In the previous article, we looked at the basic concepts of heat transfer and heat conduction. Taking the discuss forward, in this article, we will study the Heat Conduction in a cylindrical body. The most common case of heat conduction in a cylinder is that of heat transfer through a pipe with a fluid flowing inside. This system can be shown as per the figure below, by radial heat flow through a cylindrical shell.
In this analysis, we will find out the temperature distribution and the heat transfer rate in a long hollow cylinder of length L, given the inner and outer surface temperatures are and , respectively and there is no heat generation. The temperature at the boundaries is not a function of time, so the conduction equation in this case will take the form:
Integrate the eqn (1) once with respect to radius gives us,
A second integration of the above equation will give us,
Now let us apply the boundary conditons to find out the constants of integration:
Similarly for ,
The temperature distribution can be written in a dimensionless form as,
The rate of heat transfer by conduction through the cylinder of length L is given by,
In terms of thermal resistance, we can write,
Now the value of thermal resistance in this case will be,
The principles for a plane wall with conduction and convection in series can also be applied to a long hollow cylinder like a pipe or a tube. Check out the figure below to understand a thermal circuit of such nature.
Now if we follow equation (4) and find out an expression for the rate of heat flow, we will get the following expression.