Heat Conduction in a Cylinder

Abstract

The equation of heat conduction in cylindrical coordinates (r, θ, z) is expressed as follows:

$$\rho C\frac{{\partial T}}{{\partial t}} = \frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\lambda \frac{{\partial T}}{{\partial r}}} \right) + \frac{1}{{{r^2}}} \frac{\partial }{{\partial \theta }}\left( {\lambda \frac{{\partial T}}{{\partial \theta }}} \right) + \frac{\partial }{{\partial z}}\left( {\lambda \frac{{\partial T}}{{\partial z}}} \right)$$

(5.1)

The case where temperature depends only on r, z and t is considered. There is thus a symmetry of revolution about the Oz axis. The first equation becomes

$$\rho C\frac{{\partial T}}{{\partial t}} = \frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\lambda \frac{{\partial T}}{{\partial r}}} \right) + \frac{\partial }{{\partial z}}\left( {\lambda \frac{{\partial T}}{{\partial z}}} \right)$$

(5.2)

where r is the radial abscissa.