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One Dimensional Conduction
1D heat conduction, as the name suggests, involves the transmission of heat depending on just 1 variable.
The 1D heat equation can be written as follows:
However, if the system is in steady state, and there is no heat generation, qgen, then it can be written as follows:
As there is no change in temperature over time (steady-state), and no heat generation.
Analysis of a Plane Wall
Heat traveling through a plane wall has the linear temperature distribution shown in the diagram below:
Using the initial conditions shown, the heat rate can be derived:
Thermal Resistance
Thermal resistances are analogous to resistors in circuits and so multiple thermal resistances can be added in the same way as resistors. Previously, we found that Rt,cond=L/(kA) and Rt,conv=1/(hA). So when you are calculating the total resistance, these resistances can be summed:
Non-constant thermal conductivity can also be modeled in this way:
Analysis of Various Geometries
A Tube Wall (radial conduction):
With the equations given above it is possible to derive an expression for the heat rate and the conduction resistance. This is done by assuming that the process is at steady state, no generation and that it is axisymmetric. For this you assume a constant k, as well.
A Spherical Shell (radial conduction):
Just as for the tube wall, it is possible to derive an equation giving the heat rate and the conduction resistance inside a spherical shell; also assuming a constant k from the heat generation equation.
Video on Thermal Resistance
Video on Conduction through a Tube Wall
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