Computers them. Engng, Vol. 17, No. 9, pp. 879-884, 1993

Printed in Great Britain. All rights reserved 0098-I 354/93 $6.00 + 0.00

Copyright 0 1993 Pergamon Press Ltd

MULTIPLE LIQUID-VAPOUR EQUILIBRIUM FLASH CALCULATION BY A GLOBAL

APPROACH

E. ECKERT and M. Kursfikc Department of Chemical Engineering and Department of Mathematics, Prague Institute of Chemical

Technology, 166 28 Prague 6. Czech Republic

(Received 10 April 1992;finaI revision received 2 October 1992; received for publication 8 January 1993)

Abstract-A global approach to the calculation of nonadiabatic-isobaric multiple liquid-vapour equi- librium flash is suggested. The structure of the system of equations is adaptively changed with respect to suggested engineering-oriented rules. The resulting set of equations is solved by the Newton-Raphson method. Results of two examples (equilibrium mixtures of the type G-L-L and G-L-L-L) show the efficiency of the algorithm.

1. INTRODUCTION

The appearance of more liquid phases is generally an undesirable phenomenon in distillation processes; however, it is a reality in practice. The problem of three-phase distillation has been discussed in the literature, a good survey can be found in the paper by Cairns and Furzer (1990). The prediction of the appearance of more liquid phases by using a minimiz- ation of Gibbs energy (e.g. Gautam and Seider, 1979; Michelsen, 1982) seems to be superior to fugacity matching because the latter condition does not guar- antee that the solution possesses a global minimum. Ohanomah and Thompson (1984) performed a com- prehensive study of several techniques and found the Gibbs energy miminization method not to be competitive for multiphase equilibrium calculations_

Nelson (1987) suggested an algorithm for isother- mal-isobaric multiple-phase flash calculation based on two nested iteration loops. Ratios of phases and their number are corrected in the inner iteration loop, compositions of the phases are iterated in the outer loop. Poorly chosen initial guesses can produce phys- ically unacceptable situations in the course of iter- ations. Some improvement of this algorithm in these cases was suggested by Eckert (1989a). Recently Biinz et al. (1991) used Nelson’s method for the calculation of high-pressure ternary phase diagrams of two three- component mixtures. A modification of this method for a nonadiabatic flash needs an outer iteration loop also for temperature, which may cause convergence problems. We found no quotations of an example using a generalization of Nelson’s method for the problems with the possible occurrence of more than two coexisting liquid phases.

A global Newton-Raphson method was rec- ommended many times for simulation of rectification columns where only one liquid phase occurs, e.g. Kubieek et al. (1976). Eckert (1989b) also suggested the use of the global approach for three-phase rectifi- cation columns with a variable dimension of the problem depending on the number of existing phases. A modified Nelson’s algorithm is used to determine the number of coexisting liquid phases on individual trays after each Newton-Raphson iteration.

In this paper we present an algorithm for nonadi- abatic-isobaric flash calculations where the dimen- sion (the number of equations) remains constant. However, the structure of equations is changed adap- tively dependent on the number of coexisting phases. The algorithm is formulated for an arbitrary number of coexisting liquid phases.

2. THE ISOBARlC FLASH

Consider the equipment schematically drawn in Fig. 1, usually called the isobaric flash separation unit. Assume that the feed specifications (feed flows Fkr compositions z,,+, . . , zNc and enthalpies I+*, k=l,... , m), heat supply (Q) and pressure in the equipment are given. The steady-state operation of the unit can be described by the following equations:

(i) equilibrium equations for p liquid phases and a vapour phase, r is the index of a chosen reference liquid phase:

O=y,--K:xj, i=l,...,n, (la)

O=K~x~-K{x+, i=l,...,n;

j=l,...,p,j#r; (lb)

879

880 E. Ecxsar and M. KUB~CEK

I G

5

%7- Q ---_-

im . . . ti L? LP

Fig. 1. Multiple liquid-vapour flash unit.

(ii) component balances:

(iii) enthalpy balance

0=1- Gh,+ 5 LJhJ, it fih,+Q . J-1 k-1 >

(3)

Assuming that a component I does not condense or evaporate, the Zth equation in (la) is substituted by:

0 =x; (4)

or

o=y,, (5)

respectively.

A further p + 1 necessary equations have to be chosen from the following 2p + 2 equations:

O=G or 0=1-gyi, i= I

(6)

o=Lj or 0=1-g& j=l,...,p, (7) i= I

depending on the phase situation (i.e. on the number of coexisting liquid phases p and the existence of vapour phase). This choice is demonstrated in Table I for a possible coexistence of two liquid phases (p = 2). Of course, the number of cases increases rapidly with p, generally, it equals to:

(p + 1) ! c [(p + 1 -j)!j !I--‘. j- I

As a result we obtain a system of

Iv = (n + l)(p + 1) + 1

nonlinear equations:

0 = V(X), (8)

for Nunknowns y=(y,, . . . ,y,), [xj=(xj,. . ..x<). j=l,_.. ,p], G, L’, L2,. . . , Lp, T, denoted in a vector form as X. The composition of a nonexisting phase is fictitious and does not fulfill the correspond- ing summation condition in (6) or (7). Instead, equation 0 = G or 0 = LJ applies, see Table 1.

Table 1. Choice of additional equations for possible occurrence of two liquid phases @ = 2)

Existing phase(s) Case Equations Conditions

Single vapour G O=L’. O=LZ cx: -=z 1. Exx:<I,

O=l-Cy, G>O

Single liquid I LI O-G. O=L’ ~xf<l orx’=x~.

o=I-~x: EY,< 1, LIZ0

Single liquid 2 L2 O-G. O=L’ xx; < 1 or x’ = XL,

o= t-xx: CY#C 1, L’>0

Liquids I and 2 Ll-L2 O=G, O=l-xx; cJJ,< 1, L’>O,

O=l-xx: #c*>o

Vapour and liquid 2 C-L2 O=L’. O=l-_cy, ~x)cl orYP=x*,

o=l-~x: L’>O. G z-0

Vapour and liquid 1 GLI o-L2, O=l-xy, xx:< 1 or x’=x*,

o- 1-zx: L’>O, G>O

Vapour, liquids I and 2 G-Ll-L2 0=1-x)‘, L’>O, L=>o,

OPI-~X: G>O

O-I-~X:

Multiple liquid-vapour equilibrium flash calculation 881

An additional remark has to be made to the problem presented. The subsystem (1) has always a solution where:

xi’ = xf, i=l,...,n, (9)

for some t,s = I,... .p. t -c s. We shall denote such a solution as a trivial (parasitic) solulion. We can accept it only if at most one value from L’, L” is

positive (i.e. nonzero). When L’> 0, L’ z- 0 and (9) are valid then the Jacobi matrix of the system (8) is singular and the solution is not unique [the system (8) is infinitely satisfied by many values of L’, L” with the same sum L’ + L”]. This singular case can be trans- formed into a nonsingular one by the following operations: L’ + L’+ L’, 0 -+ L’ (phase s does not really exist). By repeating these operations for every t and s satisfying (9) and L’ > 0, L” > 0 we finally reach a solution where L' . L” = 0 (only such nonsin- gular solutions are considered in Table 1). The ques- tion of whether there exists another nonsingular case with a higher number of really (physically) existing liquid phases remains open.

3. THE ALGORITHM

An iteration technique must be used for solving the system of N nonlinear equations which consists of

equations (l-3) and p + 1 equations from the set (6-7). We used a Newton-Raphson method without any decomposition of the system, i.e. we used the so-called global approach. For the system (8) we thus have:

X” + ’ = Xk - I [cp ‘(Xk)] - ‘cp (XL). (10)

The N x N Jacobi matrix q’(X) can be computed in each iteration by using simple two-point difference formulas. An analytical method of evaluating partial derivatives is also possible but the formulas for K{ and hJ, may be complicated. The value E. = 1 is chosen when:

IIV(X*+‘)/I < JIcp(Xk)ll (11)

and a standard version of the Newton-Raphson method results. When the condition (11) is not fuhi- lled, the value of 1 is halved and Xk + 1 reevaluated (dumped method).

A very important problem is the choice of p + 1 equations from (67) which corresponds to an ex- pected set of coexisting phases. As we do not know this real “phase situation” beforehand, we developed an adaptive algorithm which enables us to switch to another “phase situation” in the course of the iter- ation process (always after several, e.g. two, iteration

Table 2. Decision diagram for possible switching between different cases, p = 2. In mws 14 the conditions in the last column are tested first. In the last row conditions in the first three columns are tested first. Marks /. \: in

the algorithm realization S/l means e.g. S > 0.99999, S\O means S < 0

To case

Fr0lll case G LI L2 LI-L2 G-L2 G-L1 G-LI-L2

G *

Ll

L2

LI-L2

G-L2 LV.0 -4

GLI L’\O G’xO * x.x:/1

x’ zx=

G-LI-L2 L’/O LWJ LYO *

L\O G\O

or or

G’._O x’ =x2

882 E. ECKERT and M. KUB~EEK

steps). The algorithm can be described by means of

a decision diagram. Such a diagram is presented

schematically in Table 2 for p = 2, i.e. for possible

existence of two liquid phases. After switching to

a new case it is useful to correct the vector of

unknowns so that it fulfills the conditions required

for the new case (summation conditions by normal-

ization. zero flowrates for nonexisting phases). Blank

boxes in Table 2 correspond to nongeneric switches

which have no practical sense for the algorithm.

Conditions written in the decision diagram, cf. Table

2, are based on engineering intuition and for one-

phase cases arc identical to conditions considered by

Nelson (1987).

There is a basic problem, namely the choice of

initial approximation for the Newton-Raphson

method. Not only for the unknowns but also for the

initial “phase situation”. Our experience leads to a

conclusion that the case with the highest number of

expected phases serves as a good initial guess, see the

examples below.

4. EXAMPLES

Calculation of physical properties

The values of K, h, and h, depend generally on

composition, temperature and pressure. We used for

K-values the relation:

(12)

where yi is the activity coefficient and the saturated

vapour pressure Pp is given by the Antoine equation.

The molar enthalpy of vapour mixture has been

evaluated according to:

b(T) = li Y, s T

Cdl) dt, (13) ;.. 1 To

and molar enthalpy of liquid mixture analogously:

h,(T) = 5 xi =Cpi(t) dt #=I s m

-hwMT - TciMTe, - Tc~)I~.~~. (14)

Here C, is ideal gas heat capacity, hvi is heat of

vaporization at normal boiling point TB, and T,-, is

critical temperature, T,, = 298.15 K. All necessary

data for examples presented below can be found in

the book of Reid et al. (1977).

Table 3. NRTL narameters for Exam& I

i @,I % % 4% IR 4?lZlR &,lR I 0 0.48 0.30 0 327.19 260.79 2 0.48 0 0.30 1278.85 0 998.34 3 0.30 0.30 0 -223.0 - 26.024 0

Table 4. Specifications of Example I

Feed comwsition (a) Ibb

1. Butanol

2. Water 3. Propanol Feed Rowrate (mol s ‘)

Feed temperature (K)

0.15 0.05 0.80 0.65 0.05 0.30 1 I

367 367

EXAMPLE 1

Consider the flash unit (condenser) with a pressure

of 101,325 Pa, the single feed is a vapour mixture of

butanol, water and propanol. Feed flowrate, tem-

perature and composition are presented in Table 4.

Such a mixture can form one vapour and two liquid

phases (p = 2) as was shown by Block and Hegner

(1976). Activity coefficients y, are evaluated by using

the NRTL equation (e.g. Reid et al., 1977). Binary

interaction coefficients u,, and Ag,,fR are listed in

Table 3. These values are taken or derived from the

paper by Block and Hegner (1976) for a temperature

of 364 K.

As an initial approximation the following values

have been used:

G = L ’ = L’ = (ZF)/3 [ = (CF)/(p + l)];

yi = xj = l/3 (= l/n), i = 1, 2,3;

x:=x:-o, x:= 1; T = TF

This initial guess led to convergence to a correct

(physically acceptable) solution for both examples (a)

and (b) from Table 4 and for an arbitrary value of Q

chosen from the interval (- 46,000 W, 1000 W). The

results of computations in dependence on Q are

presented for example (a) in Fig. 2. The phase

specification of the outlet mixture (the case from

Table 1) varies depending on the value of Q as we can

see from the figure. A similar dependence has also

been computed for example (b); however, there exist

only cases L 1, G-L 1 and G in the whole range of Q.

EXAMPLE 2

Consider the flash unit (reboiler) with a pressure of

101,325 Pa with three feed streams (see Table 5). Such

a mixture can form three liquid phases (p = 3) as was

shown by Null (1970). Activity coefficients yi are

evaluated by using the Van Laar equation in the form

and with values of parameters given in the book by

Feed

Table 5. Specification of Example 2

I 2 3

Feed composition

I. Nitromethance I 0 0 2. Dodecanol 0 I 0 3. Ethylene glycol 0 0

Feed flow-ate (mol s-‘) 0.3 0.3 A.4 Feed temperature (K) 295.2 295.2 295.2

Multiple liquid-vapour equilibrium flash calculation 883

0.6- L2

- 40000 - 30000 - 20000 -10000 Q 6 Fig. 2. Liquid 1, liquid 2 and gas flowrate dependent on heat supply [Example 1, column (a) from Table 41.

0.5- Ll

0.4- L2

0.3-

0.2-

L3

O.l-

CASE CASE

0 ’ Ll-L2-L3 G-U-L2

f5000 2.0000 25000 30000 61

Fig. 3. Liquids 1-3 and gas flowrate dependent on the heat supply (Example 2).

35000

Null (1970), pp. 216-219. The value of h, for dode-

canal, h, = 58,740 J mol-r, was estimated from

Fishtine’s modification of the Kistiakowski equation

(Reid and Sherwood, 1966).

We used the following initial approximation:

G = 0, L’ = L2 = L3 = l/3, y = (O,O, O), x’ = (0.005,

0.005, 0.99) x2 = (0.005,0.99,0.005), x3 = (0.99,

0.005,0.005), T = TF = 295.2. This initial guess led to

convergence to a correct solution for an arbitrary

value of Q E (0 W, 35,000 W). Computed flowrates of

products dependent on the value of Q are presented

in Fig. 3. It is obvious that the results for higher

values of Q, i.e. for higher temperatures than used by

Null (1970) can be far from reality.

Let us mention that there are 15 cases in this

example. Therefore, the decision diagram (cf.

Table 2) has 15 rows and 15 columns and contains

100 possible switches (nonblank boxes).

5. CONCLUSIONS

The suggested computational algorithm is rela-

tively simple, robust and not very sensitive to initial

approximation. The convergence is generally fast, i.e.

only several (4-8) iterations usually lead to a solution.

The algorithm can also be used for modelling of

rectification columns, because each stage can be

modelled as a flash.

Acknowledgemenr-Part of this work was supported by Alexander van Humboldt Foundation during the stay of one of the authors (EE) at the Institute for System Dynamics and Control, University of Stuttgart, Germany.

884 E. ECKERT and M. KUB~I~EK

NOMENCLATURE

F= Molar Rowrate of feed to the flash G = Molar flowrate of vapour phase h = Molar enthalpy K = Vapour-liquid distribution coefficient L = Molar flowrate of liquid phase m = Number of feed streams n = Number of components 4 = Number of possible liquid phases P= Pressure Q = Heat supply T = Temperature X = Vector of unknowns, cf. equation (8) x = Mole fraction in liquid phase y = Mole fraction in vapour phase z = Mole fraction in the feed y = Activity coelhcient

Subscripts

F=Feed G = Vapour phase

i = Component i k = Feed stream k L = Liquid phase

Superscripts

j = Liquid phase j r = Reference liquid phase

REFERENCES

Block U. and B. Hegner, Development and application of a simulation model for three-phase distillation. AIChE JI 22, 382 (1976).

Bilnz A. P., R. Dohm and J. M. Prausnitz, Three-phase flash calculations for multicomponent systems. Corn- puters them. Engng 15, 47 (1991).

Cairns B. P. and 1. A. Furzer, Multicomponent three-phase azeotropic distillation. 2. Phase-stability and phase- splitting algorithms. Ind. Engng Chem. Res. 29, 1364 (1990).

Eckert E., Calculation of rectification with possible occur- rence of two liquid phases I. Equilibrium stage. Chem. Ind. (Prague) 39, 561 (1989a) (in Czech.).

Eckert E., Calculation of rectification with possible occur- rence of two liquid phases II. Stage column. C/rem. fnd. (Prague) 39, 619 (1989b) (in Czech.).

Gautam R. and W. D. Seider, Computation of phase and chemical equilibria: part I. Local and constrained minima in Gibbs free energy; part II. Phase-splitting. AIChE Jl 25, 991, 999 (1979).

KubiEek M.. V. Hlav%ek and F. ProchPska, Global modu- lar Newton-Raphson technique for simulation of an interconnected plant applied to complex rectification columns. Chem. Engng Sci. 31, 277 (1976).

Michelsen M. L., The isothermal flash problem (parts I and II). Fluid Phase Equil. 9, I (1982).

Nelson P. A., Rapid phase determination in multiple- phase flash calculations. Computers them. Engng 11, 581 (1987).

Null H. R., Phase Equilibrium in Process Design. Wiley, New York (1970).

Ohanomah M. 0. and D. W. Thompson, Computation of multi-component phase equilibria-part III. Multi-phase equilibria. Computers them. Engng It, 163 (1984).

Reid R. C. and T. K. Sherwood. The Prozwrties ofGases and Liquids. McGraw-Hill, New ‘York (1466). _

Reid R. C., J. M. Prausnitz and T. K. Sherwood, The Properties of Gases and Liquids. McGraw-Hill, New York (1977).