numerical analysis of a cavity radiator with mutual interaction

ELSEVIER

Numerical analysis of a cavity radiator with mutual interaction

N. Ramesh, ’ C. Balaji 2 and S. P. Venkateshan

Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, India

The results of a numerical study of two thin longitudinal fins forming a conducting and mutually irradiating cauity radiating to outer space are reported. Three fin profiles, namefy rectangular, trapezoidal, and triangular, were considered. Out of these, triangular profiled fins have been subjected to a detailed analysis to elicit information pertaining to the thermal performance parameters. A two-dimensional finite difference technique is adopted to analyze the system. A new nondimensional heat dissipation parameter usefur in the design of such systems, as a function of the pertinent parameters, is presented.

Keywords: cavity radiator, conduction, radiation, mutual interaction

1. Introduction

Radiation heat transfer is the only mode of heat rejection in outer space. In a spacecraft, extended surfaces (fins) are widely employed as a means to reject waste heat from space power plants, heat exchangers, and electronic equip- ment.

Radiating fins of a radiator were a fascination for many a researcher during the 1960s. Nilson and Curry ’ devel- oped a method to determine the optimum profile of a triangular fin. The fin in this case was radiating to free space with no mutual interaction between fins. Sparrow et al.’ estimated the effectiveness of a radiating conducting wedge with rectangular profiled fins and presented an optimum fin geometry. Sparrow et a1.3 studied the heat transfer from fin-tube radiators including longitudinal heat conduction and radiant interchange between longitudinally nonisothermal finite surfaces. The finiteness of the fin length was considered in their study. Karlekar and Chao4 studied a similar problem, but with fins of trapezoidal

Address reprint requests to Dr. S. P. Venkateshan at the Heat Transfer

and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology, Madras 600 036, India.

Received 3 August 1994; revised 18 October 1995; accepted 2 November 1995.

1 Former Research Scholar and presently Graduate Student, Drexel Uni- versity, U.S.A.

* Former Research Scholar and presently Lecturer, Department of Me-

chanical Engineering, Regional Engineering College, Tiruchirapally 620 015, India.

Appl. Math. Modelling 1996, Vol. 20, June 0 1996 by Elsevier Science Inc. 65.5 Avenue of the Americas, New York, NY 10010

profile, and discussed rectangular and triangular profiled fins as special cases. The optimum fin number and their proportions were evaluated. Tip heat loss was considered to be negligible in all the above studies. Hering’ analyzed the radiative heat exchange between two conducting plates forming a wedge. The conducting plates, which formed the cavity, were of rectangular profile. Both the specular and diffuse cases were considered. Ambirajan6 studied the effect of directional surface radiative properties on the performance of a radiating conducting wedge, which irra- diate each other. The effects of metallic coatings and dielectric coatings were specifically studied.

All the above studies, with the exception of Sparrow et a1.,3 were based on a one-dimensional temperature distribution in a direction parallel to the fin height. The fins in effect were considered to be infinitely long. Recently Sridhar et aL7 discussed the effect of two-dimensionality in a radiating conducting wedge. They analyzed a space radiator with six rectangular profile fins interacting with each other and with the environment. The conditions under which two-dimensionality of temperature variation is important were discussed.

The present investigation undertakes a two-dimensional finite difference analysis with mutual irradiation between the fin surfaces. The heat loss from the sides is more realistically accounted for by the choice of appropriate boundary conditions. Aluminum has been chosen as the fin material, and Tb is maintained constant independent of z. Three fin profiles - rectangular, trapezoidal, and triangular - are considered for the analysis.

The configuration considered is a simple case of two fins of equal height sharing a common edge. Figure l(a) shows the geometry of the problem. The heat transfer is by simultaneous internal conduction and external surface radiation. The entire system radiates to outer space, which in

0307-904X/96/$15.00 SSDI 0307-904X(95)00167-0

Numerical analysis of a cavity radiator: N. Ramesh et al.

dz

dx

Figure 1. (a) Schematic of the radiator showing trapezoidal profiled fins, the coordinate axes, and the computational grid used for the finite difference formulation. (b) Energy balance for two-dimensional conduction for a typical element of the radiator.

the present analysis is assumed to be at 0 K. The following assumntions are made:

1. 2.

3.

The fin material is homogeneous and isotropic; The material properties (k and E) are independent of temperature; The base thickness is very small compared with the height of the fin, and the variation of temperature across the thickness is negligible; Heat conduction under the steady state is two-dimensional with the temperature varying with x and z; Each of the elemental surfaces is isothermal, diffuse, and gray; Radiosity and irradiation are assumed constant over an area element; and The energy incident on the fin from other planets, the Sun, planetary matter is ignored.

surface or the cavity or from any other

2. Mathematical formulation

2.1 Governing equations

Under steady-state conditions, the application of the basic law of energy conservation to a typical element of the fin, as shown in Figure l(b) yields the following:

dQ c,net + dQ,,net = 0 (1) The net conduction is found by applying Fourier conduction law as follows. We have

Q,= -&ix; (4

and by Taylor expansion, ignoring terms of order (dx12 and higher,

(3)

Hence dQ,,,,, due to conduction along the x-direction is

given by Qcx+dxj - Q,. Since A, = t(x)dz, we get

Q(x+dx) - Qx =

Similarly the net heat be derived as

-kdxdz; t(x) ; [ 1 (4)

Q (z+dz) -Q,=

conducted along the z-direction can

Approximating A, as t(x)&, the above becomes

Q (z+dz)

Therefore, the net conduction into the element is given by

(7) Net radiation from both the surfaces of the element (the top as well as the bottom) is obtained as a difference between the radiosity and the irradiation. Hence

dxdz dQ r,net = 2EGW4 -HI (8)

where the factor 2 on the right-hand side accounts for the radiation leaving the two sides of the fin. Expressions (7) and (8) may be substituted into equation (1) and rear- ranged to obtain

; tcx,g [ 1

2

+t(x)S=2 +&w4-fQ

(9) The following nondimensional variables are introduced:

tc v, = -

h

Appl. Math. Modelling, 1996, Vol. 20, June 477


‘h r Vb = - 1,=-

h rh

P=$ x=H l T3h2

NRC = b b UT,4 kth

(10)

The thickness t(x) is a function of x and is given by

t(x)=2t,+2tan cz(h+r-xX) (11)

for the most general case of a trapezoidal fin. In the nondimensional form it is given by

v([)=2~,+2tan cu(l+v,-5) (12)

With these, equation (9) takes the final form

a6 NRCvb -tan ff-=

a6 y[04-xl (13)

Along the base of the fin, the temperature is assumed to remain constant. Along the other three edges that exchange heat radiatively with free space, the conductive heat flux within the fin is balanced by the radiant heat flux away from the surface. Thus the boundary conditions in the nondimensional form are specified as

t=o Os[_<A l9=1 (14)

[=l OSIJSA -z =NR,vbe4 at

(15)

ae l=O Or(s1 -z=NR,vb04 (16)

l=A 0-<511 -ae=Nacvb04 8.5

(17)

The governing equation (13) as well as the boundary conditions involve radiation from various surfaces. The evaluation of the irradiation (or the incident radiant flux) X involves a detailed analysis of the radiant interchange among the fin surfaces and the environment. Such an analysis is presented below.

Let us assume that the temperatures are prescribed for all surfaces. The aim of the analysis is to determine the corresponding heat transfer rates. For this purpose, con- sider two elemental areas, as shown schematically in Fig- ure 2. Radiosity of area Ai consists of the following:

(1) emission from Ai and (2) the reflected portion of the incident power falling

on area Ai

B=mT4(xi, q) +pH(x,, zi) (18)

Under the assumption that the participating surfaces are gray, p is equal to (1 - E) in equation (18). With these the net radiant heat flux over the surface is given by

dQ , r net = Bi - H,

= cAxAz[ (rT4( xi, z,) - H( xi, q)] (19)

Figure 2. Nomenclature for defining view factor for diffuse radiant interchange. The illustration also shows the application of the contour integration method to typical surface elements for determining the view factor.

By a similar procedure, the radiosity over Aj may be evaluated. The fraction of the radiosity leaving Aj that is incident on Ai is given by the product of B, and the elemental view factor dF,,. Integration over the area Ai of this product yields the i&diat& over A,.

H( x1, q) = /, B,( xi, zj)dFijdAj I

In the nondimensional form the above may be

X( 6;, li) = Is /, P( tj, lj)dE;;jdSjd!Zj I I

(20)

written as

(21)

These equations along with the governing equation and the boundary conditions given earlier complete the formulation of the problem in the most general form.

2.2 View factors

In order to estimate the radiation heat loss, the view factors between elemental surface areas are required. In the present geometry there is no shadow effect between elements. The contour integration technique can be directly applied for calculating the required view factors for area elements that do not share a common edge. When area elements do share a common edge, the integral over the common edge has a singularity. Hence in these cases alone analytical formulas given by Sparrow and Cess’ and Kreith’ are used. Figure 2 shows the schematic of the geometry and the elemental areas that are considered for evolving a general formula for the evaluation of view factors by the contour integration method. For diffuse radiant interchange between two finite areas, the view factor is defined as

cos pi cos p,

n-r2

dA,ax, (22)

An alternate form of the defining equation for view factors can be obtained by replacing the area integrals by line

478 Appl. Math. Modelling, 1996, Vol. 20, June


integrals by using Stoke’s theorem as given by Sparrow and Cess8 In that representation equation (22) will be replaced by

F ‘J,.A, = &$c,~c,[h rdxidxj + In rdy;dy, I

+ In rdzidzj] (23) where

r=[(xj-xj)~+(yi-y,)‘+(zi-z~)Z]o~i (24)

which represents the distance between two points lying on the boundaries of A, and Ai. The area Ai is bounded by the contour Ci, which consrsts of the four segments indicated as I-IV in Figure 2. Integration in equation (23) is first performed along these four segments keeping fixed a point on contour Cj. If any of the coordinates remains fixed on a boundary, the corresponding differentials are zero and hence do not contribute to the integral. This is followed by an integration along contour Cj using a similar procedure. The integration procedure is simplified to a significant extent by using the coordinates x’, y’, z’ instead of x, y, z (Figure 2). The details are presented in Ramesh.” The final result is represented as

2TFA,,AZ = I, + I, -t I, + & (25)

where the Z’s are integrals as given below:

I, =~::+‘dz~ul::“ln[(~;X:,,~+y2

, +(r;-z;)Z]o’5di;-~~+11n[(x;-x~)2+y2

2,

+ (2; - z;)2]o’5dz;11 (26)

I, = ~“dx;[l~~+‘ln[( xi -xi)’ + y* I

+(zJ+~ -z~)‘]~‘~~x; -_/~‘+‘ln[(x;-x~)z+y’

+#+r -*:+,)*]“~sdX;~ (27)

I,= lE,dz;[[y+‘ln[(x:,l --_~~+~)~+y~

+(z~--z~)2]o’sdz~-~~‘~‘ln[(~~+,-~~)2+~2

+( 2; -z’1)2]o’sdz;~ (28)

~~=~I’,~;(~T:+‘ln[(~~-x;)‘+y’ +

+ ( 2; - *j)Z]“-5dx;

-~~~+~~n[(n~-~~)2+yZ+(~~-~~)2]o~5dX~~ x,

(29)

Since obtaining closed form expressions for the above is tedious, a numerical integration scheme based on Simpson’s rule was adopted. The shape factors were evaluated with an accuracy of some eight decimal places by a suitable choice of the step size in the numerical integration scheme.

3. Method of solution

A finite difference equivalent of the governing equation obtained by replacing the partial derivatives in terms of finite differences was used in solving the problem. A Lagrangian interpolation polynomial of second degree was assumed to express the variation of the dependent variable (temperature) using three-point formulas. These were dif- ferentiated with respect to space coordinates to obtain the finite difference analogs for the partial derivatives. With the step sizes along the 5 and 5 directions taken as A[ and &‘, respectively, the finite difference form of equation (13) is

2

+ (5i-5j-l)(Si-Si-l) “”

2

+ (Si+~-~~-~)(~i+l-~i) 8i+1’i

2

+ (Si+l-5i-1)(5i+l-li) Bz’i+’ I

[v,+tan a(l+v,--[,)I

[

5i - 5, + 1 -

(SC-~ - ti>( Ei-1 - 5i+l> ei-l’j

2ti-5i+l-5i-1

+ (ti-ti-l)(5i-5i+l) “‘j

X tan (Y = S{ 0z j - X,4} (30)

The boundary conditions along the common edge, being of the first kind, are specified as

c=o O,,j = 1 for 1 Ij 5 )2 (31)

The boundary conditions along the other three edges are of the same type being of the third kind. As an example, the boundary condition along the edge at 5 = A for all ,$ is



Table la. Effect of grid size on 7) for AR = 1, E = 0.98 and Table 2. Comparison of present work with Sparrow et al. I = 0.1 m. (1961).

Grid size Efficiency for Efficiency for % error finite elements infinite elements

10 x 10 77.41 76.63 1.01 12 x 12 77.21 76.63 0.75 14x14 77.08 76.63 0.58

NC Efficiency from Sparrow (1961)

(%)

Efficiency (%) (Present work)

AR=2 AR=5 AR=10

given in terms of three nodal temperatures - the boundary and the previous two nodes.

2& - L-1 - &-2

(5, - G-,1( 5, - 5,-1) (+*,

5, - Jn-2

0 80 77 77 79 0.5 60 56 57 58 1 50 46 47 48 2 39 36 37 38 3 34 31 32 32 4 30 27 28 28

+ (5,-i - 5;1--2)( J,-1 - L) C-1

5, - f;, - 1

- ( L2 - CL *)( L-2 - L) eivn-2

= NRC vbe;j (32)

In the above, i, j represents a node (as indicated in Figure 2) while k indicates the element number.

Tables la and lb give satisfactory results. It was decided however to adopt a 12 X 12, grid for all calculations reported here, considering an error criterion that the uncer- tainty in the calculated efficiency should be I 1%. It was also verified that the grid-dependent errors are the most severe in the case of triangular profiled fins, and hence the same grid size was adopted for fins of rectangular as well as trapezoidal profiles.

The solution starts with the specification of the geometry. This is done by specifying the base thickness, tip thickness, angle between adjacent fins, height of the fins, and aspect ratio. The view factors are then evaluated. Starting values for all the nodal temperatures and the nodal radiosities are now specified. With these assumed values, the radiosity equations are solved to obtain updated values for the nodal radiosities. With these updated values of radiosities, the finite difference analog of the fin equation are solved to obtain updated values for the nodal temperatures. This process is repeated until convergence occurs.

4. Results and discussion In order to choose the number of grids on each of the participating surfaces, a grid dependence study was carried out for two cases, one of a low aspect ratio of 1 and the other of a high aspect ratio of 8, for a wedge cavity using triangular profiled fins. For the case of E = 0.98, the fin efficiency was calculated for 10 X 10, 12 X 12, and 14 X 14 grids. Using a Lagrangian polynomial of second degree, the fin efficiency for the case of an infinite number of elemental areas was estimated by extrapolation. The percentage error of the efficiency for the three grid sizes used in the computation was calculated with respect to the extrapolated value. Tables la and lb gives the values of the efficiency and the percentage error for the various grid sizes. It is seen that the error is a very weak function of the grid size, and hence any of the grid sizes indicated in

In order to ascertain the validity of the present analysis, computations were carried out for a radiator with rectangular profile fins for different aspect ratios, y = 90” and E = 0.75. The heat loss from the sides was ignored by imposing adiabatic boundary conditions as was done by Sparrow et al. 2 in their one-dimensional analysis. The fin efficiency values thus obtained were compared with the one-dimensional results of Sparrow et a1.2 The reference values are strictly valid as A + ~0. Since the efficiency values calculated by Sparrow et al.2 were read from a graph, the values are accurate to whole numbers only. Hence the present values have also been rounded to the nearest whole number as shown in Table 2. It is seen that the efficiency of a two-dimensional fin is always lower than that of a one-dimensional fin. This is a consequence of the temperature variation along the length of the radiator which is totally ignored in the one-dimensional formulation. The average surface temperature for a two-dimensional fin is invariably less than that for a one-dimensional fin. However, as A becomes larger and larger, the results of the two-dimensional calculations approach those of the one-dimensional model as indeed they should.

4.1 Effect of various parameters on the thermal performance

In order to study the effect of various parameters on the heat transfer performance of the radiating wedge, results were computed for the range of parameters shown in Table 3. These calculations show interesting trends that are described below.

Table lb. Effect of grid size on q for AR = 8, E = 0.98 and I = 0.8 m.

Grid size Efficiency for Efficiency for % error finite elements infinite elements

10 x 10 78.84 77.89 1.22 12 x 12 78.65 77.89 0.97 14x14 78.50 77.89 0.78

Effect of fin profile. - Since system mass should be held to the lowest possible value in space systems, the most significant parameter is the heat dissipated per unit

Table 3. Range of parameters considered in the present study.

2b: 1.5 x 10m3 m g: 30”, 60”, 90 2t: 1.5 x 10m3 m e: 0.5, 0.75, 0.98

k: 204 W/m K AR:3<AR<8



0.03 0.0 5 0.07 0 09 0.1 1 Fin height (m)

Figure 3. Variation of heat dissipated per unit mass with fin height for three fin profiles (E = 0.98, I= 0.3 m, y = 90”).

system mass. A comparison of three different fin profiles with regard to heat dissipated per unit system mass is shown in Figure 3, for a typical case. For a given fin height, rectangular fins have the largest volume (and hence mass) followed by the trapezoidal and triangular profile fins in that order.

The total heat loss is a function of the temperature distribution and the irradiation. Both of these are the greatest at the base of the radiator and decrease monotoni- cally as we move toward the fin tip. For relatively short fins, the total heat dissipated from the system is not a strong function of the fin profile. But after a certain height, the rectangular profiled fins appear to lose more heat. The reason for this becomes clear if we observe the midplane temperature distributions along the fin height for three profiles, as shown in Figure 4. There is very little difference in the temperature profiles up to a certain value of fin height for all three profiles. Thus the fin profile does not influence the temperature and hence the heat loss (since it is proportional to the fourth power of temperature) for relatively short fins. However, as the fin height is increased further, the temperature distribution shows a strong

‘ii E j, 0.99- : E .-

D

6 2 0.98-

: 2 2 x E 0.9?-

:

,.,,I 0.00 020 0.40 0.60 0 60 l.Dil

Distance along fin height (non dimensional)

0.80 8 0.00 0.20 0.40 0.60 060 1.00

Distance along fin height (non dimensianol)

Figure 4. Variation of midplane temperature (at 5 = 0.5) along Figure 6. Variation of midplane temperature (at 5 = 0.5) along the fin height for three fin profiles (E = 0.98, I= 0.2 m, y = the fin height for different aspect ratios. Fins are of triangular 90”, A = 4). profile (E = 0.98, I = 0.3 m, y = 90”).

0 length .O 1 m 0 . :O.Zm a II =0.3m

0 Y .0.4m

2.50 4.50 6.50

Aspect rot io

8i50

Figure 5. Variation of efficiency with the aspect ratio for different radiator lengths (E = 0.98, y = 90”). The fins are of triangular profile.

dependence on the fin profile. The temperature profile is the steepest for the triangular fins and much less so for the trapezoidal and rectangular fins. Hence the rectangular fins will lose more heat compared with the trapezoidal and triangular fins when the fin height is increased. At the same time the volume of the fin material increases, the increase being the largest for the rectangular fins followed by the trapezoidal and triangular fins in that order. When the heat dissipated per unit system mass is considered, it is the triangular profiled fins that show the largest values followed by the trapezoidal and rectangular profiled fins. In view of this, the triangular fins appear to be the best choice for a space radiator, and hence the rest of the paper is restricted to this case.

Effect of aspect ratio. - The aspect ratio is defined as the ratio of the length of the radiator to the height of the fin. For a given height of fins, as the aspect ratio increases, the surface area of the fins available for heat transfer increases. It is expected that the aspect ratio should strongly



influence the efficiency. Fin efficiency is defined as the ratio of the actual heat dissipated (Q,) by the fins to that which would be dissipated if the fins were at the base temperature throughout <Qi, which is calculated by the program by taking all the nodal temperatures equal to Tb). As the fin length is increased, for a given fin height, the fin effectively tends to become ‘shorter’, and the fin efficiency increases (Figure 5). Fin efficiency is a nonlin- ear function of the aspect ratio showing a diminishing increment in fin efficiency per unit increase in aspect ratio as the aspect ratio is increased. This trend is observed for all values of y and E.

The aspect ratio also has an important role in the two-dimensionality of the temperature field. Two-dimensional effects reduce with increasing aspect ratio. For a given radiator length, as the aspect ratio is increased the fin tends to appear shorter and the temperature profiles become more “uniform”. The temperature difference between the base and the tip decreases. For example if the coolant inlet temperature (and hence the base temperature) is 350°K and A = 8, the drop in temperature between the base and the tip is only 7°K while it is about 63°K for A = 2 (Figure 6).

5. Correlation

A large amount of numerical data was obtained by varying the various parameters over the range of values shown in Table 3. When the results are analyzed in terms of suitable nondimensional parameters certain interesting features are revealed. When the fin efficiency is plotted as a function of the radiation conduction interaction parameter (NRC), for various radiator lengths, all the results fall on a single curve as shown in Figure 7. Again a plot of the variation of heat dissipated per unit system mass with fin height (Figure 8) shows that the results are insensitive to the radiator length. In fact, this insensitivity to radiator length is even more glaring when the product of heat dissipated per unit system mass and fin height is plotted as a function of fin height (Figure 9). All the data points collapse onto a

o IengIh .O.lm

T: 0 ” =O.Zm

z & ” = 0.3m

c 85 E z .-

5

J 0.00 0.10 0.20 0.30 0.40

Radiation Conduction brameter

Figure 7. Variation of efficiency with N,, for different radiator lengths. Fins are of triangular profile (t = 0.98, y = SO”).

I I 0.0 5 0.10

Fin height (m)

Figure 8. Variation of heat dissipated per unit system mass with fin height for different radiator lengths. Fins are of triangular profile (E = 0.98, y = SO”).

p 3000 -

f 2 a E 2000 -

; a Q length ~0.1

; 1000 _ .Ep . .7 :0.4 Y r

OO.--I ( J.1

Fin height(m)

6

Figure 9. Variation of the product of fin height and heat dissipated per unit system mass with fin height for different radiator lengths. Fins are of triangular profile (E = 0.98, -y = SO”).

0.00 0.50 1.00 1.50

Heat dissipation parameter (data)

Qpc (dota)

Figure 10. Parity plot of correlated Qp versus computed 0,.

single curve. These observations apply to all the cases



considered in the present study. These observations, as well as a dimensional analysis based on the Buckingham 7~ theorem, indicate that a correlation for the nondimensional heat dissipation parameter (Qrc) as a function of the aspect ratio, nondimensional fin base thickness, nondimensional profile area parameter (A,), and NRC defined previ- ously should describe the results in a form that is useful for design. The correlation is given by the following:

Qpc = 0.340A’.‘4vb0.077AOp.91NROi81 (33)

The correlation indicates that the aspect ratio, profile area parameter, and radiation conduction interaction parameter have a significant effect on the heat dissipated, while the base thickness parameter has a minor direct influence on it. However, the base thickness indirectly affects QPc through A, as well as NRC, both of which involve the base thickness. A, is defined as the ratio of the opening area of the cavity to the profile area of the fin, given by

c

y+2a

2

h cos-

A,= 2

+1 t, cos (Y cos ff

1

(34)

The cavity opening area is appropriate for two-dimensional problems since radiant heat leaves the cavity from a rectangular opening to the background between the two fins as well as through two triangular openings at the two ends. The correlation given by equation (33) is based on 118 data points. The standard error of the estimate of the dependent variable is 0.015. The correlation coefficient is 0.999, and the maximum error of the correlation is 13%. The excellent agreement between the numerical data and the correlation is seen from the parity plot shown in Figure 10.

6. Conclusions

Comparison of the performance of cavity radiators with rectangular, trapezoidal, and triangular profile fins has shown that the triangular fins are the right choice from a consideration of heat dissipated per unit system mass. Computations were confined to a radiator consisting of triangular profiled fins over a practically useful range of parameters. Two-dimensional effects are important for small aspect ratios. Irrespective of the length of the radiator, it was found that the efficiency values correlate strongly with the radiation conduction interaction parameter. The product of the heat dissipated per unit mass and the fin height plotted against the fin height indicates that all the data, irrespective of the fin length, collapse onto a single curve. Based on these observations, a correlation has been obtained for the nondimensional heat dissipation per unit mass in terms of the other relevant nondimensional parameters.

aspect ratio (l/h)

Nomenclature

Symbols

A

A, AS B

‘ij

H h k 1 m n

:

dimensionless area parameter, defined in the text surface area cm*> radiosity (W/m*) Diffuse view factor between surfaces i and j irradiation (W/m*) height of fin (m) thermal conductivity of fin material (W/m . K) length of the radiator (m) mass of the system (kg) upper limit of grid index i or j unit vector normal to a differential area element total number of elemental areas on any one of the fin surfaces, also upper limit on index k, equal to n* dimensionless radiation conduction interaction parameter (~aTzh*/kt,) total heat dissipated from the radiator (W) dili;;sionless heat dissipation parameter (Qplh/

distabnce (see Figure 3) half fin thickness (ml temperature (K) coordinate along fin height (m) coordinate along fin length (also radiator length) (m)

N RC

is,,

Greek symbols

X

half taper angle of the fin profile dimensionless radiosity (B/U Tz) also, angle between the normals and the line joining the two areas (see Fig. 2) angle between adjacent fins (degrees) hemispherical surface emissivity of fin flats dimensionless z coordinate (z/h) fin efficiency <Q,/Q,> dimensionless fin thickness (t/h) dimensionless x coordinate (x/h) density of fin material (kg/m3> Stefan-Boltzmann constant (5.667 X lo-’ W/m* . K4) dimensionless irradiation (H/a Tz)

Subscripts

actual base cold edge or tip of fin hot edge or base of fin indices ideal pertaining to the profile pertaining to the surface

References

1. Nilson, E. N. and Curry, R. The minimum weight fin of triangular profile radiating to space. J. Aerospace Sciences 1960, 27 146-147

2. Sparrow, E. M., Eckert, E. R. G., and Irvine, T. F. The effectiveness of radiating fins with mutual irradiation. J. Aerospace Sciences 1961, 28, 163-772



3. Sparrow, E. M., Jonsson, V. K., and Minkowycs, W. .I. Heat transfer from fin-tube radiators including longitudinal heat conduction and radiant interchange between longitudinally nonisothermal finite surfaces, NASA Technical Note, NASA TN D-2077, 1963

4. Karlekar, B. V. and Chao, B. T. Mass minimization of radiating trapezoidal fins with negligible base cylinder interaction. Inf. J Heat and Mass Transfer 1963, 6, 33-48

5. Hering, R. G. Heat exchange between conducting plates with specular reflection. J. Heat Transfer 1966, 88, 73-75

6. Ambirajan, A. Effect of directional radiative surface properties on the performance of mutually irradiating conducting fins. M.S. Thesis, Department of Mechanical Engineering, Indian Institute of Technol- ogy, Madras, India, 1990

7. Sridhar, K., Balaji, C. and Venkateshan, S. P. The effect of two-dimensionality in radiating conducting wedges. Proceedings of First ISHMO-ASME Heat and Mass Transfer Conference, Bombay, India, 1994, 171-176

8. Sparrow, E. M. and Cess, R. D. Radiation Heat Transfer, Revised Edition. Brooks/Cole Publishing Company, California, USA, 1966

9. Kreith, F. Radiation Heat Transfer for spacecraft and Solar Power Plant Design. International Textbook Company, Pennsylvania, USA, 1962

10. Ramesh, N. Numerical study of cavity and tubular space radiators. M.S. Thesis, Department of Mechanical Engineering, Indian Institute of Technology, Madras, India, 1995


numerical analysis of a cavity radiator with mutual interaction

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