432
Ind. Eng. Chem. Process Des. Dev. 1980, 19, 432-439
Collection of Phase Equilibrium Data for Separation Technology Wllllam Schotte Engineering Technology Laboratory, E. I. du Pont de Nemours & Company, Wilmington, Delaware 79898
Techniques for measuring phase equilibrium data in a glass equilibrium cell at pressures up to 800 psig are described. Various equations of state are reviewed and are compared on the basis of their ability to predict vapor-phase nonideality over a wide range of temperatures and pressures. Equations for the prediction of activity coefficients for multicomponent mixtures are discussed and results have been compared for three-, four-, and five-component mixtures. The effect of vapor-phase association is shown for the special case of HF.
Physical separations are used in most processes to obtain a desired product, recover unreacted materials, remove byproducts, and clean up waste streams. Quantitative information on phase equilibria is needed for design of the separation equipment. Currently, the most widely used separation techniques are distillation, extraction, absorption, and stripping for which vapor-liquid and/or liquidliquid equilibrium relationships are required. We have expanded our experimental facilities during the last few years. Measurements of vapor-liquid equilibria, liquid-liquid equilibria of partially immiscible liquids, solubilities of gases in liquids, freezing points of liquid mixtures, and densities of liquid mixtures have been carried out. Temperatures from -50 to 250 "C and pressures up to 1000 psig have been employed.
'1\I
Experimental Technique A schematic view of the main components of the equipment is shown in Figure 1. The key component is the glass equilibrium cell, developed by Slocum (1975). It consists of a 2 in. diameter by 6 in. long glass cylinder held between stainless steel flanges. A magnetically driven agitator disperses the upper phase (gas or liquid) into the lower liquid phase. The visibility afforded by the glass walls of the equilibrium cell has been vitally important in the detection of solid or liquid phases, dew or bubble points, critical points, and in the measurement of rates of phase separation. The glass cell has been used at pressures up to 800 psig; a stainless steel cell can be substituted for work at higher pressures. In most cases, two liquid compounds are fed in metered amounts to the cell, but on occasion, as many as five components have been separately metered into the cell. As shown here, degassed compound A can be drawn into a calibrated high-pressure buret from which it can be metered under constant mercury pressure into the evacuated cell. The buret is a Penberthy sight glass of 67 cm length, which was modified by attaching a scale and by replacement of the ribbed glass with a flat glass face. The displacement of component A by the mercury is read on the scale with the aid of the telescope of a cathetometer with a precision of 0.05 cm3 and an accuracy of 0.1 cm3. Compound B can be fed from another high-pressure buret or can be metered with a piston pump, as shown in Figure 1. In this case, degassed compound B is drawn into a Hoke cylinder, maintained at constant temperature, from which it is displaced by mercury for introduction into the cell. The piston pump is a Ruska positive displacement pump, which is manually operated at constant pressure. The metering precision is 0.02 cm3, but the accuracy is within 0.05 cm3. There is an additional uncertainty in the composition of the mixture in the cell due to holdup in the 1/16-in.0.d. lines (0.1 cm i.d.) and the nearby valves. This 0196-4305/80/1119-0432$01.00/0
creates an uncertainty of up to 0.3 cm3for each component. Including the metering error, the total uncertainty can be as much as 0.4 cm3 for each component. The volume of the glass cell is 265 cm3 of which at least 75% is occupied by the liquid. Known or measured liquid densities are used to convert metered volumes to weight fractions and then to mole fractions of the components. The absolute error in the mole fractions is less than 0.003. The equilibrium cell is submerged in an oil bath which contains heating and cooling coils for precise temperature control. Oil bath temperatures are maintained within 0.1 "C. Cell and oil bath temperatures are measured with calibrated thermocouples and a Doric Trendicator (digital thermometer). The Trendicator is constantly compared with a Doric platinum-resistance thermometer (accuracy within 0.1 "C), which also measures the oil bath temperature. Temperature non-uniformities within the oil bath are not more than 0.1 "C, giving an overall uncertainty in temperature of 0.2 "C. The multipoint Trendicator is also used to read temperatures of the burets and Hoke cylinders. After filling of the equilibrium cell and attainment of the desired operating temperature, the cell pressure is measured with a pressure transducer. Initially, strain gauge type transducers were used. Their accuracy is within 0.5% of full range. By changing transducers when pressures are less than one-half of full range, the error in pressure readings can be kept below 1%. Between runs the calibration of pressure transducer is checked with a Heise or Digigauge of the appropriate pressure range. The accuracy of recalibration of the pressure transducer is within 0.3%, but the main source of error is hysteresis during the run. This may amount to 1% of the actual reading, giving a total uncertainty of 1.3%. A t present, variable-reluctance type transducers are being employed. A variable-reluctance type transducer of a particular differential pressure rating (e.g., 50 psid) is used at absolute pressures from one to six times the pressure rating (50 to 300 psia) as a nulling device. The cell pressure on one side of the diaphragm is kept balanced by helium or nitrogen pressure on the other side and the gas pressure is measured with one of a set of Heise or Digigauges. This procedure has reduced the maximum error in the measured pressure to 0.8%. Another advantage of the variable-reluctance type transducer is its very low holdup of 0.07 cm3, as compared to several cm3for the more conventional strain gauge type transducers. A vapor-liquid equilibrium experiment is usually complete when the total pressure has been measured. The vapor composition is then calculated from vapor-liquid equilibrium equations and equations for the activity coefficients, as discussed below. The liquid composition is determined from the total amounts of the components
0 1980 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3. 1980 433
Figure 1. Equipment schematic.
and the small amounts which are present in the vapor phase. Occasionally,vapor samples are taken and analyzed with a gas chromatograph. The phase-equilibrium facility is equipped with a gas chromatograph, which is connected hy a heated 0.d. sample line to the equilibrium cell. Samples are injected with a heated sample valve into the chromatograph. The uncertainty of the vapor analysis is usually 1 to 2%, but can be as large as 5% for reactive compounds, such as HF. It should be noted here that our purpose of vapor-liquid equilibrium experiments is not so much the determination of specific activity coefficients as it is to determine the empirical parameters in a reliable equation for the activity coefficients. Equations for the activity coefficientsare finally used in computer programs for the design of distillation columns or other separation equipment. As a result, vapor-phase analysis is really not needed. Occasional measurements are made as a check, particularly when more than two components are present. Many experiments have been carried out with hazardous materials, such as HF, HCl, Clz, and CzF4,at elevated pressures. A transparent Teflon liner was developed for experiments with HF in the glass cell. For safety, the cell and oil bath are located inside a steel housing, which can be lowered hydraulically for maintenance and other purposes. Figure 2 is a photograph of part of the equipment with the steel housing in a lowered position. The housing contains two large viewing ports. There is a continuous nitrogen purge through the housing, which is used not only for safety hut also to eliminate external condensation during low-temperature runs. After completion of runs, the contents of the cell are released to a scrubber for removal of harmful compounds. Most experimental work has involved the determination of vapor-liquid equilibria by measuring the saturation pressures exerted by two-component liquid mixtures of known composition a t selected temperatures. Liquidliquid equilibria have usually been determined by careful metering of component A into the cell, which contained a known amount of component B, until at the limit of solubility a cloud point is reached. The liquid mixture becomes transluscent at the cloud point and contains then a little bit too much A. Very small amounts of component B are added until only a few tiny droplets of the second liquid phase remain as seen with the telescope of a cathetometer. In addition to two fluorescent lights inside the steel enclosure, there are also two submerged fiber lights in the oil bath to improve observation of the droplets. The amounts of the components fed to the equilibrium cell are used to calculate the composition at the solubility limit
Figure 2. Equilibrium apparatus.
for a binary mixture. The phase envelope of a ternary mixture can be measured in a similar manner by feeding known amounts of components A and B to the cell and by careful addition of component C until the limit of miscibility is reached. However, samples from both liquid phases have to be taken and analyzed to determine tie lines for three-component mixtures. Error Analysis Measurement errors are due to uncertainties in the liquid composition, the pressure, and the temperature. Experiments are usually made with about nine compositions ranging from 10 to 90 mol % Uncertainty in the liquid composition is less than 0.3 mol %, which may cause a 3% error at 10 mol % or a 0.6% error at 50 mol %. During subsequent least-squares fitting of parameters for the activity coefficient equations to give the best match between measured and predicted pressures, all points are given equal weight. Although measurements in the middle of the concentration range are more accurate, the parameters are better defined by the points at high and low conceutrations. It is estimated that error due to uncertainties in composition is not more than 1.6%. Pressure measurements with the variable-reluctance type transducers contribute less than 0.8% to the error. A 0.2 OC temperature uncertainty creates a 0.6% uncertainty in the vapor pressures of the pure components. Measurement errors total at most 3%. Calculation errors come from the prediction of fugacity coefficients and from the shortcomings of the equations for the activity coefficients. Fugacity coefficients can be predicted within 2%, except in the critical region. Since a ratio of two fugacity coefficients is used, the uncertainty will usually be less than 1%.The uncertainty of a reliable equation for the activity coefficients of a binary mixture of moderate nonideality should not be more than 3%. However, 10% errors for highly nonideal mixtures, such
.
434
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980
as immiscible systems, are to be expected. In general calculation errors will be less than 4%. The overall error of 7% for a typical set of measurements and calculated activity coefficients seems large. However, this is a maximum error and uncertainties will probably be no more than about 5% for mixtures of moderate nonideality. Better correlations to describe activity coefficients are needed before uncertainties can be reduced. Evaluation of Results Analysis of samples is time consuming and generally not needed. We prefer to simply measure the saturation pressures exerted by liquid mixtures of known composition in vapor-liquid equilibrium studies. At equilibrium, the fugacity of each component in the vapor is equal to that in the liquid phase
gleb (1953) to correlate the association of HF have not been adequate. We have developed a useful model for HF association, based on the vapor-phase equilibria: 2HF $ (HF),, 6HF F! (HF),, and 8HF @ (HF),. Tetramer was found to contribute very little and was of some significance only in the range of 195 to 240 K. Equilibrium constants for the above equilibria are
where fll, flz,f16,and fl, are the fugacities of monomer, dimer, hexamer, and octamer, respectively. Tamir and Wisniak (1978) have shown for a mixture of monomers, dimers, tetramers, etc., that the fugacity coefficients of all of these species are approximately equal. Then for pure HF $IOP~O
where yi = mole fraction of component i in the vapor, $i = fugacity coefficient of i in the vapor mixture, P = total pressure, xi = mole fraction of i in the liquid, y i = liquid-phase activity coefficient, c $ ~=~fugacity coefficient of pure component i, pio= vapor pressure of pure component i, ui = partial molar volume of i in the liquid, R = gas constant, and T = absolute temperature. For a binary mixture, there are two equations corresponding to eq 1. A computer program solves these equations for the total pressure, P, and the vapor-phase composition, y1 (and yz = 1 - yl). An equation of state is used to calculate the fugacity coefficients. Nonideality in the liquid state is accounted for through the use of activity coefficients. The equations for the activity coefficients contain two or three semiempirical parameters, which need to be determined by comparing calculated and measured saturation pressures of the mixtures. A nonlinear regression technique is used to obtain those values of the parameters which give the smallest deviations between the estimated and measured pressures. By carrying out experiments at different temperatures, the temperature dependence of the parameters can also be estimated. Measurements have usually been made for liquid mixtures which nearly fill the cell. Occasionally, the amount of material in the vapor space is not negligible. If necessary, our computer program will correct the liquid-phase composition for the amounts of the components present in the vapor phase. Another computer program can be used to make liquid-liquid equilibrium calculations. The equilibrium relationship is (Yixi)a = (YiXi)b (2) where subscripts a and b refer to the two liquid phases. An iteration technique is used to solve eq 2 for both components. By selecting an appropriate option, the program can calculate either solubility limits, (xi)Band (Xi)b, when the parameters of the equations for the activity coefficients are known, or the parameters when the solubility limits have been measured. Vapor-Phase Association A few compounds, such as acetic acid and HF, show strong association in the vapor phase, which must be included in the calculations. Investigators, such as Tamir and Wisniak (1978), have dealt successfully with the association of acetic acid and related compounds. However, attempts by Franck and Meyer (1959), Simons and Hildebrand (1924), Smith (19581, and Strohmeier and Brie-
= fila
+ K ~ ( f 1 ~+~K6(fll0I6 )' + KS(~I~O)'(3)
where = fugacity coefficient of HF, pl0 = vapor pressure of pure HF, and fllo = fugacity of HF monomer in pure HF. The equilibrium constants were determined by least-squares fitting of vapor density data. This gave
(1277F22947.97731)/R] K6 exp[ (41927*495 T 138.55185)/R] K8 = ex,[ ( 5012:984 165.85264)/R] K 2 = exp[
-
=
-
-
where K , = (atm)-("-') and R = 1.9869 cal/(mol)(K). Many investigators have discussed an association factor, which is the mean molecular weight of HF divided by the molecular weight of monomeric HF. This is F" = YI, + 2 ~ 1+ , 6 ~ 1+ ,
+ 8K8(f1l0)' - fl," + 2KZ(fll")2+ 6K13(fl,")~
(4)
41°P10 Most investigators have actually defined an association factor, F, in terms of the deviation from the ideal gas law Pv - 1 _
RT-F
In that case F O
F=;
where z = compressibility factor of HF. Figure 3 shows a comparison between the predicted association factor, F , of HF and the results of Vanderzee and Rodenburg (1970) as a function of pressure at 26 "C. Predicted and measured association factors of H F a t saturation are compared as a function of temperature in Figure 4. Agreement is quite good over most of the range from the freezing point (190 K) to the critical temperature (461 K), considering the scatter in the experimental data. It is obvious from these figures that HF association is very significant. When dealing with HF-containing mixtures, the vapor-liquid equilibrium equation for HF monomer becomes
where +1: = fugacity coefficient of pure gaseous monomer
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980 435 A Frajenhagan I 1933 ) 0 Simonr and Hildebrand I1924 ) 0 Vanderzee and Rcdenburg I 1970) V Jarry and Davis I 1953 ) 4 Franck and Spalthoff I1957 ) Theory
4.4
-
2.6
1
2.4 Y
2.2
.!
c
1.6
1'4 1.2 l
t
O
0
B I
01
'
1
1
0 2 0:;
04
05
Pressure
1
I
06 07
' 08
I
I
09
10
I
11
I
12
3.6 3.4
3.0 3'2
J
2.0 2'2
13
1
t
0
\
190 210 230 250 270 290 310 330 350 370 390 410 430 450
am
Tampmure,
Figure 3. Association factor of HF at 26 "C.
' K.
Figure 4. Association factor of HF at saturation.
in the standard state (saturation pressure at T ) and plls = saturation pressure of pure monomer at T. Equation 5 is not convenient for use, since the saturation pressure of pure HF inonomer is not known. However, this can be expressed in terms of the measurable vapor pressure of total HF, plotwhich leads to the expression
'% 10
/
L -o
where fl, = yl,&P = ifugacity of HF monomer in the vapor and fllo = fugacity of HF monomer in pure HF vapor. Equation 3 can be solved to give fl," and eq 6 yields then fll, The partial pressure of total HF in the vapor mixture is P1
=
fl, -t- KdlI2+ Kd2 + Kd1,8
-
(7)
41 and the total pressure is P = Pl + P2,where Pzis calculated
from eq 1 for a nonassociating component. The vapor composition, as comimonly measured on the basis of molecular weight 20.01 for HF, is Y1
=
+ 2Y1, + 6Y1, + 8Y1, -+ 2Y12 + 6YlB+ 8Y1, + Yz
Y1,
Y1,
Since y1, = f1,/41P, ~1~= K jlI2/dJ1P, etc., and y2 = P2/P S Yl = s + 41p2 where
S = fl, + 21Kj112+ 6Kdll6+ 8KSfl18 Measurements and calculations have been made for a mixture of HF and a Freon at 50 OC. The Y-x diagram is shown in Figure 5. Agreement is quite good, considering the difficulty of accuirately measuring HF concentrations by gas chromatography. Although Figure 5 shows only a few vapor composition measurements at 50 "C, similar results were obtained 70 and 90 "C. Equation of State An equation of state is needed to calculate the fugacity coefficients which correct for nonideality of the vapor phase. The most widely used equation is probably the virial equation, since reported data have generally been obtained at relatively low pressures. Good information is available only for the second virial coefficient which limits
IO
20
Measured Predick
30MaleA0% HF50 In LIquld W 70
80
W
1W
Figure 5. Vapor-liquid equilibrium of HF/Freon at 50 "C.
the use of the virial equation to relatively low pressures. During the past few years, several new or modified equations of state have been proposed which appear to give good results at both low and high pressures. They are the Peng-Robinson equation of state (1976),the Graboski and Daubert-modified Soave equation (1978), the HamamChung-Elshayal-Lu modification (1977) of the RedlickKwong equation, and the Plocker-Knapp-Prausnitz modification (1978) of the Lee-Kesler equation. However, there has been no significant comparison of these equations with regard to their ability to predict nonideal behavior of the vapor state. Most comparisons, such as a detailed one by Oellrich et al. (1977), have been based on the use of the equation of state for direct vapor-liquid equilibrium calculations. This requires that the equation of state should account accurately not only for vapor-phase nonidealities, but also for nonideal liquid behavior. Liquidphase nonidealities are more difficult to predict with an equation of state and the comparison might be more of a test of liquid-phase rather than vapor-phase predictions. In our vapor-liquid equilibrium calculations, as described previously, liquid-phase nonideality is included in the activity coefficients. Therefore, a comparison of predicted compressibility factors and fugacity coefficients, which result from vapor-phase nonideality, would be desirable. The Peng-Robinson equation of state is a p = - -RT (9) u - b u(u + b) + b(u - b) where P = pressure, R = gas constant, T = absolute temperature, u = molar volume of the vapor mixture, and a, b = parameters. The compressibility factor is the largest root of the cubic equation
436
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980
+ ( A - 3B2
2B)z - (AB - B2 - B3) = 0 (10) The equation for the fugacity coefficient of component i, 4it is bi In q+ = ~ ( - z1) - ln(z - B) z3 - (1 - B)z2
( :)
0
A
rn
-
2Cyjaji ia
z
RTcm Pcm= (0.2905 - 0 . 0 8 5 ~ ~ ) -
(14)
ucm
+ 2.414B
In ( z
2B(2112)
cm
where
-
0.414B) (11)
where z = compressibility factor, y j = vapor-phase mole fraction of component j , A = a P / R 2 P ,and B = bP/RT. The parameters ai and bi for each component are given by
ai = 0.45724(
F) i
W,
= Cyiwi;
7 =
0.25
I
where ai = [l
+ k i ( l - Tr1/2)]2, ki = 0.37464 + 1.54226wi
= acentric factor, T , = TIT,, T , = critical temperature, and P, = critical pressure. The mixing rules give the mixture parameters a CCygjaij
The compressibility factor, z,, of the mixture is found by interpolation between results for the two reference fluids
- 0.26992wi2, wi
i
l
The equation for the fugacity coefficient, 4i, of component i is
b = Cyibi 1
aij = (1 - Gij)(a;a,)l/z where 6ij = interaction coefficient. The Soave equation is based on the Redlich-Kwong equation RT aa
p=Fi-m
where a and b are functions of the critical temperature and pressure and a is a temperature-dependent function. Equations for the compressibility factor and fugacity coefficient resemble eq 10 and 11,respectively. Graboski and Daubert (1978) have improved the temperature dependency of a. The Lu-modified Redlich-Kwong equation (Hamam et al., 1977) is also based on eq 12. It leads to the same expressions for the compressibility factor and the fugacity coefficient as the Soave equation. However, 42 constants are used to express the temperature dependency The mixing rules of the a and b parameters ( a = P.5). are also somewhat more complex. Lee and Kesler (1975) developed a correlation based on the corresponding states principle. Compressibility factors for a simple reference fluid (zo), methane, and a heavy reference fluid (zJ, n-octane, are each calculated from B C D z = - PI4 = I + - + - + - + TI ur u,2 u.5
where u, = reduced volume. B, C, and D are functions of the reduced temperature. Along with C,, @, and y,there are twelve constants for the simple fluid and another twelve for the heavy reference fluid. The reduced temperature and pressure are expressed in terms of pseudo-critical temperatures and pressures P P, = -
Pcm
where j # i and k # ij. The equations needed to calculate the various terms on the right-hand side of eq 17 can be found in the paper by Plocker et a1 (1978). The four equations of state have been compared by making calculations for as many mixtures of not closely related compounds as could readily be found in the literature. There were only two mixtures, ammonia/ propane and C02/n-butane, for which fugacity coefficients were available from direct experimental measurements. For six mixtures of not closely related compounds, compressibility factors could be found. All together, the following mixtures were used: n-butane/C02 (Olds et al., 1949); n-butane/ methanol (Petty and Smith, 1955);H20/C02(Franck and Todheide, 1959); H2S/n-pentane (Reamer et al., 1953); CH4/C02 (Reamer et al., 1944); C2H,/C6H6 (Kay and Nevens, 1952); and NH3/C3H, (Antezana and Cheh, 1976). This list includes both polar and nonpolar compounds. Table I shows the average percent deviations between predicted and measured compressibility factors and fugacity coefficients of the vapor mixtures. Fugacity coefficients were calculated only for the first component of the binary mixtures, since only the fugacity coefficient of NHB was measured for the ",/propane system. In all cases, the results are shown for the optimum value of the interaction coefficient present in the equation of state. All of the equations of state gave fairly good results. The Lee-Kesler equation (abbreviated Lee), modified by Plocker et al., gave the lowest overall average deviation between predictions and measurements for the mixtures. The Peng-Robinson equation (Peng) was second best. Compressibility factors of HCl have also been predicted, because estimates for this polar compound are hard to make. As shown in Table I, the Peng-Robinson equation
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980 437
Table I Comparison of Compressibility Factor Predictions
T, " C
mixture or compound n-butane/CO, n-butane/CH,OH H,O/CO, H,S/n-pentane cH,/co, CZH, /C,H, pure HCl
pressures, psia
38-204 71-138 400-600 38-171 38-104 116-27 1 2 5-51.5
200-3000 20-500 4350-14500 100-1200 4 00-60 00 300-1400 590-1200
m o l % no. of comp. 1 points 38 34 27 23 42 35 6
17-83 25-75 20-80 14-99 20-85 15-85 100
av % dev of compress. factors Peng
Soave
1.99 1.41 3.18 3.72 1.42 2.22 10.02
4.03 1.47 4.82 3.73 1.58 2.66 14.66
Lu
Lee
Lee (S)
1.72 1.71 2.80 4.07 1.18 4.73 21.75
1.32 1.38 2.40 4.59 0.39 2.86 36.81
1.43 1.21 2.64 4.27 0.36 3.16 8.76
Comparison of Fugacity Coefficient Predictions mixture
T. " C
pressures. psia
",/propane CO,/n-butane
54-71 38-138
16 3-422 200-1150
mol % comp. 1 60-100 2-95
gave a rather high average deviation of 10% when compared with the measurements of Franck et al. (1962) and Beaume and Coulon (19681, which go up to the critical point. The reason is that the Peng-Robinson equation predicts a universal value of 0.307 for the critical compressibility factor, which for HC1 is 0.248. The comparison is worse for the Soave equation (Soave), modified by Graboski and Daubert, which predicts a universal critical compressibility factor of 0.333. It is not obvious why predictions were even poorer with the Lu-modified Redlich-Kwong equation (Lu). The Lee-Kesler eqpation (Lee), modified by Plocker et al., has an inconsistency which was responsible for the very high average percent deviation of 36.8% for HC1. Equations 14 and 15 do not predict critical pressures and temperatures of mixtures very well. When eq 14 is applied in the limit bHCl1)to HC1, a reduced pressure of 0.885 instead of 1.0 is obtained at the critical temperature. This leads to a critical compressibility factor of 0.556 vs. the measured value of 0.248. For good consistency, eq 14 should approach the critical pressures of the pure components when y approaches either 0 or 1. An obvious modification is the use of (18) zcm = Y l Z q + Y2Zcz which gives
-
P,,
=
ZcmRTcm ~
(19)
"cm
where z,, and zc, are the critical compressibility factors of the pure components. Table I shows that this modification, labeled Lee (SI, has given a much lower average percent deviation for HC1. For mixtures, the compressibility factors and fugacity coefficients were as well predicted with Lee (S) as with Lee. For general use in industrial calculations, the PengRobinson equation seems preferable. It is simple, gives acceptable results, and does not give convergence problems when properly used in computer programs. The modified Lee-Kesler equation did not always converge when applied near the critical region. Equation 13 must be used for the simple reference fluid to calculate zo and then for the heavy reference fluid to determine z,. There may be conditions of temperature and pressure where the heavy reference fluid is in the liquid state and so far from saturation that eq 13 cannot give a vapor-phase compressibility factor, 2,. Another consideration is computer time. The Lee-Kesler equations are extremely complex. This is important in process calculations where many iterations are usually required.
av % dev of fugacity coeff. of 1
no. of points
Peng
Soave
Lu
Lee
Lee (S)
7 32
0.80 1.85
1.14 1.14
1.55 1.80
1.42 0.83
0.80 0.81
Activity Coefficients The object of phase-equilibrium measurements is usually to determine activity coefficients. These can be correlated for binary mixtures by Van Laar, Margules, Wilson, NRTL, or UNIQUAC equations. We have found that the NRTL equations of Renon and Prausnitz (1968) are the most generally applicable and we have, therefore, used them in nearly all cases. The NRTL equations can also be written in multicomponent form
where rji = (gji - gii)/RT;( g j i = gi,); Gji = exp(-ajiTji);and (Cuji = " i j ) . The required parameters (gii - gii) and aji are the same as those for the binary mixtures which make up the multicomponent system. Good rules exist for selecting values of ajiwhereas the values of (gji- gii) are determined by correlating the binary mixture data. Activity coefficients in the literature or private data banks are often expressed in terms of the popular Van Lam or Margules equations. Unfortunately, there are no multicomponent Van Laar or Margules equations, except for a ternary Van Laar equation, which can be used when only binary data are available. Kohler (1960) suggested an approximate equation for multicomponent activity coefficients in terms of those for binary mixtures
n
C(xiIn yij + x j lnTji)(l - x i - x j ) -
j=1
n-1
n
j#i
The value of T ~ is, calculated from the equation of the activity coefficient of a pseudo-binary mixture which has the composition ii=
*-
xi
Xi
+ Xj'
x j
=xi
+xj
Different types of equations can be used to estimate the pseudo-binary activity coefficients. For example, the Van ~ , Laar equation can be used to calculate y12and T ~ the Margules equation for 7 2 3 and 732, and the NRTL equation for 7 1 3 and 7 3 1 .
i
438
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980
Table 11. Ethanol (l)/Benzene (2)/n-Heptane ( 3 ) a t 400 mmHg
Table IV. n-Hexane ( 1 )/Methylcyclopentane (2)/Ethanol (3)/Benzene (4) at 760 mmHg mean % deviation of activity coefficients
mean % deviation of activity coefficients A-Y~ ArZ
Kohler: Van Laar Chien-Null: Van Laar Kohler: NRTL Chien-Null: NRTL Kohler: NRTL + Van La& Chien-Null: NRTL and Van Laara a
NRTL binaries:
12.98 12.51 11.29 6.30 11.16 6.58
Ay3
5.07 4.41 5.35 3.97 5.48 3.38
6.94 5.77 6.45 3.02 6.76 3.30
overall 8.33 7.56 7.70 4.43 7.80 4.42
Kohler: Van Laar Chien-Null: Van Laar Kohler: NRTL + Van L a x a Chien-Null: NRTL and Van LaaF NRTL binaries:
Kohler: Van Laar Chien-Null: Van Laar Kohler: NRTL Chien-Null: NRTL Kohler: NRTL + Van Laara Chien-Null: NRTL and Van LaaP a NRTL binary: 1-3.
5.48 5.05 5.02 1.29 4.39 4.06
4.12 2.92 3.42 2.37 4.59 4.42
5.15 3.31 3.23 1.67 3.52 1.60
4.92 3.76 3.89 1.78 4.17 3.36
xi(CAjixj)(CRjixj)
(CAjkXj)(CRjkXj) I
(&vjkx j ) (Csjkxj) Rki
CAjixj
CRjixj I
J
+E
Aki I
1-3, 2-3, and 3-4.
mean % deviation of activity coefficients
Chien and Null (1972) recommended another relationship in which parameters from different types of activity coefficient equations can be mixed
[
9.02 8.61 5.83
Table V. Benzene (1)/Chloroform (2)/Methanol (3)/Methyl Acetate (4)/Acetone ( 5 ) at 760 mmHg
mean % deviation of activity coefficients
2 In Tk =
1.60 4.07
1-2 and 1-3.
Table 111. Methanol (1)/Carbon Tetrachloride (2)/Benzene ( 3 ) a t 5 5 " ~
I
2.54 5.48 12.18 9.33 7.38 2.48 3.80 11.89 9.45 6.91 4.41 6.29 9.66 8.77 7.30
J
(cvjix
(csjixj)
I
J
J
ski
CSjixj I
X
j)
vki
CVjixj
]
(22)
I
Equation 22 was developed in such a way that it reduces exactly to the binary Van Laar, Margules, ScatchardHamer, Fariss, and NRTL equations when the appropriate substitutions are made for Ajk, Rjk, Vjk, and s j k . It also has the advantage that it becomes identical with the multicomponent NRTL equation when
Also Ajj = 0 and Ajj/Aj,= 1. Chien and Null have compared results for various three-component systems by using either NRTL or Van Laar parameters in eq 22, but they have not compared results obtained by using both types of parameters simultaneously. It is also of interest to see how well the Kohler equation applies. We have made comparative calculations for two three-component systems, a fourcomponent system, and a five-component system. The parameters for the binary mixtures of these multicomponent systems were taken from compilations listed by Renon (1966), Gmehling and Onken (1977), and Hala et al. (1968), or from listings of the investigators of the systems. Table I1 shows results for ethanol/benzene/n-heptanefor which Nielsen and Weber (1955) made measurements at 400 mmHg. Results for methanol/carbon tetrachloride/ benzene a t 55 "C are compared in Table I11 with measurements by Scatchard and Ticknor (1952). Table IV
Multicomponent Wilson Kohler: Wilson Kohler: Van Laar Chien-Null: Van Laar Kohler: NRTL and Van L a d Chien-Null: NRTL and Van La& a
NRTL binaries:
3.11 3.73
7.57 4.41 5.13 4.79
4.85 9.65 11.39 7.29 1.28 8.09 5.09 2.53 4.88 6.37 5.42 4.86 7.20 6.11 7.75 6.17 5.62 6.57 5.94 2.54
4.37 6.24 5.40 4.90
4.85 5.41
5.22 5.59 6.66 5.54
1-3, 2-3, 3-4, 3-5, and 4-5.
shows deviations for n-hexane/methylcyclopentane/ethanol/ benzene compared to the measurements of Kaes and Weber (1962) and Sinor and Weber (1960) at 760 mmHg. Table V shows results for benzene/chloroform/methanol/methyl acetate/acetone compared with measurements of Hudson and Van Winkle (1969) at 760 mmHg. The tables indicate that the Chien-Null equation with all NRTL parameters, which is the same as the multicomponent NRTL equation, is usually the best. The Kohler equation, although acceptable, does not usually give very good results. The results from the Kohler equation do not improve significantly when some of all of the Van Laar activity coefficients are replaced by NRTL values. However, there is a noticeable improvement with the ChienNull equation when Van Laar parameters are replaced by NRTL parameters. The Chien-Null equation is, therefore, preferable to the Kohler equation when mixed parameters are used. Exclusive use of NRTL parameters appears to be the best for general purpose work, although sometimes the Wilson equation may be better for particular systems.
Summary The glass equilibrium cell has been invaluable for phase equilibrium determinations. Vapor-liquid equilibrium data can be obtained from measurements of the saturation pressures of binary mixtures at several constant temperatures. For industrial purposes, it is necessary to express the activity coefficients by means of equations which can then be used in computer programs for the design of separation equipment. In the correlation of data, we prefer the Peng-Robinson equation for the calculation of fugacity coefficients and the NRTL equation for the activity coefficients. In a few cases, the effect of vapor-phase association must be included in the calculations. Reliable predictions can be made for multicomponent mixtures from the parameters obtained during the correlation of the
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980 439
binary mixture data. In general, the NRTL equation for multicomponent mixtures is preferred. The Chien-Null equation has an adva.ntage that it permits the use of parameters from different types of activity coefficient equations. Uncertainties as high as 7 % can occur for the activity coefficients for nonideal mixtures, but larger errors should be expected for immiscible systems. Most of the uncertainty is due to shortcomings in the equations which describe the activity coefficients. Acknowledgment G . Tosun helped to improve many of the experimental techniques and computer programs. The work of E. R. Schmelzer and S. H. 'Wiggins on the development of the computer programs i13 also greatly appreciated. Nomenclature f l , = fugacity of monomer, atm fllo = fugacity of monomer in pure HF, atm F = association factor including compressibility factor F" = true association factor g . . - gii = parameter in NRTL equation, cal/g-mol di, = parameter in NF~TLequation K = equilibrium constant for vapor-phase association ( K , for dimer, K6 for hexamer, K8 for octamer), atm-(n-l) pio = vapor pressure of component i, atm pl: = saturation pressure of pure monomer, atm P = total pressure, atrn P, = critical pressure, atm Pi = partial pressure of component i, atm P, = reduced pressure R = gas constant, 0.082057 (atm)(L)/(g-mol)(K)or 1.9869 cal/ (mol)(K) T = absolute temperature, K T , = critical temperature, K T , = reduced temperature u = molar volume of vapor, L/g-mol fli = partial molar vo1u:me of component i in liquid, L/g-mol u, = critical volume, L,/g-mol u, = reduced volume xi = mole fraction of component i in the liquid yi = mole fraction of component i in the vapor Yi = apparent mole fraction of component i in the vapor z = compressibility factor z, = critical compressibility factor Greek Letters ai, = parameter in NRTL equation yi or yi, = activity coefficient of component i Tij = activity coefficient of i in pseudo-binary mixture
= interaction coefficient in equation of state 4i = fugacity coefficient of component i $io = fugacity coefficient of pure component i +116 = fugacity coefficient of pure gaseous monomer ~~j = parameter in NRTL equation wi = acentric factor of component i Literature Cited 6ij
Antezana, F. J., Cheh, H. Y., Ind. Eng. Chem. Fundam., 15, 95 (1976). Beaume, R., Coulon, R., C.R . Acad. Sci., Ser. B , 266, 1024 (1968). Chien, H. H. Y., Null, H. R., AIChE J., 16, 1177 (1972). Franck, E. U., Brose, M., Mangold, K., Progr. Int. Res. Thermodyn. Transp. Prop., Pap. Symp. Thermophys. Prop., 2nd, 159-165 (1962). Franck, E. U., Meyer, F., Z . Nektrochem., 63,571 (1959). Franck, E. U., Spaithoff, W., 2. Elektrochem., 61, 348 (1957). Franck, E. U., TGdheide, K., 2. Phys. Chem. (Frankfurt am Main), 22, 232 (1959). Fredenhagen, K., 2.Anorg. Allg. Chern., 210, 210 (1933). Gmehling, J., Onken, U., "Vapor-Liquid Equilibrium Data Collection", Vol. 1, Pari 2a, DECHEMA, 1977. Graboski, M. S.,Daubert, T. E., Ind. Eng. Chem. Process Des. Dev., 17, 443 (1978). Hala, E., Wichterie, I., Polak, J., Boublik, T., "VaporClquid Equllibrium at Normal Pressure", Pergamon Press, New York, 1968. Hamam, S. E. M., Chung, W. K., Elshayal, I. M., Lu, B. C.-Y., Ind. Eng. Chem. Process Des. Dev., 16, 51 (1977). Hudson, J. W., Van Winkle, M., J. Chem. Eng. Data, 14, 310(1969). Jarry, R. L., Davis, W., J. Chem. Phys., 57, 600 (1953). Kaes, G. L., Weber, J. H., J. Chem. Eng. Data, 7, 344 (1962). Kay, W. B.. Nevens, T. D., Chem. Eng. Prog. Symp. Ser. No. 3,4& 10&114 (1952). Kohler, F., Monatsh. Chem., 91, 738 (1960). Lee, B. I., Kesler, M. G., AIChE J., 21, 510 (1975). Nielsen, R. L., Weber, J. H., J. Chem. Eng. Data, 4, 145 (1955). Oellrich, L., Plocker, U., Prausnitz, J. M., Knapp, H., Chem. Ing. Tech., 49, 955 (1977). Olds, R. H., Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. Eng. Chem., 41, 475 (1949). Peng, D.-Y., Robinson, D. B., Ind. Eng. Chem. Fundarn., 15, 59 (1976). Petty, L. B.. Smith, J. M., Ind. Eng. Chem., 47, 1258 (1955). Plocker, U., Knapp, H., Prausnitz, J. M., Ind. Eng. Chem. ProcessDes. D e v . , 17, 324 (1978). Reamer, H. H., Olds, R. H., Sage, B. H., Lacey, W. N., Ind. Eng. Chem., 36, 88 (1944). Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. Eng. Chem., 45, 1805 (1953). Renon, H., Ph.D. Thesis, University of California, Berkeley, 1966. Renon, H., Prausnitz, J. M.,AIChE J., 14, 135 (1968). Scatchard, G., Ticknor, L. B., J. Am. Chem. Soc., 74, 3724 (1952). Simons, J., Hildebrand, J. H., J. Am. Chem. SOC.,46, 2163 (1924). Sinor, J. E., Weber, J. H., J. Chem. Eng. Data, 5, 243 (1960). Slocum, E. W., Ind. Eng. Chem. Fundam., 14, 126 (1975). Smith, D. F., J. Chem. Phys., 26, 1040 (1958). Strohmeier, W., Briegleb, G., 2.Elektrochem., 57, 662 (1953). Tamir, A., Wisniak, J.. Chem. Eng. Sci., 33, 651 (1978). Vanderzee, C. E., Rodenburg, W. W., J. Chem. Thermodyn., 2, 461 (1970).
Received for review July 23, 1979 Accepted March 24, 1980
Presented at the 24th Annual All-Day Meeting of the Delaware Section AIChE, May 8, 1979.
