For flash calcu-lations for two-phase systems, information about phase compositions is often expressed in terms of equilibrium ratios also known as K-values, K i =x i1 where x i1 and x i2
Trang 1Table 3.2: Peng-Robinson Binary Interaction Parameters [99, 18]
Component CH4 C2 C3 CO2 N2
C4 0.020 0.010 0.010 0.130 0.120
C5 0.020 0.010 0.010 0.125 0.120
C6 0.025 0.010 0.010 0.119 0.120
C7 0.025 0.010 0.010 0.100 0.120
C8 0.035 0.010 0.010 0.112 0.120
C9 0.035 0.010 0.010 0.100 0.120
C10 0.035 0.010 0.010 0.102 0.120
C12 0.035 0.010 0.010 0.095 0.120
C16 0.035 0.010 0.010 0.105 0.120
C20 0.035 0.010 0.010 0.093 0.120
in subsequent chapters Interaction parameters for CO2 are those recommended by Deo et al [18].
Otherwise Table 3.2 reports values of δ ij recommended by Peng and Robinson [99]
Eqs 3.3.7–3.3.16 provide the description of volumetric behavior needed to complete the calcu-lation of the partial fugacity of a component in a mixture Differentiation of Eq 3.3.7 gives an expression for
∂P
∂n i
That expression is then substituted into Eq 3.2.8, and the integration is performed The result, after considerable algebraic manipulation, is
ln
ˆij
x ij P
= ln ˆφ ij = b i
b m (z − 1) − ln
z
1− b m
V
+
a m
2√ 2b m RT
b i
b m − 2
a m
n c
k=1
x kj (aα) ik
ln
1 + (√
2 + 1)b m V
1− ( √2− 1) b m
V
. (3.3.17)
3.4 Flash Calculation
The result of the thermodynamic analysis and the use of an equation of state to describe volumetric behavior is a set of nonlinear equations (Eqs 3.1.30 in which each value of ˆf ij is given by an expression like Eq 3.3.17) that must be solved for the compositions of the phases The following procedure can be used:
1 Estimate composition, x ij, of each of the phases present
2 Solve Eq 3.3.7 for the molar volume, V , of each phase.
3 Use Eq 3.3.17 to calculate the partial fugacity, ˆf ij, for each component in each phase
Trang 24 Check to see if component partial fugacities are equal in all of the phases (Eq 3.1.30) If not, then adjust the estimates of phase compositions and return to step 2
While this procedure will give the compositions of phases that satisfy the requirement that component partial fugacities (or equivalently, chemical potentials) be equal at equilibrium, it should
be noted that it is sometimes possible to find solutions for which the resulting phases are not stable [3] In other words, the phase compositions make component chemical potentials equal but do not minimize free energy For such situations, the stability of phases can be tested directly See Baker
et al [3] or Michelsen [78, 80] for examples and details.
Given the nonlinearity of Eq 3.3.17, it is easy to see why flash calculations with van der Waals equation did not catch on in the 1880’s when van der Waals [122] performed the first phase equilibrium calculations for binary systems In fact, equation-of state calculations of phase equilibrium did not come into widespread use until the late 1960’s when availability of computing resources made solution of the set of nonlinear equations reasonable
Step 1 requires that some sort of initial guess of phase compositions be made For flash calcu-lations for two-phase systems, information about phase compositions is often expressed in terms of
equilibrium ratios (also known as K-values),
K i =x i1
where x i1 and x i2 are the mole fractions of component i in phases 1 and 2, typically vapor and
liquid The Wilson equation [136],
K i = x i1
x i2 =
P ci
P exp
5.37(1 + ω i)
1− T ci
T
is frequently used to estimate equilibrium K-values from which phase compositions can be estimated
by the following manipulations
From the estimated or updated K-values, the phase compositions can be obtained from a mate-rial balance on each component Consider one mole of a mixture in which the overall mole fraction
of component i is z i A material balance for component i gives
z i = x i1L1+ x i2(1− L1), i = 1, n c (3.4.3)
where L1 is the fraction of the one mole of mixture that is phase 1 Elimination of x i1 from Eq.
3.4.3 gives
x i2 = z i
1 + L1(K i − 1) , i = 1, n c (3.4.4)
Similar elimination of x i2 using Eq 3.4.2 gives
x i1 = 1 + L K i z i
An equation for L1 is obtained by noting that Eqs 3.4.4 and 3.4.5 each sum to unity, so that
n c
i=1
x i1 −
n c
i=1
x i2=
n c
i=1
z i (K i − 1)
1 + L1(K i − 1) = 0. (3.4.6)
Trang 3Eq 3.4.6, which is known as the Ratchford-Rice equation [103], can be solved for L1 by a
Newton-Raphson iteration [80, p 220] Given the value of L1, Eqs 3.4.4 and 3.4.5 give the phase
compo-sitions consistent with the K-values
The equilibrium K-values defined in Eq 3.4.2 are related to the equilibrium partial fugacity coefficients The equilibrium relations, Eqs 3.1.30, can be written using Eq 3.1.28 as
ˆi1 = ˆφ i1x i1P = ˆ φ i2x i2P = ˆ f i2, i = 1, n c (3.4.7) Rearrangement of Eq 3.4.7 shows that the equilibrium K-value is just the ratio of partial fugacity coefficients,
K i = x i1
x i2 =
ˆ
φ i2
ˆ
φ i1
The form of Eq 3.4.8 suggests a simple successive approximation scheme for updating K-values
in step 4 of the flash calculation [98, p 103][80, p 245] If the component partial fugacites are not
equal at the k th iteration, then new K-values can be estimated from
K k+1
i = K i k ˆ
k
i2
ˆk
i1
Eq 3.4.9 modifies the K-values in the appropriate direction if ˆf i2 = ˆ f i1 While iteration with Eq.
3.4.9 usually converges to solutions that satisfy the equilibrium relations even when the guess of initial phase compositions is relatively poor, convergence will be very slow for two-phase mixtures near a critical point (a pressure, temperature and composition for which the two phases become identical) For such situations, more sophisticated iterative schemes can and should be used [79, 75]
Negative Flash
The flash calculation can be performed whether a mixture forms one or two phases Whitson
and Michelsen [135] pointed out that Eq 3.4.6 can be solved for L1 equally well when only one
phase forms, a calculation that is known as a negative flash When the iteration for L1has converged
for a single-phase system, the resulting value will be in the range L1 < 0 or L1 > 1 The phase
compositions calculated with Eqs 3.4.4 and 3.4.5 will be equilibrium compositions that can be combined to make the single-phase mixture In other words, the single-phase mixture is a linear combination of the phase compositions, which means that the single-phase composition must lie
on the extension of the line that connects the equilibrium compositions on a phase diagram We
will make repeated use of the negative flash to find that line, known as a tie line, for displacement
calculations
Whitson and Michelsen showed that their negative flash calculation converges as long as L1 lies
in the range
1
1− K max
< L1< 1− K1min
where K max and K min are the largest and smallest K-values
If the single-phase composition is far enough from the two-phase region that the condition 3.4.10
is not satisfied, a modified negative flash suggested by Wang [128] can be used It is based on the
Trang 4idea that while L1can vary over wide ranges, the equilibrium phase compositions, x ij, are restricted
to lie between zero and one The mole fraction of phase 1, L1 can be eliminated by solving Eq.
3.4.4 written for component 1,
L1 = z1− x12
Substitution of Eq 3.4.11 into Eq 3.4.4 gives an expression for the phase compositions in phase
2 in which the only unknown is x12,
x i2 = z i x12(K1− 1)
z1(K i − 1) + x12(K1− K i). (3.4.12) The revised version of Eq 3.4.6 is
n c
i=1
(K i − 1) x12=
n c
i=1
z i x12(K i − 1) (K1− 1)
z1(K i − 1) + x12(K1− K i) = 0. (3.4.13)
Eq 3.4.13 can be solved for x12 by a Newton-Raphson iteration.
3.5 Phase Diagrams
It will be convenient for many of the flow problems considered here to represent the solutions as a collection of compositions on a phase diagram Accordingly, we review briefly the terminology and properties of binary, ternary, and quaternary phase diagrams
3.5.1 Binary Systems
Fig 3.1 is a typical phase diagram for a binary mixture at some fixed temperature above the
critical temperature of component 1 At pressure, P , liquid phase (phase 2) with mole fraction
x12 of component 1 is in equilibrium with vapor phase (phase 1) containing mole fraction x11 of
component 1 Those equilibrium compositions are connected by a tie line, along which a tie line
material balance like Eq 3.4.3 applies There is one tie line in Fig 3.1 for each pressure
A mixture with an overall mole fraction z1of component 1 between x12and x11forms two phases
Mixtures with z1< x12are all liquid, while those with z1> x11form only vapor For a given overall
composition, the mole fraction of phase 1 present is easily determined by rearrangement of Eq 3.4.3
to be
L1 = z1− x12
Eq 3.5.1 is a lever rule, which states that the mole fraction vapor is proportional to the distance
from the overall composition to the liquid composition locus divided by the length of the tie line
A similar statement applies to systems with any number of components Eq 3.5.1 indicates that
L1 ≤ 0 for mixtures that form only liquid, and L1 ≥ 1 for mixtures that are all vapor.
At the top of the two-phase region in Fig 3.1 is a critical point, at which the liquid and vapor phases, as well as all phase properties, are identical The critical point can be thought of as a tie line with zero length Because phase compositions are equal at a critical point, Eq 3.4.2 indicates that all K-values must be equal to one for a critical mixture
Trang 50 200 400 600 800 1000 1200 1400 1600 1800 2000
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction Component 1
a
Critical Point
Two Phase Region Liquid Phase
Vapor Phase Tie Line
Figure 3.1: Pressure-composition phase diagram for a two-component mixture
A phase diagram like Fig 3.1 can be calculated with an equation of state Fig 3.2 shows such
a diagram calulated with the Peng-Robinson equation for a two-component system, CO2/decane (C10) system Also shown in Fig 3.2 are experimental data of Reamer and Sage [104] along with the phase diagram calculated with van der Waals equation of state Critical properties used for CO2 and
C10 are given in Table 3.1, and the binary interaction parameter used in the Peng-Robinson flash
calculations was δ12= 0.102 (Table 3.2) Fig 3.2 shows that the Peng-Robinson phase compositions
agree much better with the experimental observations than do the van der Waals predictions
They should, of course, because the value of δ12 was chosen to minimize the disagreement between
calculated phase compositions and measured data Even so, with the average value of δ ij chosen
by matching experimental data at several temperatures [18], there is some disagreement between experiment and calculation near the critical point
3.5.2 Ternary Systems
A typical vapor/liquid phase diagram for a three-component system is shown in Fig 3.3, which displays phase behavior information at fixed pressure and temperature Because the mole fractions
at any composition point in the diagram always sum to one, it is useful to plot equilibrium phase compositions on an equilateral triangle On such a diagram, the three mole (or volume or mass) fractions are read from the perpendicular distances from the composition point to the three sides The corners of the diagram represent 100% of the component with which the corner is labeled, and the opposite side represents the zero fraction The sides of the ternary diagram represent binary mixtures of the two components that lie on that side For gas/oil systems, the component at the top corner of the diagram is usually the lightest component, and the heavies component is usually
Trang 60 200 400 600 800 1000 1200 1400 1600 1800 2000
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction CO2
b b b b b b b
b b
0 200 400 600 800 1000 1200 1400 1600 1800 2000
b b b b b b b
b b
b Experimental Data Van der Vaals Peng-Robinson
Figure 3.2: Phase behavior of the CO2/C10 system at 160 F Experimental data shown are from
Reamer and Sage [104]
placed at the bottom left corner
Any mixture of two overall compositions, whether or not either is single phase, must lie on a straight line that connects those compositions For example, if M moles of a mixture with overall
compositions, z i1, are mixed with N moles of mixture with compositions, z i2, the overall composition
of the resulting mixture is given by
z i = 1
Eq 3.5.2 is the equation of a straight line (the dilution line) that connects the two overall
compo-sitions
The region of overall compositions in Fig 3.3 that form two phases is enclosed by the loci of
liquid and vapor phase compositions, which taken together are known as the binodal curve Here
again, equilibrium liquid and vapor compositions are connected by tie lines In contrast with binary systems, in which there is only a single tie line at fixed temperature and pressure, a ternary system has an infinite number of tie lines A critical point on a ternary phase diagram is often referred to
as a plait point.
An example of a calculated (Peng-Robinson) ternary phase diagram that includes a plait point
is shown in Fig 3.4 for mixtures of CO2, butane (C4), and decane (C10) at 160 F (71 C) and 1250
psia (85 atm) Mixtures containing no C4 lie on the CO2/C10 side of the diagram Hence, that
side shows one tie line, at 1250 psia, from the pressure composition phase diagram in Fig 3.2.
Also shown in Fig 3.4 are experimental data reported by Metcalfe and Yarborough [76] and by Orr and Silva [87] Again, the Peng-Robinson equation of state shows reasonable agreement with
Trang 710 20 30 40 50 60 70 80 90
10 20 30 40 50 60 70 80
90 10
20 30 40 50 60 70 80 90
Comp.1
Comp.2 Comp.3
a
Ternary Diagram
Plait Point
Tie Lines
Vapor Locus
Liquid Locus
Figure 3.3: Ternary phase diagram
the experimental observations Calculated vapor compositions agree quite well with experimental observations, while calculated liquid phase compositions show slightly lower CO2 mole fractions Tie line slopes are represented quite well by the equation of state Tie line slopes turn out to be important in the analysis of flows in systems with three or more components, so it is useful that the equation of state captures that behavior well
3.5.3 Quaternary Systems
When four components are present, an equilateral tetrahedron can be used to plot phase compo-sitions in much the same way an equilateral triangle is used for ternary systems Fig 3.5 is an example of a quaternary diagram for mixtures of CO2, methane CH4, butane (C4), and decane (C10) at 160 F (71 C) and 1600 psia (109 atm) A ternary diagram like Fig 3.4 is one of the
faces of the quaternary diagram At that temperature and pressure, there are plait points in the
CO2/C4/C10 and CO2/CH4/C4 ternary faces, and there is a locus of critical points in the inte-rior of the quaternary diagram that connects those plait points The two-phase region on the
CO2/CH4/C10 face is a band of tie lines that spans that face The locus of liquid compositions is
now a surface in the interior of the diagram, as is the locus of vapor compositions Those surfaces meet at the locus of critical points, on which the compositions of liquid and vapor are identical Each composition point on the surface of vapor compositions is connected to another composition
point on the surface of liquid compositions by a tie line Thus, the space enclosed by the binodal surface is densely packed with tie lines The geometry of tie lines in the four-component diagram
of Fig 3.5 is worth a bit of study because displacement behavior is intimately connected with the geometry of surfaces of tie lines, as the analysis of subsequent chapters will show
Trang 810 20 30 40 50 60 70 80 90
10 20 30 40 50 60 70 80
90
10 20 30 40 50 60 70 80 90
CO2
C4
C10
a
a
a
a
a
a a
a
b
b
b
b b
b
b
b b
b
b
b b
b b
b b
b b
b b
b b
b
a
b
Peng-Robinson binodal curve Peng-Robinson tie lines Metalfe and Yarborough experiment Orr and Silva experiment
Figure 3.4: Ternary phase diagram (in mole fractions) calculated with the Peng-Robinson equation
of state for the CO2/C4/C10 system at 1250 psia (85 atm) and 160 F (71 C) Measured phase
compositions reported by Metcalfe and Yarborough [76] and by Orr and Silva [87] are also shown
3.5.4 Constant K-Values
Eq 3.3.17 shows clearly that ˆφ ij can depend quite strongly on the composition of phase j through the parameters, a m and b m, and it follows, therefore, from Eq 3.4.8 that K-values are also functions
of composition For example, equilibrium K-values for a phase diagram that includes a critical point must be strong functions of composition (and, of course, temperature and pressure), because K-values are all equal to one at the critical point and differ significantly from one for the remainder
of the two-phase region If the pressure is low enough for many gas/oil systems, however, a critical point will not be present, and K-values will depend only weakly on composition For those systems,
it is reasonable to assume that K-values are constant The constant K-value limit is a useful one, because the solutions for pure convection simplify considerably, and hence, we will pause for a moment to explore the properties of the phase diagrams that result from the assumption of constant K-values
If K-values are constant, then the liquid and vapor portions of the binodal curve are straight lines
on a ternary diagram Consider Eq 3.4.6 with L1 = 0, which gives an expression for compositions
on the liquid locus,
3
i=1
z i (K i − 1) =3
i=1
z i K i −3
i=1
z i =
3
i=1
z i K i − 1 = 0. (3.5.3)
Because the K i are constants, Eq 3.5.3 is the equation of a straight line (It is a bit easier to see
that it is a straight line if z3 = 1− z1− z2 is eliminated from Eq 3.5.3.) Similarly, when L1 = 1,
Trang 9C10
CO2
CH4
•
•
Figure 3.5: Quaternary phase diagram calculated with the Peng-Robinson equation of state for mixtures of CO2, methane, butane and decane at 160 F (71 C) and 1600 psia (109 atm) The
dashed lines are tie lines that lie on the four ternary faces of the diagram
the equation for the vapor phase compositions is
3
i=1
z i (K i − 1)
K i
=
3
i=1
z i −3
i=1
z i
K i
= 1−
n c
i=1
z i
K i
Again, Eq 3.5.4 is the equation of a straight line Hence, for a ternary system with constant K-values, the two-phase region must be a band across the diagram that intersects the two sides with one K-value greater than one and the other K-value less than one Two phases cannot form on a side where both K-values are greater than one or both are less than one Fig 3.6, for example,
shows a phase diagram for K1 = 1.5, K2= 2.6, and K3 = 0.01 Those K-values roughly reproduce
the behavior of the CO2/CH4/C10 face of Fig 3.5, for example The intersection points on the binary sides of the diagram can be easily obtained from Eq 3.5.3 written for two components with K-values above and below one For example, on the component 1/component 3 side of the diagram (CO2/C10),
x12= 1− K3
K1− K3, x11= K1
1− K3
K1− K3
For four–component systems, Eqs 3.5.3 and 3.5.4 indicate that the surfaces of vapor and liquid
compositions are planes Fig 3.7 is an example of such a phase diagram for K1 = 2.5, K2 = 1.5, K3
= 0.5, and K4 = 0.05 Those values are roughly equivalent to K-values for the CO2/CH4/C4/C10
system at 160 F (71 C) and 1600 psia (109 atm) in regions of the phase diagram far from the
critical locus As Fig 3.7 shows, the liquid and vapor phase planes intersect all four ternary sides
of the diagram when two K-values are greater than one and two are less than one If only one
Trang 1010 20 30 40 50 60 70 80 90
10 20 30 40 50 60 70 80
90 10
20 30 40 50 60 70 80 90
Comp.1
Comp.2 Comp.3
Figure 3.6: Ternary phase diagram for a system with constant K-values, K1 = 1.5, K2 = 2.6, and
K3= 0.01.
K-value is less than one (or only one is greater than one), the two–phase region does not intersect the ternary face on which all K-values are greater than (or less than) one
3.6 Additional Reading
Thermodynamic Background, State Functions, Chemical Potential and Fugacity van
Ness and Abbott’s text, Classical Thermodynamics of Nonelectrolyte Solutions [123], gives concise
but readable derivations of the material in Sections 3.1 and 3.2 The various equations of state
in widespread use are reviewed in considerable detail by Walas in Phase Equilibria in Chemical Engineering [126] A shorter description of phase equilibrium calculations with an equation of state
is given by Lake in Chapter 4 of Enhanced Oil Recovery [62] A description of flash calculations
and K-values can also be found there A detailed account of the fundamental thermodynamic structure of phase equilibrium as well as the many computational issues that arise in solving practical problems is given by Michelsen and Mollerup [80]
Phase Diagrams Binary and ternary phase diagrams are described in detail by Walas [126,
Chapter 5] Properties of ternary diagrams as well as approximate representations for two-phase regions in ternary diagrams are described by Lake [62, Chapter 4] Examples of phase diagrams for systems with constant K-values are reported by Dindoruk [19]
3.7 Exercises
1 Use the Peng-Robinson equation of state to calculate the molar volume and molar density of
CH4 and C10at 160 F and 1600 psia.
.... (3. 3.17)
3. 4 Flash Calculation
The result of the thermodynamic analysis and the use of an equation of state to describe volumetric behavior is a set of nonlinear... Estimate composition, x ij, of each of the phases present
2 Solve Eq 3. 3.7 for the molar volume, V , of each phase.
3 Use Eq 3. 3.17 to calculate the partial fugacity,... 0. (3. 4. 13)
Eq 3. 4. 13 can be solved for x12 by a Newton-Raphson iteration.
3. 5 Phase Diagrams
It will