12 - Conduction - One Dimensional Heat Conduction Equation

Rectangular (Cartesian) coordinates: \[\frac{\partial^2T}{\partial x^2} + \frac{\partial^2T}{\partial y^2} + \frac{\partial^2T}{\partial z^2} + \frac{\dot{g}}{k} = \frac{1}{\alpha}\frac{\partial T}{\partial t}\] Cylindrical coordinates: \[\frac{1}{r} \frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2T}{\partial \theta^2} + \frac{\partial^2T}{\partial z^2} + \frac{\dot{g}}{k} = \frac{1}{\alpha}\frac{\partial T}{\partial t}\]

The objective of deriving the heat diffusion equation is to determine the temperature distribution within the conducting body.