The Robust Mathematical Modeling Blog: RMM Example #2: Spacecraft Thermal Management

When designing Spacecrafts, one of the major issue aside for designing its primary instruments is to devise its thermal management (i,e, managing the way power produced by the spacecraft can be removed so that it does not overheat). The thermal management of Spacecrafts requires solving different sets of issues with regards to modeling. Because spacecrafts generally live in low earth or geostationary orbit, the only way to remove power generated on the spacecraft is through radiation out of its radiators. This radiator point is the lowest temperature the spacecraft will experience. If the spacecraft is well conditioned all other parts of the spacecraft will have higher temperature no matter what. The main issue of thermal modeling for spacecraft design is really making sure that all the other points of the spacecraft will be within the temperature bounds they are designed for: i.e. The thermal rating for a DC/DC converter is widely different than that of a simple CMOS or the lens of a camera. Hence computing the radiator temperature is of paramount importance and can be done very quickly with a one node analysis. Yes, you read this right, at the beginning, there is no need for Finite Element computations in spacecraft analysis except maybe for very specific components and very specific conditions. The most important computation is figuring out this one spacecraft-node analysis. In terms of modeling, it doesn't get any simpler and it is robust. The issues that tend to crop up are when one gets into the detailed power consumption and thermal energy flow within the spacecraft as more detailed constraints. are added To summarize the issues, let me try to follow the list of issues that is making up the definition of problems needing Robust Mathematical Modeling as a guideline:

1. The laws describing the phenomena are not completely known ;

In fact, in this case the laws are known but there are large uncertainties at many different levels:

each element of the spacecraft has a thermal conductance, but since one is dealing with heterogeneous elements like a CMOS or a slab of aluminum, the designer is constrained into a lumped analysis involving a delicate weighting.
the thermal contact resistances / conductances of the electronics are generally unknowns in terms of performance especially in vacuum. Most information on the electronics is given when convection is available (for ground use). Even when environment is known, electronics components are very hard to evaluate. See this very interesting thread on LinkedIn.
the thermal contact conductance of two pieces of metals connected to each other through nuts and bolts is by no means a trivial subject. The contact conductance certainly depends on how much torque was put on the washer/nuts/bolts and the level of vacuum.
the space environment produces different heating and cooling conditions that are inherently different based on the positioning of the spacecraft, its orbit, etc...
in order to regulate temperature efficiently, cloth and paints are covering the spacecraft for the duration of its life. There are uncertainties with regards to how these decay over time and most computations include Beginning Of Life (BOL) and End Of Life (EOL) estimates.
An element of confusion is a mathematical one too. Since most of the thermal power is managed through conduction, radiation transport (a nonlinear term in T^4) is generally modeled as a linear node. When temperature gets too high, the conductance node varies with temperature to follow the nonlinear T^4 term.

2. The data are missing or corrupted ;

Spacecraft are generally designed with a clear emphasis on reducing its weight at the subsystem or bus level. A GEO satellite maker would rather put one more transponder bringing revenue on its spacecraft than add additional instrumentation to provide data to the ground. Experimental data is rare in spacecraft design because real conditions are rarely fully instrumented. Tests are performed at every iteration of the spacecraft design though, but they are not total reproduction of the actual thermal environment sustained by the future spacecraft. For instance, Sun lamps only produce some subset of the wavelengths given by the Sun, so it difficult to find the thermo-optical properties of some paints or the efficiency of some solar cells. While vacuum tests get rid of the convection issue, it can do little to evaluate the performance of systems that rely on convection inside said systems such as loop-heat pipes.

3. The objectives are multiple and contradictory.

Four words: Hot Case, Cold Case. The worst thermal environments are generally sought in order to provide acceptable bounds during the lifetime of the spacecraft. It is no small secret to say that these two objectives are contradictory. One of the two cases, the cold case generally, also generate additional mass to remedy to it. Adding mass for a subsystem is not considered optimal as every kilogram in Low Earth Orbit cost about $10,000. The number is obviously higher for Geostationary Orbit. Another surprising element is that sometimes, the colder case is not an obvious one so the solver really has to go through many different types of iterations to define what that case it.
The objective to have the lightest spacecraft possible also flies in the face of thermal "equilibiuim". the less thermal mass a spacecraft has, the less capable it will be able to handle environmental swings. Cubesats, for instance, fall in this extreme category of spacecraft for which thermal fluctuations can be very large (and bring about a possible thermal "event:" such electronics board cracking, etc....)

As the design progresses the modeling is iterated upon testing (from components to whole spacecrafts) but as Isidoro Martinez points out, this really is just the beginning of a long processes fitting models with the result of the few experimentl data gathered through the lengthy Thermal Balance and Themal Vacuum Tests (TB/TV tests: From here:

Spacecraft thermal testing
Measurement is the ultimate validation of real behaviour of a physical system. But tests are expensive, not only on the financial budget but on time demanded and other precious resources as qualified personnel. As a trade-off, mathematical models are developed to provide multi-parametric behaviour, with the hope that, if a few predictions are checked against physical tests, the model is validated to be reliable to predict the many other situations not actually tested.

Final testing of a large spacecraft for acceptance is at the present limit of technology, since very large vacuum chambers, with a powerful collimated solar-like beam, and walls kept at cryogenic temperatures, must be provided (and the spacecraft able to be deployed, and rotated in all directions, while measuring).

Typical temperature discrepancy between the most advanced numerical simulation and the most expensive experimental tests may be some 2 K for most delicate components in integrated spacecraft (much lower when components are separately tested).

Picture of the large vacuum chamber at NASA JSC where the Apollo LEM was tested.

An item of considerable interest to the thermal designer is to reduce substantially the time it takes to fit the few experimental results to the models. This part is by no means trivial given the nonlinearities of radiation heat transfer. The time required to fit models to the experiments is critical as tests are expensive and lengthy. From here the timeline for the TB/TV test of the Rosetta spacecraft that took thirty days: