Environ. Sci. Technol. 1994, 28, 1331-1340
Semlempirical Thermodynamic Modeling of Liquid-Liquid Phase Equilibria: Coal Tar Dissolution in Water-Miscible Solvents Catherine A. Peters'it and Richard 0. Luthy*
Department of Civil and Environmental Engineering, The University of Michigan, Ann Arbor, Michigan 48109-2 125, and Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 This work investigates thermodynamic modeling of phase equilibria of mixtures of coal tar, solvent, and water using the nonrandom, two-liquid (NRTL) equation, a semiempiricalexcess free energy equation. Coal tar, a complex mixture of polycyclic aromatic hydrocarbons (PAHs),was represented as a pseudocomponent, permitting coal tar/ solvent/water systems to be modeled as ternary systems. Experimental ternary liquid-liquid equilibria (LLE) and coal tar/water mutual solubility data, along with literature data for solvent/watervapor-liquid equilibria (VLE),were used in a simultaneous regression procedure to optimize the representation of ternary LLE. Estimated model parameters characterize the collective chemical thermodynamic behavior of the coal tar pseudocomponent, and the calibrated liquid-phase activity coefficient equations can be used to predict solvent extraction process effectiveness. Application to a ternary system of pure identified components, acetonitrile/benzene/n-heptane, minimum data requirements. Good representation of ternary LLE was obtained with only one tie line in the regression when three binary data sets were included. When one of the binary VLE data sets was left out of the regression, the inclusion of three tie lines produced good results. Introduction With increasing recognition that many subsurface contaminants are nonaqueous-phaseliquids (NAPLs)and complex chemical mixtures, the need exists to increase our ability to model the chemistry of multicomponent twophase liquid systems. Coal tar is a NAPL that is a complex mixture of polycyclic aromatic hydrocarbon (PAH) compounds comprising a wide range of thermodynamic properties. Extraction with water-miscible organic solvents has been considered as a possible remediation technology for coal tar contaminated soils (I). The feasibility and effectiveness of such a process depends on the extent to which solvents increase the solubility of the coal tar to mobilize it for removal. A common means of characterizingthe solubility of coal tar or other multicomponent mixtures such as gasoline or diesel oil is to describe the phase equilibria of individual constituent compounds. While this provides necessary information for assessing the behavior of target compounds, the data requirements for assessing the behavior of the entire mixture are extensive. Furthermore, thermodynamic modeling of phase equilibria of complex multicomponent systems is difficult without simplifying assumptions of ideal molecular interactions. This paper presents an alternative means of characterizing liquid~~~~
~
* To whom correspondenceshould be addressed; e-mail address:
cpeters@I umich.edu. + The University of Michigan. 4 Carnegie Mellon University. 0013-936X/94/0928-1331$04.50/0
0 1994 American Chemical Society
liquid phase equilibria of a multicomponent mixture using a thermodynamic model that employs a semiempirical excess free energy equation. This type of thermodynamic modeling is made tractable by representing the coal tar as a single component, Le., a pseudocomponent. The coal tar pseudocomponent concept was introduced in Peters and Luthy (2), in which the pseudocomponent simplification was employed to make experimental observations of coal tar solubility in solvent/water solutions. The pseudocomponent simplification allows the composition of the two immiscible liquid phases to be described in terms of only three components: coal tar, solvent, and water. Thus, experimental data are conveniently presented in the form of ternary phase diagrams. The validity of the pseudocomponent simplificationfor thermodynamic modeling of coal tar/solvent/water systems depends on the extent to which the equilibrium composition of the coal tar component in the solvent/water phase (dissolved) and in the coal tar phase (undissolved) are similar. This condition is not met in systems without solvent, Le., in coal tar/water mixtures, since the partition coefficients of individual components vary by orders of magnitude. However, it was found (2) in systems with appreciable volume fractions of solvent (Le., 710%)that individual PAHs partitioned into the solvent/water phase to roughly the same extent. The plausibility of the pseudocomponent assumption was also evidenced by the lack of significantchange in the coal tar phase composition upon extraction, indicating that the compositions of dissolved and undissolved coal tar are similar. In the present paper, the thermodynamic properties of the coal tar pseudocomponent are modeled, characterizing its phase equilibria in two-phase liquid mixtures of coal tar, solvent, and water. All experimental methods and data have been published previously (2). Semiempirical Thermodynamic Modeling of Ternary
LLE While much work has been done in recent decades on the development of models characterizingphase equilibria in liquids, empiricism remains the central component of liquid mixture modeling (3). With its growing ability to relate thermodynamic properties of liquid mixtures to intermolecular forces and fundamental liquid structure, the field of solution theory has provided a basis for the development of excess free energy models, such as the nonrandom, two-liquid (NRTL) equation used in this investigation. These models make no assumptions of ideality. Semiempiricalmodels, unlike strictly empirical models such as the Margules and Redlich-Kister equations, make use of existing theory but incorporate adjustable parameters for general applicability. The semiempirical models have received the most attention for liquid mixtures because of their predictive capacity,limited data requirements, and capacity for reliable extrapolative model predictions beyond where data exist. Envlran. Scl. Technal., Val. 28, No. 7, 1994 1991
Practical application of semiempirical excess free energy models to systems with greater than three components is limited due to the large data requirements to account for all the molecular interactions and due to the increased mathematical complexity. Thus, the representation of coal tar as a pseudocomponent greatly simplifies model representation of coal tar/solvent/water mixtures. From a review of published example applications of semiempirical thermodynamic models to multicomponent liquid-liquid systems (4-14),it is clear that the success of parameter estimation and phase equilibria prediction is highly dependent on the type and quality of the data used for parameter estimation and on the specific details of the data reduction methods. The first objective of this work was to develop a parameter estimation approach that makes the optimal use of available data to represent the ternary LLE of coal tar/solvent/water systems. The next objective was then to estimate model parameters for the phase equilibria of coal tar/solvent/water systems for three solvents that have demonstrated potential for application in a solvent extraction system for coal tar-contaminated soils (2). This provides an analytical tool for interpolating and extrapolating experimental data of coal tar/solvent/ water phase equilibria and serves as a mathematical summary of the data, which can then be used in largerscale process models of solvent extraction of coal tarcontaminated soils. The three solvents studied are n-butylamine, acetone, and 2-propanol. The coal tar used for all experimental work was obtained from the subsurface at the site of the former manufactured gas plant in Stroudsburg,PA. Its properties and chemical composition have been presented previously (1,2). Another objective was to assess data requirements for the calibration of this type of thermodynamicmodel, given the parameter estimation approach developed for this investigation. To eliminate the question of the effect of the pseudocomponent simplification on parameter estimation results, a ternary system of pure components was used for this purpose. This is a system for which extensive, high-quality data are available in the literature allowing detailed investigation of the sensitivity of model predictions to the type and quality of data used to estimate parameters. Knowledge of data requirements indicates the minimal experimental effort for characterizing equilibria of similar systems, which would facilitate, for example, the screening of new solvents being considered for this or similar processes. It is apparent from a review of literature of LLE that semiempirical thermodynamic models have not been used to describe complex mixtures represented as single components. To the best of our knowledge, this research constitutes a new area of application for LLE models. The general success of the application to coal tar may indicate the degree to which this procedure is applicable to other complex mixtures of related chemicals encountered in environmental contamination scenarios. Isoactivity. The system to be modeled is a three component, two-phase liquid mixture. The three components are coal tar, solvent, and water, in which the coal tar pseudocomponent comprises all the constituent compounds in coal tar (i.e,,PAHs). The two immiscible liquid phases are referred to here as the “coal tar phase“, consisting of undissolved coal tar, solvent, and a small amount of water and the “solvent/waterphase”, consisting of solvent, water, and dissolved coal tar. 1332 Environ. Sci. Technol., Vol. 28, No. 7, 1994
The thermodynamic relations describing liquid-liquid equilibrium (LLE) are derived from the basic relations of equality of the fugacity of each of the three components in both liquid phases. For liquids, fugacity is related to composition by defining an ideal liquid solution and using activity coefficientsas a quantitative measure of departure from ideal behavior. For LLE, it is assumed that the same reference state defining ideality exists for both phases, so the equilibrium relations can be expressed as the equality of chemical activities:
ct ct
sw sw
Ys xs
(2)
pet$ = yswxsw
(3)
Ya
x,
w w
where x4 is the mole fraction of component i in phase a, yq is the liquid-phase activity coefficient of i in phase a, the subscripts ct, s, and w denote the coal tar, solvent, and water components, and the superscripts ct and sw denote the coal tar and solvent/water phases, respectively. The isoactivity criterion (eqs 1-3 together) is a necessary condition of thermodynamic LLE, but it is not sufficient. The necessary and sufficient condition of equi1ibriu.m is that the molar Gibbs energy of mixing, Ag, for the system is minimized. It can be shown (6) that the condition of equal activities follows from the condition of minimum Ag, but that equal activities can exist for hypothetical situations of the mixture for which Ag is not a minimum. This phenomenon is a result of the often-times complicated nature of the Ag function over composition (see the discussion in ref 15). Liquid-Phase Activity Coefficients. The chemical equilibria of a real solution is related to that of an ideal solution using the molar excess Gibbs energy, gE, which is dependent on the solution composition, i.e., gE = f ( x 1 , ...,xm). The partial molar excess Gibbs energy, for a component i is related to gE by a partial differential with respect to the moles of i. The activity coefficient is related to and, thus, to composition by
e,
(4)
where R is the universal gas constant, T is temperature, nT is the total number of moles, and nj = xinT is the number of moles of i. The nonrandom, two-liquid (NRTL) equation is a semiempirical model of the functiongE = f ( x 1 , ...,X m ) (16, 17). It is one in a category of models for liquid solutions that accounts for intermolecular forces of unlike molecules through the concept of local composition, introduced by Wilson (18). The theoretical and mathematical ancestry of the NRTL equation is reviewed in Peters (15). Unlike the Wilson equation, the NRTL equation has the ability to describe two-phase liquid systems. Another local composition model, the UNIQUAC equation (191,is also capable of describing phase immiscibility, but it was not selected for this work because of its dependence on additional unicomponent parameters, This information is not available for the coal tar pseudocomponent, and this would add to the empirical nature of the model. The differentiated form of the NRTL equation for a solution of m components produces expressions for the
liquid-phase activity coefficients for each component
where
Gji = exp(-a..T..) Jk Jk
(7)
and aji
- ajj
(8)
The parameters ~ j and i a j i are adjustable parameters explained below. In the limits of X j 1and X i 0, the NRTL equation simplifies to describe ideal solutions in terms of Raoult's law and Henry's law, respectively. The nonrandomness parameter a j j is intended to account for the nonrandom distribution of molecules of type i in solution, relative to molecules of type j . This parameter is considered to be independent of temperature. Theoretically, a i j is between 0.1 and 0.3 (16),but due to the semitheoretical nature of the NRTL model, the physical significance of a i j is obscured. When the NRTL model was introduced in 1968,it was intended that a j j would not be adjustable. That is, recommended values based on chemical classificationswould be used and, indeed, many textbooks present this approach (e.g., ref 20). However, many practitioners have since found the recommended values of a i j to be unsuitable (14), and a i j can be treated as adjustable. The term g j i is a measure of the intermolecular forces between molecules of type j and i , and g i i is a measure of the intermolecular forces between like moleculesof i. These terms appear in eq 5 only as the difference between them which is represented by a single adjustable parameter, T j i , which is unitless with R equal to 1.9872 cal mol-' K-l, and T in Kelvin. This value can be positive or negative depending on the relative magnitudes of the interaction energiesbetween unlike and like molecules. The difference ( g i i - g i i ) is linearlyrelated to temperature (16), so according to eq 6, Tjj is in theory constant for a given binary pair and the NRTL equation is a function of composition only. For a ternary system ( m = 31, there are nine adjustable ~ coal tar/ model parameters: ayCt+, r d S ,and T ~ for- the solvent pair; T ~ -and ~ , T ~for - the ~ solvent/water pair; and ahw,Tct-w, and Tw-et for the coal tar/water pair. The parameters for the pairs including coal tar are empirically representative of the molecular interactions of a coal tar pseudomolecule in solution. +
+
Parameter Estimation Relevant Data. The equilibrium compositions of two liquid phases of a ternary system, and the resulting appearance of the ternary phase diagram are determined by the molecular interactions between two species (binary interactions) and the simultaneous interactions between all three species (ternary interactions). While both types of interactions determine the phase equilibrium, much of the general behavior of the ternary system can be explained
in terms of the binary interactions of the three component pairs. A discussion of the general rules for explaining ternary LLE in terms of the characteristics of the binary pairs can be found in Peters (15). A very important aspect of the NRTL model is the dependence on only binary parameters, Le., parameters that describe the binary molecular interactions. The dependence on binary parameters exploits the theoretical basis of the model because it allowsprediction of the phase equilibria of ternary and higher level multicomponent mixtures without the need to experimentally study these systems. That is, in principle, experimental data of binary systems are all that are required for model calibration, and these are often available in the literature. In practice, however, the sole use of binary data for parameter estimation for characterizing multicomponent phase equilibria is often inadequate due to the inability of binary parameters to characterizeternary molecular interactions. Anderson and Prausnitz (9)have shown that incorporating some ternary data, along with the binary data, in the regression for parameter estimation improves ternary predictions. Estimation of binary parameters from ternary data alone is also not usually successful because ternary LLE may not be sensitive to the binary molecular interactions of all the pairs (&IO, 11,13). Without some binary data, the parameters may not successfully extrapolate to predict ternary LLE in regions where data were not collected. The use of binary VLE data in the regression serves to constrain the parameter values to physically meaningful values. A coal tar/solvent/water system has three pairs of components. The relevant binary data for the solvent/ water pair are the VLE data, since the solvents of interest form completelymiscible solutionswith water. The binary data for the coal tar/water pair are the mutual solubility data, since this pair forms a heterogeneous system. Binary experimental data are not available for the coal tar/solvent pairs. Ternary LLE data, i.e., tie line data, complete the data available for parameter estimation. For this application, the need to use some ternary data is determined not only by the expected inability of binary data to predict ternary LLE but also by the absence of binary data for the coal tar/solvent pairs. Simultaneous Regression Method. The parameter estimation method developed for this applicationuses both binary and ternary phase equilibrium data in a simultaneous regression. The three systems described by the data and models in the regression are binary VLE, binary LLE, and ternary LLE. The models for these systems have the same parameters because they all use the NRTL equation as the predictor equation for the liquid-phase activity coefficients. The thermodynamic relations for VLE and LLE are, however, different. This has implications in terms of the form of the objective function and in the resulting weight that is put on the different data sets, as discussed below. The parameter estimation method presented here is an updated version of that reported in Peters (15). In this version, data points and data sets were weighted to optimize their contribution to the regression to produce the best representation of ternary LLE. Ideally, a parameter estimation procedure for determining an optimum set of model parameters would be derived from the maximum likelihood principle, in which Environ. Scl. Technoi., Vol. 28, No. 7, 1994 1333
the objective function to be minimized would be expressed in terms of errors in the dependent variables, which for this model are the component mole fractions, x:, x:, etc. Examples of this approach have been published for NRTL and other excess Gibbs energy models (4,9,10,21). The disadvantage of using objective functions in terms of mole fractions ensues because the model equations cannot be written explicitly in terms of mole fractions, so numerical estimation of a set of predicted mole fractions, given a set of NRTL parameters, would have to be undertaken a t each iteration in the parameter optimization routine. Because of the computational difficulties associated with this method, it is not commonly applied (6). More commonly, an objective function is written in terms of activity coefficients (see, e.g., refs 11and 131, which is the approach used for this work. Predicted values of 74 are computed from the WRTL equation (eq 5) using experimental mole fraction data as estimators of the predicted values. The objective function for the VLE data is NVLE
Table 1. Molecular Weight and Density Data for Computation of Pure Component Molar Volumes
MW 2100
P (g/mL)
Y
(mL/mol)
coal tar 0.994* 211 water 18.02 0.9896 18.2 n-butylamine 73.14 0.737c 99.2 acetone 58.08 0.784c 74.1 2-propanol 60.10 0.782c 76.9 a Number average MW determined by vapor pressure osmometry (2). Determined using hydrometers, 30 O C (1,15). 25 O C ; source, Riddick et al. (23).
and Morisue (1975), contains 8 data points and is isothermal (35 "C, 64.4-84.7 mmHg). Only one data set, originally published by Komarov and Krichevtsov (1966), was available for n-butylaminelwater. It contains 16data points and is isobaric (760 mmHg, 77.0-95.4 "C). Experimental data for binary LLE (mutual solubility) and ternary LLE (tie lines) were measured as volume fractions (2). For a ternary solution, the volume fraction of component i, Vfi, is related to mole fraction through xivi
vfi = -
where wvle is a VLE data point weight term (discussed below) and NVLE is the total number of VLE data points. The superscripts e and p refer to experimental and predicted values, respectively. The terms in the objective function are written only for the solvent component since, in the VLE data sets available, experimental mole fraction measurements are reported only for this component. By mass balance, correlation to the water mole fraction is automatically satisfied, The experimental value of y, is computed from the following VLE thermodynamic relation:
where ys is the mole fraction of solvent in the vapor phase, P i s total pressure, and Pa"is the vapor pressure for pure solvent. The simplifyingassumption used in the derivation of eq 10is that the vapor phase is ideal,which is appropriate for moderate pressures (22). The ideal reference state is the pure compound. Although temperature does not appear explicitly in eq 10, the dependence is related through P,", which is represented by the Antoine equation, the parameters of which were obtained from Gmehling et al. (22). The predicted value of ys for binary VLE is computed from the NRTL equation:
written for only two components (m = 2). All three solvent/water VLE data sets were obtained from Gmehling et al. (22),a compilation of published VLE data. The data sets are identified here by the original authors. Where numerous data sets were available for a given pair, a single data set was selected based on reported thermodynamic consistency ratings and on whether the published NRTL parameters were reproducible. The selected acetone/water data set, originally published by Brunjes and Bogart (1943), contains 21 data points and is isobaric (760 mmHg, 56.5-92.0 "C). The selected 2-propanol/water data set, originally published by Sada 1334
Environ. Scl. Technol., Voi. 28, No. 7, 1994
3
where Vi is the molar volume of component i. The sum in the denominator is the molar volume of the mixture. Assuming there is negligible volume change on mixing, U i is computed from pure component properties using vi = MWi/pi, where MWi is the molecular weight of i and pi is its density. The assumption of volume additivity is necessary for this investigation as no volumetric thermodynamic data are available. This is not expected to contribute a large error to the Ui terms, although further studies would be required to verify this. The data presented in Table 1were used to compute molar volumes. Equation 12, written for two of the components, and a mole balance were solved for the three component mole fractions given the volume fractions of a solution. For the binary LLE, the objective function is
where wm, i are weight terms for the two mutual solubility terms. Experimentally based estimates of coal tadwater mutual solubility (2) are v c = 2 X 10-5 and v$ = 103 ( x z = 10-6 and x$= 10-2). For the ternary LLE, the objective function is BW
p
.-awt~.r1n(~)-ln(~) W t L w [In
]
2
(z) I']'):(
+
-In
YW
(14) YW
k
where for each tie line data point, k,up to the total number of tie line data points used in the regression, NTE, two terms are included: one for the solvent component and one for the water component, representing the two
components for which experimental data were collected (2). Inclusion of a term for the coal tar component in the objective function is not necessary because, by mass balance, correlation to the coal tar fractions is satisfied due to consistency with the solvent and water fractions. The w t l i terms are their respective weights. The phrase "tie lint? refers to a single ternary LLE data point, which represents the equilibrium compositions of the two liquid phases, connected on a ternary phase diagram by astraight line. (See ref 2 for an explanation of ternary phase diagrams.) For the system with n-butylamine, there are eight experimental tie lines; for 2-propanol, there are five tie lines; for acetone, there are three tie lines. For LLE, the ratio of activity coefficientsis a convenient term for the objective function because rearrangement of the equilibrium relations (eqs 1-3) provides a readily computed value for the experimental part of the objective function term
where the mole fractions are the experimental values. The mole fraction ratio is often referred to as a distribution coefficient, so the objective function is effectively minimizing the squared differences of the experimental and predicted log-transformed distribution coefficients. The predicted part of the objective function term, In ($"/rFt)p, is computed using the NRTL equation (eq 5). Simultaneous regression is accomplished by combining the three objective functions in a single objective function S = W v d h + WmSm + W t A l (16) where the W l terms are data type weight factors used to scale the relative input from each of the three types of data sets. Data Point Weighting Factors. In a simultaneous regression procedure, it is important that the objective function is designed to (1) normalize data points to dimensionless valuesthat are comparable across data types and (2) weight data points according to their experimental measurement precision. For VLE, it can be argued that the activity coefficientstend to be log normally distributed because yj has a multiplicative relationship with the variables on which it is dependent (eq lo), and consequently values vary over orders of magnitude. Similarly, ) , thus (y:w/y:t)in the LLE objective the ratio ( x ; ' / ~ ; ~and functions, tends to be log normal because it results as the quotient of terms (mole fractions) that vary over orders of magnitude. Thus, the use of log-transformed variables in the objectivefunctions servesto equalizethe magnitudes of the squared error terms. Further normalization is accomplished by dividing by the variances, Le., letting the weight terms be
wVle= l/var[ln]:y
(17)
which de-dimensionalizes the objective function terms as well as ascribing associated experimental uncertainties to the data points.
For VLE, the variance in eq 17 was estimated by propagating the errors of the VLE measurement variables. Applying first-order uncertainty analysis (24) to eq 10
where u' represents the variance of variable i, and B, and C, are Antoine constants for the solvent component. Measurement errors were not reported in Gmehling et al. (22),so experimental errors were estimated from published deviations between experimental and predicted values, and a mean deviation was used as the error for all NVLE data points. The independent variables were assumed to be error free. The estimated errors for the VLE variables for the three solvent/water data sets are for n-butylamine/ water, UT = 0.87 "C and uy,,= 0,019;for 2-propanol/water, up = 0.27 mmHg and uy8= 0.0032; and for acetone/water, UT = 0.75 "C and urn = 0.0081. For LLE, the variance terms in eqs 18 and 19 can be derived using eq 15 -
2
Uln (y:w/y;&)e
2 2 - Uln (xi&) + Uln (x:?
(21)
Assuming negligible errors in the estimates of the component and solution molar volumes in eq 12, the relative errors in the mole fractions are approximately equal to the relative errors in the volume fractions. Also, since the relative errors in the volume fractions for ternary LLE are small (21,they can serve as estimates of the variances of the log-transformed fractions
providing an expression for the variances in eq 19. The coal tar/water mutual solubility could be measured only to order of magnitude accuracy, so the relative errors in the values of the volume fractions are large and the above approximation is not valid. Since the larger relative errors are from x z and x: and not from x:: and x:, one of the variance terms can be eliminated from eq 21 for the mutual solubility data (23) nI ' (
2
(y~/y$p
- "ln2
(x$)
(24)
providing expressions for the variances in eq 18. Based on estimated errors in experimental procedures, the variances were estimated to be var [In (431 = 0.59 and var [In (x:)l= 2.3. The resulting weight factors for the mutual solubility data are much smaller than for the ternary LLE (tie lines). OptimizationProcedure. Sets of the optimum NRTL model parameters were obtained using a nonlinear least squares fitting procedure modified to account for multiple models. The optimization algorithm used (DBCLSF in ref 25) employs a modified Levenberg-Marquardt search strategy and estimates the Jacobian using a finitedifference method. The two types of parameters were not optimized simultaneously due to the inability to scale for differences in objective function sensitivity to the two types. The program was written to optimize the six ~ i parameters for given values of the three CY;, parameters. A shell script was used to run the optimization program Environ. Scl. Technol., Vol. 28, No. 7, 1994 1335
j
many times for a large range of sets of the ai, parameters. The optimum ajj values were determined based on the minimum evaluated objective function, S. If necessary, the solutions for the optimum 7ij parameters were used as new initial guesses, and the procedure was repeated until convergence. The isoactivity condition is satisfied by minimizing an objective function written in terms of differences in activities. However, since this is not a sufficient condition for thermodynamic equilibrium, a solution may or may not constitute thermodynamic stability. As discussed previously (15),the practitioner must therefore be aware of the possibility of multiple solutions. The procedure described here facilitates detection of local minima since a global search is performed by varying the aij values. Should several solutions be found, the true solution is det,ermined as the one that obtains the minimum Ag. It was found, however, that this additional step was not necessary with the present objective function. This procedure proved to be robust, producing consistent meaningful results regardless of the grid resolution for the ajj values or the initial guesses for the 7 i j parameters. This was not always true for previously reported results (15). The better results are attributed to the use of weighting factors which equalize data input in the regression, as evidenced by comparable residuals for all data points. In some cases it was necessary to decrease the weight of one of the data sets using the data type weight factors, WI, to achieve a minimum. The data type weight factors were selected as those that resulted in parameters that produced the best fit (minimum root mean square deviation) to the ternary LLE data. Flash Calculations for Tie Line Prediction. Once optimum parameters are obtained, the model is used in a flash calculation fashion to predict a tie line. The variables that must be specified are the overall system component mole fractions, (xct, x,, xw), two of which are independent since they sum to unity. The six unknown mole fractions are xBCt, x z , x::, x : ~ ,and xkw. TWO additional unknowns are the fractions of t,z" total moles in the system that are in the coaltar phase, L ,and solvent/ water phase, Lsw. Together with the three equilibrium equations (eqs 1-3), material balances for each component and mole balances within each phase contribute five additional relations. The NRTL equation (eq 5) computes the yia used in eqs 1-3. The overall system compositions used as input were the midpoints of the experimental tie lines (Le!., Ld = Law= 0.5) after conversion tomolefractions. This set of eight equations was solved for the eight unknowns using a nonlinear equation solving routine (DNEQNFin ref 25), which uses a Levenberg-Marquardt search strategy and a finite-difference approximation to the Jacobian. To quantitatively assess the goodness of fit of the predicted liquid-phase mole fractions with the experimental values, the root mean square deviation (rmsd) was calculated using the tie line data:
+
[ ( x : " ) ~- ( x : ~ )[~ ( x I: ~~) ~(
~ 3 ~ 1 ~ ) k
where NTL is the total number of experimental tie lines 1336
Environ. Scl. Technol., VoI. 28, No. 7, 1994
available. This is similar to the goodness of fit criteria used by others (9,10). For the coal tar/solvent/water data sets, the tie line predictions were converted to volume fractions using eq 12, and the root mean square deviations were computed using volume fractions instead of mole fractions in eq 25. Exploration of Data Requirements. The acetonitrile/benzene/n-heptane system was used to explore the sensitivity of model predictions to the type and amount of data used in parameter estimation. Available data (26) include acetonitrile/benzene VLE (11 data points, isothermal at 45 O C , 215-234 mmHg), benzeneln-heptane VLE (15 data points, isothermal at 45 "C, 126.50-225.10 mmHg), acetonitrile/n-heptanemutual solubility (45 "C), and acetonitrile/benzene/n-heptane LLE (9 tie lines at 45 "C). All composition data for this data set are in mole fractions. The base case for comparison was generated by using the maximum amount of available data, Le., both sets of binary VLE, mutual solubility data, and 9 ternary LLE tie lines, in the simultaneous regression procedure. To compare with the case when no tie line data are available, parameters were estimated independently from binary VLE and LLE data alone, i.e., not in a simultaneous regression, according to methods previouslyreported (15). To determine the effect of the importance of ternary LLE, additional optimizations were performed using the simultaneous regression procedure with three, two and one tie line@),using tie lines from the middle of the twophase region, as suggested in ref 8. To determine the effect of the binary VLE data, optimizations were performed without one of the VLE data sets, with various numbers of tie lines. No weighting of data was performed since experimental errors were not reported for these data. Results and Discussion
Application to Coal Tar/Solvent/Water. NRTL model parameters estimated for the three coal tar/solvent/ water systems are presented in Table 2, along with the rmsd computed in units of volume fractions and the data type weight factors. For data sets from the n-butylamine and 2-propanol systems, it was necessary to decrease the weight of the mutual solubility data to achieve a good fit to ternary LLE data. The inability to simultaneously satisfy the objective function with both LLE data types could be attributable to thermodynamic inconsistency across the two data sets. This is plausible since the validity of the pseudocomponent assumption is not expected to hold near the coal tar/water mutual solubility region (2). In a system with very low solvent concentration, the dissolved coal tar component is not expected to be composed of the same suite of compounds as the coal tar component whose phase equilibrium is being described in the remainder of the ternary phase diagram. Thus, the molecular interactions could be different. However, even though W,, values were less than unity, some contribution from the coal tar/water mutual solubility data was found to be beneficial. Experimental and predicted values of vc: are shown in Table 3. Order of magnitude agreement is observed for the n-butylamine system, and the model adequately captures the order of magnitude increase in coal tar solubility for the acetone system. The predicted
Table 2. Estimated NRTL Model Parameters for Coal Tar/Solvent/Water Systems solvent/water coal &/water coal tar/solvent akw
n-butylamine 2-propanol acetone
0.67 0.47 0.02
1.09 0.85 -5.9
1.8
6.4 7.9
0.72 0.59 0.27 0.63 0.49 0.50
n-butylamine 2-propanol acetone (none)
T
~
-
~T
~
-
~abw
sctw
Simultaneous Regression Results 0.38 2.75 0.56 2.89 0.33 4.6 0.82 1.92 4.8 0.23 0.57 1.7 Binary Data Regression Results 0.20 2.91 1.88 0.58 1.65 0.97 0.47 6.36
Table 3. Experimental and Predicted Values of
data type weights
vfi
et
rmsd
Wvie
Wm
wu
4.38 4.9 12.0
0.039 0.024 0.019
1 1 1
0.2 0.5
1 1
1
1
13.5 0
e
2-propanol 100
7(
solvent = n-butylamine solvent = 2-propanol solvent = acetone experiexperiexperimental predicted mental predicted mental predicted 0.048 0.080 0.065 0.106 0.106 0.162 0.196 0.352
0.067 0.071 0.074 0.079 0.083 0.086 0.089 0.097
0.001 0.041 0.048 0.041 0.075
0.029 0.025 0.020 0.015 0.0098
0.001 0.006 0.061
0.00073 0.00051 0.0017
b86
P
Figure 2. Modelpredictions(solidlines)of ternary LLE in volume fractbns for coal tar/2-propanol/water, compared to experimental tie lines (dashed lines). 0
acetone 100 7(
Figure 1. Model predictions(soli lines)of temary LLE In volume fractions for coal tar/n-butylaminelwater, compared to experimental tie lines (dashed lines).
values for the 2-propanol system are unacceptable since the coal tar solubility is predicted to decrease with increasing solvent concentration. Apparently, the data included in this regression does not contain sufficient information to accurately predict the phase equilibria in the ternary system. The ability to fully include coal tar/ water mutual solubility data with the acetone tie line data but not with the n-butylamine and 2-propanol data was likely due to the very small vc: for the acetone system. Thus, the solventlwater end points of the tie lines in the acetone ternary phase diagram are comparable to the very small solubility of coal tar in pure water. Tie line predictions for the three coal tar/solvent/water systems are plotted in the ternary phase diagrams in Figures 1-3 in units of volume fraction. The solid curve is a spline connecting the end points of the predicted tie lines. Experimental tie line data are plotted as boxed
20
Y \ Q
ao,
C ,
6
0
Figure3. Model predictions(solil1nes)of ternary LLE Involume fractions for coaltarlacetonelwater,comparedto experimentaltie lines (dashed lines).
points connected by dashed lines. The circles are twophase check points, which were experimentally determined overall compositions known to be within the two phase region (2). These points were determined via visual examination for the purpose of gaining information about the extent of the upper portion of the two phase region where quantitative tie line data were difficult to obtain. Environ. Scl. Technol., Vol. 28, No. 7, 1094
1337
Table 4. Estimated NRTL Model Parameters for Acetonitrile(a)/Benzene(b)/n-Heptane(h) System Using Both VLE Data Sets, Binary LLE, and Different Numbers of Tie Lines in Optimization
no. of tie lines included
acetonitrile/ benzene %.b
re-b
rbe
nine three two one
0.95 0.96 0.78 0.77
0.751 0.757 0.672 0.689
0.800 0.763 0.693 0.650
zero
1.02
0.740
0.852
benzeneln-heptane abh
Tbh
acetonitrileln-heptane
7h-b
Xi
7a.h
%-e
rmsd
0.40 0.40 0.40 0.40
2.613 2.612 2.615 2.605
2.306 2.302 2.304 2.304
0.014 0.012 0.016 0.021
0.40
2.605
2.304
0.055
%h
Simultaneous Regression Results 0.88 0.88 0.84 0.96
0.71 0.70 0.61 0.62
-0.026 0.0028 0.021 0.041
Binary Data Regression Results 1.75
The rmsd values are all less than 0.04 volume fraction unit, providing a quantitative indicator of the model representation of the ternary LLE. The best fit was obtained for the acetone system, with a rmsd of 0.02. For each of the three systems, the binodal curve accurately captures the two-phase checkpoint, indicating adequate extrapolation to the upper portion of the ternary phase diagram. For all three ternary phase diagrams, there is good agreement in the slopes of the predicted and experimental tie lines. This is expected since the experimental methods used produced more precision in the slope of the tie lines than in the position of the solvent/water end points along these lines (2). Because the model equations are strongly nonlinear, it is not feasible to estimate confidence limits for estimated parameters. However, because the optimization procedure allowed for the examination of numerous solutions (for different values of the aij terms), it was possible to qualitatively assess that the parameters for the coal tar/ solvent pairs were considerably more variable than the others. This is expected since binary data for these pairs were not included in the regressions. In fact, for the 2-propanol system, the optimization procedure failed to obtain a minimum for the aCbs parameter. This can be explained in terms of insufficient information about coal tar/2-propanol molecular interactions in the availabledata. For this ternary system, the number of adjustable coal at 0.47, tarlsoivent parameters was reduced by fixing act-s the literature recommended value. Assuming that the coal tar pseudocomponent can be classified as a nonpolar species, the literature recommended values for aij (16) are 0.47 for coal tadwater, coal tarln-butylamine, and coal tar/Bpropanol; 0.30 for coal tar/acetone; and 0.30for all three solvent/water pairs. The disparity between these values and aij that were determined through optimization supports the findings of other practitioners that the values published with the introduction of the NRTL equation are suboptimal (14). The NRTL parameter estimates optimized independently for the binary data sets are also shown in Table 2. Details of the optimization routines for binary VLE and LLE are presented in Peters (15). There is considerable discrepancy between the values of the binary parameters obtained using the simultaneous regression method and those obtained using only VLE data or only the coal tar/ water mutual solubility data. The implications are 2-fold. First, independent determination of the binary parameters for which binary data are available is not adequate for describing these ternary systems. That is, optimization of all nine binary parameters to all available data through the simultaneous regression procedure produces better model predictions to accurately represent the ternary LLE 1988 Envlron. Sci. Technol., Vol. 28, No. 7, 1994
0.700
0.204
0
benzene 100 7(
00
/I@
/‘c. / 60
40
Flgure 4. Base case model predictions(solidtie 1ines)andexperimental data (dashedtie lines)for acetonltrile/benzene/rrheptanesystem, using parameters simultaneously fit to all available data.
system. Similar observations were found for the acetonitrile/benzene/n-heptane system, as noted below. The unsuccessful attempts to reduce the number of adjustable parameters and predict coal tar/solvent/water parameters from binary data independently are discussed in detail elsewhere (15). The second implication is the interdependence, or correlation, of binary parameters for a ternary system. Correlation is normally viewed negatively in parameter estimation efforts because it often implies poor model formulation or overspecificationof parameters. However, in this case the interdependence of model parameters is advantageous because it implies that some and 78-ctis contained of the information about act.s,7Ct-B, in the data used to estimate the other binary parameters, thus compensating for the missing binary pair. Importance of Ternary LLE data. Parameter estimation and goodness of fit results for the acetonitrile/ benzeneln-heptane base case, using all available data, are listed in Table 4 in the row for nine tie lines. Predicted ternary LLE phase compositions (solid tie lines) are compared with experimental data (dashed tie lines) in Figure 4, which along with the small rmsd demonstrates the excellent representation of the ternary LLE. Parameters optimized independently for the individual binary data sets are listed in Table 4 in the row labeled “zero” tie lines, and the resulting model predictions are shown in Figure 5. The model predictionsverify published conclusionsthat the use of binary data alone overestimates the size of the two-phase region (9,12),highlighting the need for some tie line data. As with the coal tarlsolventl water systems, there is a discrepancy in the values of the
Table 5. Estimated NRTL Model Parameters for Acetonitrile(a)/Benzene(b)/a-Heptane(h) System Using One VLE Data Set, Binary LLE,and Different Numbers of Tie Lines in Optimization
VLE data set included
no. of tie lines included
aa-b
Sa-h
7b-a
abh
7b-h
7h-b
an-h
7a.h
Sh-a
rmsd
acetonitrile/benzene benzeneln-heptane benzeneln-heptane benzeneln-heptane
nine nine three two
0.96 0.88 0.95 0.75
0.71 0.80 0.81 0.63
0.83 1.13 1.03 0.73
1.04 0.85 0.86 1.11
0.77 0.63 0.67 0.59
-0.089 0.018 0.0087 0.086
0.41 0.40 0.40 0.41
2.7 2.615 2.614 2.690
2.4 2.305 2.304 2.414
0.042 0.035 0.026 0.034
acetonitrile/benzene
benzeneln-heptane
acetonitrileln-heptane
0
Zi
0
benzene 100 7(
/ f
40
c
p2
8
Q
0
Q4; Figure 5. Acetonitrlle/benzene/rrheptane LLE model predlctlonsfrom parameters fit to blnary data alone.
parameters from the simultaneous regression and from the binary data alone, implying that the molecular interactions in the binary mixtures are not representative of the molecular interactions in the ternary mixture. This seems to be especially true for the miscible pairs (a-b and b-h), and least true for the immiscible binary pair (a-h), suggesting that binary mutual solubility data are more important in a ternary simultaneous regression than are binary VLE. The effect of the number of tie lines included is seen by comparing the results in Table 4 for the cases of three, two, and one tie line(s). Comparingthe rmsd values, there is improvement as the number of tie lines is increased from one to three, but the use of more than three tie lines does not necessarily lead to better predictions. In fact, very good results were obtained from the optimizations with only two or one tie line(s). Model predictions using the parameters optimized using one tie line are shown in Figure 6. The results with one tie line corroborates published findings (9) in which acetonitrilelbenzenel n-heptane LLE were represented using a single tie line in a UNIQUAC model and an optimization procedure based on the maximum likelihood principle. These results imply that, when all three binary data sets are included, three or less tie lines contain sufficient information about the ternary molecular interactions for accurate representation about ternary LLE. Importance of Binary VLE data. The second objective was to study the effect of elimination of one of the binary VLE data sets to verify that this approach was appropriately applied to coal tar/solvent/water systems for which coal tarlsolvent binary data are not available. Parameter estimation and goodness of fit results for acetonitrilelbenzeneln-heptanein which one of the VLE
Figure 6. Acetonitrlle/benzene/rrheptaneLLE modelpredictlons from parameters fit to all blnary data and only one tle line (fourth from the bottom).
data sets was left out of the regression are listed in Table 5. While it was not possible to obtain rmsd values on the order of 0.01, in all cases the resulting fits to ternary data were better than if only binary data were used. The fact that the entire binary data set for one pair can be left out of the regression and parameters for this pair can still be found implies that sufficient information about these binary molecular interactions is present in the ternary data that are included. Again, we see that numerous tie lines are not required to adequately represent the ternary system, as is shown by the equivalence of the results for the cases when nine tie lines were used and when three tie lines were used (Figure 7). However, without the acetonitrilelbenzene VLE data the optimization failed if only one tie line was included, i.e., the objective function was completely insensitive to the parameters for this pair ((Ya-b, 7a-b) 7b-a) indicating that insufficient information was available to optimize these values. The results with acetonitrilelbenzeneln-heptaneshowed that it is possible to describe the two-phase region of the ternary phase diagram with one VLE data set, three tie lines, and binary mutual solubility data. For a ternary system with a larger two-phase region, such as coal tar/ solventlwater systems, ternary predictions are expected to be even better since the best ternary predictions are made for systems with very broad two-phase regions (9). Because the acetonitrilelbenzeneln-heptaneternary system has a small two-phase region, parameter sensitivity to regression data is likely to be more severe, and data requirements are likely to be larger. Thus, the results from this analysis can be extended to coal tarlsolventl water systems with some degree of confidence. Environ. Sci. Technoi., Vol. 28, NO. 7, 1894
1339
0
benzene loo 7(
60
40
-I\
/
Figure 7. Acetonttrite/benzene/rrheptane LLE model predicttons from parameters fR to benzene/rrheptaneVLE, acetonltrile/mheptanebinary LLE, and three tie lines.
Conclusions The representation of the complex mixture, coal tar, as a pseudocomponent makes thermodynamic modeling using excess free energy equations tractable and significantly reduces data requirements for the description of ternary LLE. Careful weighting of individual data points and data types in the simultaneous regression procedure developed for this investigation ensured objectivefunction sensitivity to all nine parameters. This was critical for the coal tar systems for which binary coal tar/solvent data were unavailable. It was shown that the ternary system can be modeled using binary VLE data for one pair, binary mutual solubility data, and only three tie lines. This implies that the screening of new solvents could be accomplished with a minimum of two or three tie lines and a two-phase checkpoint near the plait point for verification of extrapolation. Ternary compositions used for model calibration should be selected with > l o % solvent to ensure pseudocomponent applicability. The calibrated liquid-phase activity coefficient equations for coal tar/solvent/water mixtures for three solvents (n-butylamine, 2-propanol, and acetone) can now be used in larger scale models to predict solvent extraction remediation technology effectiveness. In principle, the parametrized equations can also be used in three phase distillation calculations to obtain rough estimates for the design of solvent recovery systems. Acknowledgments The Electric Power Research Institute was the primary sponsor for this research project through Contract RP
3072-2. Dr. Babu Nott was the project manager. Ad-
ditional fellowship support was provided by the Patricia Harris Government Opportunities Program. The authors thank Dr. David Dzombak for thoughtful review of this research and Dr. Walter J. Weber, Jr., for enabling the extension and completion of this work.
1340 Environ. Scl. Technol., Vol. 28, No. 7, 1994
Literature Cited (1) Luthy, R. G.; Dzombak, D. A.; Peters, C. A.; Ali, M. A.; Roy, S. B. Solvent Extraction forRemediation ofManufactured Gas Plant Sites; Final report, EPRI TR-101845,Project 3072-2;Carnegie Mellon University: Pittsburgh, PA, 1992. (2) Peters, C. A.; Luthy, R. G. Environ. Sci. Technol. 1993,27, 2831-2843. (3) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular ThermodynamicsofFluid-PhaseEquilibria, 2nd ed.;Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1986;Chapter 6. Groves, F. R, Jr. Environ. Sci. Technol. 1988,22,282-286. Sorensen, J. M.; Arlt, W. Liquid-Liquid Equilibrium Data Collection; Chemistry Data Series;DECHEMA Frankfort, Germany, 1979;Vol. 5, Part I, pp XIII-XIX. Sorensen, J. M.; Magnussen, T.; Rasmussen, P.; Fredenslund, A. Fluid Phase Equilib. 1979,3,47-82. Mattelin, A. C.; Verhoeye, A. J. Chem. Eng. Sci. 1975,30, 193-200. Prausnitz, J. M.; Anderson, T. F.; Grens, E. A.; Eckert, C. A.; Hsieh, R.; O'Connel, J. P. Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1980. Anderson, T. F.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1978, 17 (4),561-567. Varhegyi, G.;Eon, C. H. Znd. Eng. Chem. Fundam. 1977, 16 (2),182-185. Yee, D.; Simonetty, J.; Tassios, D. Znd. Eng. Chem. Process Des. Dev. 1983,22,123-129. De Fre, R. M.; Verhoeye, L. A. J.Appl. Chem. Biotechnol. 1977,27,667-679. Newsham, D. M. T.; Vahdat, N. Chem. Eng. J. 1977, 13, 27-31. Novak, J. P.; Matous, J.;Pick, J. Liquid-Liquid Equilibria; Studies in Modern Thermodynamics Series; Elsevier: Amsterdam, 1987;pp 218-238. Peters, C. A.; Ph.D. Dissertation, Carnegie Mellon University, Pittsburgh, PA, 1992. Renon, H.; Prausnitz, J. M. AZChE J. 1968,14(11,135-144. Renon, H.; Prausnitz, J. M. AZChE J. 1969, 15 (5), 785. Wilson, G. M. J. Am. Chem. SOC.1964,86,127-130. Abrams, D. S.;Prausnitz, J. M. AZChE J . 1975,21(1)116127. Henley, E. J.; Seader, J. D. Equilibrium-Stage Separation Operations in Chemical Engineering; John Wiley & Sons: New York, 1981;pp 212-215. Anderson, T. F.;Abrams, D. S.;Grens, E. A. AZChE J.1978, 24 (l),20-29. Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection,Chemistry Data Series;DECHEMA Frankfort, Germany, 1977;Vol. 1, Part 1. Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Solvents: Physical Properties and Methods ofPurification, Techniques of Chemistry Series; John Wiley & Sons: New York, 1986;Vol. 11. Morgan, M. G.; Henrion, M. Uncertainty, A Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis; Cambridge University Press: New York, 1990; Chapter 8. International Math SoftwareLibrary.ZMSL Manual;IMSL, Inc.: Houston, TX, 1987;Math Library. Palmer, D. A.; Smith, B. D. J . Chem. Eng. Data 1972,I7 (l), 71-76.
Received for review November 16, 1993.Revised manuscript received March 15, 1994. Accepted March 23, 1994." @Abstractpublished in Advance ACS Abstracts, May 1, 1994.
