MECH2700 - Computational Heat Transfer - The University of Queensland - Engineering Assignment

Internal Code: MAS5473

Engineering Assignment :

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 (MB = 1000 kg) with a uniform temperature Ta (Tb). The average heat capacity of the array and body are Ca = 753 J/(kg.K) and Cb = 897 J/(kg.K), respectively. The body and array are coupled together by two aluminium struts as indicated in Figure 2, which shows the model geometry. The coupling struts each have cross-sectional area Ac = 0.0025 m2 , length Lc = 0.1 m and conductivity kc = 205 W/(m.K). 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 LEO = 100 min. For eclipse = 35 min 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 I = 1366.1 W/m2 It is this variation in radiative heating that prevents the satellite temperature from stabilizing at a constant value. Immediately following deployment of the satellite in LEO (t = 0), the temperature of both the satellite body and the satellite are T0 = 300 K. Assume that t = 0 corresponds to the beginning of the eclipse period of the first orbit. Minor radiative heating due radiation from the Earth is to be neglected, as it radiation for and from the coupling struts. The aim of this assignment is to use a numerical solution procedure to estimate the temperature history of the main satellite components, and from this compute the temperature extremes the components experience. Mathematical Method : The general lumped-mass heat transfer equation is mC dT/dt = ?Q ? 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.irr = I(t)A irradiated, radiative heat loss to space Q ?rad = ??T4Asurface, the heat dissipated by internal components in the satellite body Q ?internal = 3 kW, and heat conduction between the body and array

Q ?conduction = ±kcAc Tb ? Ta/Lc

where the ? sign applies when Q ? conduction is used in the lumped mass heat transfer equation for the solar array. In the radiative heat loss term, A surface is the total surface area of the component being analyzed, is the radiative emissivity (which is equal to the absorptivity) and ? = 5.67 × 10?8 W/(m2K 4 ) is the Stefan-Boltzmann constant. The emissivities for the solar array and satellite body are a = 0.9 and b = 0.2 , respectively. These values are so different due to the solar array needing to absorb a high fraction of radiation to serve its purpose, while the satellite body is assumed to the wrapped in Kapton, a radiative insulator. This may play a significant role in the observed thermal behaviour of the two components. Numerical Solution:  The lumped mass heat transfer equations for the solar array and satellite body form a system of two coupled first-order ordinary differential equations (ODEs). These are to be solved numer-ically over a period of 15 orbits (tmax = 15tLEO) using a 3rd order Runge-Kutta scheme. For the general 1st order ODE y ? = f(t, y), the solution at time tn+1 (yn+1) is computed from the solution at time tn (yn) as follows: yn+1 = yn + k1 + 4k2 + k3/6 where k1 = hf(tn, yn), k2 = hf(tn + h/2, yn + k1/2), k3 = hf(tn + h, yn ? k1 + 2k2). Here the time step h = tn+1 ? tn = tmax/N, where N is the number of steps. Questions:  Task 1 Using the information given in the previous sections, write down coupled ODEs for the solar array temperature Ta and satellite body temperature Tb. These should be of the form: dTa/dt = fa(t, Ta, Tb), dTb/dt = fb(t, Ta, Tb). When these equations are solved numerically, it will be convenient to put them in vector form: dT/dt = ~f(t, T~) where T~ = [Ta, Tb]T and ~f(t, T~) = [fa(t, Ta, Tb), fb(t, Ta, Tb)]T Task 2. Show that the 3rd order Runge-Kutta scheme given in Eq. 2 is at least 2nd order accurate in time (it is 3rd order accurate, but doing the algebra to demonstrate this is the case is optional). To do this, first expand yn+1 as a Taylor series about tn: yn+1 = yn + hy ?(tn) + h 2 2 y ?(tn) + O(h) (3) Now, from the ODE y ?(t) = f(t, y), so y ?(t) = d/dtf(t, y(t)) = ?f(t, y) /?t + ?f(t, y) /?y* ?y /?t ? ft + fyf Using this result in Equation 3 gives yn+1 = yn + hf(tn, yn) + h 2 2 ft(tn, yn) + h 2 2 fy(tn, yn)f(tn, yn) + O(h3 ). (4) The goal now is to show that the approximation of yn+1 given by Equation 2 is equivalent to the above. In order to show this, you must expand k2 and k3 as Taylor series about (tn, yn). Since each k value is already multiplied by h, only the first two terms in the Taylor series are required: k(t, y) = k(tn, yn) + (t ? tn)kt(tn, yn) + (y ? yn)ky(tn, yn) + O(h 3 ). Be careful to expand the k functions sequentially, as the expansion for k2 is required in the k3 expression. Computer implementation Task 3. Write a Python function to determine the irradiance encountered by the satellite as a function of time. This should be zero in the first teclipse of each orbit, and I = 1366.1 W/m2 otherwise. Hint: there are many ways to code up this routine, but the numpy.floor function is useful for determining how many whole orbits have been completed so far: numpy.floor(t/tLEO). Using a step size of h = 5 min, plot the output of this function from t = 0 to tmax. Task 4. 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 T~ = [Ta, Tb]/T . The output should be a numpy.array ~f(t, T~) = [fa(t, Ta, Tb), fb(t, Ta, Tb)]T . Make use of the irradiance function developed in Task 3 within this function. What is the output of the function at t = 0 where T~ 0 = [300, 300]T ? Task 5. Write a Python function rk3step(f,t0,y0,h) that takes a right-hand-side function name, the cur- rent time and solution values, and the step size as inputs and returns the approximate solution at time t0 + h calculated using the 3rd order Runge-Kutta scheme given in Equation 2. Task 6. Write a Python program that uses rk3step to solve the ODE system from Task 1 from t = 0 to tmax = 15tLEO. Initially solve the system using a time step of h = 4 min. • Plot both temperatures over the period of investigation. • Is the duration of the simulation long enough for the component temperatures to tend to repeating periodic behavior during each orbit? • What are the maximum and minimum temperatures for each component over the duration of the investigation? • Explain the physical reasons for the differing behavior between the solar array and the satellite body. Task 7. Run your program with a time step of h = 0.1 s and record the maximum and minimum temperatures. Take these to be a good approximation of the exact values. Now run your program for h values of 4 min, 2 min and 1 min. How do the minimum and maximum temperatures converge towards the near-exact solutions? Determine what time step h is required for the greatest deviation from these values to be less than 0.01%.