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About the Tool

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.

Tutorial

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

Subject Review

Fig. 1 Heat conduction in a hollow cylinder

Governing Equations

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

where

T: Temperature

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.

Solutions

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

References

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.