Heat Conduction Equation

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This blog is intended to summarize the Heat Conduction Equation and its derived forms in different analyses. Prerequisite to this article is basic knowledge in Heat Transfer and therefore explanation of the used symbols in the equations, as well as explanation of formulas is excluded from this writing.

Heat conduction in a medium is a dimensional process, depending on the number of dimensions in which it acts. It also varies with time, however, based on the assumptions of steady state (when the temperature does not change with time), or unsteady state (transient), one can have two approaches.

The following flowchart is intended to help distinguishing the states and to select the best suited case with formulas provided.

Steady: no change with time at any point,

Transient: variation with time and position.

(Lumped system: variation with time but not with position.)

Figure 1. Conduction main flowchart

In the Steady State Conduction the temperature is independent of the time change. One can further distinguish one dimensional heat transfer, where the temperature change varies only in one direction. We can talk about multiple dimensions as two- and three-dimensions, which are calculated in a different manned.

In case of 1D conduction, the function depends only on the x dimension, therefore the partial derivatives are replaced by ordinary derivatives.

General form of Heat Conduction Equation (HCE), in rectangular coordinates is called: Fourier- Kirchoff equation:

The general HCE equation can be derived into different special forms, depending on the assumptions and the used boundary conditions.

Boundary and initial conditions for HCE:

There are 4 main boundary conditions used or HCE, these are:

1. Temperature of the surface (TS) at any time is given,
2. Heat flux (qS) on the boundary at any time is given,
3. Fluid temperature distribution (Tf) around the system and convective heat transfer (h) between the system and the fluid is given:
4. Balance of heat fluxes on both sides of the boundary is given, (applied between 2 solids of different thermal conductivity)
   
   

Figure 2. Steady State Conduction flowchart

Figure 3. Transient Conduction flowchart

Figure 4. Transient Conduction computation flowchart

References:

·         Yunus A. Cengel – Heat Transfer – A practical approach, second ed., 2003

·         Heat Transfer lecture notes – Maciej Jaworski, Warsaw, 2016