Month: February 2017

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)
\begin{equation}
c_p(T) = \sum_{i=0}^7 c_i T^i, \tag{1}
\end{equation}
the temperature in the j-th cell \(T_j\) is calculated from the following equation
\begin{equation}
\displaystyle \int_{T_{std}}^{T_j} \left( \sum_{i=0}^7 c_i T^i \right) dT = h_j, \tag{2}
\end{equation}
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

Study with great curiosity and let’s see the physical phenomena from a HIGHER point of view!

Interesting News – Vanadium Dioxide Violates the Wiedemann-Franz Law


It is an interesting news on conduction heat transfer.

The new article reports that vanadium dioxide conducts electricity much better than it conducts heat at near room temperature:

Abstract

In electrically conductive solids, the Wiedemann-Franz law requires the electronic contribution to thermal conductivity to be proportional to electrical conductivity. Violations of the Wiedemann-Franz law are typically an indication of unconventional quasiparticle dynamics, such as inelastic scattering, or hydrodynamic collective motion of charge carriers, typically pronounced only at cryogenic temperatures. We report an order-of-magnitude breakdown of the Wiedemann-Franz law at high temperatures ranging from 240 to 340 kelvin in metallic vanadium dioxide in the vicinity of its metal-insulator transition. Different from previously established mechanisms, the unusually low electronic thermal conductivity is a signature of the absence of quasiparticles in a strongly correlated electron fluid where heat and charge diffuse independently.

Wiedemann–Franz(-Lorenz) Law

In solids, heat is transported by vibrations of the solid lattice (Phonon contribution) and motion of free electrons (Electronic contribution). In metals, thermal energy transport by electrons predominates. Thus, good electrical conductors are also good thermal conductors as the Wiedemann–Franz law states that:
\begin{equation}
\frac{\lambda}{\sigma} = LT \tag{1}
\end{equation}

  • \(\lambda\): thermal conductivity
  • \(\sigma\): electrical conductivity
  • \(T\): absolute temperature
  • \(L\): Lorenz number (\(=2.45 \times 10^{-8}\)) [\({\rm W}\Omega/{\rm K}^2\)]

The news reports that vanadium dioxide does not obey this empirical law.

Related Topics and Refereces (Japanese)