Homework answers / question archive / This may play a signi?cant role in the observed thermal behavior of the two components

This may play a signi?cant role in the observed thermal behavior of the two components

Electrical Engineering

Share With

This may play a signi?cant role in the observed thermal behavior of the two components. The numerical solution method The lumped mass heat transfer equations for the solar array and satellite body form a system of two coupled ?rst-order ordinary differential equations (ODEs). These are to be solved numer- ically over a period of 15 orbits (tm,m = 15tLEO) using a 3rd order Runge-Kutta scheme. For the general 1st order ODE g} = f(t, y), the solution at time tn“ (ynH) is computed from the solution at time t,, (ya) as follows: 19' +4}: +16 yn+1 = 3}?! + %i (3) where k1 = hf(tn>yn)1 k2 = hfan + h/Zayn + Isl/2)) k3 = hfan +hayn _ k1 +2k2)- Here the time step h = tn+1 — tn = th/N, where N is the number of steps. Assignment Tasks Preliminary analysis Task 1. Using the information given in the previous sections, write down coupled ODEs for the solar array temperature Ta and satellite body temperature T3. These should be of the form: dTa dTb E = f?(taTme)a E = fb(t,Ta,?). When these equations are solved numerically, it will be convenient to put them in vector form: where a" = [11,11]?" and fie, a") = {320,113,}, ?,(t, Ta, To]? Computer implementation Task 2. Write a Python function to determine the irradiance encountered by the satellite as a function of time. Using a step size of h = 5 min, plot the output of this function from t = 0 to tmx. 'Ihsk 3. Write a Python function Tdot(t,T) that outputs the right-hand—side of the ODE system from Task 1. The inputs should be the current time and temperature vector T4 = [Tm Tb]T. The output should be a numpy.array f-‘(Li? = [fa(t,Ta,?), ?,(t, Tn, Tb)]T. Make use of the irradiance function developed in Task 2 within this function. What is the output of the function at t = D where if"; = [300, 30011"?

smoothly between I In,“ full and zero over a brief time. All of this behaviour can be modelled by the function 1 _ 21rt Ht) = Iqu_ [1 + tanh (2051n(—)):| . (l) 2 two It is this variation in radiative heating that prevents the satellite temperature from stabilizing at a constant value. Wb:2 m Ha=2 in solar irradiance [irradiate surfaces. are red) Figure 2: Geometry of satellite lumped model Immediately following deployment of the satellite in LEO (t = 0), the temperature of both the satellite body and the satellite are To = 300 K. Assume that t = 0 corresponds to the end of the eclipse period of the ?rst orbit. Minor radiative heating due radiation from the Earth is to be neglected, as it radiation to and from the coupling struts. The aim of this assignment is to use a numerical solution procedure to estimate the temper- ature history of the main satellite components, and from this compute the temperature extremes the components experience. The Mathematical Model The general lumped-mass heat transfer equation is 551‘ - mCE — EQ (2) where Q are heating rates in W. There are several heating rates that need to be included in this problem: Heating due to solar irradiance Q,” = eI(t)A,:,.,.m;aM, radiative heat loss to space QM; = —eaT‘1A,,,,.fm, the energy dissipated as heat by internal components in the satellite body QWem; = 3 kW, and heat conduction between the body and array Ta _Tb Q'cmdttdion = ikcAc LC where the — sign applies when dem,m is used in the lumped mass heat transfer equation for the solar array. In the radiative heat loss term, 115me is the total surface area of the component being analyzed, 5 is the radiative emissivity (which is equal to the absorptivity) and or = 5.67 x 10-8 Wi'(m2K4) is the Stefan-Boltzmann constant. The emissivities for the solar array and satellite body are 6,, = 0.9 and q, = 0.2 , respectively. These values are so different A

The University of Queensland School of Mechanical and Mining Engineering MECH2700 Engineering Analysis I (2020) Assignment: Computational Heat Transfer Introduction Thermal management of satellites such as that shown in Figure 1 can be very challenging: When in sunlight they are exposed to the Sun’s unmitigated irradiation that can drive rapid heating, but when in Earth’s shadow, they rapidly radiate heat to deep space. Thus there is potential for system failures due to temperatures both above and below the operating limits. The overall thermal balance of a satellite is a result of the interplay between incoming radiative heating (primarily from the sun), the heat radiated to space, internal heating from the satellite systems, conduction between components and the thermal mass of the system. In this assignment, you will investigate the temperature histories of the body and solar array of a satellite operating in low Earth orbit (LEO) over a number of orbits after deployment. This will allow thermal threats to system performance to be evaluated. Figure 1: Earth observation satellite (source: spacenews.com). Problem Details To assess the overall temperatures of the satellite body and solar array, you will conduct a transient (time-varying) lumped-mass thermal analysis. In a such an analysis, all components within the solar array (satellite body) are lumped into one mass ma = 100 kg (rmJ = 1000 kg) with a uniform temperature Ta (Tb). The average heat capacity of the array and body are Ca, = 753 J/(kgK) and 0;, = 897 Jl(kg.K), respectively. The body and array are coupled together by two aluminum struts as indicted in Figure 2, which shows the model geometry. The coupling struts each have cross-sectional area A; = 0.0025 m2, length Lc = 0.1 m and conductivity kc = 205 W/(mK). The dimensions of the body and array are shown in Figure 2. The thickness of the solar array is negligible. The satellite to be analyzed is in a low Earth orbit with a period of $1.50 = 100 min. For approximately half of this orbit, the Sun is eclipsed by the Earth and no solar radiation is incident on the satellite. For the remainder of each orbital period, one face of the satellite body and array surfaces are subject to the full solar irradiance of Ifuu = 1356.1 meg. As the Sun is gradually eclipsed by the Earth, the irradiance I(t) varies