OpenFOAM: sensible enthalpy/internal energy for a reactive case

mkhm · August 15, 2022, 04:17

Hi foamers,

I have a question regarding the calculation of the sensible enthalpy/interal energy.
I first start by explaining what I expect to see in the algorithem based on thermodynamic text books and after, I 'll compare it with the codes in OpenFoam.

For an inviscid/adiabatique reactive case exchanging no work with the environment, we have:
$h^{0}_{\rm f,mixture} + c_{p,\text{mixture}} (T-T_{\rm ref})+\frac{u^{2}}{2} = \text{constant}$

with $h^{0}_{\rm f,mixture} = \sum h^{0}_{f,i} y_{i}$ where $h^{0}_{f,i}$ and $y_{i}$ are, respectively, the enthalpy of formation of species i and the mass fraction of species i. In the same way, $c_{p,\text{mixture}}$ can be calculated;

$c_{p,\text{mixture}} = \sum c_{p,i} y_{i}$ However, as the nature of species change in presence of chemical reactions, for enthalpy (the same applies for internal energy)
we HAVE to take a reference. The standard reference is $T_{\rm ref}$ = 25°C and $p_{\rm ref}$ = 1 atm. As such h, is calculated based on a reference value:

$h(T,p) = h^{0}_{f}(T_{\rm ref},p_{\rm ref}) + c_{p} (T-T_{\rm ref})$

Now, my question is how this reference temperature is considered in OpenFOAM (we know that the sensible enthalpy/interal energy do not change a lot with pressure). I cannot find any track of this reference temperature in OpenFOAM.

For instance, if we take a look at temperature calculation based on Newton’s method in src/thermophysicalModels/specie/thermo/thermo/thermoI.H, we have something as*:

$T_{\rm new} = T_{\rm old} - [ hs(p_{\rm old},T_{\rm old}) - hs_{\rm fromTransportEqn}]/ c_{p}(p_{\rm old},T_{\rm old})$

However, we know that:
$hs_{\rm new}(p_{\rm new},T_{\rm new}) = c_{p,{\rm mixture}}(p_{\rm new},T_{\rm new}) (T_{\rm new} - T_{\rm ref})$
$hs_{\rm old}(p_{\rm old},T_{\rm old}) = c_{p,{\rm mixture}}(p_{\rm old},T_{\rm old}) (T_{\rm old} - T_{\rm ref})$

Therefore:
$hs_{\rm new}(p_{\rm new},T_{\rm new}) - hs_{\rm old}(p_{\rm old},T_{\rm old})$
$= c_{p,{\rm mixture}}(p_{\rm new},T_{\rm new}) (T_{\rm new} - T_{\rm ref})$
$- c_{p,{\rm mixture}}(p_{\rm old},T_{\rm old}) (T_{\rm old} - T_{\rm ref})$

We know that $c_{p,{\rm mixture}}(p_{\rm new},T_{\rm new})$ is different from $c_{p,{\rm mixture}}(p_{\rm old},T_{\rm old})$ and we can NOT have the following simplification:

$hs_{\rm new}(p_{\rm new},T_{\rm new}) - hs_{\rm old}(p_{\rm old},T_{\rm old}) = c_{p,\text{mixture}}(p_{\rm old},T_{\rm old}) (T_{\rm new} - T_{\rm old})$

to deduce:

$T_{\rm new} = T_{\rm old} + [hs_{\rm new}(p_{\rm new},T_{\rm new}) - hs_{\rm old}(p_{\rm old},T_{\rm old})]/c_{p,{\rm mixture}}(p_{\rm old},T_{\rm old})$

I would be grateful if someone could explain that to me. Am I missing a step here?

P.S.: I read carefully http://caefn.com/openfoam/temperature-calculation and tried to search a lot on different forums. But, I could not find my answer there.

* that expression is coming from Temperature calculation from sensible internal energy .

August 15, 2022, 04:17	OpenFOAM: sensible enthalpy/internal energy for a reactive case	#1
mkhm Member K Join Date: Jul 2017 Posts: 97 Rep Power: 9	Hi foamers, I have a question regarding the calculation of the sensible enthalpy/interal energy. I first start by explaining what I expect to see in the algorithem based on thermodynamic text books and after, I 'll compare it with the codes in OpenFoam. For an inviscid/adiabatique reactive case exchanging no work with the environment, we have: $h^{0}_{\rm f,mixture} + c_{p,\text{mixture}} (T-T_{\rm ref})+\frac{u^{2}}{2} = \text{constant}$ with $h^{0}_{\rm f,mixture} = \sum h^{0}_{f,i} y_{i}$ where $h^{0}_{f,i}$ and $y_{i}$ are, respectively, the enthalpy of formation of species i and the mass fraction of species i. In the same way, $c_{p,\text{mixture}}$ can be calculated; $c_{p,\text{mixture}} = \sum c_{p,i} y_{i}$ However, as the nature of species change in presence of chemical reactions, for enthalpy (the same applies for internal energy) we HAVE to take a reference. The standard reference is $T_{\rm ref}$ = 25°C and $p_{\rm ref}$ = 1 atm. As such h, is calculated based on a reference value: $h(T,p) = h^{0}_{f}(T_{\rm ref},p_{\rm ref}) + c_{p} (T-T_{\rm ref})$ Now, my question is how this reference temperature is considered in OpenFOAM (we know that the sensible enthalpy/interal energy do not change a lot with pressure). I cannot find any track of this reference temperature in OpenFOAM. For instance, if we take a look at temperature calculation based on Newton’s method in src/thermophysicalModels/specie/thermo/thermo/thermoI.H, we have something as: $T_{\rm new} = T_{\rm old} - [ hs(p_{\rm old},T_{\rm old}) - hs_{\rm fromTransportEqn}]/ c_{p}(p_{\rm old},T_{\rm old})$ However, we know that: $hs_{\rm new}(p_{\rm new},T_{\rm new}) = c_{p,{\rm mixture}}(p_{\rm new},T_{\rm new}) (T_{\rm new} - T_{\rm ref})$ $hs_{\rm old}(p_{\rm old},T_{\rm old}) = c_{p,{\rm mixture}}(p_{\rm old},T_{\rm old}) (T_{\rm old} - T_{\rm ref})$ Therefore: $hs_{\rm new}(p_{\rm new},T_{\rm new}) - hs_{\rm old}(p_{\rm old},T_{\rm old})$ $= c_{p,{\rm mixture}}(p_{\rm new},T_{\rm new}) (T_{\rm new} - T_{\rm ref})$ $- c_{p,{\rm mixture}}(p_{\rm old},T_{\rm old}) (T_{\rm old} - T_{\rm ref})$ We know that $c_{p,{\rm mixture}}(p_{\rm new},T_{\rm new})$ is different from $c_{p,{\rm mixture}}(p_{\rm old},T_{\rm old})$ and we can NOT* have the following simplification: $hs_{\rm new}(p_{\rm new},T_{\rm new}) - hs_{\rm old}(p_{\rm old},T_{\rm old}) = c_{p,\text{mixture}}(p_{\rm old},T_{\rm old}) (T_{\rm new} - T_{\rm old})$ to deduce: $T_{\rm new} = T_{\rm old} + [hs_{\rm new}(p_{\rm new},T_{\rm new}) - hs_{\rm old}(p_{\rm old},T_{\rm old})]/c_{p,{\rm mixture}}(p_{\rm old},T_{\rm old})$ I would be grateful if someone could explain that to me. Am I missing a step here? P.S.: I read carefully http://caefn.com/openfoam/temperature-calculation and tried to search a lot on different forums. But, I could not find my answer there. * that expression is coming from Temperature calculation from sensible internal energy . Last edited by mkhm; August 15, 2022 at 07:05.

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