Two-species relax2ation towards equilibrium: "relax2"
Reaction model
The reactant is consumed at constant density.
At any instant of time, the products will not necessarily be at the same
density as the reactant because of their different compressibility.
The mass reacted, its energy and the change in volume fraction
are taken into account in calculating the change in
product density and specific internal energy.
Equilibration method
The equilibration processes for pressure and temperature are applied
in operator-split fashion (i.e. at constant total volume) over a finite
time interval.
A number of substeps is specified; the time interval is divided by this
and each equilibration process applied in turn over the substep interval.
Pressure
The pressure difference between the phases is calculated.
The bulk sound speed (dp/drho at constant entropy) is used to
estimate the change in volume fraction necessary to give pressure
equilibrium.
The actual change in volume fraction applied is obtained assuming
an exponential approach to equilibrium, with some time constant tau.
The fraction of the estimated change for equilibrium is given by
f = 1 - exp(-dt / tau)
where dt is the time interval.
The specific internal energy of each phase is also adjusted in a forward-time
way by applying the p.dv work.
This adjustment is made conservative by using the (volume-averaged)
mean pressure and density in the cell.
If complete equilibration is desired, the equilibration time scale
should be set very small (not zero though).
A safety factor is applied to the change in volume fraction for
numerical stability.
If this is fs say, the number of substeps should be
at least some small integer (like 5 or 10) times 1/fs.
Temperature
The temperature difference between the phases is calculated.
The specific heat capacity constant volume is used to
estimate the heat transfer necessary to give thermal
equilibrium.
The actual change in specific internal energy applied is obtained assuming
an exponential approach to equilibrium, with some time constant tau.
The fraction of the estimated change for equilibrium is given by
f = 1 - exp(-dt / tau)
where dt is the time interval.
If complete equilibration is desired, the equilibration time scale
should be set very small (not zero though).
A safety factor is applied to the change in volume fraction for
numerical stability.
If this is fs say, the number of substeps should be
at least some small integer (like 5 or 10) times 1/fs.
Input
EOS 0
EOS 1
tau_mech tau_therm # time scales for mechanical and thermal equilibration
p_tol t_tol # "negligible" difference in p and T between phases
substeps # number of times to cycle equilibration process
sfac # relax2ation factor
Notes
- For EOS types, see the EOS documentation.
- If p and T are within the specified tolerances, the equilibration
step is omitted.
(This is for speed.)