new methodologies for the thermal modelling of …

SSC12-VIII-5

NEW METHODOLOGIES FOR THE THERMAL MODELLINGOF CUBESATS

Philipp Reiss1

Supervisors: Philipp Hager1, Malcolm Macdonald2, Charlotte Lucking2

One of the main threats for the success of a CubeSat mission is the unbalanced distribution of thermalloads caused by internal and external heat sources. In order to design an appropriate thermal subsystemthat can cope with these loads a detailed analysis is required. However, currently available thermalsoftware is considered as being less convenient for the application with CubeSats, mainly due to thecomplexity of the modelling process. This paper examines thermal engineering issues for CubeSats anddescribes the development of new methodologies to realise a more appropriate thermal modelling andanalysis specifically for such systems. This includes the utilisation of a component database and newapproaches to create thermal couplings in the modelling process. A thermal software tool based onMATLAB R© was developed which implements these approaches to build the thermal model. Furthermore,the development of specific ray-tracing algorithms is presented to compute external radiation on theCubeSat. Application of the software for the thermal analysis of the UKube-1 CubeSat showed that thecomputational and methodological approaches provide good modelling capabilities and result similartemperature distributions as the professional software ESATAN TMS R©.

1 INTRODUCTION

Thermal analysis is one of the crucial elementswithin the design process of a spacecraft as thethermal loads in orbit can easily harm the func-tionality of components and hence the entire sys-tem. Usually huge efforts are taken to simulatethis environment prior to the mission by perform-ing thermal analysis and test of the spacecraft forvarious operating cases. With the growing interestand rapid development in small satellite missionsand particularly CubeSats over the past years [1, 2],their fields of application have been continuouslyextended, ranging from basic missions for com-ponent verification (e.g. DELFI-C3, UWE-1/2/3)to complex scientific research missions (AAUSat-

1Technische Universitat Munchen, Lehrstuhl fur Raumfahrt-technik, Germany

2University of Strathclyde, Advanced Space Concepts Labor-atory, Glasgow, UK

2, SwissCube-1). Following this development, thegeneral need for simulating and analysing the sys-tem of a CubeSat prior to its operation in space hasgained importance. This also applies for the fieldof thermal analysis. Similar to conventional space-craft, CubeSats need to be designed to maintaintemperatures within the operational range. This isachieved by performing thermal analysis to sim-ulate the thermal behaviour of the satellite andpredict temperature distributions for varying op-erational use cases. Such simulation requires amethodology capable of computing the heat fluxeswithin the satellite caused by internal and externalheat sources and allows the engineer to balancethese sources in order to understand and quantifythe time-dependent temperature distribution. Com-monly used software tools for spacecraft thermalanalysis were primarily designed for conventionalspacecraft systems. Due to their complexity theyoften require expert knowledge. CubeSat design-

Reiss 1 26th Annual AIAA/USUConference on Small Satellites

ers typically lack in expertise and budget to pay forsuch professional software and hence an alternat-ive is needed to perform basic thermal analysis ofCubeSats with minimum effort.

To fulfill this task new methodologies specific-ally designed for CubeSats were developed and im-plemented to create a new CubeSat thermal soft-ware. Aiming to simplify thermal analysis for suchsystems, a novel approach to model the satellite wasrealised by providing a database of standardisedCubeSat components to select from. Being espe-cially designed around this approach, the softwareoffers a more convenient alternative to presentlyavailable thermal software with improved usabil-ity. Built in MATLAB [3] environment it offers astandalone solution to create the geometrical math-ematical model (GMM) and the thermal mathemat-ical model (TMM) of the CubeSat. It further con-tains a radiation module that computes external ra-diation on the satellite by using Monte-Carlo ray-tracing methods. The software uses standard or-dinary differential equation (ODE) solvers from theMATLAB library to solve the thermal model. Itincludes a graphical user interface (GUI) to leadtrough the pre-processing to setup the GMM andthe post-processing to evaluate the results. The cap-abilities of the software were tested with a thermalmodel of the CubeSat “United Kingdom UniversalBus Experiment” (UKube-1). A worst hot and coldcase scenario were analysed to compare the resultswith former analyses performed with the thermalsoftware ESATAN TMS [4].

2 THE NEED FOR CUBESAT-SPECIFICTHERMAL SOFTWARE

So far only few CubeSat missions have undergonea detailed thermal analysis and even less are docu-mented and publicly available. This is mainly dueto the insufficient resources of such missions, as al-most always being university projects carried outby students. Some documented examples can befound in [5] and [6]. A very detailed thermal ana-lysis and test of the OUFTI-1 CubeSat is describedby [7].

Most CubeSat designers might not have the re-quired experience in thermal engineering or thespecific knowledge of how to use the appropri-ate software. Commercially available software re-

quires time-consuming training and practice to behandled the right way and to prevent mistakes inthe complex modelling process. Therefore most ofthe thermal analyses are performed with radicallysimplified thermal models or significant approxim-ations for the thermal environment. Even if theknowledge and the skills to handle such analysistools are available, it might be hard to create an ac-curate thermal model of a CubeSat. Standard usageof commercial software rapidly reaches its limitswhen modelling custom-shaped CubeSat parts withthe available primitive shells. In such cases a com-mon but rather complicated solution is to use othertools in combination with the thermal analysis soft-ware to create a custom-shaped discretisation pat-tern which can be imported as geometrical model.However, this effort most likely exceeds the avail-able resources for a CubeSat project.

CubeSats are highly modular spacecraft, oftenconsisting of similar components and limited tocertain dimensions and materials. The most com-mon CubeSat sizes range from one unit (1U) tothree units (3U), whereas one unit is defined as be-ing 100.0 ± 0.1mm wide and 113.5 ± 0.1mm tall(340.5 ± 0.3mm for 3U) [8]. The mass is limitedto 1.33kg for 1U and 4.0kg for 3U CubeSats. Theirinternal components usually are electrical devicesmounted on circuit boards, most often these aredesigned according to the PC/104 standard. Sev-eral CubeSats also contain deployables, such assolar panels or antennas, which exceed the pre-viously mentioned standard dimensions when de-ployed. The most crucial impact on the temperat-ures of a CubeSat is caused by the external radi-ation from Sun and Earth, as well as internal heatdissipation through electronic devices. For Cube-Sats in low Earth orbit, external radiation has typ-ical values in in the range of 1, 400Wm−2 for solar,450Wm−2 for albedo (maximum value, averageover orbit is 150Wm−2) and 200Wm−2 for in-frared radiation from Earth [1]. Internal heat loadsdepend on the type and efficiency of on-board elec-tronics, whereas the electrical orbit average powergenerated by an exemplary 1U (3U) CubeSat is inthe range of 5W (20W ) with peak values reachingup to 9W (40W ) [9].

It can be questioned to what extend it is neces-sary and reasonable to conduct a thermal analysisof CubeSats at all. They usually do not carry any


highly sensitive payloads and their projected oper-ational lifetime in most cases is lower than for con-ventional satellites. Additionally CubeSat missionsusually accept higher risk levels in general. How-ever, due to the limited design variation of Cube-Sats, it is of particular importance to invest into thethermal analysis and control and to allow a most ef-ficient utilisation of its limited resources. CubeSatsoften contain highly integrated electronics, whichraises the risk of overheating as a significant part oftheir electrical power is dissipated as heat. CubeSatcomponents in most cases use commercial off-the-shelf parts that are not designed specifically for ap-plication within the space environment. This leadsto a rather small temperature range in which theycan operate and subsequently to more stringent re-quirements on the thermal control of the CubeSat.Typical operational temperatures for CubeSat com-ponents are in the range of −40... + 80◦C [2],whereas the more sensitive components are the on-board electronics with common temperature rangesof −10... + 40◦C [1]. The most delicate compon-ents usually are batteries with operational temper-ature ranges of −5... + 15◦C (some batteries canalso operate at higher temperatures), optical instru-ments and individual payloads. Furthermore, theCubeSat surface in most cases is entirely coveredwith solar cells to collect power and to re-chargethe on-board battery. This leads to a lack of pass-ive and active thermal control capabilities, as thesurface properties can only be modified in a smallrange and attaching paint or radiators is often notpossible. Since CubeSats are launched into lowEarth orbits, they are subject to a large number ofthermal cycles during their lifetime and hence fre-quently changing thermal stress is to be expected.Only a few CubeSats have an active and reliable at-titude control system, with most of them movingin a passively controlled way. In this case a priorestimation of thermal loads is only possible in alimited way, e.g. by simulating worst case scen-arios, and an active thermal control during opera-tion through altering the attitude is not applicable.

Concluding these problems, it can be said thatthermal control is indeed an important topic cru-cial for the success of any CubeSat mission. Tocope with partly undeterminable and highly vari-able thermal loads within a very limited range ofdesign and using mostly non-optimised compon-

ents is a challenging task for any thermal engin-eer. A CubeSat which fulfills these requirementscan only be realised if its thermal behaviour canbe analysed in advance. Thermal analysis helpsto identify weak and sensible spots in the CubeSatdesign, to estimate the risk of failure and eventuallyto optimise its design for increased reliability. Soft-ware specifically dedicated to the thermal analysisof CubeSats does not currently exist. Taking intoaccount the special requirements and limitationsof CubeSat missions, while avoiding the afore-mentioned drawbacks of standard thermal software,such specific software could heavily reduce themodelling effort. The previously mentioned lim-itations lead to a certain similarity in the designof present CubeSats and even to the emergence ofcompanies that specialise in the manufacturing ofready-made CubeSat components. A thermal ana-lysis software tool that allows the designer to createthe thermal model of a CubeSat modularly usingsuch frequently used components would signific-antly reduce the complexity of modelling. Ratherthan creating a new model each time a new Cube-Sat is designed, which so far is common practice,the modelling could be done once in a very detailedlevel and then simply be reused. This approachcould also be extended to create libraries of com-ponents and assemblies to be provided for futureanalyses.

3 COMPUTATIONAL APPROACH OF THECUBESAT THERMAL SOFTWARE

Any common thermal software is capable of gener-ating a geometrical and thermal model of the sys-tem to be analysed and eventually must provide thecapabilities to solve it. This includes the possibilityto compute conductive and radiative heat exchangefactors of the system and its environment. Fluidflow and hence convection is not considered in theCubeSat thermal software as these systems usuallydo not carry tanks or pipes with liquids.

Whereas internal thermal couplings can be calcu-lated by using the material properties of the model,the external heat loads depend on the orbit geo-metry and constellation of the spacecraft and thesurrounding heat sources, i.e. Earth and Sun. Thisinvolves the computation of view factors, which of-ten leads to an increased mathematical complexity.


Geometry, material properties, thermal couplings,internal heat loads

Create TMM(Generate GL,GR)

Calculate external heat loads SolveTMM

Nodal temperatures

Orbit and planetaryparameters

Solver parameters

Figure 1: Overall computational approach for the Cube-Sat thermal software

Figure 1 visualises the main steps to generate nodaltemperatures of a thermal model starting with thedefinition of geometry, material and boundary con-ditions. The methodologies and mathematical ap-proaches developed to realise the above mentionedcapabilities of the thermal software are presented inthe following.

3.1 TMM Generation

The thermal network is a mathematical descriptionof the heat transfer between the nodes of the dis-cretised thermal model. The general heat balancingequation for a system of nodes is given by the ex-pression in Eq. (1) [10, 11],

CdT

dt= GL · ∆T +GR · ∆T 4 + Qint (1)

where C [JK−1] is the capacity of the node,∆T [K] is the temperature variation due to heat in-and output, and Qint [W ] is the internal heat load.The factors GL and GR summarise the conductiveand radiative heat exchange factors between the re-spective node and its environment. The conductiveheat exchange factor is defined by the conductiv-ity k [Wm−1K−1], the cross sectional area A [m2]and the distance x [m] of the heat transfer as de-scribed by Eq. (2). The radiative heat exchangefactor is the product of the Stefan-Boltzmann con-stant σ [Wm−2K−4], the emissivity ε [−], the ab-sorptivity α [−] and the view factor F [−] as givenin Eq. (3) [10, 11].

GL = k · Ax

(2)

GR = σεαAF (3)

In Eq. (1), all variables have to be regarded asmatrices since according to Eq. (2-3) they containthe physical and geometrical properties of all nodesof the system. TheGLmatrix is symmetric and dueto the nature of the thermal model the entries onthe secondary diagonal above and below the maindiagonal are equal. Furthermore, it was found thatthe entries on the main diagonal are the negativesum of the entries on the respective row or column.Both conclusions lead to the convention describedby Eq. (4-5) for forming theGLmatrix for a modelwith n nodes:

GLij = GLji (4)

GLii = (−1) ·n∑

m=1

GLm,i

= (−1) ·n∑

m=1

GLi,m

(5)

For the GR matrix this is somewhat differentdue to the directivity of the radiative heat exchangefactor. Again the entries below and above the maindiagonal are mirrored, but the indices of the GRfactors need to be swapped. Furthermore the entrieson the main diagonal can be formed by the sum ofthe respective column:

GRii = (−1) ·n∑

m=1

GRm,i (6)

If a conductive connection between two nodes iand j is specified as active, the respective GLij canbe calculated using the material properties of bothnodes, according to the convention for series con-nection [10, 11]:

GLij = GLji =

(xi

ki ·Ai+

xjkj ·Aj

)−1(7)

If there is a radiative connection between twonodes, the respective GRij and GRji is calculatedby the following equation, according to the conven-tion introduced in Eq. (3):


GRij = σεiαjAiFi→j (8)

GRji = σεjαiAjFj→i (9)

After filling all entries in theGL orGR matricesto represent any given active thermal connection,they have to be completed by filling the main di-agonal entries. This is done by applying Eq. (5-6). Some applications might require the definitionof nodes with constant temperature, e.g. for thedefinition of the deep-space node. In this case theentries on the row representing this node are simplychanged to zero in both the GL and GR matrix toobtain dTi/dt = 0. Also for the deep-space nodein general, a modification has to be implemented.Due to its infinite emitting area, theGR from deep-space to any other node would be infinite. Hence,if one of the nodes involved is deep-space, the GRfrom this node is set to zero. For the GR to thisnode, a view factor of 1 and a formal absorptivityof 1 are used.

3.2 Thermal Network Solver

As it can be seen from Eq. (1), the heat balancingequation is an explicit nonlinear ordinary differen-tial equation (ODE) of first order. Such can eitherbe solved analytically or numerically, depending onthe type of problem and the desired computation ef-fort. Due to the complexity of the described thermalproblem it is not practical to search for an analyt-ical solution, so that numerical solvers have to beapplied. Furthermore the heat transfer equation isconsidered as being a stiff equation, based on thedefinitions of the term stiff described in [12] and[13]. Concerning the solution of such stiff equa-tions, several problems have to be considered. Themain issue is the common trade-off between accur-acy and numerical stability and computational ef-fort. Another issue when choosing small step-sizesis the impact of numerical round-off errors, whichlead to less accurate results.

MATLAB offers a set of different ODE-solversfor almost every purpose. It also includes thestiff equations solvers ode15s, ode23s, ode23t andode23tb. According to [14], ode15s presents a goodalternative to the commonly used non-stiff solverode45 with a low to medium accuracy. It is amultistep variable order solver based on numer-ical differentiation formulas and optionally uses the

usually less efficient backward differentiation for-mulas (BDFs). Alternatively ode23s can be used,which is a one-step solver with fixed order and con-sidered to be more effective than ode15s when al-lowing high tolerances [15]. The solver ode23t usesthe trapezoidal rule (or Heun’s method) for solvingmoderately stiff ODEs without numerical damping.Another approach is realised with ode23tb, whichuses a combination of the trapezoidal rule andthe BDF of second order (TR-BDF2). Similar toode23s, it can be more efficient than ode15s whileaccepting higher tolerances [14]. For the applica-tion in the thermal software, the solver ode15s wasinitially chosen as it offers the best combination ofaccuracy and speed among these solvers [14]. Al-ternatively the solver ode23t could be implementedwhich is assumed to provide improved speed whileaccepting lower accuracy. However, the solver per-formances were also compared with respect to thecomputational effort required to solve an exemplarythermal model (satellite model with 12 nodes, in-cluding internal heat load, conductive and radiativecouplings). While choosing a relative error toler-ance of 10−6 the fastest solver was ode23tb, fol-lowed by ode23t and ode15s. The latter turned outto be approximately 1.8 times slower than ode23tbfor this test case, nevertheless its speed was con-sidered as acceptable. Furthermore the comparisonproved that non-stiff solvers are not appropriate forsolving the heat balancing equation. Several testruns turned out that ode45 needed about 690 timeslonger than the fastest solver ode23tb.

3.3 Radiation Module

For the calculation of external heat fluxes the solarand planetary radiation received by the satelliteneed to be determined. This mainly depends onthe orientation of the satellite surface towards thoseheat sources, thus the view factors between bothare required. Analytical determination of viewfactors is possible in a limited range only and re-quires a comparably high computational effort. An-other means to find them is by using ray-tracingalgorithms, as commonly done to visualise com-plex geometry and light effects in computer graph-ics. This methodology can also be used to simulatethermal radiation being emitted by a surface and re-ceived by another. The mathematical approach is to


describe the path of the ray as a line and check theintersection with objects along the line. The viewfactor Fi→j can then be found by dividing the num-ber of rays fired from the emitting surface i whichhit the receiving surface j by the total number ofrays fired from surface i, as shown in Eq. (10)[16, 17]:

Fi→j =No of rays fired from surface i which hit j

No of rays fired from surface i(10)

The accurate view factor estimation is heav-ily influenced by the computational effort behindit, which again is dependent on the distributionof starting points and the direction of rays [16].Following the stochastic approach of Monte-Carlomethods, both parameters distribution and direc-tion of rays can be chosen randomly to achieve themost realistic results. In comparison with analyt-ical methods or numerical integration, Monte-Carloray-tracing offers a flexible alternative whose ap-plication is independent from the actual complex-ity of the three-dimensional geometry. While ana-lytical computation requires the knowledge of anexact solution to the specific geometry, numericalintegration has the drawback of high computationaldemands [16].

In the CubeSat thermal software the concept ofMonte-Carlo ray-tracing was mainly applied forthe determination of view factors between satel-lite and Earth. The latter was approximated as aperfect sphere, so that the view factor was foundby applying the mathematical approach to findthe intersection between a line (ray) and a sphere(Earth). Equalising both vector equations for lineand sphere results a quadratic equation with twosolutions. Eq. (11) depicts a simplified version ofthis equation, where ~sr is the start and ~dr the direc-tion of the ray, ~cs is the centre and rs the radius ofthe sphere [18]:

ir,s = − ~dr · (~sr − ~cs)

±[r2s + (~d2r − 1)(~sr − ~cs)

2] 1

2(11)

If there is any real solution for ir,s, both objectsintersect. The view factor between one surface ofthe satellite and Earth is therefore found by firinga certain number of random rays from the surface,

checking the intersection by solving Eq. (11) foreach ray and then divide the number of intersec-tions by the total number of rays fired according toEq. (10).

The view factor between satellite and Sun wasdetermined by calculating the angle of incidence ofsunrays and deriving the surface area perpendicularto the ray [11]. The view factor can be expressed asthe ratio between perpendicular and original area,so that it equals the cosine of the angle of incidenceα as shown in Eq. (12):

Fi→j = cosα (12)

Besides determining the view factors, the radi-ation module also detects if the satellite is in ec-lipse or on the dayside of the Earth. This is done bychecking if the vector between satellite and Sun in-tersects the Earth. Naturally this method results twointersection points, one on each side of the planet.Hence the distance between satellite and intersec-tion points is compared in order to find the closerintersection. If the satellite is close to the far sideintersection, it is in eclipse. Here the Earth wasapproximated as a sphere, using the mean radiusenlarged by 2% (6, 505, 700m), as advised by [19]and [20] to consider atmospheric effects which leadto an enlargement of the Earth shadow.

Other than for the Earth infrared radiation, theintensity of albedo radiation is not distributeduniformly over the sphere of Earth. While itsmaximum is at the sub-solar point, albedo de-creases continuously towards the terminator whereit reaches zero. This was adopted in the radi-ation module by overlaying the constant maximumintensity Malbedo,max with a cosine function ofthe angle β between satellite and sub-solar point,roughly following an approach to model solar radi-ation intensity described in [21]:

Malbedo = Malbedo,max · cosβ (13)

Radiation inside the CubeSat was calculatedwith an approximation based on the heat exchangebetween the internal circuit boards. Most Cube-Sats consist of a number of circuit boards stackedtogether parallel to each other and only few mod-els differ from this setup. The internal configur-ation of a CubeSat therefore can be simplified asseveral parallel plates with certain spacing. If the


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Figure 2: View factor for parallel square plates(18x18mm) and varying vertical and horizontaldisplacement

circuit boards or any other components with sim-ilar shape would be discretised with a single node,the heat exchange would only depend on the ver-tical spacing of both nodes or plates respectively.If it contained more nodes, the horizontal displace-ment between the nodes of each component wouldalso be important as the radiation from one nodeaffects all opposite nodes in the respective field ofview. To determine which nodes are affected, theview factor has to be determined depending on thevertical and horizontal displacement between emit-ting and receiving node. As the exact determinationof this requires the geometrical configuration of themodel and hence a high computational effort, a fur-ther approximation was developed. To estimate theview factors in a common CubeSat configuration, asensitivity analysis for the view factor calculationof two parallel finite plates was done. Both platessimulate nodes of two respectively arranged com-ponents. The aim was to determine the range ofinfluence of the radiation emitted by one node andreceived by nodes located opposite to it.

An analytical approach to calculate the viewfactor between two parallel rectangular plates isgiven in [22]. This approach was applied to twoparallel square plates (nodes) with an edge lengthof 18mm and varying vertical and horizontal dis-placement. The dimensions of the node werechosen assuming that most of the standard PC/104-sized PCBs (approximately 90x95mm) would be

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Figure 3: Radiation range for parallel square plates(18x18mm) and varying vertical (z) and hori-zontal displacement

discretised with 5x5 nodes. The resulting viewfactor is depicted in Figure 2. It can be seen thatit rapidly decreases with growing displacement inany direction, while for the horizontal displace-ment it falls more rapidly. Figure 3 depicts the sig-nificant impact of horizontal displacement on theview factor. It can be seen that if the displacementequals the node length, the view factor is approx-imately 0.1, mostly independent from the verticalspacing. The main radiation thus is received by thedirectly opposing node. For a vertical spacing of5mm between the PCBs (including their electron-ics assembly), the maximum view factor reachedat the opposing node is 0.6. As CubeSat compon-ents are usually very narrowly stacked, this valuewas assumed to cover most of the applications. In-dependently from the actual geometry of the PCBsurface, the standard view factor between PCBs forthe use in the CubeSat thermal software was there-fore defined to be 0.6.

4 DEFINITION OF DATABASECOMPONENTS

The discretisation of components aims to repres-ent a most detailed model of the real geometrywith a minimum number of nodes. This clearlyposes a trade-off between accuracy and design ef-fort and hence compromises have to be accepted.While in most cases the CubeSat components fol-low the design of primitive forms such as plates,


rods or cuboids, some parts have a more complexgeometry. These mainly are connections holdingdrillings and screws, as well as electrical assem-blies on circuit boards. However, the geometry ofsuch regions does not need to be modelled if therepresenting node(s) contain the same physical andthermal properties of the original geometry, suchas volume, density, heat capacity and conductiv-ity. Even mixed materials can be modelled in away that the nodal properties are put together withthe same ratio of materials being involved. Thisrefers to the linearity of the conductive heat trans-fer as described in Eq. (2). A connector consist-ing of copper contacts housed by a plastic casingcould therefore be modelled with a mixture of bothmaterials, each contributing its properties in the re-spective parts.

With the component database approach, the geo-metrical model can be assembled by selecting therequired components and connecting them at therespective connection spots. Each component canbe designed and discretised independently from theothers, according to the respective needs. To as-semble these components it requires the assignmentof potential connectors, which is done by using aparticular type of nodes. The discretisation withthermal nodes therefore distinguishes four differentnodal types: diffusive nodes, conductive and radi-ative connector nodes, and boundary nodes. Dif-fusive nodes were defined as the standard type withfinite thermal mass (capacitance) [10]. Conductiveand radiative connectors are diffusive nodes whichinteract with adjacent nodes of another compon-ent. Boundary nodes are nodes with infinite thermalmass and hence constant temperature (e.g. deep-space). The conductances between diffusive nodesinside a component are calculated manually and theconductances between connector nodes in betweencomponents are calculated automatically using thegeometrical and material properties of the respect-ive nodes. Using this approach diffusive nodes donot necessarily have to be geometrically defined.They are regarded as non-geometrical nodes withcertain material properties. Calculating the internalcomponent conductances manually brings the ad-vantage of simplified discretisation, as the thermalmodel geometry can be much more abstract thanthe original one. Only the connector nodes mustbe geometrically defined, given properties such as

the cross sectional area and distance for the con-ductive heat exchange or the emitting and absorb-ing area for the radiative heat exchange. Accordingto the methodology described by Eq. (4-9), the heatexchange factors are then automatically generatedfrom the nodal properties, so that the user of thethermal software only has to assign the connectionby selecting the respective connector nodes of twocomponents.

5 VERIFICATION OF THE RADIATIONMODULE

The results from the external radiation analysiswere compared to ESATAN TMS radiative ana-lyses to determine the level of detail. In both soft-ware the approach for computing the external ra-diative exchange factors is numerical and based onMonte-Carlo ray-tracing. While the standard con-figuration in ESATAN is to fire 10,000 rays fromeach surface, the ray-tracing algorithm of the MAT-LAB software by default uses 500 rays. This stillensures moderate computation time in the range of120 . . . 180s for one orbit and a satellite with sixsurfaces while offering a reasonably high level ofdetail. The comparison between ESATAN TMSand MATLAB ray-tracing routines was based on asatellite placed in a circular orbit with 97.79◦ in-clination, 650, 000m altitude, a RAAN of 10◦ andan argument of periapsis of 45◦ (UKube-1 orbit).The satellite was given different attitudes, beingeither Sun- or Earth-oriented.

Figure 4 shows the resulting radiation receivedby the satellite surface which points directly to-wards Sun. It can be seen that the higher num-ber of rays utilised by ESATAN leads to a muchmore continuous curve. The deviation of the ray-tracing approach realised in MATLAB is smallerfor solar radiation and larger for radiation fromEarth. Whereas direct solar radiation only deviatesless than 1Wm−2, the albedo and infrared radiationhas a maximum error of approximately 12%. How-ever, this maximum only occurs when the respect-ive satellite surface is pointing frontally towardsEarth. The higher the surface is tilted against Earth,the smaller is the error. As all environmental para-meters were equal in the comparison, this error iscaused by the view factor calculation and requiresfurther investigation. The results also showed that


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Figure 4: Comparison of external radiation on a test or-bit for the Sun-pointing surface, calculated withESATAN TMS and the CubeSat thermal soft-ware (MATLAB)

solar and albedo radiation in the MATLAB modelhave a longer eclipse duration than in ESATAN.This refers to the previously described enlargementof the Earth shadow, which apparently is not in-cluded in ESATAN TMS by default. It can furtherbe seen that the approach to model the transitionfrom full albedo radiation to eclipse matches theESATAN results almost perfectly. Only the verylast part when the spacecraft is over the terminatorseems to be more continuous for the ESATAN res-ults.

6 THERMAL ANALYSIS OF UKUBE-1

In order to test the capabilities of the thermal soft-ware and to basically verify the computational ap-proach, a thermal analysis of UKube-1 (UnitedKingdom universal bus experiment), a CubeSatcurrently under development by ClydeSpace andseveral UK universities, was performed. The res-ults were compared against former thermal ana-lyses done with the thermal software ESATANTMS. Following the special database approach ofthe software, all components of UKube-1 weremodelled first independently and then assembledfor the analysis within the software GUI. The Cube-Sat was built using 15 different models of electron-ics mounted on circuit boards, a model of the stand-ard 3U skeletonised chassis, two custom-made and

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Figure 5: Discretised model of the 3U skeletonisedCubeSat chassis with connecting nodes (or-ange) and diffusive nodes (yellow)

four standard, partly deployable, solar panels, twoantenna systems and auxiliary clips and standoffsfor fixation purpose. The discretisation was doneaccording to the needs of the respective geometryso that the entire UKube-1 model eventually con-sisted of 1560 nodes. Figure 5 exemplarily showsone of the modelled and discretised components.Where applicable the material and thermo-opticalproperties of the respective geometry were used.Regarding parts with mixed materials these prop-erties were approximated using the ratio of eachmaterial involved. For the thermal conductivity ofthe printed circuit boards, the directivity was con-sidered, so that following an approach described in[23], an in-plane- and through-conductivity wereapplied.

Two operational scenarios were selected to beanalysed, providing a worst hot and a worst coldcase. The worst hot case was defined with the mostdemanding operational power mode of the payloadand the satellite pointing with its largest surface to-wards Sun. The worst cold case was defined withthe least possible operational power mode and thesmallest surface being exposed to solar radiation.

Figure 6 shows an example output of the soft-ware, providing the minimum and maximum tem-peratures of all nodes for the worst cold case scen-ario. As expected there is a temperature gradientin z-direction, as the positive z-side of the satellite


points towards Sun. The distribution of heat loadsinside the PCB assembly leads from the centre to-wards the ends in positive and negative z-direction,as the largest heat load is carried by the centrallylocated AMAC payload (Figure 7-8). The temper-ature distribution within the PCB stack is quite dis-crete, with larger steps between the single PCBs.This is due to their poor through-conductivity invertical direction. The chassis temperatures liebetween the extreme values of the solar panels andthe PCB assembly, whereas its highest temperat-ures again are located at the top end, in positivez-direction.

The results were further compared to a formerESATAN analysis, done by the StrathSEDS stu-dent group at University of Strathclyde, which usedanother more rough but yet sufficiently compar-able thermal model. The temperatures were provento show the same distribution, although regardingthe payload PCBs being generally higher for bothworst cold and hot case scenario (Figure 7-8). Thiswas due to several differences in the design of thethermal model, whereas three main factors wereidentified which influence the difference in abso-lute PCB temperatures: Material properties, radi-ative connections and conductive connections tothe periphery. Regarding the material propertiesit was found that small changes, e.g. adding alayer of copper to the circuit board, can signific-antly decrease the resulting temperatures. This isdue to the improved in-plane conductivity, as cop-per has an unequally higher conductivity than FR4(kCu = 394Wm−1K−1 compared to kFR4 =0.3Wm−1K−1). Figure 7-8 show the temperat-ure range for the PCBs for the initial analyses us-ing circuit boards with two copper layers and anadditional analysis case where a third copper layerwas added. The resulting temperature differencebetween both configurations in average was in therange of 4.8...5.2K for the worst cold case scenarioand 9.0...9.8K for the worst hot case.

Another driving factor for PCB temperatures arethe radiative exchange factors to the environment.Narrowly stacked PCBs radiate a majority of theirheat towards the neighbouring PCB, whereas morespace in between both would allow more radiationtowards the chassis. This would in most cases bemore efficient, as the chassis can dissipate hightemperature potentials more easily by either radi-

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Figure 7: Temperature ranges for the PCBs of UKube-1with two (a) and three (b) 35µm thick copperlayers in the circuit board material, as calcu-lated with the CubeSat thermal software (MAT-LAB) for a worst cold case scenario

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Figure 8: Temperature ranges for the PCBs of UKube-1with two (a) and three (b) 35µm thick copperlayers in the circuit board material, as calcu-lated with the CubeSat thermal software (MAT-LAB) for a worst hot case scenario

ating directly to space or conducting the heat to thesolar panels. Large cut outs in the chassis furtherallow transferring heat directly from PCBs to thesolar panels, which is even more efficient. Besidesthe conductivity of the PCB material, the number ofconductive connections to its periphery is also cru-


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Figure 6: Example output of the CubeSat thermal software for the analysis of UKube-1, showing the minimum andmaximum temperatures of all nodes for the worst cold case scenario

cial. Especially the size and composition of elec-trical connectors between PCBs or the use of cableswhich are linked to solar panels or other compon-ents attached to the CubeSat surface can have a sig-nificant influence on the PCB temperature. Otherattachments such as spacers are usually less effi-cient in terms of conductivity since they are mostoften not connected to the conductive layer of thePCB.

7 CONCLUSION

As shown in this paper there is currently no thermalsoftware available which is specialised on the ap-plication for CubeSats. The methodologies and ap-proaches described in the present work provide anew way of realising such. Their implementationin a MATLAB-based software lead to the evolve-ment of a new thermal software which is special-ised on the requirements of CubeSats. It considerssimplifications and approximations which are ap-plicable for such missions to significantly reducethe complexity of the thermal analysis and its pre-and post-processing. The novel approach to build

geometrical models using a database of compon-ents proved to be applicable and convenient withthe applied thermal model of UKube-1 and hencepresents a promising solution for the future ap-plication with other CubeSats. The non-restrictiveway of choosing discretisation meshes and definingnodes leaves space for creative modelling whichcan be adapted to the special needs of each com-ponent. Its modularity further allows the simpleuse of parameterised models as it completely un-couples the design of database components fromthe actual thermal analysis process. As shown forthe case of UKube-1 the combination of a stand-ard ODE solver and a radiation module that usesray-tracing methods offers a reliable framework toperform thermal analysis. Comparisons with pro-fessional thermal software have verified the basicconcepts and proven their potential for future ap-plication on CubeSat missions.

References

[1] Volodymyr Baturkin. Micro-satellites thermal con-trol - concepts and components. Acta Astronautica,56:161–170, 2005.


[2] J. Rotteveel and A.R. Bonnema. Thermal controlissues for nano- and picosatellites. Proceedings ofthe International Astronautical Congress, IAC-05-B5.6.07, 2005.

[3] MathWorks. MATLAB v7.13 R2011b, 2011.

[4] ITP Engines UK Ltd. ESATAN Thermal ModellingSuite r3, 2011.

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[10] David G. Gilmore, editor. Spacecraft Thermal Con-trol Handbook, volume I: Fundamental Technolo-gies. American Institute of Aeronautics & Astro-nautics, Inc., 2 edition, 2002.

[11] E. Messerschmidt and S. Fasoulas. Raum-fahrtsysteme. Springer, 2 edition, 2005.

[12] E.J. Kostelich and D. Armbruster. IntroductoryDifferential Equations: from linearity to chaos.Addison-Wesley Publ. Co., prelim. edition, 1981.

[13] W.L. Miranker. Numerical Methods for StiffEquations and Singular Perturbation Problems.Springer, 1 edition, 2001.

[14] MathWorks. MATLAB Product Documentation,r2011b edition, 2011.

[15] Lawrence F. Shampine and Mark W. Reichelt. TheMATLAB ODE Suite. SIAM Journal on ScientificComputing, 18(1):1–22, 01 1997.

[16] T. Walker, S.-C. Xue, and G.W. Barton. Numericaldetermination of radiative view factors using raytracing. Journal of Heat Transfer, 132:072702(1–6), 2010.

[17] P. Vueghs, H.P. de Koning, O. Pin, and P. Beckers.Use of geometry in finite element thermal radiationcombined with ray tracing. Journal of Computa-tional and Applied Mathematics, 234:2319–2326,2010.

[18] A.S. Glassner. An Introduction to Ray Tracing.Morgan Kaufmann Publishers Inc., 8 edition, 2000.

[19] National Aeronautics and Space Administration.NASA eclipse website. http://eclipse.gsfc.nasa.gov/LEcat5/shadow.html[Accessed 12/2011], 12 2011.

[20] Paul Marmet and Christine Couture. Enlarge-ment of the earth’s shadow on the moon: An op-tical illusion. http://www.newtonphysics.on.ca/astronomy/index.html [Accessed12/2011], 12 2011.

[21] Colin R. McInnes. Solar Sailing: Technology, Dy-namics and Mission Applications. Springer, 1 edi-tion, 1999.

[22] U. Gross, K. Spindler, and E. Hahne. Shapefactor-equations for radiation heat transfer between planerectangular surfaces of arbitrary position and sizewith parallel boundaries. Letters in Heat and MassTransfer, 8(3):219–227, 1981.

[23] Electronics Cooling Editors. Conduction heattransfer in a printed circuit board. Electronics Cool-ing, 1998.


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