Equations of state
An equation of state is a constitutive equation which provides a mathematical relationship between thermodynamic state variables for a given material.
With the mass, momentum, and energy equations, it completes the mathematical representation of your fluid model.
The following state variables need to be defined:
- Density, ρ
- Dynamic viscosity, μ
- Specific heat a constant pressure, cp
- Conductivity, k
- Specific enthalpy, h
You define all of these material variables except the specific enthalpy in this software. These material variables can vary with time, temperature, pressure, or they can vary with both temperature and pressure (bivariate properties).
In the case of bivariate properties, and in the absence of the standard state models such as ideal gas, the flow solver reads the user-specified bivariate table, and then performs a bi-linear interpolation with respect to temperature and pressure in order to calculate the value of the state variables.
The flow solver uses the state variables differently depending if your fluid is a gas or a liquid.
- If the gas constant, Rg, is defined, the fluid material is a gas.
- If the gas constant, Rg, is not defined, the fluid material is a liquid.
When your fluid is a liquid, the density ρ and the specific heat cp are assumed constant. The flow solver calculates the specific enthalpy, h, from the energy equation, and uses the following equation to calculate the temperature T:
When your fluid is a gas, an equation of state is used to model the relationship between thermodynamic state variables.
The flow solver supports the following models that you specify in the software when you define the fluid material:
The flow solver calculates the specific enthalpy, h, from the energy equation, and uses the following equation to calculate the temperature T:
where cp is the specific heat at constant pressure of the gas.
Ideal gas law
The ideal gas law equation of state used by the flow solver is given by:
where:
- P is the pressure of the gas.
- T is the temperature of the gas.
- ρ is the density of the gas.
- Rg is the specific gas constant (J/kg K in SI units).
Redlich-Kwong real gas equation of state
The Redlich-Kwong real gas equation of state is given by:
where:
- P is the pressure of the gas.
- T is the temperature of the gas.
- ρ is the density of the gas.
- Vm is the molar volume of the gas.
- R is the universal gas constant (8.314472 J/mol K).
The constants a and b are defined as:
where:
- Pc is the critical pressure of the gas.
- Tc is the critical temperature of the gas.
The Redlich-Kwong equation of state is more realistic than the ideal gas law at high pressure and valid when: