2.3 Calculation of high-pressure phase equilibria
2.3.1 Bubble point-, dew point- and flash calculations
In phase-equilibria calculations different combinations of input variables are possible but, in general, the following choices are the most commonly used in order to solve typical problems.
9 For a liquid liquid mixture of given composition and at a given temperature (or pressure) the pressure (or temperature) at which the first bubble of vapour phase appears and its composition is calculated (bubble point calculation).
9 For a vapour mixture of given composition and at a given temperature (or pressure) the pressure (or temperature) at which the first drop of liquid phase appears and its composition is calculated (dew point problem).
9 For a mixture of given global composition at given conditions of temperature and pressure it can be determined if the mixture is a stable homogeneous liquid or vapour or that the mixture is heterogeneous and splits in a vapour phase and a liquid phase with different composition. If the mixture is heterogeneous the composition and the amount of the vapour phase and of the liquid phase are calculated. This calculation is known as flash calculation.
Bubble point calculations and dew point calculations can be considered as particular cases of the general flash calculation.
All these calculations with an equation of state are iterative, and in the following discussion the basic approach for the bubble point pressure and flash calculation will be described briefly.
In the case of the bubble point pressure calculation (see Fig. 2.2-1), the liquid composition, xi, and the temperature, T, are known and the vapour phase composition, yi, and the pressure, P, must be calculated. For the iterative calculations an initial guess for the bubble point
pressure and the vapour-phase composition is needed. As alternative for the vapour-phase composition an initial guess for the fugacities or the fugacity coefficients in the vapour phase can be used.
An initial guess for the pressure is assumed and the fugacity coefficient of each component in the liquid phase (~t) can be calculated. An initial guess is also assumed for the fugacity coefficient of each component in the vapour phase (#i v ), and consequently a first estimate of the vapour composition is evaluated. With this value of yi, the fugacity coefficients in the vapour phase are recalculated using the equation of state and a second estimate for yi is evaluated. This iterative procedure is continued until the difference between two successive values of the composition are below a predetermined error. At this point, the sum of yi is checked: if the sum is different from unity a new value of the pressure is assumed for a new iteration. The iterative procedure ends when the Y'. y, differs from unity by less then a given value.
With some slight modifications the same scheme can be adopted for dew point calculations.
In the case of the flash calculations, different algorithms and schemes can be adopted: the case of an isothermal, or P T flash will be considered. This term normally refers to any calculation of the amounts and compositions of the vapour and the liquid phase (V, L, yi, xi, respectively) making up a two-phase system in equilibrium at known T, P, and overall composition. In this case, one needs to satisfy relation for the equality of fugacities (eq. 2.3-1) and also the mass balance equations (based on 1 mole feed with N components of mole fraction zi ):
L + V = 1 (2.3-4)
z~ = x i L + y g V i= 1,2...,N (2.3-5)
Introducing the equilibrium conditions written in the form:
~t X~
= .-. = K~x, (2.3-6)
z~ = (L + K~V)x~ (2.3-7)
After elimination of V the following equation is obtained for xi by simple rearrangement:
x, = Z i ( 2 . 3 - 8 )
z, + Ki(1- L) or alternatively for yi:
g i z i
Yi = (2.3-9)
L + Ki(1- L)
These sets of mole fractions must sum to unity:
u u ( l _ K i ) z i
Fx =~'xii=, =Y'~L+Ki(1-L)i=, = 1 (2.3-10)
N N g i z i
Fy = i~1 = ~-'~ L + = 1 (2.3-11)
or equivalently:
iv ( l _ K i ) z i
F = Fx - Fe = Z L + Ki( 1 - = 0 (2.3-12)
The solution of a PT flash problem is accomplished when a value of L is found that makes the function F equal to zero; the advantage of using the F function instead of F,, and Fy is apparent from the fact that the function F vs. L is monotonic and which makes the numerical methods to solve the equations more efficient.
It is not known in advance what the real physical state of a system of given composition at given pressure P and temperature T is. For this reason a bubble pressure (Pb,,bt) calculation of a hypothetical liquid of composition zi at the temperature T, and a dew pressure (Pdew) calculation of a hypothetical vapour of composition z; at the temperature T, is performed.
Only for pressures between Pdew and Pbubl is the system an equilibrium mixture of vapour and liquid, and flash calulations make sense.
The iterative calculations start by assuming a guess for Ki (it is necessary to assume a value for xi and yi, -for example the values previously obtained in the dew- and bubble-points evaluation). With these values of Ki the eq. (2.3-12) is solved iteratively for L. With this value of L, using equations (2.3-8) and (2.3-9) new values of vapour- and liquid-phase compositions are calculated and new values of/2,, calculated for the next iteration. The iterations stop when satisfactory convergence is reached.
Various authors have modified this classical procedure by introducing the so-called stability test based on the Gibbs' energy minimization [ 19-23].
I~' = ~v (.T, P, y~ )I
k
ly,- y;[ no I
T, X i I
Initial guess for P 1
f
Initial guess for ~i
"-'1
~ ) i x i
Y i ~--- "-"V
First iteration ?
l n o .... [
[Y:-Y!?I
yes
New value o f P . I
~k
[ no
Id ,
- q Z Y , - 1
J yesl
Calculated B.P. pressure and composition are correct
Fig. 2.3-1 Bubble point pressure calculation