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A direct method for calculation of liquid—liquid equilibria in multicomponent systems
Shorter
2954
P u
t
Communications
friction coefficient, dimensionless length, m pressure, Pa linear velocity, m s-t thickness of the draft tube, m
Greek letters E gas hold up, dimensionless total gas hold up, dimensionless ET liquid viscosity, Pas p density, kg mm3 P d surface tension, N m- 1 Subscripts a atmospheric of the outer tube ; of the downcomer e equivalent G ofthe gas L of the liquid P at the middle point of the riser R of the riser s slip
REFERENCES
Akita, K., Okazaki, T. and Koyama, H., 1988, Gas holdup and friction factors of gas-liquid two-phase flow in an airlift bubble column. J. them. Engng Japan 21, 476482. Costa, E., 1985, Zngenierio Quimica (Edited by Alhambra), Vol. 3, p. 136. Madrid.
Chemical Engineering Saence, Printed in Great Britain
Vol. 46, No. 11, pp
2954-2958,
Chisti, M. Y., Halard, B. and Moo-Young, M., 1988, Liquid circulation in airlift reactors. Chem. Engng Sci. 43, 451457. Garcia Calvo, E., 1989, A fluid dynamic model for airlift reactors. Chem. Engng Sci. 44,321-323. Harmathy, T. Z., 1960, Velocity of large drops and bubbles in media of infinite or restricted extent. A.I.Ch.E. J. 6, 281-288. Jorgensen, J., 1969, 1974, Flow of fluids through valves, fittings and pipes. Tech. Papers 409 and 410, Crane Co., Chicago. Koide, K., Iwamoto, Sh., Takasaka, Y., Matsuura, S., Takahashi, E. and Kimura, M., 1984, Liquid circulation, gas holdup and pressure drop in bubble column with draught tubes. J. &em. Engng Japan 17,611-618. Merchuk, J. C., in Biorechnology (Edited by K. Schiigerl), Vol. 3, Chap. 3.3. VCH (in press). Miyahara, T., Hamaguchi, M., Sukeda, Y. and Takahoshi, T., 1986, Size of bubbles and liquid circulation in a bubble column with draught tube and sieve plate. Can. J. them. Engng 64, 718-725. Philip, J., Proctor, J. M., Niranjan, K. and Davidson, J. F., 1990, Gas hold-up and liquid circulation in internal loop reactors containing highly viscous Newtonian and nonNewtonian liquids. Chem. Engng Sci. 45, 651464. Verlaan, P., Tramper, J., Van’t Riet, K. and Luyben, K., 1986, A hydrodynamic model for an airlift-loop bioreactor with external loop. Chem. Engng J. 33, 43-53. Weiland, P. and Onken, IJ., 1981, Fluid dynamics and mass transfer in an airlift fermenter with external loop. Germ. them. Engng 4, 42-50.
1991.
A direct method for calculation of liquid-liquid equilibria in multicomponent
systems
(Received 19 November 1990; accepted for publication 20 March 1991)
INTRODUCTION The problem of phase equilibria, from the practica1 view point, has been succinctly stated by Prausnitz et al. (1986). It involves finding temperature (or pressure) and mole fractions of one of the two phases, given the pressure (or temperature) and mole fractions of the other phase. This requires an iterative scheme of calculations for multicomponent systems. The bubble-point and dew-point calculations are the wellknown examples of such calculations. But, analogous methods are not available for liquid-liquid equilibrium (LLE) calculations. That is, no method is available for finding mole fractions of extract phase given the mole fractions of rat&ate phase or vice versa. Such a method may be termed as “direct” method. Due to lack of the direct method, currently one has to resort to the “flash” calculations to compute LLE (Prausnitz
et al., 1980,1986). In the flash calculations, the mole fractions of both extract and raffinate phases are found from the mole fractions of a feed (two-phase) mixture from which they are formed. This is an indirect method. The direct method is a natural choice in the analysis of continuous-contact extractors and in the determination of stage efficiencies as it permits evaluation of the driving force without trial and error unlike the indirect method. It is likely to be of use in developing faster algorithms for the multistage extractor calculations. A better measure of the goodness of fit of the LLE data by activity-coefficient correlations can be obtained by comparing mole fractions of one of the phases, rather than both (which is the current practice), using the direct method. This communication presents a direct method for LLE calculations.
Shorter
Communications
PROBLEM STATEMENT We restrict ourselves to the two-phase LLE in the lowpressure region. In this region, LLE is rather insensitive to pressure variation. In other words, the work due to the change in volume is not important. Therefore, the volume drops out from the fundamental equations of the system (Callen, 1985). Then, the Gibbs phase rule takes the form F + P = C + 1, where F and P are the number of degrees of freedom and number of phases, respectively. Thus, we get F = C - l for the case under consideration. Hence, the equilibrium state of the two-phase mixture is fixed by the specification of any C - 1 independent thermodynamic variables. i in Let xi and yi denote the mole fractions of component the x- and y-phases, respectively. From the above discussion, it follows that if x,, x2, _ , x,_~ are given, we need to _ , y,_ I from the c equilibrium relacalculate T, y,, y,, tions. However, in practice, Tand x1, x2,. , xc_1are given and we need to find yt, y2, . . , y,_,. This amounts to overspecification of the state. Its implications are discussed later.
METHODS OF SOLUTION The phase equilibrium
relations
-%Y,. i = yiyy,;
,...,
as
c
(1)
where y,, i and yy, i are the activity coefficients of component in the x-phase and y-phase, respectively. We can formulate c - 1 discrepancy functions ~=yyi-y~,ix,/y,,,=o and a cth discrepancy = 1) as
i=l,2 function
,...)
(making
Scheme 1 This scheme has been conjectured based on the notion that the magnitude of the yi reflects the tendency of the component to escape into the other phase. If we accept this notion, we may then write Yi @zYx,i = KY*.;
i=l,2,...,c.
Taking the proportionality components, we get Y: = r,.&,.i
can be expressed
i=1,2
2955
initial guess vector, Y’. In the case of dew-point and bubblepoint calculations, the equality of mole fractions of the vapour and liquid phase, or the assumption of ideal solution yields an adequate initial guess vector. Unfortunately, no simple method of finding the Y’ is available in the case of LLE. Considerable difficulties are encountered in finding a suitable initial guess vector, even in the flash calculations (Ohanomah and Thompson, 1984). We have employed two mtttalisation schemes and these are presented below.
c-l
constant
I
i-l,2
to be the same for all
,...,
c.
Thus we can find Y’ given T and X.
i
(2)
use of the fact xyi
f,=iy;-t=o.
Scheme 2 This scheme is essentially the one proposed by Ohanomah and Thompson (1984) to find LLE using the flash calculations. It involves identifying the most dominant component among the y: and setting its mole fraction equal to 0.95 and the mole fractions of the rest to O.O5/(c - 1) such that 1~: = 1. Scheme 1 may be used to identify the most dominant component in the y-phase.
1
These discrepancy form as
functions
can be written
in the vector RESULTS AND DISCUSSION
F(Y) = 0
(4)
where F=(f,,f,,...,I-,)r
(5)
and Y = (Y1, Y29
. . 1YJ’.
Equation (4) can be solved using method to find Y given temperature vector X where x = (Xl, x1,.
, XJT.
The correction vector, AY, required calculations, can be obtained as AY= The elements of the Jacobian update of Y is obtained as
(6)
the Newton-Raphson T and mole fraction
in
the
iterative
-J-IF.
(7)
are given in the Appendix.
Y r+r = Y, + BAY,
The (8)
where I is the iteration number and /i is a step-limiting factor (Holland and Liapis, 1983). We have to omit the discrepancy function of one of the components as can be seen from eq. (2). Since the yy of the dominant component in the y-phase is a weak function of Y compared to the rest, we elected to delete its discrepancy function. It is found to be a good choice from the experience with the method. INITIALISATION SCHEMES The speed and the success of convergence Newton-Raphson method depends on the choice
of the of the
Performance of the method The proposed method was tested employing 21 systems (18 ternary, 2 quatemary and 1 six-component system). A total of 125 tie-line data have been tested. For each of the tieline data, one end of the tie-line was taken as the x-phase and the other end was estimated. The reverse case was also tested. Thus, a total 250 LLE calculations were performed. In the literature referred to in Table 1, the experimental tie-line data as well as those obtained by the flash calculations (obtained to check the goodness of fit of the activity coefficient correlations) are available. In this study, we employed those obtained by the flash calculations and the binary constants of UNIQUAC reported in the literature. Whereever the tie-line data were sparse, we generated the data by the Aash calculations. The details of the systems are given in Table 1. The performance of the method was evaluated using the two initialisation schemes, on a HP 9OOO/SSO super-mini system employing a tolerance criterion, xx < lO-‘o. To ensure convergence, we employed the Newton-Raphson method coupled with the step-limiting scheme proposed by Broyden (Holland and Liapis, 1983). The mole fractions of some of the components became negative during the iterations in a few cases. In such cases, the S in eq. (8) was reduced in steps of 0.1 in the range 1 and - 1 till the updated mole fractions were positive. The iterations were terminated however, if the mole fractions remained still negative. The method converged within 3-6 iterations except in a few cases with both initialisation schemes. The CPU time required to perform one LLE calculation was very small. Hence, the total CPU time required for a system involving several equilibrium calculations have been reported in Table 1. The performance of the method was about the same using either of the schemes. Scheme 1 was marginally better than
Shorter
2956 Table
S. No. 1 2
9 10 11 12 13 14 15 16 :: 19 20 21
Communications
1. Performance
Points checked
System
of the method CPU time (s) Scheme 1 2
Benzene-water-methanol Benzene-water-ethanol Cyclohexane-nitromethane-benzene Hexane-anihne-methylcyclopentane Methanol-n-heptane-benzene Cyclohexane-furfuralbenzenzene Trimethylpentane-furfural+=yclohexane Water-acrylonitrile-acetonitrile
14 10 12 10 10 12 10 10
0.17 0.18 0.17 0.18 0.22 0.17 0.16 0.20
0.18 0.18 0.18 0.21 0.19 0.19 0.14 0.18
Propylene carbonate-n-heptane-toluene Water-ethanol-toluene Water-ethanol-chloroform Water-chloroform-toluene Hexane-benzene-sulfolane Heptane-toluene-sulfolane Cyclohexane~benzene~sulfolane Furfum-trimethylpentane-benzene Acetonitrile-n-heptae-benzene Hexane-benzeneedimethyl sulfoxide Furfural-trimethylpentane-benzenecyclohexane Water-ethanol-chloroform-toluene Benzene-toluene-hexane-heptanecyclohexane-sulfolane
14 IO 10 6 12 10 10 10 10 10
0.16 0.17 0.15 0.19 0.10 0.18 0.19 0.16 0.16 0.17
16 34 10
Total CPU
Time
Convergence Scheme 1 2
Reference
(1)’ all all all all all all all
0.20 0.20 0.16 0.17 0.16 0.23 0.15 0.19 0.15 0.19
all all all all (5)’ all all (2)’ (3)f all all all all all all all all (1)’ all
all all all all all all all all all all
PI cw WI
0.16 0.17
0.18 0.28
ail (1)’
all (1)’
c91 WI
0.19
0.22
all
(2)”
c71
3.60
3.93
[::;
Cl1 Cl3 ;:; Cl1 c91
“,:3 c73 c71 c91 $3
+Converged to unstable region. +Converged onto itself ‘Converged to metastable region, IiNo convergence. (n) Number of cases.
scheme 2. Some difficulties were encountered in convergence in a few cases and these are detailed below. With scheme 1, the method converged back onto X in 4 out of 250 cases. These arc marked by “3” (see Table 1, the number of cases is given in brackets). In another 8 cases, the method converged to a point which lay inside the binodal curve. These are marked by “7”. Using the stability criteria suggested by Heidemann (1975), we have found, that these points were in the unstable region. A close examination of the trajectory of convergence in these cases revealed that the method could be forced to converge to the required solution, if the initial guess vector is adjusted slightly. We increased y; of the dominant component by a correction factor “s” and reduced yt of other two components by 2~13 and s/3 in the order of their prominence in composition. On such correction, if the mole fraction of any one of the components went out of the range 0 and 1, the value of‘s” was successively decreased by 0.05, until it fell within the range. We have used 0.3 as the initial value of the correction. This ensured convergence in all the cases for the ternary systems. With scheme 2, the method converged to the solution in all but 4 out of 250 casts (see Table 1). Two of these cases belonged to the six-component system. In these cases, the iterations were terminated as some of the mole fractions remained negative even after the adjustment, mentioned earlier. These are marked by “II”. In one case it converged to another point in the metastable region close to the binodal curve and it is marked by “$” (Table I). In systems with c 13, the order of prominence of the two most dominant components in Y’ may get reversed in Y. In
such cases, if the method fails, the replacement of the dominant component by the next dominant one may lead to convergence. Such a case was encountered in the quaternary system (see Table 1). But for these few cases, there appears to be little to choose between the two schemes. Whenever divergence is encountered with one scheme, the other scheme may be employed. Thus the two schemes complement each other and provide a fast, reliable method of LLE calculations. It may be mentioned that the difference between yx,ixi and yyVlyi, especially for the dominant component was significant (10m3) for some of the literature data. In these cases, though the method converged (Icyi - 11 -=zlo-‘“), the difference persisted for the dominant component as its discrepancy function was left out from the F. The difference was less for the calculated data as compared to the reported data. Conjecture It is of interest
to examine how good the employed conjecture is in arriving at scheme 1. The yi and y; are compared for a few of the best and the worst instances in Table 2. The conjecture was reasonably good in most cases. The conjecture may be inadequate in some cases. However, the minor adjustments in the schemes suggested above will help to overcome the inadequacy. State of the x-phase
Generally, one is assured of the fact that the given Tand X correspond to one end of a tie line in the continuous-contact and multistage extraction calculations, and in testing the
Shorter Table 2. Comparison S. No. 1
2
3
4
5
6
7
Components
2957
Communications of Y’ and Y in a few typical
X
Y
Y
Toluene/ Hexaue/ Hentanei Cy&ohexane/ Sulfolane
0.1070 0.1098 0.5360 0.1196 0.1180 0.0095
0.0117 0.0098 0.0109 0.0075 0.0117 0.9484
0.0637 0.0517 0.0272 0.0028 0.0117 0.8409
Furfural/ Trimethylpentane/ Benzene/ Cyclohexane
0.0910 0.3330 0.1080 0.4680
0.7433 0.0876 0.0847 0.0845
0.7884 0.0335 0.0894 0.0888
Benzene/ Water/ Methanol
0.5670 0.0610 0.3720
0.5611 0.0733 0.3656
0.5670 0.0610 0.3720
Water/ Ethanol/ Chloroform/ Toluene
0.6800 0.2950 0.0220
0.0077 0.0039 0.1028 0.8856
0.2563 0.4018 0.2441 0.0977
Benzene/
Water/ Methanol
0.9280 0.0044 0.0679
0.0722 0.7393 0.1885
0.0468 0.4030 0.5502
Trimethylpentane Furfural/ Benzene/ Cyclohexane
0.5552 0.0375 0.2134 0.1939
0.1358 0.4645 0.1003 0.2993
0.1864 0.1231 0.2303 0.4603
Benzene/
Required
adjustment
Water/ Acrylonitrile/ Acetonitrile
cases No. of iterations
Category
4
Best
4
Best
6
Best
8
worst
9
Worst
6
Worst
Y’ after adjustment 0.9484 0.0067 0.0448
of Y’ 0.2141 0.6963 0.0896
goodness of fit of the LLE data. However, there may be instances in which it is not evident if the set T and X corresponds to the end of a tie line. Consider, for instance, the case of ternary systems. The given T and X may lie in the homogeneous region or inside the binodal curve. In vapour-liquid equilibria too, a similar situation arises whenever overspecification is made. For instance, if the set of T, P and mole fractions of the liquid or vapour phase is specified, it is not evident whether the state corresponds to the end of a tie line (i.e. saturated condition) or not. To check the state of the x-phase, we could use the stability criteria as suggested bv Heidemann (1975) to ascertain if it correspondsto the unstable region. If the pomt lies in either metastable or homogeneous region, it could be ascertained from the amounts of the phase obtained by performing the flash calculations with the given X as the feed composition. But this requires much more effort compared to the LLE calculations. Therefore, we have examined how the method behaves for points which are not on the binodal curve. In some cases, the method converged back onto the X, which is an indication that the state belongs to the homogeneous region. In some cases, the Newton-Raphson method exhibited osciIlafions. In other cases, the method converged, but the difference in (y,x - y,y) of the dominant component as it was left out from the set of discrepancy was significant functions. If these features are encountered, it is likely that the given Tand X, correspond to a state in the homogeneous region, and could be verified by flash calculations.
0.6984 0.1734 0.1282
0.9521 0.0341 0.0138
CONCLUSIONS A direct method for LLE calculations has been presented based on the Newton-Raphson method. Two initialisation schemes have been tried. The method has been tested employing 21 systems. Scheme 1 gave convergence for all points, with a minor adjustment in a few cases, whereas Scheme 2 gave convergence in all but 3 cases. The method along with the two mitialisation schemes provide a fast, reliable method for LLE calculations. The method will facilitate the calculation of interphase mass transfer in multicomponent systems. AVIJIT
BHOWAL D. P. RAO’
Department of Chemical Engineering Indian lnstirute ofTechnology, Xanpur Kanpur 208016, India NOTATION ; F .I ii
number of components discrepancy functions column vector of the N discrepancy Jacobian matrix proportionality constant
‘Author
to whom correspondence
should
functions
be addressed.
Shorter
2958 s
T x X Y Y
Communications
correction factor absolute temperature, K mole fraction of x-phase vector of mole fraction of x-phase mole fraction of y-phase vector of mole fraction of Y-pha*
Greek letrers step-limiting factor P activity coefficient Y
for Broyden
procedure
Subscripts iteration number I components i, i x, Y phases Superscript label for the initial guess vector REFERENCES
Anderson, T. F. and Prausnitz, J. M., 1978, Application of the UNIQUAC equation to calculation of multicomponent phase equilibria. Ind. Engng Chem. Process Des. Dev. 17, 561-567. Annesini, M. C., Gironi, F. and Marrelli, L., 1985, Liquid-liquid equilibria for ternary systems containing hydrocarbons and propylene carbonate. J. &em. Engng Dam 30, 195-196. Callen, H. B., 1985, Thermodynamics and an Introdttction to Thermostatics. Wiley, New York. Cassel, G. W., Hassan, M. M. and Hines, A. L., 1989, Liquid-liquid equilibria for the hexane-benzenedimethyl sulfoxide ternary system. J. them. Engng Data 34, 328-331. Heidemann, R. A., 1975, The criteria for thermodynamic stability. A.1.CL.E. J. 21, 824-826. Holland, C. D. and Uapis, A. J., 1983, Compurer Methodsfor Soluing Dynamic &per&ion Problems. McGraw-Hill, New York. Mukhopadhyay, M. and Sahasranaman, K., 1982, Computation of multicomponent Iiquid-liquid equilibrium
data for aromatics extraction systems. Ind. Engng Chem. Process Des. Dev. 21, 632-640. Ohanomah, M. 0. and Thompson, D. W., 1984, Computatlon of multicomponent phase equilibria-part 2. Liquid-liquid and solid-liquid equilibria. Comput. them. Engng 8, 157-162. Prausnitz, J. M., Anderson, T. F., Grens, E. A., Eckert, C. A., H&h, R. and O’Connel, J. P., 1980, Computer Calculations for Multicomponent Vapour-Liquid and Liquid-Liquid Equilibria. Prentice-Hall, Englewood Cliffs, NJ. Prausnitz, J. M., Lichtenthaler, R. N. and de Azevedo, E. G., 1986, Molecular Thermodynamics ofFluid-phase Equilibria. Prentice-Hall, Englewood Chffs, NJ. Reid, R. C.. Prausnitz, J. M. and Poling, B. E., 1988, The Properties ofGases and Liquids. Prentice-Hall, Englewood Cliffs, NJ. Ruiz, F., Prats, D. and Gomis, V., 1985, Quaternary liquid-liquid equilibrium: water-ethanol-chloroformtoluene at 25°C. Experimental determination and graphical and analytical correlation of equilibrium data. J. them. Engng Data 30, 412416. Triday, J. O., 1984, Liquid-liquid equilibria for the system J. them. Engng Data 29, benzene-water-methanol. 321-324. APPENDIX
The elements
of the Jacobian
i=l,2
,_._,
c-l,j=1,2
,_..,
c
(A-l)
where bij =
1
i=j
0
i #j.
and
a!
ayi=l
j=l,2,...,c.
am-2509/91 $3.00 + 0.00 Q 1991 Pergamon Press plc
Minimum
film thickness
for reverse roll coating
(First received 10 August 1990; accepted in revised form 22 March 1991)
INTRODUCTION
Reverse roll coating operation is considered as a simple and flexible means for depositing a thin liquid film on a moving substrate. Ho and Holland (1978), Benkreira et al. (1981) and Greener and Middleman (1981) applied the lubrication approximation to predict the coating thickness. They found that the coating thickness can be reduced as the coating speed goes up, and this conclusion was supported with the experimental results. Recently, Coyle et af. (1989) carried out a theoretical analysis on reverse roll coating. They claimed that the lubrication approximation is only valid over a
limited range of parameters. The coating thickness decreases initially as the coating speed goes up, it passes a minimum value and then increases rapidly. The prediction of Coyle et al. was supported by some experimental evidence. However, an emprirical slip coefficient on the three phase line was required for their finite-element simulations and therefore the theoretical predictIon may depend on the choice of this coefficient. In this communication we supplement the work of Coyle et ai. by presenting the results based on our reverse roll coating
experiment.
We also observed
the phenomenon
of
2954
P u
t
Communications
friction coefficient, dimensionless length, m pressure, Pa linear velocity, m s-t thickness of the draft tube, m
Greek letters E gas hold up, dimensionless total gas hold up, dimensionless ET liquid viscosity, Pas p density, kg mm3 P d surface tension, N m- 1 Subscripts a atmospheric of the outer tube ; of the downcomer e equivalent G ofthe gas L of the liquid P at the middle point of the riser R of the riser s slip
REFERENCES
Akita, K., Okazaki, T. and Koyama, H., 1988, Gas holdup and friction factors of gas-liquid two-phase flow in an airlift bubble column. J. them. Engng Japan 21, 476482. Costa, E., 1985, Zngenierio Quimica (Edited by Alhambra), Vol. 3, p. 136. Madrid.
Chemical Engineering Saence, Printed in Great Britain
Vol. 46, No. 11, pp
2954-2958,
Chisti, M. Y., Halard, B. and Moo-Young, M., 1988, Liquid circulation in airlift reactors. Chem. Engng Sci. 43, 451457. Garcia Calvo, E., 1989, A fluid dynamic model for airlift reactors. Chem. Engng Sci. 44,321-323. Harmathy, T. Z., 1960, Velocity of large drops and bubbles in media of infinite or restricted extent. A.I.Ch.E. J. 6, 281-288. Jorgensen, J., 1969, 1974, Flow of fluids through valves, fittings and pipes. Tech. Papers 409 and 410, Crane Co., Chicago. Koide, K., Iwamoto, Sh., Takasaka, Y., Matsuura, S., Takahashi, E. and Kimura, M., 1984, Liquid circulation, gas holdup and pressure drop in bubble column with draught tubes. J. &em. Engng Japan 17,611-618. Merchuk, J. C., in Biorechnology (Edited by K. Schiigerl), Vol. 3, Chap. 3.3. VCH (in press). Miyahara, T., Hamaguchi, M., Sukeda, Y. and Takahoshi, T., 1986, Size of bubbles and liquid circulation in a bubble column with draught tube and sieve plate. Can. J. them. Engng 64, 718-725. Philip, J., Proctor, J. M., Niranjan, K. and Davidson, J. F., 1990, Gas hold-up and liquid circulation in internal loop reactors containing highly viscous Newtonian and nonNewtonian liquids. Chem. Engng Sci. 45, 651464. Verlaan, P., Tramper, J., Van’t Riet, K. and Luyben, K., 1986, A hydrodynamic model for an airlift-loop bioreactor with external loop. Chem. Engng J. 33, 43-53. Weiland, P. and Onken, IJ., 1981, Fluid dynamics and mass transfer in an airlift fermenter with external loop. Germ. them. Engng 4, 42-50.
1991.
A direct method for calculation of liquid-liquid equilibria in multicomponent
systems
(Received 19 November 1990; accepted for publication 20 March 1991)
INTRODUCTION The problem of phase equilibria, from the practica1 view point, has been succinctly stated by Prausnitz et al. (1986). It involves finding temperature (or pressure) and mole fractions of one of the two phases, given the pressure (or temperature) and mole fractions of the other phase. This requires an iterative scheme of calculations for multicomponent systems. The bubble-point and dew-point calculations are the wellknown examples of such calculations. But, analogous methods are not available for liquid-liquid equilibrium (LLE) calculations. That is, no method is available for finding mole fractions of extract phase given the mole fractions of rat&ate phase or vice versa. Such a method may be termed as “direct” method. Due to lack of the direct method, currently one has to resort to the “flash” calculations to compute LLE (Prausnitz
et al., 1980,1986). In the flash calculations, the mole fractions of both extract and raffinate phases are found from the mole fractions of a feed (two-phase) mixture from which they are formed. This is an indirect method. The direct method is a natural choice in the analysis of continuous-contact extractors and in the determination of stage efficiencies as it permits evaluation of the driving force without trial and error unlike the indirect method. It is likely to be of use in developing faster algorithms for the multistage extractor calculations. A better measure of the goodness of fit of the LLE data by activity-coefficient correlations can be obtained by comparing mole fractions of one of the phases, rather than both (which is the current practice), using the direct method. This communication presents a direct method for LLE calculations.
Shorter
Communications
PROBLEM STATEMENT We restrict ourselves to the two-phase LLE in the lowpressure region. In this region, LLE is rather insensitive to pressure variation. In other words, the work due to the change in volume is not important. Therefore, the volume drops out from the fundamental equations of the system (Callen, 1985). Then, the Gibbs phase rule takes the form F + P = C + 1, where F and P are the number of degrees of freedom and number of phases, respectively. Thus, we get F = C - l for the case under consideration. Hence, the equilibrium state of the two-phase mixture is fixed by the specification of any C - 1 independent thermodynamic variables. i in Let xi and yi denote the mole fractions of component the x- and y-phases, respectively. From the above discussion, it follows that if x,, x2, _ , x,_~ are given, we need to _ , y,_ I from the c equilibrium relacalculate T, y,, y,, tions. However, in practice, Tand x1, x2,. , xc_1are given and we need to find yt, y2, . . , y,_,. This amounts to overspecification of the state. Its implications are discussed later.
METHODS OF SOLUTION The phase equilibrium
relations
-%Y,. i = yiyy,;
,...,
as
c
(1)
where y,, i and yy, i are the activity coefficients of component in the x-phase and y-phase, respectively. We can formulate c - 1 discrepancy functions ~=yyi-y~,ix,/y,,,=o and a cth discrepancy = 1) as
i=l,2 function
,...)
(making
Scheme 1 This scheme has been conjectured based on the notion that the magnitude of the yi reflects the tendency of the component to escape into the other phase. If we accept this notion, we may then write Yi @zYx,i = KY*.;
i=l,2,...,c.
Taking the proportionality components, we get Y: = r,.&,.i
can be expressed
i=1,2
2955
initial guess vector, Y’. In the case of dew-point and bubblepoint calculations, the equality of mole fractions of the vapour and liquid phase, or the assumption of ideal solution yields an adequate initial guess vector. Unfortunately, no simple method of finding the Y’ is available in the case of LLE. Considerable difficulties are encountered in finding a suitable initial guess vector, even in the flash calculations (Ohanomah and Thompson, 1984). We have employed two mtttalisation schemes and these are presented below.
c-l
constant
I
i-l,2
to be the same for all
,...,
c.
Thus we can find Y’ given T and X.
i
(2)
use of the fact xyi
f,=iy;-t=o.
Scheme 2 This scheme is essentially the one proposed by Ohanomah and Thompson (1984) to find LLE using the flash calculations. It involves identifying the most dominant component among the y: and setting its mole fraction equal to 0.95 and the mole fractions of the rest to O.O5/(c - 1) such that 1~: = 1. Scheme 1 may be used to identify the most dominant component in the y-phase.
1
These discrepancy form as
functions
can be written
in the vector RESULTS AND DISCUSSION
F(Y) = 0
(4)
where F=(f,,f,,...,I-,)r
(5)
and Y = (Y1, Y29
. . 1YJ’.
Equation (4) can be solved using method to find Y given temperature vector X where x = (Xl, x1,.
, XJT.
The correction vector, AY, required calculations, can be obtained as AY= The elements of the Jacobian update of Y is obtained as
(6)
the Newton-Raphson T and mole fraction
in
the
iterative
-J-IF.
(7)
are given in the Appendix.
Y r+r = Y, + BAY,
The (8)
where I is the iteration number and /i is a step-limiting factor (Holland and Liapis, 1983). We have to omit the discrepancy function of one of the components as can be seen from eq. (2). Since the yy of the dominant component in the y-phase is a weak function of Y compared to the rest, we elected to delete its discrepancy function. It is found to be a good choice from the experience with the method. INITIALISATION SCHEMES The speed and the success of convergence Newton-Raphson method depends on the choice
of the of the
Performance of the method The proposed method was tested employing 21 systems (18 ternary, 2 quatemary and 1 six-component system). A total of 125 tie-line data have been tested. For each of the tieline data, one end of the tie-line was taken as the x-phase and the other end was estimated. The reverse case was also tested. Thus, a total 250 LLE calculations were performed. In the literature referred to in Table 1, the experimental tie-line data as well as those obtained by the flash calculations (obtained to check the goodness of fit of the activity coefficient correlations) are available. In this study, we employed those obtained by the flash calculations and the binary constants of UNIQUAC reported in the literature. Whereever the tie-line data were sparse, we generated the data by the Aash calculations. The details of the systems are given in Table 1. The performance of the method was evaluated using the two initialisation schemes, on a HP 9OOO/SSO super-mini system employing a tolerance criterion, xx < lO-‘o. To ensure convergence, we employed the Newton-Raphson method coupled with the step-limiting scheme proposed by Broyden (Holland and Liapis, 1983). The mole fractions of some of the components became negative during the iterations in a few cases. In such cases, the S in eq. (8) was reduced in steps of 0.1 in the range 1 and - 1 till the updated mole fractions were positive. The iterations were terminated however, if the mole fractions remained still negative. The method converged within 3-6 iterations except in a few cases with both initialisation schemes. The CPU time required to perform one LLE calculation was very small. Hence, the total CPU time required for a system involving several equilibrium calculations have been reported in Table 1. The performance of the method was about the same using either of the schemes. Scheme 1 was marginally better than
Shorter
2956 Table
S. No. 1 2
9 10 11 12 13 14 15 16 :: 19 20 21
Communications
1. Performance
Points checked
System
of the method CPU time (s) Scheme 1 2
Benzene-water-methanol Benzene-water-ethanol Cyclohexane-nitromethane-benzene Hexane-anihne-methylcyclopentane Methanol-n-heptane-benzene Cyclohexane-furfuralbenzenzene Trimethylpentane-furfural+=yclohexane Water-acrylonitrile-acetonitrile
14 10 12 10 10 12 10 10
0.17 0.18 0.17 0.18 0.22 0.17 0.16 0.20
0.18 0.18 0.18 0.21 0.19 0.19 0.14 0.18
Propylene carbonate-n-heptane-toluene Water-ethanol-toluene Water-ethanol-chloroform Water-chloroform-toluene Hexane-benzene-sulfolane Heptane-toluene-sulfolane Cyclohexane~benzene~sulfolane Furfum-trimethylpentane-benzene Acetonitrile-n-heptae-benzene Hexane-benzeneedimethyl sulfoxide Furfural-trimethylpentane-benzenecyclohexane Water-ethanol-chloroform-toluene Benzene-toluene-hexane-heptanecyclohexane-sulfolane
14 IO 10 6 12 10 10 10 10 10
0.16 0.17 0.15 0.19 0.10 0.18 0.19 0.16 0.16 0.17
16 34 10
Total CPU
Time
Convergence Scheme 1 2
Reference
(1)’ all all all all all all all
0.20 0.20 0.16 0.17 0.16 0.23 0.15 0.19 0.15 0.19
all all all all (5)’ all all (2)’ (3)f all all all all all all all all (1)’ all
all all all all all all all all all all
PI cw WI
0.16 0.17
0.18 0.28
ail (1)’
all (1)’
c91 WI
0.19
0.22
all
(2)”
c71
3.60
3.93
[::;
Cl1 Cl3 ;:; Cl1 c91
“,:3 c73 c71 c91 $3
+Converged to unstable region. +Converged onto itself ‘Converged to metastable region, IiNo convergence. (n) Number of cases.
scheme 2. Some difficulties were encountered in convergence in a few cases and these are detailed below. With scheme 1, the method converged back onto X in 4 out of 250 cases. These arc marked by “3” (see Table 1, the number of cases is given in brackets). In another 8 cases, the method converged to a point which lay inside the binodal curve. These are marked by “7”. Using the stability criteria suggested by Heidemann (1975), we have found, that these points were in the unstable region. A close examination of the trajectory of convergence in these cases revealed that the method could be forced to converge to the required solution, if the initial guess vector is adjusted slightly. We increased y; of the dominant component by a correction factor “s” and reduced yt of other two components by 2~13 and s/3 in the order of their prominence in composition. On such correction, if the mole fraction of any one of the components went out of the range 0 and 1, the value of‘s” was successively decreased by 0.05, until it fell within the range. We have used 0.3 as the initial value of the correction. This ensured convergence in all the cases for the ternary systems. With scheme 2, the method converged to the solution in all but 4 out of 250 casts (see Table 1). Two of these cases belonged to the six-component system. In these cases, the iterations were terminated as some of the mole fractions remained negative even after the adjustment, mentioned earlier. These are marked by “II”. In one case it converged to another point in the metastable region close to the binodal curve and it is marked by “$” (Table I). In systems with c 13, the order of prominence of the two most dominant components in Y’ may get reversed in Y. In
such cases, if the method fails, the replacement of the dominant component by the next dominant one may lead to convergence. Such a case was encountered in the quaternary system (see Table 1). But for these few cases, there appears to be little to choose between the two schemes. Whenever divergence is encountered with one scheme, the other scheme may be employed. Thus the two schemes complement each other and provide a fast, reliable method of LLE calculations. It may be mentioned that the difference between yx,ixi and yyVlyi, especially for the dominant component was significant (10m3) for some of the literature data. In these cases, though the method converged (Icyi - 11 -=zlo-‘“), the difference persisted for the dominant component as its discrepancy function was left out from the F. The difference was less for the calculated data as compared to the reported data. Conjecture It is of interest
to examine how good the employed conjecture is in arriving at scheme 1. The yi and y; are compared for a few of the best and the worst instances in Table 2. The conjecture was reasonably good in most cases. The conjecture may be inadequate in some cases. However, the minor adjustments in the schemes suggested above will help to overcome the inadequacy. State of the x-phase
Generally, one is assured of the fact that the given Tand X correspond to one end of a tie line in the continuous-contact and multistage extraction calculations, and in testing the
Shorter Table 2. Comparison S. No. 1
2
3
4
5
6
7
Components
2957
Communications of Y’ and Y in a few typical
X
Y
Y
Toluene/ Hexaue/ Hentanei Cy&ohexane/ Sulfolane
0.1070 0.1098 0.5360 0.1196 0.1180 0.0095
0.0117 0.0098 0.0109 0.0075 0.0117 0.9484
0.0637 0.0517 0.0272 0.0028 0.0117 0.8409
Furfural/ Trimethylpentane/ Benzene/ Cyclohexane
0.0910 0.3330 0.1080 0.4680
0.7433 0.0876 0.0847 0.0845
0.7884 0.0335 0.0894 0.0888
Benzene/ Water/ Methanol
0.5670 0.0610 0.3720
0.5611 0.0733 0.3656
0.5670 0.0610 0.3720
Water/ Ethanol/ Chloroform/ Toluene
0.6800 0.2950 0.0220
0.0077 0.0039 0.1028 0.8856
0.2563 0.4018 0.2441 0.0977
Benzene/
Water/ Methanol
0.9280 0.0044 0.0679
0.0722 0.7393 0.1885
0.0468 0.4030 0.5502
Trimethylpentane Furfural/ Benzene/ Cyclohexane
0.5552 0.0375 0.2134 0.1939
0.1358 0.4645 0.1003 0.2993
0.1864 0.1231 0.2303 0.4603
Benzene/
Required
adjustment
Water/ Acrylonitrile/ Acetonitrile
cases No. of iterations
Category
4
Best
4
Best
6
Best
8
worst
9
Worst
6
Worst
Y’ after adjustment 0.9484 0.0067 0.0448
of Y’ 0.2141 0.6963 0.0896
goodness of fit of the LLE data. However, there may be instances in which it is not evident if the set T and X corresponds to the end of a tie line. Consider, for instance, the case of ternary systems. The given T and X may lie in the homogeneous region or inside the binodal curve. In vapour-liquid equilibria too, a similar situation arises whenever overspecification is made. For instance, if the set of T, P and mole fractions of the liquid or vapour phase is specified, it is not evident whether the state corresponds to the end of a tie line (i.e. saturated condition) or not. To check the state of the x-phase, we could use the stability criteria as suggested bv Heidemann (1975) to ascertain if it correspondsto the unstable region. If the pomt lies in either metastable or homogeneous region, it could be ascertained from the amounts of the phase obtained by performing the flash calculations with the given X as the feed composition. But this requires much more effort compared to the LLE calculations. Therefore, we have examined how the method behaves for points which are not on the binodal curve. In some cases, the method converged back onto the X, which is an indication that the state belongs to the homogeneous region. In some cases, the Newton-Raphson method exhibited osciIlafions. In other cases, the method converged, but the difference in (y,x - y,y) of the dominant component as it was left out from the set of discrepancy was significant functions. If these features are encountered, it is likely that the given Tand X, correspond to a state in the homogeneous region, and could be verified by flash calculations.
0.6984 0.1734 0.1282
0.9521 0.0341 0.0138
CONCLUSIONS A direct method for LLE calculations has been presented based on the Newton-Raphson method. Two initialisation schemes have been tried. The method has been tested employing 21 systems. Scheme 1 gave convergence for all points, with a minor adjustment in a few cases, whereas Scheme 2 gave convergence in all but 3 cases. The method along with the two mitialisation schemes provide a fast, reliable method for LLE calculations. The method will facilitate the calculation of interphase mass transfer in multicomponent systems. AVIJIT
BHOWAL D. P. RAO’
Department of Chemical Engineering Indian lnstirute ofTechnology, Xanpur Kanpur 208016, India NOTATION ; F .I ii
number of components discrepancy functions column vector of the N discrepancy Jacobian matrix proportionality constant
‘Author
to whom correspondence
should
functions
be addressed.
Shorter
2958 s
T x X Y Y
Communications
correction factor absolute temperature, K mole fraction of x-phase vector of mole fraction of x-phase mole fraction of y-phase vector of mole fraction of Y-pha*
Greek letrers step-limiting factor P activity coefficient Y
for Broyden
procedure
Subscripts iteration number I components i, i x, Y phases Superscript label for the initial guess vector REFERENCES
Anderson, T. F. and Prausnitz, J. M., 1978, Application of the UNIQUAC equation to calculation of multicomponent phase equilibria. Ind. Engng Chem. Process Des. Dev. 17, 561-567. Annesini, M. C., Gironi, F. and Marrelli, L., 1985, Liquid-liquid equilibria for ternary systems containing hydrocarbons and propylene carbonate. J. &em. Engng Dam 30, 195-196. Callen, H. B., 1985, Thermodynamics and an Introdttction to Thermostatics. Wiley, New York. Cassel, G. W., Hassan, M. M. and Hines, A. L., 1989, Liquid-liquid equilibria for the hexane-benzenedimethyl sulfoxide ternary system. J. them. Engng Data 34, 328-331. Heidemann, R. A., 1975, The criteria for thermodynamic stability. A.1.CL.E. J. 21, 824-826. Holland, C. D. and Uapis, A. J., 1983, Compurer Methodsfor Soluing Dynamic &per&ion Problems. McGraw-Hill, New York. Mukhopadhyay, M. and Sahasranaman, K., 1982, Computation of multicomponent Iiquid-liquid equilibrium
data for aromatics extraction systems. Ind. Engng Chem. Process Des. Dev. 21, 632-640. Ohanomah, M. 0. and Thompson, D. W., 1984, Computatlon of multicomponent phase equilibria-part 2. Liquid-liquid and solid-liquid equilibria. Comput. them. Engng 8, 157-162. Prausnitz, J. M., Anderson, T. F., Grens, E. A., Eckert, C. A., H&h, R. and O’Connel, J. P., 1980, Computer Calculations for Multicomponent Vapour-Liquid and Liquid-Liquid Equilibria. Prentice-Hall, Englewood Cliffs, NJ. Prausnitz, J. M., Lichtenthaler, R. N. and de Azevedo, E. G., 1986, Molecular Thermodynamics ofFluid-phase Equilibria. Prentice-Hall, Englewood Chffs, NJ. Reid, R. C.. Prausnitz, J. M. and Poling, B. E., 1988, The Properties ofGases and Liquids. Prentice-Hall, Englewood Cliffs, NJ. Ruiz, F., Prats, D. and Gomis, V., 1985, Quaternary liquid-liquid equilibrium: water-ethanol-chloroformtoluene at 25°C. Experimental determination and graphical and analytical correlation of equilibrium data. J. them. Engng Data 30, 412416. Triday, J. O., 1984, Liquid-liquid equilibria for the system J. them. Engng Data 29, benzene-water-methanol. 321-324. APPENDIX
The elements
of the Jacobian
i=l,2
,_._,
c-l,j=1,2
,_..,
c
(A-l)
where bij =
1
i=j
0
i #j.
and
a!
ayi=l
j=l,2,...,c.
am-2509/91 $3.00 + 0.00 Q 1991 Pergamon Press plc
Minimum
film thickness
for reverse roll coating
(First received 10 August 1990; accepted in revised form 22 March 1991)
INTRODUCTION
Reverse roll coating operation is considered as a simple and flexible means for depositing a thin liquid film on a moving substrate. Ho and Holland (1978), Benkreira et al. (1981) and Greener and Middleman (1981) applied the lubrication approximation to predict the coating thickness. They found that the coating thickness can be reduced as the coating speed goes up, and this conclusion was supported with the experimental results. Recently, Coyle et af. (1989) carried out a theoretical analysis on reverse roll coating. They claimed that the lubrication approximation is only valid over a
limited range of parameters. The coating thickness decreases initially as the coating speed goes up, it passes a minimum value and then increases rapidly. The prediction of Coyle et al. was supported by some experimental evidence. However, an emprirical slip coefficient on the three phase line was required for their finite-element simulations and therefore the theoretical predictIon may depend on the choice of this coefficient. In this communication we supplement the work of Coyle et ai. by presenting the results based on our reverse roll coating
experiment.
We also observed
the phenomenon
of