Question

Problem 2(30 points) Consider the steady-state temperature distribution in a square plate with dimensions 2 m x 2 m. There is a heat generation of ġ(x.y)=6x [W/my], and the thermal conductivity of k=1[W/(m-°C)]. The temperature on the top boundary is given by a piecewise function, f(x), which is defined below. x(4- x²)+10 0<x<1 | x(4- x?) + 20, 1<x<2 The bottom boundary is insulated. The temperatures on left-handed and right-handed boundary are maintained at constants 10[°C] and 20 [°C] as shown in Figure 1, respectively. T=f(x) [°C] T=20[°C] T=10[°C] insulated Figure 1 a. In order to find the steady-state temperature distribution, T(x,y), in the plate, specify the partial differential equation and the boundary conditions that define the problem. b. Find the analytical expression of T(x,y). (Note: to receive the full points, all key steps must be shown up.)

Answer 1

pqi nor1 Solution : The tempecatuces on left handed and right handed boundary Constants 100c] to 20{c] T = -ath taxa Cat where, x=0 , TET, tot Tata > At T=Ti. when Tel 2= Gext + Gixx a = 12 [AC=12]

pq not a a steady state temperature disterbution and o The boundary Condations are id G=12 & Co 20 The dobberential ean are Analytical expression at (x,y) T= -9,4 + 12x