FcPh

State Definition

This approach describes the material state in terms of total component flow (\(F\): flow_mol_comp), total specific enthalpy (\(h\): enth_mol) and pressure (\(P\): pressure). As such, there are \(2 + N_{components}\) state variables.

Application

This approach is similar to the FPhx formulation used by many process simulators, with the exception that component flow rates are used in place of total flow and mole fractions. This changes where the bilinear terms (flow rate time mole fractions) appear in the problem structure and may improve robustness in some cases. The use of pressure and enthalpy as state variables avoids the issues related to using temperature and pressure as state variables for single component systems. However, as the user generally does not know the specific enthalpy of their feed streams, this approach requires some method to calculate this for feed streams. This can generally be done by specifying temperature of the feed, and then solving for the specific enthalpy.

This approach suffers from the following limitation which the user should be aware of:

  • If the property package is set up for multiphase flow, an equilibrium calculation is required at the inlet of each unit, as the state definition does not contain information on multiphase flow. This increases the number of complex equilibrium calculations that must be performed, which could be avoided by using a different state definition.

Bounds

The FcPh module supports bounding of the following variables through the state_bounds configuration argument:

  • flow_mol_comp
  • enth_mol
  • pressure
  • temperature

Supplying bounds for temperature is supported as these are often known to greater accuracy than the enthalpy bounds, and specifying these can help the solver find a feasible solution.

Supporting Variables and Constraints

In addition to the state variables, this definition of state creates a number of supporting variables and constraints.

Variables

  • flow_mol_phase (\(F_{mol, p}\))
  • mole_frac_comp (\(x_{j}\))
  • mole_frac_phase_comp (\(x_{p, j}\))
  • temperature (\(T\))
  • phase_frac (\(\psi_p\))

Expressions

An Expression is created for the total flowrate such that \(F = \sum{F_j}\)

Constraints

In all cases, a constraint is created to calculate component mole fractions from the component flow rates.

\[F_j = x_j \times \sum{F_j}\]

Note

If only one component is present in the property package, this is simplified to \(x_j = 1\).

If the property package supports only one phase:

\[F_{mol, p} = F_{mol}\]
\[x_{p, j} = x_{j} \text{ for all }j\]
\[\psi_p = 1\]

If the property package supports only two phases, the Rachford-Rice formulation is used:

\[\sum_p{F_{mol, p}} = F_{mol}\]
\[F_{mol, j} = \sum_p{(F_{mol, p} \times x_{p, j})} \text{ for all }j\]
\[\sum_j{x_{\text{phase 1}, j}} - \sum_j{x_{\text{phase 2}, j}} = 0\]
\[\psi_p \times F_{mol} = F_{mol, p} \text{ for all }p\]

If the property package supports more than two phases, the following general formulation is used:

\[F_{mol, j} = \sum_p{(F_{mol, p} \times x_{p, j})} \text{ for all }j\]
\[\sum_j{x_{p, j}} = 1 \text{ for all }p\]
\[\psi_p \times F_{mol} = F_{mol, p} \text{ for all }p\]

Default Balance Types and Flow Basis

The following defaults are specified for Unit Models using this state definition:

  • Material balances: total component balances
  • Material flow basis: molar flow
  • Energy balances: total enthalpy