Fluid Phase Equilibria, 3 (1979) 273-290 273 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

A VERSATILE ALGORITHM FOR CALCULATING VAPOUR-LIQUID EQUILIBRIA l

L. ASSELINEAU, G. BOGDANIC l * and J. VIDAL

Institut Francais du Pktrole, 1 et 4, Avenue de Bois-PrPau, 92502, Rueil-Malmaison (France)

(Received February 26th, 1979; accepted in revised form July Sth, 1979)

ABSTRACT

Asselineau, L., Bogdanic, G. and Vidal, J., 1979. A versatile algorithm for calculating vapour-liquid equilibria. Fluid Phase Equilibria, 3: 273-290.

A versatile algorithm is proposed for solving vapom-liquid equilibrium problems. It has been prepared so that the search procedure is generally applicable with any analytical equation of state and any kind of data. Special attention is paid to the applicability of the method in critical and high-pressure regions. Derivatives of the quantities describing the state of the system can be obtained for any equilibrium state as soon as equilibrium is determined. Results are reported for computing based on the use of a modified Redlich- Kwong equation of state.

INTRODUCTION

When applying the same equation of state to both phases, good results are obtained in the prediction or the correlation of vapourliquid equilibria, in the low, medium or high pressure range (Wenzel et al., 1971; Soave, 1972; Peng et al., 1976; Deiter et al., 1976). However, besides the problems of choos- ing the best equation of state, and of evaluating the binary parameters, some difficulties have to be solved when building an efficient calculation aigorithm. For example the diagnosis of retrograde condensation processes, which can be a solution of flash equilibria, is necessary. The so called trivial solution, with false unit equilibrium constants (Harmens, 1973, 1975; Shah and Bishnoi, 1978) must be avoided in the near critical range. Then, as we found, in the high pressure range, the convergence criterium of the iterative algorithm must be increased to obtain good accuracy in solutions, and the efficiency of first order iterative methods is doubtful as shown in a comprehensive study (Rohl and Sudall, 1967). Furthermore, if only two data define a vapour-liquid equilibrium, many kinds of problems can be proposed in common engineering

l Part of this work was presented at the Chlsa Congress, Prague, August, 1978. l * University of Zagreb, Technological Faculty.

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practice, according to the effective data such as pressure, temperature, vapor- ization ratio, volume, enthalpy, entropy so that a versatile algorithm would be useful.

Lately published works in this field (Peng and Robinson, 1977a; Hirose et al., 1978) have shown that applying a Newton--Raphson iterative technique results in a quick convergence. Furthermore, the method can be applied to the principal flash calculations (Hirose et al., 1978). In this work, we use an algorithm which can be applied to any kind of vapour-liquid equilibrium problem. Since the Newton--Raphson technique is highly initialization sensi- tive, we describe methods which give an initial guess of the unknowns, and ensure a successful iterative procedure. The values of the derivatives of the equilibrium conditions can be used to make a diagnosis of the retrograde con- densation process and a quick stepwise calculation of the coexistence curve, and may also be necessary for calculating multistage separation (Shah and Bishnoi, 1978). We show that the exact values of these derivatives can readily be obtained from the Jacobian matrix used in the iterative search.

THERMODYNAMIC MODEL

The equation of state (modified Redlich-Kwong equation) used for this work has the form

[P + [a/v(u + b)] ] (v - b) = RT (1)

where for the pure components, parameters a and b are calculated from Soave’s correlation (Soave, 1972)

l aii = g(21/3 - 1) [l +mi(l- E)] Rizi . (2)

with

mi = 0.480 + 1.547 ~i - 0.176 ~a (3)

and

bii = g (21’3 - l)(RT,i/P,,i) (4)

For the mixture, the classical mixing rules are applied. For example, for a liquid phase with composition xi, parameters a and b are calculated:

a = C C CZijXiXj where aii = Uji = fi(l - h,) (5) I j

b = C C bijXiXj where Oij = [(&ii + bjj)/2](1- k;) (6) I i

275

From this, the fugacity coefficients can be expressed as

_ -- - +1-

where the molar volume uL is obtained from the equation of state. The same expression is applied to the vapour phase with composition yi. Since P is the same for both phases, equilibrium condition

f:(T, P, [YI) = f:(T, P, [xl) @a)

can be expressed by

yi~v = Xi~L (6b)

Parameters a and b, the molar volume u and fugacity coefficients depend only on the variables T, P, x (or y), and by mathematical manipulations their partial derivatives can be obtained. In what follows, the molar fraction xi (or yi), where i = 1, . . . . n, will be considered as independent, but the stoichio- metric equations

kxi=l and gYi=l

will be added to the equilibrium solution equations.

PSEUDO-CRITICAL COORDINATES AND PHASE DEFINITION

Calculated P(u) isotherms are shown on Fig. 1 for a mixture of constant composition. Between the bubble and dew points, which are shown on each isotherm, phase splitting occurs, and calculated values have no physical mean- ing, but may be encountered during the iterative calculation of phase equi- libria. The isotherm shapes are similar for a mixture and for a pure compo- nent and we can see that a temperature range exists for which the mechanical stability condition (aP/au),. < 0 is fulfilled whatever the volume may be. The temperature limit can be calculated from the equation of state and the condi- tions

(w/au), = (a2Ppv2jT = 0 (10) As pointed out by Rowlinson (1969), the mechanical stability region lies

inside the two phase region, and the P, u, T coordinates determined in this way are not the true critical values, with the exception of pure components;

276

Log 10"hn'/mols 1

Fig. 1. P(v) Isotherms for an equimolar mixture of ethane and n-butane. A, bubble points; q , dew points; X, pseudo-critical point; Tp, = 369.58 K;Pp, = 41.79 atm; upC = 241.91 cm3 mole-l.

hereunder, they will be called “pseudo-criticals”. So, the pseudo-critical tem- perature of a mixture is the lowest temperature for which the equation of state has only one root in volume, whatever the pressure may be.

When using the equation of state (l), and conditions (lo), the pseudo- critical temperature can be calculated by solving:

e(T,,, [xl) 1

RTpc b( [xl) - 3(21’3 - 1)s = O (11)

where a(T,,, [xl) and b( [xl) are calculated from eqns. (2) to (6). The pseudo- critical volume is given by

UPC = [~(13c1)l/w’3 - 1) w3 and the pseudo-critical pressure calculated from T,, and up,, applying the equation of state.

One of the greatest difficulties encountered when an equation of state is used in phase equilibrium calculations is caused by the phase definition. If the temperature is lower than the pseudo-critical one, the liquid and vapour, ranges are unequivocally diagnosed, as can be seen on Fig. 1. For the liquid phase,

u is less than upc and for the vapour phase, u is greater than uPC. At higher temperatures, where phase equilibrium still exists, such a distinction is not obvious, except by means of the continuity of behaviour with varying tem- peratures, or by a comparison of the density of both phases. In this range, equilibrium computation can turn to the so-called “trivial solution”, for which both vapour and liquid composition are set at the same value. To avoid such a failure, we used the possibility of calculating the pseudo-critical point, as will be explained later.

CALCULATION ALGORITHM

The thermodynamic properties of a system containing vapour and liquid phases in equilibrium can be described by 2 n + 4 variables, where n is the number of components in the system, assuming that the total mole number of each constituent, iV;, in the system is known, a priori. The 2 n + 4 variables include all of the following: pressure, P, temperature, Z’, total mole number of the liquid and vapour phases, NL and NV, and the mole fraction in the liquid and vapour phases, x1, x2, _.., x, and yr, y2, . . . . y,,. Hereunder these variables will be called the system coordinates, and denoted as the [U] vector

[VI = tr e NL, NV, Xi, --., xn, Yi, .--, Ynl (13)

They are related by 2 n + 2 equations, as shown in Table 1: material bal- ance requirements for each of the n components (rows 1 to n), equilibrium condition (rows n + 1 to 2 n), the stoichiometric eqns. (9) which by combina- tion with the material balance give (row 2 n + 1)

and the overall mole balance (row 2 n + 2)

(14)

(15)

So, among the coordinates, two must be specified as data. Instead of this, the knowledge of a property of the system may also be satisfactory, if such a property can be calculated from the coordinates, for example the total vol- ume, or the liquid volume, or the vapour volume, or the total enthalpy or entropy. Such are, for example, the data in isenthalpic or isentropic processes. In any case, data will be used by the way of two specifications equations, according to rows (2 n + 3) and (2 n + 4) in Table 1, where such coordinates or properties (T, P, NV, V, I-Z, S, . ..) are denoted by CI and /3.

In the non-linear system

[Gl = 101 (1’3)

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TABLE 1

Equations (16), Gk(T, P, NL, NV, x, y) = 0, k = 1, .., 2 n + 4; for vapour--liquid equilibria

Row Equation

1 NLxl + NVyl - N; = 0

2 NLx2 + NVy2 - IV; = 0

. . . n

n+l

n+2

. . .

2n

2n+l

2n+2

2n+3

2n+4

IPYY, - P,LX, = 0

~ Xi-_yyi=O 1 *

NV+NL-NNtot=O (Y-a* =o p-p* =o 1 suchas{pT~~~~~or{P-P* =O NV-NV* ~0

or

and we now have (2 n + 4) equations, and (2 n + 4) unknowns, the coordi- nates;

It is clear that only the two last rows depend on the kind of problem. The Newton-Raphson method will be used for solving this. At any iteration step, m, we shall write:

]aG/aU], . [AU] = - [G], (17)

where G is the vector of the functions defined in Table 1, U is the vector of coordinates and AU the vector of the Newton-Raphson increments:

IU,+l = [Urn + [AU1 (18) Cqnvergence will be achieved if

IIAUII G c (19)

where E depends on the computer (here E will be 10-12). Suitable checks have to be built into the algorithm in order to obtain a

good iterative sequence. First the new values of the coordinates must have physical significance, i-e:, !L’, P, NL, NV and the molar fractions must be posi- tive, new generated phases must satisfy their definition (“liquid” or “vapour”) when such a definition is possible (7’ < T,,), and the liquid density must be higher than that of vapour. Then the test

IIG m+lll < IlGmII (20)

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First initialization

T P N‘ NV x y

b,;, ='I&'

I-~~

Calcuiate fy, f4

and derivatives

Build matrix

Unsolved problen

Fig. 2. Schematic diagram of the proposed algorithm.

must be satisfied. If any of these conditions are not fulfilled, every increment will be diminished by half of its value, up to five times.

When applying such a method, the choice of the initial guess values [U,] can be decisive and will be fully described hereunder. The schematic calcula- tion algorithm is shown in Fig. 2.

280

DERIVATIVES CALCULATION

When an equilibrium is determined by the above method, the derivatives of the coordinates Ue which describe the system in an equilibrium state with respect to any among the independant variables: T, P, NV, . . . . H, S, can be obtained readily.

Such results are useful for various applications such as, for instance, to char- acterize the well known retrograde condensation process. Considering a mix- ture such as that represented in Fig. 3, it can be seen that both the criconden- therm and cricondenbar points lie on the dew curve. This means that the same temperature can correspond to two dew pressures (isotherm AB in Fig. 3), and the same pressure to two dew temperatures (isobar DB in the Fig. 3). A dis- tinction between the dew states, for example points A, B, D, can be made from the sign of the derivatives as is shown in Table 2. In the same way the cricodentherm and cricondenbar can be located from the value of (aP/a T),,. Knowing the derivatives is also useful for computing the evolution of a mix- ture through continuously changing states in isenthalpic or isentropic com- pression and expansion. As a third example, as will be explained hereunder, derivatives were used in this work for a stepwise calculation of the whole co- existence curve, determining in this way the cricodentherm, cricodenbar and critical points.

Since an equilibrium state is defined by two data, the derivatives we can obtain are partial derivatives. If OL and /3 are two coordinates (T, P, . . . . y,) on properties (V, H, S) we shall obtain:

[(a piap),i

Here, we shall first underline that properties a and /3 can be different from

P

D

C /‘ Critical paint

Bubble carve

3

) ‘A

T

Fig. 3. Retrograde condensation (see Table 2).

281

TABLE 2

Diagnosis of retrograde condensation (see Fig. 3)

Point P T NV/&t @PlaT),v (aNVIaT)p

A pA TA 1 >o >O B pB Ts = TA 1 <o >O D PO =PB TD 1 >o <o

the data used to define the equilibrium problem. For example, after solving a partial vaporization with T and P as data, where (Y is T and fl is P, we can turn to the calculation of derivatives (a Ue/aP), where 01 is H and /3 is P.

For any infinitesimal change of [ Ue] , material balances, equilibrium condi- tions and stoichiometric relations are preserved; so, we can write:

[aG,/aU][dUe] = dGR = 0 k:1,2n+2 (21)

where [a Gk/a U] is the vector of the partial derivatives of the Gh function with respect to the coordinates U.

As we are calculating partial derivatives taking cu as constant we have

[sol/au] . [dUeI = dcu = 0

and, as the last relation, the expression of the p differential

[apjau] * [dU=] = dp (23)

where [ acu/tl U] and [ap/a U] are the vectors of the partial derivatives of (Y and fi with respect to the coordinates U.

Dividing eqns. (21) to (23) by d/3, a linear system is obtained, the unknowns of which are the partial derivatives of the equilibrium conditions

(24)

where [G] is the vector previously defined by eqns. (16) and [C,] is the vector

[Cpl = 1% 0, . . . . 0; 0.11 (25)

2n+Z

It must be underlined that the values of the derivatives [aG,/aU], k = 1, 2 n + 2 have been calculated in the last step of Newton-Raphson iterative search, and are the same whatever the data of the previously solved equilibri- um problem may be, and the kind of partial derivatives we want. Only the vectors [acu/aU] and [ap/aU] have to be calculated if these properties (or coordinates) were not used to define the equilibrium problem.

It is also obvious that, if CY or 0, or both are among the coordinates [U] , we have only 2 n + 3 or 2 n + 2 partial derivatives, and the system (24) can be reduced. Such are for example the derivatives (aP/aT),v or (aiVv/aT), which enable retrograde condensation to be diagnosed.

282

INITIALIZATION

The main problem in the Newton-Raphson iteration method is initializa- tion, i.e. providing a first set of guess solution values To, PO, N$, Ng, Xi.0. If the problem data are T’ and P’, or T* and NV * , or P* and NV * , application of the following method is found to be highly reliable.

Input data at our disposal are used to provide values of T,, and PO, To and Nx or PO and We, i.e., To = T” or PO = P* or Nz = NV* respectively, depend- , ing on the kind of problem.

Equilibrium ratios are estimated from Raoult’s law

Yi.clXi,c = Ki = F?/J’c (26)

where the “vapour pressure” values are calculated from critical data and nor- mal boiling point (or any vapour pressure value) by linear interpolation or extrapolation, whatever the value of To may be (lower or higher than TC,i):

ln fi - ln Pb i _ (~/TO) - (l/Tb,i) -2 -

ln pc.i - In pb.i (l/Tc,i) - (l/Tb,i) c-9)

By combining eqn. (26) and material balances, we obtain

Xi.0 = NT/(N,L + N,VKi)

Y i.0 = (NZKi)I(N,L + N,VKi)

(23)

(29)

Conditions (9) can be ‘written as

0 (30)

with

N,L+N;=N tot = cl Nf (31)

By solving eqn. (30), the unknown (Nr or through the K, values, To or PO) is calculated and initialization achieved by applying eqns. (28), (29) and (31).

From the guess solution obtained in such a way, the iterative procedure was successful for most problems. Equilibria with super-critical components or azeotropic systems were solved, in spite of vapour pressure extrapolation or Raoult’s law being used to obtain the first guess solution.

ALTERNATE PROCEDURES

Initialization from the pseudo-critical point

It has already been stated that a so-called “trivial solution” can be found when using the same equation of state for both phases. Diagnosis of this fail-

283

ure is easy by the equations:

xi = yi

and

i = 1, . . . . n (32)

& = “v (33)

In the last iteration of the Newton-Raphson sequence the matrix degener- ates. The right hand sides of the linear system (17) are small but calculated increments are high and erratic. Such a situation can be encountered in the high pressure range if data are T and NV, or P and NV, when the true solution is near the extrema in, respectively, T or P of the P(T) function at constant NV, or near the critical point, whatever the data may be. This failure is hardly ever encountered by the previously described initialization procedure, but when it appears it is nearly always associated with the existence of only one root of the equation of state for both phases in the very first iteration step.

When finding a trivial solution in problems in which input data were T (or P) and NV, the following initialization was attempted. The pseudo-critical coordinates of the whole mixture are calculated as was previously described by solving eqn. (11). Then, whatever the data may be, the first input guess solution for the Newton-Raphson procedure will be

TO = Tpc - AT where AT = 1 Kelvin (34)

Xi.0 = yi,e = Z{ = NT/N,,, (35)

PO = J’(To, upc) which agrees with eqn. (1) (36)

In respect to this, the equation of state will always have three roots. Fuga- cities and their derivatives will be calculated using the smallest root for the liquid phase and the largest root for the vapour phase. Compositions will quickly distinguish themselves by the evaluated increments after solving lin- ear system (17). As is obvious from the last two rows of system (17), after incrementations the temperature or pressure would be the same as T* or P* as soon as the second iteration step starts. A subrelaxation was imposed to diminish every increment, up to the fifth iteration, in order to improve the phase composition distinction. In this way, a true solution was achieved up to the very close vicinity of the critical point.

When the input data were T* and P' , and if the trivial solution failure is encountered, a calculation of the dew temperature Td corresponding to the P* data is provided. Then, starting from this point, the equilibrium is calcu- lated as described hereunder.

Initialization from an equilibrium state

Whenever an equilibrium problem is solved, with solutions [ IP], and if a new set of data, (Y*, /3*, is provided, the previously obtained [ Ue] coordinates

284

can be used as first guess values.

ru, = ue (37)

Since material balances, equilibrium and stoichiometric conditions are verified for such a starting point, eqns. (17), which give the first Newton-Raphson increments, can be written:

[~G/~~l,[A~l = (a* -a)[Gl + (P* -P>[c,~l (38)

where

Ll = r3 . . . . 0, ?,Ol (39) 2n+2

and [Cg] is defined by eqn. (25). Therefore, the [AU] values are connected to the derivatives of the equilib-

rium conditions (a U”/aol )s and (a V/a&, which, as previously shown, can be obtained from eqn. (24) and

[aCiaU][(aUVa),] = C,

since, from eqns. (24), (38), (39) and (40) we can write:

(40)

[AU] = [(a UV+](o* -a) -t r(apiap),](p* -0) (41)

So, we have two possibilities of initialization at hand. First, we can apply eqn. (37) and obtain from eqn. (17) a first set of [AU] values. Or, if the required derivatives have been calculated, we can use a first order Taylor ex- pansion.

[U’], = [ Ue] + [AU]

where AU is obtained from eqn. (41). Both ways will produce the same itera- tive search, with a one step difference.

When the new set of data is such that the coordinates of the new solution will be far from the previous set, such initialization may not result in solving the problem. Then, some steps have to be considered between the two states.

We applied this method, as mentioned above, when a problem, the data of which were P’ and T*, could not be solved in the normal initialization way. The starting points were the dew temperature at P*, or the dew pressure at T*, or the bubble temperature at P*, or the bubble pressure at T* .

The same technique was applied for calculating isenthalpic, or isentropic, or isochore evolutions and, as a last example, we shall give the calculation of the full coexistence curve (P(T) with FV = 0, or 1).

STEPWISE CALCULATION OF THE COEXISTENCE CURVE

Handling derivatives of equilibrium conditions enables a straight sweeping process to be used for the coexistence curve. Starting from a dew point calcu-

285

lation in the low pressure range, and by choosing an increment in temperature (for example AT = 10 K), the next calculation will be initiated from eqn. (38) with

CY *=T*=p+AT

and

p* = Nve = N,v” = Ntot

In such a way, convergence will be quickly achieved. From the derivative value (aP/a T),, the effective data will be choosen as T* and NV near the extremum in pressure, and P* and NV near the extremum in temperature. To obtain the full coexistence curve, AT will be positive or negative if the prod- uct (aPlaqNv . (aN"/a T)P is positive or negative. The same calculation will be used for the bubble curve, or any equilibrium curve at constant NV. Multi- ple solutions are determined in this way in the retrograde condensation range. The so-obtained family of curves goes to a single limit, which is the critical point of the mixture; in the nearest neighbourhood of that point, trivial solu- tions will be obtained, but, from the last determined equilibria for different values of the vaporized ratio, and from derivatives (LIP/i3 T),", a short and pre-

$1

1 I I ,

0.25 0.50 (x75 ;WW Fig. 4. pressure-composition diagram for ethanelcarbon dioxide system bubble point calculation using first initialization method (0) and alternative method (0). A, Critical points at 293.15 K.

286

cise extrapolation enables the critical data to be determined. It must be noted that such a stepwise calculation, described here for a con-

stant vaporization ratio, can be applied for other evaluations, at constant pres- sure or temperature or volume, for example, since the necessary derivatives can easily be calculated.

APPLICATIONS

Some illustrations of vapour-liquid calculations will be given. Figure 4 shows the equilibrium isotherms for an ethane/carbon dioxide system at

2ol

15

10

5

5

3-

O-

O-

O-

/

L

‘/otm.

c

I

__e

5

, /

0 100 200 -

8/V

Fig. 5. Vapoul--liquid equilibria calculated for a natural gas’containing (mole percent): nitrogen 4.5%, methane 7096, ethane 11.71%, propane 4,97%, n-butane 2.77%, n-pentane 1.5%, n-hexane 0.98%, n-nonane 3.57%. - Constant vaporization ratio curves, - - -, isenthalpic and e-.-*, isentropic expansions. Iteration number and solution coordinates using the data: A, T and FV, 0, P and FV; +, 2' and P. XC: critical point.

. 263.15, 283.15 and 293.15 K. The interaction constant Was adjusted from the experimental data (Fredenslund and Mollerup, 1974) and the critical coordinates calculated by a separate algorithm (Huron et al., 1978a). At 263.15 K, the first initialization method described above was successful at any composition in spite of the implicit use of Raoult’s law. At higher tem- perature and near the critical region it can be seen that the alternative method is necessary; so the full temperature and composition range corresponding to experimental data was covered.

To give a second example, a multicomponent mixture was studied. Since experimental data were not available, the values of the interaction constant are zero for each binary and the results, shown in Figs. 5 and 6, must be looked upon only as an illustration of the calculation algorithm. Figure 5 presents some samples of vapour-liquid problems, together with the number of iterations in the Newton-Raphson iterative search: data were T and FV, orPand FV, orPand T.

Isenthalpic and isentropic evolutions, as calculated in a stepwise process, are also presented. It must be underlined that for such a mixture, the bubble curve lies in a very narrow stability range, so that the calculation of such

’ P/c&m

195

190

185.

I i

Fig. 6. Illustrating critical coordinate determination for the multi-component mixture presented on Fig. 5. of = (11.5 k 0.2) OC;P, = (189.4 i 0.2) atm; (,&‘/a@), = (0.85 f 0.03) atm ‘C-l; - - - e - - -, the constant vaporization ratio curve, calculated point and tangent. .,. (

288

.

equilibria was a difficult problem. For most mixtures and conditions the num- ber of iterations is from 5 to 10. A larger scale (Fig. 6) enables the critical range to be examined. The constant vaporization ratio curves were calculated in a stepwise way, together with the (aPli3T),v values. In this way, the criti- cal coordinates are determined with a good precision.

This method is not as rigorous as the one recently presented (Peng and Robinson, 1977b) with regard to stability conditions, but it requires nothing but the functions used in the vapour-liquid equilibrium algorithm.

CONCLUSION

The method described in this paper was applied to vapour-liquid equilib- ria correlation and parameter optimization (Asselineau et al., 1978). Exten- sive use of it showed us that it is both versatile and reliable. Using a Newton- Raphson iterative technique, the precision required for solving the equilibri- um equations in the high pressure range was easily obtained up to the com- puter limit.

As has been shown, when an equilibrium is determined, the derivatives of the quantities describing the state of the system with respect to any among the possible data can be obtained, thus making possible retrograde condensa- tion diagnosis and stepwise calculation of various equilibrium curves, up to the critical point.

The algorithm is self-starting. Three kinds of initialization are built-in. Our experience is that the straightforward initialization for bubble, dew and flash calculation is effective in most cases, but if a trivial solution should be encoun- tered, it can be replaced by an alternative one.

Enclosing data as specification equations in the resolution matrix enhances the versatility of the method and also makes it possible to use the properties of the system such as total volume or enthalpy or entropy instead of the main variables P, T, NL, NV, x, y.

The algorithm is prepared in such a way that the search procedure is gener- ally applicable with any analytical equation of state, introducing nothing but expressions for solving this new equation of state and for calculating the thermodynamic properties and their derivatives. For example, alternative mixing rules were introduced without difficulty (Huron and Vidal, 197813).

NOTATION

a, b f FV G H

Ki kij

equation of state parameters fugacity vaporization ratio FV = NV/Ntot function vector, as defined by Table 1 enthalpy equilibrium ratio correction factor for the geometric average rule for ail

289

Ii P S T u V V X Y 2

number of components in the system mole numbers pressure entropy temperature system coordinates vector, as defined by eqn. (13) volume molar volume mole fraction of liquid phase mole fraction of vapour phase overall mole fractions

Greek letters

7 system property such as V, or H or S

cp fugacity coefficient e temperature

-w acentric factor

Subscripts

b bb C

d i, j m

PC tot

pure component boiling point bubhle point critical point dew point components of mixture m-th iteration step pseudo-critical point total

Superscripts

e equilibrium state L liquid state S saturated state V vapour state * specified data

Symbols

[ ] matrix 11 11 vector module

290

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Fredenslund, A. and Mollerup, J., 1974. Measurement and prediction of equilibrium ratios for the CsHgC0.g system. J. Chem. Sot., Farad. Trans. I., 70: 1653-1660.

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Hirose, Y., Kaware, Y. and Kudih, M., 1978. General flash calculation by the Newton- Raphson method. J. Chem. Eng. Jpn., 11: 150-152.

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