128
7. Integration and solution
The final chapter is subdivided in two parts. First part deals with the integration of the models for computing external heat loads and spacecraft attitude inside the thermal code, with a simple example to assess the correctness of this activity. The second part shows how the advancement of the solution in time is obtained, the numerical techniques implemented and a final quantitative validation.
7.1 Orbits and attitude integration
The tests shown in the previous chapter assess the correctness of the FDM to describe the TMM, but say nothing about Earth-orbiting satellites with prescribed attitude laws.
Before solving the complete system of Eq. (6.2), the terms Qext,i must be computed for any kind of orbit and attitude law. This is done by integrating thermal_code.m with the models discussed in Chapters 3 and 4. Ideally, during the advancement of solution in time Qext,i should be evaluated for any node i at after any time-step, but more detailed models like conical eclipse and refined albedo would be too time- expensive since they require computing fday-night and falb at any time-step.
A faster alternative is to establish a number of instants nstep along each orbit at which Qext,i is evaluated. These data are stored in a matrix like the following:
𝑸
𝑒𝑥𝑡=
[
𝑄
1,𝑡1𝑄
1,𝑡2⋯ 𝑄
1,𝑡𝑘⋯ 𝑄
1,𝑡𝑛_𝑠𝑡𝑒𝑝𝑄
2,𝑡1𝑄
2,𝑡2⋯ ⋮
⋮ 𝑄
𝑖,𝑡1⋮ 𝑄
𝑝,𝑡1⋮
…
⋱
…
⋮ 𝑄
𝑝,𝑡𝑛_𝑠𝑡𝑒𝑝]
(7.1)
where p is the total number of nodes. In the end, the column vector composed of the various Qext,i at a generic time t can be obtained by using an interpolation technique between the two columns of Eq. (7.1) in which t falls.
129 In doing so, the external heat loads throughout the mission are known before the advancement of solution. Intuitively, the higher nstep, the more precise the interpolation would be.
The simplest way to show the correct evaluation of Qext is to build a test “ad hoc”
with the following criterion: given a certain symmetry of the boundary conditions during the mission, the resulting temperature field is expected to maintain a similar symmetry.
On 21st March 2015, 10:00 (time t0), the Reference Satellite starts at ν0 = 0 and flies along a circular equatorial orbit of altitude h = 500 km. Its camera is always pointing to the Earth center, while the axis of the solar panels is aligned with the orbital velocity. The battery generates a constant heat flow Qint = 250 W and all the compartments are at unknown T.
Figure 7.1 - Orbit and attitude at t0
Setting nstep = 10, the equilibrium solution is evaluated at ten different times along one orbit. Between the generic instants tk and tk+1, the satellite covers an angular distance of 36 deg because the orbit is circular; therefore, the ten instants are equally spaced.
130 Figure 7.2 - Instants at which the equilibrium solution is evaluated
The plane perpendicular to the orbit and containing Sun direction is a plane of symmetry for the external heat loads and the satellite’s attitude; therefore, one can compare the temperature field at equilibrium for the following couples of instants: t1
– t9, t2 – t8 and t3 – t7. Actually, the Sun direction at initial time is slightly different from the one depicted above because of the use of Julian Date as input for the mission, but this effect is irrelevant.
In order to see the differences in temperature among the various part of the satellite, different ranges of T are adopted and are specified in each figure, but the center of the range (aqua-green colored) is always kept at 273 K.
131 Figure 7.3 – Temperature field at t1 and t9 – side to Sun
Figure 7.4 - Temperature field at t1 and t9 – side to Earth
In this case, setting the center of the range at 273 K allows appreciating the relatively small temperature differences (about 10 K) between the faces of the main structure directly invested by Sun or Earth and the upper and lower faces that are perpendicular to both the sources and so colder.
132 Figure 7.5 - Temperature field at t2 and t8 – side to Sun
Figure 7.6 - Temperature field at t2 and t8 – side to Earth
At instants t2 and t8, solar panels show a temperatures decreased of about 60 K w.r.t.
instants t1 and t9. In fact, with reference to t2, the satellite is going toward the terminator so the angle between axis xBF and Sun direction is reducing more and more, which means lower and lower Sun heat loads.
Because also the plane perpendicular to both the equator and the Sun direction is a plane of symmetry for the orbit, but not for external heat loads (due to eclipse), the temperature distribution at instants t3 and t7 is quite similar to the one at t2 and t8, except for the absence of albedo that makes temperatures lower.
133 Figure 7.7 - Temperature field at t3 and t7 – side to Sun
Figure 7.8 - Temperature field at t3 and t7 – side in shadow
At instants t4, t5 and t6 the satellite is in eclipse and temperatures drastically decrease. The only external input is the Earth heat load, which is absorbed by the camera, the solar panels and the upper face of the main structure. The solution at equilibrium is the same at the three instants and the maximum value of T is reached by the battery.
134 Figure 7.9 - Temperature field at t4, t5 and t6
7.2 Advancement of solution in time
Once computed Qext, all the terms of the system of equations expressed by Eq. (6.2) are known. Actually, even terms like the conductances and the nodal thermal capacitances could vary in time (e.g., if the satellite has moving parts or if there is a tank with fuel that is consumed in time), but in this context only the time-variation of the external inputs has been considered.
To convert the finite-difference equations to a set of algebraic equations that can then be numerically solved, the time derivative6𝑑𝑇𝑖
𝑑𝑡 in the left-hand member must be approximated. A way to do it is to use the Taylor series development around a
6 Obviously, the time derivative is a partial derivate since temperature varies continuously in space.
Nevertheless, here it is adopted the common symbol to express the total derivative because, at this point, the system has been already discretized and each node is isothermal.
135 generic time t, arrested to the first order and weighted in time by means of a variable- weighted implicit factor ϑ, [7]:
𝑇
𝑖(𝑡 + ∆𝑡) = 𝑇
𝑖(𝑡) + 𝜗 ∙ (
𝑑𝑇𝑑𝑡
)
𝑡+∆𝑡
∙ ∆𝑡 + (1 − 𝜗) ∙ (
𝑑𝑇𝑑𝑡
)
𝑡
∙ ∆𝑡
(7.2) Specifying 𝑑𝑇𝑖𝑑𝑡 for both t and t+Δt by means of Eq. (6.2) yields to the following algebraic equation for the generic node i:
𝐶𝑀,𝑖𝑇𝑖(𝑡+∆𝑡)−𝑇𝑖(𝑡)
∆𝑡 = 𝜗 ∙ (∑𝑝𝑗=1𝐶𝑖𝑗(𝑇𝑗− 𝑇𝑖)+ ∑𝑝𝑗=1𝐻𝑖𝑗(𝑇𝑗− 𝑇𝑖)+ ∑𝑝𝑗=1𝑅𝑖𝑗(𝑇𝑗4− 𝑇𝑖4) − 𝑞𝑒𝑚𝑖𝑡𝑡𝑒𝑑,𝑖(𝑇𝑖4− 𝑇𝑑𝑠4) + 𝑄𝑖𝑛𝑡,𝑖+ 𝑄𝑒𝑥𝑡,𝑖)
𝑡+∆𝑡+
(1 − 𝜗) ∙ (∑𝑝𝑗=1𝐶𝑖𝑗(𝑇𝑗− 𝑇𝑖)+ ∑𝑝𝑗=1𝐻𝑖𝑗(𝑇𝑗 − 𝑇𝑖)+ ∑𝑝𝑗=1𝑅𝑖𝑗(𝑇𝑗4− 𝑇𝑖4) − 𝑞𝑒𝑚𝑖𝑡𝑡𝑒𝑑,𝑖(𝑇𝑖4− 𝑇𝑑𝑠4) + 𝑄𝑖𝑛𝑡,𝑖+ 𝑄𝑒𝑥𝑡,𝑖)
𝑡 (7.3) The parameter ϑ can be adjusted along with the FDM mesh size and time-step to yield various finite-difference approximations with different local truncation errors.
Two different approximations have been successfully implemented:
ϑ = 0, the so-called forward-explicit approximation,
ϑ = 1, the so-called backward-implicit approximation.
Figure 7.10 - Explicit and implicit derivatives, [7]
136 In both cases, if f is the number of the selected boundary nodes, the number of equations to be solved is p-f. The system of equations is typically written as:
𝑻
𝑘+1= 𝑨 ∙ 𝑻
𝑘(7.4) where A is a p x p matrix, T is the column-vector for the p nodal temperatures and k is a progressive index to count the number of time-steps. For thermal models with a large number of nodes, A is typically a sparse matrix because each node is normally connected to a small subset of the total number of nodes in the model.
7.2.1 Forward-explicit method
The system of equations is solved by means of the explicit Euler’s method, which is the simplest of the Runge-Kutta methods for obtaining numerical solutions of differential equations, [40]. The solution at next time tk+1 is expressed by:
𝑻
𝑘+1= 𝑻
𝑘+ ∆𝑡 ∙ 𝑭(𝑡
𝑘, 𝑻
𝑘)
(7.5) where F is a 1 x p vectorial function obtained after manipulation of the p equations in the form of Eq. (6.2).The main advantage is that it is a one-step solver, in other words it only requires the value of Tk to compute the solution Tk+1. In fact, each equation is independent from the others, so it has only one unknown. Despite the simplicity, this method is the least accurate if compared to other fancier methods running at equivalent step-size, [40]. Furthermore, the main drawback of explicit methods is that Δt is limited by stability criteria, otherwise the solver “explodes”.
Once the solution Tk is evaluated, the maximum Δt for next k-th advancement is related to the concept of nodal thermal time constant τ, which has been already introduced in Chapter 3 for the one-node satellite. In a thermal network, for each node τi is computed as:
𝜏
𝑖=
𝐶𝑀,𝑖∑𝑝𝑗=1𝐶𝑖𝑗+∑𝑝𝑗=1𝐻𝑖𝑗+∑𝑝𝑗=1𝐿𝑅𝑖𝑗
(7.6)
137 where the term LRij is the so-called “linearized” radiative conductance between nodes i and j, which can be evaluated from the relative radiative contribution in Eq.
(6.2) exploiting the values of Ti and Tj at previous time tk-1:
𝐿𝑅
𝑖𝑗= 𝑅
𝑖𝑗(𝑇
𝑗,𝑘−12+ 𝑇
𝑖,𝑘−12)(𝑇
𝑗,𝑘−1+ 𝑇
𝑖,𝑘−1)
(7.7) The linearization of the radiative conductances is here made only for this purpose.Despite that, references like [20-39] use LRij to obtain a linearized system of equations for the thermal network. Moreover, the convective conductances should be considered only in a pre-launch environment, [7]. In the end, the value of each time-step is given by:
∆𝑡 = 0.95 ∙ min
𝑝
(𝜏
𝑖) = 0.95 ∙ 𝜏
𝑚𝑖𝑛(7.8) Hence, in using this technique the analyst is trading simplicity for potentially many small time-steps, a situation that can cause excessive execution time. In applying the forward-differencing equations, the analyst does not have to specify neither the convergence criteria nor the time-step, since these can be conveniently computed from the specified thermal data, [7].
7.2.2 Backward-implicit method
When dealing with a large number of first-order differential equations, there is the possibility that the system becomes stiff. In thermal analysis, stiffness occurs anytime there are two or more very different scales of t on which the nodal temperatures are changing. Because of the large number of nodes required in these analyses, the thermal problem is stiff by nature, [39] and explicit methods are not suitable even though they work.
The simplest implicit differencing is the one where the second term of the right-hand side of Eq. (7.5) is evaluated at the new instant tk+1 (sometimes referred to as implicit Euler’s method):
𝑻
𝑘+1= 𝑻
𝑘+ ∆𝑡 ∙ 𝑭(𝑡
𝑘+1, 𝑻
𝑘+1)
(7.9)138 This leads to an algebraic system of non-linear dependent equations whose solution requires iterative techniques. The system expressed by Eq. (7.9) can be always rewritten in the form:
𝑮(𝑻
𝑘+1) = 0
(7.10) where G is again an appropriate 1 x p vectorial function and Tk+1 is the unknown vector. Instead of using fsolve command, Eq. (7.10) is solved by means of the Newton-Raphson method extended to vectorial functions and arrested to the first order. It requires the evaluation of G and its Jacobian matrix J at time tk:𝑱
𝑘= 𝑱(𝑻
𝑘) = [𝐽
𝑖𝑗]
𝑘
= (
𝜕𝐺𝑖𝜕𝑇𝑗
)
𝑘
(7.11)
Because of the polynomial nature of Eq. (6.2), the computation of J is an easy hand- made calculus and allows to compute the temperature-vector at next time-step as:
𝑻
𝑘+1= 𝑻
𝑘−
𝑮(𝑻𝑘)𝑱(𝑻𝑘)
(7.12) The great advantage of this method is that it is unconditionally stable for any Δt. Of course, one gives up accuracy in following the evolution towards equilibrium if too large Δt are used. If the selected time-step is five to ten times τmin, it is probably too large, but this judgment depends on the problem being solved. Moreover, the number of iterations at each time-step depends on the accuracy required; the larger Δt, the higher the number of iterations necessary to satisfy the convergent criterion the analyst has to impose, and this translates into higher computational time. For some problems, the implicit scheme may not be any faster than the explicit method, [7].
7.2.3 Comparison between the methods
From the standpoint of the simulations done on the Reference Satellite and other simpler models, the use of both methods has been satisfactory, but on average faster solutions have been obtained with the backward-implicit formulation. In fact, all those nodes belonging to solar panels have so very small τi that τmin is on the order of 10-2 s. Playing with the parameters tol (to set the maximum tolerance
139 between two consecutive iterations) and mult (a number to express the time-step as a fraction of τmin), even complex models can be analyzed with reasonable time in the case ϑ = 1.
To show the differences between the methods proposed, a very simple case is discussed: a 30mm-thick flat plate, made of aluminum alloy and covered with white paint on one side and black paint on the other side, flying on a circular equatorial orbit of h = 500 km.
Figure 7.11 - Flat plate data
Figure 7.12 - Comparison between explicit and implicit methods
140 Plots in the previous figure clearly show that differences are negligible by appropriately choosing tol and mult in applying implicit method. In this case, tol = 0.01 K and mult = 4 allows saving about 30% simulation time.
7.3 Quantitative validation
Throughout the dissertation, there have been attempts to show the validity of the work done by means of tests and comparisons with literature data. The last activity is devoted to prove even a quantitative consistency of the results obtained with the model of TMM proposed. In doing so, the work of C. Scaramelli, [6], a former student of University of Pisa, has been exploited as a benchmark to try and replay its results.
The available data are:
a simplified model of satellite built and analyzed with Astrium’s SYSTEMA software;
the model of satellite realized by Scaramelli to validate ThermoCAS, a software developed during his final thesis at ISAE-SUPAERO, France.
The following table show the available data for SYSTEMA model. It is a cubic satellite covered with white paint and schematized with one volumetric node.
Table 7.1 - Features of SYSTEMA model, [6]
ThermoCAS model is slightly different and more similar to the models developed in this work: it is made of 6 nodes located at the center of the 6 faces of a cube. The table below shows how Scaramelli has chosen data in order to supply to the lack of
141 information about SYSTEMA model and make ThermoCAS model as isothermal as possible.
Table 7.2 - Features of ThermoCAS model, [6]
The model proposed to match as much as possible the previous two is called “Cube”
and is composed of 384 facet-nodes + 1 compartment-node. The reason for such a large number of nodes is that the available FEM mesh scheme is not suitable to put a node exactly at the center of the faces, but building a mesh “ad hoc” allows obtaining nodes at position that are very near to the desired one. Moreover, by comparison with the meshes shown in Chapter 6, dealing with such number of nodes is all but a problem for the duration of simulations.
Figure 7.13 - a) Cube mesh; b) orbit GEO used in quantitative validation
142 The reference orbit is a GEO (h = 36000 km) and the mission starts on 1st January 2000, 00:00. On this day, the Earth is at perihelion, so the Sun direction and the orbital plane form an angle equal to the ecliptic angle ϵ. As depicted above, the satellite’s initial position is ν0 = 0, while the simulation is done along two orbits (2 days).
A comparison between SYSTEMA and ThermoCAS models shows that the average temperature Tmean of the 6 nodes sufficiently approximates the result obtained with SYSTEMA software.
Figure 7.14 - SYSTEMA vs ThermoCAS, (adapted from [6])
Except for the transient region, which is strongly affected by initial condition on T, the maximum difference between the curves is about 2.34 °C and is measured on 1st January, 20:50. Therefore, the maximum error is about 2.3%, [6]. Below is the plot of the temperature history for the 6 nodes of ThermoCAS model, obtained by employing a fixed Δt = 300 s, [6].
143 Figure 7.15 - Nodal temperatures - ThermoCAS model, [6]
The same simulation has been done for the Cube. For each face, 4 nodes are at the same distance from the center and choosing one instead of another actually makes no difference. Below is the plot of temperatures for the 6 “central” nodes of the Cube.
Figure 7.16 – Temperatures of central nodes – Cube
144 Comparing Fig. 7.15 and 7.16, one should notice the difference of scale on the temperature axis: while the ThermoCAS nodal temperatures oscillate around the mean value -100 °C in the range [-95; -105] °C, the temperatures of the central nodes of the Cube oscillate in a more restricted interval [-98; -102] °C. This is due to two factors:
1. a much larger number of nodes has been used in building the Cube’s TMM;
moreover, the mesh is finer at the central points of each face in order to obtain results locally as accurate as possible;
2. setting mult = 3 and tol = 0.01 K in the backward-implicit method for simulating the mission produces a strong reduction of the transient region even setting initial conditions for all the nodes as Ti,0 = 0 °C (the use of the equilibrium solution has been deliberately avoided to make the curves of the two models as similar as possible).
Just after a few minutes, the solution of the Cube converges to the limit-cycle (introduced in Chapter 3), while ThermoCAS model requires about 3 hours before confining the solution in the range seen before. Then, although the nodal temperatures continue in adjusting themselves, the difference between the average value after some minutes and the one at regime conditions (e.g. at the end of simulation) is about 2 °C.
It is instructive to compare also the 6 central nodes of the Cube with the temperature possessed by an hypothetical spherical one-node satellite with the same feature of the Cube. Because of the ecliptic angle, the satellite does not experience eclipse, so its isothermal temperature is constant during the mission. This is represented by the dashed line in the figure below, obtained as a solution of Eq. (3.28) starting from equilibrium condition. It is not surprising that, at regime, the one-node temperature is a sort of mean value of the 6 central nodes.
145 Figure 7.17 - Cube vs one-node satellite
Finally, the differences in temperature history between the one-node satellite and the one-node SYSTEMA model are due to the different shapes considered. In this case, the deviation of SYSTEMA model from a constant value of T is due to the
“irregularities” introduced by the cubic shape. This is also underlined in [6], and since the resulting difference is less than 2 °C, the initial hypothesis of modelling spacecraft components with simple shapes is justified.
The purpose of ThermoCAS model was to obtain a satellite as isothermal as possible. The Cube, reducing the range of T of the central nodes, has improved the results of [6], thus obtaining a “more isothermal” satellite. Actually, this has been possible even thanks to the use of today’s laptops, which are not comparable with those of more than ten years ago.
