I n d . Eng. C h e m . Res. 1988,27, 1481-1487
1481
Packed Crisscross Flow Cascade Tower Efficiencies for Methanol-Water Separations. Experimental versus Calculated Values Based on Countercurrent Flow Correlationst A. Velaga Process Development, A T & T Technology Systems, Lee's Summit, Missouri 64063
L. J. Thibodeaux* and K. T. Valsaraj Department of Chemical Engineering and Hazardous Waste Research Center, Louisiana State University, Baton Rouge, Louisiana 70803
R. B. Eldridge and D. M. Moncada Department of Chemical Engineering, University of Arkansas, Fayetteville, Arkansas 72701
J. S. Cho Department of Chemical Engineering, Oregon State University, Corvallis, Oregon 97331
A methodology was developed for predicting the stage efficiencies of chemical separation in stripping and distillation in crisscross flow packed columns by using existing correlations for mass-transfer coefficients in countercurrent packed columns. Experimental data for methanol-water distillation in a four-stage crisscross flow packed column and for air stripping of methanol from water-methanol solutions in two- and four-stage packed sections were used to check the algorithm. Gas-liquid contacting operations like absorption, distillation, and stripping are usually carried out in one of three configurations, namely cocurrent, countercurrent or crisscross flow. In cocurrent operations, the two contacting fluids flow in the same direction, while in countercurrent operations the liquid phase runs down the packing by gravity and the gas phase is forced upward. One of the major drawbacks of a countercurrent device is the so-called "flooding" condition which happens when a critical gas velocity is reached, at which point the downward stream of liquid is so impeded by the upward flow of the gas that the void spaces of the packing are saturated and mass transfer ceases. However, below this condition high mass-transfer rates are possible. The cocurrent device, on the other hand, does not suffer from such a condition, but due to the small driving force for mass transfer, the efficiency is generally small. The third mode of operation is called the "crisscross" flow cascade. Here the liquid downflow is vertical while the gas phase is in cross flow, alternating in a back and forth direction (Figure 1). This operational mode offers certain advantages over the countercurrent tower in mechanical operation such as significantly lower pressure drops, higher gas-to-liquid ratios, and small gas-phase power requirements (Thibodeaux, 1965). However, it is a somewhat less efficient mass-transfer device as compared to a countercurrent process (Thibodeaux, 1965; Pittaway and Thibodeaux, 1980; Thibodeaux and Moncada, 1980). On the other hand, it is a more efficient mass-transfer device than a cocurrent tower. A performance comparison of a crisscross flow cascade with a countercurrent tower for stripping methanol from water has been reported by some of us (Thibodeaux and Moncada, 1980). Even though the raw efficiencies were slightly lower than the conventional countercurrent column operated under identical conditions, the cascade cross flow stripper was found to be superior when the efficiencies were based on pressure drop, energy expended, or volume of packing. We have also
* To whom
correspondence should be addressed. Presented a t the 194th National American Chemical Society Meeting before the Division of Industrial Engineering Chemistry a t New Orleans, LA, August 30-September 4, 1987.
0888-5885/88/2627-1481$01.50/0
reported previously a design approach for crisscross flow cascade by utilizing the theoretical transfer units concept (Thibodeaux et al., 1977), and the fluid dynamic observations on this device (Thibodeaux, 1980). The overall operation in a crisscross tower is still countercurrent. We therefore felt that it would be worthwhile to investigate whether the abundant information available on mass-transfer coefficients in countercurrent columns can be profitably used to predict crisscross flow efficiencies. A methodology for the above is described and a comparison made between the experimental and calculated stage efficiencies for the methanol-water distillation data of Eldridge (1981) and the methanol stripping data of Moncada (1978).
Theory The theory of crisscross flow packed columns has been described in detail in some of our earlier publications (Thibodeaux, 1965;Thibodeaux et al., 1977). We start here with the defmition of the width of the packed stage (Figure 2) in the direction of the gas flow as (Thibodeaux, 1965) w = NogHog (1) where Hogis the overall height of a transfer unit and No is the number of overall gas-phase transfer units. Hogand Nogare given by Hog = G/(Kog4 (2) and No, =
Jhyoa
& j
(3)
The determination of Nogby using eq 3 is possible only after the gas-phase mole fraction, y*, in equilibrium with the liquid-phase mole fraction, x , in the tower element is known. The liquid-phase concentration in the element varies with both w and z. The only known data about the liquid-phase concentrations are the average inlet ( x h ) and outlet (xouJ mole fractions. Through experience with cross flow cooling tower calculations, Baker and Shyrock (1961) suggested that the simplest procedure would be to assume that the entering potential difference exists throughout the 0 1988 American Chemical Society
1482 Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 Liquid In
L.
","
L.h,
Figure 2. Single stage in crwscurrent cascade and volume element dz-dw-1. Liouid
In
Liquid Out
Figure 1. Four-stage crosscurrent packed column
unit volume. They noted that this driving force is always greater than the true average. Hence, an alternate method was proposed hy them which was to average the reciprocals of the entering and exiting conditions, hut this corresponds to parallel flow, and the average will he too low. The true mean value lies between the two methods. They recommended that the potential difference he averaged, instead of the reciprocals, to give a value that, although smaller than the former and greater than the latter, more closely approximates the true value. If the equilibrium is expressed in general form, y* = f ( x ) , the potential difference may he expressed as
Fcv) = Y*
- Y = mi") - Yi"l/2
+ VX(J,
- Y,,t1/2
(4)
Thus,
No, =
I I I
, I
Liquid Out
Figure 3. Drift angle 0 definition,
The drift angle can he obtained by using the relations proposed by Pittaway and Thihodeaux (1980):
- Yi" Fb)
Yovt
(5)
Using the two-resistance theory, we can obtain Hog as (Treyhal, 1980)
Hog = H ,
+
(m$Hl
where Hg= , G / ( k p ) ,Hl = L / ( k , a ) ,and m is the slope of the equilihrlum line. The Murphree efficiencyfor a single stage is defined as E ( M w . ) = Cyou-t Y~.)/[F(~,J
- yi.l
(7)
The flow of gas across the liquid induces a "drift- on the liquid in the direction of gas flow at an angle 0 (Figure 3) from the vertical plane. This "drift" creates a dry portion in the packing. The fraction of the wetted portion can he calculated as fi
= 1- (t tan 8)/zw
(8)
where h, = liquid holdup, f is the friction factor, A, = characteristic area (ftz),e = void fraction, g = void fraction without liquid, and Cz = constant. For the calculation of Hoghy using eq 6, one needs t o have the correlations for Hgand HI for crisscross flow in packed columns. Since the overall operation in a crisscross flow cascade is still countercurrent, in the present work existing correlations for mass transfer in packed columns measured under countercurrent conditions are used. The liquid-phase-transfer unit was calculated by using the Sherwood-Holloway (1940) correlation:
This fraction, which is also called the "irrigation factor", changes the effective length of the gas-phase flow:
wen= wfi = H,JV%
(9)
where s = 0.5 and a and n are different for each packing.
Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 1483 Table I. Summary of Equations Used for Physical Property Estimations property equation used
ref
gas-phase density gas-phase diffusivity
Chen and Othmer. 1962
gas-phase viscosity
Bromsley and Wilke, 1951 Schiebel, 1959
where gas-phase viscosity (mixture) liquid-phase diffusivity liquid-phase density where
Schiebel, 1959 King et al., 1965 Yen and Wood. 1966
E* = -3.2826 + 13.637+jc+ 107.48iif:
- 384.2iif: for f c 5 0.26 E * = 60.2091 - 402.063fc + 501.05f: - 384.211f: for f, > 0.26 G* = 0.93 - E*; f, = xx,PciVci/RTci; V, = x ~ i V , i ;T, = xxiTCi; T,= TIT,
B' +T-T'
liquid-phase viscosity
log pL = A'
where
Ai = -1.56688 (for water), -1.6807 (for methanol) B1= 230.298 (for water), 354.876 (for methanol) T' = 146.797 (for water), 48.584 (for methanol) p1/3 = xzi/.til/3
liquid-phase viscosity (mixture)
In the present case of 'Iz in. dumped Raschig rings, CY = 280 and n = 0.35. For the gas-phase-transfer unit, the correlation proposed by Yoshida (1955) was chosen:
Hg = a(G/pg)bLCScgd
(12)
where a, b, c, and d are constants which depend upon the packing type. For the present case, a = 0.54, b = 0.33, c = -0.31, and d = 0.67. A number of other correlations can also be used, some of which are described in detail in a recent work (Velaga, 1986). In order to use these equations, one has to obtain the thermal and transport properties in both phases. For the digital computer calculations, several empirical equations developed by previous workers were used. These are tabulated in detail in Table I. All these properties have to be calculated as conditions change from stage to stage. The values calculated using these combinations are within 10% of the experimental values for the physical properties in most cases.
Algorithm for Stage-to-Stage Calculation The data from the experiments described in the next section consisted of the following: the inlet liquid-phase concentration on the top stage; the outgoing gas-phase concentration from the top stage; the superficial mass velocities for both phases; the operating pressure in the column; the length, width, and height of each stage. The flow rate of both phases remained constant throughout the column. With this information and the equations described in the previous section, stage-to-stage calculations were performed by using the algorithm described in the flow sheet diagram given in Figure 4. The computer program developed for this purpose is available as supplementary material. Results and Discussion Eldridge's (1981) experimental data on methanol-water were used as the basis of calculation for the distillation operation. Experiments were performed on a four-stage crisscross device, and concentrations of methanol in liquid
Reid et al., 1977
Perry and Chilton, 1973
Table 11. Methanol-Water Distillation Data" flow rate x 4 E(Murp.) lb/(ft'.h) run L G (expt) (computed) (computed), % 0.3595 30.1 1 6.74 3.0914 0.347 0.3673 31.1 2 6.74 3.0914 0.377 3 6.74 3.036 0.340 0.3502 31.1
x4
"Average value of Hg= 0.7525 ft; average value of HL = 0.4635 ft; packing, '/'-in. Raschig ring (ceramic); pressure, 1 atm; feed concentration, 50% (mass percent methanol); stage height, 1.10 ft; stage length, 0.40 ft; stage width, 0.27 ft. Table 111. Methanol-Water Distillation Data" flow rate, x4 x 4 E(Murp.) lb/(ft'-h) run L G (expt) (computed) (computed), % 1 12.893 3.632 0.314 0.3205 34.8 0.3195 32.9 2 12.893 3.632 0.301 0.3123 32.2 3 12.893 3.883 0.316 aAverage value of Hg= 0.6656 ft; average value of HL = 0.5782 ft; packing, '/'-in. Raschig ring (ceramic); pressure, 1 atm; feed concentration, 45% (mass percent methanol); stage height, 1.10 ft; stage length, 0.40 ft; stage width, 0.27 ft.
and vapor samples at the inlet and outlet of each stage were measured. Experimental details are available in the thesis by Eldridge (1981). Data for the top stage (stage 1) were not used because of the vapor condensation due to the unsaturated feed condition. Hence, the calculation was done from the second stage through the bottom stage. Liquid composition (xl) from the first stage and vapor composition ( y z ) from the second stage were obtained from the experimental data, and the compositions at each succeeding stages were calculated by the previously described algorithm (a sample output is available along with the supplementary material). Eldridge (1981) performed 13 separate experiments, and computations were carried out for each in this work. The bottom product concentration (x4) from the calculation was compared with Eldridge's experimental data. They showed remarkably good agreement for all the data (Figure 5). The Murphree
1484 Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988
Select Xou+
Calculate Y , " , X o v s yovs I
Call Subroutine To Obtain
Thermal B Transwrt Properties 1
I
I
I
8
I
I
I
2
3
4
X4 (Experiment 1 Figure 5. Calculated vs experimental exit concentration for distillation.
Print, X,,,
1-
xinXout for next stage
1
Figure 4. Flow sheet for the design algorithm. Table IV. Methanol-Water Distillation Data' flow rate, x4 x 4 E(Murp.) lb/(ft2*h) run L G (expt) (computed) (computed), % 1 7.391 2.287 0.149 0.1534 33.6 2 7.391 1.989 0.142 0.1355 35.2 3 7.391 1.016 0.137 0.1402 35.0 'Average value of H, = 0.6476 ft; average value of HL = 0.4389 ft; packing, '/2-in. Raschig ring (ceramic); pressure, 1 atm; feed concentration, 25% (mass percent methanol); stage height, 1.10 ft; stage length, 0.40 ft; stage width, 0.27 ft. Table V. Methanol-Water Distillation Data' flow rate, x4 x4 E(Murp.) lb/(ft2*h) run L G (expt) (computed) (computed), % 1 14.747 2.696 0.172 0.1597 39.0 2 14.747 2.504 0.175 0.173 38.7 3 14.747 2.495 0.160 0.1504 39.5 aAverage value of H, = 0.5534 ft; average value of HL = 0.5620 ft; packing, '/&. Raschig ring (ceramic); pressure, 1 atm; feed concentration, 25% (mass percent methanol); stage height, 1.10 ft; stage length, 0.40 ft; stage width, 0.27 ft.
efficiencies also increased with an increase in the ratio of liquid and gas flow rates (Figure 6). All the relevant data used for these calculations are given in Tables 11-VI. The heights of mass-transfer units for both phases changed with the flow rates. When the L I G ratio increased, the ratio of H,/H, decreased. This result is not surprising since this mass-transfer operation would be controlled more by the liquid phase as the ratio LIG increased. For a given operating condition, the variation in drift angle for each run
I
2
3
4
5
6
L/ G Figure 6. Murphree efficiency vs L / G . Table VI. Methanol-Water Distillation Data' flow rate, x, x4 E(Murp.) lb/(ft2.h) run L G (ex& (commted) (comDuted). % 1 8.633 3.257 0.075 0.0961 32.93 'Average value of Hg= 0.6877 ft; average value of HL = 0.4470 ft; packing, '/*-in. Rashig ring (ceramic); pressure, 1 atm; feed concentration, 18% (mass percent methanol); stage height, 1.10 ft; stage length, 0.40 ft; stage width, 0.27 ft.
ranged from 1' to 3". For this range of drift angles, the irrigation effect would be very small. For the operation of air stripping of methanol from water, Moncada's (1978) experimental data were used as the basis. Moncada (1978) conducted several experiments on air stripping methanol from water on a four-stage and a two-stage crisscross flow device. Experimental details and analytical methods are available in the thesis by Moncada (1978). In these experiments, the bottom product was recycled as the feed; hence, the whole process was an unsteady-state batch operation. The feed concentration therefore decreased during the experiment. A schematic of the process in given in Figure 7. To compare the experimental results with the computed values under
Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 1485
f
Go iyout
r"l
01
4
1 I I
4
I I
Col;nit
I I
I
0
* 0 a
I
-z5 u
Figure 7. Schematic of cross flow air stripping equipment.
steady-state conditions, it was necessary to calculate the overall mass-transfer efficiencies with the varying feed concentration at constant G and L. The overall masstransfer efficiency for the stripping operation in this case was defined as Xin - Xout Xin
3
a
xi"
K=
05
04
4
in 9
_ _ _ _ L_i ,_ _ _ _J
L
0 3
Figure 8. Feed concentration vs overall mass-transfer efficiency for the methanol-water stripping.
Lo 1 XOUt
r----------
0.2
FEED CONCENTRATION ( M o l e Fraction )
2
x
Xin'xout Xin
- 1
(13)
A material balance on the solute in the collecting tank (see Figure 7 ) yields
% (Experimental 1
Figure 9. Computed vs experimental overall mass-transfer efficiency-stripping operation. 4
0
0 3 a a
L
where Lo and Li are the molar liquid flow rates and M is the moles of water present in the tank. We shall now make the following assumptions: (i) Li = Lo = L at all t > 0; (ii) M = Mo at t 2 0. Notice that this last assumption is only applicable at low liquid rates at which the liquid holdup in the column is negligible. Using eq 13 in 14 yields
E
0 I
$ 2
x
The above equation can be integrated using the following conditions: (iii) x h = x , at t = 0; (iv) x h = x at t 1 0. We can thus obtain In ( x / x , ) = -(KL)t/Mo
(16)
The above equation suggests that a plot of In ( x / x , ) versus time would give a straight line, the slope of which can be used to calculate the overall mass-transfer efficiency for the stripping experiments. The experimental masstransfer efficiencies for the stripping operations were thus obtained and compared with the calculated values. These are listed in Table VII. The results plotted in Figure 8 indicate that the overall mass-transfer efficiencies in the
2
i
x'~i:out
t
3
4
%(Experimental)
Figure 10. Computed vs experimental overall mass-transfer efficiency-stripping.
column remained constant at the dilute feed concentration. Figures 9 and 10 show comparison of calculated and experimental overall mass transfer efficiencies at two different gas flow rates. In general, the computed values were always slightly higher than the experimental values. In contrast to the distillation operation, the height of the liquid phase transfer unit was several times larger than
1486 Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 Table VII. Air Stripping of Methanol from Water"
L , kg/m2
G, kg/m2 0.789 0.789 0.789 0.789 0.789 0.270 0.789 1.033 1.033 1.033 1.033 1.033 1.033
3.5 4.0 4.5 5.0 6.0 6.5 7.0 4.0 4.5 5.0 6.0 6.5 7.0
H g , ft
HI,, ft
K (expt), %
0.3160 0.3056 0.2967 0.2890 0.2760 1.2645 1.2977 0.3314 0.3218 0.3134 0.2994 0.2934 0.2880
1.0181 1.0668 1.1117 1.1535 1.2295 1.6900 1.6000 1.0668 1.1117 1.1535 0.2295 1.2644 1.2977
2.45 2.10 1.80 1.90 1.79 1.59 1.44 3.05 2.83 2.05 1.95 1.80 1.70
4 stage (calcd) 3.03 2.48 2.35 2.02 1.90
K (expt), % 2.22 1.80 1.55 1.43
2 stage (calcd) 2.61 2.26 1.93 1.76
3.05 2.99 2.76 2.20 2.03 1.89
3.01 2.29 1.96 1.81 1.73
2.85 2.43 2.22 1.93 1.87
"Packing, 6/s-in. Pall ring (plastic); pressure, 1atm; temperature, 298 K; feed concentration, 0.1 (mole fraction of methanol); stage height, 0.755 ft; stage length, 0.463 ft; stage width, 0.503 ft.
I LL
aa
-
20
G
3 1
I
3
4
I
I
5
6
5
4
6
7
kq/m2sec) Figure 12. Calculated drift angle vs L for the stripping operation. L (
7
L ( kg/m2sec)
Figure 11. H L / H , vs L for the stripping operation.
that of the gas phase. This is shown in Figure 11. Also it was observed that the overall mass-transfer efficiencies were strongly affected by the mass velocity of the liquid phase. The drift angle (computed) reached 3 4 O when the gas flow rate was 1.033 kg/(m2s) (Figure 12). Such large drift angles can cause large dry regions in the packing section and can considerably reduce the efficiency of a stripping operation. Conclusions The results show that, in general, the mass-transfer correlations which are most commonly used for countercurrent packed tower operations can be used for satisfactory predictions of a crisscross flow packed tower operation. This means that, without performing separate experiments to obtain the mass-transfer coefficients, the equations and values obtained from experiments for the countercurrent operations can be used, with slight modifications for crisscross flow tower operations. A tentative design algorithm has been proposed for a crisscross flow operation, and calculations based on this algorithm are supported by the experimental results. The effects of drift angle as given by the proposed algorithm were shown to be an important factor that adversely effects the masstransfer efficiencies in a crisscross operation. Further experiments to test such effects are recommended. It is also recommended that the applicability of this algorithm be tested for systems other than methanol-water.
Acknowledgment We thank two anonymous reviewers for some helpful comments. Nomenclature A = component A in a mixture of A and B B = component B in a mixture of A and B a = interfacial area, ft2/ft3 a, = wetted area, ft2/ft3 ap = surface area of packing, ft2/ft3 D = diffusivity, ft/h2 f = friction factor, dimensionless fi = irrigation factor, dimensionless G = superficial gas mass velocity, lb/ (ft2.h) h, = total liquid holdup, dimensionless H = height of mass-transfer unit, f t 12 = mass-transfer coefficient, (lb.mol)/(ft2.h) 1 = length of stage, f t M = molecular weight, lb/(lb.mol) m = slope of equilibrium line P = total pressure, atm R = gas constant T = temperature, K V = volume, ft3 w = width of stage, f t x = mole fraction in liquid phase y = mole fraction in gas phase z = height of stage, f t Greek Symbols t = porosity 0 = drift angle, deg
Ind. Eng. Chem. Res. 1988,27, 1487-1493 p
= density, lb/ft3 = viscosity, lb/(ft-h)
SubscriDts
A = component A B = component B c = critical g = gas phase i = component i 1 = liquid phase o = overall t = total Registry No. Methanol, 67-56-1; water, 7732-18-5. Supplementary Material Available: The Fortran program for the algorithm described in the text (3 pages). Ordering information is given on any current masthead page. Literature Cited Baker, D. R.; Shyrock, H. A. J. Heat Transfer 1961,83,339. Bromsley, L. A.; Wilke, C. R. Ind. Eng. Chem. 1951,43,1641. Chen, N.H.;Othmer, D. H. J. Chem. Eng. Data 1962,7,37. Eldridge, R. B. M.S. Thesis, University of Arkansas, Fayetteville, 1981. King, C. J.; Hsuek, L.; Mao, K. W. J.Chem. Eng. Data 1965,10,348.
1487
Moncada, D. M. M.S. Thesis, University of Arkansas, Fayetteville, 1978. Perry, R. H.; Chilton, C. H. Chemical Engineer’s Handbook, 5th ed.; McGraw-Hill: New York, 1973. Pittaway, K. R.; Thibodeaux, L. J. Znd. Eng. Chem. Process Des. Deu. 1980,19,40. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. Sherwood, T. K.; Holloway, F. A. L. Trans. AZChE 1940,36, 39. Schiebel, E. G. Znd. Eng. Chem. 1959,46,1959. Thibodeaux, L.J. Chem. Eng. 1965. June 2. 165. Thibodeaux; L. J. Znd. Eng. Chem. Process Des. Dev. 1980,19,33. Thibodeaux, L.J.; Moncada, D. Paper presented a t the symposium on “Recent Advances in Separation Technology“, National Meeting of the American Institute of Chemical Engineers, Chicago, IL, Nov 16-20, 1980. Thibodeaux, L.J.; Daner, D. R.; Kimura, A,; Millican, J. D.; Parikh, R. J. Ind. Eng. Chem. Process Des. Deu. 1977,16, 325. Treybal, R. E. Mass Transfer Operations, 3rd ed.; McGraw-Hill: New York, 1980. Velaga, A. Ph.D. Thesis, University of Arkansas, Fayetteville, 1986. Yen, L. C.; Woods, S. S. AZChE J. 1966,12,95. Yoshida, A. Chem. Eng. B o g . Symp. Ser. 1955,51,59.
Received for review April 23, 1987 Revised manuscript received November 20, 1987 Accepted April 15, 1988
GENERAL RESEARCH Cyclopentadiene-Dicyclopentadiene Phase Equilibria and Reaction Rate Behavior Using the Transient Total Pressure Method Colin S. Howat* and George W. S w i f t Kurata Thermodynamics Laboratory, Department of Chemical and Petroleum Engineering, University of Kansas, Lawrence, Kansas 66045
This article presents new phase equilibria and dimerization rate data for the cyclopentadiene-dicyclopentadiene binary system using the transient total pressure experimental method. This experimental method can be applied to reacting systems to determine simultaneously phase equilibria and reaction rate behavior without phase sampling when it is coupled with a suitable experimental design method. Phase equilibria and reaction rate data are reported a t 313 K. Reaction rate parameters based on an autocatalytic, second-order model are also reported. The initial rate of dimerization is equivalent to those reported in the literature. Cyclopentadiene (CPD) is a contaminant and catalyst poison in the process for the polymerization of isoprene. The acceptable limit for CPD in polymerization-grade isoprene is approximately 1ppm. Raw isoprene feedstocks contain about l0-20% CPD. Measurement of the phase equilibria of chemical mixtures containing CPD is complicated by the spontaneous dimerization of CPD to dicyclopentadiene (DCPD). Relative volatilities of CPDisoprene mixtures are near unity, which mitigates against using the traditional method of sampling coexisting phases for reacting systems. Also, the uncertainty in the phase equilibria is increased further by the chemical-reactioninduced nonequilibrium. Consequently, a new experimental/numerical method was required to measure the phase equilibria of CPD-containing mixtures. This article presents a numerical method founded upon maximum likelihood estimation (MLE) which can be applied to transient total pressure data to simultaneously 0888-5885/88/2627-1487$01.50/0
determine the phase and kinetic behaviors. This analysis procedure is used in conjunction with the experimental design procedure of Howat and Swift (1983) to ensure that systematic differences between the resultant model and the data are minimized. In this new experimental procedure, the pressure transients of samples with different starting compositions are measured over a suitable time interval. The transient data are then analyzed by using maximum likelihood estimation to determine the rate constants, solution model parameters, and vapor pressures. The experimental design is then reanalyzed to minimize systematic differences between the model and the data. Discussion of t h e Method The governing equation for vapor-liquid equilibria is y.@P $ 2 = 3tiyiPioq+” exp(ui(P- P,O)/RT) (1) provided that (1) the component partial molar volume is 0 1988 American Chemical Society
