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RADIANT-I"'I'ERCHANGE CONFIGURATION FACTORS FOR SPHERICAL AND CONICAL SURFACES TO SPHERES

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N A T I O N A L AERONAUTICS A N D SPACE A D M I N I S T R A T I O N W A S H I N G T O N , D. C. A P R I L 1968

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TECH LIBRARY KAFB, NM

RADIANT-INTERCHANGE CONFIGURATION FACTORS FOR

SPHERICAL AND CONICAL SURFACES TO SPHERES

By J a m e s P. Campbell and Dudley G. McConnell

Lewis Research Center Cleveland, Ohio

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sole by the Clearinghouse for Federal Scientific ond Technical Information Springfield, Virginia 22151 - CFSTI price $3.00

RA DIANT-I NTE RC HANGE CONFIGURATION FACTOR S FOR

SPHERICAL AND CONICAL SURFACES TO SPHERES

b y James P. Campbell a n d Dudley G. M c C o n n e l l

Lewis Research Cen te r

SUMMARY

Geometric configuration view factors have been calculated for a sphere and for cones radiating to a sphere. A computer program was set up to give configuration view factors not only for the entire radiating body but, also, for bands of the radiating body which permit the determination of the configuration view factor between any part of the radiating body and the sphere.

The range of configurations consisted of equal radii spheres, and right-circular cones of semiapex angles of 15', 30°, 45', and SO0 radiating to a sphere on the cone axis with radius equal to the cone base radius. Separations for all cases ranged from zero to 10 sphere radii.

within the limits of the study. Curves a r e presented that give the configuration view factor for any configuration

INTRO DU CTlON

Many chemical- and nuclear -rocket propulsion systems use cryogenic liquids for propellants. A s a result, on long-time interplanetary missions, thermal protection of these liquids is imperative to prevent excessive boil off.

be the solar heat flux. Smolak, Knoll, and Wallner (ref. 1) have shown that shadow shields can greatly reduce solar heating under such conditions. In particular, inflated shadow shields promise low structural weight and, therefore, high shielding efficiency in t e rms of boiloff weight to shield weight.

storage tank, the configuration view factor is needed. The equation for the configuration view factor between finite areas is

During the planet transfer portion of the flight, the largest external heat source will

In the analysis of the radiant heat t ransfer between a shadow shield and a cryogen

For the differential a r ea view factor f rom ana rea element to a finite area, the equation is

A

Solutions of these equations are available in closed form for only a few simple configur- ations.

Hamilton and Morgan (ref. 2) compiled the configuration view factors that had been previously published and added several for different geometrical shapes. They were limited to a few simple configurations consisting of flat surfaces, concentric bodies, o r infinite parallel surfaces. Since then, the digital computer has made more complex sur - faces amenable to analysis. Consequently, configuration view factors for three- dimensional surfaces, such as spheres, cones, cylinders, e tc . , a r e now appearing in the literature.

shielding in a space environment, shapes that were attractive, because they afforded intuitively small view factors and could be easily fabricated, were the cone and the sphere. However, a review of the l i terature showed that many of the desired configura- tion view factors were not available.

Of the few available sphere-sphere (ref. 3) and the cone-sphere (ref. 4) configura- tion view factors, the pertinent parameters investigated were limited and none were useful for analyzing the effect of applying high emittance zones o r bands (ref. 5) and for variation of temperature over the radiating surface. A computer program was therefore set up to give configuration view factors from a sphere, o r any part of a sphere, to a sphere and from a cone, o r any part of a cone, to a sphere over a range of separations. The computer program w a s set up by use of the method presented in the appendix.

computing program. Although the work presented herein does not include every possible sphere-sphere and cone-sphere configuration, the actual cases studied a r e quite com- plete. In addition, because configuration view factors a r e presented for any part of a radiating body, radiant heat-transfer calculations can be made for cases where the emittance and temperature a r e not constant over the entire surface.

In setting up an experimental program to determine the effectiveness of shadow

This paper is a compilation of the configuration view factors which resulted from this

2

SYMBOLS

surface area

radiative heat-transf e r configuration view factor

unit vectors along x, y, z axes, respectively

distance between dA1 and dA2

unit vector normal to dA

radius of sphere

base of cone

distance from center of sphere to elemental surface a rea o r distance from apex of cone to elemental surface a rea

separation between two spheres or cone and sphere

axes in Cartesian coordinate system

angle from z-axis to position vector r and cone semiapex angle

angle in the x-y plane

angle between the normal to an elemental a rea and a line connecting this

-

elemental a rea to another (angle between and L)

Subscripts :

1 shield

2 tank

DESCRIPTION OF CONFIGURATIONS

S phe re-S phe r e Con f ig u ra t ion

The sphere-sphere configuration view factors were determined for separation dis- tances varying from zero to 10 tank radii (in steps of one tank radius) and were limited to equal radii spheres. At each step, the configuration view factor w a s determined for the complete range of spherical-cap half angles from zero to 90' as shown in figure 1.

Cone-S phe r e Con f i g u r a t i o n

The configuration view factor for four different half-angle (15O, 30°, 45O, and SOo)

3

right-circular cones radiating to a colinear sphere were determined. The cone-sphere separations varied from zero to 10 tank radii. The cone base radius ran the complete range from zero to the sphere radius as shown in figure 2.

DESCRl PTlON OF CHARTS

Sphere -Sphere

a r e shown in figure 3 for the whole and A1FA1-A2 The values for both FA1-A2 sphere radiating to a sphere of equal radius. The configuration view factors for this particular case (R1 = Rz) w a s one of those reported by Jones (ref. 3). However, the Values for F A ~ - A ~ that come out of this study are twice those of Jones, because he took the entire surface a rea of the sphere for A1, while this study considered only that part of the sphere that was radiating to the other sphere. This same criterion for establishing the value of A1 was used for all the values in the present report.

may be determined by use of figure 4.

AIFA 1-A2 of 0 is simply the difference between the values for each 0 . The differential a rea view factor Fa 1-A2 for any point on the surface of the radiating sphere is given in figure 5. If the configuration view factor for an extremely narrow band is desired, greater accuracy should result from the use of the differential area view factor over the desired area.

The results, when only a part of the first sphere is radiating to the second sphere, The curves shown indicate the value of

for the spherical caps, but the value of A1FAl-A2 between any two values

Cone-Sphere

for right-circular cones of semiapex angles and FA1-A2 The values for A1Fa1-A2 of 15', 30°, 45' and 60 radiating to a spherical body with the same radius as the cone base a r e shown in figure 6 over a range of separations from zero to 10 tank radii. configuration view factors determined by Nothwang, Arvesson, and Hamaker (ref. 4) for configurations that matched those of the present study a r e indicated by points on the appropriate curve.

The configuration view factors, for cases where only part of the radiating cone is considered, may be determined by use of figure 7. These curves indicate the value of

A lFA 1-A2 configuration, the value of A1FA1-A2 fo r bands around the cone is simply the difference between the values of A lFA 1-A2 for the two cones with base radii equal to the radii of

The

for the cone caps as indicated, but, as in the case of the sphere-sphere

4

the bottom and top of the band. As in the sphere-sphere configuration, the differential a rea view factors FdA1-A2 for cones have been included (fig. 8) for greater accuracy for narrow bands. 45', and 60'.

cones with semiapex angles different from the 15O, 30°, 45', and 60' reported. have not been included because of the great number of combinations of base radius and separation possible with the reported data.

This view factor is given for cones with semiapex angles of 15', 30°,

Plots can be drawn that would permit interpolation of the configuration view factors for These

Lewis Research Center, National Aeronautics and Space Administration,

Cleveland, Ohio, November 24, 1967, 124- 09-05- 09 -22,

5

APPENDIX - CONFIGURATION FACTORS

Configuration Factor Between Two Spheres

The calculation for the configuration view factors reported herein were performed with a high-speed digital computer. The necessary equations f o r these computations a r e derived in this appendix.

The method is that of Nothwang, Arvesen, and Hamaker (ref. 4) and Jones (ref. 3) in which the coordinate positions of points on each body, the distance between the points, and the unit normals to the body surface at the points a r e expressed in vector notation and applied to the equation for the diffuse radiation configuration view factor.

written as In the sphere-sphere case, the geometry is as shown in figure 9 and the vectors a r e

-c -+ -c

r1 = rl( i sin 8 cos cp + j sin .Q1 sin cpl + k cos 8,) 1 1

r2 = r2(i sin O2 cos cp + j sin O2 sin cp2 + k cos 0,) 2

nl = i s in e l cos ‘pl + j s in e l sin q1 + k cos el

+ - c -c + n2 = i sin O2 cos q + j s in 0 sin q2 + k cos O2 2 2

- c - c

R1 = krl

- c - c

R2 + kr2

4 4

S = k s

The distance vector between the two points is

+ - - - -c

L = r R - S - R 1 - r 1 2 - 2

The angles ql and q2 between the distance vector and the normals to the surfaces may be obtained from the scalar product relations:

6

cos G2 = - ' In21 ILI

4

The elemental areas located at T1 and r2 are given by

dA1 = r1 2 s in el de1 d q l

dA - r 2 s in O2 de2 dq2 2 - 2

Substituting into

cos *l cos l&2 dA1 dA2

yields the configuration view factor.

Configuration Factor Between a Cone And a Sphere

In the cone-sphere case the geometry is as shown in figure 10 and the vectors a r e written the same as in the sphere-sphere case except for the normal to the conical sur - face which is

4 + - + n1 = i cos 0 cos 'pl + j cos el sin q1 - k sin 0 1

The distance vector between the two points is

1 L = r - R - S - r 2 2

The angles ql and q2 and the elemental a rea dA2 a r e the same as in the sphere- sphere case, and the elemental a rea of the conical surface is

7

dA1 + rl sin O1 d r 1 dql

The configuration view factor is then found with the same substitutions as before.

8

REFERENCES

1. Smolak, George R. ; Knoll, Richard H . ; and Wallner, Lewis E. : Analysis of Thermal- Protection Systems fo r Space-Vehicle Cryogenic-Propellant Tanks. 1962.

NASA TR R- 130,

2. Hamilton, D. C. ; and Morgan, W. R. : Radiant-Interchange Configuration Factors . NACA TN-2836, 1952.

3. Jones, L. R. : Diffuse Radiation View Factors Between Two Spheres. J. Heat Transfer, vol. 87, no. 3, Aug. 1965, pp. 421-422.

4. Nothwang, George J. ; Arvesen, John C. ; and Hamaker, Frank M. : Analysis of Solar- Radiation Shields for Temperature Control of Space Vehicles Subjected to Large Changes in Solar Energy. NASA TN D-1209, 1962.

5. Jones, L. R. ; and Barry, D. G. : Lightweight Inflatable Shadow Shields for Cryogenic Space Vehicles. J. Spacecraft and Rockets, vol. 3, no. 5, May 1966, pp, 722-728.

9

Figure 1. - Sphere-cap - sphere configuration. Range of angles f r o m z-axis t o position vector, 0' to 90'; sphere-sphere separation, 0 to 10 sphere rad i i (in increments of 1 radius).

Figure 2. - Cone-sphere configuration. Range of cone semiapex angles , 15'. 30°, 4 5 O , and 60'; cone base radius. 0 to rad ius of sphere; cone-sphere separation. 0 , 1, 2 , 4, 6 , 8, and 10 sphere radii.

10

. I

. 0 1

00 1

i \ \ \

\ \ \ \ \

\

\

/ \

\

\

\ \

\

\

\ \

2 I I I I I 4 6

Separation to radius ratio, s/R

Figure 3. - Configuration view factor and dimensionless configuration view fac tor t i m e s radiating area for sphere to sphere configuration.

I

11

...

100

10-1

0 2

I

60 ao 100

Spherical cap angle, 0 . deg

Figure 4. - Dimensionless configuration view factor t imes radiating area for sphere cap to sphere .

12

1 oc

10-1

lo -*

I

10

i 30

1 I

1 7 - Separation - to radius ~

ratio,

s/R

\ -> \ - 2

\ 4

\ \

-6

'. \ -8

- 10

I

Differential a rea latitude, i J , deg

'\

\ '\

'. '. \ \

Figure 5. - Differential a rea view factor of any point on surface of sphere to sphere of equal radius.

13

10

10-

lo-'

l o - - 2 3 4

I l l I ! !

semiapex angle

0 , deg

4 15-

-\30-

t= I

5 6 Separation to r ad ius r a t io , s / R

7

I '-t-

A i F ~ i -A!

R2 - -___

FA1-A2

0 Ref. 4

8 9 10

Figure 6 . - Configuration view factor and dimensionless configuration view fac to r t i m e s radiating a r e a fo r cone to equal r ad ius sphe re on cone axis .

14

. 2

- id s4 id M

.* +

v 3 X

4 h

+ id 1

3 .- c .d + ri .3 M

2 u - m m aJ C 3

.d 0

C

2 a

. 0 8

.06

. 0 4

. 0 2

. O l

, 0 0 8

, 006

. a04

,002

,001

0

/

/

/

/ =

i

I

/

i

/

/

/ /

t i n e ;

I

I

/

/

/

/

/

/

/-

I

0

0

-/

I

r C o n e base

I

I Isc: 1 I I I I I I 1 I I I I . 3 . 4 . 5 . 6 . I . 8 . 9 1.0

Cone base radius. R, R

(a) 15’ cone.

F igu re 7 . - Dimens ionless configuration view fac tor t imes radiating a r e a for cone to sphe re on cone axis.

15

I I

I Linea r I s ca l e

~

3 . 4

I

I

0

i

I

/

/

- -

I

I

I

I

I

I

I I Separation- to r a d i u s -

- 1 1 I I I I I I I

I

I

I

0

0

0

i

rCoiie base

1 1 1 1 1 . 6 .I . 8

i

ratio, -

s/R

7-==

z I I

$ I I

1 I

3 1 1 1

9 1.0 Cone base rad ius , Rc R

(b) 30’ cone.

Figure 7. - Continued.

16

1. I

. f

. f

N p: ' 1

M .A .Of c

3 . O f v

X - b . 0' c

rd I

m 2 . O O t 4 C .: . O O E C aJ E 6" ,004

002

,001

0

-

/

i

/

/'

/

i /'

t o r ad ius ratio,

I I

0

/

/

/

/

/

/

t #inear

I 1 Iscallel 3 . 4 . 5

I

0

/

/

/

i

/

6 .7 . 8 . 9 1.0 Cone base radius . R, R

(c) 45' cone.

F igu re 7. - Continued.

17

1.0

.8

. 6

. 4

. 2

. 1

. 08

.06

.04

.02

. O l

008

006

004

002

00 1

0 .1

ra t io ,

Separation to r ad ius

sca l e I I I I I

~

~

7

I

I

I

/

/

/

i l l i l . 6 . I . 8

(3 I i l

. 9 3 Cone base radius. Rc R

(d) 60’ cone.

F igu re I. - Concluded.

18

d

i m

10-1

S

Differential a r ea location, R, R

(a) 15' cone.

Figure 8. - Differential a rea view factor of any point on surface of cone to sphere on cone axis.

tw 0

. 1 . 2 . 3 . 4 . 5 . 6 .I . 8 . 9 1.0 Differential a r e a location, R, R

(b) 30' cone.

Figure 8. - Continued

5 .3

100, \

A, ratio,

I I I I I I I I I I Separation- to radius

s 'R + 10-

>

-- Id 4 -- - ----- ------ k r j - I

Differential a r e a location, Rc R

(c) 45' cone

Figure 8. - Continued

N N

N

P 3 d

[r

3

3 i) - i

d -- -------- ----- ---- --- 3

41

-- 4 3

(d) 60' cone.

Figure 8. - Concluded.

Z Z

Figure 9. - Sphere-sphere geometry. Figure 10. - Cone-sphere geometry.

N W

I

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