Partial Differential Equation Toolbox |

Heat Transfer

The heat equation is a parabolic PDE:

It describes the heat transfer process for plane and axisymmetric cases, and uses the following parameters:

- Density
- Heat capacity C
- Coefficient of heat conduction k
- Heat source Q
- Convective heat transfer coefficient h
- External temperature Text

The term h · (Text - T) is a model of transversal heat transfer from the surroundings, and it may be useful for modeling heat transfer in thin cooling plates etc.

For the steady state case, the elliptic version of the heat equation,

is also available.

The boundary conditions can be of Dirichlet type, where the temperature on the boundary is specified, or of Neumann type where the heat flux, n · (k(T)) is specified. A generalized Neumann boundary condition can also be used. The generalized Neumann boundary condition equation is n · (k(T)) + qT = g, where q is the heat transfer coefficient.

Visualization of the temperature, the temperature gradient, and the heat
flux kT is available. Plot options include isotherms and heat flux vector field plots.

Example

In the following example, a heat transfer problem with differing material parameters is solved.

The problem's 2-D domain consists of a square with an embedded diamond (a square with 45 degrees rotation). The square region consists of a material with coefficient of heat conduction of 10 and a density of 2. The diamond-shaped region contains a uniform heat source of 4, and it has a coefficient of heat conduction of 2 and a density of 1. Both regions have a heat capacity of 0.1.

Using the Graphical User Interface

Start the `pdetool` GUI and select the application mode Heat Transfer. In draw mode, set the x- and y-axis limits to `[-0.5 3.5]` and select the Axis Equal option from the Options menu. The square region has corners in (0,0), (3,0), (3,3), and (0,3). The diamond-shaped region has corners in (1.5,0.5), (2.5,1.5), (1.5,2.5), and (0.5,1.5).

The temperature is kept at 0 on all the outer boundaries, so you do not have to change the default boundary conditions. Move on to define the PDE parameters (make sure that the Heat Transfer application mode is selected) in the PDE mode by double-clicking each of the two regions and enter the PDE parameters. You want to solve the parabolic heat equation, so make sure that the Parabolic option is selected. In the square region, enter a density of 2, a heat capacity of 0.1, and a coefficient of heat conduction of 10. There is no heat source, so set it to 0. In the diamond-shaped region, enter a density of 1, a heat capacity of 0.1, and a coefficient of heat conduction of 2. Enter 4 in the edit field for the heat source. The transversal heat transfer term h · (Text - T) is not used, so set h, the convective heat transfer coefficient, to 0.

Since you are solving a dynamic PDE, you have to define an initial value, and the times at which you want to solve the PDE. Open the Solve Parameters dialog box by selecting Parameters . . . from the Solve menu. The dynamics for this problem is very fast -- the temperature reaches steady state in about 0.1 time units. To capture the interesting part of the dynamics, enter `logspace(-2,-1,10)` as the vector of times at which to solve the heat equation. `logspace(-2,-1,10)` gives 10 logarithmically spaced numbers between 0.01 and 0.1. Set the initial value of the temperature to 0. If the boundary conditions and the initial value differ, the problem formulation contains discontinuities.

Solve the PDE. By default, the temperature distribution at the last time is plotted. The best way to visualize the dynamic behavior of the temperature is to animate the solution. When animating, turn on the Height (3-D plot) option to animate a 3-D plot. Also, select the Plot in x-y grid option. Using a rectangular grid instead of a triangular grid speeds up the animation process significantly.

Other interesting visualizations are made by plotting isothermal lines using a contour plot, and by plotting the heat flux vector field using arrows.

Visualization of the Temperature and the Heat Flux

| Conductive Media DC |  | Diffusion |