I n d . E n g . C h e m . R e s . 1987, 26, 2129-2134 Smith, K. J.; Anderson, R. B. Can. J. Chem. Eng. 1983, 61,40. Smith, K. J.; Anderson, R. B. J. Catal. 1984,85,428. Supp, E.,paper presented a t the 78th Spring National AIChE Meeting, New Orleans, April 1986. Taylor, R. J . Chem. SOC.(London) 1934,1429. Vedage, G. A.; Himelfarb, P.; Simmons, W. G.; Klier, K. Prepr. Pap.-Am. Chem. SOC.Pet. Chem. Div. 1983,28,1261.
2129
Villa, P. L.; Forzatti, P.; Buzzi-Ferraris, G.; Garone, G.; Pasquon, I. Ind. Eng. Chem. Process Des. Dev. 1985,24,12. Villa, P. L.; Del Piero, G.; Cipelli, A.; Lietti, L.; Pasquon, I. Appl. Catal. 1986,26,161.
Received f o r review July 23, 1986 Accepted J u n e 2, 1987
A Simplified Flash Procedure for Multicomponent Mixtures Containing Hydrocarbons and One Non-Hydrocarbon Using Two-Parameter Cubic Equations of State Bjarne H. Jensen and Aage Fredenslund* Znstituttet f o r K e m i t e k n i k , Technical University of Denmark, DK-2800 Lyngby, Denmark
An economical two phase (P,T) flash procedure for cubic equations of state containing two parameters (a and b ) is developed. T h e geometric mean is used as the mixing rule for the a parameter for hydrocarbon-hydrocarbon interactions, but a deviation parameter is used for non-hydrocarbonhydrocarbon interactions. When there is only one non-hydrocarbon present, this permits reduction of the flash problem to solving five equations in five unknowns, irrespective of the number of components in the mixture. For multicomponent flash calculations, this implies substantial savings in computer time and storage requirements compared with standard flash procedures. The new procedure is advantageous t o use in, e.g., simulation of COSand N2 flooding of oils in reservoirs.
For hydrocarbon-hydrocarbon interactions, the oil characterization procedure of Pedersen et al. (1985) uses the geometric mean to describe the a parameter in the Soave-Redlich-Kwong equation of state (SRK-EOS) (Soave, 1972). The deviation parameter (termed Ki,) is equal to zero. The correlations used to describe the a and b parameters for the hydrocarbon fractions are developed such that K i j = 0 for hydrocarbon-hydrocarbon interactions gives the best results for the thermodynamical properties of the reservoir fluids. When Ki, = 0 for all combinations of i and j , Michelsen (1986) has shown that the two-phase (P,T)flash problem reduces to solving three equations in three unknowns, for any multicomponent mixture. This significantly reduces computational costs and storage requirements. In some important practical applications, significant amounts of non-hydrocarbons are present. Examples of this are oil reservoir flooding with CO, or N2. In these cases, the characterization procedure of Pedersen et al. (1985) still permits Ki; = 0 for all hydrocarbon-hydrocarbon interactions. However, for acceptable results, the C02-hydrocarbon or the N2-hydrocarbon interactions must be different from zero. This paper formulates an economical flash algorithm for cases where significant amounts of non-hydrocarbons are mixed with multicomponent hydrocarbon mixtures. The flash algorithm is outlined by using the Peng-Robinson equation of state (Peng and Robinson, 1976) as an example, but it can be used with any other two-parameter cubic equation of state, such as the SRK-EOS.
Peng-Robinson Equation of State The Peng-Robinson equation of state, eq 1 (Peng and Robinson, 1976), has two parameters, a and b, defined as shown in eq 2, where the critical coefficients, a, and b,, are 0888-5885/87/2626-2129$01.50/0
given by eq 4 and obtained from the condition stated by eq 5. a p = - -RT (1) u - b (U + b)U + b(U - b) a =
m ( T ) = (1 +
a,T,2R2m(T ) b = -b C T 3 (2) p, PC (0.37464 1.54226~0.26990')(1 - (T/T,)'/'))' (3)
+
a, = 0.45724 b, = 0.07780 [dP/dU],= [d2P/du2],= 0
(4) (5)
For mixtures, the EOS parameters are evaluated through mixing rules, as shown in eq 6 and 7. The parameter Ki, describes the deviation from the geometric mean. N N
a =
C Cnixj(aiaj)'/2(l- Kij) i=1;=1
(6)
N
b = Cxibi i=l
(7)
The equilibrium relations for multicomponent systems are given in terms of fugacity coefficients, which are calculated after having solved the compressibility equation
where the largest or the smallest root is chosen, depending on the phase considered.
0 1987 American Chemical Society
2130 Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987
For the Peng-Robinson EOS,the fugacity coefficient is given by eq 9. In
@i
bi = -(Z - 1) - In
b
We introduce the partial derivative
Equations for Flash Calculations Given a flash drum feed stream which contains nf moles of component i (i = 1,2,...,N), the stream may split into two phases: a vapor phase n, = Onf and a liquid phase nl = (1 - P)nf, where /3 is the vapor fraction. The feed stream composition is z, and the two resulting phases have compositions x and y, respectively. For N components, the mass balances and equilibrium relations are
+ nvyi z i = (1- @ ) x i + byi nfzi = n1xi
1d[n,2aI ai* = n, ani
which becomes
@:xi N
ai* = 2Cx;(aia;)'/2(1 - Kij)
(10)
= ; 1
i = lfl
i = 1,N i = l,N
= @.'y. I 1
(20) (204 (21)
In addition, the following summation is an independent relationship:
bi is given by eq 2. i=l
The Simplified B -Mixing Rule We are given a mixture with N components, of which only one (here denoted as component 1)is a non-hydrocarbon. When only deviations from the geometric mean for the non-hydrocarbon and the remaining N - 1 hydrocarbons are taken into account, the mixing rule for the a parameter can be formulated as N
a = x:a1
N
+ 2x1a1'/2[Cxjaj'~2(l - K1j)] + [CXjUj'/2]2 ;=2 ;=2 (11) N
a,* = 2(x,a,
+ a11/2Cxjaj1/2(1 - K,j)) j=2
(12)
Michelsen (1986) shows how these 2N + 1 equations in 2N + 1unknowns (p, x , y) can be reduced to three equations in three unknowns, when all the a parameter K, values are equal to zero in eq 6. This is the case irrespective of the number of components. By use of a similar approach with the modified mixing rule, eq 11, allowing non-zero interaction parameters between a single component (number l) and the rest of the components in the mixture, the isothermal flash problem can be expressed in five equations in five unknowns. Multiplication of the mass balance, eq 20a, by and summation over the last N - 1 components in the mixtures gives
N
ai* = 2(xl(aia,)'/2(1 - K,i)
i
+ ail~zCxja;'~z) ;=2 Similarly for ai1/2Kli,
= 2,3,...,N
N
N
Defining the parameters
CZiU?I2K1i = (1- P)CxiaiKli i=2
r=2
N
a1 = Cx.a.1/2 I 1 ;=2
(13)
N
+ PCyi~i'/'K,i i=2
(24)
and for the b parameter,
N
dl = Cxjaj1/2Klj ;=2 N
gl =
Cxibi i=2
the parameter a and its partial derivative (ai*) (eq 11 and 12) are given by a =
x12al+ 2x1a11/2(al- dl) + a12
+ 2al1i2(al- dl) ai* = 2x1(alai)1/2(1- Kli) + 2ai1l2al 2 Ii IN a,* = 2x1a1
By substituting the equilibrium relations, eq 21, into the mass balance, eq 20a, for the non-hydrocarbon component in the mixture (component denoted as one) and expressing eq 23-25 in terms of al, dl, and gl defined by eq 16-18, the resulting five flash equations are
(16) (17)
With the simplified mixing rules described above, the use of cubic equations of state may become more economical. In the evaluation of parameter a, the computational work, using eq 11,is increasing nearly linearly with the number of components, whereas it increases quadratically when normal mixing rules, eq 6, are used. For the b parameter, there is no simplification possible.
(26)
dlf = (1 - @)dl1+ pdl,
(27)
df = (1 - Pkll + Pglv 21 = (1- P ) X l + BY1 4JllXl = @IVY,
(18)
The b parameter is given by eq 7, which can be reformulated in terms of gl: b = rlbl + gl (19)
alf = (1 - p)all + pal,
(28) (29)
and N C ( X i i=l
-yJ = 0
The variables in the flash problem are the phase split,
p; the mole fraction of the non-hydrocarbon component in the phase considered as vapor (component one), yl; the dl parameter evaluated from mole fractions, y,, j = 2,N, dk; the corresponding gl parameter, gl,; and the parameter al,. This gives a total of five variables.
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 2131 The corresponding parameters for the liquid phase are al, dll, gll, and xl. They are obtained from eq 23, 24, 25, and 20a: all =
The stability test is performed as a constrained minimization of eq 41, with solution given by the constraints
alf - Pal,
i=2
1-0
i=l
N
N
i=2
YiBi - gl
i=l
N i=2
(33)
(43)
Yi N
YiAi1I2K1j - d l C Yi i=l
(44)
N
Y1 - u l C Yj
g4:
(45)
i=l
(34)
The solution is given by the vector To obtain values of the mole fractions x and y , the parameters al,, dl,, gl,, yl, and p, the corresponding parameters all, dl1, gll, and x1 are evaluated through eq 31-34. The fugacity coefficients for all N components in both phases & and &", i = 1,2, are evaluated from eq 9 by use of eq 1619, and the equilibrium constants Ki, i = are evaluated.
...a,
la,
Ki = r p ? / r p i V
(35)
The mole fractions in the two phases are evaluated by usding eq 36 zi
xi
yi = XiKi
= 1 + @(Ki- 1)
(36)
To avoid numerical troubles, the equation of state parameters a and b may at specified (T,P)be scaled using (37)
In the Appendix, the above shown equations are outlined with scaled parameters (A$), and in the following, (A,B) replace (a,b). Calculational Procedure. 1. Stability Test. Given a mixture at specified (P,T),a stability test is performed prior to a flash calculation in order to confirm the possible presence of more than one phase. The Michelsen (1982a) tangent plane method is used. This method can be reformulated for use in connection with the present flash problem formulation. Given the distance k in Gibbs free energy between the tangent plane to the feed composition z and an alternative composition u which is parallel to the tangent plane, the mole numbers Yi are defined as In ui+ In $i(u)- In zi - In rpi(z) = k (38) Yi = exp(-k)ui In Yi = (In zi + In rpi(z) - In rpi(u))
(39)
i = 1, N
(40)
The feed stream to the flash drum is stable if the value of k is positive for all compositionsu; i.e., the feed mixture has a tangent plane in the Gibbs space, which does not intercept the Gibbs free-energy surface. Thus, the global Gibbs free-energy minimum is given by the overall composition z. This is seen in the Yi summation: N
f = 1-CYi i=l
which gives positive or zero values of eq 41 when the feed composition is stable, and negative values when the feed composition is unstable at the specified ( P , T ) . After having found negative values off (eq 41), a flash calculation can be performed.
h* = (al*, dl*, g1*, ul*)
(46)
To solve eq 42-45, Michelsen (1982a) recommends a procedure, where a Newton-Raphson iterative method is used in search of two solution vectors. Initial values of the variable vector (denoted bo)are taken as vectors h representing the heavy and light end of the feed, respectively. In practice the initial vector ( h o )is evaluated as ho = ((1- yle)Aj, (1 - yle)(AjK1j),(1 - yle)Bj,yle)
(47)
where j indicates the lightest or the heaviest component in the mixture, respectively. The mole fraction yle is found as Yle = ZlKl
(48)
Yle = zdK1
(49)
in the light case and in the heavy case. K1 can be found from
K1 = p,l P ex.(
5.42( 1 -
:)
(50)
Either one or two nontrivial solutions (y # z ) to eq 42-45 are observed in case of instability. In case of stability, the feed stream represents the minimum searched, and the trivial solution is found, irrespective of the initial guess vector 4 O. By use of initial guesses for the initial vectors, bo,as described above for mixtures where the heaviest or lightest component contributes very little to the overall properties, (described through the mixing rules), initial vectors ( hO) far from the vector representing the feed (h? are seen. Iterations starting from vectors h far from the trivial solution, h f, may converge very slowly toward the trivial solution, h * = h f. It is therefore recommended to trace the iterations moving toward the trivial solution by comparing the Newton-Raphson iterative step at iteration number k, Altk, to the distance between the solution vector in iteration number, hk,and the trivial solution vector, h f, by calculating the ratio rk (eq 51) at each iterative step.
I.k=
(hk- hq Ahk
When the ratio, rk, reaches values in a small region around unity, an indication of iteration against convergence in the trivial solution in the steps k + 1, k + 2, etc., is found. By use of this distance approach with a region 1 f 0.00001 as the criterion for termination of iterations in step k, significant savings of computer time in evaluation of the stability tests are found.
2132 Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 Table I. Mixture of 75% C02 (Simon et al., 1978) pc
+ 25% Oil B
0.75333 CO, c1 0.09675 0.01078 c2 0.00788 c3 0.00738 c4 0.00673 c5 0.00678 C6 C7+Q 0.10973 total 0.099933
304.2 190.6 305.4 369.8 425.2 469.6 507.4
72.8 45.4 48.2 41.9 37.5 33.3 29.3
(56)
i=2
KU
9
T,,K atm
X
N
gfl = CuiAi1/2 - al"
w
0.225 0.008 0.098 0.152 0.193 0.251 0.296
N
(hydrocarbon-COJ 0.00 0.093 0.128 0.123 0.136 0.125 0.131
gfz = C ~ i A i ' / ~ K-1 dl" i
(57)
i=2
N
gf, = i=2
- gl"
(58)
gf4 = u1- Y1
(59)
N
"C7+ properties: M , = 239.1 g/mol; SG at 60
O F
(60)
= 0.8496;
"API = 35.05.
When one or two nontrivial solutions are found, giving negative values off (eq 41), initial estimates of the equilibrium constants (Ki, i = 1,N) for the following flash calculation can be generated from the solution vector, h *. If one nontrivial solution is found, the mole numbers, Yi, corresponding to the solution h * are calculated from eq 40 and K ivalues from eq 52.
Ki = YJZi
(52)
When two nontrivial solutions are obtained, two Yi vectors are calculated from the two solution vectors, h * (which are found by iterations from the heavy and the light end of the feed mixture) by using eq 40, and initial Ki value estimates are obtained from eq 53. K~=
yi(light)
yL(heavy)
(53)
2. Initial Estimates for the Flash Routine. Initial estimates for the phase split (P) are found by solving the Rachford-Rice equation
(54)
As suggested by Michelsen (1986), the phase split estimate, 0,should be replaced by (1- 6) if the solution to eq 54 exceeds a value of 0.5, replacing the vapor fraction by the liquid fraction. The initial estimate of variable vector hfo should be evaluated from the mole fraction, xi, and the other independent variables, al, dl, and gl should represent the phase denoted 1 (considered as a liquid phase). Otherwise, the initial solution vector, hf", is evaluated from mole fractions, yi, and variables al, dl, and gl represent the phase considered as a vapor phase: xi
=
zi
1
+ P(Ki + 1)
yi = XiKi
(55)
3. Flash Routine. The five variables in the flash routine are phase split (P), non-hydrocarbon component mole fraction (yl), interaction parameter variables (dl) and equation of state parameters (a1 and gl). The check functions which must be zero at the flash solution are
The procedure is as follows. 1. From initial estimates of y1 ( x , in the case where solution, P, to eq 54 exceeds 0.5 and is replaced by (1- P)), alv, dl', P, and gP are evaluated. 2. Corresponding values of the variables in the other phase, all, dll, gll and xl,are calculated from eq 31-34. 3. For both phases (v and l), fugacity coefficients (eq a9) and analytical derivatives of the fugacity coefficient with respect to the five variables in vector hf,k (P, yl, al,, dl,, gl,) are calculated. 4. The mole fractions, x and y , are evaluated from eq 55. Check functions 56-60 and their analytical derivatives with respect to the five variables are evaluated. 5. A Newton-Raphson iterative step is performed to gain new variable values al,, P, dl,, yl, and gl,. If the length of iteration vector (hfk-'- h f k )is greater than the stop criterion value, return to 2. Step limitation is recommended in two levels. The phase split parameter (P) is step limited to the highest level to steps of a maximum of lAPl 50.1. The mole fraction y1 is limited to steps that keep it greater than lo4 and less than unity. A method to bypass divergence, if it should occur, is to solve the Rachford-Rice equation (eq 54) once, using the K values obtained in iterative step k . The resulting phase split parameter, 0, can then be used as the updated value, maintaining all the other variables constant for iteration step k + 1. This method is easy to implement, and it works for most cases.
Results As an example, an oil mixture given by Simon et al. (1978), mixed with COz (25% oil + 75% C02),with the composition and the critical properties shown in Table I is used. The interaction parameters (Kij)for interactions between COz and hydrocarbons are from Lin (1984). The heavy end is characterized by using the method of Pedersen et al. (1985) into four pseudocomponents, where the estimated properties shown in Table I1 are used. The interaction parameters (Kij)between C02 and the pseudocomponents are estimated to be Kij = 0.13. Similar calculations are performed using the same input parameters but with another flash program, allowing interaction coefficients between all components (Michelsen, 1982a,b). For the hydrocarbon-hydrocarbon interactions,
Table 11. Characterization of the C7+ Fraction of Oil B (Simon et al., 1978) into Four Pseudocomponents X T,,K Pc, atm w Mw, g b " SG c10 Cl7 C26 C42 c7+
0.05181 0.03005 0.01675 0.01113 0.1097
604.5 717.8 798.2 903.8
26.14 16.14 11.31 9.327
0.5437 0.9679 1.285 1.644
134.9 239.6 360.7 613.7 239.1
0.7795 0.8355 0.8766 0.9267 0.8496
K, 0.13 0.13 0.13 0.13
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 2133 Table 111. Flash Separation (Oil B, Mixture 3, Simon et al. (1978)) at T = 397.05 K and P = 205.44 atm this paper
exptl
co2 c1 c2 c3 c4 c5 C6 c7 C8 c9 c10
c11 c12 C13 C14 C15 C16 C7-Cl6 C17+
B
Y 0.84445 0.11052 0.01052 0.00628 0.00533 0.00420 0.00329 0.00417 0.00556 0.00223 0.00164 0.00094 0.00051 0.00023 0.00008 O.ooOo3 0.00001 0.01540 0.00001 0.679
X
0.50572 0.06628 0.01315 0.01170 0.01326 0.01298 0.01332 0.02110 0.03333 0.01853 0.1843 0.01835 0.01829 0.1245 0.02231 0.01582 0.01448 0.19309 0.17050
Y 0.84014 0.10887 0.01084 0.00720 0.00593 0.00479 0.00417
0.01599 0.00207 0.726
Michelsen, 1982a,b
Michelsen, 1986
X
Y
X
Y
X
0.52517 0.06488 0.01064 0.00971 0.01123 0.01189 0.01369
0.84014 0.10867 0.01084 0.00720 0.00593 0.00479 0.00417
0.52518 0.06489 0.01064 0.00971 0.01123 0.01189 0.01370
0.81897 0.12163 0.01147 0.00742 0.00610 0.00492 0.00441
0.70811 0.07941 0.01031 0.00821 0.00828 0.00801 0.00845
0.14680 0.20602
0.01599 0.00208 0.726
0.14679 0.20599
0.01975 0.00533 0.412
0.07434 0.09488
however, the Kij values are taken equal to zero here too. Finally the calculations are made using all interaction coefficients equal to zero (Michelsen, 1986). From Table 111,it is seen that the Peng-Robinson EOS represents the experimental data fairly weel, when all hydrocarbon-hydrocarbon interaction parameters are equal to zero, and non-zero interaction parameters are used to describe hydrocarbon-C02 interactions. The vapor fractions are somewhat overestimated compared to the experimental points and so are the vapor-phase mole fractions of the hydrocarbons. By use of the interaction parameters equal to zero between C02 and the hydrocarbons, a somewhat poorer result is obtained for the vapor-phase composition, and unacceptable results are seen for the phase split and liquid-phase composition. For a calculation at P = 272.65, the feed composition was erroneously found to be stable where all Kij interaction parameters were set equal to zero. The flash calculation of using the method of this paper is compared to the result obtained when another flash calculational principle using the general dominant eigenvalue method (Michelsen, 1982a,b) is used (Table 111). The inputs to the two flash programs were identical, and the result from the two programs are almost identical too. Three flash calculations were performed by using the flash principle of Michelsen (1982a,b) and the principle of this paper, in order to discover deviations in computing time between the two principles. The mixtures used contained 7, 10, and 14, components respectively. The mixtures and the executing times for the various flash calculations on an Olivetti M24-SP personal computer equipped with a Intel 8087 coprocessor, using a Watcom Fortran-77 compiler, are shown in Table IV. It is seen that the reduction is about 37% with 7 components, and increases to about 45% in calculations with 14 components. With many components, larger savings in computing time should be expected. From table IV, the storage requirements of the two methods are illustrated by the array area occupied in a calculation with the 14-component mixture. Here significant savings in array area are observed when the procedure of this work is used.
Generalized Approach The flash algorithm outlined in this paper is limited to two-parameter cubic equations of state and to mixtures
Table IV no. of mixture components 1 7 2 10 3 14 storage requirement: array area 14
execution time, s this Michelsen, paper 1982a.b 5.3 8.4 5.7 9.6 6.9 12.6 3.39 K
20.82 K
Mixtures Calculated at T = 300 K and P = 20 atm mixture 1: 30% Cop, 20% C1, 10% C2, 10% i-C4, 10% i-C5, 10% C6, and 10% C8 mixture 2: 20% cop, 20% c1, 10% C2,5% C3,5% i-C4,5% n-C4, 5% i-C5, 10% C6, 10% C7, and 10% C8 mixture 3: 20% cop, 20% c1, 10% c2, 5% c3, 5% i-c4, 5% n-C4, 8% i-C5, 2% n-C5, 1% C6, 6 % C7, 4 % C8, 8% C9, 4% N2, and 2% H,S
containing one dislike component only. The principle can be generalized to cubic equations of state with more parameters (NP parameters) (Jensen, 1987). It can further be extended to mixtures containing any number of dislike components (NH dislike components among a total of N components). The number of variables observed in the general case is 2NH + NP + 1 (Jensen, 1987). For a four-parameter equation of state and with mixtures containing both hydrocarbons, water, and COz, the flash problem can be solved in nine equations in nine unknowns, irrespective of the number of hydrocarbons present.
Conclusion A simplified flash procedure for hydrocarbon mixtures mixed with one non-hydrocarbon component is developed. The isothermal flash problem is reduced to solving five equations in five unknowns, irrespective of the number of components, when deviations from the geometric mean (K..)are zero for hydrocarbon-hydrocarbon interactions ana non-zero for non-hydrocarbon-hydrocarboninteractions. Significant savings in computer time and storage requirements are obtained when compared to flash routines using general dominant eigenvalue methods. The procedure is suited for use in simulaton of, e.g., COz or N2 flooding processes.
I n d . Eng. Chem. Res 1987,26, 2134-2139
2134
Nomenclature a = equation of state parameter a(T)= a A = scaled equation of state parameter a1 = summation parameter defined by (13) b = equation of state parameter
B = scaled equation of state parameter dl = summation parameter defined by (14) EOS = equation of state f = criteria function in stability test, eq 4 1 g = constrain functions in stability test gf = flash equation gl = summation parameter defined by (15) h = variable vector in stability test hf = variable vector in flash problem k = distance between two tangent planes in stability test k = iteration number K = equilibrium constant K i . = interaction parameter m(T) = temperature correction to parameter a in EOS n = mole number N = number of components NH = number of non-hydrocarbons in mixture N P = total number of EOS parameters P = pressure r = distance parameter in stability test R = universal gas constant T = temperature u = mole fraction in iterative steps, resulting from h u = molar volume V = total volume x = mole fraction in phase 1 y = mole fraction in phase v Y = mole number of alternative phase in stability test z = mole fraction in feed Z = compressibility factor Greek Symbols p = fraction of v phase (n,jnf) = fugacity coefficient w = acentric factor C$
Subscripts c = critical state f = feed i = component number
j = component number or equation number (g, gf, dl) k = equation number k = EOS b-parameter number
1 = 1 phase t = total v = v phase
Superscripts e = estimated k = iteration number 1 = 1 phase v = v phase * = at solution, indicates partial derivative when used with subscript i Appendix For the Peng-Robinson EOS,the equations for the flash algorithm with sraled parameters A and B and compressed parameters al, dl, and gl evaluated from A and B are Bi(Z - 1) In @i = - In (Z - B ) B
N
a1 = C X ~ A ~ ~ / ~
(A-2)
i=2
N
dl = CxiAi1/2K,i
(A-3)
i=2
N
gl = CxiBi
(A-4)
i=2
A = x12A1+ 2~,A,'/~(al - dl)
B = x,B1
+ a12
+ gl
(A-6)
A,* = 2A11/2(xlA11/2 + a1 - dl) Ai* = 2x,(A1A,)1/2(1- Kl,)
+ 2AL1I2al for
(A-5)
(A-7)
2 Ii I N (A-8)
Registry No. COP,124-38-9; N, 7727-37-9. Literature Cited Jensen, B. H. Ph.D. Dissertation, Technical University of Denmark, Lyngby, 1987. Lin, H.-M. Fluid Phase Equilib. 1984, 16, 151. Michelsen, M. L. Fluid Phase Equilib. 1982a, 9, 1. Michelsen, M. L. Fluid Phase Equilib. 198213, 9, 21. Michelsen, M. L. Ind. Eng. Chem. Process Des. Deu. 1986,25, 184. Pedersen, K. S.; Thomassen, P.; Fredenslund, Aa. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 948. Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundum. 1976,15,59. Simon, R.; Rosman, A,; Zana, E. SOC.Pet. Eng. J . 1978, Feb, 20. Soave, G. Chem. Eng. Sci. 1972, 27, 1197.
Received for review October 27, 1986 Revised manuscript received June 30, 1987 Accepted July 23, 1987
Multiphase Behavior of Supercritical Fluid Systems: Oil Solutions in Light Hydrocarbon Solvents Maciej Radosz E x x o n Research and Engineering C o m p a n y , Annandale, N e w Jersey 08801
Multiphme equilibria have been experimentally studied for oil solutions in light hydrocarbon solvents. Phase diagrams for such solutions have been found to depend primarily on the degree of molecular size asymmetry. Specifically, low molecular weights of solvents but high molecular weights of oils favor large VLL regions. T h e equilibrium two-phase compositions are explained by utilizing retrograde phenomena concepts and regular P-X and T-X diagrams. P-T phase diagrams for such oil solutions, qualitatively similar to those known for simple binary mixtures of hydrocarbons, allow constructing H-T enthalpy diagrams which are also discussed in this paper. Phase equilibria in binary mixtures containing hydrocarbons having similar molecular sizes are usually close to 0888-5885/87/2626-2134$01.50/0
those expected based on simple solution theories. For example, such similar hydrocarbons are completely mis0 1987 American Chemical Society
