- 2 For Steady Heat Flow The Net Change In The Amount Of The Energy Stored Is Zero Therefore The Sum Of The Net Rate O 1 (272.2 KiB) Viewed 11 times
2. For steady heat flow, the net change in the amount of the energy stored is zero. Therefore, the sum of the net rate of enery flow is zero. That is () + () + ) a at a ar a ar + = 0 (5) ar дх ду ay az дz Equation (5) governs the steady conduction of heat in a solid. Assuming the thermal conductivity to be constant, Eq (5) simplifies to VT=0 (6) which is the Laplace equation. In two dimensional Cartesian coordinates, this equation can be written as = 0 (7) Equation (7) is an elliptic partial differential equation for which the solution T(x, y) is governed by boundary conditions specified at each point on the boundary of the physical domain. Consider a rectangular flat plate of height h = 1.0 m and width w = 1.0 m. The temperature of the plate is held at 100 C on the top edge, whereas the bottom edge is held at 50 C (i.e., Dirichlet conditions on these boundaries). The left and the right boundaries are adiabatic, i.e., at/dx = 0 (Neumann boundary conditions). Solve the Laplace's equation numerically [to obtain T(x,y)] for the given domain and the boundary conditions. Specifically you are required yo perform the following: Tax +Tyy • Obtain a finite difference equation (FDE) using second order central-difference finite dif- ference approximations of the partial derivatives. Assume that the rectangular domain is equally spaced in x and y directions with increments of Ar and Ay, respectively. • Discuss the nature of the algebraic FDE you have obtained and propose a numerical solution technique for the solution of the FDE. • Apply the given boundary conditions and determine the numerical solution of the temper- ature values using a: - 11 x 11 node grid - 21 x 21 node grid - 41 x 41 node grid • For each numerical solution plot lines of constant temperature over the computational domain (use contour plot capability of Matlab) and observe how the grid density affects the numerical solution. • Observing the two-dimensional numerical solution obtained by adiabatic boundary condi- tions (at/dx = 0) on the right and left boundaries, what would be an easier way to solve the same problem? Is modeling the problem as one-dimensional an option? If yes, how does that solution compare to the two-dimensional solution?