THERMAL MODELING FUNDAMENTAL

The goal of the thermal engineer who develops the thermal model is to achieve the best possible accuracy with the relative lower cost. The cost can be divided into two different aspects: the cost of developing the model, and subsequently the cost of using it for analysis.

The thermal model is a network, of variable complexity and dependent on various factors, from which the distribution of temperatures and thermal gradients are derived.

The laws governing the dynamics within thermal models derive from the fundamental laws of electrical systems: Ohm's law and Kirchhoff’s law. This characteristic is called electrical-thermal analogy, and allows to adapt the partial differential equations that numerically regulate electrical phenomena, to the

resolution of thermal numerical problems simply by replacing the electrical variables with the appropriate thermal ones, as listed in the table.

Quantity Electrical variable Thermal variable

Potential 𝐸 𝑇

Flow 𝐼 𝑄̇

Resistance 𝑅 𝑅

Conductance 1

⁄ 𝑅 𝐺

Capacitance 𝐶 𝐶

Table 1:Electrical-Thermal analogy

These correlations between the variables allow the analogy for Ohm's law in the electrical and in the thermal form:

𝐼 = 𝐸 𝑅⁄ 𝑄̇ = 𝐺 ∙ 𝑇

4.1.1 Nodes

The construction of a thermal network, at the basis of a model for solving numerical problems, begins with the subdivision of the system to be analysed into

35 finite-sized sub-volumes called nodes. The subdivision is called Nodalization, and it allows to concentrate the thermal characteristics at the centre of the nodal volume, generating a lumped parameters model. These characteristics are temperature and capacitance.

Figure 12: Nodalization (Credits: NASA, “Thermal Network Modelling Handbook”)

By concentrating the thermal potential (temperature) and the thermal mass (capacitance) at the central point of the sub-volume, a linear distribution of the temperature between adjacent nodes can be considered. By interpolation, the temperature at an intermediate point with two nodes is known.

Generally, software dedicated to thermal analysis allows the use of three different types of nodes for the construction of a model. The nodes classification is described below:

 Diffusion nodes: characterized by a finite and non-zero capacitance, they are the most commonly used nodes, as they describe the thermal behaviour of any type of material that is employed. The temperature of these nodes depends on the incoming and outgoing heat flows involving it, the time of exposure to the flows, and the heat capacity of the material. The thermal behaviour is described by the following equation:

𝛴𝑄̇ −𝐶∆𝑇 𝑡 = 0

 Arithmetic nodes: they are nodes that are found in small numbers within the models. They do not represent real elements, but are a mathematical artifice which, representing a node with zero thermal capacitance, allows to facilitate and speed up the thermal simulation of some real elements of the system. For example, in the case in which there is an element characterized

36 by a very small thermal capacitance compared to that of the other elements of the system, modelling it with arithmetic nodes with zero capacitance avoids having to solve a Stiff numerical problem, significantly saving on resolution times of the problem. Mathematically, an arithmetic node is described by the following expression:

𝛴𝑄̇ = 0

 Boundary nodes: Unlike arithmetic nodes, boundary nodes are

characterized by an infinite thermal mass. This allows them to be used for the modelling of elements whose temperature does not vary under the influence of thermal loads, for the duration of the phase to be simulated. In space applications, typically the deep space temperature sink is modelled by a boundary node, since its temperature is constant regardless of the loads related to the analysed spacecraft. It follows that the law that characterizes these nodes is:

𝑇 = 𝑐𝑜𝑠𝑡

Nodes and their associated sub-volume usually have simple shapes, such as rectangular-shaped nodes, for ease of calculation. The size and therefore the number of nodes that make up a network depend on how accurate the model is to be,

considering the performance of the computer solving the thermal problem. The more nodes there are, the greater the accuracy, the computation time and the dedicated memory space.

4.1.2 Conductors

A conductor is a network element of the thermal mathematical model, which implements a heat flow path through which heat flows from one node to another.

Once the conductors have been introduced, it is possible to make a schematic representation of the thermal network in analogy with electrical systems.

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Figure 13:Thermal Network electrical scheme (Credits: NASA, “Thermal Network Modelling Handbook”)

The scheme generically represents the conductor element that allows the

transmission of heat from the element with a higher temperature to that with a lower temperature. More specifically, there are three types of heat transmission and

therefore of conductor: conduction, convection and radiation. The first and the last are typical phenomena of heat transport present in space systems operating outside the atmosphere, while convection is not characteristic in thermal problems concerning space, as it implies the presence of a fluid that transports the heat while moving.

The following figure represents the schematic of conductive and radiative type conductors.

While the heat flow between two nodes by conductive or convective way is a linear function of the temperatures of the two nodes i and j:

𝑄̇ = 𝐺_𝑖𝑗(𝑇_𝑖− 𝑇_𝑗)

In the case of radiative heat transport, it is no longer linear, but depends on the fourth order power of the two temperatures:

𝑄̇ = 𝐺_𝑖𝑗(𝑇_𝑖⁴− 𝑇_𝑗⁴)

Most thermal analysis computer programs linearize the radiation term before performing thermal equilibrium at each time step.

Figure 14: Conductive and radiative conductors (Credits: NASA, "Thermal Network Modelling Handbook")

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