Investigation of approximate methods for calculation of the diffuse radiation configuration view factor between two spheres

Pergamon Press Printed in the United States

IN HE~T AND MASS TRANSFER Vol. 3, pp. 513 - 522, 1976

INVESTIGATION OF APPRCXIMATE METHODS FGK CALCULATION OF THE DIFFUSE RADIATION CONFIGURATION VIEW FACTOR BETWEEN T~40 SPHERES

Niels H. Juul Department of Mechanical Engineering State University of New York at Buffalo Buffalo, New York 14214

(C~cated

ABSTRACT

by J.P. Hartnett and W.J. M i n k ~ z )

The view factor calculated by a numerical integration procedure is used to evaluate t~e error involved by using the view factor obtained by various approximate methods, such as: a) the well known spherical point source approximation, b) a spherical point source approximation utilizing the reciprocity relationship, and c) the closed form integration over the spherical surface, where the cones from any point subtend the other sphere. The approximation (c) is shown to be identical to (b) if the viewed sphere is replaced by an infinitesimal sphere. The regions where the approximate method is accurate to within i% are determined. Detailed results obtained by numerical integration are presented for the region close to zero separation where all the approx'_mate methods are inaccurate.

Introduction The diffuse radiation configuration view factor between two spheres has been calculated previously by numerical integration utilizing a digital computer, see for example [I], [2] and [5]. However, the numerical integration procedure is expensive due to the large amount of computer time needed for the small grid size required for suitable accuracy.

Therefore, approximate methods, which are expressed by simple analytical

equations, are important and very useful.

Approximate methods also often give ad-

ditional physlcal insight into the various mechanisms which interact in the exchange proce ss. The point source approximation is obtained by replacing the radiating sphere with a spherical point source. evaluated in [5].

The error involved by using this approximation is

The error for all configurations with a radius ratio less than 513

514

N.H. Juul

Vol.

one tenth is shown to be less than 3.8%, erKor

increases

drastically

as the separation crease

is increased

the error

Therefore,

spacings when the radius when the radius

is rapid

methods

for diffuse

radiation

to spherical

satellites

applied

radius ratio.

radius

coordinate

as the contribution

cone which vertex

to the extreme

second approximation.

obtained

by the closed

area on the earth

is used

approach,

the equation

the closed

approaches

in [3] and [4j result in identical

are excellent.

satellites.

Furthermore

in [3] and [4], are surprisingly

subtends

The point source approximation,

cases the above approx-

to the view factor obtained

utilizing

It is an excellent approximation

configuration

given geometric reciprocal surfaces

configuration

radius

I' and 2'

ferent values

ratios,

Figure

i represents

one for surfaces

of radius

by replac-

is developed

radius

in this

than one,

I shows the curves

for constant

of the

ratios.

The

two cases of radiation with

i and 2, and one for the interchanged

The two cases have identical

ratio is equal to one.

sets of view factor but dif-

ratio and spacing ratio; except, when the radius Therefore,

factor at point A from surface reciprocating

I.

two spheres

in Figure

reciprocity,

presented

relationship.

for radius ratios greater

by the aid of Figure

view factor between

that the

for the view factor

ing the satellite with a point source and using the reciprocity

is demonstrated

system in the

it is shown here that the results

identical

is a

by the first one.

It turns out,

For these extreme

sate-

the earth.

form integral

coordinate

expressions

of the

is defined by the

introduced

for the view factor.

The view factor

to the spherical

with the defining

to derive

from the earth to spherical

cases for radiation

and which

It tends to cancel the error

[4] uses a unit sphere

to a small one

form integration

the integral

to evaluate

satellite,

which

is good for all

system is fixed in the earth.

is in the center of the Satellite

The area of the earth which

paper.

This de-

ratios but is very

from a large sphere

The area of the earth used to evaluate

imations

the error decreases

[3~ and [4].

exact view factor from a differential

Watts

for small

if two spheres with

radios are less than one, but only for very large spacing

In [3] the defining

llite.

However,

the point source approximation

are available and have been successful

is approximated

is 31%.

the

ratio is larger than one.

Approximate

from earth

For example,

for spheres with constant

in error, with separation,

slow for large ones.

but as the radius ratio is increased

as would be expected.

a radius ratio of I0 are touching

3, No. 6

relationship.

as illustrated

i to 2 is related

However,

view factor at B for the interchanged

in Figure

i the view

to the view factor at E by

the view factor at E is equal surfaces

i' and 2'.

to the

Thus the relation-

VOI. 3, No. 6

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Vol. 3, No. 6

ship between the view factors at A and B are expressed exactly by (FI2) A = (F21) E A2/A I = (FI,2,)B A2/A I for a given geometric configuration.

For a con-

stant radius ratio, the curve representing the view factor FI2 can be generated from the curve for FI,2, or visa versa by using the above equation for corresponding spacing ratios.

Next, if the point source approximation is used for inter

changed surfaces then (FI,2,)B is approximated by (Fps,2,) D.

Finally, the point

source approximation utilizing reciprocity relationships is expressed by FI2 = Fsp,l(R/r)2.

The error incurred by this approximation is theoretically iden-

tical to the error obtained by the point source approximation for the reciprocal radius ratios, R/ro

The error is rather small (R/r < I) and decreases

rapidly with separation, as mentioned earlier. The purpose of this paper is to evaluate the merits of the various approximate methods for calculating the view factor between two spheres.

Furthermore,

the regions are determined where the approximate methods give errors of less than I% when compared to numerical calculations. Analysis a)

The point source approximation.

The diffuse radiation view factor for a point source to a sphere of radius R is expressed by Fps 2

=

½ (i - J i - (R/c)2)

(i)

where c is the spacing. b)

The point source appr~imation utilizes

the reciprocity relati0ns.

Using reciprocity, the view factor from a sphere I with radius r to a sphere 2 with radius R which is replaced by a point source sp' is expressed by FI2 ~ Fsp,l A2/A I c)

=

½ (R/r) 2 (i - J I - (r/c) 2)

(2)

The close d form iDte~ration aPProximation.

In Figure 2, the view factor from the differential surfa:ce dA 1 at latitude ~I (less than the limiting angle ~ILI ) to surface 2 is obtained by closed form integration and expressed by FdA IA 2

=

R 2 (c cos ~i

-

r)/(r 2 + c 2

-

2rc cos ~i )3/2

!

An approximate value of the view factor FI2 is then obtained by integration of FdAIA 2 over that part of the surface i which faces sphere 2 within a cone gen-

Vol. 3, No. 6

VIE~FACTOR~TWD

SPHERES

517

erated by the cross tangents to the spheres and is expressed by , FI2

=

I f diLl 2~r sin sir d~ I 4~r 2 --o FdAIA2

=

½ (R/r) 2 (I - (I - r/c cos SiLl)/J (r/c)2 + i -

2 r/c cos SiLl)

(3)

Substituting the limit, cos SiL 1 = (R + r)/c, is into equation (3) we obtain F{2

=

½ (R/r) 2 (I - (I - (r/c) 2 - rR/c2)/~ I - (r/c) 2 - 2 Rr/c 2)

(4)

The view factor FI2' is the closed form contribution in the range 0 ~ s I _< SlL 1 to the total view factor FI2 (i.e. the first term of equation (I) in [5]).

In the

range SIL 1 ~ s I ~ alL the contribution to FI2 can only be evaluated by numerical integration (second term of equation (I) in [5]). SlL , SiL 1 and SlL 2 are presented.

In Figure 2 the angle limits

The relative size of the two ranges of s I men-

tioned above can be observed on Figure 2 and indicate how good the closed form approximation is.

For example it is observed that for r/R less than one, SIL 1 is

small compared to SlL thus the closed form integration is only a small part of the total and therefore is a poor approximation.

If r/R is greater than one, SiL 1

approaches SlL , and the closed form approximation improves. is sufficiently ~ r g e

For example, if r/R

then rR/c 2 = (r/c)2R/r << (r/c) 2 and equation (4) becomes

identical to equation (2).

Furthermore, if sphere 2 is replaced by an infinite-

simal sphere, the limiting angle for the integration becomes SiLl = SlL 2 = cos-l(r/c). Substituting this angle into equation (3) results in equation (2). In [3], the limit of integration is also SlL 2 and the equation obtained can be shown to be identical equation (2).

That is, the method in [3] is identical

to the approximation (b) which is an interesting observation. The following approximation is obtained using a binomial series expansion of equation ~I), Fsp 2 = ½ (I - (I - ½(R/c) 2 - I/8(R/c) 4 - 1/16(R/c) 6 + etc.)) for R/c << I we obtain Fsp 2

=

~(R/c) 2

(5)

On the other hand equation (5) can also be obtained if equation (2) is expanded by a binomial series and r/c << I. Thus for large separation ratios the view factor is approximately equal to the ratio of the projected area of the incident sphere and the area of a sphere with a radius equal to the spacing c. This appr~imation is excellent for radiation between interplanetary bodies.

518

N.H. Juul

Vol. 3, No. 6

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VOI. 3, No. 6

VI~WFACIDR~~D

SPHERES

519

Results The error involved in using the approximate methods, is evaluated by dividing the difference between the view factors obtained by the numerical method and the approximate method with the value obtained by the numerical method.

The re-

suits are presented in Figure 3. Figure 3 shows that the percentage error for all approximations decreases with increasing separation ratio.

The error using the point source approximation

(equa-

tion (I)) is large for radius ratios larger than one but small if the radius ratio is less than one.

The error in using the point source approximation utilizing

reciprocity relations

(equation (2) is small for radius ratios greater than one

and decreasing with increasing radius ratios.

Theoretically,

the error in using

equation (I), for radius ratios less than one, is identical to the error obtained by using equation (2) and interchanging the surfaces for the same geometric configuration.

For example, in Figure 3 the curve for r/R = 0.i representing the

point source approximation and the curve for r/R = I0.0 representing the point source approximation utilizing reciprocity show nearly

the same error at corres-

ponding separation ratios~ i.e., the curves are closely

affine

ordinate.

in respect to the

The small discrepancy observed is due to slightly different error in view

factor for r/R = 0.I and r/R = i0.0 when calculated by the numerical method. The error involved using the closed-form approximation is 100% for zero separation, but as the separation increases the error decreases. errors involved by the different approxlmatlons~

A comparison of the

shown in Figure 3, indicates

that the point source approximation is superior for r/R < 1.0 whereas the point source approximation utilizing reciprocity is best for r/R > 1.0.

For separation

ratios close to zero all approximate methods are inaccurate. Figure 4 presents the regions where the respective approximate methods apply with an accuracy of 1% or better.

In region I the values of the view factor with

an accuracy of 1% or better can only be obtained by the numerical methods.

The

boundaries between region I and respectively regions II and III are determined by the condition that the error incurred by using equations is about 1%.

(I) and (2) respectively

The numerical values of the view factor in region I are presented

in Table I, for an accuracy of 1% or better. Conc l u s i o n The two a p p r o x i m a t e are

found to be superior

configuration

view factor

methods based and the regions between

on s p h e r i c a l where

two s p h e r e s

point

they apply

source are

approximations

determined.

c a n now b e o b t a i n e d

for

all

The con-

520

N.H. Juul

Vol. 3, No. 6

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Vol. 3, No. 6

VIEW F A C T O R ~ T W O

SPHERES

figurations to an accuracy of 1% or better by using Table I.

521

For configurations

not presented in Table I, equation (i) is an accurate representation for radius ratios equal to or less than one.

For radius ratios greater than one equation (2)

is accurate to within one percent. Nomenclature A

surface area

F

diffuse radiation configuration view factor distance between dA 1 and dA 2

r

radius of sphere 1

R

radius of sphere 2

S

separation between the spheres

C

spacing, distance between centers of the spheres angle between normal to surface and direction of line

oe1

latitude, angle between line c and location of dA I

Subscripts

1

surface I

2

surface 2

ps

point source

L

limit of angle

'

interchanged surface References

I.

L . R . Jones, "Diffuse Radiation View Factors Between Two Spheres", Journal of Heat Transfer, Aug. 1965, pp. 421-422.

2.

James P. Campbell and Dudley G. McConnell, 'Radiant-Interchange Configuration Factors for Spherical and Conical Surfaces to Spheres", NASA TN D-4457, 1968.

3o

J . A . Stevenson and J. C. Grafton, '~adiatlon Heat Transfer Analysis for Space Vehicles", ASD ~R-61-I19, Wright Patterson Air Force Base, Ohio, 1961, pp. 80.

4.

Ro G. Watts, '~adiant Heat Transfer to Earth Satellites", Journal of Heat Transfer, Aug. 1965, pp. 369.

5.

Niels H. Juul, "Diffused Radiation Configuration View Factors Between Two Spheres and Their Limits", Letters in Heat and Mass Transfer, Vol. 3, pp. 205212, 1976.