· Web viewThe view factors required for radiative heat transfer were computed by Monte Carlo Ray...

* Corresponding Author

External Heat Fluxes For The Transient Thermal Analysis Of Satellites

111EQUATION CHAPTER 1 SECTION 1

A COMPLETE METHODOLOGY FOR THE COMPUTATION OF EXTERNAL HEAT FLUXES FOR THE TRANSIENT THERMAL

ANALYSIS OF SATELLITES

Ahmet Bilge Uygur*

Türk Havacılık ve Uzay Sanayi A.Ş., 06980 Ankara, Türkiye,

[email protected]

Accepted:

ABSTRACT

This paper describes a complete methodology for the computation of transient external heat fluxes (i.e. solar, albedo and earth) for a given set of orbital parameters such as altitude, solar constant etc. and orientation of the satellite. A software module based on this methodology was developed and its performance was assessed by comparing its predictions with the ones obtained by a commercial software on the same test problem. Comparisons revealed that the external fluxes can be computed accurately with the developed module. It was then used in conjunction with an in-house thermal analysis code based on Thermal Network Method (TNM) and Monte Carlo Ray Tracing Technique (MCRT) for the transient simulation of the same test problem. It was shown that the temperature predictions obtained by the in-house code and the commercial software were in excellent agreement with each other and the developed software is a reliable, accurate and complete tool that can be used in the thermal analysis of satellites.

Keywords: Solar Heat Flux, Albedo Heat Flux, Earth Heat Flux, Satellite Thermal Analysis, Indirect Heat Fluxes, Shadowing Effect.

UYDULARIN ZAMANA BAĞLI ISIL ANALİZİ İÇİN DIŞ ISI AKILARININ HESAPLANMASINA YÖNELİK BÜTÜNSEL BİR YÖNTEM

ÖZET

Bu makalede, zamana bağlı dış ısı akılarının (güneş, albedo ve dünya kaynaklı) irtifa, güneş katsayısı, uydunun yönelimi vb. yörünge parametrelerine bağlı olarak hesaplanabilmesi için bütünsel bir yöntem anlatılmaktadır. Bu yönteme dayanan bir yazılım geliştirilmiş ve yazılımın başarımı aynı işi yapan ticari bir yazılımın sonuçlarıyla kıyaslanarak değerlendirilmiştir. Yapılan kıyaslamalar, geliştirilen yazılımın dış ısı akılarını doğru bir şekilde öngörebildiğini ortaya koymuştur. Bu yazılım daha sonra, Isıl Ağ Metodu (IAM) ve Monte Carlo Işın Takibi Metodu'na (MCITM) dayalı bir ısıl analiz yazılımı ile birleştirilerek aynı test probleminin benzetiminde kullanılmıştır. Birleşik yazılım ile elde dilen sıcaklık öngörüleri ile ticari yazılımdan elde edilen sonuçların çok iyi bir uyum içinde olduğu ortaya konulmuş olup, uzay araçlarının ısıl analizinde kullanılabilecek güvenilir, hassas ve bütün bir yazılım elde edilmiştir.

Anahtar Kelimeler: Güneş Isı Akısı, Albedo Isı Akısı, Dünya Isı Akısı, Uydu Isıl Analizi, Dolaylı Isı Akıları, Gölgeleme Etkisi.

1. INTRODUCTION

Thermal analysis software packages are extensively used in the design phases of the satellites. Their utilization makes it possible to simulate the extreme

thermal environment of space which is critical for the design of an efficient and accurate thermal control system. Since the space is vacuum, thermal analysis of a satellite is carried out by the solution of energy equation in the absence of convective heat transfer

UYGUR

JAST, 2018;11(1):X-XX

mailto:[email protected]


and by taking into account conduction, radiation, internal and external energy sources [1]

For the transient solution of described energy equation, a computer code [2] based on Thermal Network Method (TNM) [3] and Method of Lines [4] was developed to be used for the thermal analysis of spacecrafts and space-born equipment. The view factors required for radiative heat transfer were computed by Monte Carlo Ray Tracing Technique (MCRT) [5] and the external heat fluxes were obtained from a commercial software package namely, THERMICA [6] The accuracy of the code was assessed by applying it to a simple test problem and comparing its predictions with those obtained by the commercial software on the same problem. Excellent agreement obtained with the two sets of results have led to the further development of the code by a recent research effort [7] In this study, a module for the computation of external heat fluxes (i.e. solar, albedo and earth fluxes) was developed and incorporated into the existing code in order to eliminate its dependency to any external software. The predictive accuracy of the external flux module was evaluated on a cubic satellite with prescribed orbit parameters. The results obtained with the module were compared with THERMICA predictions and it was found that the results mimic each other very closely on this simple test problem.

In an effort to assess the performance of this module on a more realistic problem, it was applied to a test case which involves deployed solar panels and complex geometric structures such as antennas. The comparison of external fluxes obtained by the module with those of THERMICA indicated that the accuracy was deteriorated for this problem. Upon investigation of the possible causes for this degraded accuracy, it was concluded that the formulations used in the module did not take into account the shadowing due to the solar panels and hence the solar fluxes were overpredicted. Moreover, it was also found that the formulations did not account for the indirect fluxes which occur mostly due to multi-reflections between the solar panels and the satellite.

There are limited number of studies in the open literature which address the computation of external fluxes in conjunction with the thermal analysis of satellites. In one of these studies, Liu et al. [8] performed numerical simulations on an antenna in solar simulator environment which is subject to solar flux only. Their research focused on the effects of non-uniformity and instability on the temperature predictions rather than the validation of the results. Li et al. [9] carried out the thermal analysis of composite

solar arrays at low earth orbit (LEO) and geosynchronous orbit (GEO) without considering the effects of indirect fluxes and shadowing. In another study by Liu et al. [10], the effects of indirect fluxes and shadowing were included in their analysis. However, the test problem was not realistic enough in the sense that it did not involve an orbit and attitude definition which determine the position and orientation of the satellite, respectively.

The purpose of this study is to eliminate the drawbacks of the above studies by presenting a complete methodology for the computation of external fluxes which takes into account the effects of shadowing and indirect fluxes and can be applied to any satellite geometry and orientation regardless of its complexity. In accordance with this, the paper is organized as follows: First the methodology is explained in detail and then the mathematical model and numerical solution technique employed in the existing thermal analysis code is described briefly. Afterwards, the predictive accuracy of the module developed based on the methodology is demonstrated on a realistic problem which involves a satellite on a LEO. Finally, the thermal analysis of the same test problem performed by incorporating the module into the existing code are presented.

2. METHODOLOGY FOR THE COMPUTATION OF EXTERNAL FLUXES

The external fluxes within the scope of this paper are illustrated in Figure 1. Here, the solar flux is the radiative energy emitted by the sun, albedo flux is the solar flux reflected by the earth and earth flux is the infrared radiation emitted by the earth due to its temperature. In the next sections, computation of each of these items will be described in detail.

UYGUR


Figure 1. Schematic illustration of the types of external fluxes acting on a satellite

2.1. Computation of Solar Heat Flux

Computation of direct solar flux incident on node i is carried out by the following expression

22\*MERGEFORMAT ()

where is the solar flux absorbed by node i, Cs is

the solar constant, is the radiative area of node i,

is the absorptivity of node i in the solar

spectrum. In this expression, is the cosine of the angle between the normal of the surface and the solar beam coming from the sun, as depicted in

Figure 2. In this expression, the angle is the only parameter which is not a constant and its calculation requires the position of the sun with respect to node i throughout the orbit. For this purpose, and in-house orbit module based on SGP4 [11] is utilized. The details of the orbit module is beyond the scope of this paper but in essence, the module takes orbit parameters as input and calculates the position of the sun and the earth with respect to the geometrical model reference frame, as output.

Figure 2. Schematic illustration of direct solar heat flux acting on the satellite

2.2. Computation of Albedo Heat Flux

Computation of direct albedo flux incident on node i is carried out by the following expression

33\*MERGEFORMAT ()

where is the albedo flux absorbed by node i, a is

the albedo coefficient, is the radiative area of

node i, is the view factor from node i to the

earth, is the absorptivity of node i in the solar

spectrum. Last, is the cosine of the angle between the line from the center of the earth to node i and line from the center of the earth to sun, as illustrated in Figure 3.

The angle is computed by the help of aforementioned orbit module. For the computation of view factor from node i to the earth, a previously developed view factor module [2] based on Monte Carlo Ray Tracing Technique [5] is utilized. Basically, the module sends a specified number of rays in randomly selected directions from node i. Then, each ray is traced whether it hits the target node or not. The view factors are then calculated by taking the ratio of the number of rays hitting the target over the total number of rays sent.

UYGUR


Figure 3. Schematic illustration of albedo heat flux acting on the satellite.

2.3. Computation of Earth Heat Flux

Computation of direct earth flux incident on node i is carried out by the following expression

44\*MERGEFORMAT ()

where is the earth flux absorbed by node i, is

the earth temperature, is the radiative area of

node, is the view factor from node i to the

earth and is the emissivity of node i in the

infrared spectrum. The only variable in Eq., is computed by using the view factor module discussed earlier.

2.4. Effect of Indirect Fluxes and Shadowing

As mentioned earlier, the accuracy of the methodology described in this paper stems from the fact that not only it computes these fluxes in the classical sense [7] but it also takes into account the effect of indirect fluxes and shadowing. The concept of indirect flux is illustrated in Figure 4. It is defined as the flux reflected from node j and absorbed by node i. For the sake of simplicity, only the indirect flux due to sun is depicted in the figure. However, it has to be noted that the concept is also valid for albedo and earth fluxes.

Figure 4 Schematic illustration of the indirect fluxes acting on the satellite.

In view of this, computation of solar, albedo and earth fluxes by indirect means are carried out by the following expressions, respectively.

55\*MERGEFORMAT ()

66\*MERGEFORMAT ()

77\*MERGEFORMAT ()

where is the reflectivity of node j in the solar

spectrum, is the reflectivity of node j in the

infrared spectrum and is the Gebhart factor [12]

between node i and j. can be defined as the fraction of radiant energy emitted by node j that is absorbed by node i by considering all possible paths

(multi reflections). The calculation of requires the knowledge of view factor amongst the nodes composing the complete system. The view factors are made available by using previously mentioned

MCRT module and is obtained by the solution of the following linear equation

88\*MERGEFORMAT ()

where

UYGUR


99\* MERGEFORMAT ()

1010\*MERGEFORMAT ()


If we use the assumption that all surfaces composing

the system are diffuse, gray and opaque ( ) together with Kirchoff's law ( ) [13], we can further simplify Eqns.5-7 to the following expressions

1212\* MERGEFORMAT ()



Having calculated the indirect fluxes, the second effect that has to be considered for an accurate thermal simulation is the shadowing. Depending on the position of the radiative energy source (e.g. sun) with respect to the satellite, some surfaces of the satellite shadow others inevitably. This effect is illustrated in Figure 5 for the direct solar heat flux. As depicted by the figure, node j shadows node i (colored with gray) due to their orientation. In order to take this effect into account, one has to calculate the illumination factor (IF) of each node for each orbital position and orientation of the satellite. For the computation of the IF, a vector originating from the center of node i and extending to the sun (which can be taken as a point since it is far away from the satellite) is created. Then this vector is traced for possible intersection with any node j in the system: if an intersection occurs, it is concluded that node i is

under the shadow of node j and the illumination factor for node i designated as zero (0); if there is no intersection, it is concluded that the solar radiation can reach node i without any obstruction and IF is designated as one (1). The above arguments are equally valid for albedo and earth fluxes. However,

since the term present in Eqns.3-4 inherently possess the illumination factor, no additional effort is spent. For this reason, the shadowing effect can only be distinguished in solar flux computations which will be seen during the discussion of the results.

Figure 5. Schematic illustration of the shadowing effect on node i.

3. MATHEMATICAL MODEL AND THE NUMERICAL SOLUTION TECHNIQUE

The energy equation which can be used to govern the heat transfer occurring in the vacuum conditions of space can be written as


where is the density, is the specific heat, T is the temperature and t is the time. The first term on the

right-hand-side of the equation is the rate of energy addition per unit volume by heat conduction,

whereas second term is the rate of energy addition per unit volume by radiative heat transfer.

UYGUR


The last term is the source term per unit volume and accounts for the internal heat dissipations and absorbed external fluxes.

Application of Thermal Network Method in conjunction with Method of Lines to Eqn.15 yields the following discretized version of the energy equation

∂T i

∂t≈ 1

(mCp )i [∑j=1

N A i , jC λ i , j

Li , j( T j−T i )+∑

j=1

N

σ Air εi γi , j (T j

4−T i4 )+qi]


where is the heat capacity of node i, is the

conduction area between nodes i and j, is the effective thermal conductivity between the nodes i

and j, is the distance between the nodes i and j.

Following MOL approach, the right-hand-side of Eqn.16 is evaluated by the present time step values which results in a set of ordinary differential equations (ODE) in the form of


The set of ODEs given by Eqn.17 are integrated in time using readily available stiff/non-stiff implicit ODE solvers, yielding the temperature distribution in time. By this way the code not only benefits from the simplicity of the explicit formulation but also the errors due to quasi-linearization of radiative heat transfer term in a classical implicit solution of Eqn.17 is avoided. Furthermore, with the selection of a quality ODE solver, it is possible to use large time steps for the time integration without compromising from the stability of the solution. In the present study, ROWMAP [14] was used as the ODE solver.

4. RESULTS AND DISCUSSION

The test problem under consideration consists of a cubic satellite with two solar panels and representative equipment as shown in Figure 6.

Figure 6. Geometric model of the satellite and the node structure .

The main body of the satellite is a cube composed of six aluminum honeycomb panels (main body) having a dimension of 0.50 x 0.50 x 0.01 m. The dimensions of the aluminum honeycomb solar panels are also 0.50 x 0.50 x 0.01 m. The solar panels are assumed to be conductively decoupled from the main body. Thermo-physical and thermo-optical properties of all surfaces making up the system are listed in Table 1. The parameters defining the orbit are summarized in Error: Reference source not found In addition, the -Z panel always points towards the center of the earth (nadir pointing) and the velocity vector is in +X direction.

Table 1 Thermo-physical and thermo-optical properties

(kg/m3) (J/kg.K)

(W/m.K)

Main Body

158.9 883.7 5.39 0.82

0.65

Solar Panel

158.9 883.7 2.79 0.82

0.75

Table 2 Orbital parameters

Type of the orbit Sun-synchronousAltitude (km) 700Epoch day 07 December 2013Epoch time 07:22Right ascension of the ascending node (deg) 120

True anomaly (deg) 264Solar constant (W/m2) 1416Albedo coefficient 0.35Earth temperature (K) 254

UYGUR


4.1. Numerical Solution Parameters

For the numerical solution of the described test problem, main body and the solar panels were divided into 4 x 4 nodes whereas all equipment were represented by a single node. The computation of view factors necessary for radiation exchange factors, albedo and earth fluxes were carried out by firing 1000 rays from each node. All nodes were assumed to be at 273.15 K initially and space temperature was taken as 4.15 K. The simulations were performed for 1 orbital period which is approximately equal to 6000 s with a time step of 100 s. Benchmark solutions for external fluxes and temperature distributions were obtained by THERMICA software package.

4.2. External Heat Flux Predictions

In this section, the predictive accuracy of the module for the computation of solar, albedo and earth fluxes is presented in comparison with the benchmark predictions obtained with the commercial software for the following cases:

a) direct flux only,b) direct flux with the effect of indirect flux

only,c) direct flux with the shadowing effect only,d) direct flux with the combined effect of

indirect flux and shadowing.

Since the calculated fluxes are periodical, results are given for 1 orbital period which is approximately equal to 6000 s. The eclipse phase of the defined orbit corresponds to the period of time between 1250 and 3000 s. Among the nodes present in the model (see Figure 6) the effects of indirect fluxes and shadowing are best observed at the nodes located on the main body panels shadowed by the solar panels (i.e. +Y and -Y panels). Hence, the proceeding discussions will be carried out based on a node selected from main body -Y panel.

shows the transient absorbed solar fluxes for the aforementioned cases. As can be seen from the figure, when only the direct fluxes are considered, the trends exhibited by the benchmark solution cannot be

captured and the only overlapping portion of the results is the eclipse phase. With the incorporation of indirect fluxes, the results are improved slightly and the maximum fluxes are well predicted. However, the results still indicate that there is absorbed solar flux from 4000 s and onwards despite the fact that benchmark result is nearly zero. On the other hand, when the results obtained for case (c) is examined, it is seen that the effect of shadowing is the principle cause of the absence of solar flux that could not be estimated by prior cases. When both indirect fluxes and shadowing effects are taken into account (case (d)), best agreement between the module and the benchmark predictions is obtained. The minor discrepancies observed for this case can be attributed to the difference in the treatment of the shadowing effect by the two solutions.

The transient absorbed albedo fluxes are illustrated in Figure 8. For reasons explained in Section 2.4 (shadowing effect inherently included in the computation of albedo flux), the predictions for case (a) and (c) are identical. When indirect fluxes are incorporated to the solution (case (b) and consequently case (d)), the accuracy of the predictions are improved, in particular for the initial and final phases of the orbit.

The transient absorbed earth fluxes are shown in Figure 9. The major difference between the earth flux and the others is the fact that its presence is not affected from the eclipse phase. Hence it is always greater than zero for the node under consideration as can be seen from the figure. Moreover, it is constant throughout the orbit since the view factors from the

satellite to the earth ( ) are constant due to the nadir pointing. Like in the computation of albedo fluxes, comparison of the results obtained for case (b) and (c) reveals that the effect of indirect fluxes is the only factor that improves the accuracy. Therefore predictions for case (b) and (d) are identical.

For the sake of completeness, transient absorbed total flux is presented in Figure 10. An overall comparison of the Figure 10 with reveals that the solar flux is the dominant external flux which determines the behavior of the total flux.

UYGUR


Figure 7. Effects of indirect fluxes and shadowing on the absorbed solar flux.

Figure 8. Effects of indirect fluxes and shadowing on the absorbed albedo flux.

UYGUR


Figure 9. Effects of indirect fluxes and shadowing on the absorbed earth flux.

Figure 10. Effects of indirect fluxes and shadowing on the absorbed total flux.

UYGUR


4.3. Temperature Predictions

The transient temperature predictions obtained by using the external fluxes computed by the methodology described in this study are presented in Figure 11. When predictions obtained for case (b) are compared with those for case (c), it can be concluded that the shadowing effect is the predominant factor that increase the accuracy of the results. This finding is consistent with the outcome of the solar flux predictions owing to the fact that magnitude of solar fluxes are approximately 10 times greater than those of albedo and earth fluxes. On the other hand, the incorporation of indirect fluxes in addition to the shadowing effect (case (d)) further enhances the accuracy of the results as expected.

In order to provide the reader an overall view of the accuracy of the complete methodology in terms of all

nodes present in the system, mean deviation was calculated using


where is the mean deviation, N is the total

number of nodes in the solution, and are the temperature predictions obtained for node i by THERMICA and the in-house code, respectively. The variation of mean deviation with respect to solution time is plotted in Figure 12. As can be seen, mean deviation varies between 1 and 2 K which is the range suggested by European Space Agency standards [15] for thermal mathematical model correlation.

Figure 11. Transient temperature profiles.

UYGUR


Figure 12. Mean deviation as a function of solution time

5. CONCLUSIONS

In this study, a complete methodology for the computation of time dependent solar, albedo and earth fluxes for a given set of orbital parameters was described. The performance of the software module based on this methodology was demonstrated on realistic test problem by comparing its predictions with the benchmark solutions obtained by a commercial software package on the same problem. The effects of indirect fluxes and shadowing on the predictive accuracy were analyzed and it was concluded that shadowing is the predominant factor for the accuracy of the solar flux predictions whereas indirect fluxes is the only factor affecting albedo and earth flux predictions. The external fluxes computed by the module were then utilized in conjunction with an in-house thermal analysis code based on Thermal Network Method (TNM) and Monte Carlo Ray Tracing Technique (MCRT) for the transient thermal analysis of the test problem under consideration. Analysis of temperature predictions obtained by the in-house code and the commercial software revealed that the mean deviations calculated at each time step fell in the range suggested by the standards for the correlation of thermal mathematical models.

6. REFERENCES

[1] Gilmore, D. G., “Spacecraft Thermal Control Handbook,” (Second Ed.) The Aerospace Press, 2002.

[2] Isik, H. G., Uygur, A. B., Omur, C., Solakoglu, E, “The thermal analysis of a satellite by an in-house computer code based on thermal network method and Monte Carlo ray tracing technique,” In: 5th International Conference on Recent Advances in Space Technologies, 978-983, 2011

[3] Oppenheim, A. K., “Radiation analysis by the network method”, Transactions of the American Society of Mechanical Engineers, Vol.78, 725-735, 1956.

[4] Uygur, A. B., Tarhan, T, Selçuk, N, “Mol solution for transient turbulent flow in a heated pipe,” Int. J. Therm. Sci., Vol.44, No.8, 726-734, 2005.

[5] Modest, M. F., “Radiative Heat Transfer” (Second Ed.), McGraw-Hill, 2003.

[6] ASTRIUM, Systema software package, http://www.systema.airbusdefenceandspace.com/, 2017.

[7]. Uygur, A. B., Isik, H. G, Karaismail, F. N, “Incorporation of an external heat flux computation module to an in-house code for the thermal analysis of satellites”, In: 6th International Conference on Recent Advances in Space Technologies, 541-544, 2013.

[8] Liu, Y, Hui, Li, G. Jiang, L., “Numerical simulation on antenna temperature field of complex structure satellite in solar simulator,” Acta Astronautica, Vol.65, 1098-1106, 2009.

[9] Li, J., Yan, S., Cai, R., “Thermal analysis of composite solar array subjected to space heat flux,” Aerospace Science and Technology, Vol.27, No.1, 84-94, 2013.

[10] Liu, Y, Hui, Li, G. Jiang, L., “A new improved solution to thermal network problem in heat-transfer analysis of spacecraft,” Aerospace Science and Technology, Vol. 14, No.4, 225-234, 2010.

[11] Hoots, F. R., Roehrich, R. L., “Models for propagation of norad element,” Tech. Report, U.S. Air Force Aerospace Defense Command, Colorado Springs, CO, 1980.

[12] Gebhart, B., “Surface temperature calculations in radiant surroundings of arbitrary complexity for gray, diffuse radiation,” Int. J. Heat Mass Trans., Vol.3, 341-346, 1961.

UYGUR


[13] Bergman, T. L., Lavine, A. S., Incropera, F. P., Dewitt, D. P., “Fundementals of Heat and Mass Transfer” (Fifth Ed.), John Wiley & Sons, 2011

[14] Weiner, R., Schmitt, B. A., Podhaisky, H., “Rowmap - A ROW-code with Krylov techniques for large stiff ODEs,” Tech. Report, FB Mathematik und Informatik, Universitaet Halle, 1996.

[15] ECSS, “Space engineering, Testing (ECSS-E-10-03A),” 2002.

Curriculum Vitae

Dr. Ahmet Bilge Uygur graduated from Chemical Engineering Department of Middle East Technical University (Ankara, Turkey) in the year 2000. He received his M. Sc. and Ph. D. degrees from the same department under the supervision of Prof. Dr. Nevin Selçuk in the years 2002 and 2007, respectively. His principle research interests are Computational Fluid Dynamics, Spacecraft Thermal Control and Design, Thermal Mathematical Model Development, Spacecraft Environmental Testing. He has published 19 academic papers. He currently works as a senior test engineer in Turkish Aerospace Industries, Inc.

UYGUR

· Web viewThe view factors required for radiative heat transfer were computed by Monte Carlo Ray...

Documents

Transcript of · Web viewThe view factors required for radiative heat transfer were computed by Monte Carlo Ray...