ДВУХМЕРНАЯ ТЕПЛОВАЯ МОДЕЛЬ СИСТЕМЫ ТЕРМОРЕГУЛИРОВАНИЯ КОСМИЧЕСКИХ АППАРАТОВ НЕГЕРМЕТИЧНОГО ИСПОЛНЕНИЯ

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Introduction. Satellite communication systems are one of the fastest growing varieties of space information systems that are widely used in various areas of human activity [1; 2]. Every year in the world there is an ever more intensive development of satellite communication systems for various purposes. Among many, two main types of systems can be distinguished: connected systems for civil (commercial) use and military communications systems. Every year the information flow becomes more and more and this requires the appropriate development of communication systems. In this regard, satellite commu- nication systems have a great future. One of the indispensable conditions for reliable func- tioning of the spacecraft, its service systems and payload equipment is to provide the necessary thermal regime for all its elements [3; 4]. However, this task under the conditions of outer space has its own specifics: for the most part of the operational period, various external radiation heat fluxes (thermal radiation from the Sun and the Earth), which can vary over a wide range (in general, the temperature at different points of the surface of the spacecraft at the same time can be in the range from -150 to +150 °C), operate on the spacecraft [4]. In addition, the thermal conditions of the spacecraft are influenced by the optical properties of the surface of the apparatus, its orientation in outer space, the power of the fuel-generating airborne equipment (which, as a rule, varies depending on the operation modes of the spacecraft), and the thermal-radiative thermal bonds in the spacecraft. In connection with this, the thermal load is nonstationary [5; 6]. At the same time, satellites are equipped with various equipment and devices that have a strictly limited tem- perature range of operability, and this raises the problem of providing this range. Therefore, modern spacecraft is unthinkable without a special on-board system - a thermal control system. Statement of the research task. The thermal regime of the thermal control system is determined by the positional heat load from the spacecraft instrumentation, evenly distributed over the outer skin by the solar heat flow, by radiation into the open space, and by convective heat and mass transfer in the liquid loop of the thermal control system [7]. Fig. 1 shows the calculated two- dimensional thermal model of the liquid loop of the thermal control system with N and S (N - North, S - South) cell panels and liquid circuits embedded in them. Let us consider the case of a two-dimensional problem, when the emerging temperature gradients in the transverse (y-axis) and longitudinal directions (the x-axis) are taken into account, while conductive heat transfer inside the skin and along the X axis of the the liquid loop of the thermal control system profile is neglected. Fig. 1. Calculated two-dimensional thermal model of the thermal control system: 1 - compressor; 2 - heat pipes; 3 - throttle valve; 4 - honeycomb panel Рис. 1. Расчетная двухмерная тепловая модель СТР: 1 - компрессор; 2 - тепловые трубы; 3 - дроссельный вентиль; 4 - сотопанель Fig. 2. Isothermal surfaces for the southern honeycomb panel Рис. 2. Изотермические поверхности для южной сотопанели Consider the process of two-dimensional complex heat transfer in the form of heat balance balance from the southern panel (QS), internal heat sources ( Qint ) and heat release from the northern panel (QN). The balance is a particular case of the energy conservation equation This is emitted back into the open space [8]: 1 0 12i 1 (-) e s DF T 4 , where ε1 is the degree of blackness of radiation from the radiation surface; σ0 = 5.67 · 10-8 - radiation constant, QS + Qint = QN , (1) W / m2×grad4; F12 is the area of the radiation surface, m2; T1 is the temperature of the radiation surface, K. where QS is the heat flux passing successively through the isothermal surfaces of the southern panel (fig. 2): 1 - the outer surface of the southern panel with the temperature The heat flow is diverted inside the honeycomb panel by heat conduction: (-) l12DF12i (T - T ) , T1; 2 - inner surface of the southern honeycomb panel with temperature T2; 3 - conditional inner surface of the d12i 1i 2i liquid circuit with temperature T3; 4 - the surface along the average cross-section of the channel of the liquid circuit with the refrigerant temperature T4. where λ12 is the thermal conductivity of the honeycomb material; δ12 - distance between surfaces 1-2. The heat balance for surface 1 will be written as [9]: It is obvious that the heat flux QS, passing through the surfaces 1-2-3, is accumulated by the mass flow of the AS × S0 × DF1i ×sina - e s F T - 4 Подпись: 1 0 12i 1i liquid circuit, in series (integrally) along the contact - l12DF12i (T - T ) = 0, (2) length l1S of the liquid contour of the southern panel. It d12i 1i 2i should be noted that surfaces 1-2-3-4 are formed con- structively by various boundary surfaces, thermodynamic properties of materials and types of heat transfer. In addi- tion, the temperatures T -T are indicative values, that is, formally, the heat flux at the surface element ∆F12i is de- termined like this: DQ = A × S ×DF ×sina - e s F T 4 = 1 4 Si S 0 1i 1 0 12i 1i necessary for comparison with the maximum permissible = l12DF12i (T - T ). (3) values. For example, the temperature of the refrigerant T4 is necessary to evaluate the cavity supply of the liquid d12i 1i 2i circuit. Mathematical model. In the finite-difference form, the heat transfer at the i-th section of the southern panel (fig. 2) is determined by the following system of process equations [8]: 1. The heat flux from the Sun enters the surface 1 of the panel S: (+) As × S0 ×DF12 ×sin a , 2. The heat flux from the honeycomb panel is fed to the isothermal surface 2 by heat conduction d 1i 2i (+) l12DF12i × (T - T ) , 12i and is also taken away by the thermal conductivity to the inner surface (3) (fig. 2) of the liquid circuit l DF where AS is the absorption coefficient of solar radiation; 2 (-) 23 23i (T2i - T3i ) , d23i S0 - solar constant, W / m ; F12 is the area of the radiation surface; α is the angle between the normal of the radiation surface and the direction to the sun. where λ23 is the thermal conductivity of the honeycomb material; δ23 is the distance between surfaces 2-3. The heat balance for surface 2 is expressed by the equation Given T40 - at the entrance to the liquid contour of the southern panel, given the values of thermophysical, geol DF l DF metric and regime determining parameters at the integra- 12 12i (T1i - T2i ) - 23 23i (T2i - T3i ) = 0 . (4) tion step, we calculate the system (6), (7) with respect to d12i d23i the unknown temperatures T1i, T2i, T3i, T4i, along the x(i) 3. The heat-conducting heat flux considered in (4) is fed to the surface 3 d 2i 3i (+) l23DF23i (T - T ) , 23i and the convective heat flow is already diverted into the liquid circuit (-)ai DF34i × (T3i - T4i ) , (fig. 1) at the length of the thermal contact of the liquid line with the honeycomb panel liS. Obviously, the integral heat output from the southern panel, including radiation and internal heat sources, is determined from an expres- sion similar to (7) with regard to (1) QSS = QS + Qint S = mɺ Cp × (T4n - T40 ) , (8) where T40 is the temperature of the refrigerant at the inlet, and T4n is the temperature of the refrigerant at the outlet where αi is the heat transfer coefficient. of the liquid circuit, Qint S = Qint is the heat from the in- The equation of balance over the surface 3 takes the form ternal sources supplied to the liquid circuit along the length of the thermal contact liS (fig. 1) from the southern l23DF23i (T - T ) - a DF (T - T ) = 0 . (5) panel side, QS is the radiation component of the heat d23i 2i 3i i 34i 3i 4i input. It should be noted that heat from internal sources of Q is physically formed from two components: We group the equations of balances on surfaces 1-2-3 into the system: int S Qint S = Qint.fr + Qint. HP. , (9) 1 0 12i 1i 4 AS × S0 ×DF1i ×sina - e s F T d 1i 2i - l12 DF12i (T - T ) = 0, 12i where Qint.fr is the frictional loss in the liquid circuit, turn- ing into the heat of the coolant; Qint.HP is the zonal thermal power transmitted by the heat pipe from the working de- vices of the spacecraft. The temperature of the heat pipe, l12DF12i (T - T ) - l23DF23i (T - T ) = 0, (6) the area of the contact zone and its coordinates along the d12i l DF 1i 2i d23i 2i 3i length lS are to be determined in the calculation scheme [11] (position 2 in fig. 1) 23 23i (T2i - T3i ) - ai DF34i (T3i - T4i ) = 0. d23i It should be noted that (6) at a known coolant tem- perature at the i-th element T4i is completely determined by the number of unknown variables - T1i, T2i, T3i, the system (6) is the basis of the marching algorithm when integrating along the length of the liquid contour of the southern panel [10]: å i=4 QS = DQSi. i=0 The temperature change T1i, T2i, T3i, T4i forms the pro- jection of the temperature gradient on the transverse y(j) axis (fig. 2). The projection of the temperature gradient on Let us consider the zonal heat input from the space- craft through the heat pipe contact calculation case. We assume that, as in the case of the radiation con- stant, the zonal thermal power is given and uniformly distributed over the contact area of the heat pipe at the finite length of the liquid loop ∆liS = ∆lHP [12], then the input thermal power at the integration step is defined as: ∆qHP·∆F65. Let us pay attention that in the design scheme additional isothermal surfaces modeling heat transfer in a zone of contact of a heat pipe are entered: surfaces 6 and 5 (fig. 3). Accordingly the system is supplemented with two equations: l DF the longitudinal axis x(i) is formed by the balance of the q × DF - 65 65i × (T - T ) = 0 ; thermal power received during the heat transfer through the lateral surface (3) of the elementary calculated HP l DF 65i d65i 6i 5i i-volume (fig. 2) and the difference in the thermal power of the refrigerant flow through the cross-sections at the output and input of the liquid circuit elementary calcu- lated volume in step ∆Xi: 65 65i × (T6i - T5i ) - xi DF54i × (T5i - T4i ) = 0 . (10) d65i The system ((6) and (10)) is completely determined by the number of unknowns. The zonal internal heat flux at DQSi = mɺ Cp (T4i - T4i+1 ), (7) the integration step is determined by one of the terms where mɺ is the mass flow rate of the coolant in the liquid circuit, and Cp is the heat capacity. (10), for example: DQ = x DF × (T - T ) = l65 DF65i × (T - T ) In this case, the temperature at the entrance to the next calculated volume will be determined as or int .HPi i 54i 5i 4i d65i 6i 5i T = DQSi + T . l DF 4i+1 mɺ C 4i DQ = q ×DF = 65 65i × (T - T ) (11) p int .HPi HP 65i d65i 6i 5i Fig. 3. Zonal heat input from the spacecraft through the heat pipe contact calculation case Рис. 3. Расчетный случай с учетом зонального теплопритока от приборов КА через контакт тепловой трубы The heat influx from internal friction at the integration l23DF23i (T - T ) - a DF (T - T ) = 0; step is equivalent to frictional losses [13]: d23i 2i 3i i 34i 3i 4i Dx J2 DQint.fr.i = DH fr × mɺ = l fr × i × av × mɺ (12) T3i = T5i ; di 2 q ×DF - l D F (T - T ) = 0; where λfr is the coefficient of hydraulic friction; ∆xi is the TT 65i 65 65i 6i 5i integration step; di is the hydraulic diameter; J av is the average flow refrigerant velocity in the liquid path; mɺ is the mass flow. Taking into account (3), (11), (12), the total heat flux at the integration step is written as ∆QSSi = ∆QSi + Qint.fr i + Qint.HP i ; Si 0 1i 1i 1i DQ = As × S DF sina - e ×s× DF ×T 4 = = l12DF12i (T - T ); (16) DQS i = DQSi + DQint .fr .i + DQint. HP i (13) d12i 1i 2i Then, as in the case of a simple radiation heat load Q = q × DF = l65 DF65i (T - T ); (13) from the balance of thermal power DQiS = mɺ Cp × (T4i - T4i+1 ) (14) int .HP i HP 65i d65i Dx J2 6i 5i Qint.fr.i = l fr × i × av × mɺ ; the temperature at the entrance to the next calculated (elementary) volume is determined by di 2 DQ DQiS = mɺ Cp × (T4i - T4i+1 ) DT4i+1 = iS + T4i . mɺ Cp ∆QSS = QS + Qint.S = mɺ Cp (T4Sn - T4S0 ) . Taking into account the fact that the temperature along the perimeter of the cross-section of the inner surface of the channel of the liquid circuit is a mean-integral value [14], the last equation in (10) is replaced by T3i = T5i, then the record (10) becomes simpler With the known determining parameters and thermo- physical properties of materials, the set of equations (16) allows numerical integration over the length l1S of the thermal contact of the liquid contour of the southern panel with the final result ∆QΣS (8) and temperature field Tij qHP ×DF65i - l65DF65i × (T6i - T5i ) = 0 ; technically accessible for the measurement. For the northern panel, we will retain all the methodo- T3i = T5i (15) logical approaches and designations associated with con- Taking into account all the above-mentioned notations (1...15), we finally write down the heat balance equation for the southern panel: A × S DF sina - e ×s×DF ×T 4 - l12DF12i (T - T ) = 0; structive (isothermal) surfaces: T1 is the temperature on the outer surface of the honeycomb panel; T2 is one on the inside, etc. Solar radiation decreases (in our calculation case it is reset). Integration is conducted in the direction of the average flow rate along the thermal contact line of S 0 12 12i 1i d12i 1i 2i the liquid contour ∆l1N from the side of the north panel. l DF l DF We perform the necessary inversion of the signs and the 12 12i (T1i - T2i ) - 23 23i (T2i - T3i ) = 0; specified additions we transform (16) into the thermal d12i d23i energy complex of the heat balance on the north panel: -e ×s×DF ×T 4 + l12DF12i (T - T ) = 0; mine the area of efficiency and the area of optimum per- 12i 1i d12i 2i 1i formance under certain performance criteria (for example: the ratio of mass of the thermal control system to power - l12DF12i (T - T ) + l23DF23i (T - T ) = 0; of diverted heat flow). d12i 2i 1i d23i 3i 2i References - l23DF23i (T - T ) - a DF (T - T ) = 0; 1. Testoyedov N. A., Dvirnyi V. V., Morozov E. A., d23i 3i 2i i T3 = T5 ; 34i 4i 3i Dvirnyi G. V., Eremenko N. V. [Improving the durability of spacecraft]. Vestnik SibSAU. 2015, No. 2, P. 430-437. qTT ×DF65i - l65D F65i (T6i - T5i ) = 0; 2. Dai G., Chen X., Wang M., Fernandez E., Nguyen T. N., Reinelt G. Analysis of Satellite Constella- ∆Qå Ni = -∆QNi + Qint.fr.i + Qint.HP i ; (17) tions for the Continuous Coverage of Ground Regions. Journal of Spacecraft and Rockets. 2017. DQ = e ×s×DF ×T 4 = l12DF12i (T - T ); Vol. 54, No. 6, P. 1294-1303 (In Russ.). URL: Ni 12i 1i d12i 2i 1i https://doi.org/10.2514/1.A33826. 3. Krushenko G. G., Golovanova V. V. [Perfection of Q = q × DF = l65 DF65i (T - T ); the system of thermal regulation of spacecraft]. Vestnik int .HP i HP 65i d65i x J2 6i 5i SibSAU. 2014, No. 3 (55), P. 185-189 (In Russ.). 4. Gilmore D. G. Spacecraft thermal control hand- book. The Aerospace Corporation Press. 2002, 413 p. Qint.fr.i = lfr i i avi × mɺ ; di 2 Qå Ni = mɺ Cp (T4i - T4i+1 ); ∆Q = Q + Q = mɺ C (T - T ). 5. Meseguer J., Perez-Grande I., Sanz-Andres A. Spacecraft thermal control. Woodhead Publishing Lim- ited, Cambridge, UK, 2012, 413 p. 6. Delkov A. V., Kishkin A. A., Lavrov N. A. et al. SN N int . N p 4 N 0 4 Nn Analysis of efficiency of systems for control of the ther- mal regime of spacecraft. Chemical and Petroleum Engi- It is necessary to pay attention to the fact that if external heat sources are insignificant, then on the northern panel there is an unambiguous decrease in temperature. With the joint integration of heat and power balances ((16) and (17)), a mandatory condition neering. 2016, No. 9, P. 714-719. 7. Delcov A. V., Hodenkov A. A., Zhuikov D. A. Mathematical modeling of single-phase thermal control system of the spacecraft. Proceedings of 12th Interna- tional Conference on Actual Problems of Electronic In- ∆QSN + ∆QSS = 0, (18) strument Engineering, APEIE, 2014, P. 591-593. 8. Zigel R., Khauell D. Teploobmen izlucheniem (see the last equations in (16) and (17)) is satisfied when the temperature differences are equal T4Sn - T4S 0 = T4 N 0 - T4 Nn , or in another presentation - the temperature of the coolant at the output of the thermal contact of the liquid circuit of one panel basically determines the temperature at the in- put to the other: [Heat exchange by radiation]. Moscow, Mir Publ., 1975, 934 p. 9. Burova O. V., Romankov E. V. Tsivilev I. N., Mi- nakov A. V. [Investigation of the impact of the effective coefficient of the thermal conductivity of the reflector on the temperature distribution]. Vestnik SibSAU. 2014, No. 4 (56), P. 25-32 (In Russ.). 10. Kishkin A. A., Delkov A. V., Zuev A. A. et al. T4Sn » T4 N 0 ; T4S 0 » T4 Nn , (19) [Project optimization of heat engineering systems operatclarification is possible with a specific topology of the hydraulic circuit of the liquid circuit outside the thermal contact lengths on the panels and hydraulic losses in the electric pump unit and the control throttle (fig. 1) [15]. Conclusion. The considered system of thermal bal- ances of the thermal control system of the spacecraft on the characteristic surfaces of constant temperatures is re- duced to the form allowing to conduct a numerical solu- tion: the number of equations corresponds to the number of detected temperatures along the north and south panels and is closed through the coolant temperature of the liquid circuit. The system of equations makes it possible to in- vestigate the thermal state of the spacecraft of a leaky design in the stage of preliminary design with varying mode (the angle of inclination of the radiation surfaces to the sun, the heat release of the service module and the payload module, etc.) and the design parameters (the spe- cific dimensional topology of the object, diameter of pipe cross-sections, refrigerant flow, etc.), in order to deter-

Об авторах

Ф. В. Танасиенко

Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева

Email: spsp99@mail.ru
Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31

Ю. Н. Шевченко

Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева

Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31

А. В. Делков

Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева

Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31

А. А. Кишкин

Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева

Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31

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