Answer Happy

3 (CT)-(.)(7) ) + ( + az 1. Three-dimensional unsteady conduction of heat in a solid is governed by the partial differential equation given by a () a ar მ ar a ar k + at (1) ar ar ду ду az where k is the thermal conductivity, p is the density and C is the specific heat of the solid material. When the thermal conductivity, density and specific heat are all constant, Eq. (1) simplifies to T+ = a (T. + Tyy +Tuz) (2) where a = k/(PC) is the thermal diffusivity, T+ = at ſæt, and Tzz = ə?T/8x2. Consider the plate of width L as shown in the figure below. The plate is infinitely long in y and z directions and the problem can be considered one-dimensional in space. Therefore, the governing equation [Eq. (2)] can be further simplified to T, = 0Tz2 (3) The solution to the above Equation is the function T(t, x), which must satisfy an initial condition at time t = 0, T(0, x) = F(2) and a set of boundary conditions at x = 0 and x = L. Plate T(0,t) T(L,t) 0 L Figure 1: One-dimensional heat conduction in a plate. Assume that the plate illustrated in the figure has a thickness L = 1.0 cm and a thermal diffusivity a = 0.01 cm²/s and the initial temperature distribution in the plate is specified by 1 T(0, 2) = (4) tanh(7) tanh [T sin (TT)
where T is measured in degrees centigrade (C). Also assume that the temperatures on the two faces of the plate are held at 0.0 C for all time. That is, T(t,0.0) = T(t, 1.0) = 0.0 = = 2 Your task is to solve the one-dimensional diffusion equation numerically (to obtain T(t, x)) for the given initial and boundary conditions. Specifically you are asked to perform the following a • Obtain a finite difference equation (FDE) using a first order forward difference approxima- tion for the time derivative and a second order central-difference approximation of the space ative for the one-dimensional dif sion equation. Assume that the domain is equally spaced in t and X. Does this discretization result in an Implicit or an Explicit solution technique? = = = • Obtain the numerical solution (T(t, x)) for Ax 0.1 cm and At 0.1 sec. Plot the solutions at t = 0.5, 1.0, 4.0, 10.0 sec. Discuss how the numerical results change. • Repeat computations using Ax 0.1 cm and At 0.5, 1.0 sec and plot results at t= 10.0 sec. Discuss how the selection of timestep affects the accuracy of the numerical results. Explain how increased timestep is affecting the stability of the numerical solution. = = =