6. The thermal network

100

6. The thermal network

These last two chapters are devoted to the construction of the TMM for a generic spacecraft configuration and the development of numerical methods for evolution in time of the resulting temperature field. Although there have been attempts in proposing completely analytical approaches to the computation of temperatures, such as the work of [34] where a solution for a thick-walled cylindrical space vehicle is obtained, the most popular way to represent a TMM is through the “thermal network”, which is discussed starting from Par. 6.2.

Figure 6.1 - Topic of Chapters 6 and 7 (yellow box)

The rest of the work is englobed in a Matlab^® script called thermal_code.m, for which various versions have been generated according to the various intermediate results obtained.

6.1 Pre-processing

With reference to Fig. 5.9, all the information collected are stored in three files, which are appropriately read on Matlab^®. The only missing information is about the right position of the doubled facets w.r.t. their real ones, which the Mesher set by default

101 at a distance of 1 cm, independently of the material of the wall. To evaluate the correct view factors among the facets, the correct thickness of the material is supplied and a new MSH file called test_mod.msh is written and passed to the VFS to obtain view factors. This operation must be done any time a new model is to be analyzed.

At the end of the pre-processing phase, during which the geometric centroids of the facets are computed through their vertices, all the elements to build the TMM are available.

6.2 The nodal structure

The pillar on which the construction of the thermal network is based is the “lumped- parameter method”, [7-35-36]. This method has its conceptual origins in the thermal- electric analogy in which temperature corresponds to voltage and heat flow to current flow. In fact, thermal and electrical systems are analogous since they are represented by similar equations and boundary conditions, and the equations describing the behavior of one system can be transformed into the equations for the other by simply changing symbols of the variables, [7]. The table below, [7-37], shows definitions and symbols for the complete analogy (In the next paragraph, different symbols are used to describe quantities of the thermal system).

Table 6.1 - Thermal-electrical system analogy

102 The analogy between thermal and electrical systems allows the thermal engineer to utilize widely known basic electrical laws such as Ohm's Law and Kirchhoff's Laws, which are used for balancing networks. Numerical techniques used to solve the Partial Differential Equations (PDEs) describing such electrical systems have been conveniently adapted to computer solutions of thermal networks, thus enabling the engineer to readily compute temperature distribution and heat flows of complex physical thermal networks, [37].

To develop a thermal network and apply numerical techniques, first step is to discretize the continuous thermal system into finite sub-volumes called “nodes”. The thermal properties of each node are considered to be concentrated at the central nodal point of each sub-volume. Each node represents two thermal-network elements, a temperature T (potential) and a thermal mass CM (more properly, thermal capacitance), while the thermal path among the nodes is described by means of conductances, also called “couplings”.

Figure 6.2 - Representation of a generic node (subscript x) in the generalized thermal network

Because a node represents a "lumping" or concentration of parameters at a single point in space, the temperature distribution through the sub-volume implied is linear, and not a step function, [7]. This concept should be kept in mind when discussing results, since two adjacent nodes with relative large temperature difference would

103 require a local refinement to return acceptable data. Nevertheless, since the temperature gradient inside a node is negligible, nodes are usually referred to as isothermal, [19-20].

Three different types of nodes can be assigned in building a thermal network, [7]:

 Diffusion nodes: they possess finite CM and represent sub-volumes whose T can change as a result of the net heat flow into the nodes. The gain or loss of potential energy depends on CM, the net heat flow into the node, and the time during which the heat is flowing;

 Arithmetic nodes: because they have zero thermal mass, they are not physical real quantities. Their temperature responds instantaneously to its surroundings, so they should be introduced only when surface temperatures need being computed or to describe thermal-system elements whose CM is very small in comparison to the large majority of the other nodes;

 Boundary nodes: they are opposite to arithmetic nodes, since they have infinite thermal mass. Therefore, they are used to represent boundaries or sinks (e.g. deep space) whose T are set and will not change independently of their net heat flow. In addition, boundary nodes may represent thermal- system components that have a very large CM in comparison to other nodes.

Up to now, the thermal network is supposed to be volumetric because of the association “nodes = sub-volumes”. On the contrary, the objective of the analysis is the evaluation of surface temperature field and the GMM is mostly composed of 2D facets, except for 3D solid compartments that are expected to be much less than facets. In order to provide the tridimensional piece of information to facets, the thermal network is built by using only diffusion nodes, with the following criteria:

 a node is put at the centroid of each single facet, except for facets belonging to a wall separating two solid compartments;

 double facets require two nodes, one at the centroid of the real facet and the other at the centroid of the doubled facet; these two categories are called

“facet-nodes”;

104

 a node is put at the centroid of each solid compartment; these are called

“compartment-nodes”.

Figure 6.3 - Position of nodes for Reference Satellite

Arithmetic nodes have been deliberately avoided, although they could lead to considerable computational time savings. In fact, the replacement of small- capacitance diffusion nodes with an arithmetic node must be preceded by computations to verify that the capacitance-conductor effects are such that the node in question will essentially reach steady-state temperatures during the time-step required by the larger nodes, [7]. This requires careful analysis, since arithmetic nodes could suffer from instability problems during the advancement in time of the numerical solution, as discussed in the next chapter.

Boundary nodes are here used on either facets or compartments to impose constraints given as input; despite that, they could be also set by choice of the thermal analyst, [7]. For example, the camera of the Reference Satellite is sometimes kept at constant T: this means that the relative compartment-node centroid is at prescribed T, while conduction taking place inside the compartment creates a ΔT between the center of the compartment and its walls. In this sense, the compartment-nodes denies the definition of “isothermal node” in place of a better physical description of the heat transfer. Nevertheless, a trick to maintain the whole

105 compartment at constant T (if required) is to put boundary nodes to both the solid compartment and all the facets belonging to its walls, thus thermally insulating it.

Once the nodes are established, the nodal structure is created in order to pass information from facets and compartments to nodes: this permits to write equations that are formally identical for both kinds of elements coming out from the Mesher.

The following double-entry table conceptually describes the nodal structure, where nodal properties can be named by means of Matlab^® struct command. Some quantities are read during pre-processing phase, other quantities are derived from the available data.

Table 6.2 - Example of nodal structure

106 About the mass M, its computation has been always reduced to the expression

𝑀 = 𝜌𝑉

(6.1) where V is:

 for facet-nodes whose facets are double, the product between facet area and half the wall total thickness (which is not expressed by the voice “thickness”

in the previous table);

 for facet-nodes whose facet is single, the product between facet area and the wall total thickness;

 for compartment-nodes, the difference between the volume in input (which does not take into account the presence of materials at the walls) and the volume of the walls surrounding it.

6.3 The nodal equations

From the standpoint of computation, the discrete approximation introduced by the lumped-parameter method has advantages in numerical as well as experimental work, and lends itself to simple hand calculations as well as to computerized solution.

A continuous system obeys a differential equation, whereas a networked model obeys a system of algebraic equations in a finite number of variables that can be solved by standard techniques, [35]. This passage is obtained transforming the FEM mesh of the GMM into a Finite-Difference Model (FDM) of TMM by means of the method previously described.

Figure 6.4- Comparison between FEM and FDM meshes, [35]

107 Nevertheless, thermal codes like NASTRAN and ANSYS, which are mainly concerned with structural analysis of solids, make use of the FEM scheme even for generating solution of the TMM. Advantages and disadvantages of both methods are discussed in [7-35].

Starting from Eq. (2.1), adding considerations about the three mechanisms of heat transfer and after the discretization process, a set of p equations can be written, one per each node, [7-35]. Generally, p is the sum of n and m, the number of facet-nodes and compartment-nodes respectively. Under the convention that heat power Q is positive if entering a node and negative if exiting, the energy-conservation equation for a generic node i rewrites as:

𝐶

_𝑀,𝑖^𝑑𝑇𝑖

𝑑𝑡

= ∑

^𝑝_𝑗=1

𝐶

_𝑖𝑗

(𝑇

_𝑗

− 𝑇

_𝑖

) + ∑

^𝑝_𝑗=1

𝐻

_𝑖𝑗

(𝑇

_𝑗

− 𝑇

_𝑖

) + ∑

^𝑝_𝑗=1

𝑅

_𝑖𝑗

(𝑇

_𝑗⁴

− 𝑇

_𝑖⁴

) − 𝑞

_{𝑒𝑚𝑖𝑡𝑡𝑒𝑑,𝑖}

(𝑇

_𝑖⁴

− 𝑇

_𝑑𝑠⁴

) + 𝑄

_{𝑖𝑛𝑡,𝑖}

+ 𝑄

_{𝑒𝑥𝑡,𝑖}

(6.2) where the nodal couplings (positive by nature) are identified as follows:

 Cij, [W/K] is the conductive conductance between nodes i and j and the relative term is the heat power exchanged by the two nodes by conduction;

 Hij, [W/K] is the convective conductance between nodes i and j and the relative term is the heat power exchanged by the two nodes by convection;

 Rij, [W/K⁴] is the radiative conductance between nodes i and j and the relative term is the heat power exchanged by the two nodes by radiation.

Figure 6.5 - Schematics of a) conductive, b) convective and c) radiative conductances

108 The following two considerations, which hold in these discretized TMMs, allow reducing the computational time for conductances:

 each node cannot exchange heat by itself, so Cii = Hii = Rii = 0;

 the net heat flux exchanged between two nodes in any of the three heat transfer mechanisms must have the same value (except for the sign) if the equation is written for either node i or j . This translates in Cij = Cji, Hij = Hji and Rij = Rji.

The other terms have been widely discussed in Chapter 3, but are now specified for each node:

 qemitted,i, [W/K⁴], plays the same role as Rij between node i and the deep space. Its definition obeys the precise choice to define a node only for elements coming from the GMM, [19-20]. It is non-zero only for external nodes, those that “see” space with a view factor Fi-ds that can be computed by exploiting Eq. (2.14) and the knowledge of all the view factors between node i and the other nodes. So its expression is:

𝑞

_{𝑒𝑚𝑖𝑡𝑡𝑒𝑑,𝑖}

= 𝜎𝜀

_𝑖

𝐴

_𝑖

𝐹

_{𝑖−𝑑𝑠}

(6.3)

 Qint,i is the internal heat produced by a compartment-node, such as the

battery, so it is zero for facet-nodes. As input from the JS file, it is constant, but it could be potentially any function of time, provided that it is specified as input to the thermal code;

 Qext,i is the total external input absorbed by external nodes:

𝑄

_{𝑒𝑥𝑡,𝑖}

= 𝑄

_{𝑆𝑢𝑛,𝑖}

+ 𝑄

_{𝐸𝑎𝑟𝑡ℎ,𝑖}

+ 𝑄

_{𝑎𝑙𝑏,𝑖}

(6.4) It depends on satellite’s orbit and attitude, so it varies in time. It is obtained by specifying Eq. (3.2) – (3.15) – (3.17) for each node:

{

𝑄

_{𝑆𝑢𝑛,𝑖}

= 𝛼

_𝑖

𝐽

_𝑆𝑢𝑛

𝐴

_𝑖

∙ (−𝒓

_𝑆𝑢𝑛

∙ 𝒏

_𝑖

) ∙ 𝑓

_{𝑑𝑎𝑦−𝑛𝑖𝑔ℎ𝑡}

(𝑡) 𝑄

_{𝐸𝑎𝑟𝑡ℎ,𝑖}

= 𝜀

_𝑖

𝐽

_{𝐸𝑎𝑟𝑡ℎ}

𝐴

_𝑖

∙ (−𝒓

_𝑠𝑎𝑡

∙ 𝒏

_𝑖

) 𝑄

_{𝑎𝑙𝑏,𝑖}

= 𝛼

_𝑖

∙ 𝑎𝑙𝑏 ∙ 𝐽

_𝑆𝑢𝑛

𝐴

_𝑖

∙ (−𝒓

_𝑠𝑎𝑡

∙ 𝒏

_𝑖

) ∙ 𝑓

_𝑎𝑙𝑏

(𝑡)

(6.5 – 6.7)

where:

109 o Sun-satellite direction is the same as Sun-Earth direction under the

“parallel rays” assumption (given by Eq. (3.3) with algorithm defined in Chapter 4);

o Earth-satellite direction is given by Eq. (3.5);

o ni is the node’s normal in GRF (normals in MSH file are defined in MF, this also explains the “minus” before rSun and rsat).

It is important to note that Eq. (6.5 – 6.7) show no reference to possible obstructions in absorbing external inputs due to satellite’s geometry. For example, the Reference Satellite could be oriented in such a way that part of the solar panels could be shadowed by the main structure (see Fig. 5.3). Consequently, QSun should not be an input for the relative shadowed facets, but this relatively important effect has not been considered in this work. A careful observation of the MeshLab outputs in Par.

6.4 confirms this shortcoming.

The only exception to the use of Eq. (6.2) is represented by a boundary node, which acts as a temperature constraint in the thermal network and whose equation is:

𝑇

_𝑖

= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

(6.8) As already stated in Chapter 2, convection regards a very small class of spacecraft, therefore the convective terms in Eq. (6.2) have been neglected. Moreover, the modelling of Hij terms is rather complex, as widely discussed in [37].

6.3.1 Calculus of conductive conductances

Conductive conductances are non-zero only among nodes that are adjacent. Here, conduction is assumed to take place both in perpendicular and parallel directions, w.r.t. the normal of the node in question. In both cases, Cij can be the result of conduction between two dissimilarly shaped nodes (e.g., conduction between facet- node and compartment node) or two nodes of dissimilar materials (e.g. conduction inside multi-layer material). The equivalent value of conductance is given by the series of the conduction paths, [7]:

𝐶

_𝑡𝑜𝑡

=

¹

∑ ¹

𝐶𝑘 𝑙𝑘=1

(6.9)

110 where l is the number of layers involved.

Eq. (6.9) easily applies in perpendicular conduction inside a wall composed of double facets. In this case, if l is the number of the layers of the wall material between the two nodes at both sides of the wall, [15], one has:

𝐶

_𝑖𝑗

=

^𝐴𝑖

∑ (^𝐿

𝑘)

𝑘 𝑙𝑘=1

(6.10)

where Lk is the thickness of the k-th layer, kk is its thermal conductivity and Ai = Aj. If the wall material has the minimum number of layers, which is two for walls with double facets, Eq. (6.10) reduces to the example in [37], Fig. 2.16.

Figure 6.6 - Example of conduction through a wall made of a three-layer material

Conduction between a compartment-node and its walls is perpendicular in the sense that, starting from the external facet of the wall, the first conduction path is assumed perpendicular to the wall surface. With reference to Fig. 6.7, the conduction path between facet-node i and compartment-node j is the series of two conductances.

The former is the conduction inside the multi-layer material and has been discussed above, the latter is the conduction inside a solid volume and takes into account the relative position of the nodes by means of the angle φ. The knowledge of φ allows

111 modifying the conductive conductance inside the compartment, indicated as Cj, introducing the following quantities:

 Lj is the distance between compartment-node j and its wall, which ends with the inner layer of material; the distance between points D and E is given by the ratio of Lj and the cosine of φ;

 the projection of Ai along the direction DE is given by the product between Ai

and the cosine of φ.

Figure 6.7 - Conduction between a compartment-node and its walls

To sum up, the searched conductive conductance is computed as:

𝐶

_𝑖𝑗

=

^𝐴𝑖

∑ (^𝐿_𝑘)

𝑘+ ^𝐿𝑗

𝑘𝑗(cos 𝜑)2 𝑙−1𝑘=1

(6.11)

A negligible error is committed if the angle φ^’ is used in place of φ. In fact, φ^’ suggests what should be the direction (blue line) along which the conduction path should lie.

112 By considering a pyramid of basis Ai and height given by the line connecting i and j, a more refined modelling of conduction could be obtained.

In a similar fashion, conduction between two adjacent compartment-nodes is modelled. The only difference is the use of the area Aij of the wall in common to both the compartments (which is known as input from the JS file) in place of information about the facets.

Figure 6.8 - Conduction between two adjacent compartment-nodes

113 Attention must be devoted, inside the Mesher, in selecting the direction of the normals to the common facets compatible with the definition of the common wall material, whose first and last layers are zero-thick. The conductive conductance can be evaluated as:

𝐶

_𝑖𝑗

=

^𝐴𝑖𝑗

𝐿𝑖

𝑘𝑖(cos 𝜑𝑖)2+∑ (^𝐿

𝑘)

𝑘+ ^𝐿𝑗

𝑘𝑗(cos 𝜑𝑗)2 𝑙−1𝑘=2

(6.12)

In the end, parallel conduction among adjacent facets is slightly more complex, but it can be reduced to a 1D conduction in which the conductance depends on the shape of the facets. The picture below shows the geometric model for one facet.

Figure 6.9 - Geometry for parallel conduction among adjacent facets

The resistance, inside facet i, to heat propagation toward facet j is due to the portion of material inside the pink volume. Therefore, the common expression of conductance inside facet i rewrites:

𝐶

_𝑖

=

^{𝑘𝑖𝐴𝑝𝑎𝑟,𝑖}

𝑙_𝑖

(6.13) where li and Apar,i are quantities defined in the figure above. For each facet i adjacent to facet j, a “shape parameter” pi is defined in the MSH file, given by:

𝑝

_𝑖

=

^2𝑙𝑖

𝑎_𝑖+𝑑_𝑖

(6.14)

114 where also ai and di are defined in the figure. On the other hand, Apar,i is the average area of the surfaces of the pink volume perpendicular to the direction toward facet j:

𝐴

_{𝑝𝑎𝑟,𝑖}

=

^{(𝑎𝑖+𝑑𝑖)𝑡𝑖}

2

(6.15) therefore, comparing Eq. (6.14) and (6.15) one has:

𝐴_{𝑝𝑎𝑟,𝑖} 𝑙_𝑖

=

^𝑡𝑖

𝑝_𝑖

(6.16) To conclude, parallel conduction between facets i and j is given by the series of the conductance of Eq. (6.13), specialized for both facets:

𝐶

_𝑖𝑗

=

^𝑝𝑖

𝑘_𝑖𝑡_𝑖

+

^𝑝𝑗

𝑘_𝑗𝑡_𝑗

(6.17) 6.3.2 Calculus of radiative conductances

Radiative conductances are non-zero only among those nodes that see each other.

By their nature, compartment-nodes do not radiate since they are completely surrounded by walls.

The generic form of Rij (radiation of node i towards node j) is easily obtained by comparison between Eq. (2.10) and the radiative heat exchanges in Eq. (6.2):

𝑅

_𝑖𝑗

= 𝜎𝜀

_𝑖

𝐴

_𝑖

𝐵

_𝑖−𝑗

(6.18) Many references like [6-15-18] underline the importance of distinguishing between enclosures and non-enclosures in computing Bi-j between two nodes. In fact, the reflective contribution of other nodes can becomes significant in enclosures with high-reflecting surfaces. The most common way to account for effects of other nodes is to compute Bi-j with the Gebhart’s method, [7-18-37]. A main drawback of using this method is that the irradiation is assumed uniform, a hypothesis that can be easily invalidated, [4]. A simpler approach to the problem is to use the following formula:

𝐵

_𝑖−𝑗

= 𝜀

_𝑒𝑓𝑓

𝐹

_𝑖−𝑗

(6.19) where εeff is the so-called “effective emittance”, [19-20-38], and whose most general expression is, [7-37]:

115

𝜀

_𝑒𝑓𝑓

=

^{𝜀𝑖𝜀𝑗}

1−𝐹_𝑖−𝑗𝐹_𝑗−𝑖(1−𝜀_𝑖)(1−𝜀_𝑗)

(6.20) The use of εeff is recommended only between surfaces having high emissivity. The lower the values εi and εj, the larger the error committed. Nevertheless, their simplicity of computation make them preferable in this context, since the Gebhart’s factors would require iterative techniques to be computed.

6.4 Post-processing and static tests

At the end of each simulation, all the Matlab variables are saved and then loaded in a script called postprocessing.m, in which results are generated. There are two useful ways to visualize them:

1. Generation of a PLY file with information to rebuild the mesh and assign the correct temperature value to each mesh element. 3D visualization is then obtained with MeshLab software;

2. Plot of the temperature history in time for each node.

Before proceeding with the solution in time of the thermal network, a series of tests with the purpose of verifying, at least qualitatively, the correctness of the model have been done. The term “static” refers to the fact that the system of equations described by Eq. (6.2) is solved at a fixed time; therefore, the thermal inertia term vanishes and the system reduces itself to a set of non-linear algebraic equations. Matlab^® command fsolve has been successfully employed with the reasonable initial condition Ti,0 = 0 °C for each node (fsolve interprets this initial condition as the starting point for finding a solution).

To simplify as much as possible the tests, the only effect of Sun as external input has been considered (except for the first test). Its direction is specified for each test.

6.4.1 Effect of optical properties

The role of the ratio α/ε has been discussed in Chapter 3, with reference to the isothermal temperature of a satellite. To give an idea of how much the optical properties of the surfaces of a satellite can influence its temperature field, it is

116 sufficient to consider Eq. (3.30) in a simple case: a dawn-dusk orbit of h = 500 km.

The satellite is covered with:

 a black paint (α = ε = 0.9);

 a white paint (α = 0.13, ε = 0.9).

All the external inputs are taken into account, plus a constant Qint = 50 W.

Figure 6.10 - Effect of optical properties

The result is that different coatings drive to extremely different results in terms of T values. In fact, ΔT ≈ 84 °C for this simulation along one orbit. Therefore, in the following tests, attention must be devoted mainly to ΔT variations rather than absolute values, which could appear “strange” because of the approximated values adopted in defining materials for the Reference Satellite.

6.4.2 Effect of thermal conductivity

From now on, the effects of materials and temperature constraints are visualized on the Reference Satellite model, which is tested in two limit-cases:

 satellite all made of conductive material, e.g. 7075-T6 aluminum alloy (k = 121.6 W/m*K), [7];

117

 satellite all made of insulating material, e.g. a nickel alloy (k = 25.5 W/m*K);

To see the effects of different values of k, the model is tested with all compartments at unknown T. The resulting temperature field is shown in the range [140; 210] K.

Figure 6.11 - Effect of thermal conductivity - aluminum alloy

Aluminum is highly conductive, so the Sun heat input is easily shared throughout the satellite, as confirmed by the maximum values of ΔT ≈ 21 K that is quite low.

118 Figure 6.12 - Effect of thermal conductivity - nickel alloy

On the contrary, the nickel alloy is low-conductive, so the Sun heat input finds a

higher resistance to its propagation throughout the satellite, as the maximum ΔT ≈ 58 K suggests. As expected, Tmax and Tmin have respectively increased and

decreased.

119 6.4.3 Effect of a compartment at constant T

To see the effect of a temperature constraint on the whole system, the model is tested with the camera compartment at fixed T = 273 K. Neither external nor internal heat inputs are considered. Moreover, also the radiation to deep space is suppressed in order to simply show how the temperature field rearranges itself.

Results are shown in the range [130; 273] K.

Figure 6.13 - Effect of a compartment at constant T

120 For conductive materials, such kind of compartment makes average temperature of the system higher than in the insulating case. The reason is that the constraint is

“shared” throughout the structure if the material is a good conductor, while the constraint is “shared” only in the proximity of the compartment if the material is insulating.

6.4.4 Effect of a compartment generating a constant Q

The effect of a compartment producing a constant heat flow is discussed separating the battery from the rest of the Reference Satellite. The compartment-node is assumed to generate a constant Qint = 50 W, which is transmitted to its walls by conduction. Such heat generation is balanced by radiation to space of the facet- nodes belonging to the surrounding walls. No external heat inputs are considered.

Figure 6.14 - Battery mesh scheme, separated from the Reference Satellite

Figure 6.15 - Comparison of temperature distributions on the battery

121 The battery is roughly schematized as a parallelepiped; so temperatures are lower on those walls that are farther from the compartment-node. This is clearly visible if the battery is made of insulating material, since the resistance encountered by heat propagation is larger. In both cases, the temperature of battery compartment-node peaks.

To sum up, such kind of compartment makes temperature of the system uniformly distributed if the material is conductive, while it enlarges the range of temperatures if the material is insulating.

6.4.5 Effect of facets at constant T

As already shown, the Mesher gives the possibility to set constant values of T also for facets. By default, this value is inherited also by the doubled facet if the wall has two normals (e.g. solar panels), thus creating a not physical wall at uniform T. This shortcoming can be tolerated for the purpose of the test.

The role of this constraint is expected to be similar to what described in Par. 6.4.3.

As a proof, consider the same conditions described in Par. 6.4.2 and see how things change by setting three double facets of the colder solar panel at T = 200 K, which is a relatively high value since the range of T was [140; 210] K.

122 Figure 6.16 - Effect of facets at constant T

The value T = 200 K is near to Tmax obtained in the previous test without temperature constraint, so also regions of the structure farther from the constrained facets are influenced. This is clear in the following figures, where situations with and without the constraints are compared.

Figure 6.17 - Comparison with and without constrained facets - conductive material

123 Figure 6.18 - Comparison with and without constrained facets - insulating material

6.4.6 Mesh size effects

Another important role of the equilibrium solution is to provide a measure of the

“goodness” of the FEM mesh. As a rule of thumb, after setting a maximum value of temperature difference between two adjacent nodes, indicated as ΔTthreshold, one can

solve iteratively the system of Eq. (6.2) in order to adjust the mesh unless ΔT < ΔTthreshold for each couple of nodes. Obviously, such activity would require too

much time; moreover, it is mainly related to the whole field of thermal design, which is out of the scopes of the present work.

In some cases, previous pictures show a very large gradient of T between adjacent facets, as indicated by relative colors. Even the Reference Satellite tested with appropriate materials has shown similar problems for mesh schemes similar to the ones shown in the previous tests. In particular, at the interface between solar panels and the body of the satellite local ΔT = 60 K has been measured and this represents a not physical situation. To ensure that such large gradients only depend on a coarse mesh, a series of tests have been done to try to improve the results and reach a maximum local ΔT < 5 K, thus obtaining more pleasant visual outputs. The model is

124 tested in the more general case described in the figure below (neither Earth nor albedo inputs). The range of T is [230; 330] K.

Figure 6.19 - Reference Satellite configuration to evaluate mesh size effects

First, the mesh size has been gradually reduced accepting the relative increase of simulation times.

Figure 6.20 - Global mesh reduction - 3592 nodes

125 The figure above shows a uniform mesh with 3592 nodes. Nevertheless, the critical region at the panel-body interface maintains a too large value of local ΔT ≈ 30 K, which is half the previous value. The required simulation time is about 1h:15m, employing a laptop with a Quad-Core processor of 2.4 GHz and a RAM of 8 GB.

The second attempt is to strongly reduce the mesh size at the interface and accept a coarse grid elsewhere, in order to maintain the same simulation time.

Figure 6.21 - Strong local mesh reduction – 3501 nodes

This time the model has 3501 nodes, but the precision of temperature field is increased at the interface. Consequently, ΔT ≈ 20 K is diminished, but it is again too high. Despite that, the trend of reducing the size seems to improve the results.

Before switching to relatively long simulations, an alternative approach has been proposed, based on splitting the system and performing local tests only. The critical panel-body interface is separated from the rest of the model and tested apart with the same Sun input as in Fig. 6.19. The advantage of this choice is that the mesh can be refined maintaining an acceptable number of nodes and restrained simulation times. On the contrary, absolute values of T cannot be compared anymore with those of the whole model.

126 Figure 6.22 - Panel-body interface, separated from the Reference Satellite

The figure above employs 1824 nodes, while the total simulation time is about 35m.

The result is encouraging: ΔT ≈ 8 K at the interface is now a reasonable value.

Such “local” tests have been repeated for different simplified models to show that the code is able to manage geometric models different from the Reference Satellite without singularities, thus assessing the generality of the code itself.

The final proof that the problem is only due to mesh size and not to errors in the code is shown below. The temperature field appears quite continuous and the imposed

ΔTthreshold = 5 °C is never overcome. This is obtained by combining a global mesh

reduction and a local refinement where the gradients of T were known to be quite harsh. The result is an extra-fine mesh with 5951 nodes; consequently, the simulation time is almost doubled.

127 Figure 6.23 - Example of "good" mesh for the Reference Satellite - 5951 nodes

The previous simulation times are mainly concerned with the creation of the nodal structure and the solution of the system at equilibrium. When advancing solution in time, the nodal structure is computed one-time at the beginning, but the system must be solved at least once at any time-step. Such refined mesh schemes would require unaffordable times and are therefore put aside in favor of simpler meshes, even accepting “unattractive” outputs.