Sunshade of a passive cryogenic detector-cooling system for a geostationary satellite

A. I. Abrosimov, A. A. Verlan, and V. K. Sysoeva�

S. A. Lavochkin Scientific Manufacturing Organization, Khimki, Moscow Oblast�Submitted August 19, 2009�Opticheski� Zhurnal 77, 45–50 �May 2010�

This paper presents a physical model of a passive cryogenic system �a radiative cooler� forcooling a detector equipped with a sunshade to protect it from the sun when it is mounted on ageostationary spacecraft. The results of a numerical simulation of heat transport in the coolerare given. It is shown that the sunshade has an insignificant effect on the cooling throughput ofthe system and the detector temperature. Values of the thermal fluxes are given that characterizethe heat release of the sunshade radiators into outer space and the interaction of the sunshade withthe detector radiator. © 2010 Optical Society of America.

Some versions of systems for cryogenic cooling of ra-diation detectors in outer space are passive radiative heatexchangers1–3 that use only the absorptivity of outer space.Such systems are constructed using a two-stage setup, whilethe attainable temperature of the detector in geostationaryorbit is 73 K in the long-term operating regime.2

By comparison with a system that includes a cryogenicengine that operates by the split-Sterling cycle, a passivesystem has a virtually unlimited service life, is characterizedby the absence of detector oscillations, and is not a source ofelectromagnetic noise. At the same time, a passive systemhas large dimensions and mass and also must be placed onthe spacecraft so that no solar radiation is incident on itsradiators and illumination of the radiators by the earth and bythe components of the spacecraft is minimized or, better yet,completely eliminated.

The problem of protection from the sun was solved onthe geostationary spacecraft Electra-1 by turning the space-craft around an axis perpendicular to the axis of the radiativecooler �RC� by 180°. In other words, in flight, the sun isalways in the same half-space relative to the plane of theradiators of the passive cryogenic system. Such rotationswere carried out once per half-year.

Even though the operation of rotating the spacecraft wassynchronized with the operation of cleaning the entrancewindow of the detectors from cryogenic deposits, it would beeven better to avoid them altogether. This would make itpossible to avoid losses of observation time and increase thefunctioning reliability of the space complex.

This paper discusses the functioning layout of a spaceobservation system, according to which protection from thesun is provided by rotating the sunshade. Numerical simula-tion is used to study how the sunshade affects the thermalregime of the cooling system by noting the temperaturevariation of the detector-mounting site and the loss of cool-ing throughput.

Figure 1 shows the layout of a two-stage RC consistingof a series of reflective screens of cylindrical and conicalshape attached to each other by special polyamide inserts andfour metallic radiators �two on the RC and two on the sun-shade�. It is assumed that the heat comes from the base,whose temperature is 300 K, and is released into space,whose temperature is 10 K. Besides this, heat can be given

331 J. Opt. Technol. 77 �5�, May 2010 1070-9762/2010/050

off when the detector located on the second stage of the RCis operating, from the signal cable that leads up to it and thecontrol cable �it was assumed that the heat influx along it isconstant and equals 60 mW�. One more heat source in thesystem is the sun. It is assumed that it is located in the planeof maximum and minimum generatrices of the sunshade. Inorder to take into account the most unfavorable version, it isassumed that the solar radiation is perpendicular both to thevertical boundary of the shell and to the maximum generatrixof the sunshade, as shown in Fig. 2.

First of all, let us consider a physical model of the sun-shade of the RC. To calculate the heat exchange in the sun-shade and how it affects the radiators of the RC, it is requiredto determine the areas of all the conical surfaces and radia-tors of the RC and to determine the angle coefficients of thelast cone with respect to the radiators.

The sunshade is a system of concentric conical reflectivesurfaces cut by an oblique plane. This plane is perpendicularto the plane in which the sun lies, and its slope angle is �=23° �Fig. 3�. For simplicity, we assume that the generatrices

123

4

5

6

7

8

0

9

P

BP

B1

T = 300 K

B2B3

B4

10

FIG. 1. Layout of the radiative cooler with designations of the elementswhose temperatures were calculated �P is the surface of the SVHI of theshell, BP is the surface of the SVHI of the sunshade, 0 is the shell, 1–6 and9–10 are screens, B2 and B4 are the sunshade screens, 7 is the first stage, 8is the second stage, and B1 and B3 are the bodies of the sunshade�.

331331-05$15.00 © 2010 Optical Society of America

make an angle of �=20° with the vertical and that the lengthof the minimum generatrix is l=50 mm. Introducing a coor-dinate system as shown in Fig. 3, we write the equation ofthe XZ plane that describes how the height of the cut-off partof the sunshade depends on the x coordinate and the condi-tions described above,

z = tan �x + l cos � + tan ��r0 + l sin �� . �1�

Cones B1 and B2 are equipped with radiators that lie inthe intersecting plane. All the cones are connected with eachother by inserts.

Let us consider the surface areas of the cones. It is con-venient to write the equation of the surface area of the conein cylindrical coordinates in which the Z axis and the coor-dinate origin coincide with the Z axis and the coordinateorigin of the Cartesian coordinate system �see Fig. 3�,

z = �r − r0�/tan �, x = r cos �, y = r sin � ,

where r0�r�R,

0 � � � 2� , �2�

and R is determined from the condition

z � tan �x + l cos � + tan ��r0 + l sin �� . �3�

From this we find

R = �r0 + tan �b�/�1 − tan � tan � cos ��, 0 � � � 2� ,

�4�

where b is determined by Eq. �1�.

= 20°

FIG. 2. Direction of incidence of the sun’s rays assumed in the calculations�to take into account the worst versions�.

X

Z

O

r0

FIG. 3. Sunshade and introduced coordinate system �the Y axis is perpen-dicular to the plane of the page�.

332 J. Opt. Technol. 77 �5�, May 2010

Taking into account the symmetry relative the XZ plane,we get a formula for calculating the area of the lateral coni-cal surface

S = ��r0 + tan �b�2

��0��1 − tan � tan � cos ��−2d� − �r0

2�� sin � .

�5�

The formula is the same for all cones. When computingthe next nested cone, only the value of r0 changes. It isobtained from the preceding r0 by subtracting �ins /cos �,where �ins is the length of the inserts.

Let us calculate the area of the radiators of the sunshade.These radiators are strips of a some specified width �rad,lying in the intersecting plane. The equation of the innercurve R1��� of each strip is thus Eq. �4�, while the equationof the outer curve is obtained from it by using the conditionthat the width of the strip is constant; it is given by

R2 = R1 + �rad/�1 + tan2 � cos2 � . �6�

Taking into account the symmetry relative to the XZplane, we get the formula for finding the area of the radiatorstrip,

Srad = �rad�1 + tan2 ��

0

�

�2R1���/�1 + tan2 � cos2 �

+ �rad/�1 + tan2 � cos2 ��d� . �7�

Let us calculate the heat exchange between the last coneand the radiators of the RC. We denote the emissivity of thecone as �k and the emissivity of the radiator as �p, and weconsider the interaction of elementary area dAp of the radia-tor and elementary area dAk of the cone. The radiation inten-sity of element dAp in the normal direction is ipn

=�ribn.

Since it is assumed that the radiation intensity of thesurfaces obeys the cosine law, like the radiation of an abso-lute blackbody, the directed emissivity ��= i� / i�n= �in cos �� / �ibn

cos ��= in / ibnis independent of angle � and

identical to the hemispherical emissivity,

� =e

eb=� id�� ibd = 2�in�

0

�/2

cos � sin �d��2�ibn�

0

�/2

cos � sin �d�� = in/ibn. �8�

The energy emitted in unit time by surface dAp in thedirection of the solid angle in which surface dAk is seen willequal

d2Qp = �pibn1cos �1d1dAp, �9�

where �1 is the angle between the normal to dAp and thedirection to dAk. Since d1=cos �2 /s2, where �2 is the anglebetween the normal to dAk and the direction to dAp, and s isthe distance between the elementary areas, we get that

332Abrosimovet al.

d2Qp = �pibn1cos �1 cos �2s−2dApdAk. �10�

It is assumed that the surfaces are grey, and thereforeKirchhoff’s law is valid for them: the absorptivity equals theemissivity of the surface �at the same temperature, but thetemperature dependence is neglected in this simplified for-mulation�. Therefore, the heat absorbed by an element of thecone is expressed by

d2Qp−k = �p�kibn1cos �1 cos �2s−2dApdAk. �11�

According to the Stefan-Boltzmann law, ibn1= � /��T4,

and therefore

d2Qp−k = �p�k�/��Tp4 cos �1 cos �2s−2dApdAk. �12�

The rest of the heat is reflected and can at some anglesbe reflected several more times inside the cone �without re-turning, however, to the radiator�. It would be too compli-cated to take this into account strictly; therefore, we assumethat all the rest of the heat escapes into space. In the sameway, we find the expression for the energy emitted by ele-ment dAk to element dAp,

d2Qk−p = �p�k�/��Tk4 cos �1 cos �2s−2dApdAk. �13�

As a result of the radiant heat exchange from elementdAk to element dAp, the following amount of heat is trans-ferred:

d2Q = �p�k�/���Tk4 − Tp

4�cos �1 cos �2s−2dApdAk. �14�

Now, in order to calculate the total heat flux from thecone to the radiator, it remains to integrate the expression ford2Q over the area of the radiator and over the area of thecone. Taking into account that �p, �k, Tk, and Tp are constantover the area, we get that the total heat flux is expressed by

Q = A�p�k�Tk4 − Tp

4� , �15�

A =� � �cos �1 cos �2/�s2�dApdAk. �16�

Since it was not assumed that the size of the sunshadevaries, while the calculation of all the areas and coefficientsdescribed above takes time, they were calculated with a sepa-rate program and were then incorporated into the main pro-gram as starting data.

Having analyzed the physical model of the sunshade, weconsider its mathematical model. Taking into account the as-sumptions for calculating the temperatures of the screens, wewrite the algebraic heat-balance equations, which constitutethe Cauchy problem �this is done from considerations of con-venience of the subsequent calculations, as well as to make itpossible when necessary and with the minimal changes touse the program to calculate the nonsteady-state operatingregimes of the RC�. We designate the temperature T of thescreens of the elements of the sunshade by appending to itthe symbols of the items shown in Fig. 1 �TS is the tempera-ture of space� and we designate the heat flux Q betweenelements likewise. Since the program makes it possible tocarry out calculations for various versions of the design, in

333 J. Opt. Technol. 77 �5�, May 2010

fact different systems of equations are solved. Let us con-sider the most complete version, in which the influence ofthe sun on the shell and the influence of the sunshade aretaken into account. Then the mathematical model, taking intoaccount the connection of the temperature of the elements ofthe sunshade and the thermal fluxes between these elements,is the following system of equations:

• for the surface of the screen-vacuum heat insulation�SVHI� of the sunshade,

QPBS�TS4 − TB0

4 � + QPB1�TB1 − TB0� + QPBSun = 0, �17�

• for the first cone of the sunshade,

QB12Q�TB24 − TB1

4 � + QB12�TB2 − TB1� + QPB1�TB0 − TB1�

+ QB1S�TS4 − TB1

4 � = 0, �18�

• for the second cone of the sunshade,

QB12Q�TB14 − TB2

4 � + QB12�TB1 − TB2� + QB23�TB3 − TB2�

+ QB23Q�TB34 − TB2

4 � = 0, �19�

• for the third cone of the sunshade,

QB23Q�TB24 − TB3

4 � + QB23�TB2 − TB3� + QB34Q�TB44 − TB3

4 �

+ QB34�TB4 − TB3� + QB23S�TS4 − TB3

4 � = 0, �20�

• for the fourth cone of the sunshade,

QB34Q�TB34 − TB4

4 � + QB34�TB3 − TB4� + QB8Q�T84 − TB4

4 �

+ QBKQ�T74 − TB4

4 � + QB4S�TS4 − TB4

4 � = 0. �21�

The coefficients of the equations are computed from thefollowing formulas:

QPS = 2�r0lCT�SVHI , �22�

QSun = 2r0lCTA�S, �23�

QKS = ���rk32 − rk2

2 � − QBK�sc/��p−1 + �S

−1 − 1� , �24�

QBKQ = QBK�p�sc�, �25�

QB8Q = QB8�p�sc�, �26�

QPBS = �S1 + SRB1��SVHI , �27�

QPB1 = �S1 + SRB1�/q , �28�

QPBSun = SSA�S, �29�

QB12Q = �S2 − f insnvb�/��sc−1 + �S2/S1���sc

−1 − 1� , �30�

QB12 = f insnvb�ins/�ins, �31�

QB1S = SRB1/��p−1 + �S

−1 − 1� , �32�

QB23Q = �S3 − f insnvb�/��sc−1 + �S3/S2���sc

−1 − 1� , �33�

Q = Q , �34�

B23 B12

333Abrosimovet al.

QB34Q = �S4 − f insnvb�/��sc−1 + �S4/S3���sc

−1 − 1� , �35�

QB34 = QB12, �36�

QB1S = SRB2/��p−1 + �S

−1 − 1� , �37�

QB4S = �S4 − QB8 − QBK�/��p−1 + �S

−1 − 1� . �38�

The conventional symbols of the parameters and theirvalues used in the calculations are as follows:

QPBS is the heat flux between the surface of the SVHI of thesunshade and space,QPBSun is the heat flux between the surface of the SVHI ofthe sunshade and the sun,=5.7536�10−8 W /m2K4 is the Stefan-Boltzmann con-stant,TS=10 K is the temperature of space,�S=1 is the emissivity of space,�0=0.245 is the emissivity of the shell and the base,nvb=48 is the number of inserts that attach the elements ofthe sunshade to each other,�SVHI=0.64 is the emissivity of the SVHI mats,�S=0.4 is the absorption coefficient of the solar radiation ofthe SVHI mats,q=20 m2K /W is the thermal resistance of the SVHI mats,�sc=0.05 is the emissivity of the screens,�p=0.92 is the emissivity of the radiators,r0 is the radius of the lower base of the outer cone, equal tothe radius of the shell,f ins=12.57�10−6 m2 is the area of the cross section of aninsert,�ins=0.004 m is the length of an insert,�ins=0.16 W /mK is the thermal conductivity of an insert,Si is the area of the lateral surface of the ith cone �I=1,2 ,3 ,4�,S1=0.7 m2, S2=0.69 m2, S3=0.684 m2, S4=0.675 m2,SRB1=0.063 m2 is the area of the radiator of the outer cone,SRB2=0.0276 m2 is the area of the radiator of the third cone,SS is the area of the projection by the visible sun of part ofthe outer cone onto a plane perpendicular to the direction ofthe sun’s rays �computed�.

METHOD OF SOLUTION

The resulting system of equations was solved by the pre-ferred nonlinear block method of Seidel.

The internal iterations for the systems of algebraic equa-tions were carried out by the Newton–Kantorovich method,and explicit difference schemes were written for solving thedifferential equations. The 300-K temperature of the hot wallof the base was assumed as the initial approximation for allthe temperatures.

To carry out the calculations, a Windows application wascreated, consisting of the basic calculational module, writtenin FORTRAN, a user shell, created in the Delphi medium inPascal, and a number of service files.

334 J. Opt. Technol. 77 �5�, May 2010

RESULTS OF THE CALCULATIONS

A series of calculations was carried out, using the math-ematical model developed here. A version was first consid-ered in which the sunshade was eliminated from the calcula-tion scheme. The optimum ratio of the radii of the radiatorswas found for this version. In this case, the external radius ofthe first-stage radiator was considered fixed and equal to360 mm, the radius of the second-stage radiator was variedfrom 180 mm to 349 mm, and the inner radius of the first-stage radiator was taken to be 3 mm greater. It was foundthat the cooling throughput is highest when the radius of thesecond-stage radiator is 310 mm. This version was taken asthe basic one �version 1�, and a study was then carried out,taking into account the influence of the sunshade on the ther-mal regime of the cryogenic system.

In version 2, a distributed heat source with a power of250 mW was introduced on the shell, simulating the heatoutput of the electric drive of the sunshade. In version 3, theinfluence of the sunshade was taken into account in the cal-culated scheme in addition. In all the versions, the heat out-put of the detector and the heat influx over the cables �thesignal cable and the cable that controls the heat regime� weretaken to be equal to 100 mW and 60 mW, respectively.

The main results of the calculations are shown in Table I.

TABLE I. Temperatures of the main elements of the cryogenic system, K.

Elements Version 1 Version 2 Version 3

Second stage 73.21 73.25 75.15Screen 5 121.13 121.32 121.77Screen 4 134.59 134.83 135.20First stage 145.42 145.69 146.01Screen 3 224.97 225.48 225.55Screen 2 254.77 255.39 255.43Screen 1 277.74 278.44 278.45Screen 6 283.06 284.96 284.97Screen 10 280.72 285.68 285.69Screen 9 278.98 286.21 286.22Shell 277.88 286.54 286.54Surface of the SVHIof the cooler

272.81 272.95 272.95

Screen B4 — — 125.76Body B3 — — 138.68Screen B2 — — 164.08Body B1 — — 185.20Surface of theheat-release SVHI

— — 313.91

TABLE II. Powers of the heat fluxes between the elements of the sunshade.

Item No.

Power of heat fluxes, W

conductive�along the inserts� radiative

B1−B2 0.5096135E+00 0.4626766E+00B2−B3 0.6127166E+00 0.3595738E+00B3−B4 0.3119266E+00 0.1200726E+00

334Abrosimovet al.

The temperature of the detector-attachment site �the sec-ond stage� is insignificantly increased by placing the sun-shade drive on the shell �Table I�. The installation of thesunshade increases the temperature more substantially. How-ever, it should be pointed out that this increase does notexceed 2° and can be considered allowable.

TABLE III. Powers of the heat fluxes of the radiation from the radiators ofthe sunshade into space.

Radiator No.Power of heat

fluxes, W

B1 0.3931758E+01B3 0.5402911E+00

TABLE IV. Thermal loading �by radiation� from screen B4 to the radiators.

RadiatorPower of heat

fluxes, W

to the radiator of the first stage −0.1875354E−01to the radiator of the second stage 0.4397776E−01

335 J. Opt. Technol. 77 �5�, May 2010

The influence of the sunshade on the temperature of thesecond stage can be reduced by increasing the thermal resis-tance of the conductive heat transfer of the elements �theinserts� that attach the sunshade screens to each other �TableII�, as well as by increasing the radiators installed on screensB1 and B3 �Tables III and IV�.

CONCLUSIONS

The calculated results have thus confirmed that a rotatingsunshade can be implemented for a passive system of cryo-genic cooling of photodetectors in geostationary orbit.

a�Email: sysove@laspace.ru

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3A. I. Abrosimov, A. A. Verlan, G. M. Polishchuk, K. M. Pichkhadze, andV. K. Sysoev, “Optimizing a cryogenic radiative refrigerator,” Prikl. Fiz.No. 2, 129 �2009�.

335Abrosimovet al.