INT. COMM. HFATMASSTRANSFER 0735-1933/86 $3.00 + .00 Vol. 13, pp. 423-432, 1986 ~PergamonJournals Ltd. Printed in the United States

RADIANT-INTERCHANGE CONFIGURATION FACTORS INSIDE SEGMENTS OF FRUSTUM

ENCLOSURES OF RIGHT CIRCULAR CONES

Joseph C.Y. Wang Sui Lin Centre of Building Studies Mechanical Engineering Dept.

Concordia University, Montreal, Canada H3G IM8

Pai-Mow Lee School of Engineering

Lakehead University, Thunderbay, Canada P7B 5E8

Wei-Liang Dai You-Shi Lou Mechanical Engineering Dept. Department of Mathematics

Concordia University, Montreal, Canada H3G I M8

(C~q~tnicated by C.L. Tien)

ABSTRACT

Radiant-interchange configuration factors inside segments of frustum- enclosures of r ight circular cones are numerically determined and graphically presented.

Introduction

The solution of practical thermal radiation problems depends frequently

on the ava i lab i l i t y of interchange configuration factors. The interchange

configuation factors for many practical geometries have been presented [ i -6 ] .

The important groups of geometries which were not presented are the cases of

two segments cut from the two bases of a frustum-enclosure of r ight circular

cone by a plane parallel to the axis of the frustum-enclosure, and an

isosceles trapezoid and a segment of a disk being perpendicular to the

isosceles trapezoid and having a common edge with the isosceles trapezoid.

The purpose of this paper is to present the results of the configuration

factors for these two groups.

423

424 J.C.Y. Wang, et al. Vol. 13, No. 4

Determination of the Configuration factors

I t is well known that the conf igurat ion factor , FA~_A2, under the

assumption that the magnitude and surface distr ibut ion of the radiosity is

uniform over A~, can be expressed by

i [ [ cos 81c0s 62 dA1 dA2

= JA JA ( I ) FAI-A2 ~ I 2 xr2

where 61 and 62 are the angles formed by the normals of the elements dA1 and

dA 2 and the connecting l ine between the elements dAi and dA2. In Eq. (1) r

represents the length of the connecting l ine. I

Case 1. For the determination of the configuration factors for radiant inter-

change between two segments cut from the two bases of a frustum- enclosure of

r ight c ircular cone by a plane parallel to the axis of the frustum-enclosure,

a schematic diagram, Fig. 1, shows the coordinate syste, for the relat ive

position of the two segments. The contour of the segments Aland A 2 can be

expressed, respectively, by

y l = _+~a 2 - x~ (2)

The angles 61 and 62 can be obtained as d cos 61 = cos 62 = F (4)

where

r =~/(xl - x2) 2 + (Yl - Y2) 2 + d2 (5)

The areas of the elements dAi and dA2 can be expressed as

dAl = dxl dyi (6)

dA2 = dx2 dy2 (7)

and the total surface area of segment 1 is

= [cos- T - c 2 (8)

Substituting Eqs. (2) to (8) into Eq. (1) gives

FAI_A = ~ dx dx 2 dy i (9)

c Vb2_x~ ~ [ (xl-x2) 2+(yl-Y2) 2+d2 ] 2

Eq. (9) can be reduced to a double definite integration with constant lower

and upper l imi ts as

VOI. 13, NO. 4 RADIANT-INTERCHAN(~CGNFIGt~ATICN~RS 425

where

Y2 d2 [a b I [yl~n_1(~) . Y2~n.1(l(_) ] ~2

FAI"A2 = ~-~lJc dxl c ~ (1o)

X =~d 2 + (x I - x212 (111

Y1 x2 +V T. x2 (12) 1 2

Eq.

to 2.3.

FAI_A 2 ca lcu la ted for the case of a = b and c = -a (the two segments under

th is consideration become two disks having the same radius) was compared with

that obtained from the solution of radiant interchange between two bases of a

r igh t c i rcu lar cyl inder [1-6 ] . The comparison shows that both results agree

to 6 places of decimal for 1.0 < d/a < 100 while 5 places of decimal for

0.1 <_d/a < - l .

Case 2. For the determination of the configuration factors for an isosceles

trapezoid radiat ing to a segment of a disk being perpendicular to the

trapezoid and having a common edge with the trapezoid, Fig. 3 shows the

coordinate system for the re lat ive posit ion of the trapezoid and segment. The

contour of the segment can be described by

Y2 = + ~ + z l la - z) 1141

The contour of the legs of the isosceles trapezoid is

b 1 - b Yl = +- ( T X +~) (15)

The angles 1)i and f)2 can be written as _ z ( 1 6 ) c o s 111 - ?

Y2 = ~ - x2 - % / ~ - x2 (131 1 2

(10) was numerically integrated. The results ape shown in Figs. 2.1

In order to evaluate the accuracy of the numerical integrat ion,

x (17) cos P2 = ~ where

r =%/X 2 + (Yl - Y212 + z2 (18)

The areas of the elements dA1 and dA 2 ace expressed by

dA 1 = dx dy I (19)

dA2 = dY2 dz (20)

426 J.C.Y. Wang, et al. Vol. 13, No. 4

The total area of the trapezoid is

b I + b AI = (---Z--) • c (21)

Substituting Eqs. (14) to (21) into Eq. (1) gives

~b I - b ~/(b 2

FAI_A 2 = x-~-lJ'Co x dx[ zdz I dy I Jo b b I _ b J~j/(b 2 + r4

• / T X ~ ~T z)(a - z)

+ z)(a - z)

(22)

Eq. (22) can be reduced to a double definite integration with constant lower

and upper limits as

I I i Y2 Y3 Y~

1 c z (~)+Y2tan -I (~ ) Ystan -I (~r-)-Y4tan- I(~T-) ]dz FAI"A2=i~-'~-I o x dx ~-~-[3 Yltan-1 -

(23)

where

X =~/x 2 + z 2 (24)

b - b I +lu/( b2 Y1 = ( T x - ~) ~-~ + z)(a - z) (25)

Y2 = ( T x + ) - ~-~+ z)(a - z) (26)

. . . .

Y3 = ( ~ x - ) - ~-~+ z)(a - z) (27)

Y4 : (---Z~---x + ) (T~ + z)(a - z) (28)

The double definite integration of Eq. (23), with constant lower and upper

l imits, was numerically calculated. The results are shown in Figs. 4.1 to

4.4.

I t should be noted that there is a singularity existing in the evaluation

of Eq. (23) at x = z = 0. To accomplish the integration of Eq. (23), the

lower l imi ts of x and z in the integration are modified to very small values

instead of zeros. The modified lower l imits are so chosen that i t gives

s a t i s f a c t o r y accuracy to the value of FAI_A 2. Results indicate that a

modified lower l im i t having a value of 10 -8 w i l l give a good accuracy for

FA1-A2"

Vol. 13, No. 4 RADIANT-INTER2HANGE CC~FI(~RATICN F~I~ORS 427

Appl icat i on Example

As an example, we consider the case that a frustum-enclosure of r ight

c i rcular cone be cut by a plane parallel to the axis of the frustum-enclosure

as shown in Fig. 5. By making use of the configuration factors presented in

Figs. 2.1 to 2.3 and 4.1 to 4.4, and in combination with the configuration

factors for radiation interchange between a disk and a segment of a parallel

concentric disks [7], the configuration factors for radiation interchange

between any two of the ten surfaces (surfaces 1,2,3,4,5, 6,7,(1+2),(3+4),

(5+6)) as shown in Fig. 5, can be determined.

References

1. R. Siegel, J.R. and Howell, "Thermal Radiation Heat Transfer", 2nd. Ed.

McGraw-Hill, New York (1981).

2. E.M. Sparrow and R.D. Cess, "Radiation Heat Transfer", McGraw-Hill, New

York (1978).

3. H.C. Hottel and A.F. Sarofim, "Radiative Transfer", McGraw-Hill, New York

(1967).

4. H.Y. Wong, "Handbook of Essential Formulae and Data on Heat Transfer for

Engineers", Longman, New York (1977).

5. J.A. Stevenson and J.C. Grafton, "Radiation Heat Transfer Analysis for

Space Vehicles", SID 61-91, North ~erican Aviation, (ASD TR 61-119, pt.

1), (Dec. 1961).

6. J.R. Howell, "A Catalogue of Radiation Configuration Factors", pp.

89-242, McGraw-Hill, New York (1982).

7. S. Lin, P.M. Lee, J.C.Y. Wang, W.L. Dai and Y.S. Lou, "Radiant-lnterchange

Configuration Factors between a Disk and a Segment of a Parallel

Concentric Disk", Int. J. Heat Mass Transfer (in press).

428 J.C.Y. Wang, et al. Vol. 13, No. 4

z dA 2

b (x2 Y2'd) X2

4o/

A1 o

(xI,Yl,O)

FIG.1 Geometric configuration for radiant interchange between two segments cut from the two bases of a frustum enclosure of right circular cone by a plane parallel to the axis of the frustum enclosure.

F A I-A2

I

c/a-O.O

0.0- oo

0 , 4 - -

0.3-

0.2--

0.1-

0 . 0 -

10-2 v I vo d/a FIG.2.1

FIG.2.1--2.3 Configuration factors, F A -A ' for radiant interchange between

two segments cut from the two bases of ~ f~ustum enclosure of right circular cone by a plane parallel to the axis of the frustum enclosure as a function of d/a with b/a as a parameter. FIG.2.1 for c/a=O.O, FIG.2.2 for c/a-0.5, FIG.2.3 for c/a=0.99.

VOl. 13, NO. 4 RADIANT-~CC~I(~RATICNF2~S 429

FA I-A2

1

B.

~o

B . B -

1 g - 2

~ .,~, ~/.-o.s

\

~~ ~/-~.,oo \\\\\"~,~

, \\~\

I g - I I I ~ FIG.2.2

d/a

F A i-A2

1 .~--

g.8 =

~ .7~

~.6 =

~.5--

~ . 4 -

z. 2-:"

~.I~

Ig-2 I I I I I I I

10-I

[ b / a - I00

×\ \ \ \ \

I~ 2 d/a I I l l l l

FIG.2.3

c /a -0 .99

10

430 J.C.Y. Wang, et al. Vol. 13, No. 4

J -dA 1

(x,Yl,0)

FIG.3 Geometric configuration for an isosceles trapezoid radiating to a segment of a disk being perpendicular to the trapezoid and having a common edge with the trapezoid.

I" A I - A 2

O.

O, ~ ~ ~ b |Ib=O.O

0 .

2:2 10 I I 10 | 0 2 c/b

FIG.4.1 FIG.4 .1 - -4 .4 Conf igurat ion f a c t o r s , F A A ' for r a d i a n t in terchange between

an isosceles trapezoid and a segment ofla ~isk being perpendicular to the trapezoid and having a common edge with the trapezoid as a function of c/b with a/b as a parameter. FIG.4.1 for bl/b:O.O, FIG.4.2 for bl/b=O.5, FIG.4.3 for b l /b=l .O, FIG.4.4 for bl/I~=2.0

Vol. 13, NO. 4 RADIANT-INT~RCHAN(~ CONFI(~JRATION FACIDRS 431

F' A I-A2

\ \ \ \ \ \

i:iii \ \;\~'

1 0 ~-I

FA l-A2

1 10 !

FIG.4.2

0 .

0 .

0 .

0 .

0 .

0 . 1

0 . 1

0 . 0 0 -

10

~ bllb=l'O

\\'~ \\\\\\ ~o

, ) \ \ \ ~ \ \ \ \

\\\\ \I \\\\\ \

clb

FIG.4.3 I 1 0 c/b

432

F A I-A2

0.

0 . 2 5 -

0 . 2 0 -

O.1

O.1

0,

0.00

10 -I

J.C.Y. Wang, et al. Vol. 13, No. 4

, \ \ L. ~ ~ ~ X. XIX XXX

1 1 0 clb

FIG.4.4

1 2

5 6

FIG.5 Geometric configuration for the application example --- determination of configuration factors for radiant interchange between any two of the ten surfaces. ( l , 2, 3, 4, 5, 6, 7, (I+2), (3+4), (5+6)).