Keywords: ccx

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### Heat transfer

In a heat transfer analysis, triggered by the *HEAT TRANSFER procedure card, the temperature is the independent degree of freedom. In essence, the energy equation is solved subject to temperature and flux boundary conditions ([10]). For steady-state calculations it leads to a Laplace-type equation.

The temperature can be defined using the *BOUNDARY card using degree of freedom 11. Flux type boundary conditions can consist of any combination of the following:

1. Concentrated flux, applied to nodes, using the *CFLUX card (degree of freedom 11)
2. Distributed flux, applied to surfaces or volumes, using the *DFLUX card
3. Convective flux defined by a *FILM card. It satisfies the equation

   | (11)

   where is the a flux normal to the surface, is the film coefficient, is the body temperature and is the environment temperature (also called sink temperature). CalculiX can also be used for forced convection calculations, in which the sink temperature is an unknown too. This applied to all kinds of surfaces cooled by fluids or gases.

4. Radiative flux defined by a *RADIATE card. The equation reads

   | (12)

   where is a flux normal to the surface, is the emissivity, is the absolute body temperature (Kelvin) and is the absolute environment temperature (also called sink temperature). The emissivity takes values between 0 and 1. A zero value applied to a body with no absorption nor emission and 100 reflection. A value of 1 applies to a black body. The radiation is assumed to be diffuse (independent of the direction of emission) and gray (independent of the emitted wave length). CalculiX can also be used for cavity radiation, simulation the radiation interaction of several surfaces. In that case, the view factors are calculated, see also [15] for the fundamentals of heat transfer.

For a heat transfer analysis the conductivity coefficients of the material are needed (using the *CONDUCTIVITY card) and for transient calculations the heat capacity (using the *SPECIFIC HEAT card). Furthermore, for radiation boundary conditions the *PHYSICAL CONSTANTS card is needed, specifying absolute zero in the user's temperature scale and the Boltzmann constant.

Notice that a phase transition can be modeled by a local sharp maximum of the specific heat. The energy per unit of mass needed to complete the phase transition satisfies

| (13)

where is the specific heat and is the temperature interval within which the phase transition takes place.

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