FDPDEType: FDPDE_ENTHALPY

Advection-diffusion(-reaction) equation using the Enthalpy method

The Enthalpy Method is used for modelling the equilibrium thermodynamics of melting of a multi-component system (i.e., Alexiades and Solomon [1993], Katz [2008b]). A general coupled system for conservation of energy (with enthalpy as primary energy variable) and composition, together with pointwise closure equations for temperature, porosity, and fluid/solid compositions provided by the Enthalpy Method:

where the unknowns are $H$ - enthalpy, $T_p$ - primary temperature variable (for earth applications, $T_p$ is potential temperature), and $C$ - bulk composition. Other variables: $T$ - secondary temperature variable, $\phi$ - porosity, and $C_{f,s}$ are liquid and solid compositions. The variables in red are coefficients that need to be defined by the user in FormCoefficient().

Enthalpy method

The number of chemical components $N$ (default is a 2-component system) is specified with

FPDEEnthalpySetNumberComponentsPhaseDiagram(fd,N);

The closure equations using the Enthalpy method are specified with

FDPDEEnthalpySetEnthalpyMethod(fd,FormEnthalpy,description,usr);
Form_PotentialTemperature(T,P,&TP,usr);

FormEnthalpy() has a similar structure and idea as FormCoefficient(), but it is cellwise and is supposed to work in the following way: enthalpy method(H,C,P,&T,&phi,CF,CS,ncomp,user);. It has inputs (H,C,P) and outputs (T,phi,CF,CS).

And to update the thermodynamic state, do:

FDPDEEnthalpyUpdateDiagnostics(fd,dm,x,&dmnew,&xnew);

Time-stepping schemes

The following time stepping schemes are implemented

TS_FORWARD_EULER
TS_BACKWARD_EULER
TS_CRANK_NICHOLSON

and are set with

FDPDEEnthalpySetTimeStepSchemeType(fd,timesteptype);

Advection schemes

The following advection schemes are implemented

ADV_UPWIND
ADV_UPWIND2
ADV_UPWIND_MINMOD
ADV_FROMM

and are set with

FDPDEEnthalpySetAdvectSchemeType(fd,advtype);

Please consult the Manual for theory and details on the time-stepping and advection schemes.

See Tests for examples with FDPDE_ENTHALPY

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