Solve the following problem:
Q: A plane wall (p = 4000 kg/m3, c p = 500 J/kg · K, k = 10 W/m · K) of thickness L = 20 mm initially has a linear, steady-state temperature distribution with boundaries maintained at T1 = 0°C and T2 = 100°C. Suddenly, an electric current is passed through the wall, causing uniform energy generation at a rate q = 2 X 107 W/m3. The boundary conditions T1 and T2 remain fixed.
(a) On T-x coordinates, sketch temperature distributions for the following cases: (i) initial condition
(t < 0);="" (ii)="" steady-state="" conditions="" (t="" ??),="" assuming="" that="" the="" maximum="" temperature="" in="" the="" wall="" exceeds="" t="" 2="" ;="" and="" (iii)="" for="" two="" intermediate="" times.="" label="" all="" important="" features="" of="" the="" distributions.=""
(b)="" for="" the="" system="" of="" three="" nodal="" points="" shown="" schematically="" (1,="" m,="" 2),="" define="" an="" appropriate="" control="" volume="" for="" node="" m="" and,="" identifying="" all="" relevant="" processes="" derive="" the="" corresponding="" finite-difference="" equation="" using="" either="" the="" explicit="" or="" implicit="">
(c) With a time increment of t = 5s, use the finite-difference method to obtain values of Till for the first 45 s of elapsed time. Determine the corresponding heat fluxes at the boundaries that is, q"s: (0, 45 s) and q"x (20 mm, 45 s).
(d) To determine the effect of mesh size, repeat your analysis using grids of 5 and 11 nodal points (it = 5.0 and 2.0 mm, respectively).
