# Heat Sink Equivalent Thermal Test Method and Its Application in Low-Orbit Satellites
## 2 Principle of the Heat Sink Equivalent Thermal Test Method
See Formula (1) for the calculation method of _Ki_. \[K_{i}=\frac{A_{i}\sum\limits_{j=1,j\
eq i}^{n}\abs{q_{i}-q_{j}}}{C_{i}M_{i} \Delta T}\leq n \tag{1}\]  
where \(q_{i}\) and \(q_{j}\) are the heat flux of region \(i\) and \(j\), \(C_{i}\) is the equivalent specific heat capacity of the satellite in the region, \(A_{i}\) is the equivalent area of the satellite in the region, \(M_{i}\) is the equivalent mass of the satellite in the region, and \(\Delta T\) is the allowable error of this test. After completing the calculation step S2, proceed to the judgment step S3. If the heat flow can be completely simulated with the heat sink, skip directly to step S5 to normalize the heat flow. Otherwise, go to step S4 and calculate the control point temperature of the heat flow simulator. In step S4, the external heat flow simulation device of the area \(j\) uses the radiant plate for temperature control in the thermal test, and the temperature of the control point is calculated by Formula (2). \[T_{j}=\sqrt[4]{\frac{\alpha_{j}q_{solar-j}+\alpha_{j}q_{albedo-j}+\varepsilon_{ j}q_{Earth1R-j}}{\varepsilon_{j}\sigma}} \tag{2}\]  
where \(q_{solar-j}\) is the solar incident heat flux arriving in region \(j\); \(q_{albedo-j}\) is the Earth-reflected shortwave radiation flux arriving in region \(j\), and \(q_{Earth1R-j}\) is the Earth-emitted longwave radiation flux arriving in region \(j\); \(\sigma\) is the Stefan-Boltzmann constant; \(\alpha_{j}\) is the solar absorptivity of the satellite surface in region \(j\); and \(\varepsilon_{j}\) is the infrared absorption (emission) rate of the satellite surface in region \(j\). Step S4 completes the heat flow simulation setup of the simulation device. After setting the control point temperature of the heat flow simulation device, proceed to step S5. Apply the heat sink to simulate the heat flow in area \(i\), and normalize it with Formula (3). \[q_{avg}=\sum\frac{A_{i}(\alpha_{i}q_{solar-j}+\alpha_{i}q_{albedo-j}+ \varepsilon_{i}q_{Earth1R-i})}{A_{S}} \tag{3}\]  
where \(q_{avg}\) is the normalized satellite absorbed heat flux, which is used to calculate the heat sink setting temperature in the thermal test; \(q_{Solar-i}\) is the solar incident heat flux arriving at region \(i\), \(q_{albedo-j}\) is the Earth-reflected shortwave radiation flux arriving at region \(i\), and \(q_{EarthIR\_i}\) is the Earth-emitted longwave radiation flux arriving at region \(j\); \(\alpha_{i}\) is the solar absorptivity of the satellite surface in region \(i\); \(\epsilon_{i}\) is the infrared absorption (emission) rate of the satellite surface in region \(i\); and \(A_{S}\) is the superficial area of the satellite. After the normalization of heat flow is completed, calculate the heat sink setting temperature in the heat sink simulation area according to Formula (4), that is, step S6. \[T_{0}=\sqrt[4]{\frac{q_{avg}}{\sigma\epsilon_{S}}} \tag{4}\]  
where \(T_{0}\) is the heat sink temperature of the vacuum chamber and \(\epsilon_{s}\) is the equivalent emissivity of the satellite surface. The derivation process of Formula (4) is as follows. The cosmic background temperature is about 3 K, ignoring the cosmic background radiation, the thermal balance equation of the satellite in orbit is  
\[Q_{solar}+Q_{allbedo}+Q_{EarthIR}+Q_{n}=Q_{o} \tag{5}\]  
where \(Q_{solar}\) is the solar heat radiation absorbed by the satellite, \(Q_{allbedo}\) represents the Earth-reflected shortwave radiation absorbed by the satellite, \(Q_{EarthIR}\) is the Earth-emitted longwave radiation absorbed by the satellite, \(Q_{n}\) is the internal heat source of the satellite, and \(Q_{o}\) is the outward radiant heat of the satellite. Assuming that the satellite has fully convex surfaces, the average heat flow of each surface in orbit is \(q_{avg}\). After the satellite reaches thermal equilibrium, the in-orbit equilibrium equation can be expressed as:  
\[A_{S}\epsilon_{S}\sigma T_{a}^{4}-A_{S}q_{avg}=Q_{n} \tag{6}\]  
where \(T_{a}\) is the satellite surface temperature in orbit.