Thermodynamic Description of Heat Transfer

In continuum mechanics, a domain Ω is submitted to variations of its kinetic energy due to some external forces according to an equation of motion. The study of such phenomena is covered by solid mechanics and fluid mechanics and the theories behind can be found in the Structural Mechanics Module User’s Guide and CFD Module User’s Guide. From an energy point of view, the aforementioned description is incomplete because it does not include heat as another form of energy transfer due to microscopic vibration and interactions of particles. The laws of thermodynamics introduce several concepts to define heat transfer consistently with mechanical energy. In the next paragraphs, a concise presentation of the theory adapted to the use of COMSOL Multiphysics is given. More materials and details are provided in the references listed in the References section.

A homogeneous fluid taking place in a domain Ω is characterized by the knowledge of three extensive parameters:

The internal energy, EΩ (SI unit: J), is an extensive state function of these three variables. It measures the amount of energy in the system excluding kinetic energy and potential energy from external applied forces and is the subject of conservation laws more detailed in The Heat Balance Equation section. To fit with the finite element method solved by COMSOL Multiphysics, specific quantities per unit mass are preferred:

The specific internal energy, E (SI unit: J/kg), is then a function of specific entropy, S, and specific volume, ν, related to EΩ by:

For a solid, the specific internal energy, E(S, F), is a function of entropy and deformation gradient, F.

Internal energy is related to the enthalpy, H, via the following for a fluid:

or the following for a solid (7.33 in Ref. 1):

Compared to the internal energy, the enthalpy also includes the pressure-volume potential energy, p ⁄ ρ, necessary for instance in volume expansion after an isobaric transformation.

The variations of internal energy correspond to variations of entropy and volume according to:

First-order parameters are partial derivatives of the specific internal energy. They correspond to the thermodynamic definitions of temperature and pressure:

These lead to the fundamental thermodynamic relation:

	Temperature is the measurable quantity that gives a phenomenological description of heat transfer. When expressed in kelvin (K), the Second Law of Thermodynamics ensures that T can only take positive values.

Here, the counterpart of the fluid pressure is the first Piola-Kirchhoff stress tensor, P.

Second order parameters correspond to second partial derivatives of the specific internal energy and provide a various number of thermodynamic coefficients. These are usually given as material properties of the domain material. Among them, the heat capacity at constant pressure and the coefficient of thermal expansion are most often provided. For a fluid, these are

and for a solid, the definitions become:

Specific heat capacity at constant pressure is the amount of energy required to raise one unit of mass of a substance by one degree while maintained at constant pressure. This quantity is also commonly referred to as specific heat or specific heat capacity.

The heat capacity at constant pressure and coefficient of thermal expansion are related to the enthalpy, seen as a function of T and p (or P), according to:

The enthalpy can then be retrieved from Cp and αp (or α) by:

where r is the integration vector variable, containing temperature and pressure or stress tensor components:

The starting point, r0, is the value of r at reference conditions, that is, pref (1 atm) and Tref (293.15 K) for a fluid. The ending point, r1, is the solution returned after simulation. In theory any value can be assigned to the enthalpy at reference conditions, Href (Ref. 2), and COMSOL Multiphysics sets it to 0 J/kg by default. The integral in Equation 4-5 is sometimes referred to as the sensible enthalpy (Ref. 2) and is evaluated by numerical integration.