The tool FT-08 computes and plots steady state temperature and heat flux in a hollow circular cylinder that is subject to a pointwise heat source and three types of boundary conditions.
In addition, a tutorial on how to use the tool in computation is provided and a subject review on the governing equations and analytical solutions for the heat conduction problem is presented.
Example: Heat condution in a hollow cylinder
The input in the tool is
After clicking "Run", the results will be shown in another page.
The analytical solutions are also shown as follows.
Analytical Expression of Temperature Distribution-Steady State
If ri ≤ r < rg, T(r) = 15.73168015 * ln(0.5*r) + (0);
If rg ≤ x ≤ ro, T(r) = -8.26831985 * ln(0.5*r) + (9.73116259)
Analytical Expression of Heat Flux Distribution-Steady State
If ri ≤ r < rg, q(r) = -k*(dT(r)/dr) = -(0.78658401) / r;
If rg ≤ x ≤ ro, q(r) = -k*(dT(r)/dr) = -(-0.41341599) / r
Fig. 1 Heat conduction in a hollow cylinder
For an infinite hollow circular cylinder shown in Fig. 1, the heat conduction with axis-symmetry and at steady state is described by the differential equation
r: Radial coordinate
k: Thermal conductivity
g: A pointwise heat source at location rg
ri: Inner radius
ro: Outer radius
The heat flux of the cylinder is
The pointwise heat source can be expressed as
where δ(r) is the Dirac delta function. The boundary conditions at the inner and outer circumferences are of the following three forms:
Type 1. Prescribed temperature
where Ti and To are prescribed temperatures at the boundaries.
Type 2. Prescribed heat flux
where pi and po are given heat flux at the boundaries, and they are in the positive radial direction.
Type 3. Convective boundary conditions
where hi and ho are the heat transfer coefficients and Ti,∞ and To,∞ are the fluid temperatures at the boundaries.
The solution of the heat equation (1) with the pointwise heat source described by Eq. (3) can be written as
where A and B are constants that are determined by the boundary conditions of the cylinder. The heat flux of the body by Eq. (7) is given by
The temperature and heat flux satisfy the following conditions on temperature continuity and heat flux jump
1. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford University Press, Oxford, 1959.
2. D. W. Hahn and M. N. Ozisik, Heat Conduction, John Wiley & Sons, Inc., 3rd Ed., New Jersey, 2012.