Heat Equation in Cylindrical and Spherical Coordinates

Heat Equation in Cylindrical and Spherical Coordinates

In engineering, there are plenty of problems, that cannot be solved in cartesian coordinates. Cylindrical and spherical systems are very common in thermal and especially in power engineering. The heat equation may also be expressed in cylindrical and spherical coordinates. The general heat conduction equation in cylindrical coordinates can be obtained from an energy balance on a volume element in cylindrical coordinates and using the Laplace operator, Δ, in the cylindrical and spherical form.

Cylindrical coordinates:

heat equation - cylindrical coordinates

Spherical coordinates:

heat equation - spherical coordinates

Obtaining analytical solutions to these differential equations requires a knowledge of the solution techniques of partial differential equations, which is beyond the scope of this text. On the other hand, there are many simplifications and assumptions, that can be applied to these equations and that lead to very important results. In the next section we limit our consideration to one-dimensional steady-state cases with constant thermal conductivity, since they result in ordinary differential equations.

 
References:
Heat Transfer:
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  2. Heat and Mass Transfer. Yunus A. Cengel. McGraw-Hill Education, 2011. ISBN: 9780071077866.
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Nuclear and Reactor Physics:

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Advanced Reactor Physics:

  1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
  2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
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  4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See above:

Thermal Conduction