## Temperature calculation from energy variables in OpenFOAM

The energy conservation equation is expressed in terms of internal energy $$e$$ or enthalpy $$h$$ in OpenFOAM and the temperature field is calculated from these solution variables. More precisely, we can specify the energy variable in the energy entry in the thermophysicalProperties file and the available options are the followings:

• specific sensible enthalpy $$h {\rm [J/kg]}$$ (sensibleEnthalpy)
• specific sensible internal energy $$e {\rm [J/kg]}$$ (sensibleInternalEnergy)

If we select the hePsiThermo thermophysical model, the temperature field $$T$$ is calculated from the solved energy variable in the following function, where the compressibility $$\psi$$, dynamic viscosity $$\mu$$ and thermal diffusivity $$\alpha$$ are also calculated.

Source Code

The calculation procedure of the temperature field depends on the selected energy variable, so the following THE function accordingly switches the called method.

If we choose the sensible enthalpy as the energy variable, the following THs function is called to calculate the temperature from the sensible enthalpy.

The calculation of the temperature is done iteratively using the Newton-Raphson method.

If the specific heat capacity at constant pressure $$c_p$$ is expressed in the form of temperature polynomial function (hPolynomial)

c_p(T) = \sum_{i=0}^7 c_i T^i, \tag{1}

the temperature in the j-th cell $$T_j$$ is calculated from the following equation

\displaystyle \int_{T_{std}}^{T_j} \left( \sum_{i=0}^7 c_i T^i \right) dT = h_j, \tag{2}

where $$h_j$$ is the sensible enthalpy value in the j-th cell. In general the equation will be nonlinear, the iterative solution technique is implemented.

 Compressible Flow Solvers

The above function calculate() is called by rhoPimpleFoam, rhoSimpleFoam and sonicFoam etc. from the line “thermo.correct()” after solving the energy conservation equation EEqn:

After updating the compressibility field $$\psi$$, the pressure Poisson equation pEqn is constructed and solved.

The density field $$\rho$$ is calculated with updated pressure and compressibility fields.

 Summary
 Energy variable Function used to calculate temperature sensible enthalpy THs absolute enthalpy THa sensible internal energy TEs absolute internal energy TEa

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