Equifugacity equations
The equality of chemical potentials has been introduced before you came to this page to be the condition for phase equilibria. Calculations are actually to solve the equality, which in fact consists of N simultaneous equations (N is the number of components in the system).
However, it is more practical to use the residual properties to do the calculations. Particularly when we use an equation of state, we will not use the chemical potentials, but instead the fugacity coefficients, the logarithm of which are the residual chemical potentials:
R.T. ln φi = μi - μiig = μiR
The superscripts "ig" and "R" are for the ideal gas system and residual property, respectively. The ideal gas chemical potential can be written as:
μiig = R.T. ln {xi.P.ƒi(T)}
where xi and ƒi(T) are the mole fraction of component i and some temperature-dependent function, respectively. Therefore substituting this ideal gas chemical potential to the equality of chemical potentials for two-phase equilibrium, say vapor (V) and liquid (L), leads to a new set of equations:
xi(V).P.φi(V) = xi(L).P.φi(L)
fi(V) = fi(L)
which is called the equifugacity equations. There are N simultaneous equations in this set. For vapor phase, the mole fraction is usually written as yi, while xi is usually used for liquid phase. But we will use xi with the phase explicitly indicated in the superscripts.
For special cases, for example, at low pressures or for systems containing similar components, we can simplify the equifugacity equations and end up with easier calculations. Click here to see these special cases.
Fugacity coefficients
While the pressure can be cancelled out in the equifugacity equations, the fugacity coefficients have still to be calculated. They are functions of T, P, and {xi}. To calculate them, we recall that the chemical potentials are actually the partial molar Gibbs energy, so that:
R.T. ln φi = {∂GR/∂ni}T,P,nj≠i = GiR
From the differential:
dG = -S.dT + V.dP + Σμidni
we can conclude that volume:
V = {∂G/∂P}T,n
Introducing the compressibility factor Z = P.V/(n.R.T) and ideal gas equation Zig = P.Vig/(n.R.T) = 1, it is straightforward to show that:
GR = n.R.T.0∫P{(Z-1)/P}.dP (constant T and n)
The fugacity coefficient is calculated by taking the partial molar property of the residual Gibbs free energy:
ln φi = 0∫P[{∂nZ/∂ni}T,P,nj≠i - 1].dP/P (constant T and n)
Note that the integral and the partial derivative can be interchanged, because the integral variable (P) is kept constant in the partial derivative. Also n may enter the integral as it is constant for the integral.
However, many equations of state (EoS) are expressed in P, or equivalently Z, as an explicit function of V, for example, in cubic EoS. Therefore, the integral in P is practically converted to an integral in V before calculating. The conversion is:
ln φi = -∞∫V[{∂nZ/∂ni}T,V,nj≠i - 1].dV/V - ln Z (constant T and n)
The term ln Z on the right-hand side appears from the conversion of the ideal gas part. Also note that the small pressure and large volume of the ideal gas are represented by P = 0 and V = ∞.
For cubic EoS, we can carry out the integration right away, and the result is:
Note that in deriving the fugacity coefficient we have used the partial molar of the parameters based on their mixing rules:
ai = 2 ∑xjaij - a
bi = bi
Go to cubic EoS to see the meaning of the symbols.
Flash calculation
When we need to solve the equifugacity equations for the compositions of the phases {xi(V)} and {xi(L)} at T and P, we have 2N unknowns. While we have N equifugacity equations, we still have a set of N mass balance equations:
xi(V).V + xi(L).L = zi
where zi is the known feed mole fraction of component i, while V and L are 2 additional unknowns, which are the mole fractions of vapor and liquid phase, respectively.
Altogether we have 2N+2 unknowns with 2N equations. However, the mole fractions have to satisfy the consistency equations:
∑xi(V) = 1
∑xi(L) = 1
so that we now have 2N+2 equations to solve for 2N+2 unknowns.
This is what the flash calculation doing. It is to solve the simultaneous equations. The example above is the T-P flash to calculate the equilibrium compositions. It is also possible that some of the equilibrium compositions are known, while P or T has to be calculated, for example if we need to determine the bubble points or the dew points.
Bubble point
Bubble points are equilibrium points where the gas phase starts emerging (when the process is going from the liquid phase) or the gas phase is disappearing (when the process is going from a condition containing the vapor phase). At this condition:xi(L) = zi
These N equations will remove the mass-balance equations, as now L = 1 and V = 0. With the remaining one consistency equation of the disappearing phase and the equifugacity, we now have N+1 equations. If T is known, then the calculation will give us the bubble-point pressure and the composition of the disappearing phase. If P is known, then the calculation will give us the bubble-point temperature and the composition of the disappearing phase.
Dew point
Dew points are equilibrium points where the liquid phase starts emerging (when the process is going from the vapor phase) or the liquid phase is disappearing (when the process is going from a condition containing the liquid phase). At this condition:xi(V) = zi
These N equations will remove the mass-balance equations, as now L = 0 and V = 1. With the remaining one consistency equation of the disappearing phase and the equifugacity, we now have N+1 equations. If T is known, then the calculation will give us the dew-point pressure and the composition of the disappearing phase. If P is known, then the calculation will give us the dew-point temperature and the composition of the disappearing phase.
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