Consider a large plane wall of thickness L = .3m, thermalconductivity k=2.5W/m C

Question

Consider a large plane wall of thickness L = .3m, thermalconductivity k=2.5W/m C, and surface area A=12m2. Theleft side of the wall at x=0 is subjected to a net heat fluxq0=700 W/m2 while the temperature at thatsurface is measured to be T1= 80 C. Assuming constantthermal conductivity and no heat generation in the wall, (a)express the differential equation and the boundary conditions forsteady one-dimensional heat conduction through the wall, (b) obtaina relation for the variation of temperature in the wall by solvingthe differential equation, and (c) evaluate the temperature of theright surface of the wall at x=L. The answer for (c) is -4C. I need help on setting up the diff.equation.

Explanation / Answer

a) -d(k*dT/dx)/dx = -k*d2T/dx2 =0              (Recall that cp*dT/dt = -div(Flux) + generation. There is nogeneration and T is invariant with time for a steady state problem=>    0 = -div(Flux) + 0 =>     -div(Flux) = 0 Flux = -k*dT/dx by Fourier's Law     =>    -div(-k*dT/dx) = 0 =k*d2T/dx2 ) -k*dT/dx = 700 W/m2    at x = 0 T = 80C                        at x = 0 b) -k*d2T/dx2 = 0 d2T/dx2 = 0 dT/dx = C1 T = C1*x + C2 Apply T = 80C, at x = 0 80 = C1*0 + C2 C2 = 80 => T = C1*x + 80 Apply -k*dT/dx = 700 -k*dT/dx = -k*C1 =>   C1 = -700/k =-700/2.5 = -280 T = -280*x + 80 At x = 0.3, T -280*0.3+80 = -4 C TA-DA PM me if you need more help with this problem.

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