Question

11. Consider a thin, infinitely long rectangular plate that is free of heat sources, as shown below. For a thin plate, is negligible, and the temperature is a function of x and y only. The solution for this problem is best obtained by considering scaled temperature (ie. 1-T - To, where To is the absolute temperature at T-0) variables, so that the two edges of the plate have "zero-zero" boundary conditions and the bottom of the plate is maintained at T. The steady state temperature distribution of the plate must satisfy the Laplace heat equation: V?T=0 and the boundary conditions: † = 0 at x = 0 for all values of y † = 0 at x = L for all values of y T = T, at y = 0 for 0 SXSL 1 = 0 at y = 00 for 0 SXSL a. Determine the partial differential equation. b. Apply the method of Separating Variables, where, 1 = x(x)Y().Reduce the PDE into 2 ODEs and report the BCs in terms of f = x(x)Y(). c. Report the General solution of the ODEs and the boundary conditions. Determine functions representing X(X) and Y(y) and leave the integration constants as arbitrary variables. d. Bonus: find the particular solution by applying boundary conditions

Answer 1

Given, *1 0 2x F (o, y) = 0 conditions. D²i = 0 T = T-Tot is function of xxy 227 ay? partial differential equation where, (,0) : & Boundary TEL, y) =0 7 (x,x)=0 homogeneous governing equation and boundary conditions. b) Applying separation variables - x(x) Yly). ²7 d²x d²y ay² dy? of 7 so, 22 axa yeye 38 3 and, ON d2x dx² + X dy² +21=0 225/3x² - 37/ay? Y d²y -03 dy -2² Y dy² both sides function of two independent variable. +2 x 50 12x O Х d x² are d2x dx² and d²Y -x²y =o. dy² two ODGS and, x(0) = 0, X (L) - 0 y Coo) -0 Boundary ar conditions.

e) the ODES- d2x dx² + os solving 2x20. x (02-23) -0, D = tia x = a exp (2x) + b exp (-idx) c, cos Ax) + C sim (2x). X(0)=0 cicos (0) + Casin (0) C + ₂ O we can say (O. as (₂ 70 sin sin (2L)=0 and, X (L) -0 (₂ sir (XL) or Oo sin (ure) = sin(a) or ana ure/ where, nao, 1,2... x. so X C₂ sin sin (ure. 2) and , solutior of 1st ODE d²y +22y. dyz d²Y -X?y=0 dy? y CDP-250. D-12. Y = Cz. exp (ay). Cu exp (ny) 04

at, y=x, Y(0) = 0 . C exp (6) + cu exp(-x) a Cz eep (6) C3=0, You Cu exp(-xy) exp(-xy) = Cheap threy) -solution of 2nd ODE d) solution for F- I. X (+), (4) o Tao (C.Cu) sim (nre 8/) exp (niet) Cn. sim Lure A) expl-nre I). and T = I Tu - I cn sin (ure & (ure & exp (ure & no at, y=0, for all x, + (0, x) = 7, 8 con sin (inte & exp (nie-o) Ti e con sin (mrex/L). 71/ sin (uter/) Cu a constant value.