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461662REPORT NO. RS TR-65-1

SOLUTION OF TRANSIENT HEAT TRANSFI R > lOBLEMS

FOR FLAT PLATES. CYLINDERS. ANP Si ERFS

BY FINITE-DIFFERENCE METH(

by

W. 6. Burleson

R. Eppes, Jr.

March 1965

a U S ARMY MISSILE COMMANDREDSTONE ARSENAl, ALABAS^A ^--

D D <

hi ' ■ ® ' J IDUCIRA l

»0«M 138?. I JAN 63 PREVIOUS EDITION IS OBSOLETE

I FOR ERRATA

AD 461662

THE FOLLOWING PAGES ARE CHANGES

TO BASIC DOCUMENT

■

-

!■ I* I I -'- II »IWIIIII -■■IHM !■■■ .■...■^I™I Mil I ■ ■■-.■ Ml > II ^»11 11 III jl ,. »MI^PMMiMMMaa . JSK.

Jp ¥<* I iji --

HEADQUARTERS U. S. ARMY MISSILE COMMAND Redstone Arsenal, Alabama

In Reply Refer To AMSMI-RAP

HOC

JUL6

lioiA a 22 June 1965

SUBJECT: Errata for Report No. RS-TR-65-1, Subject: Solution of Transient Heat Transfer Problems for Flat Plates, Cylinders, and Spheres by Finite-Difference Methods, dated 15 March 1965

TO: Recipients of Subject Report

CD CD

CD

It is requested that the following changes be made in your copy of the subject report.

1. Equation 15, page 8 should be:

2. Equation 16, page 8 should be

2kaAt(R - ETn + f) T1 = T + n n

MV + f KVa» + (R • £Tn - TKCbV (T - T ) + n-i n

u^lMvLx) (R - K + T) ^,CaTa' + (R • ^'n ■ r) <'bCbV]

(T . - T ) n+i n'

/'

0

0) 00 (8 a.

§ •rl 4J

3 W 0)

» Ü

in

I I 4J

0

« -I

l

U

W i

<

3 op

^■*

a t-

W i

^

a

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B

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e6

M i

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a i«

i

B

W

a +

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0)

Ü I H

I

m

1 ■

4. Equation 25, page 13 should be:

W. G. BURLESON Stress & Thermo Anal Branch Struct- res & Mechanics Lab, DR *D

DOC AVAILABILITY NOTICE

Qualified requesters may obtain copies of this report from DDC.

DISPOSITION INSTRUCTIONS

Destroy this report when it is no longer needed. Do not return it to the originator.

DISCLAIMER

The findings in this report are not to be construed as an official Department of the Army position, unless so designated by other authorized documents.

15 March 1965 " REP0RT ^ RS-TR-66-1

SOLUTION OF TRANSIENT HEAT TRANSFER PROBLEMS

FOR FLAT PLATES, CYLINDERS, AND SPHERES

BY FINITE-DIFFERENCE METHODS

by

W. G. Burleson

R. Eppes, Jr.

DA Project No. 1A013001A039

AMC Management Structure Code No. 501611844

Stress and Thermodynamics Analysis Branch

Structures and Mechanics Laboratory

Directorate of Research and Development

U. S. Army Missile Command

Redstone Arsenal, Alabama

ABSTRACT

Presented in this report are finite-difference (forward) heat transfer equations applicable to transient, radial heat flow in spheres and cylinders and applicable to transient, one-dimensional heat flow in flat plates. By very minor manipulations, a single heat transfer procedure can easily be utilized to determine transient heat flow in all three basic structure configurations.

For each skin configuration the, accuracy of the finite-difference procedure, compared with exact analytical methods, depends on optimum selection of the calculation time increment and the incremental distance between temperature nodes in relation to the material thermal properties and on the closeness of the approximate temperature gradients to the true gradients.

11

TABLE OF CONTENTS

Page

Section I. INTRODUCTION 1

Section II. FLAT-PLATE, CYLINDER, AND SPHERE PROCEDURES . . 2

1. Background 2 2. Transient, One-Dimensional Heat Transfer for

Flat Plates 3 3. Transient, Radial Heat Transfer for Cylinders .... 6 4. Transient, Radial Heat Transfer for Spheres., 11 5. General Equations Applicable to Flat Plates,

Cyllnaw -s, and Spheres 15 6. Finite-Difference Results and Procedure

Selection 19

Section III. CONCLUSIONS 20

LITERATURE CITED 21

SELECTED BIBLIOGRAPHY 22

Appendix A. DERIVATION OF AVERAGE AREA EQUATIONS FOR SPHERICAL CONDUCTION 23

Appendix B. LIMITS OF THE DIMENSIONLESS MODULUS, 0, FOR FLAT PLATES, CYLINDERS, AND SPHERES 27

HI

■■■■■■^^w^^^" ,.,'m.,'j „,.III-„^;.i" ,"', "" ' ",i""u ',■.,■.'—.i,--^ ...™-

LIST OP SYMBOLS

A *, area

9 - angle In radians

9 - solid angle in steradians

ß - dimensionless modulus

R - radius of cylindrical or spherical section

ETn - distance from outer surface of cylindrical or spherical section to temperature point n

e - surface emissivity

At - time increment for computation

T - temperature ■

k - thermal conductivity of material

C - specific heat of material ■

p - density of material

h - heat transfer coefficient

T - Incremental thickness for each material

Tr - radiation sink temperature

Te££ - effective gas temperature (boundary layer)

q - convective heating

q - radiative heating

%oUr ' solar heatin8

qcon<j - heat conducted

'»stored " 8tored ener8y

a • Stefan-Boltzmann constant

iv

LIST OF SYMBOLS (Cencludtd)

Subtcrlpf or Superscript!

a - material MA"

b - material "B"

be - backside or Internal surface

0 - external surface ,«,

1 - Internal surface

r - radiation sink

1 - condition existing atter the lapse of one (1) At

n - nodal point

x - nodal point

•

Stetion I. INTRODUCTION

The U. S. Army has had great success using finite-difference (forvard) techniques to accurately determine transient heat transfer in homogeneous and composite wall missile structures. Heating effects on most Army missile structures have been analyzed by use of finite- difference methods applicable to transient, one-dimensional heat flov in flat plates. This flat-plate technique is generally quite adequate for structure analyses of missiles such as JUPITER, PERSHING, REDSTONE, and SERGEANT because most of the airframe exposed to inflight aero- dynamic heating has a relatively large radius of curvature in relation to the skin thickness. Thus the curved structure actually approached a flat plate in relation to heat storage and heat flow.

Contrarlly, a flat-plate solution may not suffice for analysis of small hemlsphericsl tips, small hemicyllndrical leading edges, small caliber vehicles, large or small cylindrical and spherical solid propellent motors exposed to environmental thermal shocks, and many other missile components and sections. Thus, accurate, flexible procedures for at least three basic structure configurations are necessary for proper thermal analyzation of present and future Army

missile systems.

One purpose of this report is to focus attention on the similarities existing in finite-difference heat transfer equations for radial heat flow in spheres and cylinders and one-dimensionsI heat flow in flat plates. Another purpose of this report is to present one general heat transfer procedure which can be used to determine transient heat transfer effects in three basic configurations, i.e., sphere, cylinder, and flat plate.

mmmmmmmmmm

Section II. FLAT-PLATE, CYLINDER, AND SPHERE PROCEDURES

1. Background

Transient, finite-difference (forward) heat transfer methods have occupied a vital role in the development of several of the Army's most prominent, reliable missile systems. During the historical heat protection material development program for the JUPITER nose cone, flat-plate, finite-difference, heat transfer methods were used success- fully in analyses of reentry simulation test results1 and in analyses of inflight aerodynamic heating effects.z'3 This same heat transfer method, in conjunction with a simultaneous ablation routine,4 was used exclusively to determine the thermal protection requirements for the PERSHING reentry body. Later, calculated inflight temperatures were found to be in good agreement with temperatures measured on the reentry body of PERSHING missile 308.5 Agreement of finite-difference results with exact solutions is discussed briefly in Paragraph 6 of this section.

With the advent of large solid-propellant power plants for use in severe environments, as well as increased demand for hypersonic, low- altitude flights by small vehicles or large vehicles having small lead- ing edges or tips, the Army has become increasingly involved with heat flow problems which are not adequately solvable by flat-plate techniques. Recently, therefore, transient, radial heat transfer procedures for cylinders and spheres, in finite-difference form, have been derived. Using the radial procedure for a cylinder, good agreement between measured and calculated temperatures of a PERSHING XB-1 motor subjected to severe thermal shock in a cooling chamber were obtained.6

Procedures for determining -radial heat flow in cylinders and spheres have many applications in missile design and analysis, since in numerous cases the assumption of only radial flow is sufficiently accurate. For instance, transient, radial heat flow in a small, solid hemispherical tip may be adequate for materials with low thermal diffusivities (reinforced plastics, for example) even when the heat input varies with external surface station. Transient, radial heat flow at the midsection of a cylindrical solid-propellant motor having a large L/D ratio may be adequate in most cases because the present solid propellants have small thermal diffusivities and end effects do not influence the motor midsection for many hours.

The conduction equations presented for each of the three basic structure configurations are applicable to any number of material layers; however, only two layers are illustrated because this is the maximum number of materials necessary for the application of all

equations for thermally thick skins. Equations for thermally thin skins and thick-thin combinations have been derived for flat plates, cylinders, and spheres to account for all practical material arrange- ments. These equations are not presented due to similarity of the derivations to those of the thick-thick wall cases discussed.

Many statements made throughout the discussion of the flat-plate procedure are also applicable to the other tvo procedures, and are not repeated to preclude repetition. Since the average area of a sphere across which heat flows Is not a linear function of the radius, the derivation of average areas for the spherical configuration is shown in Appendix A. Shape factors have been purposely omitted from the radia- tion terras throughout the report.

2. Transient, Ont-Dimtniienal Heat Tramftr lor Flat Platts

One-dimensional, flat-plate heat transfer in a homogeneous material can be determined by solving heat balance equations at the exposed surface, unexposed surface, interior nodes, and interfaces. The forward finite difference method was used. It was assumed that the incremental thickness (T) can be selected sufficiently small to give accurate temperature gradients between adjacent nodes and that the incremental time (At) is small enough to neglect any effect on regions more than one r from the node in question.

conv 0

q"d0

^solar

TiO

Ta/2

I ' T ' I T9I ^-l •

t • I

VVAVVVVV^VV'AVNVVVKVVVVVVVAW"^

IIAII

k- Tb-H q Mconvj _ 1 _ I - I T • T„ 1 T_4.1 1 in+a 1 1bs-i 1 •■n 1

1 1

K- V2

"B'

Figure 1

a. Exposed Surface

From Figure 1 the heat balance at the exposed surface is

^COUVQ ^solar ' rad0 4cond 1— 2

'stored (1)

BBBV

1^2

where

^convo"^ (Teffo - TO

^solar " ^eÄt received from sun per unit area

qrad0" eo^ <!? - TrM

q - M (T1 - Ta) ncond T ■a

H8to paLa 2 At 1

The area, A, is uniform for one-dimensional, flat-plate heat transfer. Rewriting Equation (1) we have

ho^££0 - I.) + Vlar + *< - TA + m^Jil . ^li m_lii)

(2)

A f- '• , let Y » —r- . and SOIVP fnr T.1

pCTC

n = ^(1 - 2Yah0 - 2ßa) + 2ßaT8 + 2Yah0Teff + 27.^(1* - T?) +

Multiply by 2At ■ . let S - kAt ^p^ v a -At a , 1 r m. PacaTa OCT

2 ' pCr ' solve for Ti

2Y O a^solar (3)

The coefficient of ^ in Equation (3) must be > 0 otherwise V will depend on ^ in the negative sense which is not reasonable. Therefore set the coefficient of T, equal to zero and solve for ßa. Note after Equation (2) that Y ■ ß(T/k). a

1 " 2Yaho " 2ßa - 0

ß i 3 2 + 2(hoTa) (4)

ka

Tin Eouatlon'm^ ^^ CritlCal ^ ^ i8 the UPPer limit of ** ß in Equation (8) for convergence. It is readily seen that if ß in Equation (8) is a maximum of 0.5 (to make the coefficient of T - 0)

nn!?M C0*fficient of Ti in EcIuation (3) is negative because^Yh is positive. Therefore, ß must be equal to or less than 1

2 + 2(hoTa)

ka

'«■majw.iu.

The maximum h0 expected or calculated for the specific case for analysis should be used In determining the critical 0.

b. Interface Equation

At the Interface between material "A" and "B" (T„. FlRure 1) the energy balance Is n» » /

qcond "cond " ^stored

n-i-*n n-»n+i n (5)

or

ka<Tn-x - Tn) + kb(Tn+1 - Tn) Tb

(PaVa+PbVbX^ -Tn) 2At

Rearrange and solve for T' n

TA " Tn + (!„., - Tn) 2k At a

.Ta<PacaTa + PbcbTb)_

(Tn+l " Tn) 2kbAt

Tb<PacaTa + PbcbTb)

(6)

c. Internal Nodes

The heat balance at any interior point of a homogeneous wall (Figure 1) may be written

^cond n^n+i

^cond n+i-* n+2

^stored n+i

(7)

or

kb(T" - W + MV^iW (T^ - T^) Tb Tb PbcbTb Tt

Solving for T'+1 wlt'i 3 - -^~ gives per

TiUi -Vi^1 ■ 2M + M^+W (8)

__-^^_^ M^HMHIH1.

V

d. Backside Surface

The energy balance at the backside surface, T. , may be written as

q,-q -q m a cond ^com^ ^radi ^stored .QV

bs-l-^bs bs

or

V\l ' ^^ + hi^f£i - Tb8) + e^ - Tb*8) -

^W^s - Tbs) 2At

Rearrange and solve for T'

Tbs " Tbs<1 " 2Ybhi " 2^b) + 2ßbTbS.i + ^iTefft +

^hH^ti " Tbs)

(10)

To find the limiting 0 for the backside material, equate the coefficient of Tbs (Equation 10) to zero and solve for ß,.

h = —ITT (ID 2 + —L2

kb

3. Transient. Radial Heat Transfer for Cylinders

a. External Surface Equation

Consider a cylindrical segment heated as shown In Figure 2.

From the energy balance at the peripheral surface

^conv ^solar ~ ^rad„ "* ^cond " ^stored o o 1-2 1

or

"I l>o<Tef£() " Ti) + »» 1,oUr ' R9L «9» & ' V "

(i - ^)ei^ (t, -1.) - (» - ^)9i(capa i) 2Llä

Let ß ■ ^-r- and Y ■ «-— , rearrange, and solve for Ti. pCr PtT

Ti = 1 - 2ß

(12)

4 \ , 'aqsolar

1-Iä

Equating coefficient of Ti to zero, since T^ cannot depend on the previous T], in the negative sense;

0 = 1 l-lä

4R L

Yaho + ^a (^ " ä)

(13)

Since Y ■ ß T , the critical or limiting ß at the external surface is

h- 1- Ta

1 . Ja + hoTa 2R k„

(14)

For R-*» and T finite or T/R« 1. Equation (14) approaches the one- dimensional flat plate case(Equation 4).

b. Interface Equation (Figure 2)

At point Tn (interface between Material "A" and "B") the energy balance becomes

"cond "cond ^stored n-i-»n n-»n+i n (15)

m^^m^^^m

Figure 2

or

^M^)<v,-T„)^MTn4)(T:

(R - IT ) (paCaTa+PbVb)(T' " Tn)

Tn)

solving for T^

K T + ^M^) n V^^nHPaVa+PbCbV (Tn-1

2kbAt(R-2:Tn3)

Tb(R " ITn)(PacaTa + PbcbTb) (Tn+1 " Tn)

Tn) +

(16)

c. Interior Equation (Figure 2)

At point Tn+l (Material "B") the energy balance becomes

^cond 'cond n-»n+l n+l-*n+8

Stored n+l (17)

or

, , ^^^^^^___-_ _._^^_

^^•^mm ■■"■■ P" ' i

b b

(Tn+i " Tn+x)

n+l 'ih n+a " Tn+i)

(R -iT^Xp^Tb)

kAt .„J v _ _A£

At

Let ß - -^ and Y per pCT

Solve for T' n+l

T;+l = Tn+1 1 - ßb I(R ■£Tn+i ^)+(K. • ^Vi 4)1 R -1 Vi

- + T ßbjTn R.£T n+l n+8 R - £T n+l

(18)

^cond

bs-i •*• bs

d. Backside Surface Equation (Figure 2)

At point Tbs (Material "B") the energy balance becomes

^rad^ ^ 'conv-i ^stored bs (19)

or

Th (R ■ lTbs+ T) ^Tbs-i - Tbs>+ ei^R - ^^x^i ■ T

;S> +

hi(R - lTbs)(Teffi - Tbs) - (p.C, ^) (R - ITbs + 'i)^^)

Let ß=^ and Y - -^ PCT2 PCT

Solve-for T^

^■^^M ■■■•■'■l" ■' ,M " ■ ■■— -——

T1 1 " 2ßb R -IT. +^ bs 2

R-lTbs+^ 4

R " ITbs ^Vbhi —^ | Tbs +

R -ETb8 +

29bT^Tt>^- + ^ MR - lTbs)

R - ET. o + lb bS 4

Leff.

2\Ha •> R - L' bs

R - ETK + ^ bs 4

(T4 - T4 )

Equating coefficients of T. to zero bs

0=1- 2Ybh R-E' bs

1 R-ET, +Tb bs T 20,

*-ETbS + ^

R-ETb8+i

(20)

but Y = ß ^ , thus

0 = 1 - 2ßb ^ hi R-E' bs

R - ETbs + i 23^

R ■ ETbs + lb

Solving for ßb the critical ß at the inner surface yields

*b =

R-ZTHfi-Hi 4 J

(R-ETbs + M+(R-ETbs)Mb (21)

For small T^s and small E^g's, the one dimensional flat plate case

(Equation 11) ic approached as R becomes relatively large.

10

""

e. Observations

For finite T'S and R-^-oo, cylindrical temperature Equations (13), (16), (18), and (20) revert to flat-plate temperature Equations (3), (6), (8), and (10), respectively.

4. Transient, Radial Htat Transfer for Sphorts

solar

qconv0

^rad.

Figure 3

a. External Surface Equation

Consider a spherical segment heated as shown above. From the energy balance at the pheripheral surface

qconv0 + qsolar " qrad0 ' ^nd ~ qstored

1-^2 1

(22)

or

11

■ ,.:.,.■ ■■■rl..■■■■■■!, ,;, ■

<PR8ho<Teff0 - Ti) + ^aq8olar " ^WW " ^ '

Note: See Appendix A for derivation of average areas.

Let ß , AtiL. and _4t pc-r per

Then

CaTaJTj - W 2AT

Ti - Tx 1 - 20 (1 - Ta/2R)2 + ^/12Ra

a (1 - Ta/4R)a + Ta/48Ra 27 3 (1 - Ta/4R)3 + T|/48R2

2S2

(1 - T./2R)2 + l*/12Ra

\2 , _a //ona

'Isol ar

(1 - Ta/4R)*i + T^MSR3

27.

+ T, eff. 27, 2^-2 //.o«3 (1 - Ta/4Rr + T|/48R'

(1 - T./4R)3 + T^^SR3 + (nn - Tj) 27.

€0a

(1 - Ta/4R)a + T|/48R3

(23)

Set the coefficient of Ti equal to zero and solve for the limiting

1

3 when h is a maximum. a O

0. 2 . 2 ,,„„2 (1 - T-/4Rr + Tt/48R'

(1 - T /2R)3 + Tf/12R2 +ioI§ k Ka

(24)

For finite T^s Equation (24) approaches the flat-plate Equation (4) as R becomes large.

b. Interface Equation

At point Tn (interface of material "A" and "B") the energy balance is

12

mm

Mcond

n-i n

^cond

n"**n+i

'stored

n (25)

or

'a

|oaCaT, (R -ETn + ^j) + ^fgj + pbCbTb (R - E Tn - ^ + ^Ljj

Solve for T' n

/v; PacaTa [(R - ETn + ^) + ^]+ p^T, [(R - ETn - ^) + ^.jp

(Tnn " Tri) 2^b[(R.£Tn.^)8^]

)Tb| eacaTa [(R - ETn ^y + ^1] + BbcTb [(R . E.n - $ + lL]ji

(26)

c. Interior Node Equation (Figure 3)

At point Tn+1 (Material "B") the energy balance is

^cond "cond * ^stored

n— nfi n+i"*n+8 n+i (27)

or

13

,. ,".v. ■'..,■■-■ . ...

Let 0 - -^j , then pcrr

J, D ^-^n.^)3^]-^^^)8^]

(R- lTnJ8+.

12 (R- ^TnJ +^-

12 12 /

(28)

d. Backside Surface Equation

At the backside surface (Tbs) the heat balance Is

^cond ■ ^radi " ^convi = qstoreu

bs-i-*bs bs ^

or

1 T * T

* Th (Tbs-i " TbS) [(K - 2>bs + ^) + T2 J+ v ^i^Tri4 * V)(R - ^Tbs)2 +

. hiCTeff, - Tbs)(R - libsf - cp (^k) (T'8 - Tbs)[(R - E^ + ^) + ^3]

^ and Y = 4L per PCT

Let 0 = -^f and V - ^ , then

14

■ .■ ■ HM,-.

Tu = Th« < 1 bs OS

29b [(I- 2Tb8+-|) +:Y2J

\ D8 L' 48

aVbh^R - lTb8)S

(R.ETb8 + ^+ife: V D8 4/ 48

\ 2Vbeia(R- lTb8)8

Equate the coefficient of Tbg to zero and 3olve for the limiting 0b

when h. is a maximum.

(30)

0b

L(R-IT-^)J [(R - ETbs + ^J + ^p (R - ETbs)

a| (31)

As with the cylindrical Equation (21) this ßb approaches the flat- plate limits (Equation 11) when Tb is finite and R is relatively large.

c. Observations

For finite T'S and R —», spherical temperature Equa-

tions (23), (26), (28), and (30) revert to the flat-plate temperature

Equations (3), (6), (8), and (10), respectively.

5. General Equations Applicable to Flat Plates, Cylinders, and Spheres

Due to striking similarities between finite-.iifference heat

transfer equations for flat plates, cylinders, and spheres, general equations adaptable to all three configurations are derived and presented as Equations (32) through (37). The coefficients G and m and the exponent m are inserted to perform manipulations resulting in equations applicable to the desired skin configuration. The following

table gives values assigned to G and m.

Skin Configuration Flat Plate Cylinder Sphere

m

15

^mmmimmmmm

<U CO M ß u "0 m 0 -H ra !-< ß M-4 .-i <u o M PQ CX'H f* /*» 0

1 >>u --4 4J X v eg m u -H co to 3 co CO -o ^ c tr C -a <« c (4-1 O 0) 0 C 91

•rl H co t^ 0. Q) U U 4J cu a j: co a» CO "^ < « 3 «-( 3 <t

a* co C CM co C co (U C OJ •rl Vi n - CO (U >-< M —i r-l -rl "0 tJ CO -u CO «^«»i c ^ 1-t «a co (U (U U QJ CO ^ C co C •d- 00 3

(u a) QJ r-l ß U /*» /-s box W) •H CO r-l

r^ CO 4-> -rl CO n <U r-< QJ C •'-1 -o •rl

U ^y X «0 Ä o E 4J U 4J •rl -rl 0) QJ

J2 -H 4-1 r-l Vl 4J

60 0 M c CO CO J3 3 -U CU •H 3 CU ^- OCX! crxi co VJ -H ax) W -M 3 Ü)

JZ CO QJ «^ l-l 3 •U --l U QJ 3 O

<u CO CO Ü "0 QJ -^ II XI 1-4 :u c o c C^ H 0. <a-H E QJ n E co v^ OJ 00 en i •o • u C TS -rl

en C ro CO •rl C JC 0 C co 4J CO 4J

o x; !M •r-l

•i-l CM a E -^ i-i ^ ^ 4J CO II •rl CM 0

CO II ^ t-(

3 00 E X w s-^

cro ™ Q) O. 4-1 c w n -O > CO -rj

m CO c CO M E 0 ^ 0 P- CO X OO-rl ■rl

CO CO 1-4 4-1

MC' CM CO U CO

0) O H QJ CO QJ C -^ H II M PM X Q) 4J 3 4J w WJ 3 C Ü T3 •>

4J 0 Q) H M QJ (1) -t-l T-t c U H C U

x: -u w OJ O 0 CO

w « u X V4 C H M XI QJ ? O, 0 4J

0 3 CO ■rl CO 3 4J en TJ r-l 4J 3 en c ^ QJ co o cr

•rl 0 c U QJ QJ T) •H ■rl cn QJ

r-l (N W) CO w^ u W c 4-1 QJ QJ 0

II .c o Xi j; Qj M-I a. a- ■fi 0 0- 4-i en c

n co 4-i en co C o w VJ CO <u r-l CO •rl

-a WJ 3 M 'Ü Cu U 4-J *• d co a* co C 4J CO r-i

CO co M CJ CO 4J u CO

-1 CO 4-1

r-l r-l CO ?« 0 /"■N CO ^ QJ •H QJ

II - -H x; r^. o x: •4-1 4-1

H 4-1 am •rl QJ c w OHO a u x: v 0 3 M 3 'a w x; ü

m C "O tn t* C 4-1 r-l CO o o c CO ZJ ■rl tn c

•H 0 M 0 r-l CO M-l •r-l

C. -t-l U a M X 0 r* QJ

0 u PL, X O U) m •H QJ --4 4J C C QJ

o 4-J C/3 CO '. QJ 0 O u 3 U H rl X «) ^ ■rl

U C •'-> H n 4J CO 4-1 tn

•H -^ V) QJ 3 CO

u "0 C tn QJ M r-l X co to «j C 0 C O 0 X C -H •H o C QJ CO X 3 0 -< 4-1 •H •rl E u

Cfl -H > , u 4-1 W CO M (0

4-t U QJ CO CO QJ QJ

CO en 3 D, 3 OJ cr a o J-i

crx 'H W X u 0 (1) 4-1 0 v-' 4-1 P. M-l

3 ^^ n

B OS I- 00

+ 0 B

X ?- >^s (N r-l

X ?■ CM

o

I OS

00

U

H5

0 14-1 14-1 4)

H

I

B OS 00

Ü

IBJ

o

1

B ^ ^-^

R e R OS H Di t- ri 1- w

r-l -*

Ü

!1) r4

H

II

O

1^

in

ill H

00

o

H5

/-s DJ

o

II

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18

6. Finite-Difference Results and Procedure Selection

Calculated results from the three finite-difference heat transfer procedures presented in this report have been compared with the exact solutions.7»8 Excellent agreement, within 0.3 percent, was obtained in each case for a material initially at uniform tempera- ture (T0) and with one of its surfaces maintained at a constant ' temperature different from T0 after time zero. The accuracy "of any finite-difference heat flow equation depends on the proper selection of the time interval (At) and distance increment (T) as well as the close-

r x. . i. TX " TX+1 , ness of the approximate temperature gradients, T " '" to the true gradients.

For the majority of missile structure analysis, the simple flat- plate procedure may be sufficiently accurate. For instance, examination of the cylindrical and spherical conduction equations (Equations 20 and 30) at the inside surface (where the maximum deviation from a flat plate obviously will occur) shows that a ratio of R/Sfbs (radius/wall thickness) of 100 gives results approximating, within 0.6 percent, the flat-plate solution (Equation 10). Thus, for adequate engineering analyses and structural design, the simple one-dimensional flat-plate procedure is advantageous in many investigations because of the required computer time.

19

m^mmmmm

Section III. CONCLUSIONS

One general finite-difference heat-transfer procedure can be used with minor manipulations, to calculate transient, radial heat flow in spheres and cylinders and to calculate one-dimensional heat flow in flat plates.

Excellent agreement between finite-difference calculations and exact analytical results is obtained when proper time and distance increments are chosen in relation to material thermal properties.

For hollow or solid spheres and cylinders composed of one homogeneous material»the limiting modulus,/? , may be determined by parameters at the internal surface even though the external surface ma^ be the only surface exposed to convective heating.

Comparisons oi finite-difference calculations with exact analytical results for several special heat transfer cases are necessary for final determination of the accuracy of the finite-difference methods. Results of these additional comparisons will be published in a future document.

20

1

LITERATURE CITED

1. N. J. Hurst, Measuremeiyts of Heating to a Thermally Thick Spherical Copper Calorimeter, ABMA Report RSM-TM-1-60, October 1960.

2. W. B. Schölten, Evaluation of the Thermal Performance of the Ablation Materials on Recovered Jupiter IRBM Reentry Nose Gongs (U). ABMA Report DSD-TR-6-60 (Secret), January 1960.

3. W.G. Burleson, Comparison of Calculated and Inflight Measured Data for Metallic and Fiber Reinforced Materials on Hypersonic Vehicles (U), ABMA Report DSD-TM-10-60 (Secret), June 1960.

4. N. J. Hurst, Calculations of the Simultaneous Ablation Procession and Associated Temperature Distribution in a Heterogeneous Wall Exposed to a Time-Variable Heat FIUK, ABMA Internal Note #2, February 1960.

5. B. M. Kavanaugh, Aerodynamic Heating Analysis of a Tactical Pershlng Reentry Body Having a Reduced Insulation Thickness (U), USAMICOM Report RS-TM-63-4 (Confidential), October 1963.

6. W. G. Burleson, et al, Thermal Analysis of a Pershlng XB-1 Motor Subjected to a Cold Environment (U), USAMICOM Report RS-TR-62-3 (Confidential), November 1962.

7. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Second edition, Oxford University Press, 1959.

8. M. Jakob. Heat Transfer, Volume I, John Wiley & Sons, Inc., 1959.

21

—'

SELECTED BIBLIOGRAPHY

1. Heat-Transfer Calculations by Finite Differances, G. M. Dusinberre, 1961, International Textbook Company.

2. "Numerical Methods for Transient Heat Flow," G. M. Dusinberre, Transactions of the ASME, November 1945.

3. Graphical Presentation of Difference Solutions for Transient Radial Heat Conduction in Hollow Cylinders with Heat Transfer at the Inner Radius and Finite Slabs with Heat Transfer at One Boundary, J. E. Hatch, et al., NASA TR R-,56, 1960.

4. The Calculation of Transient Temperature Distributions in a Solid Cylindrical Pin, Cooled on the Surface, J. Heestand, D. F. Schoeberle, and L. B. Miller, Argonne/National Laboratory Report 6237, October 1960.

5. The Solution of Transient Heat Conduction Problems by Finite Differences, Purdue University /EES RS 98, March 1947.

6. J. E. Medford, "Transient Radial Heat Transfer in Uncooled Rocket Nozzles," Aerospace/Engineering, Vol. 21, No. 6, October 1962.

7. G. R. Gaumer, "Stability of Three Finite Difference Methods of Solving for Transient Temperatures," ARS Journal, Vol. 32, No. 10, October 1962.

8. A. G. Elrod, Jr., "New Finite-Difference Technique for Solution of the Heat-Conduction Equation, Especially Near Surfaces with Convective Heat Transfer," Transactions of the ASME, Vol. 79, 1957.

9. R. D. Geckler, "Transient, Radial Heat Conduction in Hollow Cylinders," Jet Propulsion, Vol. 25, January 1955.

1L

W"

Apptndix A

DERIVATION OF AVERAGE AREA EQUATIONS FOR SPHERICAL CONDUCTION

Since the surface area subtended by a solid angle on a sphere is not a linear function of the radius (R) the average area (S) through which heat flows and the average area for determining heat storage volume ( T! ) must be found.

1. External Surface

The average area from 1 to 2 (Figure 3) is found from

cp R2dR

^R-T

9 (■■f)-fi (38)

To find the average area for determining the heat storage volume from R to T

R - _ä the following equation is used. 2

'I R

R2dR

A = R-T /2

a T /2 a

9 / T \2 T 2 -

(R-f)+t8J (39)

Results of the integration in Equations (38) and (39) are reflected in Equation (22) of the text.

2- Interface

The average area from n-i to n (Figure 3) is

.R-IT_+T,

9

3- JR

RER

-= cp ''-k + rl^ (40)

and the average area from n to n+i is

H

„

9 R3dR

R- ET -T. n t

9 ('-£v4)34 (41)

The average area for finding the heat storage voluTtie at an interface (n) is n

■ R-ET+T a/2

i n a cp I R dR

R:ET -T./2

T + Tu * a b

2 2 T \ T

ET +-*) + A n 4 48 (42)

ET -^, T. X2 T. 2

"n 4/48

Results of the integration of Equations (40), (41), and (42) are used in E-quations (26).

3. Interior

The average area through which heat flows from n+i to n+2 is given by Equation (43). The Ä between n and n+i is found by using

+ T /2 in Equation (43).

cp I R2dR *VET -T.

T n-fl b A = ~ • 9 lR " ^n+l ■ ^) + ^2

(43)

To obtain the heat storage volume at node n+i the following average

area is used.

R- ETn.1+Tk/2

9 I RadR

* ■('■^4] (44)

Results of the integration in Equations (43) and (44) are

inserted in the interior Equation (27).

Z4

B^^^^^^H

■T— "—^— ■ ■■■.■-.-,■■ RiSÄ»'!' ■ •- ..■"MIWIWII

4. Backiidt Surfact

The average area through which heat is conducted from nodes

bs-i to bs (Figure 3) is

I cp I R dR

VET

R- ^Tb8+Tb a.

bs 9 R ^.4)4.] (45)

and the area determining the heat storage volume is

/•R- LTbs+Tb/2

cp I R2dR

y f-Hs4)a4] (46)

The results of the integrations of Equations (45) and (46) are reflected

in Equation (29).

25

Apptndix B

LIMITS OF THE DIMENSIONLESS MODULUS, ß , FOR

FLAT PLATES, CYLINDERS, AND SPHERES

In all conduction equations presented in this report, the modulus, ß, has limits which are critical to the stability of the equations. This is due to the necessity of coefficients of a particular temperature point being equal to or greater than zero because it is unreasonable for a temperature to depend on a previous temperature in the negative sense. The following equations show the limits of ß under conditions where the heat transfer coefficient (h) equals zero.

1. Heat Transfer Coefficient (h) = 0

a . Flat Plate

The equations of ß applicable to the external and internal

surface (Equations 4 and 11) reduce to the following upper limit

when h0 = 0 and h^ = 0.

M < 1 (47)

b. Cylinder

(1) External Surface

The limiting ß equation (Equation 14) at the external

surfaces reduces to the following when h0 = 0.

ß^ ^R - T IT a_

.a - T /4 a

(48)

which has values approaching 0.5 and 0.75 for R» Ta and Ta-* R

respectively.

(2) Internal Surface

The limiting ß at the internal surface or center of a cylinder (Equation 21) reduces to the following when hj, = 0.

^

1

bs

bs

+ V2 \ + V* /

(49)

27

The values of Equation (49) approach 0.25 and 0.5 as2n,8—••R and when R» STb8 respectively. Thus when l^- 0 the limiting modulus ß of a cylinder composed of one homogeneous material may be determined by Equation (49) rather than Equation (48). The limiting^ varies radically from 0.5 only when R - 2Tbg is less than one T.

c. Sphere

(1) External Surface

The critical ß in Equation (24) reduces to the follow- ing when hn = 0.

1 / T x5 T2^ I1 "it) +12F 7—'T

48r

(50)

The values of ß in this equation approach 0.5 and 0.875 when R»Ta and when Ta—►R respectively.

(2) Internal Surface

The limiting ß in Equation (31) reduces to the following when h. = 0.

^

1

(R-2Tbs+T)

M^r).

(51)

The ß in this equation has values approaching 0.125 and 0.5 when IT -+K and when R» 2Tb respectively. Thus the overall

critical ß for solid or near solid spheres composed of one homogeneous material may be determined by Equation (51) rather than Equation (50). The lower ß limit of 0.125 is approached rapidly after R - 2Tbg becomes less than one r.

2. Htat Transftr Cotfficitnt #0

To determine the limits of ß for each of the three configurations when h ^ 0 the/J must be determined for the maximum heat transfer coefficlent. Both external and internal surfaces /J's should be calculated to determine upper, limits which are valuable in insuring stability or convergence of the equations.

28

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SOLUTION OF TRANSIENT HEAT TRANSFER PROBLEMS FOR FLAT PLATES, CYLINDERS, AND SPHERES BY FINITE-DIFFERENCE METHODS

4 DESCRIPTIVE NOTES (Type ol report and inclusive deles)

Progress Report 5 AUTHORfS; (Last name, lirst name, initial)

Burleson, W. G. and Eppes, R. , Jr.

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15 March 1965

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13 ABSTRACT

Presented in this report are finite-difference (forward) heat transfer equations applicable to transient, radial heat flow in spheres and cylinders and applicable to transient, one-dimensional heat flow in flat plates. By very minor manipulations, a single heat transfer procedure can easily be utilized to determine transient heat flow in all three basic structure

configurations.

For each skin configuration the accuracy of the finite-difference procedure, compared with exact analytical methods, depends on optimum selection of the calculation time increment and the incremental distance between temperature nodes in relation to the material thermal properties and on the closeness of the approximate temperature gradients to the true

gradients.

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