The heat transfer and diffusion equations are parabolic partial differential equations, . The heat equation describes the heat transfer for plane and axisymmetric cases in a given region over given time:
where ρ is density, C is heat capacity, k is the coefficient of heat conduction, Q is the heat source, h is convective heat transfer coefficient, and Text is external temperature.
The term h(Text – T) is a model of transversal heat transfer from the surroundings. You can use it to model heat transfer in thin cooling plates. You can use Dirichlet boundary conditions specifying the temperature on the boundary, or Neumann boundary conditions specifying heat flux . You also can use generalized Neumann boundary conditions , where q is heat transfer coefficient.
The generic diffusion equation has the same structure as the heat equation:
where c is the concentration of particles, D is the diffusion coefficient and Q is the source density (source per area).
You can use Dirichlet boundary conditions specifying the concentration on the boundary, or Neumann boundary conditions specifying the flux . You also can specify a generalized Neumann condition , where q is transfer coefficient, and g is flux.
|PDE||Solve partial differential equations in 2-D regions|
Perform a heat transfer analysis of a thin plate.
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Solve a heat transfer problem with different material parameters.
Use PDE app to solve a heat equation that describes the diffusion of heat in a metal block with a rectangular cavity.
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Mathematical definition and discussion of the parabolic equation