The first law of thermodynamics states that the variations of macroscopic kinetic energy, KΩ, and internal energy, EΩ, of a domain Ω are caused either by the mechanical power of forces applied to the system, Pext, or by exchanged heat rate, Qexch (2.3.53 in Ref. 4):

Mass and momentum balance are needed to complete the description of the system. The mechanical laws, either for solids or fluids, generate the following balance equation between variation of kinetic energy, KΩ, stress power, Pstr, and power of applied forces, Pext (2.3.64 in Ref. 4):

This equation involves quantities of the macroscopic level where the variation of the kinetic energy due to some forces applied to it reflects a sensible displacement. In COMSOL Multiphysics, the Solid Mechanics or Single-Phase Flow interfaces are examples of physics interfaces that simulate the macroscopic level described by Equation 4-10.

Combining Equation 4-9 and Equation 4-10 yields the so-called heat balance equation (2.3.65 in Ref. 4):

This time, the equation involves quantities of the microscopic level (exchanged heat rate, Qexch, and internal energy, EΩ) more concerned with the atomic vibrations and similar microscopic phenomena that are felt as heat. The presence of the stress power, Pstr, in both Equation 4-10 and Equation 4-11 stands for the fact that such power is converted into heat by dissipation. The Heat Transfer interfaces, described in the next sections, simulate the heat exchanges described by Equation 4-11.

In this paragraph, the different terms of Equation 4-11 are more detailed to obtain the localized form of the heat balance equation.

The equations given in the previous paragraph holds for a given macroscopic continuous domain Ω where the internal energy is defined using the specific internal energy (per unit mass), E, as:

In these last relations, ρ is the density, and dv denotes an elementary volume of Ω. Contrary to the constant elementary mass, dm, the elementary volume changes by expansion or contraction of the domain. Recall that the derivation operator d/dt under the integrals is in the material frame (see Time Derivative in the Frames for the Heat Transfer Equations section).

The stress power, derived from the Continuum Mechanics theory, is defined by (2.3.59 in Ref. 4):

where σ is the Cauchy stress tensor and D is the strain rate tensor. The operation “:” is a contraction and can in this case be written on the following form:

Note that in fluid mechanics, the Cauchy stress tensor is divided into a static part for the pressure, p, and a symmetric deviatoric part, τ, as in:

so that Pstr becomes the following sum of pressure-volume work and viscous dissipation:

Finally, the exchanged heat rates, Qexch, account for thermal conduction (see Fourier’s Law at Equation 4-8), radiation and potentially additional heat sources. Joule heating and exothermic chemical reactions are such examples of domain heat source. The different kinds of exchanged heat are summarized by the equality below:

Recall the following notations used above: q for the heat flux by conduction, qr for the heat flux by radiation, Q for additional heat sources, and n for the external normal vector to the boundary ∂Ω.

With all these elements, the heat balance equation (Equation 4-11) becomes:

which leads to the following localized form in the material frame:

or equivalently in the spatial frame:

This verbally means that variations of internal energy in time are balanced by convection of internal energy, thermal conduction, radiation, dissipation of mechanical stress and additional volumetric heat sources. In the next sections, Equation 4-15 will be derived to obtain the heat transfer equations in different media.