Stationary Heat Equation
The (time) stationary heat equation is not dependent on time. In other words, it is assumed conditions exist such that:
This condition depends on the time constant and the amount of time passed since boundary conditions have been imposed. Thus, the condition is fulfilled in situations in which the time equilibrium constant is fast enough that the more complex time-dependent heat equation can be approximated by the stationary case. Equivalently, the stationary condition exists for all cases in which enough time has passed that the thermal field u no longer evolves in time.
In the stationary case, a spacial thermal gradient may (or may not) exist, but if it does, it does not change in time. This equation therefore describes the end result in all thermal problems in which a source is switched on (for example, an engine started in an automobile), and enough time has passed for all permanent temperature gradients to establish themselves in space, after which these spacial gradients no longer change in time (as again, with an automobile in which the engine has been running for long enough). The other (trivial) solution is for all spacial temperature gradients to disappear as well, in which case the temperature become uniform in space, as well.
The equation is much simpler and can help to understand better the physics of the materials without focusing on the dynamic of the heat transport process. It is widely used for simple engineering problems assuming there is equilibrium of the temperature fields and heat transport, with time.
Stationary condition:
The stationary heat equation for a volume that contains a heat source (the inhomogeneous case), is the Poisson's equation:
In electrostatics, this is equivalent to the case where the space under consideration contains an electrical charge.
The stationary heat equation without a heat source within the volume (the homogeneous case) is the equation in electrostatics for a volume of free space that does not contain a charge. It is described by Laplace's equation:
where u is the temperature, k is the thermal conductivity and q the heat source density.
Read more about this topic: Heat Equation
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