2.1. Description
This study aimed to assess the possibility of using PCMs in space to stabilize the temperatures of components and to reduce the size of the radiator in order to reduce weight. The system considered, shown in
Figure 1, was based on a PCM contained in a house, which was attached to the electronics whose temperature is to be controlled and the radiator. Every surface but the radiator was assumed to be covered by multi-layer insulation (MLI) blankets to avoid heat leaks. Moreover, the radiator was assumed to operate just when the heat needed to be released into the space. The device was considered, therefore, to be insulated from the environment to make the study conditions more unfavorable.
The key idea behind this PCM thermal control concept is to convert the thermal energy into a phase change reaction, storing heat when it is produced and releasing this energy when the electronics is switched off. Since the phase change process occurs at almost constant temperature, such thermal control means that the system temperature does not change significantly during the melting/solidification, so that if the melting point is appropriate, the electronics can be efficiently protected.
In general terms, micro-satellites (between 10 and 100 kg in weight) operate on low Earth circular orbits (LEO, 450–1200 km) with a wide range of NASA
β angles, and on high elliptical orbits, and are exposed to the Sun, albedo and infrared Earth radiation. Typical maximal incident fluxes for 550 km orbit for flat surfaces with normal to nadir trajectories are Q
IR (infrared) ~200 W/m
2 and Q
AL (albedo) max ~450 W/m
2 (averaged over orbit <150 W/m
2). The eclipse time can vary between 0.5 h (circular) and up to several hours for elliptic orbits. According to Baturkin [
21], typical requirements of average heat generation inside the satellite are in the range of 15–40 W. This power is produced mainly by housekeeping equipment (on board computer, transmitter, the attitude and control system, batteries). Peak heat generation coincides with payload operation and can reach up to 200 W.
In our case, the specifications that were to be followed are those stated by the ESA [
8]. There were requirements for functional and performance (FPR), interface (IR), environmental (ER), operational (OR), design (DR) and verification and testing (VTR) operations. However, the main requirements are those related with thermal capacity and mechanical behavior, listed below:
The device must be capable of absorbing 30 W for 45 min;
Operational temperature range must be (−20/+40 °C);
The device’s mass shall be less than 0.50 kg;
The device’s first resonance frequency must be higher than 140 Hz;
The device shall sustain a mechanical environment characterized by dynamic loads.
2.2. Governing Equations and Numerical Methods
It is very common for thermal control in spacecraft to describe the heat transfer process through the thermal lumped method (TLP). Details about the method can be found elsewhere [
22,
23,
24].
The set of nonlinear equations that describe the temperatures of a transient thermal mathematical model is (for a node
)
where
is the number of nodes of the thermal mathematical model (TMM),
is the conductive conductance (W/m) between nodes
and
,
is the Stefan–Boltzmann constant (5.67 × 10 − 8 W/(m
2·K
4)),
is the radiative conductance (m
2) between nodes
and
,
and
are the temperatures (K) of nodes
and
,
is the product of the
node mass (kg) times the heat capacity (J/(kg·K) and
is the power (W) that enters into node
. The subscripts
and
go from 1 to
. It is usual to ascribe thermal inertia to the product
as it describes the “opposition” to changing the temperature of
node when subjected to a power input.
The time derivative of the temperature of node
can be approximated by
For a node
and for a general time step
, it is possible to write
In order to simplify the notation, the following will be used
Additionally, the following set of equations is obtained
Each equation represents the thermal instant equilibrium of a node. An in-house-developed computer program called TK was used to solve the set of non-linear equations. These equations are solved for each time step where temperatures of the different nodes are calculated. The heat power (W) that goes from one node to another is also calculated, as well as the heat power (W) that goes into each node, which is employed in increasing its temperature. The TK computer program was modified to be able to deal with phase change materials. In the following, we will assume that the initial state of each node is solid and that the phase change will be melting.
For those nodes made of PCMs, additional information must be supplied to the computer program. The PCM has a latent heat , measured in J/kg. The mass of each node (kg), as well as the specific heat (J/(kg·K)) of the material are also known. By multiplying the mass of the node by the latent heat , the program can obtain the total energy needed by the node when passing from a solid to liquid phase.
The program uses a predictor–corrector method to take into account the latent heat of nodes made of PCM. For each time step, a set of temperatures is calculated (predicted) with the previously explained equations. Then, the program checks each node made of PCM. If the temperature predicted for this node is higher than the starting temperature of the change of state from solid to liquid, then the energy used (J) is calculated, the temperature of the node is corrected and the accumulated energy used in that node is calculated, as is the liquid fraction of PCM in that node. These calculations are performed taking into account the product of the mass times the specific heat of the node. If the calculated liquid fraction is lower than 1, the program continues with the next node. When the total energy accumulated in one node is higher than the total energy that the node needs to change state, the state of the node is considered liquid and, in the following time steps, it will not have influence over the predicted temperatures.
It is clear that time steps short enough must be considered when phase change takes place, otherwise significant errors can appear. It is not unusual to need to make several trials before fixing an appropriate time step length. Finally, the procedure for considering the change from liquid to solid is similar, but with decreasing temperatures.