'

I "

ARMY RESEARCH LABORATORY

_ Derivation of-Two-Dimensional (2-D) Conduction Equation in Generalized

Coordinates With Constant and Anisotropic Physical Properties

Paul J. Conroy

ARL-MR-219 Mayl995

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4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Derivation of Two-Dimensional (2-D) Conduction Equation in Generalized Coordinates With Constant and Anisotropic Physical Propezties

PR: 1Ll62618A1FL 6. AUTHOR(S)

PaulJ. Conroy

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U.S. Anny Research Laboratory ATIN: AMSRL-WT-PA ARL-MR-219 Aberdeen Proving Ground. MD 21005-5066

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13. ABSTRACT (Maximum 200 words)

The purpose of this paper is to provide a worlcing report for potential future modifications to existing heat conduction codes. A generalized form of the two-dimensional (2-D) arbitrary geometry axisymmettic heat conduction equation has beeit derived from first principles. This has been further generalized by allowing nonidealized anisotropic behavior of the material properties.

·,

14. SUBJECT TERMS

heat conduction, potential function, variable properties, arl>itrary geometry, heat transfer, transformations (mathematics)

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...

ACKNOWLEDGMENT

Appreciation is given to Mr. Nathan Gerber, U.S. Army Research Laboratory (ARL), for his

explanation of the variable property fonn of the energy storage tenn .

iii

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TABLE OF CONTENTS

ACKNOWLEDGMENT iii

1. INI'R.ODUCfiON .............................................. . 1

2. DERIVATION OF 2-D HEAT CONDUCTION EQUATION ................ . 1

3. TRANSFORMATION TO GENERALIZED COORDINATES ............... . 3

4. DERIVATION OF 2-D AXISYMMETRIC ARBI1RARY GEOMETRY CONDUCTION EQUATION FOR ANISOTROPIC MATERIALS .......... . 8

5. SUMMARY .................................................. . 11

DIS'fR.ffiUTION LIST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

v

INTENTIONALLY LEFf BLANK.

vi

1. INTRODUCTION

This report concerns the derivation of the two-dimensional (2-D) heat conduction equation in

generalized axisymmetric coordinates for both constant and nonidealized anisotropic material properties.

The purpose of this report is to provide a working paper for potential future modifications to existing heat

conduction codes.

2. DERIVATION OF 2-D HEAT CONDUCI'ION EQUATION

Beginning with a control volume description in nonnal coordinates as shown in Figure 1 and applying

the typical Taylor series expansion to Fourier's heat conduction law over the control volume enables one

to prefonn the energy balance. The cross-sectional areas and volume of the control volume in the

axisymmetric coordinate system are

r

-kdA dT zrz---- au dt

Figure 1. Energy balance control volume.

1

dT -kdAz­

dZ

z

dAr= rd8dz,

dAz = rd8dr,

and

dV = rd8drdz . (1)

Balancing the energy produces the cylindrical heat conduction equation through the following steps.

Balancing the stored energy with the difference in the flux across the boundaries produces

where

dV ()u = kdA ()T + kdA ()T - kdA ()T - kdA ()T dt rdr ZTz ZTz rdr

-i ( -kdA, -~}r -~ ( -kdAz ~ )dz ,

~ i(krdedz ~)dr + i(krdOdr ~ )dz'

~ i (krdedz) (~ )dr + krdOdzdr ~

+ ""L (krd8dr) ~ dz + krd8drdz - • a (dT). n az az az2

Assuming constant properties reduces this to

(2)

(3)

p Cpd8dzdr ~ = kd6dzdr (~ )dr + krd6dzdr a2-r + krd8drdz a2-r . ( 4) Ol ar df2 dZ2

2

Dividing through by kdV produces the familiar constant property cylindrical heat conduction equation

1 aT 1 ar a2T a2T --=--+--+--, a at r ar ar2 az 2

(5)

where the diffusivity, a, is

k a=--.

peP (6)

3. TRANSFORMATION TO GENERALIZED COORDINATES

Transformation of the orthogonal cylindrical coordinate system to a more generalized coordinate

system with body geometry included is preformed to allow for a body-contoured grid scheme while

maintaining orthogonality for computational purposes. Demonstrated in Figure 2 is a generalized

axisymmetric body with global coordinate system of r and z as well as the local body coordinate system

of 11 and ~. The inner and outer surfaces are defmed by ~(z) and Ro(z) respectively.

r ~ .... ..... .-r-...

-r--"- f-'""' ...... r--.

-~-.. ..,..f-.- -~I--" ... -r--.

r-.--~~ T)

. ~(z)

z Figure 2. General axisymmetric body.

3

The transformations from r, z, 9, to 11, ~, cj) are

(7)

~ = z (8)

and

cp = 9 . (9)

The local thickness of the body is represented as

(10)

Writing out the chain rule for each of the diffusion terms of equation (5) in the transformed region with

'I' being a scaler potential function (for instance, temperature is a scaler potential function) produces the

following terms:

o'l' o'l' ihl o'l' ~ Tr=dil"df+~Tr' (11)

(12)

(13)

4

.

and

The following simplifications are used to reduce the transfonned tenns

a; = o Tr .

drl = 1 ' az

drl = -_!_( dRi + 11 d(tk) J' az tk dz dz

5

(14)

(15)

(16)

(17)

(18)

and

For consolidation, let

a2rt _ L d(tk) az 2 - tk 2 (iZ"

L = ( dRi + 11 d(tk) J . dz dz

(19)

(20)

Substituting the previous tenns into equations (11) through (14) results in the following transfonned

diffusion tenns:

(21)

(22)

a'¥ a'¥ L a'¥ dz =~- tk all'

(23)

6

and

a2'P L d(tk) =--+--

a~2 tk2 dz

(24)

Substituting diffusion tenns, equations (21) through (24), into the original conduction equation results

in the following transfonned conduction equation with temperab.lre as the potential function. This relation

is valid for any arbitrary axisymmetric shape with constant material properties, where 0 s; 11 ~ 1.

1 aT = a2-r L d(tk) (aT) a d1 a~2 + tk2 "dZ dil

_ _!_ ( d 2Ri + 11 d

2(tk) _ ..!:. d(tk) J err

tk dz2 dz2 tk dz dr1

la1' 1 aT +--+

tk 2 art2 tk{(Tt)tk + Ri) dTt . (25)

Regrouping the right-hand side in tenns of the temperature derivatives, replacing the variable,

transfonning ~ to Z, L, and noting that the cross derivatives are the same for a well-behaved function

leaves the following (relatively) concise fonn

7

L

(26)

This function is valid for both nonnal axisymmetric geometries in which the superfluous tenns drop

out as well as arbitrary axisymmetric geometries.

4. DERIVATION OF 2-D AXISYMMEfRIC ARBITRARY GEOMETRY CONDUCfiON EQUATION FOR ANISOTROPIC MATERIALS

Equation 1, rewritten here, is general enough to use as the starting point for this derivation.

dV au = a ((krdSdz) aT ) dr + krdSdrdz al-r dt dr l dr ar2

+ :r (krdSdr) :r dz + kdSdrdz - , a ( MJ n az az az2

(27)

The principle difference from the previous derivation is that now the conductivity is functionally

dependent, which disallows it to be simply pulled out of the spacial derivatives. Expanding the newly

dependent portions of the diffusion tenns reveals the difference,

a ( ) ak az krdSdz = rdSdz dz,

a ( ) ak - krdBdz = rdBdz- + kdSdz. ar ar

(28)

8

·I

The conductivity could be assumed to be only temperature dependent at this point and the chain rule

applied to transfonn the spacial gradients of the conductivity to temperature gradients. This would enable

some simplification but may reduce the generality and treatable anisotropic behavior of certain materials.

Generality will be maintained by directly substituting these tenns into equation (27) without the

aforementioned assumption which results in

rdOdldz ~ ~ = ( rdOdz ~ + kdOdz) ( ¥r) dr + krdOdldz ~

ak err n + rd9dr """'- ~ dz + kd9drdz _ .

oz oz m.z (29)

Dividing through by the volume and the conductivity leaves

(30)

This is essentially the same function as before except for the obvious addition of the first two tenns on

the right -hand side. Transfonning these tenns into generalized coordinates requires the use of the

derivatives defined in the previous section

and

(31)

Inserting these tenns transfonned using equation (8) into the right-hand side of equation (26) produces

the following generalized 2-D axisymmetric conduction equation with variable physical properties

9

-~------ --~-

I I

L

·(1]2a2'f 1 OT + tk drl2 + tk( ('ll)rk + Ri) dij"

·(~r ~~ ·~~ -~(~~·t~J-

Substituting in L and regrouping produces

.

(32)

+ ~ ( 1 + (~ + 11 d(tk)J2

] a'T _ 2. (dRi + 11 d(tt)J a_ (crrJ. <~< ~ tk 2 dz dz drt2 tk dz dz . dil dz dz dz

1 (dRi , d(tk) J2

ak a-r _ 1 (dR.-i + , d{lk) )· (dk-_ err + aJc. ()T J · + tk 2 dz + 'I dZ dTl dTl tJc dZ 'I di , dZ dfl dtl dz • (33)

It is understood that the conductivity k and the specific beat are usually I$1'Ctions of temperature.

However, given new material processing practices, gradient or grossly aniSQ~ propemes may result.

This form of the equation is general enough to handle either thermal or spacial deviations of the

conductivity and specific heat

10

5. SUMMARY

A generalized form of the 2-D arbitrary geometry axisymmetric heat conduction equation has been

derived from first principles. This has been further generalized by allowing anisotropic and temperature

dependent behavior of the material properties. The relation provided could be incorporated into current

and future heat conduction models in ARL. Extension of this derivation to three dimensions requires the

addition of the azimuthal spacial term and expanding, with proper substitutions similar to the radial terms.

11

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ATIN E FREEDMAN BURLINGTON VT 05401 2411 DIANA RD BALTIMORE MD 21209-1525 1 MBR RESEARCH INC

ATTN DR MOSHE BEN REUVEN 601 EWING ST SUITE C 22 PRINCETON NJ 08540

19

NO. OF NO. OF COPIES ORGANIZATION COPlES ORGANJUTIQN

1 OLIN CORPORATION 1 SCIENCE APPLICATIONS lNTRNTNL CORP ATTN FE WOLF A'ITN M P.Al.MER BADGER ARMY AMMO PLANT 2109 AIR PARK'RD BARABOO WI 53913 ALlM.JQ'l.JiSkQUE NM Pr/106

3 OLIN ORDNANCE 1 SOUTHWEST RSRCH INSTITUTE ATTN E I KIRSCHKE ATTNIPIUEOS.. AFGONZALFZ 6220~It0AD D W WORTHINGTON PO DRA Witt 28510 PO BOX 222 SAN ANTOMfO TX 78228-0510 ST MARKS Fl. 32355-0222

1 SVERDRUP TECHLGY INC 1 OLIN ORDNANCE ATTN DR JOBN DEUR

ATTN H A MCELROY 2001 AEROSPACE PARKWAY 10101 91H STREET NOR1H BROOK PARK OH 44142 ST PETERSBURG FL 33716

3 THIOKOL CORPORATION 1 PAUL GOUGH ASSOC INC ATIN R WJLLER

ATTN P S GOUGH R BIDDI.J! 1048 SOUTH ST TISCH LI'DRARY PORTSMOUTII NH 03801-5423 ELKTON~ON

PO BOX241 1 PHYSICS INTERNATIONAL LffiRARY ELKTON MD 21921-0241

ATTN H WAYNE WAMPLER PO BOX SOlO 1 VERITAY TECHLGY INC SAN lEANDRO CA 945TI-0599 ATINEFMISR

A CRICKJENBERGER 1 PRINCETON COMBUSTION RSRCHLABS INC I BARNES

ATTN N A MESSINA 4845 MILLERSPORT HWY PRINCETON CORPORATE PLAZA EAST AMHER.Sl' NY 14501-0305 11 DEERPARK DR BLDG IV SUITE 119 MONMOUTil JUNCTION NI 08852 1 UNIVBR.SAL PROPULSION COMPANY

ATIN H J MCSPADDEN 3 ROCKWElL INTRNTNL 25401 NORTH CENTRAL AVE

ATTNBA08 PHOENIX AZ 85027-7837 · I FLANAGAN I GRAY 1 SRI INTERNATIONAL RBEDELMAN ATIN TECH UBRARY ROCKETDYNE DIV PROPULSION SCieNCES DIV 6633 CANOGA AVE 333 RAVENWOOD AVE CANOOA PARK CA 91303-2703 MENLO PARK CA 94025-3493

2 ROCKWElL INTRNTNL SCIENCE CI'R ATTN DRS CHAKRA V ARmY DRS PALANISWAMY 1049 CAMINO DOS RIOS POBOX 1085 1HOUSAND OAKS CA 91360

20

NO. OF COPIES ORGANIZATION

ABERDEEN PROVING GROUND

1 CDR USACSTA ATTN STECS U R HENDRICKSEN

32 DIR USARL ATTN AMSRL Wf P A HORST

AMSRLWfPA -TMINOR TCOFFEE GWREN ABIRK IDE SPIRITO A JUHASZ I KNAPTON CLEVERITT MMCQUAID WOBERLE PTRAN KWHITE L-MCHANG JCOLBURN P CONROY (5 CP) GKELLER DKOOKER MNUSCA T ROSENBERGER

AMSRLWfPB ESCHMIDT MBUNDY B GUIDOS

AMSRLWfPC R FIFER J VANDERHOFF RBEYER MMILLER

AMSRL Wf PD B BURNS

21

INI'ENTIONALLY LEfT BLANK.

22

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