Question

(10 points) A semi-precious alloy rod of length 25cm is used in laboratory experiments in a varying temperature regime. The metal has the thermal diffu- sivity coefficient1.156 (0.34)2cm2/sec. The temperature at time t 0 at point x, 0 < < 25, is described by the Heat-conduction PDE α2uzz-u, The temperature at the end-points of the rod is maintained to be T1 = 10 and T2 - -6, by using a special thermostatic device. Every morning a tech- nician switches the thermostate on, starting the whole thermal process. The experiments begin when the temperature at the mid-point of the rod reaches a steady-state value v(12.5). The initial temperature distribution of the rod is 17 where v(x) is the steady-state temperature and 17 is a room temper- ature in the morning (al temperatures are in C, not F) (a) 1 pnt. Determine the steady-state temperature function v(r) (b) 1 prnt. Write down the formula for the value of the temperature u(r, t) (c) 1 pnt. Specify the steady-state and transient parts of the solution u(r, t) (d) 1 pnt. Check that u(r,t) tends to the steady-state solution. It meas (e) 1 pnt. Calculate that the Fourier-sin coefficients C. = 68 for n odd and (f) 1 pnt. Check that u(r,t) at the mid-point of the rod is of the form at point r at a given time t (it should have both) that the transient part tends to 0 0 for n even 1.156-(2k-1)2t/625 00 2+68 Σ e. (-1)*+1, which yields an alternating series. (g) 1 pnt. Estimate the time when the temperature at the mid-point differs from its steady-state value by less than 1% of a centigrade. Just replace the series with its first term. You may use a calculator. My calculation yields somewhat above 8 min (h) 1 pnt. Explain how you obtained the formula for u(x, t) by using the separation of variables and the form of the initial condition v(r)17 (i) 1 pnt. One morning the technician finds the rod missing (stolen or mis- placed; anyway, he can't find it, and it's his fault) Can he use an aluminium rod instead, to prepare the equipent for the coming expeiment? Yes No Delete accordingly and provide an explanation G) 1 pt. How long wl take to prepare the same temperature 2 at the middle of the aluminium rod f the itial temperature is of the same form v(r) + 17? You will have to look at the Internet for the thermal diffusivity of

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Answer 1

GerDato x alone aE T ie a fo of Talane rs dt C becomes 2S

25 2 S 25 싯는. 25 hence becomes G25

25 25 2S 2 25 34 25 25 54 [1.cosmT)-3TG-(..呗 2S ワIT