Now that we have talked about conduction in 1D, steady-state examples, it is time to talk about 2D, steady-state conduction!
Let's start out with heat diffusion equation.
$$ \frac{\partial}{\partial x}(k \frac{\partial T}{\partial x}) + \frac{\partial}{\partial x}(k \frac{\partial T}{\partial y}) + \frac{\partial}{\partial x}(k \frac{\partial T}{\partial z}) + \dot{q} = \rho C_p \frac{\partial T}{\partial t} $$
Since we are in steady-state, $ \frac{\partial T}{\partial t} = 0 $, so the right term goes away! And since we are working in 2D, we will only need two of the three parts to the temperature gradient. Also taking in account the thermal conductivity is constant, we are left with Eq. (1).
[ 1 ]
$$ \frac{\partial^2 T}{\partial x}^2 + \frac{\partial^2 T}{\partial y}^2 + \frac{\dot{q}}{k} = 0 $$
Now with any second order differential equation, it's going to take some creativity to solve it! There are in fact a few different approaches to solve this:
With each of these methods, they have their own pros and cons. For exact, analytical solutions, they are limited to simple georemtries and boundary conditions. For approximate graphical approaches, you run into delineation of isotherms and heat flow lines and limited value for quantitative considerations. However, you get quick insight into establishing if the problem makes physical sense. Now as for approximate numerical approaches, they are the most useful and adaptable to any level of complexity.
We will talk more about these methods as we go and we will walk through some examples as well!