Question

a) Develop a finite difference equation, based on an energy balance for a node on the upper-left corner of a domain, as in the figure below: Too, h Ti Ay k,q Ax Note that both the top and left surfaces are exposed to an ambient temperature of Too and require a convection boundary condition, and that there is constant heat generation per volume (q) throughout the domain. The control volume surrounding the node, on which the energy balance should be enforced, is denoted with a dotted line and has a size of half the nodal spacing in both directions. Do not assume that Δχ-Δι. Your expression should relate the three labeled nodal temperatures (including the corner node with index /), along with parameters k,h, q, Ax, Ay, and Too. Please scale all terms in your equation so that the coefficient multiplying the value TE is 1 b) The difference equation you wrote in part (a) is one in a system of equations, which can be solved by writing and solving a matrix equation. If I is the index of the corner node, what is the All (or A(1, Г)) entry in the corresponding matrix equation?

Answer 1

Too, h Assuming unit thickness TE Ti hax Ta-Ti) ay 2. Similarly 92# hay (Toh) 2 Ts 2 24y Similarly kay.(Te-Ti) a3 = 24χ Applying Energy Balance to CV {for steady state? Ein-Eout-Esen = 0 taz t a.ta4 )-0+94% ay 1:0 h 4 △Y (Too-T) + kay 2. 2 24χ 24y "(讐 4 _ (h4χ th4y + kay + kax).ax.Tj Ts 4x.dL A KAy 2K4Y 2K

Nodal Equation (b) In the matrix equation AUI,I) Coefficient of T is the nodal equation of 1th node ie For our hode I in (a) part AU.) coefficient of Ti in nodal equation found in part (4)