HUNGARIAN JOURNAL OF INDUSTRY AND CHEMISTRY Vol. 47(1) pp. 25–32 (2019) hjic.mk.uni-pannon.hu DOI: 10.33927/hjic-2019-05 THERMAL MODEL DEVELOPMENT FOR A CUBESAT NAWAR AL HEMEARY *1,2 , MACIEJ JAWORSKI3 , JAN KINDRACKI3 , AND GÁBOR SZEDERKÉNYI1 1Faculty of Information Technology and Bionics, Pázmány Péter Catholic University, Práter u. 50/A, Budapest, 1083, HUNGARY 2Electromechanical Engineering Department, University of Technology, Al Senaha St., Baghdad, 10066, IRAQ 3Faculty of Power and Aeronautical Engineering, Warsaw University of Technology, Nowowiejska 24, Warszawa, 00-665, POLAND CubeSats provide a cost-effective means of several functions of satellites due to their small size, mass, relative simplicity and short development time. Therefore, CubeSat technologies have been widely studied and developed by space organi- zations, companies and educational institutions all over the world. These satellites have certain drawbacks. Small surface areas are a consequence of their small size which often imply thermal and power constraints. A novel development of CubeSats known as PW-Sat has been developed by Warsaw University of Technology. A control-oriented lumped ther- mal model of this satellite containing a fuel tank in the form of nonlinear ordinary differential equations is proposed in this paper. The model is able to simulate the thermal behavior of the surface and fuel tank of the satellite in its orbit. For the PW-Sat to operate reliably, the temperature of the fuel tank has to be maintained within given safety limits. Because of the limited power, passive thermal control is assumed in this case. Several simulation results are presented for different surface compositions to determine whether they are able to guarantee the prescribed temperature range throughout the entire orbit or not. Keywords: PW-Sat, TMM, thermal behavior, propellant tank, dynamical modeling 1. INTRODUCTION In recent years, interest in Cube-Satellites (CubeSats) has grown tremendously within the space community from space agencies as well as in industry and academia. Two factors have influenced this spurt of interest, namely the low-cost nature of access to space and the utilization of commercial off-the-shelf (COTS) technologies in the de- sign architecture. These two factors have led to a sig- nificantly low overall cost of a CubeSat mission [1]. A CubeSat is cubic in form with edges 10 cm in length and a mass of up to 1.33 kg. The PW-Sat is a CubeSat that has been in development for more than a year by differ- ent teams at Warsaw University of Technology [2, 3]. At present, PW-Sat has never been flown with an onboard propulsion system. Due to the significant and growing in- terest in CubeSat mission capabilities, several propulsion systems have been rapidly developed for use in Cube- Sats such as cold gas propulsion systems, solar sails, elec- tric propulsion systems and chemical propulsion systems [4, 5]. Cold gas propulsion systems are relatively sim- ple solutions in CubeSats. Gas from a high-pressure gas cylinder is simply vented through a valve and nozzle to produce thrust [6, 7]. *Correspondence: al.hemeary@itk.ppke.hu The goal of this paper is to propose a lumped dynam- ical model of temperatures during orbital motion in the main parts of a CubeSat that contains a fuel tank. The model is built in the form of nonlinear ordinary differen- tial equations (ODEs). Although advanced thermal sim- ulation tools exist that utilize detailed distributed math- ematical models, this simple form of a model has been chosen since it is intended to be used in the model of temperature control design. The overwhelming majority of control design techniques require models in the form of ordinary differential equations [8]. Moreover, in the case of nonlinear models such as the CubeSat system studied, a low dimensional model is preferred due to the computa- tional complexity of control design. This approach is also supported by control theory and practice, namely that in general such simple models are sufficient for controller design [9]. As a first step in terms of the regulation of temperature, passive control in the form of the appropri- ate composition of materials covering the surfaces of the satellite is used. During the construction of the model, the standard principles of thermal modeling [10, 11] and their application in aerospace engineering are followed [12, 13]. Relevant results can be found in the literature con- cerning the thermal modeling and analysis of small satel- mailto:al.hemeary@itk.ppke.hu 26 AL-HEMEARY, JAWORSKI, KINDRACKI, AND SZEDERKÉNYI lites in the form of ODEs. In [14] a simple thermal dy- namical model of a CubeSat containing two differential equations is presented. The two lumped balance volumes are the surface and internal parts of the satellite, respec- tively. It is shown that the problem is mathematically analogous to the forced vibration of a damped mechan- ical system. In [9], new theoretical results on the qualita- tive behavior of spacecraft thermal models are provided that contain several nodes (compartments). It is proven that such models exhibit a unique asymptotically stable equilibrium in the positive orthant with constant external disturbance inputs which leads to a stable limit cycle dur- ing orbital motion. The analysis concerning the frequency domain of a multi-compartmental model of a satellite is conducted in [15]. The ODEs are linearized around the equilibrium points which permit the use of Fourier anal- ysis. The structure of the paper is as follows: the descrip- tion starts with the derivation of a simple mathematical model to simulate the transient thermal behavior of the fuel tank as well as the satellite faces in orbit. Then, by changing either the optical properties of the surface of the PW-Sat or the solar cell ratio of the satellite surface, sev- eral simulations are presented to illustrate how different configurations satisfy the given temperature limits. 2. SYSTEM DESCRIPTION AND ASSUMP- TIONS The intended use of the model developed in this paper is twofold: 1) to study the effect of different surface compo- sitions (including solar cells) on temperature, 2) to eval- uate the possibility of installing a propellant tank in the CubeSat. With regard to the modeling, the following as- sumptions are made: 2.1 Structural Assumption PW-Sat is a cubic structural bus with a total mass of 1 kg composed of six faces as walls. The basic structure of these faces is composed of the aluminum alloy 6061-T6 with various optical surface properties. These properties are based on uncoated surfaces for one experiment and coated with a magnesium oxide-aluminum oxide paint for the others. The nitrogen fuel tank, made of stainless steel with a diameter of 5 cm, is planned to be installed in the center of this satellite as shown in Fig. 1a and is assumed to contain an internal gas subject to 100 bars of pressure at an initial temperature of 298 K. The solar cells will be attached to sides 1, 2 and 4 of the PW-Sat, as shown schematically in Fig. 1b, because these faces will be exposed to solar radiation due to the assumed orbit of this satellite. 2.2 Orbital Assumption The PW-Sat is designed for a circular low Earth orbit (LEO). The total orbital period (P ) is 1.5 h. However, (a) (b) Figure 1: PW-Sat Structure ((a) PW-Sat spherical fuel tank set, (b) PW-Sat solar cell arrangements). the motion of PW-Sat is assumed to be identical when exposed to solar radiation and during shadow passage at an altitude of 300 km and an inclination of zero. Face 3 is directed towards the Earth throughout its orbit. Faces 1, 2 and 4 are exposed to the sun with regard to the orbital motion of the satellite. Finally, faces 5 and 6 are directed towards the space along the satellite orbit as shown in Fig. 2. 2.3 Thermal Assumption The six faces of the PW-Sat are considered to have a uni- form temperature distribution. The conductive heat trans- fer between the fuel tank and satellite is ignored to sim- plify the thermal modeling calculations. Only face 3 is exposed to infrared radiation from the Earth in this or- bit and the albedo during the luminous orbit intervals Figure 2: PW-Sat orbital motion. Hungarian Journal of Industry and Chemistry THERMAL MATHEMATICAL MODEL 27 [14, 16]. The thermal rate of power dissipation generated from the operation of elements from the satellite is as- sumed to be 2 W. The thermal limits of the fuel tank are 228 K and 338 K, and for the surface area of the satellite are 173 K and 373 K. 3. THERMAL MATHEMATICAL MODEL (TMM) The problem concerns the formulation of the model, which is on the one hand simple enough to limit the ex- penditure, on the other hand, detailed enough to present an adequate description of the physical surroundings [17]. The main purpose of these calculations is to divide the periodic motion of the satellite (with period P ) into three intervals (parts) as shown in Fig. 2. The first inter- val (P1) starts with an initial time of t = 0 s when face 1 faces the sun and ends at a time of t = 1350 s when face 4 faces the sun. The second interval (P2) is an eclipse in- terval between t = 1351 s and t = 4050 s. The third interval (P3) starts at t = 4051 s when face 2 faces the sun and ends at the end of the satellite period at t = 5399 s. 3.1 First Interval Equations Interval P1 : t = 0 → t = P 4 (1) The satellite spends a quarter of its orbital period in this luminous part. During this time interval, the surface of the satellite receives direct solar and albedo radiation de- pending on the position of the satellite due to its motion and the satellite emits thermal IR radiation into space; however, only face 3 receives additional infrared radia- tion from Earth because it faces the Earth [18]. The rate of heat transfer between face 1, the external environment and the spherical fuel tank during the first interval can be described by (mAl Cp + msc C sc p ) dT1 dt = Gsa Al-sc s Acos ( 2πt P ) + Q̇ + Q̇F1 −ε Al-sc IR σAT 4 1 (2) where mAl denotes the mass of aluminum, Cp stands for the specific heat of aluminum (980 J/(kg·K)), msc repre- sents the mass of the solar cell , Cscp is the specific heat of the solar cell (1600 J/(kg·K)), T1 denotes the temper- ature of face 1, Gs stands for the solar constant (1367 W/m2), and aAl-scs represents the average solar absorp- tance of aluminum and the solar cell which is calculated as (Al %·aAls +sc %·ascs ), where Al % and sc % denote the percentages of aluminum and solar cells in the cover, respectively. A stands for the surface area of the face (0.01 m2), Q̇ represents the thermal rate of power dis- sipation, Q̇F1 denotes the radiative heat transfer between face 1 and the tank, εAl-scIR is the average infrared emis- sivity of Al and sc which is calculated as (Al %·εAl+sc %·εsc) and σ stands for the Stefan-Boltzmann constant (5.669 ·10−8W/m2 K4) [6]. The rate of heat transfer between face 2, the external environment and the spherical fuel tank during the first interval can be described by (mAl Cp + msc C sc p ) dT2 dt = Q̇ + Q̇F2 −ε Al-sc IR σAT 4 2 (3) where T2 denotes the temperature of face 2 and Q̇F2 stands for the radiative heat transfer between face 2 and the tank. The rate of heat transfer between face 3, the external environment and spherical fuel tank during the first inter- val can be described by mCp dT3 dt = AF ·GsaAls FsEAcos ( 2πt P ) + Q̇ + Q̇F3 + a Al IRσAT 4 E −ε Al IRσAT 4 3 (4) where m denotes the mass of the face (0.04 kg), FsE stands for the view factor between the face of the satellite and the Earth which is almost one [18], T3 represents the temperature of face 3, Q̇F3 is the radiative heat transfer between face 3 and the tank, AF denotes the factor on the albedo (0.28), aAls stands for the solar absorptivity of aluminum, aAlIR represents the infrared absorptivity of alu- minum, TE is the reference temperature of the Earth (255 K) and εAlIR denotes the infrared emissivity of aluminum. The rate of heat transfer between face 4, the external environment and the spherical fuel tank during the first interval can be modeled as( mAl Cp + msc C sc p ) dT4 dt = Gsa Al-sc s Asin ( 2πt P ) + Q̇ + Q̇F4 −ε Al IRσAT 4 4 (5) where T4 denotes the temperature of face 4 and Q̇F4 stands for the radiative heat transfer between face 4 and the tank. The rate of heat transfer between face 5, the external environment and the spherical fuel tank during the first interval can be described as mCp dT5 dt = Q̇ + Q̇F5 −ε Al IRσAT 4 5 (6) where T5 denotes the temperature of face 4 and Q̇F5 stands for the radiative heat transfer between face 5 and the tank. The rate of heat transfer between face 6, the external environment and the spherical fuel tank during the first interval can be described as mCp dT6 dt = Q̇ + Q̇F6 −ε Al IRσAT 4 6 (7) where T6 denotes the temperature of face 6 and Q̇F6 stands for the radiative heat transfer between face 6 and the tank. 47(1) pp. 25–32 (2019) 28 AL-HEMEARY, JAWORSKI, KINDRACKI, AND SZEDERKÉNYI 3.2 Second Interval Equations Interval P2 : t = P 4 → t = 3 4 P (8) These concern the duration of an eclipse. The satellite spends half of its orbital period in an eclipse. During this interval, the surface of the satellite receives neither direct solar nor albedo radiation, whilst face 3 still receives IR radiation from the Earth because it faces it. The satellite emits thermal IR radiation into space. Therefore, the rates of heat transfer of the faces of the satellite (1, 3 and 4) have slightly changed in their equations compared to the first interval. The rate of heat transfer between faces 1, 3 and 4 as well as the external environment during the second inter- val can be described by the following equations (mAl Cp +msc C sc p ) dT1 dt = Q̇+Q̇F1 −ε Al-sc IR σAT 4 1 (9) mCp dT3 dt = Q̇ + Q̇F3 + a Al IRσAT 4 E −ε Al IRσAT 4 3 (10) (mAl Cp + msc C sc p ) dT4 dt = Q̇ + Q̇F4 −ε Al-sc IR σAT 4 4 (11) 3.3 Third Interval Equations Interval P3 : t = 3 4 P → t = P (12) The satellite spends a quarter of its orbital period in this second luminous part. During this time interval, the sur- face of the satellite receives and emits thermal radiation in a similar manner to during interval 1 with only a slight change in the equation of face 2. The rate of heat transfer between this face, the exter- nal environment and the spherical fuel tank during the third interval can be modeled as (mAl Cp + msc C sc p ) dT2 dt = Gsa Al-sc s A ∣∣∣∣sin ( 2πt P )∣∣∣∣+ Q̇ + Q̇F2 −ε Al−sc IR σAT 4 2 (13) 3.4 The transient heat transfer of the spheri- cal propellant tank The rate of heat transfer between the faces of the satel- lite and the fuel tank can be described by the following equation (ms C s p + mgCV ) dTt dt = − 6∑ 1 Q̇Fn (14) Table 1: Material properties of PW-Sat. Al 6061-T6 uncoated Al 6061-T6 coated Solar cells Specific Heat [J/(kg·K)] 980 980 1600 Emissivity (thermal) 0.08 0.92 0.85 Absorptivity (solar) 0.379 0.09 0.92 Table 2: Material properties of fuel tank. Stainless Steel Nitrogen Specific Heat [J/(kg·K)] 504 743 Mass [kg] 0.0926 0.0074 Table 3: Average of optical surface properties of partially covered surfaces. Cube Face ε a Coverage 1,2 and 4 0.89 0.33 70 % Al, 30 % sc 0.87 0.67 30 % Al, 70 % sc 3,5 and 6 0.92 0.09 Al painted where ms denotes the mass of stainless steel, Csp stands for the specific heat of stainless steel (504 J/(kg·K)), mg represents the mass and CV the specific heat of nitrogen (743 J/(kg·K)), and Tt is the temperature of the tank. The radiative heat transfer between each face and the tank Q̇Fn , depending on which face it applies to, can be described as Q̇Fn = FftεσA(T 4 t −T 4 n) (15) The view factor between the face in question and the fuel tank is given by Fft = 1 (1+H)2 (see, e.g. [10, 11]), where H denotes the ratio of the distance between the spherical surface of the tank to the surface of the internal face (h = 0.025 m) in terms of the radius (r = 0.025 m), which is expressed as H = h r . The mass of the solar cell msc per unit area is on av- erage 850 g/m2, thus, the mass of the solar cell as a pro- portion of the total mass of the face is determined by the equation (msc = 850 g/m2 · A · sc %). The total mass of the tank mt is assumed to be 0.1 kg, so the mass of nitrogen gas was calculated by assuming the initial tem- perature and total pressure of the tank. The optical surface properties are shown in Table 1, and the masses of both nitrogen gas and the tank are shown in Table 2. Hence, it is necessary to calculate the properties of the average materials to conduct a thermal analysis. The emissivity of infrared radiation from these faces can be calculated as the average of the emissivity of infrared radiation from aluminum and the emissivity of infrared radiation from the solar cell in addition to the absorptivity of these faces as shown in Table 3. Hungarian Journal of Industry and Chemistry THERMAL MATHEMATICAL MODEL 29 (a) (b) Figure 3: The thermal behavior using 100 % Al during one orbital period ((a) temperatures of the faces and fuel tank, (b) temperature of the fuel tank) (T1, . . . , T6) refers to the temperatures of faces 1, . . ., 6, respectively, and (Tt) denotes the temperature of the tank. 4. COMPUTATIONAL RESULTS In this section, the faces and investigations into the ther- mal behavior of the tank are presented for several cases based on: uncoated surfaces, surfaces coated with mag- nesium oxide-aluminum oxide paint and different feasi- ble options of the ratios of solar cells from the PW-Sat to simulate the temperature of the fuel tank with different optical properties of the surface materials and solar cell ratios. 4.1 The PW-Sat faces composed of 100 % uncoated aluminum In this case, the thermal simulations of the faces and fuel tank were conducted according to the assumption that the faces of the satellite are composed of the aluminum alloy 6061-T6. Starting with the equations of the intervals and by using the ODE45 solver in MATLAB (the simulation time step was 1 s), the thermal behavior during the orbital motion of the satellite was computed. 1 - The thermal simulation of the faces and spherical fuel tank during one orbital period (time span from 0 s to 5399 s) is shown in Fig. 3a and the simulation of the tem- perature of the tank during this orbital period is shown in Fig. 3b. It can be seen that the predefined temperature limits are not adhered to in this case, since the minimum (a) (b) Figure 4: Thermal behaviors using 100 % Al during 8 or- bital periods ((a) temperatures of the faces and fuel tank, (b) temperature of the fuel tank). temperatures of the faces of the satellite and fuel tank ex- ceed 460 K during its orbital period. 2 - The thermal simulation of the faces and spherical fuel tank during several orbits (8 orbital periods with a time span of 12 h to illustrate long-term operations) is shown in Fig. 4a, and the simulation of the temperature of the tank during these orbital periods is shown in Fig. 4b. 4.2 The faces of the PW-Sat are composed of 100 %-coated aluminum The thermal simulations of the faces and fuel tank were conducted according to the assumption that the surface of the satellite was composed of aluminum coated with magnesium oxide-aluminum oxide paint. By using the as- sumed time span of each interval, the results are shown below: 1 - The thermal simulations of the faces and spherical fuel tank during one orbital period (time span from 0 s to 5399 s) are shown in Fig. 5a, and the simulation of the temperature of the tank during this orbital period is shown in Fig. 5b. It can be seen that all the defined temperature limits are adhered to in this case. 2 - The thermal simulation of the faces and spheri- cal fuel tank over 8 orbital periods (time span of 12 h) is shown in Fig. 6a, and the simulation of the temperature of the tank during these orbital periods is shown in Fig. 6b. 47(1) pp. 25–32 (2019) 30 AL-HEMEARY, JAWORSKI, KINDRACKI, AND SZEDERKÉNYI (a) (b) Figure 5: Thermal behaviors using coated Al during one orbital period ((a) temperatures of the faces and fuel tank, (b) temperature of the fuel tank). 4.3 The faces of the satellite exposed to the sun during its orbit are covered with 70 % aluminum and 30 % solar cells In this case, these three sides of the PW-Sat are assumed to be composed of 70 % aluminum and 30 % solar cells and the other faces are coated with magnesium oxide- aluminum oxide paint. The simulation results are as fol- lows: 1 - The thermal simulation of the faces and spheri- cal fuel tank during one orbital period (time span of 0 s to 5399 s) is shown in Fig. 7a and the simulation of the temperature of the tank during this orbital period is also shown separately in Fig. 7b. The results show that the temperature of the tank varied between a maximum of 281.9 K and a minimum of 265.4 K, while the tempera- tures of the faces also remained within the given temper- ature limits. 2 - The thermal simulation of the faces and spheri- cal fuel tank over 8 orbital periods (time span of 12 h) is shown in Fig. 8a, and the simulation of the temperature of the tank during these orbital periods is shown in Fig. 8b. (a) (b) Figure 6: Thermal behaviors using coated Al over 8 or- bital periods ((a) temperatures of the faces and fuel tank, (b) temperature of the fuel tank). 4.4 The faces of the satellite exposed to the sun during its orbit are covered with 30 % aluminum and 70 % solar cells The results, according to the assumption that three sides of the PW-Sat are composed of 30 % aluminum and 70 % solar cells while the rest of them are coated with mag- nesium oxide-aluminum oxide paint, are shown below: 1 - The thermal simulation of the faces and spherical fuel tank during one orbital period (time span from 0 s to 5399 s) is shown in Fig. 9a, while simulation of the tem- perature of the tank during this orbital period is shown in Fig. 9b. The results show that the temperature of the tank varied between a maximum of 302.4 K and a minimum of 270.6 K, while the temperatures of the faces were within thermal limits. 2 - The thermal simulation of the faces and spheri- cal fuel tank over 8 orbital periods (time span of 12 h) is shown in Fig. 10a, while the simulation of the tempera- ture of the tank during these orbital periods is shown in Fig. 10b. 5. Conclusion A thermal mathematical model was constructed and stud- ied to compute the temperatures of the surfaces and fuel tank of the PW-Sat. Several cases were presented using various surface compositions and the results show that the proposed TMM is able to calculate the radiative heat Hungarian Journal of Industry and Chemistry THERMAL MATHEMATICAL MODEL 31 (a) (b) Figure 7: Thermal behaviors with 30 % sc during one orbital period ((a) temperatures of the faces and fuel tank, (b) temperature of the fuel tank). (a) (b) Figure 8: Thermal behaviors with 30 % sc over 8 or- bital periods ((a) temperatures of the faces and fuel tank, (b) temperature of the fuel tank). (a) (b) Figure 9: Thermal behaviors with 70 % sc during one or- bital period ((a) temperatures of the faces and fuel tank,(b) temperature of the fuel tank). (a) (b) Figure 10: Thermal behaviors with 70 % sc over 8 orbital periods ((a) temperatures of the faces and fuel tank, (b) temperature of the fuel tank). 47(1) pp. 25–32 (2019) 32 AL-HEMEARY, JAWORSKI, KINDRACKI, AND SZEDERKÉNYI that the PW-Sat would encounter during its assumed or- bit. Initially, the surfaces of the PW-Sat were assumed to be composed of 100 % of the aluminum alloy 6061-T6. The corresponding results suggest that the temperatures of the surfaces and fuel tank would be too high. There- fore, additional finishes applied to the surfaces were taken into consideration. The first choice of finish was to coat the entire surface of the satellite with magnesium oxide- aluminum oxide paint. The obtained results show that the temperatures of the surfaces and fuel tank dropped be- cause of the increasing emissivity and decreasing absorp- tivity of the surfaces. Further simulations were performed of cases in which the faces were exposed to the sun when partially covered with solar cells. The results indicate that the case described in Subsection 4.4, which delivers the most electrical power due to the highest percentage of solar cells, still satisfies the temperature limits of the fuel tank and surfaces of the satellite. The results also suggest that it is possible to install a fuel tank inside the PW-Sat which could be the first step required to add a propul- sion system that can generate thrust for this CubeSat. Fu- ture works will include validation of the model using ad- vanced thermal simulation tools as well as active control design to precisely regulate the temperature of the fuel tank. Acknowledgement This work has been partially supported by the European Union, co-financed by the European Social Fund through the grant EFOP-3.6.3-VEKOP-16-2017-00002. The sup- port of the University of Technology of Baghdad and Warsaw University of Technology is also acknowledged. REFERENCES [1] Tummala, A. R.; Dutta, A.: An overview of cube-satellite propulsion technology and trends, Aerospace, 2017 4(4), 58. DOI: 10.3390/aerospace4040058 [2] Mehrparvar, A.: CubeSat design specification Rev. 13, The CubeSat Program, Cal Poly SLO, 2015. http://www.cubesat.org/resources [3] PW-Sat description (CubeSat Warsaw University of Technology). https://directory.eoportal.org/web/ eoportal/satellite-missions/p/pw-sat [4] Lemmer, K.: Propulsion for CubeSats, Acta Astronaut., 2017 134, 231–243. 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B.: A thermal analysis and design tool for small spacecraft, Master’s Thesis, 2008. https://scholarworks.sjsu.edu/etd_theses/3619 Hungarian Journal of Industry and Chemistry https://doi.org/10.3390/aerospace4040058 https://doi.org/10.3390/aerospace4040058 http://www.cubesat.org/resources https://directory.eoportal.org/web/eoportal/satellite-missions/p/pw-sat https://directory.eoportal.org/web/eoportal/satellite-missions/p/pw-sat https://doi.org/10.1016/j.actaastro.2017.01.048 https://doi.org/10.1016/j.actaastro.2017.01.048 https://doi.org/10.1007/s11071-010-9890-4 https://doi.org/10.1007/s11071-010-9890-4 https://trs.jpl.nasa.gov/bitstream/handle/2014/41627/10-1646.pdf https://trs.jpl.nasa.gov/bitstream/handle/2014/41627/10-1646.pdf https://doi.org/10.1117/12.666177 https://doi.org/10.1016/j.applthermaleng.2008.12.038 https://doi.org/10.1016/j.applthermaleng.2008.12.038 https://doi.org/10.1016/j.applthermaleng.2017.07.033 https://doi.org/10.1016/j.applthermaleng.2017.07.033 http://sma.jaxa.jp/en/TechDoc/Docs/E_JAXA-JERG-2-310_08_RE.pdf http://sma.jaxa.jp/en/TechDoc/Docs/E_JAXA-JERG-2-310_08_RE.pdf https://scholarworks.sjsu.edu/etd_theses/3619 INTRODUCTION SYSTEM DESCRIPTION AND ASSUMPTIONS Structural Assumption Orbital Assumption Thermal Assumption THERMAL MATHEMATICAL MODEL (TMM) First Interval Equations Second Interval Equations Third Interval Equations The transient heat transfer of the spherical propellant tank COMPUTATIONAL RESULTS The PW-Sat faces composed of 100 % uncoated aluminum The faces of the PW-Sat are composed of 100 %-coated aluminum The faces of the satellite exposed to the sun during its orbit are covered with 70 % aluminum and 30 % solar cells The faces of the satellite exposed to the sun during its orbit are covered with 30 % aluminum and 70 % solar cells Conclusion