Analysis of conduction heat transfer problems in steady, 2D regions with circular holes using a special boundary integral method

INT. OOMM. H~kT MASS TI~ANSFER 0735-1933/85 $3.00 + .00 Vol. 12, pp. 179-189, 1985 ~ _ r g a m D n Press Ltd. Printed in the United States

ANALYSIS OF CONDUCTION

HEAT TRANSFER

2D R E G I O N S W I T H C I R C U L A R SPECIAL B O U N D A R Y

PROBLEMS

IN S T E A D Y ,

H O L E S USING A

INTEGRAL

METHOD.

Masood Parang, Rao V. Arimilli, and Satish P. K e t k a r Mechanical and A e r o s p a c e Engineering D e p a r t m e n t University of Tennessee, Knoxville Knoxville, Tennessee 37996-2210, U.S.A.

(Cc,t.L~nicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT A special boundary integral method developed for two-dimensional regions containing circular holes is used to calculate temperature and heat transfer on the boundaries of several selected regions. The geometrical configuration of the region is arbitrary and convective boundary conditions are assumed. An important feature of the method is analytic representation of temperature and its normal derivative on the interior circular holes in the form of a harmonic series. This makes the applieation of the boundary integral method convenient and free from conditioning problems associated with small interior boundaries. Heat transfer from cireular isothermal interior holes are calculated for several illustrative examples using three terms of the harmonic series representation for heat transfer at eaeh of the circular boundaries. The results are presented and discussed.

Introduction Problems governed by Laplace's equation have been solved successfully, and with considerable e f f i c i e n c y , using boundary integral methods [1,2]. t h e s e problems t h a t contain interior c i r c u l a r boundaries.

There is a subolass of

Such problems arise in many

p r a c t i c a l applications as in t h e t h e r m a l analysis of dies and moulds containing c i r c u l a r coolant lines, and heat t r a n s f e r from underground utility lines.

For such problems the

boundary i n t e g r a l method would require fine d i s c r e t i z a t i o n on the internal boundaries in order to a d e q u a t e l y c a p t u r e the local field variations.

This in turn may result in

numerical conditioning problems. These problems were e l i m i n a t e d by Barone and Caulk

179

180

M. Para~lg, R.V. Arimilli and S.P. Ketkar

Vol. 12, No. 2

[3] with a formulation in which the integrals around the circular interior boundaries were evaluated analytically using harmonic series expansions for boundary values of the dependent variable and its normal derivative, and by a clever choice of kernel functions. Arimilli and Parang [4,5] used this formulation to determine the c o n d u c t i o n heat t r a n s f e r from circular tubes embedded in s e m i - i n f i n i t e medium with c o n v e c t i v e boundary conditions. Barone and Caulk applied the formulation, using only the constant term of the expansion (called zeroth-order approximation), to the problem of a twodimensional (2D) r e c t a n g u l a r die with interior coolant lines. The solution procedure was then used to solve an optimization problem [6]. In the present communication we develop a numerical formulation of the more general problem of a 2D mould with an arbitrarily-shaped mould cavity surface.

This is

in contrast to the plane mould surface considered in reference [6]. Further we consider both the constant term and the first-harmonic term of the series expansion (called firstorder approximation) of Barone and Caulk [3]. Thus the problem considered is one of 2D analysis of steady conduction heat transfer from a finite body having, as its external boundaries, three normal plane surfaces and an arbitrarily-shaped fourth surface. interior of the body contains finite number of circular boundaries maintained

The at

constant temperature. At each of the external boundaries either the temperature or a convective boundary condition may be specified.

A number of illustrative problems

were solved and the results are discussed. Analysis

Consider a two dimensional region R containing M number of holes where ath hole is c e n t e r e d at ~Q with radius a% Laplace's equation.

The t e m p e r a t u r e ~P in this region is governed by

If the values of t e m p e r a t u r e or its normal derivative are only of

interest on the boundaries of region R, an a l t e r n a t i v e integral method can be employed based on Green's identity [7]

~.d~(y)+

OR

~--

a n -- g a n

where 1

=

y(R

½ ye {0R,0ca} 0 y~{aR,~c~}

and

=l

ace

dpaOg-g-~n an

ds=O

(I)

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HEAT TRANSFER IN 219 REGIONS

1

g(x.y) = - 2"-'~LoglX - YL

181

(2)

The solution for temperature or its normal derivative on the boundary of region R is obtained by evaluation of this integral over discrete intervals on this boundary. We express the temperature and its normal derivative on the surface of circular holes (aco) in a harmonic series [3]

~v

(3) m=l

oo

On

o o --q =qo+

~

(o,qlmStnmOa+q .

osmoo)

(4)

rn.=l

where the coefficients of the above series can be obtained by employing a special kernel f u n c t i o n in E q u a t i o n 1 [3]:

Q

go (x)= g(x'[a)'

--

g,,,,

sin ,nOa ]x _ [al,n

,

Q

g2,n

--

-

cos mOa ix _ ~a[rn

(5)

After Equations 3 through 5 are substituted in Equation 1, the integrals can be evaluated analytically over the circular holes thus avoiding entirely the discrelization of these boundaries.

In this analysis we assume, following Reference

4, constant-temperature

hole

surfaces,

~(1

Q

= ~b0

(6)

H o w e v e r , we include both z e r o t h and f i r s t - o r d e r t e r m s of h a r m o n i c expansion of q ( i . e . , m = I)

cl

qa = qo ÷ qatsinOa + qc.osOz

Q

,7)

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M. Parang, R.V. Arimilli and S.P. Ketkar

Vol. 12, No. 2

Therefore with m = I in Equations 5 and substitution of Equations 2, 5, 6 and 7 into Equation l results in a set of (N + 3M) equations where N is the number of discrete intervals on the external boundary of region R and M is the number of holes in this region [3]. This set of equations can be solved for (N + 3M) unknowns consisting of temperature on N discrete intervals on the external boundary, dPl, qb2, ... qSN, and three coefficients of the harmonic expansion for normal temperature gradient for each hole: qoQ("zeroth order"), qla and q2a ("first order").

Several examples of interest are chosen for region R containing circular holes. The algebraic equations developed for the boundary of region R were then solved numerically for t e m p e r a t u r e on the boundary and z e r o t h - and f i r s t - o r d e r n o r m a l derivatives of t e m p e r a t u r e on the circular holes.

The examples used for region R not

only include r e c t a n g u l a r regions (as in [6]) but also regions having as their external boundaries three normal sides enclosed on the fourth side by an arbitrary curved boundary r e p r e s e n t i n g mould cavity surface.

Specifically for illustration purposes two

curved boundaries, a semi circle and an S-shaped contour,

are chosen for the cavity

boundary. In all examples a convective boundary conditions with arbitrary Blot numbers are assumed on the external boundaries.

The program developed can compute the

unknowns on the boundary for arbitrary location and d i a m e t e r of each hole. However, for the illustrated examples used in this communication, all holes were assumed to have equal diameter, a. Results

The numerical solution was tested for accuracy by comparing it with available analytical solution for heat t r a n s f e r in a circular annulus where i n n e r and o u t e r boundaries are at d i f f e r e n t constant t e m p e r a t u r e s .

The numerical solution for heat

t r a n s f e r showed excellent a g r e e m e n t with a n a l y t i c a l results (maximum deviation of 0.3 percent.)

This adds confidence to the applicability and accuracy of numerical method

employed and the computer code developed in this study. Several geometries of interest were assumed for region R.

The n u m e r i c a l

computation of zeroth- and first-order coefficients in the harmonic expansion (Equation 7) were computed for various c o n v e r t i n g boundary conditions imposed on region R. The solution of t e m p e r a t u r e on the external boundaries of the following examples were computed but are not presented due to brevity of space.

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~T 8~F-'---[~, LT

qo

1.0 0.5

w - - L --,.e 0.4

O.I

O.05

o.3

0.22

,

,

0

~

5

,

,

10

,

,

15

,

,

l

20

i O/a

FIG. 1 Dimensionless "zerotb order" normal t e m p e r a t u r e

gradient,

q0, as a f u n e t i o n

of

dimensionless distanee D/a for various Blot numbers on top surfaee (Bi = 0.05, L/a = 25)

,

~qo

qos

3.B

/ i00

3.~

c/z

I- VF 0.5

3.0

2.6 0.1 2.2

0.05

1.8

1.4

I.o

,

i

1!

|

s

8

I

I

12

I

i

15

t

I

20

i

i

I

S/a

FIG. 2.1 Dimensionless "zeroth order" normal temperature gradient, Eq0 / q0s as a funetion of dimensionless bole spaeing s/a for various Biot numbers (L/a = 25).

184

M. Parang, R.V. Arimilli and S.P. Ketkar

Iq 2 l qo

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,oI O.S

0.6 10.0 0.4

1.0 0.5

0.2

0.} Bi=O.05 i ]

I 5

I

i 9

i

I

~

|3

I 17

I

I 21

I S/a

FIG. 2.2 Ratio of normal t e m p e r a t u r e gradients q2 / q0 as a function of dimensionless hole spacing s/a for various Blot numbers (L/a = 25).

First, the case of a square region with one circular hole l o cat ed on its v e r t i c a l axis of s y m m e t r y was considered,

Each side of the square was assumed to be L/a = 25 and

the c e n t e r of hole was taken to be a distance D above the lower horizontal side of the square (see inset of Fig. 1).

All sides of the square were subjected to the same

c o n v e c t i v e boundary condition and hence are at equal Blot numbers, Bi (here set to 0.05) e x c e p t the top boundary of the square which is subjected to a d i f f e r e n t variable Blot number, Bi.

The results for qo as a function of D/a for various Blot numbers are

p res en t ed in Figure 1.

It is observed t h a t the n o n - s y m m e t r y of t h e c o n v e c t i v e

conditions imposed on the boundaries r e s u l t s in a non- s y m m e t r i c p r o f i l e f o r qo especially for large values of Blot numbers, Bi.

Next ex am p le considered was the problem of two holes in the square region of the previous problem. The c e n t e r of two holes were assumed a distance S apart (see inset of Fig. 2.1).

All sides of the square were again subjected to the same Blot number.

The

sum of values of qo for both holes was normalized using single tube results (qos) and is shown in Fig. 2.1 as a function of S/a for various Blot numbers. The c o m p u t e d values of qt are found to be zero due to horizontal s y m m e t r y of the problem. are normalized in r e s p e c t to % vaiues and are shown in Fig 2.2.

The results for q2

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0.5! qo

.; 0.~:

0.35

~o,

0.25

~ - 0 1 o 5 0.15

' ~

;~'

" ~

i

•

•

,6

|

•

,o

:

.

i

,

,s D/a

FIG. 3.1 Dimensionless "zeroth order" normal temperature

gradient, qo, as a function of

dimensionless distance D/a for various Blot numbers (sla = 15, L/a = 25).

-0.2 ql qo

-0.16 Bi=0.05 -0.12

-0.08

0.1 I0.0 1,0 0.5

-0.05

i

i

I

I

8

I

t

I

12

I

16

|

I

20

I

t

21~

'

f

O/a

FIG. 3.2 Ratio of dimensionless normal

temperature

gradients ql / q0 as a function of

dimensionless distance D/a for various Blot numbers (s/a = 15, Lla = '25).

186

M. Parang, R.V. Arimilli and S.P. Ketkar

Vol. 12, No. 2

Iq21 qo

i. o. 30

. . . . .

~

"

~

I0.0 1.0

0.21

0.5

0.18

0.12 0. I

O. 06

Bi=O.05 0

I

i

i

4

i

8

I

i

i

12

~

16

i

20

•

i

24

i

i

Ola

FIG. 3.3 R a t i o of d i m e n s i o n l e s s normal t e m p e r a t u r e

gradient

q2 / q0 as a f u n c t i o n

of

dimensionless distance D/a for various Blot numbers (s/a = 15, L/a = 25).

Next example chosen was a region composed of t h r ee sides of L/a=25 enclosed by a s e m i - c i r c l e and containing two circular holes with a fixed spacing of S/a distance apart (see inset of Fig 3.1).

All boundaries are subject to same Blot number.

The computed

values of qo as a function of D/a and Blot number are shown in Figures 3.1. Figures 3.2 and 3.3 show the results for q~ and q2 (normalized using %) as a function of D/a for various Blot numbers.

These results indicate that the "first order" results can be

significant (up to 30% of %) for large Blot numbers. The last example chosen was the r e p l a c e m e n t of the s e m i c i r c u l a r c a v i t y su r f ace of the previous problem with an S-shaped contour as shown in Fig 4. The results for qo ' ql and q2 are tabulated and shown in Table 1. The examples discussed above are illustrations of application of the g e n e r a l program developed for this study.

[n this program the hole size, c o n v e c t i v e boundary

conditions, boundary of the region of interest, temperature imposed on the circular holes, and the number of holes can all be easily for other 2D applications.

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II o

T ,...4

L

II

©

© D Bi = 0.i

L

t-.

,4

FIG. 4 S k e t c h of t h e p r o b l e m a n a l y z e d w i t h t h e r e s u l t s s h o w n in T a b l e 1 ( L / a = 25).

Table 1 Values of qo, q l , and q2 for the two holes in the region s h o w n as a function of D/a ( s/a = 15).

U=J,

"1"

D/a

qo

qt

q2

qo

qz

q2

4

0.357

-0.033

-0.041

0.359

-0.033

0.039

6

0.341

-0.019

-0.048

0.341

-0.018

0.043

8

0.334

-0.010

-0.052

0.330

-0.010

0.046

10

0.332

-0.003

-0.054

0.325

-0.005

0.048

12

0.335

+0.003

-0.055

0.322

-0.001

0.047

14

0.345

0.013

-0.055

0.321

-0.001

0.045

16

0.368

0.022

-0.057

0.322

÷0.003

0.041

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M. Parang, R.V. Arimilli and S.P. Ketkar

Vol. 12, No. 2

Conclusion A special boundary i n t e g r a l method for t w o - d i m e n s i o n a l regions c o n t a i n i n g circular holes was applied to regions with arbitrary boundaries and convective boundary conditions. Several illustrative examples were solved and presented.

The advantage of

boundary integral method in conjunction with analytic representation of t e m p e r a t u r e or its normal derivative on the circular hole boundaries makes the method especially powerful in solving design and

o p t i m i z a t i o n problems in numerous applications.

Examples may include design and d e t e r m i n a t i o n of the location of optimal cooling lines in moulds and the d e t e r m i n a t i o n of location of buried powerlines or heat exchanger tubes.

Nomenclature

a o

radius of tube a

Oc

hole boundary

g

harmonic function in Green's identity

M

number of holes

n

unit normal vector at boundary

q

normal derivative of t e m p e r a t u r e (see Eq. 4)

qo% ql % q2 a R, OR

defined in Equation 7

ds

differential length on a boundary

X,Y

position vector of points on the boundary

Q

tube identification number

region R and its boundary

position vector of c e n t e r of tube a OQ

azimuthal angle of tube o

d~

temperature

~bOQ

constant t e m p e r a t u r e on the surface of holes

References I.

M. A. Jaswon and G. T. S y m m , Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, N e w York (1977).

2.

C. A. Brebbia, J. C. F. Tetles, and L. C. Wrobel, Boundary Element Techniques Theory and applications in Engineering, Springer-Verlag, N e w York (1984).

Vol. 12, No. 2 3.

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189

M. R. Barone and D. A. Caulk, Quai'. J. Mechanics and App. Math., XXX!v, Pt. 3,

265 (198l). 4.

R.V. Arimilli and M. Parang, AIChE Symposiu[/i Series, No. 225, Vol. 79, 121 (1983).

5.

R.V. Arimilli, M. Parang, and P. R. Surapaneni, to be presented at the 1984 A S M E Winter Annual Meeting, N e w Orleans.

6.

M . R . Barone and D. A. Caulk, Int. J. Numerical Methods in Engineering, 18, 675

(1982). 7.

W.D. MeMillan, The Theory of the Potential, McGraw-Hill, New York (1930).