## Thermal modelling

Thermal mathematical models are a very important part of a mission. Such models are often used to develop and validate an initial design, and are refined as construction and testing proceeds. As new information becomes available, the effects on the spacecraft and its components can be evaluated with the model, and corrective design or procedure changes developed.

Temperature records are a major element of spacecraft housekeeping telemetry; electronic failure is often associated with overheating (as a cause or an effect) and the correct quantitative interpretation of the temperature evolution in terms of local power dissipation may require a thermal model (e.g. is battery 3 getting warm because it is on the Sun-facing side of the spacecraft, or has it shorted out as well?). Similarly, experiments to determine thermal properties of the planetary environment may require a detailed understanding of the heat flows inside the vehicle.

Commercial computer codes such as SINDA are generally used (since the overall problem, that of setting up nodes with specified heat transfer paths between them and appropriate boundary conditions, and propagating the corresponding differential equations forward in time, is constant; only the details of the nodes, links and conditions change from spacecraft to spacecraft). Each node is associated with a heat capacity, and the heat paths between them will depend on view factors and surface emissivities (for radiative transfer) and on the length, cross-section and conductivity (for conductive transfer).

Note that while spacecraft engineers are generally very familiar with the purely radiative and conductive heat transfer settings that occur in vacuum, the free (thermally driven) and forced (wind-driven) convective transfer that may occur between components or between components and the environment are less familiar and less easy to estimate, and can often only be determined with confidence by direct testing. Heat transfer coefficients of around 0.25Wm-2K-1 were determined for the transfer between equipment boxes in the Huygens probe and its internal atmosphere at an altitude of 150 km (where the ambient pressure is around 3 mbar), while near the surface with a pressure of 1.5 bar, the heat

transfer from equipment boxes is estimated at 3.5 Wm K (Doenecke and Elsner, 1994). In contrast, in the thicker Venusian atmosphere, convective coupling were extremely strong, 150-1000Wm-2K-1, such that the outer surfaces of the Pioneer Venus and Venera probes were essentially at ambient temperature. It is believed that the internal heat transfer of the Galileo probe was underestimated, leading to higher than planned temperatures and rates of change during descent; factors that required recalibration of the scientific instruments.

If we consider a spacecraft to be made of i plate-like elements of mass mi, with a heat capacity ci, then each block can radiate heat, have heat conducted into it, absorb heat, and generate heat internally. These heat transfer rates, PR, PC, PA, PI, alter the element's temperature history as;

The amount of heat radiated from each element depends on its temperature Ti, cross-sectional area Ai, the Stefan-Boltzmann constant9, a, and the element's emissivity, e1 as:

Considerable effort goes into choosing materials and surface finishes to control the heat balance of a spacecraft by passively rejecting or absorbing heat where necessary. If the ith element is joined to another element, marked by index j, with

Figure 8.4. Geometry of the thermal model discussed above.

temperature Tj, and a thermal connection K, then the conduction term for the element i is

The term describing power absorbed by the element is more complicated, as a spacecraft lander may be exposed to different radiant sources; the Sun, and hopefully at least one nearby planetary body. Other parts of the spacecraft which can be 'seen' by the element are neglected here for simplicity.

In Figure 8.4, n is the unit normal vector of the element, and R and h are the direction vectors to the Sun and a nearby planetary body, respectively. The first term in Equation 8.6 is the radiant heat absorbed from the Sun, giving an intensity, E, at the spacecraft's position. The second term shows the power absorbed from a nearby body of temperature TP, with eP being the emissivity of that body and ai the plate's albedo. The third term is the power absorbed via reflection from that object. Here, aP is the albedo of the reflecting planet, and f is a view factor that accounts for the variable amount of lit planet that the spacecraft element can 'view'. Examples of how this view-factor varies with the spacecraft-to-planet separation can be found in Wertz and Larson (1999).