2 ^ 2 • X 2 ) ]

Vl ( Xn> = B i l W ' V » ' V = B i 2 i B ^ = h ^ / k , B i 2 = h2JLA, L = H/tQ, R = r / r Q , X = x/S,

l ^ S R K v • -"--.-.V.ii



9.1.27 33

o

Finite cylinder with a linear time dependent temperature on one end. t = t^ 0 < r < r Q, 0 < x < l, T = 0.

t. + b(t„ -i 0 0, 0 < T b. g^ = 0, 0 < r < r Q , x = I, T > 0. Convection boundary h, t . at r = r 0 '

t = f(r)-

™ Bi J o ( 0 n R ) cosh [e n L( l - X)] 4

n 4 . fe2 + B i 2 ) j (B ) cosh (3 L) p d

n=l V n / O n n

3 Bi J . ( B R ] cosh fe L( l - xfl m o n L m J

J^n £. ( B V + &\ fel + Bi 2 ) J 2)])

-X

h.t,

J L Pd = r 2 b/a , ••b 3 0

j

(2m + l)ir, 3 n J , (3 n ) = Bi J n ( 3 J n l n O n

Fo = orrAn, Bi = h r 0 / k , L = £/a, R = r / r Q , X = x/J,,

9.1.28 33 Case 9 . 1 . 27 with convection

boundary h 2 , t f at x = 0.

fcf - t . ^ I " b < f c o " t . ) T ,

0 < T < 1/b. t = t Q, T > 1/b. Convection boundary h w t.: at r = r„.

Solution identical to that given in case 4.1.25 except multiply the second summation series by

Pd {exp | ^ T (X: + KL/l>~)\ - 1} [h K • *4



9.1.29 65

to i

IO

Inf in i t e rod attached . . to an i n f i n i t e p la te . t c " V*! fc s fc

0» 0 £ * ! « ' 1 0

b

0 £ r < <*>, T = 0 .

Solution

" V k i _ _,_ x 2 „ _,_ i 2 v ( - D n , : q 0 b 2 3 7i2 4 , n 2

0 <_ C < » , T = 0 . • ^ = q~, x = 0 , T

> 2ir 2 P o ) ,

__i ©

n=j. X = 1 , R •+ «° .

<fc - V k l [<* - V k l 1 / k 2 P 2 C 2 \ V 2

q o b = L q o b J ^ ~ W pi<V

Y P f 4 i

2 f f» \ + . i 2 , ^ 4 2 r P + RS) ' J f m ^ T x ^ Po 4 1 erfc ( ^ + 4 i erfc (^=-j - 4 i erfc [ 2 yPo J

_|o U n=0 \ ZM . i 2 e r f c ([iî^]-)) - | (erfc W + erfc ( ^ )

c f c L f ^ P o J - e r f c ^ ^ ^J ; { , X = l r R = 0 .

= o t ^ / b 2 , X = x /b , RQ = r Q / b

erfc

Po



Section 9.1. Solids Bounded by Cylindcical Surfaces—No internal Heating.

Case Ho. References Descciption Solution

VD i to

JJ!*.!***^-' •" ".•WWSj



9.2.1 9, p. 204 Infinite cylinder with steady ,t _ t ) k ~ Jn(RX ) exp (-X2 Fo) internal heating and constant 0 = I ( 1 _ R

2 ) _ 2 ) — j — ^ -surface temperature. _2_,,, 4 £-i ' , l , l 3

V0 i

to

*fo' n=l I n n

W = ° See Pig. 9.5

9 .2 .2 9, p. 205 In f in i t e cylinder with 90 steady internal heating

and convection boundary. t = t f , 0 < r < r Q , T = 0.

q . . . = q t . . .

(t - t f ) k

*n1, i" K + 2/Bi) - 2 Bi >

0^0 ^ X 2 (Bi 2 + X 2 ) J„(X ) n=l n \ n/ 0 n

BiJ0(Xn) = V l ( V ' See Fig. 9.10.

Section 9 .2 . Sol ids Bounded by Cylindrical Surfaces—With Internal Heating.


9 . 2 . 2 . 1 74, p. 372 Case 9 .2 .2 with g 1 " = q ^ ' e t = t,., 0 < r < r Q , x = 0.

i n = în^-bT fc - *-i_ _ po fr J Q ( VSdR) a R) I

pJ^V^dJ exp (-Pd Po)

0 0

„ i - Pd - X n=l n

2 Bi J„(X ) X (r n' n

W (Xn + B l J X n

I to

9 .2 .3 9, p. 204 In f in i t e cylinder with exponential time-dependent internal:heating. t = t„> 0 < r < r„, T = 0.

( f c - V k exp t-br) J 0(RVPd)

r

2 q " • ^0^0 Pd ^(VPo")

2_ Pd

J exp ( -FoX^JpCRXj

n=l ^ n ( x J / P d - l ) J l (X n)



9.2.4 3, p. 363 Infinite pipe with steady Pipe temp: fluid flow and sudden heat generation in the wall. (t^ - t..)h P

en

generation in cne wan. It - c. jn r. / i \ -XT -E r * Fluid: t - t „ T - 0. — E i i- = XT + (j-rr) (1 - e A T ) - e [(XT - 1) Pipe: t = t., T = 0. g 0 "p Fluid velocity = V. Wall heating: q'" = q"' T > 0. U

x +(EfT*) - X*(5,T*) + (j^-rr) e" X T A(C,T*,X)] , t > (X - 1)5,

No axial conduction. Fluid temp: ( t - t . ) h P. _x r

Insulated £ ± - = XT - (1 - e ) - e * 1 outerwall 1 0 " A n ( X " 1 ) / X

*0 " p x

< » - y J - V x [{XT + X - 1)I|I(5,T ) - e" A T A(C,T rX)J , T > (X - 1)5 . — - v t f, h -<

_~ (nl) 2 -i

* ,T

6 e " 6 d5 n=0 ( n l > ' 0

t, = D f / X , A = X + B f / D p , T = D p / (X - X/V) , ^ ^ *

b f = h V P f ° f A f b 2 = h P/PpOpAp, + ( E r t * , = J - ^ ~ 2 / fin+le"6 d 6

rt 1**1 / n n=0 < n l>' 0 A, = flow area, A = pipe section area, f P P. = pipe inside perimeter. ^ J!, r,_,, ,,-jn /•* " l A See Figs. 9.6a, 9.6b, and 9.6c. ^ (n!)'

n=u u



9.2.5 9, p. 405

I W -J

Infinite cylinder with linear temperature-dependent internal heating.

(t ~ t Q)k _ ^ J 0(RVP5) ^_ ~ exp l(-\l + Po)Fo] J Q(R\) B J Q(VPO) " Po + 2 2/

n=l o' • 'r 9 0 r0 X n ( P o " Xn) Jl<V

J Q(X n) = 0 , Po = Brjj/k, R = r/rQ

2 If 3 > 0, steady state solution exists for Po < A .

9.2.6 19, Infinite cylinder of p. 3-29 infinite conductivity,

variable specific heat, convection boundary, and uniform heating.

See case 8.2.11, set L = 2, radius = I.

9.2.7 19, Case 9.2.6 with a short p. 3-29 cylinder of length 2A.

See case 8.2.11, set L = 3, radius = I.

Section 9 .2 . Sol ids Bounded by Cy l ind r i ca l Surfaces—With I n t e r n a l Heat ing.

Case No. References Descr ip t ion Solut ion

9 .2 .8 63 Hollow i n f i n i t e cyl inder .2 ~ 2 2 .. with exponent ia l ly space ( t ~ ^ m „ \ WW / 2 fc\ varying heat genera t ion . 2 Z , , 2 . , . T 2,„ , 0% n \ n / t = t . , r . < r < r Q , T = 0. m r l n=l W ~ J 0 ( W t = t ^ T ) , r = s i r T > 0. , r F o

t = t 2 ( x ) , r = r 2 , T > 0. q ' " = mc exp[-6( r - r ^ ] .

r ^ < , ( t - t ) k3 2 a x x R = r / r l f Fo = a r / r ' , T (T) -

x ( / h ( T ) v w " T i ( T i l e x p W F o ) d ( P o )

0

r F o I j f (Xn) ( B r ^ 2 / mx exp ( x 2 Foj d(Fo) |

mr

/

R 2 exp ["Sr^R - 1)] RB0(XnR)dR

W > " J0(XnR)VV " W W J 0 ( W W * W V W



9.2.9 74, p. 375 Infinite cylinder with a steady surface heat flux

(t - t ^ k

and steady internal heating. q ' ' ' r Q

t = t A , 0 < r < r Q , x = 0. q " . = q . " o < r < r Q , T > 0. + s

q r = V r = r 0 ' T > °*

Po +

on

f- [ 2 Fo - J(l - 2R2) - 2 J n=l

J 0(X nR) cap ( A )

X J„(X ) n 0 n' ]

I to

j ' (X ) = 0 0 n

9.2.10 74, p. 375 Case 9 .2 .9 with ' (1 - R ) .

(t - t . )k § - = i FO + 4

a'' ' r q 0 r 0

V [ 2 Ji ( Xn j - W l Z ^3,2, n=l

XJJ„(X ) n Cr n

x J Q (XnR) [ l - exp (-X 2 Fo)] + 4>

$ given in case 9 .2 .9



9.2.11 74, p. 375 Case 9.2.9 with q'" "%"a + br).

(t - t )k v 5 — = FO ( 1 + ~ Pa Fo) +

III. 2 \ * ' %' 'ro given in case 9.2.9

9.2.12 74, p. 376 Case 9.2.9 with r l • I a ~l I I q'" exp (-be).

( t " V k 1 r l ,.,,2 Pd

I

given in case 9,2.9 V r o

9.2.13 74, p. 376 Case 9.2.9 with .III _ „• II q • •' cos (br).

(t - t )k -±— = ±-r sin (Pd Fo) + If

q'"r ,,,2 Pd

4> given in case 9.2.9

9.2.14 74, p. 376 Case 9.2.9 with ( t - y _ (PdFo)n

q-rj FO + n + 1

given in case 9.2.9



9.2.15 74, p. 393 Infinite cylinder with pulse heating on a cylindrical surface and convection boundary h, t f, 0 < r < r Q, T = 0 Instantaneous pulse heating

itrength Q r l r T = 0.

of strength Q. occurs at

i w

Q, "IT Z ~ (BI2 + x2) J 2(X ) n=l \ n/ 0 n

J 0<V = 0 , R = r ' r 0 For Bi •* °°:

<t -- V p c r o «1

00

• J 2 - V X n R > J 0 ( X ^ ) exp K F°) - V p c r o «1

00

• J 2 - *?<*»> Mean temp:

<** ~ V P c c o oo

n=l

A n W l ) B n e x p (" • X 2

n Fo) ~ V P c c o oo

n=l J?<V

4 B i 2

X 2 (X 2 + Bi 2)

n v n '

A A v*> cylindrical Surfaces-. •„„ o 2 Solids Bounded by cyim Section 9-*- *"•"•

,-With internal Heating.

VD I W M



10.1.1 2, p. 239 Sphere with steady 74, p. 126 surface temperature.

t = t i f 0 < r < r Q , T = 0. t r* = f y "*~H e x P i - ^ 2 Po) s i n (mrR) t - t TT ^_ n

n=l OD _

= X - R 2 L e C f C ( 2 R 2 Vfe" / ~ e C f C \ 2 y f e + R)J ' Po < 1 . L \ z TT jro ' ^ ^Tiro ' j n=l

Mean temp: CO

t - t . ~m 10 6 V 1 . 2 2 _ v

9 I i I l 0 IT , n .'_. \ I n=l

Cumulative heat rate: CD

,n+l

0 w n=l n

See Fig . 10.1



10.1.2.1 74, p. 254

Sphere with convection °° , ._ n _, „„„ / ,2 \ -1 . r . , t— A s in (A R) exp |-A_ Fol I _ boundary-general case _ „ V n n V n / I r_ _ 1 t = f ( r ) , 0 < r < r n , t " fcf " 2 Z fX - s in (X ) cos (Xn)"| R / L M r ' c f J

n=l L 0 x = 0. x s i n (X R) dR . n

tan{Xj V (1 - Bi) See Table 14.2 of roots

10 .1 .2 .2 3 , Sphere with convection p. 298 boundary.

t = t i f 0 < r < r Q , T = 0. i ^ . = 2 y t, - t , 2 Z

Bi s i n (X ) exp (-X Fo) s i n (X R) n \ n / n X - s in (X ) cos (X )X R n n n n

n=l X cos (X ) = (1 - Bi) s i n (X ) n n n

Cumulative heat rate:

k- I [ s in (XR) - X n cos (X f l)] [l - exp (-X* Fo)]

n-1 X n On " s i n <V C O S M Q0 = f TVrJpc(ti - t f ) See Figs . 10.2a, 10.2b, and 10.2c



10.1.3 9, p. 233 Sphere with initial temp £(r) and surface temp 4>(t), general case.

n»l

•If

2 2 exp(-n IT To) s in (mrR)

Rf (R) s in (mrR)dR - mr(-l) " - / '

e x p ( n 2 w 2 Po X)()>(A)dX

o 10.1.4 9, p. 235 Sphere with surface temp u proportional to time.

t t , 0 < r < r Q , t brt + t . , r

I V T " °-

t - t t 0 - t . \ i +

6 p 0 ; v

See Table 10.1 and Fig. 10.8

3 Fo R , n 3

n-1

^— exp(-n2Tf2 Fo) s i n (mtR)

lo

WRBm.



10.1.5 9, p. 235 Sphere with variable init'-' " itial temp. ! ~...*° = J L V ± expl*-*2^ + 1) 2 Pol sin |"(2n + 1) TIR! = <*c " V <c0 " r ) /( r0 + fc0> fcc " fc0 T,3R £ (2n + I) 3 L J L J r» II' - ii i l l ill' rr n n K n=0 0 < r < rQ, T = 0.

* = V r = r 0 ' T * °'

I

10.1.6 9, p. 236 Sphere with variable initial t - t ™ n-1 temp (parabolic). 9_ = i-L. V 1=11 s i n ( n 1 TR) exp(-n it Fo) . _ ,. . , /.2 _2^ ,/_2 .. \ t _ - tn „3„ Z- „3 t = ^ c - V (ro2 " r 2 V ( r Q + fco) 0 < r < cQ, T " 0. t = tQ, r = r Q f T > 0.

c "0 7T**R *-J n n=i.



10.1.7 9, p. 237 Sphere with variable initial _ temp (trigonometric). i

0 < r < r Q , T = 0.

o l

U1

t = t„ + ( t < / c ) s i n ( 1 r r / r 0 ) , t Q - t Q

1 2 — s i n (HR) exp(-u Fo)

10.1.8 9 , p . 237 Sphere with variable i n i t i a l temp (exponential) . _ _ fc " t 0 2ir V n , 2 2 „ , t - t Q + (tc - V eXp[b(r - r03, tprt; - T 1 - ^ T T B V e X P t " n ^

x s in (nTO) [ ( - l ) n + 1 ( B 2 - 2B + n 2 t r 2 ) - 2B exp (-B)"j 0 < r < r Q , T

t Q , r = r Q , T > 0. B = br„



10.1.9 9, p. 236 Sphere with two in i t ia l temperatures.

H O

t ' t 0 H " fc0

2 MTR

CO

sin (nirRj) nn

Rj sin (nirRj_)J n=l

2_2 x sin (nTiR) exp(-n IT Fo)

10.1.10 9, p. 237 Sphere with polynorainal in i t ia l temperature. ^ _ 2 _ = 2 V s i n < n l t R ) L i - < - l ) n + 1

+ - ^ [ ( n V - 2) t = h - ^ + V • b 2 r 2

+ b 3 r 3 fcl ' fc0 £ » * *

+ . . . . , 0 < r < r 0 , T = 0. t = t Q , r = r Q , T > 0.

b

nX , , , n+1 _T . 2 . 3 3 , „ . , , , n+1 A 3

x (-1) - 2J + -£-£ (n u - 6nii) (-1) + ~^j

[24 - (n 4n 4 - 12n2ir2 + 24) ( - l ) n + . . . . ]} exp(-n2ir2 Fo)

b[ = b r„/H, b* = b2rJ/R, bj = b ^ / R ,



10.1.11 9, p . 238 Sphere with convection boundary 89 and changing f lu id temperature,

t = t , , 0 < r < r , T = 0. t f = bx + t . r T > 0.

o

( t " V " „ , R 2 Bi - 2 + Bi . 2 Bi 2 " F o + Til + — br!

^ , s in (RXJexp (-Fo X R )

n-T K \Xl + B i < B i " 1 »] s i n < X J n=i n u n J n X cot(X ) + Bi = 1 n n

Mean temp:

F O " 15 I1 " 5i) + 6 B l 2. exp (-X 2 Fo)

brr ~ X 4 (X 2

+ B i 2 - Bi) n=l n \ n ' See Fig. 10.9

Section 10 .1 . Sol ids Bounded by Spherical Surfaces—No Internal Heating.


1 0 . 1 . 1 1 . 1 7 4 , Case 10.1 .11 with t - t . R Bi s in (VpaR) exp (-Pd Fo) P ' 1 7 t f = t f m - ( t f m - t . ) e . fcfm ~ fci ~ " ( B i " x > s i n f"v^?a" , + ^ c o s ( V ? a )

_ A R s in (A R) exp 1-X" Fo) _ 2 B i V _S 1JL_ V -S Z.

~ ( l - X2/P6) X n=l \ n / n

A = n ( - l ) n + 1 [\2

n+ ( B i - l ) 2 ] 1 / 2

(Xl + B i 2 - B i ) Mean temp:

fcm " fci _ 3 Bi [tan ( Vpd) - VPHD exp (-Pd Fo) t . - t . Pd RBi - DtanCVPd') + \PdJ

rm l

6 B i 2

exp (-X2 Fo)

^ X 2 ( X 2 + B i 2 - B i ) ( l - X 2/Pd) n=l n \ n ' v n '

Section 10.1. Solids Bounded by Spherical Surfaces—Mo Internal Heating.


10.1.12 9, p. 238 Sphere with convection boundary and changing fluid temp, t = t i # 0 < r < r Q, T = 0. t f = (tm - tj) sin (OJT + e), T > 0. t = max f l u i d temp, m

fc ~ fci 2 Bi t - t . " Pd R m i

X rX I V Pdl-Pd S i n ( E ) " C O S (£)1 ( B i " 1 ) S i n ( R V n=l [ ( X n / P d ) + 1 l K + B i 2 - B i ) cos (X n,

x exp f-Fo \ ?

n \ + 2 ^ - s i n (COT + z + ^ + 4f2)\

Pd = (at /a

X cot (X ) + Bi = 1 n n

A e = s inh (to ) cos (to ) + i cosh (to ) s i n (to )

i ( t > 2 A e = to, cosh (to_) + (Bi - 1) s inh (to )

w = R "VPd/2, u = (1 + i ) Y i d / I



10.1.13 9, p. 240 Sphere enclosed in a finite medium of infinite conductivity. t = t., 0 < r < r , T = 0. t = V ri K r < V T = ° q t = 0, r = r 2, T > 0.

O

Temp of the sphere: t " t C fc ' fc0 1 2M V M K t. - t ° M + 1 " 3R Z „2

M 2X* + 3(2M + 3)X2 + 9 h sin ^ n ' MfX* + 9(M + l)Xn n=l n n

x sin (X ) exp (-X2 Fo.jJ , 0 < R < 1 • Temp of the surrounding medium:

co

7 ~~ = „ * , - 6M y -r-r exp (-X2 Po. J , 1 < R < t. - t Q M + 1 Z, M2 X2 +

P ^ n l) n=l n '

P2°2 /3 \ 3 X n M = — - (R - l) , R = r/r , tan (Xn) = j

Pl°l w ' 1 n 3 + MX_ See Fig. 10.3


Case No. -References Description Solution

10.1.14 9, p. 241 Case 10.1.13, except the sur- Temp of surrounding medium rounding medium transfers m

heat by convection to a fluid at temp t . t t t , 0 < r < r Q , T = 0. t = t Q , r > r„ , x = 0. Exterior surface area = A_

M„

fci " 2 (3 Bi R2MX;) exp (-XJ; F o J

o i

*"? M2X4 + 3X^(3 + 3M - 2M Bi R_) - 9 Bi R , ( l - Bi R_) n=i n n z z i. P2°2 ( - J " 1 ) ' r / r ,

tan(X ) 3X

3(1 - Bi R„) + MX" ^ n

K



10.1.15 9, p . 242 Sphere with steady surface heat f lux , t = t . , 0 < r < r Q , T = 0. g r = V r = V T > °'

i to

(t - t . )k 5 R 2 _ 3 = 3 Fo +

V o 10 2. V S i n ( V n ) e x p ("Xn F ° )

~ 8 Z 2 n=l

X* s i n (X,,) n n tan(X n) = Xn

See Pig. 10.4

10.1.16 9, p., 246 Hollow sphere with two surface temperatures T__!i ( V T i ~ R i ) R

2 v T o / T i c o s ( n i r ) ~ R i t = f ( r ) , t i < r < r Q , T = 0. T^ = R + R + Rrr 2J n t = t . , r = \ . , T > 0. n = 1

fc " V r • r 0 ' T > °" * 2 2 ~ i V * x s i n (fflTR ) exp(-n IT FO) + y s i n (rnrR )

n=l

x exp(-n 2 ir 2 Fo) / (R*+ R)f (R*) s in (mm*) dR* /

R = (r - r . ) / < r . - r.) , R = r / r n , Fo = onr/(rn - r . ) 2

l i " ^ 0 ' 1 ' u 0 * i '



10.1.17 9, p. 247 Hollow sphere with steady surface flux, t = t i r ci < c < cQt T = 0. t = t . , r = r , , T > 0. - 9 r - q 0 , r = r Q , T > 0.

3 I

( t - t . ) k <R0 - R) 2 " ( l + \ 2

n ) 1 / 2 s i n g n R Q -\nR)

n-1 X n [ R 0 + ( R 0 " 1 ) X n l V i RQR • R

x e x p (-A Fo)

tan (A R„ - A ) + A = 0 n 0 n n

R = r / r . , Fo = orc/r. I I

10.1.18 9, p . 350 Sphere of i n f i n i t e conductivity enclosed by s h e l l of f i n i t e conductivity.

t - t,

l 0 Q_ = 4K

~ R

sin [ A n ( l - R)] exp (-A 2 Fo)

~ x 2K(1 - R.)A n + 4R.An s in [ A n ( l - R.)] - K s in [ 2 A R ( 1 - R.)]

R. < R < 1 . i

R.K A cos [A (1 - R.)l = ( x 2 R 2 - K ) s in [A (1 - R.)] . i n L n i J v n i ' L n i J

2 K = 3 P 2

c

2

/ / p l ° l ' R = r/ft0' F o = 0 f T / ' r 0



10.1.19 9, p. 351 Composite sphere. t - t n 2Rn t = t i f 0 < r < r Q , T = 0. * ~ *0 = ^0 £ _±_^ s i n ^ j s i n {X^ s i n [ A ( R O _ 1 } x j

t o t Q , r = r Q , T > 0. fci " fc0 R

n = 1 n

x exp (-X* FOj) , 0 < R < 1 .

* ~ t o 3 V - I - s i n 2 (X) sin[A (R - 1) X J n=l

x exp (-X 2 P o 2 ) , 1 < R < RQ .

• (Xn) = K X n s i n 2 [A X n (RQ - 1)]+ A X n(RQ - 1) s i n 2 (Xn>

+ [(1 - KA)/A X ] s i n 2 (Xn) s i n 2 [A X R {RQ - 1)] .

X n cot[A X n (RQ - 1)] + 1/A + K [ X n cot (Xn) - l ] = 0 .

Fo = crt/r^, A = VSjA^, K = k ^ , R - r / r . , A(RQ - 1) i rrat ional



Sphere with radiation cooling See Fig. 10.6 10.1.20 45 19. t = t , 0 < E < r , x = 0 p. 3-57

q = a tf"(T4 , - T 4 \ ^r \ r=r x s/ T = sink temp, s

r = r 1*

10.1.21 72 Prolate spheriod with steady surface temp t = t., throughout solid, T = 0. t = t , on surface of solid, w' T > 0.

t - t. See Figs. 10.7a & b for - — t vs Po

w i



10.1.22 74, p. 320 A sphere with convection and evaporation boundary, t = t £, 0 < r < r Q, t = 0.

m = nue" = evaporation rate.

t - t . x__ t c - t . f I

1 - R M Bi s in ( VPd R) exp (-Pd Fo) (Bi - 1) s i n (-VPd) + YPd cos ( V P 9 )

T [l ^ T J A R s in (X R) exp (-X 2 Fo) H L i -(pd/x 2u n n v n ' n=l

( - l , n + 1 B i [X 2

+ (Bi - 1 ) 2 ] 1 / 2

X 2 + B i 2 - Bi n

, M = Ym 0 /h( t f - t 1 )

X cot(X ) = 1 - Bi n n



o t

Section 10 ,1. solids Bounded by Spherical Sucfaces-No internal Heating.

o I i-1

00



10.2.1 1, p. 30? Sphere with steady surface temp and internal heating. t = t . , 0 < r < t f l , x = 0.

t - V r - t Q , x > 0 .

', 0 > r > r„, x > 0.

<t - t„)k

,,, c2 o 6 i 3 * , £ V " ^

2 2 x s in (rniR) exp (-n TT Fo).

See Pig . 10.5 for condition t . = t .



10.2 .2 9 , p. 243 Sphere with l inear internal heating d is tr ibut ion .

0.

10 .2 .2 .1 74, p. 366

t = t n , 0 < r < r Q , T t = V c = v T " °-_ i i i _ „ i i i

(t - t )k 2 3

-—-f- = i j (1 - 2R2

+ R 3 > *0 C 0

9 ' " ( r 0 - r ) / r 0 < r < r Q , T > 0.

^ n=0 < 2 n + 1 ) 5

.2_2 „ T 0' x exp [-(2n + 1) TT2 Fo] s in [(2n + 1) ITR]

Case 10.2 .2 with

b t

, , T b / 2

X j> U jr 1 / £ f • • •

t i # 0 < r < r Q , x = 0.

V r = c 0 ' T > °'

00 fc " t i V 1 r . / 2 n - l - _ R \ . / 2 n - l + _ R , j l

i-^ r- E- = 2 R [ e c f c \ 2VFO J " e c f o V 2VF5 />J n=l

Po Po | i - r ( , • * ) * • • 1 + (b/2) 00

V 1 Tjb+2 * /2n - 1 - R\ .b+2 _ /2n - 1 + R\"ll

n=l

Po = q ' - ' T b / 2 r 2 / b ( t - t i )



10.2.3 9, p. 243- Case 10.2.2, with .. » .. ... ... (2 2\ , 2 ( t ~ V * 1 „ D 2 W , - D2, 12 V (-1)""1

0 < r < r„, T > 0. q0 0 n=l '0' 2 2 x exp(-n TT Fo) sin (mtR)

10.2.4 9, p. 244 Case 10.2.2, with

o l CO

( t - V k 1 r 2 T q'V'r- j — = — [l - expf-ir Po)] sin (TtR) q " ' "-^—^ s in ( i i r / r 0 ) , q ^ ' r * TTZR

0 < r < r Q , T > 0.

Case 10 .2 .2 with (t - t )k , , _ , , , , q . ' . = q M - e x p [ b ( r - r 0 ) I . _ - £ _ . ± j ( i _ | _ ( i - - ) exp(BR - B) - ff (± - l)

00

2 V s in (MTR) , 2 2 _ , " H 1 , 2 2 , 1 e x p ( " n * P o )

n = 1 n(n 7T + B )

e" B

x [ ( - l ) n ( 2 B - n 2 ir 2 - B 2) - 2Be~ B J. B - r 0 b .

Section 10 .2 . Sol ids Bounded by Spherical Surfaces—With Internal Heating.


10.2.6 9, p . 245 Case 10 .2 .2 with

q ' " = %"*'*. (t - t„)k ~ V * _ 1_ -br [ s i n (RVPd) .1 _2_

,,lv2 ~ Pd e LR s i n ( V W " ""J „ 3 n

V c 0 7T R

£ ( - 1 ^ s in (nirR) e x p ( _ n 2 , r 2 F o ) >

n=l n<n"V - Pd)

Pd = r*b/a .

f 10.2.7 9, p . 245 Case 10.2 .2 with g 1 " = q 0 " e ' r i < E < V

0, 0 < r < r , .

(t - t Q )k g-br = R Pd • i t 2 a 1 "r q 0 E 0

I fs in ( R V W ) "I s i n (V"Pd - RVPd) ) \ [ s i n (VW) " R J s i n (VPd") f

|R cos (R^ Pd) - -^sz- s i n

Y""PcT ^VPdj ]

+ 2_ V L^ L ( _ i ) n + i - s in (mtR.)

n=l n(n"7T" - Pd)

V] 2 2 R cos (mrRn) | sin (mrR) exp (-n"ir' Fo)

Pd = r*b/a .



10.2.8 9, p. 245 Case 10.2.2 with ft - t )k g ' " = f ( r ) , 0 < r < r n , __0— _ _^_ y __ ey^[_n-1!- P o ) s i n { m R ) . R i f ( R i ) s i n (rniR')dR'

' t > °- r 0 * R n=l " -0

r 1

y ~ exp( -n 2 i r 2 Po) s i n (rniR) / R ' f (R 1 ) n=l n 0

| / R ' 2 £ ( R ' ) dR' + J R'f (R') dR' - | R , 2 f ( R ' ) dR'

io.z.9 9, p. 245 Sphere with steady internal ,...,, <=° •,-,*.. /,2 „ \ S 90 heating and convection ( t ~ V k 1 M „2 ,.,„,., . at V 5 1 n ( X n R ) e x p ("Xn F o ) .J, boundary. .,, 2 = 6 ( 1 ~ R + 2 / B l ) * 2 ft" Z .2/.2 ^ „ ;2 n J\ _,_ , u t = t £, 0 < r < r Q, T = 0

q*" = <^'\ 0 < r < r Q, X n cot C y = 1 - Bi T > 0

q'"r 2 6 R n t l x J ( x ; + B i a - B i ) . l n (XB)

See Fig. 10.10 h.t f



10.2.9.1 74, p. 366 Case 8.2.9 with q,.. „ q^"e _ b r r.

fc- fci . Po t„ - t, " ^ ~ pa

x exp(-Pd Fo)

r Bi sin (V"Pd~R) "1 L R [(Bi - 1) sin fv Pd)+ V Pd cos ("V Vd)Tj

A sin (A R) exp (-X Po) ti_ __ n \ n /

n=l n RX

2 [sin (X ) - X cos (X j ] i- n n n -1

\ ~ X - sin (A) "cos (X ) ' t a n {K] = 1 -"B! n n n

10.2.10 9, p. 246 Sphere with parabolic initial o I to

(t - tn)k temp and convection boundary. i r " C 1 , K _ 1 n „2 |"l_ _1 1 T 1 _ 2 Bi

0 < r < r , T = 0 (temp dist. for steady state with const, surface temp = t.). q'*' = 1^", 0 < r < r Q, x > 0.

^ sin (X R) exp I-A Fol

£. (Xn + Bi2 " B i) sin ( V ]

q' " r 2

X n cot (Xn) = 1 - Bi, Po = k ( t ° _ Ij



10.2.11 9, p. 350 Sphere of infinite con- . J f c

ductivity enclosed by a ( ~ o' _ 1 / 1 _ 1_\ _ 4K shell of finite ,,, 2 3 \R ~ R Q/ ~ R conductivity qn rn

i w Ol

t = t Q , 0 < r < r 0 , T = 0.

q . . . = q ^ . i , o < r < r Q ,

T > 0.

q ' " = 0, ct < r < r Q ,

T > 0.

" s in [X n (R Q - 1)] s in [X n (R Q - R)] exp (-X^ Fo)

n-1 X n | 2 K ( R 0 - 1 ) X n + 4 X n s i " 2 CXn<R0 " 1 } ] " K s i n [ 2 V R 0 " ^) '

KXn cos [X n<R 0 - 1)] = (X 2 - K) s in [X n(RQ - 1)] , 1 < R < R ( ) .

K = 3 P 2 C 2 / P 1 C 1 ' R = r / r i ' P ° = V A i

10 .2 .12 19, pp. 3-29 Sphere of i n f i n i t e oon- See case 8 .2 .11 , s e t L = 3 , radius duct iv i ty , variable spec i f i c heat, convective boundary, and steady heating.



10.2.13 74, p. 370 Sphere with a steady internal heating and steady surface flux.

(t - t,)k Po + $

ft - f t Po 3 Fo - - (3 - 5R2) - 2 ^ n=l

sin (X R) exp (-A2 Fo) n \ n ' 10

tana ) = X n n n

O sin (X ) n n

10.2.13.1 74, Case 10.2.13 with p. 370 q'" = q'"(l - r/r„). (t - t,)k

a' •'r~ H0 0 P-F"2 2 - (2 + X 2) cos (X J sin (X R) v n/ n n

n=l X R sin (X ) n n x [l - exp(-X2 Fo)] + ft

ft given in case 10.2.13.



10.2.13.2 74, Case 10.2.13 with P. 370 g l I 1 = q , , , ( 1 _ R 2 ) , <fc ~ V k 2 Po sin (XnR) [l - exp (-X2, Fo)]

a" "r q0 r0 + *

n=l X R sin (X ) ra m $ given in case 10.2.13.

10.2.13.3 74, Case 10.2.13 with p. 370 q'" = q'" exp(-bR).

o

10.2.13.4 74, Case 10.2.13 with p. 370 q'" = q^"(l + bT).

( t " fci)k 3 Fo [2 - exp(-b) (b2 + 2b + 2)] q'"r 2 40 0

" 2Xn - exp(-b) (2 + b 2 + 2b + X 2 ) sin (XJ sin (XnR) + 2 b 2. 72 . 2 ,. . A 2 ^ .2\2 _ ~ X sin (X_) (X_ + b ) R n=l n n \ n /

x [l - exp(-X2 Fo) ] + *

$ given in case 10.2.13.

(t - t.)k , x— = Fo (1 + ~ Pd Fo) + $

• • • 2 V 2 ' a M l r q0 0 $ given in case 10.2.13.

10.2.13.5 74, p. 371

Case 10.2.13 with qiii _ q > i > e x p ( - b T ) ,

(t - t.)k , -. -1— = r [l - exp(-Pd Fo)J + *

3"'r 2 d0 0

Pd




10.2.13.6 74, p. 371

Case 10.2.13 with q » " = q'< 'cos(br).

(t - t.)k . ± — = |^ sin (Pd Po) + *

given in case 10.2.13.

10.2.13.7 74, Case 10.2.13 with p. 371 q l I , = q^.tn:".

q... r2

o l IO 03




10.2.14 74, p. 390 Sphere with pulse heating on a spherical surface and a convection boundary, t = t f, 0 < r < r Q, T = 0. Instantaneous pulse of strength Q occurs at r = t x , T = 0 .

, 1 / 2 ( t - t f ) p c r * x ^ [(Bi - l ) 2 + X 2 ] A R s i n ( X ^ ) s i n {XnR)

Q | ~ 4rT Za Bi R. R i 2

* e x p ("Xn F ° ) * n=l

An = ™ n+1

2 Bi [ X 2 + (Bi - l ) 2 ] 2-11/2

2 2 X + Bi - Bi n

tan(X ) n For B i -*• °°:

»' rî i ' R = r / r o

(t - t )pcr i v 1 2 2 5 = 2? Z R~R S i n ( n l I H l ) s i n ( n 1 T R ) e x P ( ~ n T F ° )

For r = 0 n=l

3 » , 2 ( t - t f ) p c r j j i \ - X n S i n ( X n R ) e x p ( n F o )

QI 2TTR ZJ ~ - s i n (X ) c o s ( X ) l , n n n n=l

Mean temp:

<*,, - y p « 0 , ^ i- , , .V2 RTBIS D 8 1 " 1 ' 2 * ^ V n s i n <W

B =

4TTR

6 Bi

n=l

x exp « ">)

n X 2 (X 2

+ B i 2 - B i ) n v n /



Section 11.1. Change of Phase—Plane Interface.


11.1.1 9, p. 285 Initial surface maintained at Solid temp: constant temp: freezing liquid.. t = t Q, x = 0, T > 0. t = t , x > 0, T = t = melting temp, m

0.

i I w

(solid) (liquid)

* • " fc0 = erf K) l/erf (A), 0 < x < w,

Liquid temp: t„ - t ' f t ~ c ] / * \ , t _ t

= erf c/FOj^J/erf c (X M a s / < \ ) , x > w .

XYVT exp(-X2) (tm ~ V H *** (~X V a f t ) c s ( t m " V ' e r f ( A ) ( t m " t 0 ) k s e r f o ( X )

w = 2Xya T s

11.1.2 9, p. 287 Solid region maintained at melting temp: freezing liquid, t = t m, x < w, T > 0. t = t x, X > 0, T » 0. t, < t , t 1 m m

melting temp.

Liquid temps

= \YTetCcfeo^/expl-X ) , x > w .

w = 2XVa„T

X exp(X 2) erfc(X) = ( t m - t ^ C^/VVTT



11.1.3 9, p. 287 Melting of a solid in contact Solid temp: with another solid. t = t , x > 0, T = 0. t ' h > V x = °' T > °-t = melting temp.

fcB- fc0 t m " fc0

erfc (Fo" ) / e r f c (XYa. /a ) , x > w. xs ** s

Liquid temp:

''SL, \ - \ - \ | t e . a B . k K

(solid) I (liquid) (solid)

fc0 ~ fci * — - = erf (Fo „) /erf (X) , 0 < x < w . t - t , xJl m

e t f ( X > + k £ V 5 * ( t i " S* e r f c ( X V V 5 s " ) C * ( t l " t m >



11.1.4 9, p. 288 Liquid freezing on solid-properties of solidified liquid different from original solid.

0.

0. t = t Q f x < 0, T t = t . , x > 0, T t = melting temp.

Orig ina l s o l i d :

V A fcsl - fcQ t - t „ VA"+ K erf (X) m o

[ l + erf (Fo* g l ) ] , x < 0 .

Solidified liquid:

fca2- fc0 ^ * K e r f ( F < 4 2 > t - t„ ~ VA + K er f (A) m u

, 0 < x < w .

V ^ r k s i ' W k s 2 (solid) (solid)

V V k , (liquid)

Liquid:

fc

m-fci e r f c ( F o ^ J / e r f c (XV o t g 2 / < y , x > w .

XYVJ _ K exp(-X 2) , " l ^ a * * , ~ t l ) M I P ( - V & 2 / C V 3 s 2 ( t m " V V I + K erf (X) k

s 2 V T A ( t m - t Q ) e r f c (XVa s 2 /a A )

w = 2X V S ^ r , K = k A „ 2 , A - a g l / a s 2



11.1.5 9, p. 289 Melting of a solid in contact w = 2AV"<V^ AT

with another solid—properties of liquid different than original solid. t = t, > t , x < 0, T = 0. 1 m t = tQ, x > 0, T = 0.

i

XYVTT k^VHêxp {-A ) cA ( tl " V " h ^ l + " a l ^ e t f ( X )

, k B 2 ^ t ( t 0 - fcm e x P ( - X V a s 2 ) k * ^ ( t i " V e r f c ( X > ' 1 S 7 5

S ? See case.11.1.4 for temperatures t = melting temperature ,

11.1.6 9, p. 289 Solid melts over temp range For case 11.1.1, change equation for A to

<2 o f fcml t o W V fcm2' * > °' T - °'

e x p [ ( a s - y A / a J e r f c t W a ^ ( t ^ - t m l )k & VcT erf (A) l tml ' V k s * \

For case 11.1.4, change equation for A to :

K e * p l > s 2 - y X 2 / g J erfc ( X V ^ T y ( t m 2 - t m l ) k f c V a ^ V7T+ K erf (A) <V " V **T

Use c« for c„ in the above equations and solutions given in cases li.l.i and 11.1.4.



11.1.7 9, p. 290 Change of volume during solidification, t = tQ, x = 0, T > 0. t = tj, x > 0, t = 0.

t = melting temperature.

Sa-h * X

A ( P S - P I [ / 5 °XJl + P r*$-4b®->-

exp (-X2) ( t l - V * J ^ e x p ( - ^ y p f r l ) AY V¥ erf (X) " <tm - t 0 ) k a V a ^ e r f c < X p g V T r / p ^ ) " e ^ ^ - t Q )

w = 2XY~a~T" s

r t 0

| w

(solid) (liquid)

11.1.8 9, p . 291 Constant heat flux at original solid-liquid boundary. t - t , x > 0, T = 0.

m SJ = q 0 ' X = ?' T > °* t = solid temp, t < t . s , s m

s

a p ; s = * - : V + 2 F o x > + ^ T ^ + 1 2 P o x + " ?<Q S^S" 2x

2i 1 2 x 4

, 0 < x < w .

W = QfT - | Q 3 T 2 + | Q 5 T 3

Q = V W ' F ox " asX/x ' T = V t = melting temp, m



11 .1 .9 9, p. 292 Convection boundary at original so l id - l iqu id boundary. t = t v x > 0, T = 0, t x > t m . t = melting temp, m

•x

w

t - t P BiJ; F Bi* t - t j . X 2 X 31 X ~m

, 0 < x < w .

v\, te

)

P 2 B i 3

2

w = F Bi F o x J - ^ (1 + F) FOx +

F = k s ( t m - t f ) / a g p y , P o x = cyr/x* , B i x = hx/k s

(solid) (liquid)



11.1.10 55 19. p. 3-87

Solidification on plane semi-infinite solid with convection coefficient between solid-solid and liquid-solid, t = t Q , at < 0, X = 0.

c = 0. t = melting temperature.

h s pyk

V i N 2

i X N • - 1 pyk

V i N 2 H l K B i + 1) - 1

N = V f co " t > m Bi -

t ) ' B l • a

h w s

" k s See P i g . 11 .1

(Solid) t ,h .

(Solid)

P. c

(Liquid)

Vo

11.1.11 56 19, p. 3-88

Case 11.1.10 with solidified layer of finite heat capacity and heat flow at x = 0 given as q = bt , where t is temp w w at x <" 0.

See Figs. 11.2a and b



11.1.12 19, 77 p . 3-88

Case 11.1.10 with heat flow See P igs . 11.3a and b 4

at x = 0 given as q = b t w <

11.1.13 19, p. 3-88

H I

Decomposition of a semi-infinite solid, t = t Q, x >. 0, T = 0.

t , x w

0, T > 0. t. = decomposition temp, d I = bVT.

Char

© Virgin

h-f •

©

Char layer:

t - t erf (FO* )

"w "d

Virgin layer

t - t , erfc Q Ka)

S, ' fcd ~ e r f c ( b / 2 V S 2 )

, x >_ £ „

exp erf

(-b 2/"«l) /p^PÂd- ^ V ^ K ^ ) V^Px/c^! Yb (b/2VaJ) " V P ^ ! * ! \ t w ~ t d / / erfc (b/2 /cSp " 2 ( t w - t ( J )

See F igs . 11.4a and b



11.1.14 57 19, p. 3-93

Ablation of a semi-infinite See Figs. 11.5a, b and c solid by convection heating, t = t Q, x > 0, T = 0.

t, = decomposition temp.

fcd" fc0 t • •_ - = 1 - exp [(h/k) ore ] erfc [(h/k) VoTj]

t , = Time for free surface to reach temp t . d d

l to l

i

(Char) (Solid)

Steady state ablation velocity and temp distribution:

h< f ca - V "ss ~ p §<t d - t Q) + a]

fc- fco r v s s , , . i ' d 0

H = heat of ablation



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

Ablation velocity!

2

'ss _ p[c(td - t Q) + H]

Section 11.1. Change of phase—Plane interface.


Section 11.1. Change of Phase—Plane interface.


V to

Section 11.2. Change of Phase—Nonplanar Interface.


11.2.1 9, p. 29 Liquid freezing on a cylinder. t = t , 0 < r < R . m t = t., r + », t < t . 1 1 m t = melting temp.

.J® , (Solid) (Liquid)

Liquid temp:

fc»-fci a El (-Fo2) /E i <-X ) , r > r s .

X 2 exp(X 2) Ei (-X 2) + c A ( t m - t ^ / Y = 0 . r g - 2X VSJ7

11.2.2 9 , p. 295 Liquid freezing on a sphere. t = t , 0 < r < R . m' t « t r r + - , t l < t m .

(Solid) (Liquid)

Liquid temp:

2X t - t m 1 exp(-X 2 ) - xVi" erfc (X) (2 Fo

exp «l) rJl

" J e r f c r > r ("-)} • X 2 exp(X 2) | exp(-X 2 ) - Xu erfc (X)j = ~ ( t m - tj ,

r = 2X V5£T

Section 11.2. change of Phase—Nonplanar Interface.


11.2.3 9, p. 296

Steady line heat sink t = t , r > 0, T = 0, t , > t . sinK rate = q„. 1 m u t = melting temp.

Sol id tempt ( t s - V k s 4lt Ei (~2) Ei(-X^), 0 < r < r .

r = 2X V a t a s

Liquid temp: ( t c - t . )

> r s " m x

a x p ( - X 2 ) / 4 i r * ~ ^ - t m , e x p ( - X 2 o l g / a A ) / E 1 ( - X 2 a s / a J x V Y / q 0 . •U

11.2.4 9, p. 296 Freezing on steady temp cyl inder. * - V r = V T > °' fc0 < V * " V r > V T = °-t = melting temp, m

Solid temp:

f-^~T - a n ( r / r n ) / £ n ( r s / r 0 ) r r Q < r < r s

IR 0

2 r s ^ s ' V " r 2 + a 2 = 4 k s ( t m - tQ)T/YP

(Solid) (Liquid)

Section 11 .2 . Change of Phase—Nonplanar Interface.


11 .2 .5 19, Freezing inside a tube, 2 2

p. 3-87 tube resistance neglected, ( m " a' _ Bi ("A . l \ .. 0 2 , _2 0 AVI „ , , convection cooling at r = r Q . p^k " 2 LUi + 2 ) ( 1 " R ' " R * n W J ' ~ ' t = t m , r = r 1 . Bi = h 8 A , R = r

t/*0

t = melting temp.

Liquid

Solid

11.2.6 19, Case 11.2.5 with freezing on 2 p. 3-87 outside of tube and convec- < n, ~ a 1 « S i T / l | \ 2 _ + R 2 £ ]

t ion cooling on ins ide . pyk 2 |_VBi 2 / l J _

11.2.7 19, Case 11.2.5 with freezing 2 _ p. 3-88 inside a sphere. m^ a = B|_ gj_ _ ^ ( 1 _ R 3 ) + | ( 1 _ R 2 ) " J f R < x

S^^f^s?-'''?,':--^?'•



11.2.8 19, Case 11.2.6 with freezing 2 p. 3-88 on outside of sphere. ' m " V _ Bi 2 f/l_ . ,\ ,_3 ,, 3_ , p2 "1 ^k r L\Bi + 1) ( B " 1 } " 2 ( R " 1 } J ' R *

i

Section 11.2. Change of Phase—Nonplanar interface.


i



Section 12.1. Traveling Boundaries.


12.1.1 9, p. 389 Semi-infinite solid with t - t Q . linear! initial temp, time - — — 7 - = D(l - U) + G + erfc [x(l - U)] dependent surface temp 1 ~ 0 and internal heating t = t "+ dx, x > 0, T = 0. t » t ' + tit,-* = 0, T > 0. Solid increasing at velocity u.

10 1 H

+ i exp(40X 2) erfc [ x ( l + 0)] + f ( § + D + § )

x 1 (1 + U) exp(4UX2) erfc [ x ( l + U)] - (1 - U) erfc [x( l - U)]j .

B - b T / ( t x - t 0 ) , D = d x / ( t 2 - t f l ) , G = q ' - 'OT/M^ - t Q )

H = ux/x, X = x/2Vat

12.1.2 9, p. 389 Semi-infinite solid with convection boundary, t = tQ, x > 0, T = 0 . Solid increasing at velocity u. +

t _ ° n = I ( e r f c [X( l - Oj] + F o

F ° . B f p exp(U/Fo) erfc [ x { l + 0)]

2 Fo Bi -_0 e x p ^ B i ( 1 _ 0 + F o B i j ] e r f c £ X ( 2 p 0 Bi + 1 - U ) ] . Po Bi

Bi = hx/k, Po = ax/x , X = x/2 Var, u * UT/X

Section 12.1. Trav îng Boundaries.


12.1.3 9, p. 389 Semi-infinite solid with steady periodic temp, t = (tQ - tm)cos(ayr + 9), x = 0, x > 0. Solid increasing at velocity u.

t - t f- = exp[ux/2a - xVE cos (<l>/2)] cos [tox - xVb s in (<|>/2) + e]

fc0 ~ m

0— + j& = b exp(i<t.) 4a

to I

12.1.4 9, p. 389 Cylindrical boundary traveling in an infinite solid, t = t £, r > r Q, T = 0. t = t Q, r = r Q l T > 0. Cylinder traveling at ve loc i ty u.

t - t„ t - „ n _ e I <0)K <UR) cos (n9) 0 _ X n n n

fc0 " fci = ^ V°> n=u

n=0

£h + | V exp(-U 2 P o ) e n cos (n6)I n(U)

n=

" exp(-Fo X2) [j (XR)Y (X) - Y (XR)J (X)l XdX L n ii ij n J

(X2 + u 2 ) [a 2(X) + YJ(X>]

e = 1, 6 = 2 i f n > 1 , Fo = orr/rj;, R = r / r n

U = ur / 2 a



12.1.5 9, p. 391 Thin rod with fixed end temps and convection cooling, t = t , 0 < x < S., x = 0. t = t , x = £, T > 0. t = t , x = 0, T > 0. The end boundaries traveling at velocity u. p = rod per imete r . A. = rod sec t ion a rea .

s inh L \ U 2 X 2 + Bi X 2 J exp(0X - U) sinh

GO

+ 2 l y e x p ( U X - D ) S ( - X ' n n

2

5 i n <"«)

n=l 2 2

B i + U + n 71

x exp [-Fo (Bi + a 2 + n 2Tt 2)]

Bi = hpS,2/kA, Fo = on /A 2 , 0 = ufc/2a, x = x/S.

to I

t 0 .h

A il

Solution



i

1.0 , , p^B = 0-j r

FIG. 1.1. Temperature distribution and mean temperature in a porous plate (case 1.1.4, source: Ref. 2, p. 221, Fig. 9.2).

0 0.2 0.4 0.6 0.3 1.0 x/B

FIG. 1 .2 . Temperature d is tr ibut ion in an i n f i n i t e plate with internal heating and temperature dependent conductivity (case 1 . 2 . 3 , source: Ref. 3 , p . 132, Fig. 3 . 2 3 ) . F_x

o

FIG. 1.3. Teit,perature distr ibution in a heat generating porous p late (case 1 .2 .6 , source: Raf. 2, p. 223, Fig. 9 . 3 ) .

,p 0.60

0.45 -0.40

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 e l e c t r i c a l c o i l of rectangular cross sect ion (case 1 .2 .8 , source: Ref. 1, p . 180, Fig . 1 0 . 5 ) .

F-2

h FIG. 1.5. First-term approximation to the maximum temperature in a solid rectangular rod with internal heating, ^ = VqJjT^TSk 2b, ty2 ~ "Vqo /Bk 2a (case 1.2.10, source: Ref. 2, p. 198, Fig. 8.16).

F-3

b/a = 0.2 b/a = 0.4 b/a = 0.6

FIG. 1.6a. Maximum temperature variations on the cooled surface of a f l a t p la te having equally spaced adiabatic and constant temperature s t r i p s on the opposite surface (case 1 .1 .30 , source: Ref. 27) .

F-4

PIG. 1.6b. Heat flux through slabs held at a uniform temperature on one surface and having equally spaced constant temperature strips on the other (case 1.1.30, source: Ref. 88).

F-5

102

101

b/a = 0.5 b/a = 0.7 b/a = 0.9 s

w/a = 0.05

a

FIG. 1.7 . Maximum temperature variation of the cooled surface of a f l a t plate having alternating adiabatic and constant heat-f lux str ips on the opposite surface (case 1 .1 .31 , source: Ref. 28).

F-6

10 3 E

10'' v//////////////////m^/m////////M

o

1 — i — i i i 1111 r—i—i i i 111

Insulating spot-v i >„,H b

mwL.

1 1 I I I I I I 1 1 1 I I I I I I 1 3

Insulated surface

hw/k PIG. 1.8. Temperatures at the spot center of a spot-insulated plate having a uniform internal heat source (case 1.2.13, source: Ref. 29).

10d

o

" I—I—I I I I 111 1—I—I I I I 111 1—I—I I I I I I I 1—I—I I I 111,

Insulating spot-v

^ 0 ^ H

mm.

J i ' ' ' ' "

10d

hw/k FIG. 1.9. Temperatures on the cooled surface and at the spot center of a spot-insulated plate having a constant temperature heat source on one face (case 1.1.32, source: Ref. 29).

F-7

0.4 0.6 0.8 (y/t)/(1/Bi+1)

FIG. 1.10. Hotspot temperatures along plate/rib centerline (Bi = hl/k) (case 1.1.39, source: Ref. 19, p. 3-126, Pig. 77).

10 9 8

1 1 1 Adiabatic wall—v.

0 0.5 1.0 1.5 2.0 2.5 3.0 Ratio b/a

0 0.5 1.0 1.5 2.0 2.5 3.0 Ratio b/a

FIG. 1.11. Conductive shape factors for a rectangular section containing constant temperature tube (case 1.1.44, source: Ref. 87).

F-8

1.6

_- 1.0

r o / r i

FIG. 2 . 1 . Heat loss from i n s u l a t e d tubes (case 2 . 1 . 3 , source : Ref. 3 , p . 123, F i g . 3 .18) .

F-9

5 ! 1 ' " I l \ '

1

4 — l \ 12:25 1 \

•— =^^^ tm 1 \ -

p 3 — 1 \ + 1 \

£ - V b T 0 = 2.00 1 \ 1 \ -

+ 1)

/

2 1 \ S. i

. s

1

" 1.75 1 ^ V

\ — 1 1.50

1 ^ V \ —

0s.Y 0 1 — I 0.00— >

+ o

+ E

0.2 0.4 0.6 • 0.8 1.0 R

FIG. 2.2. Temperature distribution and mean temperature in a cylinder with temperature dependent heat source (case 2.2.4, source: Ref. 2, p. 189, Fig. 8.11).

100

FIG. 2.3. Temperature distribution in an infinite plate with a cylindrical heat source, B =V(hx + h 2 ) / lw r 0 , t 0 = t at r 0 , ^ = t at t = » (case 2.2.10, source: Ref. 2, p. 175, Fig. 8.2).

F-10

0.1 -

0.01 -

0.001

;o 0.0001 °" 0.01

0.001

0.01

0.001 0.05 0.4 0.5

FIG. 2.4. The inside surface temperature of an infinite tube with temperature dependent conductivity and heating {case 2.2.17, source: Ref. 16, Fig. 1). Equation (5) given in case solution.

F-ll

CM V)

2

s/rn

FIG. 2 .5 . Maximum (hot spot) temperatures in the cross sec t ion of a heat-generat ing so l id (cases 2.2.18 and 2 .2 .19 , source: Ref. 64, F ig . 3).

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1/0 = b/ro

FIG. 2.6. Shape factor for a cylinder with two longitudinal holes (case 2 .1 .44 , source: Ref. 73, Fig. 6 ) .

E-12

100 o(e, + £>)rft£/k5

FIG. 3.1. Steady temperature of thin, nonrotating spherical shell in uniform radiation field (case 3.1.10, source: Ref. 19, pp. 3-112, Fig. 68).

F-13

12

10 -

I I 6

4 -

2 -

I 1 I 1 B = 10 / l

I

- / 5S\ V -

/ / 2 X

/ / S 1 V

= - 5 ^ 1 I I I

" • = ;

= - 5 ^ 1 I I I = - 5 ^ 1 I I I I -2 -1

x/b

FIG. 4 .1 . Surface temperature of a semi-infinite sol id with heating on the surface over width 2b, which moves at veloci ty u (case 4 .1 .8 , source: Ref. 9, p . 270, Fig. 34).

|-« Heat source *»-| | width |

3 -

2 -

1 -

-3

' 1 > 1 1 1 1 - i — i

-Is—^"0^ = 0

/ / H = O.lXX L = 1

-

--~~~^\s' y^. H = i.o ^

-

1 1

Leading edge — ^ i i , i 7"---VI ^ ^ t-

-2 x/fi

FIG. 4.2. Surface temperature of a convectively cooled semi-infinite solid with a traveling strip heat source (case 4.1.9, source: Ref. 30).

F-14

-0.10 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0- rad

PIG. 4 .3a. Surface temperature of an i n f i n i t e cylinder with a r o t a t i n g surface band source, $ = 0.01 (case 4 .1 .10 , source : Ref. 30).

1 1 T T T

J I I I l VZX I I -0.10-0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08

0 - rad

PIG. 4.3b. (Same conditions as for P ig . 4.3a),

F-15

OIR.tt (4/Pe) 2.0 - I -

Nu = 2.0

Pe = 20.0

— — Pe = 2.0

g(R,f) (4/Pe)

Nu = 0.2

Pe = 20.0

— Pe = 2.0

J L -1.5

(b)

-1.0

FIG. 4.4. Temperature distribution in a cylinder with a ring heat source (case 4.1.11, source: Ref. 34), 9 = (t - t f)2irr 0k/Q 0.

9(R,f) Nu = 0.2 R 0=1.25

- Pe = 20.0 - Pe = 2.0

9(R.f)

PIG. 4.5. Temperature distributions in a hollow cylinder with an inside ring heat source (case 4.1.12, source: Ref. 34). 9 defined in Pig. 4.4.

0.9 -

0.8

0.2

0.1 —

2.0 3.0 Hv/h/kYb

4.0 5.0

FIG. 5.1. Perfocnance of pin fins (cases 5.1.2, 5.1.14, and 5.1.16 through 5.1.18, source: Ref. 8 and Ref. 7, p. 55, Fig. 3.14.

F-18

0.2 -

^ _ $ \ |n = 1/2;y = ;u. UKVWkVb

[n=1/3:y = y b (x /« 1

4 '2/3l ub' 0 =

3 u b ' - 1 / 3 ( u b ) jn = 0;y = yb(x/K);ub =2Es/h/kyb _ \ 2 M u b> , u b l 0 t u h) , . |n = -1;y = v b ( x / t ) 3 / 2 , u b =4t^hlkyb

I 4 '2<ub' 10=

"b ' l ( U b l

n = + „ ; y = y b (x/e) 2 ;u b =Cv/h/kVb

1 + V l - 4 u b

1.0 2.0 3.0 4.0 5.0

Sv/fiTEVb"

FIG. 5 .2 . Performance of straight fins (cases 5 . 1 . 4 , 5 .1 .5 , 5 .1 .7 , 5 . 1 .8 , source: Ref. 8 and Ref. 7, p. 56, Fig. 3.15).

F-19

1.0

0.9

0.8

0.7

0.6

<t> 0.5

0.4

0.3

0.2

0.1

1.0

I I

Vb 0 = l 1(u e)/K 1(u e)

t\/h/kyb

ue = ub(xe/xb)

,x f /x h = 1.0

2.0 3.0 4.0 5.0

B\/h7kVb

FIG. 5.3. Performance of circumferential fins of rectangular cross section (case 5.1.10, source: Ref. 8 and Ref. 7, p. 57, Fig. 3.16).

F-20

FIG. 5 .4 . Curve for calculating dimensions of circular fin of rectangular profi le requiring le&st material (case 5.1.10, source: Ref. 1, p. 234, Fig. 11 .11) . a = B/irhx£(t0 - t f ) .

F-21

I I

_ 4 [ i -2/3 ( u b) + P'2/3("b)|

1-(u e /u b ) 4 / 3 ] | | l /3K) + ^l-1/3Kj

^=- |-2/3("e)/'2/3(«e) 28Vh7i^T

I 1.0 2.0 3.0

£Vh7k^r 4.0 5.0

FIG. 5 .5 . Performance of cylindrical fins of triangular profi le (case 5.1.11, source: Ref. 8 and Ref. 7, p. 58, Fig. 3 .17) .

F-22

<re*-r 0K/2h7k^

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, Fig. 2.22). r| = (2/W)s.

F-23

<re*-r0)V2h7k6;

PIG. 5.7. Efficiency of an inf in i te f in heated by equilateral triangular arrayed round rods (case 5.1.20, source: Ref. 11 and Ref. 10, p. 136, Pig. 2.23). r* = ( Z W T T ) 1 / ^ .

P-24

7 IO

FIG. 5.8. Fin parameter as a function of Z for a straight fin radiating to free space (case 5.1.24, source: Ref. 10, p. 208, Fig. 4.3).

Z = T 0/T e

FIG. 5.9. Fin efficiency for a straight fin radiation to free space (case 5.1.24, source: Ref. 10, p. 209, Fig. 4.4).

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Profile number £(!<.,Tjj/2kb)1/2

FIG. 5.10. Heat flow relationship for a straight fin of rectangular profile radiating to nonfree space (case 5.1.25, source: Ref. 10, p. 216, Fig. 4.8).

0.4 0.8 1.2 1.6

Profile number £ ( r ^ T j ^ k b ) 1 ' 2

FIG. 5.11. Efficiency of a straight f in of rectangular prof i le radiating to nonfree space (case 5 .1 .25 , source: Ref. 10, p. 216, Fig. 4 . 9 ) .

F-26

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Profile number K^TJ^/kbo

FIG. 5.12a. Fin efficiency for the longitudinal radiating fin of trapezoidal profile with a taper ratio of 0.75 (case 5.1.26, source: Ref. 10, p. 223, Fig. 4.12).

Profiie number I^T^/kb,,

FIG. 5.12b. Fin efficiency for the longitudinal radiating fin of trapezoidal profile with a taper ratio of 0.50 (case 5.1.26, source; Ref. 10, p. 224, Fig. 4.13).

3 F-27

Profile number K 1 T§B 2 /kb 0

FIG. 5.12c. Fin efficiency for the longitudinal radiating f in of trapezoidal prof i le with a taper ratio of 0.25 (case 5 .1 .26 , source: Ref. 10, p. 224, Fig. 4 .14) .

Profile number K 1 T^R 2 /kb 0

FIG. 5.12d. Fin eff ic iency for the longitudinal radiating f in of triangular prof i l e (case 5.1.26, source: Ref. 10, p. 225, Fig. 4.15).

F-28

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Temperature ratio Z = T 0 /T e

PIG. 5.13. Profile number of constant-temperature-gradient longitudinal radiating f in as a function of base-to-t ip temperature ratio (case 5 .1 .28 , source: Ref. 10, p. 233, Fig. 4 .18) .

F-29

0.4 0.8 1.2

Profile number S 2 K 1 T^/kb

FIG. 5.14. Efficiency of constant-temperature-gradient longitudinal radiating fin as a function of prof i le number (case 5 .1 .28 , source: Ref. 10, 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 0.80 Radius ratio p = r n/r e

FIG. 5.15a. Radiation fin efficiency of radial fin of rectangular profile. A

Taper ra t io , X = 1.00; environmental factor, Kj^râ = 0 , 0 ° * c a s e 5 « 1 - 2 9 , source: Ref. 10, p. 250, Fig. 4.24).

F-30

0.25 0.35 0.45 0.55 0.65

Radius ratio p = r 0/r e

0.75

FIG. 5.15b. Radiation f in eff ic iency of radia l f in of rectangular p r o f i l e . Taper r a t i o , X = 1.00; environmental fac tor , K J / K J T . = 0.20 (case 5 .1 .29 , source: Ref. 10, p . 251, F ig . 4 .25) .

I

0.25 0.35 0.45 0.55 0.65

Radius ratio p - rQ/r e

0.75

FIG. 5.15c. Radiation f in eff ic iency of radia l f i n of rectangular p r o f i l e . Taper r a t i o , X = 1 . 0 0 ; environmental factor , K /K.T* source: Sef. 10, p . 251, F ig . 4 .26) .

G.40 (case 5 .1 .29 ,

F-31

0.25 0.35 0.45 0.55 0.65 0.75 Radius ratio p = rQ/re

FIG. 5.15d. Radiation fin efficiency of radial fin of trapezoidal profile. Taper ratio, \ = 0.75; environmental factor, K./K.T. - 0.00 (case 5.1.29, source: Ref. 10, p. 252, Fig. 4.27).

0.25 0.35 0.45 0.55 0.65 0.75 Radius ratio p = r n/r e

FIG. 5.15e. Radiation f i n efficiency of radial f in of trapezoid*" prof i le . Taper rat io , \ = 0.75; environmental factor, K /K T = 0.20 (case 5.1.29, source: Ref. 10, p. 252, Fig. 4.28).

F-32

FIG. 5.15f. Taper rat io ,

0.25 0.35 0.45 0.55 0.65

Radius ratio, p = rQ/r e

0.75

Radiation f in eff ic iency of radial f in of trapezoidal pro f i l e . \ = 0.75; environmental factor, K-ZK-.T': = 0 . 4 0 (case 5 .1 .29,

source: Ref. 10, p. 253, Fig. 4.29) .

!

c o

0.25 0.35 0.45 0.55 0.65

Radius ratio, p = rQ/r e

0.75

FIG. 5.15g. Radiation f i n eff ic iency of radial f i n of trapezoidal p r o f i l e . Taper r a t i o , X = 0.50; environmental factor, K /K T? = 0.00 (case 5 .1 .29 , source: Ref. 10, p. 253, Fig. 4 .30) .

F-33

•D ra

DC

0.25 0.35 0.45 0.55 0.65 Radius ratio, p = rQ/r e

0.75

FIG. 5.15h. Radiation fin efficiency of radial fin of trapezoidal profile. Taper ratio, X = 0.5C; environmental factor, S A T : =0.20 (case 5.1.29, source: Ref. 10, p. 254, Fig. 4.31).

0.25 0.35 0.45 0.55 0.65

Radius ratio, p = r 0/r e

0.75

FIG. 5.15i. Radiation fin efficiency of radial fin of trapezoidal profile. Taper ratio, X = 0.50; environmental factor, K./K^T!: = 0.40 (case 5.1.29, source: Ref. 10, p. 254, Fig. 4.32).

F-34

0.25 0.35 0.45 0.55 0.65 Radius ratio, p = rQ/re

0.75

FIG. 5.15j. Radiation fin efficiency of radial fin of triangular profile. Taper ratio, X = 0.00; environmental factor, K /K 1* = 0.00 (case 5.1.29, source: Ref. 10, p. 255, Fig. 4.33).

CO

'•5

1.0

0.8

0.6 -

0.4

1 1 u ^ ^ ^ ^ ^ -

-

TÂ^Z// f/ / / / ^- Profile number

-

^ / / r , i . i , i i 0.25 0.35 0.45 0.55 0.65

Radius ratio, p = rn/re

0.75

FIG. 5,15k. Radiation f in ef f ic iency of radial f in of triangular prof i l e . Taper ra t io , X = 0.00; environmental factor, K / s i source: Ref. 10, p. 255, Fig. 4 .34) .

0.20 (case 5 .1 .29 ,

F-35

0.25 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 = rQ/r e

FIG. 5.151. Radiation f i n eff iciency of r a d i a l f in of t r iangular p r o f i l e . Taper r a t i o , X = 0.00; environmental fac tor , K /K T = 0.40 (case 5 .1 .29 , source: Ref. 10, p . 256, F ig . 4 .35) .

o •a

o

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Base parameter ml

FIG. 5.16a. Efficiency of the side of a capped cylinder f in (case 5 .1 .30, source: Ref. 10, p . 276, F ig . 5 .6 ) .

F-36

Base parameter mB

PIG. 5.16b. Efficiency of the top of a capped cylinder f in (case 5 .1 .30 , source: Ref. 10, p. 277, Fig. 5 . 7 ) .

F-37

0.4 0.8 1.2 1.6 2.0 2.4 Parameter mil

2.8 3.2 3.4 3.6

FIG. 5.17. Heat flow ratio qj/qg as a function of mb and temperature excess ratio (t! - tfj/ftg - ff) for the doubly heated rectangular fin (case 5.1.32, source: Ref. 10, p. 410, Fig. 8.10).

F-38

1.0 — 1 | 1 1 1 1 |

0.9 ^ /

i i I >

•V <6

1 I

-

X •e- 0.8 w* / > o c »N' a j

fici

0 7 <0 — H -a> _c ^ S iZ 0.6

0.5

0.4

-

| 1 1 1 1 1 I

1 0.6

0.5

0.4 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1 m x l b l

FIG. 5.18a. Efficiency of the vertical section of a straight, single Tee fin for u = v (case 5.1.32, source: Ref. 10, p. 398, Fig. 8.4).

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Parameter m x 1 b 1

FIG. 5.18b. Efficiency of the horizontal section of a straight, single Tee fin for u = v (case 5.1.32, source: Ref. 10, p. 399, Fig. 8.5).

F-39

FIG. 5.19. Effectiveness of the concave parabolic f in radiating to non-free space (case 5.1.37, source: Ref. 31).

0 0.5 -*

FIG. 5.20. Effectiveness of the convex parabolic f in radiating to non-free space (case 5 .1 .38 , source: Ref. 31).

F-40

s

100 90 80 70 60 50 40 30 20 10 0

V I I I I I I

— V, —

I I A/2R

—

\ \

^ - 1 . 5 ^ - 0 - 3 . 0

^<S-<^—5.0 -

I ^ - _ -

I I I I ,

1-)^ FIG. 5.21. Effectiveness of sheet fin with square array tubes (case 5.1.40, source: Ref. 83, p. 294, Fig. 2).

J J i 0.5-

-^No.0 5\ kO.01

1

_a2NA\ -

— 1 * v v ^

-

~ 2 -

- 5

No i ^ s ^ :

FIG. 6.1. Temperatures in an infinite region of which the region |x| < b is initially at temperature tg (case 6.1.1, source: Ref. 9, p. 55, Fig. 4a).

F-41

1

0.1N

o!2 s

^ 5 *

^

V 0.01 = Fo 1

0.5

0.5

\ 1 -

2 i i ^S§§?

-

FIG. 6.2. Temperatures in an inf ini te region of which the region r < r 0 i s i n i t i a l l y at temperature t 0 (case 6 .1 .4 , source: Ref. 9, p. 55, Fig. 4b).

1.0

I 'c 0.5

0.05s 1 3.02S Ao.01 = Fo ' 1

o.ioN

0.20 -

0.50

1.00

1_ I X S ^ 5 *

FIG. 6 .3 . Temperatures in an inf ini te region of which the region r < r 0 i s i n i t i a l l y at temperature t 0 (case 6 .1 .5 , source: Ref. 9, p. 55, Fig. 4c) .

F-42

= 0.4

FIG. 6.4. Temperatures in an in f in i t e region with steady temperature tg on the surface r = r 0 (case 6.1.18, source: Ref. 9, p. 337, Fig. 41).

log10(Fo)

FIG. 6 .5 . Temperature in a cylinder of in f in i te conductivity, i n i t i a l l y at temperature t±, i n an inf in i te medium ind i t ia l ly at t 0 . Numbers on the curves are values of 2p 0 c o /PiCi (case 6 .1 .21, source: Ref. 9, p. 342, Fig. 45).

F-43

a

I 0 . 2 -

PIG. 6 .6 . Temperature in a cylinder of in f in i te conductivity, i n i t i a l l y at temperature t u , in an in f in i t e medium. Numbers on the curves are values of 2pQCo/Plci (case 6.1.22, source: ref. 9, p. 343, Fig. 46).

Embedded sphere

10 20 100 200 1000 2000 10,000 Fo->-

PIG. 6.7. Temperature response of solid sphere 0 < r < r l f with kj_ •*• °> and initially at tx,o» embedded in an infinite solid (r > trf initially at t 2 0 (case 6.1.26, source: Ref. 19, pp. 3-64, Fig. 37).

F-44

o 0.6

0.001 0.005 0.010 0.050 0.100 0.500 1.000 5.000 10.000 Fo->

FIG. 6.7. Temperature response of an infinite solid with a constant spherical surface temperature of t 0 (case 6.1.13, source: Ref 74, p. 430, Fig. 10.9).

Fo„

0.6 (III) 0.5

0.4 •M T

4? 0.3

0.2

0.1

0 0.02 0.04 0.06 0.08(1)0.1, I "*K*. N c-

l l l \ - i i \

III V II

I I i " I s

1.0 10.0

(II) (I) 0.9 0.99

0.8 0.98

0.7 0.97

0.6

0.5 0 0.2 0.4 0.6 0.8(11)1.0 0 40 80 120 160(111)200

F°x 20 40 60 80 100 10 I I I I U.2b

0.08 0.20

0.06 - \ - 0.15

0.04 — 0.10

0.02 —

I I I I

0.05

0 0 200 400

Fo

600 800 10 00

FIG. 7.1. Temperature distribution in the semi-infinite solid having a steady surface temperature (case 7.1.1 b = 0, source: Ref. 74, p. 95, Figs. 4.4 and 4' 5>- F-45

I

0.01

FIG. 7.2a. Temperature distribution in the semi-infinite solid with convection boundary condition (case 7.1.3, source: Ref. 5, p. 82, Fig. 4.6).

F-46

1.0000

0.8875

0.3222 0.2579 0.2031 0.1573 0.1198 0.0897

0.01 0.05 0.10 0.50 1.00

BL\ /Fo7 -

5.00 10.00 50.00

FIG. 7.2b. The dimensionless excess temperature vs the number B i x ( F o x ) ^ ' ^ a n d

various Fourier numbers for semi-infinite solid with a convection boundary (case 7 .1 .3 , source: Ref. 74, p. 207, Fig. 6 .2) .

2y/lTQTQ

FIG. 7 . 3 . Temperature distribution in a semi- inf ini te region when the surface temperatur« i s harmonic (case 7 .1 .16, source: Ref. 7, p. 99, Fig. 4-19) .

F-47

o o

PIG. 7.4. Function f Qt-tf )/(tc-tf)] (case 7.1.22, source: Ref. 4, p. 84, Fig. 4.15).

FIG. 7.5. Temperature distribution across a heated strip of width 2a on the surface of a semi-infinite solid (case 7.1.24, source: Ref. 9, p. 265, Fig. 33).

F-48

1.4 1.6

FIG. 7.6. Temperature response of semi-infinite so l id (x >_ 0) with surface temperature t s suddenly increasing as power function of time, t s = t^ + b t n , (case 7 .1 .9 , source: Ref. 19, pp. 3-66, Fig. 39).

1 1 Semi-infinite solid

q 0 = const

2.4 F°:

FIG. 7 .7 . Temperature response, temperature gradient, and heating rate in a semi-infinite s o l i d after exposure to a constant surface heat flux q n

(case 7.15, source: Ref. 19, pp. 3 -81 , Fig. 49).

F-49

Semi-infinite solid

"o = < W / D

FIG. 7 .8 . Temperature response of a semi-infinite so l id after sudden exposure to a l inear ly increasing surface heat flux for a duration D, Q = D q m a x / 2 , qg = q m a x x/t) (case 7 .1 .14, source: Ref. 19, pp. 3-82, Pig. 50).

F o * 2 , x 2 / 2 v ^ r

FIG. 7 .9 . Temperature response of an inf in i te conductivity plate and semi-i n f i n i t e so l id composite with a constant heat f lux at x^ = 0 (case 7 .1 .30 , source: Ref. 19, pp. 3-84, Fig. 51).

F-50

Vy/V-0.6

0.5

-S. 0.4

0.3

^ 0.2

0.1

[

- 2 \

= q b" e * x

r** i^

—V2-

= - 1 / 4 -

1

Opaque—*\ 1 , 1 ,

0.4 0.8 1.2

Fot -*

2 0

FIG. 7.10. Temperature distributions in a semi- inf ini te sol id with exponential heating (case 7 .2 .9 , source: Ref. 19, pp. 3-85, Fig. 52).

FIG. 8 . 1 . Temperatures i n a slab with a l inear i n i t i a l temperature distribution (case 8 . 1 . 3 , source: Ref. 9 , p . 97, Fig. 10) .

F-51

1.0

0.8

0.6-

~ 0.4

O . f f P ^ I I I

0.04 ->

O10 «

^0.40 Cv^

— VV^ —

1.0

I I 0.2 0.4 0.6

X 0.8 1.0

FIG. 8.2. Temperatures in a slab with a parabolic initial temperature distribution (case 8.1.4, source: Ref. 9, p. 98, Fig. 10).

f 0.6 -

FIG. 8.3. Temperatures in the infinite plate with a constant intial temperature and steady surface temperature (case 8.1.6, source: Roc. 9, p. 101, Fig. 11).

F-52

I

1.000

0.100 -

0.010 -

0.001 0 1 2 3 4 10 20 30 50 70 90 110 130 150 300 500 700 Fo

FIG. 8.4a. Midplane temperature of an infinite plate of thickness 2S, and convectively cooled, (case 8.1.7, source: Ref. 12, p. 227 and source: Ref. 5, p. 83, Fig. 4-7).

0.0001 0.0005 0.0010 0.0050 0.0100 0.0500 0.1000 0.50001.000 Fo

FIG. 8.4b. Surface temperature of an infinite plate of thickness 21 and correction boundary (case 8.1.7.2, source: Ref. 74 p. 226, Fig. 6.6c).

0.0001 0.0005 0.0010 0.0050 0.0100 0.05000.1000 0.50001.0000 Fo

FIG. 8.4c. Surface temperature of an infinite plate of thickness 2% and convection boundary (case 8.1.7.2, source: Ref. 74, p. 226, Fig. 6.6d).

F-54

100.0

FIG. 8.4d. Temperature as a function of midplane temperature for an infinite plate of thickness 2l and convectively cooled (case 8.1.7, source: Ref. 5, p. 86, Fig. 4-10).

10"5 10~4

FIG. 8.4e. Relative heat loss from an infinite plate of thickness 21 and convectively cooled (case 8.1.7, source: Ref. 5, p. 90, Fig. 4-14 and Ref. 13).

F-55

FIG. 8.5. The function F(a) (case 8.1.12, source: Ref. 1. p. 301, Fig. 14-3).

FIG. 8.6a. Variation of amplitude of the steady oscillation of temperature in an infinite plate caused by harmonic surface temperature (case 8.1.13, source: Ref. 9, p. 106, Fig. 13).

F-56

-600

- 5 0 0 -

-400

•o - 3 0 0 -

FIG. 8.6b. Variation in phase of the steady osc i l l a t ion of temperature in an inf ini te plate caused by harmonic surface temperature (case 8 .1 .13 , source: Sef. 9, p. 107, Fig. 14).

F-57

o u. I o CJ '— o

"a i Ul CD

0.3 I I I 1 Fo=joo

/0.3 ma.2-i » 7 0 - 1 5

JS/iO.1 J///Jom

0.2 J////JQJa-J/////l*XA

ff////fl003

0.1 V///SM/J °01'

0 0.5 jfll&Zi &££>' 1.0 _

-0.1

1 I I I I

1.4

1.2

" I I I I

/ -

1.0 — -

=o 0.8 —

0.6 -

0.4 — / -

0.2

I i i

-

0.2 0.4 0.6

Fo 0.8 1.0

PIG. 8.7. Temperature distribution in an infinite plate with no heat flow at x = 0 and constant heat flux q 0 at x = S, (case 8.1.18, source: Ref. 9, p. 113, Fig. 15).

FIG. 8.8. Temperatures in an infinite plate with constant internal heating q' and surface temperature tj (case 8.2.1, source: Ref. 9, p. 131, Fig. 20).

FIG. 8.9. Quasi-steady state temperature of an infinite conductivity plate with a harmonic fluid temperature (case 8.1.49, source: Ref. 19, Fig. 17b, p. 3-31).

F-59

FIG. 8.10. Plate temperature response for prescribed heat inputs (case 8.2.15, source: Ref. 19, p. 3-35, Fig. 18).

F-60

(c#tjj/pC6)

FIG. 8 .11 . Temperature response of thin generating plate suddenly exposed to constant heat input with surface reradiation: (a) heating, (b) cooling (case 8 .2 .16 , source: Ref. 19, p. 3-36, Fig. 19) .

F-61

FIG. 8.12. Maximum temperature of thin insulated plate suddenly exposed to circular heat pulse with surface reradiations (case 8.1.50, source: Ref. 19, p. 3-38, Fig. 20).

1.00

0.04 0.08 0.12 0.16 0.20 Fo (a)

FIG. 8.13. Temperature response of an infinite plate with uniform internal heating, & = half thickness (case 8.2.6, source: Ref. 19, p. 3-43, Fig. 24),

F-62

0.1 0.2 0.3 Ka.max-V^f-V

FIG. 8.14a. Insulation weight for substructure protection to t 2 m a x in heating duration D, W^ = Pi&x < c a s e 8.1.51, sources Ref. 19 p. 5-48, Fig. 27).

F-63

* 0 . 4 -

0.5 1

Fo i (a)

M = n + (n + 1)/Bi 1 n = p 2 C 2 5 2 / p 1 C 1 5 1

1.0r

5 10 20

B^ ' h S , / ^

10 20 Fo,

(b)

FIG. 8.14b. Temperature response of thick plate (0 £ x <_ 6 1 ) convectively heated at t = 0 and in perfect contact at i t s rear face x = 6 with a thin plate (0 <. x 2 £ 6 2 ) insulated at x 2 = 6 : (a) x /5 = 0. (b) X. /6 . = 1 (case 8 .1 .51 , source: Ref. 19, p . 3-46, Fig. 26) .

F-64

50 100 50 100

50 100

0.02 Heating: t Q / t s = 1/4 x/5 = 1

I 0.5 1

Fo (dl 50 100

FIG. 8.15. Temperature response of thick plate (0 <_ x <_ 6) with insulated rear face x = 6 after sudden exposure to uniform radiative environment t g at x = 0: (a) heating, t[j/ts = 0, x/6 = 0 , (b) heating, tnAs = °' x/fi = 1' ( c> heating, t 0/t a = 1/4, x/6 = 1, (d) heating, t 0/t s = 1/4, x/6 = 1 (case 8.1.52, source; Ref. 19, pp. 3-50, Fig. 29).

50 100

t.„ = t,

0.5 1 50 100

500

500 (f) Fo

FIG. 8.15 (Cont.). Temperature response of thick plate (0 £ x £ 6) with insulated rear face x = 6 after sudden exposure to uniform radiative environment t s at x = 0: (e) heating, tg/ts = 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.

500

500

FIG. 8.15 (Cont.). Temperature response of thick plate (0 <_ x £ 6) with insulated rear face x = 6 after sudden exposure to uniform radiative environment t- at x = 0: (k) cooling, t 0/t s = 2, x/6 = 0, (1) cooling, t 0/t s = 2, x/6 = 1.

F-67

500

500

FIG. 8.15 (Cont.). Temperature response of thick plate (0 <_ x <_ 6) with insulated rear face x = 5 after sudden exposure to uniform radiative environment t s at x = 0: (i) cooling, t 0/t s = 4, x/6 = 0 , (j) cooling, t 0/t s = 4, x/S = 1.

F-68

FIG. 8.16. Temperature response of thick plate (0 £ x £ 6) with surface temperature t„ at x = 0, increasing l inearly with time and rear face x = 6 insulated or exposed to uniform convective environment tg (case 8 .1 .8 , source: Ref. 19, pp. 3-60, Fig. 33) .

FIG. 8.17. Surface temperature response of a plate of thickness 2ft exposed to a steady heat flux qg on both s ides for a time duration D (case 8 .1 .18, source: Ref. 19, pp. 3-67, Fig. 40) .

F-69

a 0

J?

50 100 (a)

50 100 Fo (b)

FIG. 8.18. Temperature response of a plate with steady heating qg at x = 9. and convection boundary at x = 0 to to (a) x/H = 1, (b) x/H = 0 (case 8.1.17, source: Ref. 19, pp. 3-69, Fig. 41).

F-70

to O &

Q D / 2 J I S 2

(a)

3 -

2 -

1 Thick plate

I I aDIlnb2—<

1 1 1

x/8 = 1 0.1—v _ — -=— — 0 . 6 ^ \ 0 - 4 - x \ îwir •2§5ô. i 0.3 - \ N % o / / 0 0 8

- 0.2-\\\i O.16-M0 ^ ^ / 0 . 0 6 -

0.14-Tyigav 'ss/]/'am

o.i2-*/2s2S3i0 / O ^ / 0 . 0 4 — Thin plate-'~~~~^4425<2&Zs ^r ^r

^55«P>0 * V > ^ ^0 .03 ^55«P>0 -^^^"C" 0.025 C ^ ^ r - o . 0 2 L * ^ —•r^ZJ—nnu

(b)

0 1 2 3 4 5 6 7 27T7-/D

FIG. 8.19. Temperature response of an infinite plate exposed to a circular heat pulse cfogjj sin 2 (ITT/D) at x = 0 and insulated at x = 6 (a) x/6 = 0 , (b) x/6 = 1 (case 8.1.54, source: Ref. 19, pp. 3-70, Fig. 42).

F-71

FIG. 8.20. Temperature response of an infinite porous plate after sudden exposure to a constant heat flux at x = 0 and cooled by steady flow through plate from x = 6 of a fluid initially at tQ (a) x/6 = 0 , (b) x/6 = 1 (case 8.1.55, source: Ref. 19, pp. 3-74, Fig. 44).

F-72

FIG. 8.21. Temperature response of an infinite plate with a constant heat flux q 0 at ^ = 0, in contact with a plate at x-± = 6± and insulated on exposed surface of second plate x 2 = 6o ( c a s e 8.1.56, source: Ref. 19, pp. 3-76, Fig. 45).

F-73

9 5 W 1 - x 1 = 6 1

1 i > i Two-layer plate (kj/k^Vo^/aij

1

= 4 / A i ' ' 1 '

- \ V s 1 / 4 ^

"^

\ XT?12' — 2 " " ^ > i

/ 4 - K , - 0 *

^ _ — 1 -— ——* 4°°~~"

1/2 __^

r.t t i l l J i i - — o — 1 .

-r.t t i l l J i i - — o — 1 . , 1 ,

0.2

0.4

- 0.6

- 0.8

0 1 2 3 4 5 W> F o ,

6 7 8 1.0

FIG. 8.21. (continued).

F-74

FIG. 8.22. Temperature response of an infinite plate exposed to a constant heat flux q " at xi = 0 and in perfect contact at xx = &i with a plate of thickness 62 insulated at x 2 = 6 2 (case 8.1.57, source: Ref. 19, pp. 3-78, Fig. 46).

F-75

T Two-layer plate

n = p 2 C 2 5 2 / p 1 C , S 1 = 0 . 1

, m a x s i n 2 ( * r / D ) . k 2 — o teqb-tO = 5 1 % , a x F ° ' / 2 ( " + D k ,

Fo\/2ir = a,D/27rS2

FIG. 8.23. Temperature response of an infinite plate exposed to a circular heat pulse at x 1 = 0 and in perfect contact with an infinite conductivity plate of thickness 6 2 (case 8.1.58, source: Ref. 19, pp. 3-78, Fig. 47).

F-76

1000

100 -

o T 10

0.1

0.01

I I I I Two-layer plate

i P 2 C 2 f i 2 /p 1 C 1 S 1 = 1

— -0,0/2*5*

W// / / 1

t 1 - ,

0 2 4 6 8 10 12

27rr/D -»

1000

100

0.01

Two-layer plate P 2 C 2 8 2 /P iC 1 6 1 = 10 a, D/2TT62

FIG. 8 .23 . (continued),

F-77

I I 43"

0.001 0.005 0.01

FIG. 8.24. Temperature response of an inf in i te plate surface having steady temperature and convection boundaries (case 8 . 1 . 2 5 . 2 , source: Ref. 74, p. 240, Fig. 6 .12) .

FIG. 8 .25. Temperature of a f l a t plate having a ramp surface temperature equal to t s = t j + bt (case 8 .1 .33 , source: Ref. 74, p. 305, Fig. 7 . 1 ) .

F-78

2Fo 1 + 2/Bi

2Fo 1 + 2/Bi

FIG. 8.26. Transient surface temperature of a slab convectively coupled to a l inearly changing environment temperature equal to t f = t^ + be (case 8 .1 .33 , source: Ref. 89) .

F-79

ha i CD

0.001

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).

FIG. 9.1b. Temperature as a function of axis temperature in an inf in i te cylinder of radius r Q (case 9 .1 .3 , source: Ref. 5, p. 87 and Ref. 12).

1.0 0.9 0.8 0.7 0.6

lO 5 0.5 0.4 0.3 0.2 0.1

0

a o o o *-5 ^ 5 5 ^ 5 5 5 -y 4< * *

FIG. 9.1c. Dimensionless heat loss Q/QQ of an infinite cylinder of radius r 0 with time (case 9.1.3, source: Ref. 5, p. 90, Fig. 4-15 and Ref. 13).

F-81

0.11 I . I i I . I ' I • M 0.0011 I I I 0 0.2 0.6 1.0 1.4 1.8 2.2 2.2 3 4 5

mBi{Foj

FIG. 9 .Id . Center temperatures for plates , cyl inders, and spheres for small values of h (case 9 .1 .3 , 8.1.7 and 1 0 . 1 . , source: Ref. 5, p. 89, Fig. 4-13 and Ref. 12) , £ i s half thickness of radius.

FIG. 9 .2 . Temperature distribution in an inf in i te cylinder with i n i t i a l temperature fcj and steady surface temperature tg (case 9 . 1 . 1 , source: Ref. 9, p . 200, Fig. 24).

F-82

FIG. 9.3. Temperature in an infinite cylinder with constant heat flux at the surface (case 9.1.8, source: Ref. 9, p. 203, Pig. 25).

F-83

T = S(X)

t w '

Semi-infinite solid

T = S(X,)S(X 2)

Quarter-infinite solid

T = S(X 1)S(X 2)S(X 3)

Eighth-infinite solid

T = P(Fo) T = P(Fo)S(X) T = P(Fo)S(X,)S(X2)

Infinite plate Semi-infinite plate Quarter-infinite plate

T = P(F 0 l)P(Fo 2)

^ ^ ^ \S

U-

, * 2

T=P(Fo,)P(Fo 2)S(X)

V -25, Infinite rectangular bar Semi-infinite rectangular bar

T = P(Fo1)P(Fo2)P(Fo3)

+ ' «3

i !

yA->

2S,

•25, Rectangular parallelepiped

T = C(F 0 l )

Infinite cylinder

T = C(Fo1)S(X)

x i-.

T = C(F 0 l )P(Fo 2 )

-2r, -2r,

Semi-infinite cylinder Short cylinder

S = semi-infinite solid P = plate C = cylinder

Fo 1 = at/6? = ar/r,2

X| = x i /2 v /aT= Fo*

T = ( t - t w ) / ( t 0 - t w '

FIG. 9.4a. Product solutions for internal and central temperatures in so l ids with step change in surface temperature. S(X) given in Fig. 7 .2 , P(Fo) given in Fig. 8 . 4 , C(Fo) given in Fig . 9.1 (case 7.1 .19, 7 .1 .20 , 8.1.22, 9 .1 .16 , and 9 .1 .17 , source: Ref. 74, pp. 3-65, Fig. 38).

F-84

0.01

0.001

Fo, arlh\

FIG. 9.4.b. Central temperatures in an infinite plate, an infinite rod, an infinite cylinder, a cube, a sphere, and a finite cylinder of length equal to its diameter, with all surfaces at temperature tg (case 7.1.29, 7.1.20, 9.1.16, 9.1.17, and 8.1.22, source: Ref. 2, p. 248, Fig. 10-8).

FIG. 9.5. Temperature distribution in an infinite cylinder with steady internal heating and a surface temperature tg (case 9.2.1, source: Ref. 9, p. 205, Fig. 26). F _ 8 5

"a i

CO

100.0

0.4 0.8 2.0 3.0 4.0 £ore£

FIG. 9.6a. (case 9 .2 .4 , source: Ref. 3, p . 356, Fig. 7-5) .

5.0 0 1 2 3 4 5 6

FIG. 9.6b. (case 9 .2 .4 , source: Ref. 3, p. 364, Fig. 7 -8) .

0.1 1 10 50 «/(X-1)

FIG. 9.6c. (case 9 . 2 . 1 , source: Ref. 3, p. 365, Fig. 7-9),

F-87

0.001 0.005 0.010 0.050 0.100 0.500 1.000 5.000 10.000

Fo, QT/TQ

FIG. 9.7a. Temperature response of a hollow cylinder with a steady inside surface temperature equal to the initial temperature, t^ (case 9.1.10, source: Re£. 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, UT/TQ

FIG. 9.7b. Temperature response of a hollow cylinder wit,, a steady outside surface equal to the i n i t i a l temperature, t± (case 9.1.10, source: Ref. 74, p. 157, Fig. 4.25).

F-86

i 1 1 r

FIG. 9 .8 . Temperature of an in f in i t e cylinder having a ramp surface temperature equal to t s = t i + br (case 9 .1 .2 , source: Ref. 74, p. 312, Fig. 7 . 3 ) .

4Fo 1 + 2/Bi

4Fo 1 + 2/Bi

FIG. 9.9. Transient surface temperature of a cylinder convectively coupled to a l inearly changing environment temperature equal to tf = t^ + br (case 9 . 1 . 6 , source: Ref. 89) .

F-89

100 - I—I—I I I 11 I 1—I—I I I I I I 1—I—I I I 11

• 1 -I

Fo

FIG. 9.10. Central temperature of an inf inite cylinder with uniform internal heating and convection boundary (case 9 .2 .2 , source: Ref. 90).

F-90

i

1.0

0.8

0.6

0.4

0.2

0

^ f f C ^ :

,300' | |0.400 0.2 0.4 0.6 0.8 1.0

(1-R) 0.004 0.010 0.100

FIG. 10.1. Temperature distribution in a sphere of radius r 0 with initial temperature t^ and surface temperature t 0 (case 10.1.1, source: Ref. 74, p. 127, Figs. 4.15 and 4.16).

1.000

•3 I ID

3. 0.050 •3" I

0.010

0.005 0.003

0.001 0 1.0 2.0 3.0 5.0 7.0 9.0 15.0 25.0 35.0 45.0 70.0110.0 170.0 250.0

Fo PIG. 10.2a. Center temperature of a conveotively cooled sphere (case 10.1.2, source: Ref. 5, p. 85, Fig. 4-9 and source: Ref. 12).

10 nr

0.02 0.1 0.5 2.00 5.00 20.00 100.00

1/Bi

FIG. 10.2b. Temperature as a function of center temperature of a convectively cooled sphere (case 10 .1 .2 , source: Ref. 5, p. 88, Fig. 4-12, and Ref. 12) .

ala° °-5 ~

10 10* 10 3 10 4

Bi2Fo

FIG. 10.2c. Dimensionless heat loss Q/QQ of a convectively cooled sphere (case 10 .1 .2 , source: Ref. 5 , p. 91, Fig. 4-16, and Ref. 13).

F-93

FIG. 10.3. Temperature of an infinite conductivity medium surrounding a spherical solid (case 10.1.13, source: Ref. 9, p. 241, Fig. 30).

o u. m

o

"2- -0.1

-0.2

-0.3 0.2 0.4 0.6 0.8

Fo

FIG. 10.4. Temperature in a sphere caused by a steady surface heat flux (case 10.1.15, source: Ref. 9, p. 242, Fig. 31) .

F-94

0 - 2 0-6 1 -0 R

FIG. 10.5. Temperature distribution in a sphere with steady internal heating (case 10 .2 .1 , source: Ref. 9, p. 244, Fig. 32).

F-95

0.010 0.100 Fo .

1.000 10.000

10.00

FIG. 10.6. Temperature response of solid sphere (0 £ r £ rtf cooling by radiation to a sink temperature of 0°R: (a) r/rx = 0, (b) r/rj = 1/2, (c) r/rx = 1 (case 10.1.20, source: Ref. 19, pp. 3-57, Fig. 30).

T - T 0 . 6 -

T w - T i

0.02 0.06 0.10 0.40 0.80 OCT

Fo,— I*

FIG. 10.7a. Transient temperature distribution at the center point (x = y = z = 0) for various prolate spheroids (case 10.1.21, source: Ref. 72, Fig. 1),

1.0

T-T, 0.6

T w - T i

1 1 i-;tPs Cylinder,

0.01 0.04 0.08 0.20 0.601.00

FIG. 10.7b. Transient temperature distribution at the focal point (x = y = 0, z = L2) for various prolate so l ids (case 10.1.21, source: Ref. 72, Fig. 2 ) .

F-97

0.8

1 1 1 1 1

- ^ 0 . 8 0 ^

1 1 1 1

0.8

1 1 1 1 1

- ^ 0 . 8 0 ^ y^\^Q-^^^-<^^

0.6

b-JQ 0.4 -[////^*

0.2

nn \i4y\ i i i i

t, - t, + bT

t i l l 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fo

FIG. 10.8. Temperature of a sphere having a ramp surface temperature equal to t s = t A + br (case 10.1.4, source: Ref. 74, p. 310, Fig. 7 .2 ) .

0.15 -

« o 0.10

0.05 -

I I

y—Bi = 100

< l ^ B i = 10 ~~ Bi = » ^ s v - " y—Bi = 100

< l ^ B i = 10 ~~

Bi= 10 - ^ / ^ - B i = 1 B i = i — * / * * S / N v >X _

- ffl\ ^ * — B i = 0

I

B i = 0 ^

R=1 1

\v\-

0.5

= 0.4

- 0.3

- 0.2

0.1

6Fo 1 + 2/Bi

FIG. 10.9. Transient surface temperature of a sphere convectively coupled to a linearly changing environment temperature equal to tf = t± + br (case 10.1.11, source: Ref. 89).

F-98

100.0

10.0

C M Q

T 1 I I I I I 1/Bi :

1000.0

FIG. 10.10. Central temperature of a sphere with uniform internal heating and convection boundary (case 10.2.9 , source: Ref. 90).

2 ' 3 (h2/p7k)(tm-t)r •s>f- '"'I'm

FIG. 11.1. Solidification depth in semi-infinite liquid for which solidified phase has negligible thermal capacity and is exposed to convective environments t a at free surface x = 0 and t 0 at interface x = w (case 11.1.10, source: Ref. 19, pp. 3-88, Fig. 54).

F-99

0.01 0.05 0.1 0.50 1 5 10

(b2/pck)r->-

50 100 500 1000

s

0.05 -

0.01 0.005 0.01

(h?/pck)r

100

5 10 50 100

50 100 500 1000

FIG. 11 .2 . Temperature response and so l id i f icat ion depth in semi-inf inite l iquid i n i t i a l l y at t,„ (a) surface temperature for convective cooling q = bt„ at x = 0, (b) so l id i f icat ion depth (case 11.1.11, source: Ref. 19, pp. 3-89, Fig. 55) .

F-100

0.01 0.05 0.1 0.5 1 5 10 {b2t%/pck)r

50 100 500 1000

5 10 50 100

50 100 500 1000

FIG. 11.3. Temperature response and solidification depth in semi-infinite liquid initially at t,„ with radiative cooling q = bt$ at x = 0 (a) surface temperature at x = 0, (b) solidification depth (case 11.1.12, source: Ref. 19, pp. 3-90, Fig. 55).

F-101

1.0

0.8

0.6

* 0.4

0.2

T — r

Semi-infinite solid t w = const

I L

b/Zv^"

0.0001 0.0005 0.001 0.005 0.01 0.05 0.1 0.5 1 Fo: x1

•? 0.6 -

0.01 0.05

F°x2

PIG. 11 .4 . Temperature response of decomposing semi-infinite so l id after sudden change in surface temperature from t 0 to t„: (a) 0 _< x £ H, (b) x < SL (case 11 .1 .13 , source: Bef. 19, pp. 3-91, Fig. 56) .

F-102

-ww

(b)

Semi-infinite solid

{h/pH)(t a-t d)V r^7a"=1

10

0.01

w«w

Semi-infinite solid

(h/pH)(t a-t d)Vyd"=2

0.1 0.5 1

r / r d - 1 10

FIG. 11.5. Ablation depth of semi- inf ini te so l id (x >_ 0) after sudden exposure to oonvective environment t_(> t d ) : (a) (h/pH) ( t a - t a ) ?d/<» = 1/2, (b) (h/pH) ( t a - ta) Szpa = 1, (c) (h/pH) ( t a - t d ) /Ta/Ot = 2 (case 11.1 .14, source: Ref. 19, pp. 3-94, Fig. 58) .

F-103

10

5

•a

i—rm—i I i ' i ii 1—r—n—i I i i i ii 1—m—i i I I i

Semi-infinite solid q" = const

5 10 50 100

Wr d ) -1

FIG. 11.6. Ablation depth of semi-infinite sol id (x > 0) after sudden exposure to constant surface heat input q" [case 11 .1 .15 , source: Ref. 19, pp. 3-96, Fig. 59)i -:: v - - - —

F-104

TABLE 1.1. Mean and maximum temperature excesses and their ratio for electrical coils of rectangular and circular cross sections, 8 = t - t,

J m mean I 8„ = t - t„ (case 1.2.8, source: Ref. 3, p. 220, Table 10-3) . 0 max 0

Cross S e c t i o n , Rectangular C i r c u l a r ,

b/a GO 10 5 2.5 1.5 1 r = 2a// iF

( 2 k / q ' " a 2 ) 9 m 0.66 0.625 0 .58 0 . 2 8 0.32

( 2 k / q I " a 2 ) e Q 1.00 1.00 1 .00 0 .59 0.64

* " 6 n / 9 0 0.66 0.625 0 .58 0.52 0.485 0 .475 0.50

TABLE 1.2a. Values of (t - t ) / ( t . - t . ) on the surface y = 0 of a semi-inf inite strip (case 1.1.29, source: Ref. 15, Table 2 ) .

x/w hw/k hw/k hw/k hw/k hw/k

= 100 = 10 = 1 = 0.10 = 0 . 0 1

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 rates through the surface y = 0 of a semi-inf inite s tr ip (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/ke 8 .603 6 .827 5.984 3 . 9 7 1 3.107 0.792 0 .097 0 .01 0 .001

% Error 1 .8 0 . 5 0 .3 0 .08 0.C5 0.02 0 . 0 2 0.02 0 .02

P-105

TABLE 1 .3 . Conductance data for heat flow normal to wall cuts in an in f in i t e plate (case 1.1.37, source: Ref. 19, p. 3-124). These groups of data are for the four kinds of wall e la tes shown in case 1.1.37.

(a) K/K uncut

(b) K/K uncut

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 0.747 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/K uncut (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

Ui/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 0.242 1/2 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/K 'uncut b/a = 1/4 1/8

0.507

0.479

0.362

0.338

0.329 0.213

1/16

0.310

0.254

0.141

F-106

TABLE.2.1 Ref. 64, Table 2).

Values of A. for a triangular cooling array of cylinders (case 2.2.18, source:

I

B/lQ > 1 [•/'•) ;-h & 3 A 4 *5 A 6 A 7

4 . 0 -5.05072(10~ 2) -8.089K10" 4 ) -1.262(10" 5 ) -1.97<10" 7) -3 .08(10~ 9 ) - 4 . 8 ( 1 0 " n ) - 7 (10" 1 3 ) 2 . 0 -5.04988 dO" 2) -8 .0542(10 - 4 ) -1.217(10" 5 ) - 1 . 6 4 ( l O - 7 ) -1.33(10" 9 ) 2 . 8 ( 1 0 " U ) 2(10" 1 2 ) 1 .5 -5.02447(10~ 2) -6.9920(10" 4 ) -1.374(10" 6 ) 8.34(10~ 7) 5.23(10~ 7) 2.4-(10"9) 8 ( 1 0 - 1 1 ) 1 .3 -4.90792(10~ 2) -2.1203(10* 4 ) 6.354(10" 5) 5.42(10"^) 2.99(10~ 7) 1.3(10" 8) 4(10" 1 0 ) 1.2 -4.69398(10~ 2) 6.7812(10" 4) l.W " 4 ) 1 .31(10 - 5 ) 6.27(10~ 7) 1 . 7 ( 1 0 T 8 ; ) - K 1 0 " 1 0 ) 1.175 -4.60073 (10" 2) 1.0619(10" 3) 2.2!i. :i.0~ 4) +1.56(10~ 5) 6.32(10" 7) 9.8(10*7 1 0) -2(10" 9 ) 1.15 -4.48345(10~ 2) 1.5384(10~ 3) 2.736(10" 4) 1.76(10-9) 4.2K10" 7 ) -5 (10" 8) -7(10~ 9 ) 1.10 -4.15694(10" 2) 2.8029(10~ 3) 3.782(10" 4) 1.08(10"9) -2.40(10" 6 ) -5 (10" 7) -5(10" 9 ) 1.05 -3 .68059(10 - 2 ) 4.3358(10 ~ 3 ) 3.169(10~ 4) 5 .67 ( lO - 5 ) -1.51(10~ S ) -1,7(10T 6 ) -9(10" 8 )

TABLE; 2 . 2 . Values of 6. 3

for a square cooling array of cylinders (case 2 .2 .19 , source: Kef. 64, Table i ) .

s / E o «1 i:i 6 3 «4 h 8 6 «7,

4 . 0 -1.25382 (10"X) -1.0583(10" 2 ) -6.120(10" 4 ) -3 .898(10 _ 5 ) -2.42(10" 6 ) -1 .5(10" 7 ) - 8 - 0 (10 -)

2 . 0 -1.25098 (10" 1) -1.0428(10" 2 ) -5.713(10" 4 ) -3.177(10" 5 ) -1.39{10~ 6 ) -2 ( 1 0 - 8 ) 3 do" 9) 1 . 5 -1.22597(10" 1) -9.060!(10~ 3 ) -2.08 (10~ 4) 3.33 ( 1 0 - 5 ) 8.1 (10~ 6) 1 ( 1 0 - 6 ) 1 (10" 7) 1 .3 -1.17022 (10" 1) -5.995 (10" 3) 6.13 (10~ 4) 1.83 ( 1 0 - 4 ) 3 .1 (10~ 5) 4 ( 1 0 - 6 ) 4 do" 7) 1.2 -1.10421 (10 - 1 ) -2.387 (10"3) 1.566(10" 3) 3.50 (10 - 4 ) 5.3 (10~ 5) 6 (10" 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 dO" 8) 1.1 -9.3721 (10"2) 3 .6596(10 - 3 ) 2.91K10" 3 ) 4.655(10" 4) 2.44(10~ 5) - 1 . K 1 0 " 5 ) - 5 . 3(10~ 6) 1.05 -9.0358 (10~ 2) 7.310 (10 _ 3 i 3.210(10~ 3) 2.06 (10"4) -1.20(10" 4 ) -5 .8(10" 5 ) - 1 . 6 dO" 5)

TABLE 8.1. Values of (t - t,)/(t - t.) for an infinitely wide plate 1 s x

proportionally (case 8 .1 .8 , source: Ref. 1, p. 268, Table 13.5),

whose surface temperature increases proportionally to time, t = t . + br

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 0.955 1 1 1 1 1 1 1 1 1 1

TABLE 8.2. Values of the function £ (case 8.1.12, source: Ref. 1, p. 299).

X

a 0 1/8 2/8 3 / 8 4 /8 5/8 6 /8 7/8 1

0 I 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 0.05 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 a> 0 0 0 0 0 0 0 0

F-108

TABLE 9 . 1 . Values of (t - t . ) / ( t - t . ) 1 s i

for an in f in i t e ly long cylinder whose surface increases proportionally to time (t - t . = CT) (case 9 . 1 . 2 , source: Ref. 1, p. 269).

R Fo

0.08 0.16 0.32 0.80

0 0.016 0.123 0.354 0.691 0.33 0.054 0.191 0.420 0.725 0 .5 0.122 0.287 0.505 0.768 0.66 0.268 G.443 0.628 0.828 0 .8 0.470 0.621 0.755 0.888 1 1 1 1 1

TABLE 10 .1 . Temperatures for a sphere whose surface temperature increases proportionally to time ( t Q = t . + bx) (case 10 .1 .4 , source: Ref. 1, p. 269, Table 13-7).

R Fo

0.016 0.08 0.16 0.32 0.80

0 0.00 0.054 0.219 0.506 0.792 0.33 0.00 0.090 0.290 0.560 0.725 0.) 0.00 0.162 0.385 0.626 0.844 0.66 0.02 0.312 0.529 0.722 0.884 0 .8 0.14 0.516 0.686 0.819 0.925 1 1 1 1 1 1

F-109

F- l lO

SECTION 13. MATHEMATICAL FUNCTIONS

13.1. The Error Function and Related Functions.

e r f < x > = y f f exp(-z ) dz

_ 2 _ Y ( - l ) n ( x ) 2 n + 1

V? ^ {2nil)nl ' f O C S n , a 1 1 V a l u e S ° f X

2 f o t

» ( - » ) / l - k _ + L . . 3 _ _ l _ . 3 _ \ large * I* 2x3 2 2 x 5 2 3 x 7 " 7 ' ! f U e S

. 2 1 _ S2E

of x

erf(O) = 0 , erf (<») = 1

erf (-x) = - erf (x)

erfc(x) = 1 - erf (x)

- f * i erfc(x) = / i" •'erfc(z)dz , n = 1, 2,

i erfc(x) = erfc(x)

1 2 ierfc(x) =.y=exp{-x ) - x erfc(x)

i 2 e r f c ( x ) = j [erfc(x) - 2x ierfc(x)]

2n i n er fc (x ) = i n - 2 e r f c ( x ) - 2x i n _ 1 e r f c ( x )

^ e r f ( x ) = ^ e x p ( - x 2 )

d _, , 4x , 2 . — 2 erf (x) = - r s exp(-x } dx

13-1

r* M M « N M m

13.2. Exponential and Hyperbolic Functions.

e = 2.71828 . . . exp(u) = e

e u - e" u

sinh (u) = 5 , sinh (0) = 0 , sinh (<») = <*•

u u cosh (u) = -—s-S- , cosh (0) = 1 , cosh (») = <*>

e 2 u - l tanh (u) = — , tanh (0) = 0 , tanh (<») = 1 e Z U + 1

13-3

TABLE 13.2. Exponential functions (source: Re:?. 2, pp. 367-9) .

e~ u

0.00 1.000 1.000 0.02 1.020 0.980

0.04 1.041 0.961 0.06 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 0.631 0.48 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

0.60 1.822 0.549 0.62 1.859 0.538 0.64 1.896 0.527 0.66 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 0.398 0.94 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 0.353 1.06 2.886 0.346 1.08 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

13-4

1.20 3.320 0.301 1.22 3.387 0.295 1.24 3.456 0.289 1.26 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 0.219 1.54 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 0.190 1.68 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

TABLE 13 .2 . (Continued).

u e u e" u u e u e~ u u e u e" u

1.80 6.050 0.165 2.40 11.023 0.091 3.00 20.08 0.050 1.82 6.172 0.162 2.42 11.246 0.089 3.10 22.20 0.045 1.84 6.296 0.159 2.44 11.473 0.087 3.20 24.53 0.041 1.86 6.424 0.156 2.46 11.705 0.085 3.30 27.11 0.037 1.88 6.554 0.153 2.48 11.941 0.084 3.40 29.96 0.033

1.90 6.686 0.150 2.50 12.18 0.082 3.50 33.11 0.030 1.92 6.821 0.147 2.52 12.43 0.080 3.60 36.60 0.027 1.94 6.959 0.144 2.54 12.68 0.079 3.70 40.45 0.025 1.96 7.099 0.141 2.56 12.94 0.077 3.80 44.70 0.022 1.98 7.243 0.138 2.58 13.20 0.076 3.90 49.40 0.020

2.00 7.389 0.135 2.60 13.46 0.074 4.00 54.60 0.018 2.02 7.538 0.133 2.62 13.74 0.073 4.20 66.69 0.015 2.04 7.691 0.130 2.64 14.01 0.071 4.40 81.45 0.012 2.06 7.846 0.127 2.66 14.30 0.070 4.60 99.48 0.010 2.08 8.004 0.125 2.68 14.58 0.069 4.80 121.51 0.008

2.10 8.166 0.122 2.70 14.88 0.067 5.00 148.4 0.007 2.12 8.331 0.120 2.72 15.18 0.066 5.20 181.3 0.006 2.14 8.499 0.118 2.74 15.49 0.065 5.40 221.4 0.004 2.16 8.671 0.115 2.76 15.80 0.063 5.60 270.4 0.004 2.18 8.846 0.113 2.78 16.12 0.062 5.80 330.3 0.003

2.20 9.025 0.111 2.80 16.44 0.061 6.00 403.4 0.002 2.22 9.207 0.109 2.82 16.78 0.060 6.50 665.1 0.002 2.24 9.393 0.106 2.84 17.12 0.058 7.00 1096.6 0.001 2.26 9.583 0.104 2.86 17.46 0.057 7.50 1808.0 0.001 2.28 9.777 0.102 2.88 17.81 0.056 8.00 2981.0 0.000

2.30 9.974 0.100 2.90 18.17 0.055 8.50 4914.8 0.000 2.32 10.176 0.098 2.92 18.54 0.054 9.00 8103.1 0.000 2.34 10.381 0.096 2.94 18.92 0.053 9.50 13360 0.000 2.36 10.591 0.094 2.96 19.30 G.052 10.00 22026 0.000 2.38 10.805 0.093 2.98 19.69 0.051

13-5

13.3 . The Gamma Function.

x > 0

CO

T(x) = / u X _ 1 e" u du, •'o

r W » x x e~xM li + - i - + - i - - -122 5 7 1 _ "I V x L 1 2 X 288X 2 5184X 3 2488320x 4 " " J '

for large posit ive values of x T(x + 1) = xT(x)

TABLE 13.3. The gamma function (source: Ref. 17, p. 136).

X T(x) X T(x) X TW 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.C9 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 U.C9 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 Q.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 0.38 964 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

the posit ive sign used i f -TT/2 < arg u < 3ir/2 and the negative sign used i f -31T/2 < arg u < n/2 .

I (u) - I (u) v i \ —

n ' u ' = 2 s in (nil)

For n any pos i t ive integer:

Kn(u) = ( - l ) n + 1 | iln (u/2) + y | I n (u) + J C ¥ ~ J (u/2) n+2j JMn+j)!

j=0

n+j j

X r a " 1 + X m" , m = l m = l

i 2 ™: -n+2j

(u/2) j=0

C - 3 - Ul

where for j=0, \ m = 0 m=l

For large values of u:

K (u, = Jh e"u { i + rt^ + ( 4 n" - n ( 4 f - n * o( U - 3 ) l n

2 , 2 , . 2 . 2 , , , 2 , 2 ,

H8u 2!(8u)'

£ - J (u) = J (u) - £ J (u) = £ J (u) - J (u) du n n-l u n u n n+1

h V U ) ' V l ( U ) " 5 V U ) = l! V U ) " Vlli..«,n*M V U ) Z jinn + j + 1)

3=0

is ceal and u may be complex

J (u) = (-1) J (u) , if n is an integer n ~n

Jn(u) cos (nwi - J_n(u) Y (u) = — : n sin (nTf)

,Yn(u) - 2 Un (u/2) + y) Jû) - J ^ ^ " f >", n+2j

3=0

3+n 3 I "-1 * I "-1

n-1 Y (u/2)" n + 2 j (n - j - 1) 1 3=0 .m=l m=l

where y = 0.5772... is Euler's constant, and n is any positive integer

!vu> {in (u/2) + Y}JQ(u) + (u/2)2 " f 1 + j) "^ L (21)'

("7*l)^ (3!)'

n+2j x ( u ) . y wr" V u ) L jir(n + j + 1)

3=0

For large values of u:

v • - < - 1 * - • < - • - ;; | , ( ;f - 3 2 ) * • (»-!)l ( 2!(8u) j

$ - exp [-u ± (n + J) iri] j 1 + 0 ( i f 1 ) ) , V2TTU

13-8

TABLE 1 3 . 4 . 1 . Zeco and first-order Bessel functions of the f i r s t kind (source: Ref 2, pp. 369-71).

u J0(u) Jj/U) u J0(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 -0.1386 1.3 0.6201 0.5220 4.1 -0.3610 -0.1719 1.4 0.5669 0.5419 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 -0.3371 2.2 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 0.4416 5.7 0.0599 -0.3241 2.8 -0.1850 0.4097 5.8 0.0917 -0.3110 2.9 -0.2243 0.3754 5.9 0.1220 -0.2951

13-9

TABLE 13 .4 .1 . (Continued.)

u J0(u) J x(u) u J0(u) JL(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 0.0684

7.0 0.3001 -0.0047 10.0 -0.2459 0.0435 7.1 0.2991 0.0252 1C.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.C419 0.2731 11.5 -0.0677 -0.2284 8.6 0.0146 0.2728 11.6 -0.0446 -0.2320 8.7 -0.0125 0.26S7 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

TABLE 13.4.1. (Continued.)

_u JQ(u) Jû)

12.0 0.0477 -0.2234 12.1 0.0697 -0.2158 12.2 0.0908 -0.2060 12.3 0.1108 -0.1943 12.4 0.1296 -0.1807

12.5 0.1469 -0.1655 12.6 0.1626 -0.1487 12.7 0.1766 -0.1307 12.8 0.1887 -0.1114 12.9 0.1988 -0.0912

13.0 0.2069 -0.0703 13.1 0.2129 -0.0488 13.2 0.2167 -0.0271 13.3 0.2183 -0.0052 13.4 0.2177 0.0166

13.5 0.2150 0.0380

u JQ(u) Jû)

13.6 0.2101 0.0590 13.7 0.2032 0.0791 13.8 0.1943 0.0984 13.9 0.1836 0.1165 14.0 0.1711 0.1334

14.1 0.1570 0.1488 14.2 0.1414 0.1626 14.3 0.1245 0.1747 14.4 0.1065 0.1850 14.5 0.0875 0.1934

14.6 0.0679 0.1998 14.7 0.0476 0.2043 14.8 0.0271 0.2066 14.9 0.0064 0.2069 15.0 0.0142 0.2051

13-11

TABLE 13.4 .2 . Zero and first-order Bessel functions of the second kind (source: Ref. 2 p . 371-3).

u V u ) ^(u) u v> ^(u)

0.0 ^00 -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

13-12

TABIS 13.4 .2 . (Continued.)

u V u > Vj/u) u * 0 ( u ) ^(u) 9-0 0li499 6 i 6 -0.2882 -0J1750" 9-0 0li499 0.1043

6 . 1 -0>2694 -0.1998 9 . 1 0>2383 0.1275 6 . 2 -0:2433 -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 -0.2945 9 .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 C.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 0.1446 8 . 2 0.2501 -0.1072 11.2 -0.1977 0.1243 873 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 0.2659 0.0544 11.8 -0.2322 -0.0118 8 . 9 0.2592 0.0799 11.9 -0.2298 -0.0347

13-13

TABLE 13.4.2. (Continued.)

_u Y0(u) Y1(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 r 0(u) Y1(u)

13.5 Q.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

13-14

TABLE 13 .4 .3 . Zero and f i r s t order modified Bessel functions of the f i r s t kind (source: Ref. 2, p. 373-4).

u I 0 ( u ) IjÛ) u I 0 (u ) Iû)

0.0 1.0000 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 3.301 5 . 8 56.04 50.95 2.9 4.503 3.613 5 . 9 61.38 55.90

13-15

TftBLE 13.4.4. Zero and first order modified Bessel functions of the second kind (source: Ref. 2, p. 374-5).

0 . 0 m oo 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

13-16

13.5. Legendre Polynomials

The Legendre polynomial of degree n, of the first kind:

„ , , . ,,n/2 l . 3 • 5 •»« (n - 1) r. n(n + 1) 2 V U ) = (- 1 } 2 • 4 • 6 ..• n L 1 " 2, u

+ n<n - 2 M n J 1 M , * 3) + ...."] , „ = 2, „, 6

P (a) = (-1) , n" 1 ) / 2 X * 3 ' 5 •" " Tu - < n - C * 2> u3 V ' l ' 2 • 4 • 6 ••• (n - 1) L u 31 u

+ (n-lM»-3)M2 Mn t4) u5 + . . . . " J f n . 3 f 5 f 7 , u | < 1

PQ(u) = 1, Px(u) = u, P2(u) = (3u2 - l)/2 ,

P,(u) = (5u3 - 3u)/2r P.(u) = (35u4 - 30u2 + 3)/8 , 3 4

P5(u) = (63u5 - 70u3 + 15u)/8, |u|<l

The Legendre polynomial of degree n, of the second kind:

n . , , ,.(n+l)/2 2 » 4 * 6 "» (n - 1) f, n(n + 1) 2 Q n C ) = (-D 1 » 3 • 5 — n L 1 2 7 " ^ u

^ n(n - 2) (n + 1) (n + 3) 4 T , c , + - 1 " - J j — " u u + ••••_] , n = 3, 5, 7, ...,

Q (u) » (• = ,_,xn/2 2 ' 4 ' 6 •" n r _ (n - l)(n + 2) 3 "n

..n/2 2 ' 4 * 6 •" n r (n - 1) ( x' 1 • 3 • 5 ••• (n - 1) L u 31

+ ( n - l M n - 3 M n , 2 M n + 4) u5 + . . . . - | t n s 2 > 4 t 6

13-17

Q0(") H ^n(î).|u|<l

Qx(u) = QQ(u) Pû) - 1,

Q2(u) = Q„(u) P2(u) - 3u/2

Q3(u) = Q0(u) P3(u) - 5u2/2 + j,

Q4(u) = QQ(u) P4(u) - 35u3/8 + 55u2/24 ,

Qn(u) =Q0(u, Pn(u, -IflLr-iLpû, . ^ i L , ^ -

13-18

TABLE 13.5. The first five Legendre polynomials of the first kind (source: Ref. 2, p. 375-7).

u Pj/U) P 2(u) P3(u) v»> P5(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 O.OSOO -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 (1.2600 -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

13-19

TABLE 13.5. (Continued.)

u px(u) P2(u) P3(u) P4(u) P5(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 -0.0753 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

20

TABLE 13 .5 . (Continued.)

u Px(u) P2(u) P,(u) P 4(u) P5(u)

0.60 0.6000 0.0400 -0.3600 -0.4080 -0.1526 0.61 0.6100 0.0582 -0.3475 -0.4146 -0.1772 0.62 0.6200 0.0766 -0.3342 -0.4200 -0.2014 0.63 0.6300 0.0954 -0.3199 -0.4242 -0.2251 0.64 0.6400 0.1144 -0.3046 -0.4270 -0.2482

0.65 0.6500 0.1338 -0.2884 -0.4284 -0.2705 0.66 0.6600 0.1534 -0.2713 -0.4284 -0.2919 0.67 0.6700 0.1734 -0.2531 -0.4268 -0.3122 0.68 0.6800 0.1936 -0.2339 -0.4236 -0.3313 0.69 0.6900 0.2142 -0.2137 -0.4187 -0.3490

0.70 0.7000 0.2350 -0.1925 -0.4121 -0.3652 0.71 0.7100 0.2562 -0.1702 -0.4036 -0.3796 0.72 0.7200 0.2776 -0.1469 -0.3933 -0.3922 0.73 0.7300 0.2994 -0.1225 -0.3810 -0.4026 0.74 0.7400 0.3214 -0.0969 -0.3666 -0.4107

0.75 0.7500 0.3438 -0.0703 -0.3501 -0.4164 0.76 0.7600 0.3664 -0.0426 -0.3314 -0.4193 0.77 0.7700 0.3894 -0.0137 -0.3104 -0.4193 0.78 0.7800 0.4126 0.0164 -0.2871 -0.4162 0.79 0.7900 0.4362 0.0476 -0.2613 -0.4097

0.80 0.8000 0.4600 0.0800 -0.2330 -0.3995 0.81 0.8100 0.4842 0.1136 -0.2021 -0.3855 0.82 0.8200 0.5086 0.1484 -0.1685 -0.3674 0.83 0.8300 0.5334 0.1845 -0.1321 -0.3449 0.84 0.8400 0.5584 0.2213 -0.0928 -0.3177

0.85 0.8500 0.5838 0.2603 -0.0506 -0.2857 0.86 0.8600 0.6094 0.3001 -0.0053 -0.2484 0.87 0.8700 0.6354 0.3413 0.0431 -0.2056 0.88 0.8800 0.6616 C.3837 0.0947 -0.1570 0.89 0.8900 0.6882 0.4274 0.1496 -0.1023

13-21


u Px(u) P 2(u) P3(u) P4(u) P5(u)

0.90 0.9000 0.7150 0.4725 0.2079 -0.0411 0.91 0.9100 0.7422 0.5189 0.2698 0.0268 0.92 0.9200 0.7696 0.5667 0.3352 0.1017 0.93 0.9300 0.7974 0.6159 0.4044 0.1842 0.94 0.9400 0.8254 0.6665 0.4773 0.2744

0.95 0.9500 0.8538 0.7184 0.5541 0.3727 0.96 0.9600 0.8824 0.7718 0.6349 0.4796 0.97 0.97C0 0.9114 0.8267 0.7198 0.5954 0.98 0.9800 0.9406 0.8830 0.8089 0.7204 0.99 0.9900 0.9702 0.9407 0.9022 0.8552 1.00 1.0000 1.0000 1.0000 1.0000 1.0000

13-22

13.6. Sine, Cosine, and Exponential Integrals.

Sine integral: Si(x) = 0

S i {<*•) = TT/2

, / SAM. d u , J.

r 0 0 r x

Cosine integral: Ci(x) = - I C 0 S J U ) du =8, n(yx) - I * "x •'o

Jin y = 0.577215. . . (Euler's constant)

Exponential integral: Ei(x) = - I —— du, <0, » > x > 0 x

CD

/ S j - du, <0,

U(x ) = / Logarithmic integral: S,i(x) = / . " = Ei(An x) 0

£ i ( e x ) = Ei(x)

For small values of x:

Si(x) = x ,

U(x) = -x/Zn(l/x)

Ci(x) a Ei(-x) a i l ( x ) = Jln(l/yx) = y + S,n(x) - x + {X2/A) +

For large values of x:

Si(x) = ir/2 - cos (x)/x

Ci(x) = s in (x)/x

Ei(x) = (e*/x) (1 + l l / x + 2 l / x + 3! /x + • • • ) , | x | » 1

13-23

TABLE 13.6 . Values of the s ine , cosine, logarithmic, and exponential integrals ,

(source: Ref. 18, p. 6-9.)

Si(x) Ci(x) Ei(x) Ei (-x)

0.00 +0.000000 -CO —00

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

G.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 -0.5485 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

-4.0379

-3.3547

-2.9591

-2.6813

-2.4679

-2.2953

-2.1508

-2.0269

-1.9187

-1.8229

-1.7371

-1.6595

-1.5889

-1.5241

-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

13-24


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 -0.8815 0.32 +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 0.36 +0.3574 -0.4767 -0.04926 -0.7745 0.37 +0.3672 -0.4511 -0.00979 -0.7554 0.38 +0.3770 -0.4263 +0.02901 -0.7371 0.39 +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.«U4 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


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 +0.03026 +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


x Si ix) Ci(x) 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 +0.3206 +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.Q4890

2.1 +1.6487 +0.4005 +5.3332 -0.04261 2.2 +1.6876 +0.T751 +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

13-27


x Si(x) Ci(x) Ei(x) Ei (-x)

3.1 +1.8517 +1.8514

+0.08699 +10.5263 +11.3673

-0.01149 3.2

+1.8517 +1.8514 +0.05526

+10.5263 +11.3673 -0.01013

3.3 +1.8481 +0:02468 +12vl610 -0.028939 3.4 +1.8419 +0.004518 +13.0121 -0.027890 3.5 +1.8331 -0.03213 +13.9254 -0.026970

3.6 +1.8219 -0.05797 +14.9063 -0.026160 3.7 +1.8086 -0.08190 +15.9606 -0.025448 3.8 +1.7934 -0.1038 +17.0948 -0.024820 3.9 +1.7765 -0.1235 +18.3157 -0.024267 4.0 +1.7582 -0.1410 +19.6309 -0.023779

4.1 +1.7387 -0.1562 +21.0485 -0.023349 4.2 +1.7184 -0.1690 +22.5774 -0.022969 4.3 +1.6973 -0.1795 +24.2274 -0.022633 4.4 +1.6758 -0.1877 +26.0090 -0.022336 4.5 +1.6541 -0.1935 +27.9337 -0.022073

4.6 +1.6325 -0.1970 +30.0141 -0.021841 4.7 +1.6110 -0.1984 +32.2639 -0.021635 4.8 +1.5900 -0.1976 +34.6979 -0.021453 4.9 +1.5696 -0.1948 +37.3325 -0.021291 5.0 +1.5499 -0.1900 +40.1853 -0.021148

6 +1.4247 -0.06806 +85.9898 -0.033601 7 +1.4546 +0.07670 +191.505 -0.031155 8 +1.5742 +0.1224 +440.380 -0.043767 9 +1.6650 +0.05535 +1037.88 -0.04_1245 10 +1.6583 -0.04546 +2492.23 -0.054157

11 +1.5783 -0.08956 +6071.41 -0.051400 12 +1.5050 -0.04978 +14959.5 -0.064751 13 +1.4994 +0.02676 +37197.7 -0.061622 14 +1.5562 +0.06940 +93192.5 -0.075566 15 +1.6182 +0.04628 +234 956 -0.071918

13-28

TABLE 1 3 . 6 . (Continued.)

x Si (x) C i (x)

20 +1.5482 +0.04442

25 +1.5315 - 0 . 0 0 6 8 5

30 +1.5668 - 0 . 0 3 3 0 3

35 +1.5969 - 0 . 0 1 1 4 8

40 +1.5870 +0.01902

45 +1.5587 +0 .01863

50 +1.5516 - 0 . 0 0 5 6 3

55 +1.5707 - 0 . 0 1 8 1 7

60 +1.5867 - 0 . 0 0 4 8 1

65 +1.5792 +0.01285

70 +1.5616 +0.01092

75 +1.5586 - 0 . 0 0 5 3 3

80 +1.5723 - 0 . 0 1 2 4 0

85 +1.5824 - 0 . 0 0 1 9 3 5

90 +1.5757 +0.009986

95 +1.5630 +0.007110

100 +1.5622 - 0 . 0 0 5 1 4 9

110 +1.5799 - 0 . 0 0 0 3 2 0

120 +1.5640 +0 .004781

130 +1.5737 - 0 . 0 0 7 1 3 2

x Si(x) Ci(x)

140 +1.5722 +0.007011 150 +1.5662 -0.004800 160 +1.5769 +0.001409 170 +1.5653 +0.002010 180 +1.5741 -0.004432

190 +1.5704 +0.005250 200 +1.5684 -0.004378 300 +1.5709 -0.003332 400 +1.5721 -0.002124 500 +1.5726 -0.0009320

600 +1.5725 +0.0000764 700 +1.5720 +0.0007788 800 +1.5714 +0.001118 900 +1.5707 +0.001109 10 2 +1.5702 +0.000826

10 4 +1.5709 -0.0000306 10 5 +1.5708 +0.0000004 10 6 +1.5708 -0.0000004 10 7 +1.5708 +0.0 » 1/2TI 0.0

13-29

13-30

SECTION 14. ROOTS OF SOME CHARACTERISTIC EQUATIONS

TABLE 14.1. First six roots of X tan \ = C. (Source: Ref. 74, p. 217.)

c Xl X2 X3 X4 X5 \ 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 15.7086 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 16.1675 9.0 1.4149 4.2694 7.1806 10.1502 13.1660 16.2147 10.0 1.4289 4.3058 7.2281

14-1

10.2003 13.2142 16.2594

TABLE 14.1 . (Continued.)

\

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 10.7832 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 100.0 1.5552 4.6658 7.7764 10.8871 13.9981 17.1093

OD 1.5708 4.7124 7.8540 10.9956 14.1372 17.2788

14-2

TABLE 14.2. F irs t f ive roots of 1 - \ cot \ = C. ( s o u r c e : Ref. 20 p . 442.)

c \ X2 h X4 X5 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 14.0683 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 14.0712 0.080 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 14.2764 4.000 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 1.0.000 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

14-3

TABLE 1 4 . 2 . (Continued.)

c \ x 2 h h X5 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 15.4090 1U1.000 3.1105 6.2211 9.3317 12.4426 15.5537

TABLE 1 4 . 3 . F i r s t f i v e r o o t s of J (X ) = 0 . ( S o u r c e : Ref. 2 0 , p . 443 . ) in n

m X X_ X X X 1 2 3 4 5

0

1

2

3

4

2.4048 5.5201 8.6537 11.7915 14.9309 3.8317 7.0156 10.1735 13.3237 16.4706 5.1356 8.4172 11.6198 14.7960 17.9598 6.3802 9.7610 13.0152 16.2235 19.4094 7.5883 11.0647 14.3725 17.6160 20.8269

14-4

TABLE 14.4. First six roots of X J (X ) - CJft(X ) = 0. (Source: Ref. 74. n x n u n

p. 217)

c h X2 X3 0 0 3.8317 7.0156 0.01 0.1412 3.8343 7.0170 0.02 0.1995 3.8369 7.0184 0.04 0.2814 3.8421 7.0213 0.06 0.3438 3.8473 7.0241 o.oe 0.3960 3.8525 7.0270 0.1 0.4417 3.8577 7.0298 0.15 0.5376 3.8706 7.0369 0.2 0.6170 3.8835 7.0440 0.3 0.7465 3.9091 7.0582 0.4 0.8516 3.9344 7.0723 0.5 0.9408 3.9594 7.0864 0.6 1.0184 3.9841 7.1004 0.7 1.0873 4.0085 7.1143 0.8 1.1490 4.0325 7.1282 0.9 1.2048 4.0562 7.1421 1.0 1.2558 4.0795 7.1558 1.5 1.4569 4.1902 7.2233 2.0 1.5994 4.2910 7.2884 3.0 1.7887 4.4634 7.4103 4.0 1.9081 4.6018 7.5201 5.0 1.9898 4.7131 7.6177 6.0 2.0490 4.8033 7.7039 7.0 2.0937 4.8772 7.7797 8.0 2.1286 4.9384 7.8464 9.0 2.1566 4.9897 7.9051 10.0 2.1795 5.0332 7.9569 15.0 2.2509 5.1773 8.1422 20.0 2.2880 5.2568 8.2534 30.0 2.3261 5.3410 8.3771 40.0 2.3455 5.3846 8.4432 50.0 2.3572 5.4112 8.4840 60.0 2.3651 5.4291 8.5116 80.0 2.3750 5.4516 8.5466

100.0 2.3809 5.4652 8.5678 00 2.4048 5.5201 8.6537

\ 10.1735 13.3237 16.4706 10.1745 13.3244 16.4712 10.1754 13.3252 16.4718 10.1774 13.3267 16.4731 10.1794 13.3282 16.4743 10.1813 13.3297 16.4755 10.1833 13.3312 16.4767 10.1882 13.3349 16.4797 10.1931 13.3387 16.4828 10.2029 13.3462 16.4888 10.2127 13.3537 16.4949 10.2225 13.3611 16.5010 10.2322 13.3686 16.5070 10.2419 13.3761 16.5131 10.2516 13.3835 16.5191 10.2613 13.3910 16.5251 10.2710 13.3984 16.5312 10.3188 13.4353 16.5612 10.3658 13.4719 16.5910 10.4566 13.5434 16.6499 10.5423 13.6125 16.7073 10.6223 13.6786 16.7630 10.6964 13.7414 16.8168 10.7646 13.8008 16.8684 10.8271 13.8566 16.9179 10.8842 13.9090 16.9650 10.9363 13.9580 17.0099 11.1367 14.1576 17.2008 11.2677 14.2983 17.3442 11.4221 14.4748 17.5348 11.5081 14.5774 17.6506 11.5621 14.6433 17.7272 11.5990 14.6889 17.7807 11.6461 14.7475 17.8502 11.6747 14.7834 17.8931 11.7915 14.9309 18.0711

14-5

TABLE 14.5. First five roots of J 0(X n)Y 0(CX n) = Y Q(X n) J 0(CX n). (Source: Sef. 9, p. 493.)

1.2 15.7014 31.4126 1.5 6.2702 12.5598 2.0 3.1230 6.2734 2.5 2.0732 4.1773 3.0 1.5485 3.1291 3.5 1.2339 2.5002 4.0 1.0244 2.0809

_h \ 5_ 47.1217 62.8302 78.5385 18.8451 25.1294 31.4133

9.4182 12.5614 15.7040 6.2754 8.3717 10.4672 4.7038 6.2767 7.8487 3.7608 5.0196 6.2776 3.1322 4.1816 5.2301

14-6

TABLE 14.6. F irs t s ix roots of tan(Xn> = -X /C. (Source: Hef. 74, p. 322.)

c \ X2 X3 X4 X5 \ 0 1.5708 4.7124 7.8540 10.9956 14.1372 17.2788 0.1 1.6320 4.^335 7.8667 11.0047 14.1443 17.2845 0.2 1.6887 4.7544 7.8794 11.0137 14.1513 17.2903 0.3 1.7414 4.7751 7.8920 11.0228 14.1584 17.2961 0.4 1.7906 4.7956 7.9046 11.0318 14.1654 17.3019 0.5 1.8366 4.8158 7.9171 11.0409 14.1724 17.3076 0.6 1.8798 4.8358 7.9295 11.0498 14.1795 17.3134 0.7 1.9203 4.8556 7.9419 11.0588 14.1865 17.3192 0.8 1.9586 4.8751 7.9542 11.0677 14.1935 17.3249 0.9 1.9947 4.8943 7.9665 11.0767 14.2005 17.3306 1.0 2.0288 4.9132 719787 11.0856 14.2075 17.3364 1.5 2.1746 c.0037 8.0385 11.1296 14.2421 17.3649 2.0 2.2889 5.0870 8.0962 11.1727 14.2764 17.3932 3.0 2.4557 5.2329 8.2045 11.2560 14.3434 17.4490 4.0 2.5704 5.3540 8.3029 11.3349 14.4080 17.5034 5.0 2.6537 5.4544 8.3914 11.4086 14.4699 17.5562 6.0 2.7165 5.5378 8.4703 11.4773 14.5288 17.6072 7.0 2.7654 5.6078 8.5406 11.5408 14.5847 17.6562 8.0 2.8044 5.6669 8.6031 11.5994 14.6374 17.7032 9.0 2.8363 5.7172 8.6587 11.6532 14.6870 17.7481 10.0 2.8628 5.7606 8.7083 11.7027 14.7335 17.7908 15.0 2.9476 5.9080 8.8898 11.8959 14.9251 17.9742 20.0 2.9930 5.9921 9.0019 12.0250 15.0625 18.1136 30.0 3.0406 6.0831 9.1294 12.1807 15.2380 18.3018 40.0 3.0651 6.1311 9.1986 12.2688 15.3417 18.4180 50.0 3.0801 6.1606 9.2420 12.3247 15.4090 18.4953 60.0 3.0901 6.1805 9.2715 12.3632 15.4559 18.5497 80.0 3.1028 6.2058 9.3089 12,4124 15.5164 18.6209 100.0 3.1105 6.2211 9.3317 12.4426 15.5537 18.6650

00 3.1416 6.2832 9.4248 12.5664 15.7080 18.8496

14-7

-1

\4-i

SECTION 15. CONSTANTS AND CONVERSION FACTORS.

15.1. Mathematical- Constants.

e = 2i7182818. Hn 10 = 2.3025851.

•n = 3.1415926. Y = 0.5772156. (Euler's constant)

15 .2 . Physical Constants.

Standard acceleration of gravity

Joule's constant

g Q = 9.80665 m/s = 32.1742 f t / s 2

= 1 . 0 N»m/J = 778.16 f f l b f / B t u

Stefan-Boltzmann constant a = 1.355 x 1 0 " 1 2 cal /s 'cm 2 -K 4

= 1712 x 1 0 ~ 1 2 Btn/hr'f t 2 -R 4

= 5.673 x 10" 8 W/ra2'K4

Universal gas constant R = 8.3143 J/mol«K

= 0.08205 &*atm/mol*K =1.9859 Btu/lbm'mol'R = 1545.33 ft*lbf/lhm*mol*R = 6.41 x 10~ 4 J/kg-mol-K

15-1

15.3. Conversion Factors.

TABLE 15.1. Conversion factors for length.

1 m 1 cm lUm 1 A 1 in. 1 ft 1 yd

ym A in. ft yd

1 100 106 1 0 1 0 39.37 3.280 1.0936 0.01 1 104 10 8 0.3937 0.0328 0.0109 io- 6 10" 4 1 104 0.3937 x 1 0 - 4 0.0328 x 10" 4 0.0109 x 10" 4

ID" 1 0 10" 8 io- 4 1 0.3937 x 1 0 - 8 0.0328 x 10~ 4 0.0109 x 1 0 - 8

0.0254 2.540 25.4 x 104 2.540 x 10 8 1 0.0833 0.0277 0.3048 30.48 30.48 x 10 4 30.43 x 10 8 12 1 0.3333 0.9144 91.440 91.440 x 10 4 91.440 x 10 6 36 3 1

Ul I to

TABLE 15.2. Conversion factors for area.

cm 2 m . 2 in. ft2 y« 2

1 cm 2 1 lO" 4 0.1550 1.07639 1.1960 x 1 0 - 4

1 m = 104 1 1550 10.7639 1.1960 1 in.2 = 6.4516 6.4516 x 10" 4 1 6.9444 7.7160 x 1 0 - 4

1 f t 2 = 929.034 929.034 x 1 0 - 4 144 1 0.11111 1 yd 2 = 8361.307 8361.307 x 10" 4 1296 9 1

TABLE 15.3. Conversion factors for volume.

i

can in.3 ft3 ml liter ;gal

1 cm 3 = 1 610.23 x 10~ 4 35.3145 x 1 0 - 4 999.972 x 1 0 - 3 999.972 x 10" -6 264.170 x 10~ 6

1 in. 3 = 16.3872 ;i 5.7870 x 1 0 - 4 16.3867 16.3867 x 10" -3 432.9"00 x 10" 5

1 f t 3 = 283.170 x 102 1728 1 28.3162 x 10 3 28.3162 7.4805 1 ml = 1.000028 610.251 x 1 0 - 4 353.15 1 x 1 0 - 7 1 0.001 264.178 x 10~ 6

1 liter = 1000.028 61.0251 353.154 x 10" 4 1000 1 264.178 x 10~ 3

1 gal = 3785.434 231 133.680 x 1 0 - 3 3785.329 3.785329 1

TABLE 15.4. Conversion factors for mass.

lb slugs g kg ton 1 lb = 1 0.03108 453.59 0.45359 0.0005 1 slug = 32.174 1 1.4594 x 10 4 14.594 0.016087 1 g = 2.2046 x 10' -3 6.8521 x 10~ 5 1 io- 3 1.1023 x lo - 6

1 kg = 2.2046 6.8521 x 10~ 3 103 1 1.1023 x lo - 3

1 ton = 2000 62.162 9.0718 x 10 5 907.18 1

TABLB 15 .5 . Conversion factors for density.

lhm/ft 3

1 lbm/f t 3 = 1 1 s l u g / f t 3 = 32.174 1 lbni / in . 3 = 1728 1 lbm/gal = 7.4805 1 g/cra3 = 62.428

s lug / in . lbm/ii

0.03108 5.787 x 1 0.01862 53.706 1 0.2325 4.329 x 1.9403 0.03613

i. lbm/gal g/cm

1 0 - 4 0.13386 0.01602 4.3010 0.51543 231 27.680

10~ 3 1 0.11983 8.345 1

15-4

TABLE 15.6 . Conversion factors for pressure.

lbf/in.2 2 dyne/cm kgf/cm2 in. H(j nun Hg in. H O atm bar

1 lbf/in.2 = 1 689.473 x 10 2

0.07031 2.0360 51.715 27.71 0.06805 0.06895

1 dyne/cm2 = 145.0383 1 101.972 295.299 750.062 4.0188 986.923 lO"6

-7 X 10 ' x io" 8 -7 x 10 ' x 1 0 - 6 -4 x 10 * -9

1 kgf/cm2 = 14.2234 980.665 x 10 3

1 28.959 735.559 394.0918 967.841 x 10 3

980.665 x 10" 3

1 in. Hg = 0.4912 338.64 0.03453 1 25.40 13.608 0.03342 0.03386 15- x 10 2

Ul 1 ram Hg = 0.01934 1333.223 1.3595 -3

x XQ

0.03937 1 0.5358 1.315 x 10~ 3

1.333 x 10

1 in. H 2o = 0.03609 24.883 2.537 0.0735 1.8665 1 2.458 2.488 1 in. H 2o x 10 2 x 10" 3 x 10" 3 x 1 0 - 3

1 atra = 14.6960 101.325 x 10 4

1.03323 29.9212 760 460.80 1 1.01325

1 bar = 14.5038 106 1.01972 29.5299 750.0617 401.969 986.923 „ -.n-3

1

TABLE 15.7. Conversion factors for energy.

f f l b f abs joule int joule c a l I.T. cal

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 int joule = 0.737682 1.000165 1 0.239045 0.238889 1 cal 3.08596 4.18401 4.1833 1 0.99934 1 I .T. cal = 3.08799 4.18676 4.18605 1.000657 1

1 Btu 778.16 1055.045 1054.866 252.161 251.996 1 int kWhr = 265.567 x LO4 360.0612 x 1 0 4 360.000 x 1 0 4 860.565 x 1 0 3 860.000 x 10 3

1 hp«hr = 198.0000 x 1 0 4 268.4525 x 1 0 4 268.082 x 1 0 4 641.615 x 1 0 3 641.194 x 10 3

1 l i t er 'a tm = 74.7354 101.3278 101.3111 24.2179 24.2020

Btu int kW hr hp'hr liter*atm

1 f f l b f 1 abs joule 1 int joule 1 cal 1 I.T. ca l 1 Btu 1 int kWhr 1 hp'hr 1 l i ter 'atm

128.5083 x 10 947.827 x 10~ 6

0.947988 x 10" 396.572 x 10~ 5

396.832 x 10~ S

1 3412.76 2544.46 0.09604

-5 -9 376.553 x 10 277.731 x 10~ 9

2.777778 x 10" 116.2028 x 10" 116.2791 x lo" 293.018 x 10 - 6

1 0.74558

505.051 x 10 -9 -9

281.718 x 10 -7

372.505 x 10 3.7256 x lo" 7

155.8566 x lo" 155.9590 x 10 293.010 x 10" 1.3412 1 377.452 x 10"

-8

133.8054 x 10" 986.896 x 10 - 5

9.87058 x 10~ 3

412.918 x 1 0 H

413.189 x i o H

10.4122 255.343 x 10 2

264.935 x 10 2

1

Table 15.8. Conversion factors for specific energy.

abs joule/h cal/g r.T. cal/gm Btu/lb f f lb£/ lbm int . kW hr/g hp hr/lb f t V

abs joule/g . 1 0.2390 0.2388 0.4299 334.53 2.777 x 10"7 1.690 x ltf* 10763 cal/g = 4.184 1 0.9993 1.7988 1399.75 1.162 x 1 0 - 6 7.069 x 10 4.504 x 10 4

l .T. cal /g - 4.186 1.0007 1 1.8 1400.69 1.163 x 1 0 - 6 7.074 x 10 4.506 x 10 4

Btu/lb = 2.326 0.5559 0.5556 1 776.16 6.460 x 10~ 7 3.930 .< 10** 25,037 ff lbf/ lbra = 2.989 x 10~ 3 7.144 x 10"* 7.139 X 10~ 4 1 285 x 10~ 3 1 8.302 X l O - 1 0 5.051 x lo" 7 32.174 int . kWhr/g - 3.610 x 10 6 860,565 860,000 1 548 x 10 6 1.2046 x 10 9 1 608.4 3.876 x 1 0 1 0

hp'tar/lb f t W -

5919 S.291 X 10~ S

1414.5 2.220 x 10" 5

1413.6 2.219 X 10" 5 3

2545 994 x 10" 5

1.9B0 X 10 6

0.03108 0.001644

2.580 x 1 0 " 1 1

1 1.567 x io" 8

6.370 x 10 7

1

Table 15.9. Conversion factors for specific energy per degree.

abs jou le /g»K Cal/g«K I .T. c a l / g •K B t u / l b ' R W"s/kg*K

abs joule/g*K = 1 0 .2390 0 .2388 0 .2388 1 0 3

cal /g*K » 4 .184 1 0 .9993 0 .9993 4184

I . T . ca l /g*K = 4 .186 1.0007 1 1 4186

b t u / l b ' H =

Ws/kg«K =

4 .186

l O " 3

1.0007

2 .390 x l o ' -4 2

1

388 x 10" -4 1

2 .388 x 1 0 - 4

4186

1

Table 15.10. Conversion factors for thermal conductivity.

cal/s , cm-°C B t u / h r - f f °P Btu/h r . f t

2 . ' "F/in. W/ m-°C

1 eal / s 'cnr C 1 Btu/hr ' f t - F 1 B t u / h r ' f t 2 - F/ in . 1 W/m C II

n n

II 1

4.13 x 1 0 - 3

3.45 x 1 0 - 4

23.89

241.9 1 0.0833 5780

2903 12 1 69350

0.04.183 1.73 x i o - 4

1.44 x 1 0 - 6

1

ui xaoie J.: >.XJ.. tonvecsii co

f t 2 / h r stokes m2/hr m2/s

f t 2 / h r = 1 0.25806 0.092903 2.58 x 1 0 - 5

stokes = 3.885 1 0.36 io - 4

m 2/nr = 10.764 2.778 1 1.778 x 10~ 4

m / s = 38,750 1 0 4 3600 1

TABLE 15.12. Conversion factors for heat flux.

Btu W k c a l c a l

f t 2 « h r 2 m

. 2 hr«m

2 s«cm

B t u / f t 2 ' h r = 1 3 .154 x 1 0 " 8 2 .173 7.536 x 1 0 ~ 5

W/ra2 = 3 .170 x 1 0 7 1 8 .600 x 1 0 7 2389 2

kcal /hr-m = 0.3687 1 .163 x i o " 8 1 36000 2

ca l / s*cm = 13277 4 .1868 x 1 0 ~ 8 2 .778 x 1 0 - 5 1

TABLE 15.13. Conversion factors for heat transfer coefficient.

Btu W k c a l c a l

h r « f t 2 « ° F m 2«°C 2 o„ s*cm • C hr'ra 2«°C

B t u / h r - f t 2 « ° F = 1 5 .678 x 1 0 " 8 1.356 x 1 0 " 4 4.883

W/m 2'°C 1.761 x 1 0 7 1 2391 8.600 x 1 0 7

c a l / s e c - c m - ° C = 7376 4 .186 x 1 0 " 4 1 36000

k c a l / h r - m 2 - ° C = 0.2049 1 .163 x 1 0 " 8 2 . 7 7 8 x 1 0 " 5 1

15-9

Is-10

SECTION 16. CONVECTION COEFFICIENTS

16.1. Forced Flow in Smooth Tubes.

16.1.1 Fully Developed Laminar Flow (Source: Ref. 21):

— = 4.364 (constant heat rate)

— = 3.658 (constant surface temp)

16.1.2 Fully Developed Turbulent Flow (Source: Ref. 21):

~ = 6.3 + 0.003 (Re Pr), Pr < 0.1 (constant heat rate)

— = 4.8 + 0.C03 (Re Pr), Pr < 0.1 (constant surface temp)

jp = 0.022 Pr°' 6Re°" 8, 0.5 < Pr < 1.0 (constant heat rate)

— = 0.021 Pr ' Re , 0.5 < Pr < 1.0 (constant surface temp)

jp = 0.0155 Pr°" 5Re°" 9, 1.0 < Pr < 20

f = 0.011 Pr°- 3Re 0- 9, Pr < 20

Pr - Prandtl number, Re = Reynolds number

16.2. Forced Flow Between Smooth Infinite Parallel Plates.

16.2.1. Fully developed laminar flow (Source: Ref. 21):

hs — = 4.118 (constant heat rate on both sides) hs — = 2.693 (constant heat rate on one side, other side insulated)

16-1

hs — = 3.77 (constant surface temperature on both sides) — = 2.43 (constant surface temperature on one side, other side insulated)

s = spacing of plates

16.3. Forced Flow Parallel to Smooth Semi-Infinite Flat Plates Laminar Flow. 16.3.1. Laminar flow: — = 0.332 P r 1 / 3 R e x

1 / 2 [l - 'tX0/X) 3 / 4 ] ~ 1 / 3 , (constant surface temperature)(Source: Ref.7)

X = unheated starting length X = distance from leading edge

I 5 = 0.453 P r 1 / 3 R e x1 / 2 [l - ( X Q / X ) 3 / 4 ] ~ 1 / 3 , (constant heat rate)

(Source: Ref. 21) 16 .3 .2 . Turbulent flow (Source: Ref. 21):

| * = 0.0295 P r ° ' 6 R e x

0 - 8 [ l - (XQ/X) 9 / - L 0 ] ~ 1 / 9 , (constant surface temperature)

hx k

0.6 0.8 T 9/10~l-l/9 = 0.0307 Pr Re [_1 - (X /X) J , (constant heat rate)

16.4. Fully Developed Flow in Smooth Tube Annuli.

16.4.1. Laminar flow (Source: Sef. 21):

h i ( d o - d i > Nu. . 11

k 1 -

V d o - di>

«W 9t Nu oo

k 1 - (q./q )9* ni! ô o

(Subscripts i and o refer to inner and outer surfaces, q is surface heat flux, Nu., and Nu _ are inner and outer surface Nusselt numbers when only one

11 OO surface is heated and 6 is an influence coefficient given in Table 16.1.)

16-2

TABLE 16.1 . Tube-annulus solutions for constant heat rate in ful ly developed laminary flow and temperature pro f i l e s .

_ — r.r Nu.. Nu 9 . ® i o 11 oo l o 0

0.2

0.4

0.6

0.8

1.0

16 .4 .2 . Turbulent flow (Source: Ref. 22):

7 (d - d.) - 0.023 Re°" 8 Pr°" 4 (d / d . ) ° ' 4 5 , R e A j 1 0 4 , & d = d - A. k o i Ad o l Ad o l

CO 4 .364 CD 0

8.499 4 .883 0 .905 0 .1041

6 .583 4 .979 0 . 6 0 3 0 .1823

5 .912 5 .099 0 . 4 7 3 0 .2455

5.580 5 .240 0 . 4 0 1 0 .299

5.385 5 .385 0 .346 0 .346

16.5. Forced Flow Normal to Circular Cylinders.

16 .5 .1 . Local Coefficients (Source: Ref. 22):

" - 1 . " P r 0 * 4 H ^ - 5 [ l - ( 6 / 9 0 ) 3 ] , 0 < 9 < 80°

6 = cylinder angle from stagnation point

16 .5 .2 . Average Coefficients (Source: Ref. 7 ) :

J* = 0.43 + C Re™ P r ° - 3 1

K a

TABLE 16.2 .

R e d C m

1-4,000 0.533 0.500 4,000-40,000 0.193 0.618

40,000-400,000 0.0265 0.805

16-3

16.6. Forced Flow Normal to Spheres.

1 6 . 6 . 1 . Average Coefficients (Source: Ref. 7 ) :

r^ = 0.37 Re°' 6 P r 0 , 3 3 , 20 < Re , < 150,000 k d a

^ = 2 + 0.37 Re°"6 P r 0 ' 3 3 , Re,, < 20 K a d

16-7. Free Convection on Vertical Plates and Cylinders.

16.7.1. Local Coefficients (Source: Ref. 7):

jp = 0.508 (Pr)1/2(0.952 + P r ) ~ 1 / 4 ( G r x ) 1 / 4 , GrPr < 10 4

jj 5 = 0.0295 (Pr) 7 / 5 [l + 0.494 (Pr) 2 / 3]" 2 / 5(Gr) 2 / 5, GrPr > 10 4

16.8. Free Convection on Horizontal Cylinders.

16.8.1. Average Coefficients (Source: Ref. 23):

f = C (Grd Pr,1

TABLE 16.3.

Gr dp r C m

0-10~5 0.40 0 IO~ 5-IO _ : L 0.97 1/16 10 - 1-10~ 4 1.14 1/7 10 4-10 9 0.53 1/4 10 9-10 1 2 0.13 1/3

16-4

16.9. Free Convection From Horizontal Square Plates.

16.9.1. Average Coefficients (Source: Ref. 23):

•^ = C (Gr Pr) m, L = plate dimension K L

TABLE 16.4.

Condition Gr pr it

Upper surface heated or lower surface cooled Lower surface heated or upper surface cooled Upper surface heated or lower surface cooled

10 5 - 2 x 10 7 0.54 1/4

3 x 10 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 Coefficients (Source: (Ref. 5):

— = 2 + 0.43 (Gr. P r ) 1 / , 4f 10° < Gr. Pr < 10 5

K a a

16.11. Free Convection in Enclosed Spaces.

16.11.1. Enclosed Vertical Air Spaces (Source: Ref. 1)i

r 1 Gr s < 2,000

0.18 (Gr ) 1 / 4 ( s / L ) 1 / 9f 2,000 < Gr < 20,000

s s I 0.065(Gr ) 1 / 3 ( s / L ) 1 / 9 , 20,000 < Gr < 10 7

s s

16-5

k e = effective thermal conductivity s = width of air space L = length of air space

16.11.2. Enclosed Horizontal Air Spaces (Source: Ref. 1):

0.195 (Gr ) 1 / 4 , 10 4 < Gr < 4 x 10 5

5 S 0.068 (Gr ) 1 / 3 , 4 x 10 5 < Gr

s s

16.12. Film Condensation.

16.12.1. Vertical Plates—Average Coefficients (Source: Ref. 5)i

P a(P a - P v)g

1/3 -1/3 1.47 Re^ , Re. < 1800

<VpA " <V g

1/3 = 0.007 Rejj°'4, Re f c > 1800

Re. = (4hAt )/i„ u„ X, v vw £v H& At = temp difference between saturated vapor and wall in = heat of evaporation Po,ko,po = viscosity, thermal conductivity and density of liquid

at saturation vapor temp

16.12.2. Horizontal Tubes—Average Coefficients (Source: Ref. 5):

^ = 0.725

1/4

16-6

16.13. Fool Boil ing,

_ ^ p _ ^ _ i 1 / 2 . r^f™ i ( s o u r c e : R e £. X9,

At = temp difference between wall and saturated vapor c. ,ut,Pn,Prj, = specific 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 Table 16.5)

TABLE 16.5. Values of C and n (source: Ref. 19).

Surface-fluid combination Csf n

Water-nickel 0.006 1.0 Water-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 ii-Pent ane- chromi urn 0.015 1.7 Ethyl alcohol-chromium 0.0027 1.7 Isopropyl alcohol-copper 0.0025 1.7 35% K CO -copper 0.0054 1.7 50% K C03-copper 0.0027 1.7 n-Butyl alcohol-copper 0.0030 1.7

16-7

• ' • i .

Ib-S

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 = (IT/4) [l + 85(P/S) 0 , 8] k = thermal conductivity of the gas phase P = contact pressure r = height of a micro-element of roughness plus the height 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 = 0.315 ( V A T T ) 0 , 1 3 7

A = interface area (one side) I = effective gap thickness, 3.56(4. + i. ) if (S, + J, ) < 280

uin. (smooth contacts), or 0.46 (JL + t, ) > 280 uin. (rough contacts)

£..,£., - mean (or rms) depths of surface roughness k. = equivalent conductivity of interstitial fluid, for liquids use

k f = k evaluated at t = (t + t,)/2, for gases:

k k = 2 K f 1 + 8 y (v/v) (aL + a 2 - 3 j - a 2)/Pr(y + 1 ) ^ + a 2

4CTAG E 2t 3

+ £1 + E 2 " E 1 E 2

17-1

k_ = fluid conductivity 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 emissivity o = Stefan-Boltzmann constant K = conductivity number, I: (k + k )/2k k k. = conductivity of first solid evaluated at

k_ = conductivity of second solid evaluated at fc =[ f c2 + ( kl f cl + k 2 f c 2 ) / ( k l + k 2 } J / 2

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 Ref. 26 showing the effects of machining processes and surface matching is given in Fig. 17.3.

17-2

TABLE 17.1. Interface conditions for contact data given in Fig. 17.2.

RMS surface Mean contact Curve Material pair finish (Uin.)' Gap material temp (°P)

1 aluminum (2024-T3) 48-65 -4 vacuum (10 mm Hg) 110 2 aluminum (2024-T3) 8-18 —4 vacuum (10 mm Hg) no 3 aluminum (2024-T3 6-8 (not flat) —4 vacuum (10 mm Hg) 110 4 aluminum (75S-T6) 120 air 200 5 aluminum (75S-T6) 65 air 200 6 aluminum (7SS-T6) 10 air 200 7 aluminum (2024-T3) 6-8 (not flat) lead foil (0.008 in.) no 8 aluminum (75S-T6) 120 brass foil (0.001 in.) 200

-J 1 9 stainless (304) 42-60 —4 vacuum (10 mm Hg) 85 W 10 stainless (304) 10-15 —4 vacuum (10 mm Hg) 85

11 stainless (416) 100 air 200 12 stainless (416) 100 brass foil (0.001 in.) 200 13 magnesium (AZ-31B) 50-60 i (oxidized) vacuum (10 mm Hg) 85 14 magnesium (AZ-31B) 8-16 (oxidized) —4

vacuum (10 mm Hg) 85 15 copper (OFHC) 7-9 vacuum (10~ mm Hg) 115 16 stainless/aluminum 30-65 air 200 17 iron/aluminum - air 80 18 tungsten/graphi te - air 270

TABU! 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 10CQ-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 Milled 125-125 Air 200 Parallel cuts, rusted e Cold rolled steel Milled 125-125 Air 200 Parallel cuts, 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 1 416 Stainless Ground 30-30 Air 400 m Stainless Milled 195-195 Air Clean

* Sexier 1

5X10"1

FIG. 17 .1 . Thermal contact coeff ic ient from theory (source: Ref. 25) .

17-5

20

10

/ > ' V h i ~ p 2 / 3 J ' I i i i il i i ' I i i 11 I I I I 11.

5 10 50 100 200 500 1000

p - lb/in. 2

FIG. 17 .2 . Thermal contact coefficient data (source: Ref. 19),

17-6

0.016

0.014

0.012

0.010 r-

0.008

0.006 -

c o 0.004

0.002

c o o

0 100 200 300 400 500 Contact pressure — lb/ in. 2

Parallel cuts

Perpendicular cuts

FIG. 17.3. Thermal contact coeff ic ient for bare s t ee l surfaces, (source: Ref. 26, sect ion G502.5, p. 5) .

17-7

ft-%

Tlarmal prgpattlea of M 1 K M « Mti ' . i

"I - » ' Uq/ifi . io" J MAa-u - 10 " m/««ii « in

JUiBlnm Fata 1.70 O . U 1 . 4 1.01 1034 (4.51 Ca, l ,St KQI 3.TT O . M 1 . 1 0.9*

901* (4 .U Hg. 0.51 Mol 3.15 0.10 1 . 1 0.91

tot* ii.i\ * ? , o.*» %%\ 3.70 O . M 3 . 1 0.1)

•arUIuMi rare 1.15 1.14 3 . 0 0.9)

a v a i l * i Puta 7.1* 4.54 » . H

Copp»ti tan l . M 0.34 4 . 0

CCMMcelal tuans* ( t « *"1 I . H 0.41 I . I 0.90

Il l icit MM* (3St 1st 1.47 0.41 1 . 1 0.19

• m l b i n ( l i t in , 0.1* 5tii I . U 0.31 0.75 0.33

Sanaa l U n t (31* «". 15» Nil 1.59 0.40 0.35 D.D7

C*tonlc*«l (10t Ml. 3t la) i . l i 0.30 1 . 5 0.44

conatantan (404 Mil l.fD 0.43 0.21 O.Df

Mmiaanln 1131 Rn, n Ml 1.11 0.40 0.13 0.07

Soldi Fora H-12 O . U 1 . 3 1.17

Irani run 7 . n 0-43 o.ao

Mcauaht two. i < a . » c> i . n O.tt O.ID 0.17

Stay caat Iron (3.01 C, O.tt 311 7.15 D.42 0.31 0.10

SAI 10H ataal (D.ll C, 0.3* Mnl 7 . U 0.47 0.44 a.u

AMI U4D ataal 11.M HI. 0.TI CrJ 7 . H 0.44 0.3* 0.10

tflckal atsal IM Ml, O.tt Hnj 7 . H 0.44 0.29 o.oa Invar (3U VI, SI Co) • .00 0.4C O.I4

SAI 4130 i t H l (11 Cr, 0.51 Hn> 7, It 0.44 0.41 o.u M i l M4 auLalaa* (1)» Ci. 1M HI) 7 . W 0.30 0.15

MSI H i atalnlasa ,U» Cr, U t Mil 0.00 0.44 0.14

l u d i Part 11.34 0.13 0.11

Soldat IK* lb. 101 SB) 10.30 0.37

Soldar ISO* Pn> SOI So) I . H 0.11 0.44 0.3*

Lit bloat n r a 0.53 1.17 0.77 0.37

•UoaaaiBBi f u n 1.74 1.00 1.9 . 0.10 (ft U. It 11) 1.14 1.00 O.70 0.10 (131 AI, » Si) 1.11 o.»i i

O.St 0.11

Moljbdanuai Pur a ID. 33 0.39 1 . 4 0.5*

•Ickali Fata 4.10 0.49 0.91 OmanlcUl I4.H Alt 1.2* 0.94 0.19 0.04 Nonal (301 Cu, L41 Fa) 1.14 0.4) 0.23 0.0} Inoootl X-TS0 l i l t Cr, I t Fa) 4.51 a.M n.ia ti.OJ

MlduoM (10« Cr) 1.40 0.45 0.11 0.0) NlcUca* V (241 Fa, 1** Cr] O . U 0.44 0.10 0.01

Mfobluai Poia 1.97 0.37 0.94 PlltltaU Plica 31.45 0.13 0.71 Plateoliau Fota 1*.I4 0.19 0.(7 0.3] ttarlaai Fata 31.04 0.14 0.41 0.1*

IllVMt FBI* 10.41 0.34 4 . 3 1.71 Starling CT.lt Cm 10. Jl 1 . 5

•atactic ( l i t Cu] 10.04 1 . 3

Tim Fur. 7.10 0.33 0.a"7 0.41 Soldu (401 Fbi 1.44 0.11 0.91 0.13

Mtta ldu Fun l i . l 0.14 0.91 0.3S YnngitFnt I u » M . J 0.11 I . t 0.73 Tltanliai Fu» 4.IL 0.90 0.33 0.10

A-U0 ft (SI Al, 3-S» tat 4.41 0.91 0.075 0.0}

Oranlnai Fata IF.07 0.13 O.ll 0.13 Ilnei Puta 7 . U O.lf 1 . 2 0.43

l » V n .o t o « 30 90 ioo ion 4os too Mo IOOO uoo

115. 117. 13.) 3.0 2.4 1.4 1.1 7.1 0.01 0.11 0.40 0 . (1 1.0 1.4 0,00 0.17 0.40 D.*J 0.11 O.M 1.7 3.1 3.1 3.0

II.0 14.0 40.0 9.9 3.01 1.41 1.1* 1.07 D.19 0.7) 1.) t.O 3.3 1.4. 1.1 0.(7 0.11 0.T1 3.45 0.(1

IK. 105. 13.3 4.1 I .I 3.1 1.1 l.T l . f 1.* O.IB 0.43 0.11 1-3

0.51 0.14 1.1 1.5 D.ll 1.01

O.011 0.OT 0.15 0.17 0.30 0.11 0.30 0.11 D.l l 0.01 0.15 0.17 0.19

O.015 0.011 3.012 0.11 0.17 0.1B

34.3 19.0 4.3 3.5 ] . ] 1.1 3.0 3.1 3.0 3.C T.l 13.0 l.T 1.) O.M 0.(1 0.51 a .U 0.11 0.30

0.51 0.11 0.)1 9.31 0.34 0.11 0.33 0.30

D.DI 0.33 0.14 0.43 0.34 D. l l 0.11 0.11 O.J* 0.10 0.34 0 . ) ) 0.14 0.13 a.00 0.13 0.14 0.11 0,73

0.0( 0.1C 0.1T 0.37 0.001 0.031 D.Of 0.04 0.13 0.17 0.2a 0.33 0.31 0.1> 0.001 D.031 O.M 0.09 9.13 0.15 0.11 0.11 0.3) 0.35

1.71 0.51 0.44 0.40 0.37 0.14 O.lt

f . l 7.2 3 . ) 1.1 0.(1 0.71

11.7 11.1 >.» 1.7 1.* V.5 1.5 1.5 0.50 0.(3 O.U

0.10 0.44 O.fl

1.45 3.77 1.0 l . ( 1.4 1.1 1.) 1.1 1.1 l.D

(.0 l . i ) .4 l . t 1.1 D.IO 0.44 0.(7 0.T3 D.T(

0.04 0.14 0.17 0.30 0.010 0.035 0.0(5 0.045 O.10 0.1) 0.17 0.21 0.35 0.2*

0.14 0-i.J O.U 0.35 0.31 O.U 0.15 0.11 0.33 O.K

3.3 3.1 0.71 0.51 0.51 0.19 0.11 0.41 0.44 0.4* U . ) 4.1 1.1 0.71 0.72 0.73 0.71 0.74 0.71 0.13

O.U 0.41 0.10 14.0 1.4 O.M 0.40 0.51 0.44 0.44 0.44 0.41 0.4(

1(1. II . 7.0 4.5 4.1 4.3 4.1 .1 1.7 1.1

9.15 6.73 D.tt 0.94 0.93

O.S1 o.st 0.10 0.9) 0.9) 0.(0 9.41 3 . 4 2 . 0 1 . * 1 . 4 1 . 3 1 . 2 1 . 1

O . U 0.39 0.39 0.0*1

0.1) O.U

0.30 0.14

0.21 0.33

0.22 0.35 o.ia O . M a.M 0.44 0.4*

18-1

http://CT.lt

TABLE I B . 2 . Thermal p c o p e r t i e s o£ misce l l aneous s o l i d s (sourcei ReEs. 35 t o 3B).

P c o p e r t i e s a t 300 K Thermal c o n d u c t i v i t y fk) , (W/m'K) x 10

(kg/m3) (m /s) x 10 10 K 400 600 BOO 1000 1200

00 I to

Ceramics) ( F o l y c r y s t a l U n e , 99.5% pur i ty ,96% s o l i d )

(high p u r i t y fused)

BeO HgO SiO Th0 2

T i o 2

zca 2

Glassesi Boco&Uicate (Pycex) Soda l ime (75% S i 0 2 ) V i t r e o u s s i l i c a (1001 S i c y Zinc ccown (65% Si(> 2)

I n s u l a t i o n s : (Foe high temps.) Alumina, fused (90% M ^ ) Asbestos paper ( laminated) Diatomaceoi-a e a r t h s i l i c a (powder) F i r e b r i c k (50% SiO , 37% A1 2 0 3 J

Magnesi te (S5% MgO)

Micro q u a r t z f i b e r (Hlanket)

Rockwool ( loose) '

Z i r o o n i a (g ra in )

I n s u l a t i o n s : (Foe lew temps.) Expanded g l a s s (Foaaglas) F i b e r g l a s s (board) F i b e r g l a s s (b lanke t ] S i l i c a ae roge l (powder) Po lys ty rene foam ( labnabs) Po lys ty rene foaa ( I D - atmabs)

3.B4 2.97 3.21 2.21 9.58 3 .91 5.2B

2 .21

2.52 2 .21 2.60

2.70 0 .35

0.79 1.00 0.92 0.75 0.23 Q.71 0.46

0 .71 0.66 1.00 0.67

G.B3 D.B4

0.B1 0.92 0.19 1.13 0.05 0.84 0.16 1.81 0.90

0.17 0.18 0.05 0.08 0.08 0,08

36.0 272.0

4B.0 1.4

13.0 6.0 1.6

1.1

0 .9

1.4

1.1

5.2 0.046

0.12 0.92 0.16 0.008 0,06 0.03

0.007

0.007 0.005 0.006 0.006

0.02 0.002

0.34 0.005 0.050 0.002 0.036 0.005 0.046 0.18 0.002

0.062 0.040 0.038 0.022 0.038 0.017

550, 260. 160. 100. 80. 4240. 1960. 1110. 700. 470. 750. 350. 220. 140. 90. 11. 15. 18. 22. 29.

1B0. 100. 70. 50. 40. 100. 70. 50. 40. 30.

17. IB. 19. 19.

8. 11. 11. 15. 9. 13.

0.73

0.79 l.U 1.3

0.C1 5.2 6.6 8.4 10.

0.48 0.66

0.89 1.0

1.3 1.3

1.7

1.9 2.1 2.2 2.8 3.5

0.31 0.51 0.12 0.24

0,17 0.12 0.24

0,12

TABI£ IB .2 . (Continued.)

Propert ies at 300 II Thermal conduct iv i ty (k ) , <W/m*K) x 10" 2

P. c . k. a. Material ( k g / a 3 ) x 1 0 " 3 (J/kg'R) x 1 0 _ U K/m'K | n 3 / s ) * 10* 10 K 20 50 100 200 400 GOO BOO 1000 1200

P l a s t i c a c A c r y l i c , PMKA (P lex ig la s ) 1.18 1.4G 0.16 0.001 0.6 0.7 1.2

Nylon 6 1.16 1.59 0.25 0.001 0.3 0 .9

Po lyv iny l ch lor ide (r ig id) 1.40 1.00 0.15 0.001 1.5 1.6 1.5

Tef lon , PTPB Polye thy lene , h igh den a i t y

Rocks:

2.16 D.95

1.05 2 .30

0.40 0.E0

0.002 0.002

1.0 1.5 2 .0 2.S 4 . 5 5.5

Granite 2 .60 0.79 3.4 0.017 30 . 24.

Harble 2.50 0.88 1.8 O.00B 17. 11 .

Sandstone 2 .20 0.92 5.3 0.026 4 4 . 30.

Shale

Hoods: (across g r a i n , oven dry)

Sa laa

2. GO

0.16

0 .71 1.8

0.059

0.010 15 . 14.

I-" Douglaa £ l r 0.46 2.72 0.11 0.001

f Oak, red 0.67 2.38 0.17 0 .001 W P ine , v h l t e

Redwood

Misce l laneous:

Carbon black (powder)

0.40 0.42

0 .19

2 . B0 2.90

0.84

0 .10 0.11

0.021

O.0D1 0.001

0.001

Carbon (petroleum coke)

Graphite (*TJ|

2.10 0.84 1.9 0.011 21 . 25. 2B. 30.

(H t o grains) * 1.73 0.B4 129.0 0.89 5. 25. 170. 580. 1200. USD. 950. 770. 640.

( 1 t o grains) Graphite ( c y t o l y t i c )

1.73 0.B4 98.0 0.67 4. 19. 130. 420. B60. 900. 730. 590. 490.

(n t o gra ins) 2 .20 0.84 2000.0 10.8 810. 4200. 23000. 50000. 32000. 14600. 9300. 6800. 5300.

( 1 to graina) Concrete Hlca Rubber (hard) Gypaun board Sand (dry)

2.20 2.20 1.96 1.19 0.82 1.52

0.84 0.67 0.88 1.88

0 .80

9.0 l .B 0.43 0.16 0.11 0 .33

0.05 0.012 0.002 0.001

0 .003

270. 1100. 1000. 390. 150. 70. 40. 30. 20.

TABIiE 1 8 . 3 . Thermal p rope r t i e s of some metals a t 10 K above t h e i r melting

po in t (source: Refs. 35 and 38) .

mp. k, PF c. Y Metal K W/m'k kg/m3 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

18-4

TABLE 18.4. Room temperature total emissivities (source: Ref. 37).

Silver (highly polished) 0.02 Brass (polished) 0.60 Platinum (highly polished) 0.05 Oxidized copper 0.60 Zinc (highly polished) 0.05 Oxidized steel 0.70 Aluminum (highly polished) 0.08 Bronze paint 0.80 Monel metal (polished) 0.09 Black gloss paint 0.90 Nickel (polished) 0.12 White lacquer 0.95 Copper (polished) 0.15 White vitreous enamel 0.95 Stellite (polished) 0.18 Asbestos paper 0.95 Cast iron (polished) 0.25 Green paint 0.95 Monel metal (oxidized) 0.43 Gray paint 0.95 Aluminum paint 0.55 Lamp black 0.95

18-5

TABLE 18.5. Total emissivities of miscellaneous materials (source; Ref. 37).

Temp, Emis- Temp, Emis-

Material °c s i v i t y Mate r i a l °C s i v i t y

Alloys I ron , rus t ed 25 0.65 20Ni-25Cr-55Fe, oxidized 200 0.90 wrought, d u l l 25 0.94

500 0.97 oxidized 350 0.94 60Nl-12Cr-2BFe, oxidized 270 0.69 Lead, unoxidized 100 0.05

560 0.82 oxidized 200 0.63 B0Ni-20Ct, oxidized 100 0.87 Mercury, unoxidized 25 0.10

600 0.87 100 0.12 1300 0.S9 Molybdenum, unoxidized 1000 0.13

Aluminum, unoxidized 25 0.022 1500 0.19 100 0.028 2000 0.24 500 0.60 Monel meta l , oxidized 200 0.43

oxidized 200 0.11 600 0.43 600 0.19 Nickel , unoxidized 25 0.045

Bismuth, unoxidized 25 0.048 100 0.06 100 0.061 500 0.12

Bra s s , oxidized 200 0.61 1000 0.19 600 0.59 oxidized 200 0.37

unoxidized 25 0.035 1200 0.85 100 0.035 Platinum, unoxidized 25 0.037

Carbon, unoxidized 25 0.81 100 0.047 100 D.81 500 0.096 500 0.81 1000 0.152

Chromium, unoxidized 100 0.08 1500 0.191 Coba l t , unoxidized 500 0.13 S i l i c a br ick 1000 0.80

1000 0.23 1100 0.B5 Columbian, unoxidized 1500 0.19 S i l v e r , unoxidized 100 0.02

2000 0.24 500 0.035 Copper, unoxidized 100 0.02 S t e e l , unoxidized 100 0.08

l i q u i d 0.15 l i qu id 0.28 oxidized 200 0 . 6 oxidized 25 o.eo

1000 0 . 6 200 0.79 calor ized 100 0.26 600 0.79

500 0.26 Stee l p l a t e , rough 40 0.94 ca lor ized , oxidized 2 0 0 0.18 400 0.97

600 0.19 c a l o r i z e d , oxidized 200 0.52

P i c e br ick 1000 0,75 600 0*5? Gold, unoxidized 100 0.02 Tantalum, unoxidized 1500 0.21

500 0.03 2000 0.26 Gold enamel 100 0.37 Tin, unoxidized 25 0.043 I r o n , unoxidized 100 0.05 100 0.05

oxidized 100 0.74 Tungsten, unoxidized 25 0.024 500 0.84 100 0.032

1200 0.89 500 0.071 c a s t / unoxidized 100 0.21 1000 0.15

l i q u i d 0.29 1500 0.23 c a s t , oxidized 200 0.64 2000 0.28

600 0.78 Zinc, unoxidized 300 0.05

c a s t , s t rongly oxidized 40 250

0.95 0.95

18-6

TABLE 18.6 . Electrical re s i s t iv i ty of sane canon metals (source: Ref. 37).

Resistivity, Temp. Specific Melting (ifl'cm coefficient gravity, point.

Metal 20°c 20°C g/cm °c Advance. See constantan Aluminum 2.824 0.0039 2.70 659 Antimony 41.7 .0036 6.6 630 Arsenic 33.3 .0042 5.73 — Bismuth 120 .004 9.8 271 Brass 7 .002 8.6 900 Cadmium 7.6 .0038 8.6 321 Calido. see nichrome Climax 87 .0007 8.1 1250 Cobalt 9.8 .0033 8.71 1480 Constantan 49 .00001 8.9 1190 Copper: annealed 1.7241 .00393 8.89 1083

hard-drawn 1.771 .00382 8.89 — Eureka. See constantan Excello 92 .00016 8.9 1500 Gas carbon 5000 -.0005 — 3500 German silver, 18% Ni 33 .0004 8.4 1100 Gold 2.44 .0034 19.3 1063 Ideal. See constantan Iron, 99.98* pure 10 .005 7.8 1530 Lead 22 .0039 11.4 327 Magnesium 4.6 .004 1.74 651 Manganin 44 .00001 8.4 910 Mercury 95.783 .00089 13.546 -38.9 Molybdenum, drawn 5.7 .004 9.0 2500 Monel metal 42 .0020 8.9 1300 Nichrome 100 .0004 8.2 1500 Nickel 7.8 .006 8.9 1452 Palladium 11 .0033 12.2 1550 Phosphor bronze 7.8 .0018 8.9 750 Platinum 10 .003 21.4 1755 Silver 1.59 .0038 10.5 960 Steel, E. B. B. 10.4 .005 7.7 1510 Steel, B. B. 11.9 .004 7.7 1510 Steel, Siemens-Martin IB .003 7.7 1510 Steel, manganese 70 .001 7.5 1260 Tantalum 15.5 .0031 16.6 2850 Therlo 8 47 .00001 8.2 — Tin 11.5 .0042 7.3 232 Tungsten, drawn 5.6 .0045 19 3400 Zinc 5.8 .0037 7.1 419

"Trade mark.

18-7

XABI£ 18.7. Thermal conductivity integrals of miscellaneous materials (source: Hef. 76).

A 7- -2 0 = 7 Ik.) r k = J k d t , W/m x 10

C a p p e r 8

A 1 u B

2024-T4

l i n u m

6063-T5

3 t e e l

G l a s s a n d p l a s t i c s H a t e TeffPr

O.F.E i . e . H i - p u r i t y

annealed Brass (Pb)

A 1 u B

2024-T4

l i n u m

6063-T5 SHE 1020

S t a i n

l e s s

Znconel (annealed)

S o f t s o l d e r

G l a s s a n d p l a s t i c s H a t e r i a l K O.F.E i . e .

H i - p u r i t y

annealed Brass (Pb)

A 1 u B

2024-T4

l i n u m

6063-T5 SHE 1020

S t a i n

l e s s

Znconel (annealed)

S o f t s o l d e r J la s s nylon perspex Constantan Ag s o l d e r

6 G. 1 166 0 .053 0 .060 0. ,850 0 .088 0 .0063 0 .0133 0.425 2. .11 x 1 0 - 3 0. .32 x 1 0 - 3 1 .18 x 1 0 " 3 0.024 0 .059 8 1 4 . 5 382 0 .129 0 .197 2. 05 0 .231 0.0159 0 .0348 1.05 4. .43 0. .B0 2 . 3 8 0 .066 0.14B

10 25 . .2 636 0 .229 0 .347 3. .60 0 .431 0 .0293 0 .0653 1.03 6, . 81 1, .48 3 . 5 9 0 .128 0 .26S

15 6 1 . .4 1270 0 .594 0 .872 9. ,00 1.17 0 .0816 0.1B2 4.IB 13 .1 4, ,10 6 .69 0 .375 0.6B8

JO 110 1790 1 .12 1 .60 16. ,5 2 . 2 2 0 .163 0 .356 6 .86 20. , 0 0 .23 1 0 . 1 0 .753 1 .25

25 160 2160 1 .81 2 .51 25. ,8 3 .52 0 .277 0 .592 9 .66 27. ,9 13. .9 14 .4 1.24 1 .92

30 22 B 2410 2 .65 3 . 6 1 36. .5 5 .02 0 .424 0.B82 1 2 . 5 36, ,8 20. ,8 1 9 . 6 1 .81 2 .67

35 285 25BD 3 .63 4 . 9 1 46 . 8 6 .74 0 .607 1.22 15 .3 47. ,1 29 . ,0 2 5 . 9 2 .44 3 .52

40 338 2700 4.76 6 . 4 1 62. .0 8 .67 0 .824 1.60 1 8 . 1 58. .6 38. .5 3 3 . 0 3 . 1 2 4 . 4 7

50 426 2060 7.36 9 . 9 1 89. ,5 1 3 . 1 1.35 2 .47 23 .4 84. .6 60. .4 4 9 . 5 4 .57 6 .62

60 496 2960 1 0 . 4 1 4 . 1 117 1B.1 1.98 3 .45 28 .5 115 85 . 9 6 8 . 3 6 .12 9 . 1 2

70 554 3030 1 3 . 9 IB .9 143 23 .6 2 .70 4 .52 33 .6 151 113 BB.5 7 .75 12 .0

16 5B6 3070 1 6 . 2 2 2 . 0 158 2 7 . 1 3 .17 5 .19 36 .7 175 131 1 0 1 . 8 . 7 5 1 3 . °

90 606 3090 1 7 . 7 2 4 . 2 167 29 .5 3 .49 5 .66 3 8 . B 194 142 110 . 9 .43 1 5 . 2

90 654 3140 2 2 . 0 3 0 . 1 190 3 5 . 5 4 .36 6 .85 4 4 . 1 240 173 132 . 1 1 . 1 1 8 . 7

100 700 3180 2 6 . 5 3 6 . 3 211 41 .7 5 .28 8 .06 4 9 . 4 292 204 155. 12 .8 22 .6

120 788 3270 3 6 . 5 5 0 . 1 253 5 4 . 5 7 . 2 6 1 0 . 6 6 0 . 3 40B 269 200. 16 .2 3 1 . 1

140 874 3360 47 .8 6 5 . 4 293 67 .5 9 .39 1 3 . 1 7 1 . 4 542 336 247. 1 9 . 7 40 .6

160 956 3440 6 0 . 3 8 2 . 1 333 8 0 . 5 1 1 . 7 1 5 . 7 8 2 . 6 694 405 294 . 2 3 . 2 51 .0

180 1040 3520 ' 7 3 . 8 100 373 9 3 . 5 1 4 . 1 1 8 . 3 9 3 . 8 050 475 342 . 2 6 . 9 6 2 . 2

200 1120 3600 BB.3 119 413 107 1 6 . 6 2 1 . 0 105 1030 545 3 9 0 . 30 .6 7 4 . 0

290 1320 3800 128 171 513 139 23 .4 2 8 . 0 133 1500 720 510 . 40 .6 105

300 1520 4000 172 229 613 172 30 .6 35 .4 162 1990 895 630. 51 .6 138

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R - 6 . U.S. Government Printing Office: 1980/10-798-002/5513

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