### 4.15. NUMERICAL METHODS FOR STEADY-STATE CONDUCTION IN TWO DIMENSIONS

#### 4.15A. Analytical Equation for Conduction

In Section 4.4 we discussed methods for solving two-dimensional heat-conduction problems using graphical procedures and shape factors. In this section we consider analytical and numerical methods.

The equation for conduction in the x direction is as follows:

Equation 4.15-1

Now we shall derive an equation for steady-state conduction in two directions x and y. Referring to Fig. 4.15-1, a rectangular block Δx by Δy by L is shown. The total heat input to the block is equal to the output:

Equation 4.15-2