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XT, P,y)
(8) (9)
K, = r/(T^)ps-(T)/P
(10)
K,=ps'{T)/P
(11)
K, = y/ (T,x')/ri"(T,x")
(12)
Eq. 8 uses an equation of state to obtain the fugacity coefficients, ^>,. p o r both phases while Eqs. 9 - 1 1 involve varies approximations for activity coefficients, y,, and vapor pressures, P '. For liquid-liquid equilibria, Eq. 12 would be used. Figure 1 highlights the work-flow and data-flow with respect to the service role of the properties (and property models). Note that for a TP-flash simulation, first the mass balance Eq. 1 together with the constraint (equilibrium condition) Eq. 3 is solved; for each evaluation of Eqs. 1 & 3, the property models are called. When convergence is achieved, the property models are called one more time for the evaluation of the energy balance. In this case, since T and P are fixed, the energy balance computes Q (the energy added/removed) needed to make the operation possible.
Figure 1: The service role of properties and property models This is a typical simulation problem where the property models (Eqs. 4-12) are used in the service role because every time an evaluation of Eqs. 1-3 needs to be made, there is a request
31 for the properties, for which Eqs. 4-7 are called. Another example of the service role is to ask for the density, either to convert the composition in mole fractions to moles/cm or to calculate the volume of the (separation) tank or the height of the liquid in a tank. In each case, only Eq. 7 will be called. 2.2.2 Service/Advice Role In process design and synthesis, property values are often regarded as explicit or implicit target values for a synthesis/design algorithm to satisfy by finding values for the unknown intensive variables. In section 2.2.1, the problem definition assumed (or specified) that a two phase system would exist at the specified T and P. Consider now that we do not know if a two phase system would exist at the specified T and P and also that we are only interested in removing a chemical through precipitation (crystallization). At any assumed (or specified) T, P, and z, we check if the liquid stream F would be stable at T & P. If not, there would be two phases, at least. Also, if the assumed T is above the melting point of at least one of the chemicals (for example, chemical A) in the system and below the melting point of the others, then there is a good chance that this chemical would precipitate, such as if the assumed T is below the saturation temperature of the feed mixture. The advice role here is to determine if solid-liquid based separation is feasible and also which chemical would be obtained as a solid. The design problem is to determine the temperature T of operation that would match the amount of separation (or recovery) that is desired. Figures 2a and 2b show two examples of this advice role. In each case, a solid-liquid equilibrium based saturation diagram is shown for an aqueous electrolyte system and a non-electrolyte system. It should be noted that the feed composition, z, and temperature, T, indicate which solid chemical would be separated.
Figure 2a: Saturation point compositions of naphthalene in a binary mixture of naphthalene-benzene as a function of temperature (y-axis)
Figure 2b: Saturation point compositions of sodium chloride in an aqueous solution as a function of temperature
32
Another example of the advice role can be found in solvent design, where the properties have explicit target values such as solubility and selectivity for a particular solute at a specific condition of operation. Here, the chemical structure of candidate solvents is manipulated until the solution properties match the target values. The process design problem (defined above) and the chemical product (solvent) design problem may be solved as part of a two-step procedure (irrespective of whether they are solved sequentially or simultaneously): Step I - generate alternatives {e.g., T'or solvent) Step II - determine properties (e.g., number of phases, solubility, or selectivity) and screen/verify alternatives through simulation. In Step I (generation), properties play a service/advice role while in Step II (verification) properties play only a service role. Strategies for the advice role attempt to eliminate infeasible solutions, or at least reduce the size of the search space and/or mathematical problem of a given situation, by providing the necessary insight. For example, an initial list of candidate solvents may be generated by searching for substances having similar solubility parameters as the solute. For each candidate solvent-solute combination, solid solubility diagrams (see Figures 2a-2b) would provide advice for selection of the operating temperature. The list of solvents would be reduced as candidates which do not meet the criteria are identified. For final verification, a simulation is needed (that is, solving Eqs. 1-6 for solidliquid equilibrium systems). Other examples of the advice role can be found in Gani & O'Connell[l]. The service/advice role is highlighted in Figure 3. Based on the problem specification, advice role in the generation step checks for feasible solid-liquid separation. If yes, then the service role is played for the verification (simulation) step. Note that the advice role also helps to design the condition of operation (choice of temperature and amount of solvent to be added or removed). Computer-aided molecular design techniques [2] are an example of properties used in their service/advice role, that employs different versions of the product model are employed.
Figure 3: Service/advice role of properties (property models) Often the advice role for properties is less demanding on the property models than is the service role. For example, sometimes the advice role for different product models can be
33
played with only pure component or infinite dilution properties of the compounds present in the mixture; the service role usually needs properties over the entire composition range. 2.2.3 Integration Role The most comprehensive role for properties in the solution of process-product design problems is that of integration. In process/tools integration and in graphical (visual) design techniques, properties actually can define an integrated solution strategy. Typical examples of this role can be traced back to graphical design techniques for separation processes (see for example, Henley and Seader [3] who describe various types of graphical design techniques) and for heat integration based on pinch technology [4]. In the case of distillation column design, the mass balance equation (Eq. 1) and the constraint equation (Eq.3) are represented in a two-dimensional plot of saturation vapor composition on the y-axis and the corresponding liquid composition in the x-axis. Each data point corresponding to the constraint equation represents a liquid in equilibrium with the vapor at the system pressure (usually constant) and temperature which varies over the column. In the case of heat integration, only Eqs. 2, 5 & 6 are needed. The two variables considered are the cumulative enthalpies for the hot and cold streams and the corresponding temperature. In the case of distillation design, the relation between saturation temperature and composition provides integration while in the case of heat integration, the relation between temperature and enthalpy does the integration. Using the process model defined by Eqs. 1-6, the integration role can be visualized through the following simple problem: • •
Given a contaminated product stream with a flowrate of F kmol/h having a composition fc of contaminant (for example, 0.02 mole fraction of phenol), Design a process through which the product stream becomes essentially free of the contaminant (mole fraction of phenol less than l.OxlO"6).
Through the traditional service or service/advice roles, the problem would be solved as follows - find candidate solvents that will create another liquid (solvent) phase and includes as much as possible of the phenol. For each solvent candidate, simulations are made to establish the process design. Now consider the following alternative solution technique called the reverse approach 1. Use Eq. 1 to calculate the required (or design target) solubility. Since the problem information actually provides both inlet and outlet concentrations of the product stream, this must be feasible. Note that since solubility (property) is actually the known variable, the solution of this separation problem does not require a property model. 2. Use Eqs. 3,11 and appropriate activity coefficient models to estimate solubilities for a solute in candidate solvents at selected temperatures. As long as the solubilities match the desired (target) values from step 1, the mass balance equation does not need to be solved.
34
It is interesting to note that in the traditional solution approach, using the generation and test paradigm, a simulation problem needs to be solved for every alternative. For many alternatives, however, a property model may not be available as part of the process model. If the generate and test steps are performed simultaneously (that is, include the simulation as part of an optimization loop), only one process model can be used, thereby severely restricting the application range since only one property model can also be used (unless multiple optimization problems are solved). With the reverse (integrated) approach, however, any number of property models can be used for step 2 while in step 1, property models are not needed at all. The procedure is to first solve a reverse simulation problem to determine the design target in terms of a set of properties, and then solve a reverse property estimation problem to determine the solvents and the operating conditions (for example, temperature). Multiple solutions are obtained, which only need to be ordered according to a performance index to identify the optimal solution. The property (in this case, solubility) plays the integration role to connect the simulation and design problems. On the other hand, from a point of view of the solution of the model equations, it plays the role of decomposition (separation of the balance and constraint equations from the constitutive equations). More details on this reverse approach can be found in [5, 6]. Figure 4 highlights the principal difference between the traditional or forward approach and the reverse approach (with the integration role for properties). Recently, Bek-Pedersen and Gani [7] have defined a driving force which is also a function of the saturation temperature and the corresponding phase compositions. Using the driving force for integration, Bek-Pedersen and Gani [7] have shown how simultaneous design and simulation can be performed for any two-phase separation process. Here, the property model plays all roles; service, advice and integrated solution of the simulation and design problem. The use of the driving force to integrate simulation and design is highlighted in Figure 5.
Figure 4: Integration role of properties and property models.
35
Note that in Figure 5, once the maximum driving force has been located at Fm and a reflux ratio (RR) is selected, by making the two operating lines to intersect on the D-Dx vertical line, the number of stages and the column profile (T, x & y_) can be back-calculated [7]. This means that the design and simulation of the distillation column correspond to the property target (maximum driving force). By definition, at the maximum driving force, the energy consumption is the minimum because this is the external medium that creates the twophase system. In the above examples, the relations among thermodynamic properties and intensive variables generated the integrated algorithm or solution strategy in addition to providing the solutions. "Intelligent" manipulation of the process and property model equations and variables using physicochemical and mathematical insights identified a small number of intensive variables as the unknown process model variables to represent the essential information about the problem. This tools integration, which is most valuable in applying integrated algorithms for synthesis, design and/or control, can be appreciated by considering the roles of properties in different process-product engineering problems through their independent, intensive variables (Table 1).
2.2 PROPERTY ROLES AND PROPERTY MODEL SELECTION In all three roles of properties, the proper selection of property models is important. Depending on the type of the problem, the consequences of property model choice can differ, even for the same property. The use of an inappropriate property model and/or model parameters may not be limited to only wrong numerical results that cause bottlenecking and over-sizing; even wrong process configurations can be found [8, 9].
Figure 5: Driving force based simulation and design (integrator is the driving force, a function of Py (T, P, x, y_) providing integration.
36 Table 1: Relationship between problems and properties in process-product engineering. Purpose
Problem
Determine
Remarks
Synthesis
Generate feasible process alternatives
Effects of T, P, x on process model
Design
Obtain condition of operation
Values of T, P, x that satisfy constraints
Properties affect process model results and play all roles, depending on the solution approach Properties provide target values and can play all the roles, depending on the solution approach
Control
Design control system
T, P,x sensitivities
Property models provide derivative information via service and advice roles
Energy analysis Environmental impact
Determine energy needs Verify that environmental constraints are satisfied
Enthalpies from T, P, x Component flows in effluent streams
Property models are essential via service and advice roles
Economy
Minimize cost of operation and equipment
Cost as a function of capital and operation
Property models affect process model results via service and advice roles
Property models are needed via service and advice roles
Often the user of a property model does not have the knowledge and experience to choose "wisely" among the myriad of options. The selection is commonly based on familiarity, hearsay or ease of accessibility. However, consideration of the numerous property models and comprehension of the complexities of current and future property models can be difficult even for the experienced user. Issues associated with this aspect are discussed by Gani and O'Connell [10]). 2.3.1 Service Role Since property models provide requested property values for quantitative evaluations of conservation of mass and energy, their quantitative accuracy must be evaluated. Some of the important issues related to the choice of the appropriate property model are the following: problem type, system (mixture) type, availability of property model parameters and computational complexities. Note that the estimation of the secondary and/or functional properties such as fugacity or activity coefficients, mixture densities, mixture enthalpies, etc., generally require pure component single value primary properties along with secondary/functional properties. Also, when the property model is used in the service role for a simulation problem, the derivatives of the properties with respect to the unknown variables (usually also the intensive variables T, P and/or composition) are needed when numerical methods like Newton-Raphson are used. Because of the importance of quantitative accuracy, tuning of the model parameters is an issue, especially since it depends on the availability of experimental data. In this case, a sensitivity analysis, that is, variations of the property of interest with respect to changes in
37
model parameters is valuable to fine-tune a small number of parameters. This is particularly useful when only limited experimental data are available. Such analyses show that pure component vapor pressures make a significant difference in the estimated vapor-liquid equilibrium (VLE) compositions for non-ideal systems, while the pure component critical (primary) properties are important for high pressure VLE calculations. Solid-liquid equilibrium (SLE) calculations are very sensitive to heats of fusion. The ratios of the activity coefficients of each component in the two liquid phases have direct impact on liquid-liquid equilibrium (LLE) compositions. Errors in densities effect the equipment sizing (volume and area) parameters. Since only one property model for each property may be used in process models such as the one represented by Eqs. 1-3, only carefully selected property models with fine-tuned model parameters and continuous derivative properties within a wide range of intensive variable values, should be used. Note that in the service role, only one property model may be used for the necessary properties during any simulation; this can be a disadvantage during simultaneous simulation and optimization of process-product problems since process and product models are solved simultaneously. For simulation problems from section 2.2.1 involving only the mass balance (Eq. 1) and the equilibrium condition (Eq.3), it can be noted that any one of Eqs. 8-12 may be appropriate for the equilibrium constant, Kh depending on the type of the problem, system, etc. Interesting issues with respect to the choice of models for Kt are the computational scale and time. If the fugacity and/or activity coefficients need to use complex relations based on groups at the atomic level (such as the SAFT EOSfll]), the models are usually computationally intensive and expensive. On the other hand, interactions at molecular levels (such as the SRK EOS[12]) are commonly less computationally intensive and relatively inexpensive. Consequently, calculation schemes where repetitive solution of Eqs. 1 & 3 are needed, computationally cheap models are usually preferred while for schemes where only a few calculations of Eqs. 1 & 3 are needed, and quantitative accuracy is important, computationally intensive models may be implemented. One alternative in this case is to use the reference and local models. The properties are generated by a reference model {e.g., SAFT EOS) and then matched with a computationally cheaper local model {e.g., the SRK EOS) to create a local, system-problem specific model. This idea is similar to property model simplification techniques proposed some time ago [13, 14]. The difference is that simplified local models needed to be generated for every iteration of a simulation. Here, system/problem specific local models need to be generated only once and can be used for all types of simulation problems for the same chemical system and properties. 2.3.2 Service/Advice Role In addition to the service role, for properties to play the advice role, the corresponding property model must be qualitatively as well as quantitatively correct. This is because the design problem (where advice is needed) usually involves generation and screening of different alternatives and verification of the feasibility of a desired operation (separation or reaction). Therefore, there is more emphasis on the qualitative accuracy. Of course, when the condition of operation (such as the temperature of operation) is to be found, quantitative accuracy is also essential.
38 In the case of equilibrium based separation processes, the advice role depends on the known (or generated) phase equilibrium (needing the solution of Eqs. 3 and 4 only). While these need to be calculated only once for each case, there can be many different chemical systems. Therefore, the choice of the property models (for Eqs. 8-12) needs to consider the application range not only in terms of intensive variable values but also in terms of chemical systems. This is particularly important for designing molecules with desired properties, where the selected property model needs to be predictive with respect to chemical systems to generate and screen large numbers of feasible alternatives. Unfortunately, if solvent selection is done simultaneously with the optimization of the process performance, only one property model per property can be used, restricting the search space for the problem solution. The best property model selection strategy is to choose a group or atomic contribution method (see chapters 3-4) for the advice role because of both predictive ability and qualitatively correctness for a wide range of chemical systems. Then, when a chemical system has been selected by such a method, measured and/or generated data should be used to regress (or fine tune) the model parameters of a local model with limited predictive capabilities. The last step also validates the property model for use in the service role. 2.3.3 Integration Role An important issue in the integration role of properties is how to bring together the special features of property models for process model-based calculation schemes. That is, it is necessary to identify the appropriate properties and the intensive variables. For a number of separation processes, including all of the 2-phase equilibrium separation processes, a driving force has been identified as the integrator (figure 5 and Section 2.2.3). Then the process model equations can be decoupled for integration of simulation and design. Note that the driving force can be calculated from physical equilibrium constants (as in Eqs. 8-12) and the location of the maximum is a function of the intensive variables for the same physical equilibrium constant values. The chemical identities provide an estimate of the chemical equilibrium constants from which the driving force diagram is calculated. Since the design target of minimum energy consumption (or solvent rate, etc.) can only be achieved if the design is based on the maximum driving force, the choice of property models and their accuracy in representing the variations of properties on the intensive variables is very important. In the context of the reverse approach, any number of property models can be used, so it is possible to select the appropriate property model based on their application range. This means that the design and simulation results based on the location of the maximum driving force and the driving force values would be valid for any number of separation problems (chemical systems).For reactive systems (with or without separation), a similar definition of driving force can also be employed to generate a similar diagram as the one showed in Figure 5. Hildebrandt et al. [15] call this driving force-based diagram the attainable region, which forms the basis for reactor design and simulation. Another useful feature of the reverse approach is that it often allows the use of the property models under the same conditions they have been developed and verified. That is, property models are usually validated by model developers through comparison with measured experimental phase equilibrium data not
39 through process simulation results. The driving force diagram involves precisely the same information.
2.3 DERIVATIVE ANALYSIS: A CONNECTION BETWEEN MODEL GENERATORS AND MODEL USERS One final issue, which enters into all property roles, is the question of the derivatives of properties with respect to the intensive variables which illustrates the differences between developers and users. Consider the problem shown in Figure 6. The numerical solution of this simulation/optimization based design problem requires the following derivatives (where 6 is any property and % is any intensive variable):
Figure 6: Simulation & optimization of a PT-flash operation (minimize the amount of heat to be added/removed, keeping the ratio of V/F constant while manipulating the operating temperature and pressure) If the selected property model, i.e., Eqs. 4, 5, 6, have not been tested for the existence of these derivatives, convergence failures are likely to occur. If the derivatives are inconsistent, the Gibbs-Duhem conditions (see chapter 1) will not be satisfied and so the numerical results will not be thermodynamically consistent, even when all other solution criteria may be
40
satisfied. Model developers (generators and implementers) need to recognize this issue and test derivatives for existence and consistency and to provide this information to users. Because the objectives, organizations and styles of property model generators, implementers and users are not usually the same, their interconnections are typically weak. In particular, generators usually seek generalized models to have as large an application range as possible, while they ignore computational speed and derivative issues . On the other hand, property model users commonly implement models having high computational efficiency and as wide spectrum of applications as possible. But they usually do not provide developers with much information on the limitations of their models. Better communication is needed between property model generators and users to deal with these concerns for the most effective impact of models in process-product design applications.
2.4 CONCLUSIONS Three different roles for properties and their corresponding models have been highlighted. Property models can play a much wider role than the traditional service role, for which they are mostly known (or receive credit for). Using them in the advice role can improve the design, widen the search space and increase the efficiency of the solution method. When property models become difficult to use in forward simulation approaches because of complexity, possibilities of using a reverse approach should be investigated. In most design and/or simulation problems, the "exit" conditions are usually known but not actually used in the simulation; they merely verify the simulation results or the design. By using both the known "inlet" and "exit" conditions, property integration can be identified and target values assigned. Many interesting process-product design problems can be solved more easily through this reverse approach. Finally, for property models to play their roles efficiently and for more advanced models to find applications in process-product design, better communication between property model generators, implementers and users is necessary.
REFERENCES 1. R. Gani, J. P. O'Connell, Computers & Chemical Engineering, 25 (2000) 3. 2. J. P. O'Connell, M. Neurock, "Trends in property estimation for process and product design", in: Proceedings of FOCAPD'99, Breckenridge, USA, 1999. 3. J. D. Seader, E. J. Henley, Separation Process Principles, John Wiley & Sons, Inc., New York, USA, 1998. 4. B. Linhoff, J. R. Flower, AIChE J., 24 (1978) 633-642. 5. R. Gani, E. N. Pistikopoulos, Fluid Phase Equilibria, 194-197 (2002) 43-59. 6. M. R. Eden, S. B. Jorgensen, R. Gani, M. M. El-Halwagi, Chemical Engineering & Processing, 43(2004) 595-608. 7. E. Bek-Pedersen, R. Gani, Chemical Engineering & Processing, 43 (2004) 251-262. 8. Y. Xin, B. W. Whiting, Ind & Eng Chem Res, 39 (2000) 2998.
41 9. E. A. Brignole, R. Gani, J. A. Romagnoli, Ind & Eng Chem Process Des & Develop, 24 (1985) 42-48. 10. R. Gani, J.P. O'Connell, Comp. Chem. Eng., 13, (1989) 397-404. 11. J. Gross, G. Sadowski, Ind & Eng Chem Res, 41 (2002) 1084-1093. 12. G. Soave, Chem Eng Sci., 27 (1972) 1197. 13. S. Macchietto, E. H. Chimowitz, T. F. Andersen, L. F. Stutzman, Ind Eng Chem Process Des Develop, 25 (1986) 674-682. 14. J. Perregaard, E. L. Sorensen, Computers & Chemical Engineering, 16S (1992) 247254. 15. D. Hlidebrandt, D. Glasser, C. Crowe, Ind & Eng Chem Res, 29 (1990), 49.
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Part II: Models for Properties
3. Pure component property estimation^ Models & databases Jorge Marerro & Rafiqul Gani 4. Models for liquid phase activity coefficients-UNIFAC Jens Abildskov, Georgios M. Kontogeorgis & Rafiqul Gani 5. Equations of state with emphasis on excess Gibbs energy mixing rules Epaminondas C. Voutsas, Philippos Coutsikos & Georgios M. Kontogeorgis 6. Association models — The CPA equation of State Georgios M. Kontogeorgis 7. Models for polymer solutions Georgios M. Kontogeorgis 8. Property estimation for electrolyte systems Michael L. Pinsky & Kiyoteru Takano 9. Difussion in multieomponent mixtures Alexander Shapiro, Peter K. Davis, J. L. Duda 10. Modeling of phase equilibria in systems with organic solid solutions Joao Coutinho, Jerome Pauly, Jean-LucDaridon 11. An introduction to modeling of gas hydrates Eric Hendriks & Henk Meijer
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Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B.V. All rights reserved.
45
Chapter 3: Pure Component Property Estimation: Models & Databases Jorge Marrero and Rafiqul Gani 3.1 INTRODUCTION Pure component properties are needed for many process and product design calculations. They may be needed to study the behavior of the product (such as the solubility of drug in water), behavior of a chemical under the conditions of operation of a process (heat of vaporization of a refrigerant or process fluid in a closed cycle), dimensioning of equipment (density of the chemical in a tank), the physical state of the product (melting point and/or boiling point to identify solid, liquid or vapor state) and many more. For the estimation of mixture properties also, the pure component properties are employed in different mixture property models. For example, the well-known SRK equation of state employs the critical properties while an ideal mixing model for liquid density employ only the pure component liquid densities of each chemical species present in the mixture. The objective of this chapter is to provide the reader with a set of pure component property models for a corresponding set of frequently used properties in process-product design. These models have been tested and evaluated against a wide range of chemical species by the authors. A good collection of pure component property models can also be found in many specialized property estimation books, journal papers, commercial software and databases. It is beyond the scope of this chapter to name all the references as well as methods. The calculation methods outlined in each section of this chapter should provide some guidance in terms of the important steps related to estimation of a pure component property.
3.2 MODELS FOR PRIMARY PROPERTIES As described in Chapter 1, primary properties are classified as those, which can usually be determined only from the molecular structural information and have a single unique value. In this chapter, only a set of primary properties that are needed for a wide range of processproduct design calculations are presented through one property estimation approach. This does not mean that the property models used below are the best or have the highest accuracy. These models are, however, frequently used and their details, including the model parameter tables, are readily available. The property model presented below can be classified as an additive method using a group-contribution+ approach. The property estimation methods will be highlighted through the molecular structure of Glycine (CAS No. 000056-40-6). Only the estimated primary and the secondary properties of
46 Glycine are given. When the estimated value for a secondary property for Glycine is not given, it means that not all the dependent properties are available. When experimental data for the property is available, it is also given. The chemical formula, group assignment and 3D molecular structure is given in Figure 1.
Figure 1: Molecular structural details for Glycine 3.2.1
Primary Property Models
All the properties listed below are only functions of the molecular structural information described in terms of first-order, second-order and third-order groups. Note that all molecules must be completely described by first-order groups and may or may not have second- and third-order groups. The estimation steps are as follows: 1. 2. 3. 4.
Identify the groups (first-, and if necessary, second- and third-order groups) Determine how many groups of each type are needed to represent the molecule Retrieve the parameters from the model parameter tables for the property of interest Sum the contributions and use the corresponding property model function
Properties & Models The following property models have been proposed by Marrero and Gani [1,2]. In each case, the summation terms having the following expression Contribution of first-order groups, Sum.Groups.I = £i ni C; for i = 1, NCi Contribution of second-order groups, Sum.Groups.II = Ej rrij Dj for j = 1, NC2 Contribution of third-order groups. Sum.Groups.Ill = Zk Ok Ek for k = 1, NC3
(1) (2) (3)
In the above equations, n;, rrij, Ok are the number of first-, second- and third-order groups of types i, j and k, respectively. Q, Dj and E^ are the contributions for the selected property for first-, second- and third-order groups of types i, j and k, respectively. NCi, NC2 and NC3 are the total numbers of different types of first-, second- and third-order groups representing the molecule. Critical temperature, K Tc = 231.239*Iog(Sum.Groups.I + Sum.Groups.II + Sum.Groups.III)
(4)
47
Glycine: 1028.0 K (experimental: 1028) Critical pressure, bar Pc = l/(Sum.Groups.I + Sum.Groups.II + Sum.Groups.III + 0.108998)2 + 5.9827
(5)
Glycine: = 67.4 bar (experimental: 67.4) Critical volume, cm3/mol Vc = 7.95 + Sum.Groups.I + Sum.Groups.II + Sum.Groups.III
(6)
Glycine: = 234.01 cmVmol (experimental: 234.0) Normal melting point, K T m = 147.450*log(Sum.Groups.I + Sum.Groups.II + Sum.Groups.III)
(7)
Glycine: = 535.63 K (experimental: 535.15) Normal boiling point, K Tb = 222.543*log(Sum.Groups.I + Sum.Groups.II + Sum.Groups.III)
(8)
Glycine: = 710.97 K Standard Gibbs free energy of formation. kJ/mol Gf = -34.967 + Sum.Groups.I + Sum.Groups.II + Sum.Groups.III
(9)
Glycine: -300.1 kJ/mol (experimental: -300.1) Standard Enthalpv of formation at 298 K, kJ/mol H r = 5.549 + Sum.Groups.I + Sum.Groups.II + Sum.Groups.III (10) Glycine: -388.49 kJ/mol (experimental: -392.1) Enthalpv of vaporization at 298 K. kJ/mol Hv = 11.733 + Sum.Groups.I + Sum.Groups.II + Sum.Groups.III (11) Enthalpv of vaporization at TH, kJ/mol HVb - a + Sum.Groups.I + Sum.Groups.II + Sum.Groups.III (12) Glycine: = 43.0 kJ/mol Heatoffusionat298K, kJ/mol HfuS = -2.806 + Sum.Groups.I + Sum.Groups.II + Sum.Groups.III (13) Glycine: 28.4 kJ/mol (experimental: 28.4)
48 3.3 MODELS FOR SECONDARY PROPERTIES As defined in Chapter 1, secondary properties are those that cannot be explicitly calculated only from structural information and usually require the knowledge of other properties. Most of these models have been derived from the principle of corresponding states, although, a number of empirical correlations also exist. There is available, a wide range of methods for prediction of secondary properties. Many books and handbooks provide methods for calculating these properties. Poling el al. [3] provides a good collection of many of the needed methods. Horvath [4] also provides a large number of methods for primary as well as secondary properties. In this section, a set of pure component properties that have a single value is listed together with a corresponding property estimation method. 3.2.1 Secondary Property Models The following steps may be followed in the estimation of pure component secondary properties. 1. For the secondary property of interest, select an estimation method 2. For the selected estimation method, identify the properties (data) needed to use the model and verify the application range of the method in terms of chemical species (type) 3. Retrieve from a database or predict the necessary properties (to be used as input) 4. Calculate the property through the selected method For the secondary properties listed below, the following properties are needed as input. All these properties are also defined below. For each property, first the generic form of the equation in terms of dependency on other properties/variables is given, followed by the method for calculation, the model equations, and finally, the calculated value for the chemical used as an example. Hfus (kJ/mol). Tb (K), Tc (K), Pc (bar), Vc (cmVmol), SolPar (MPa°5), Ss (MPa°5), nD, Dm (debye), ps(bar), Mw (g/mol), Ws (mg/L) Properties & Models For each property, the name of the property, the representation of the property in terms of its dependence on other properties, the method used and the equations involved are presented. Heat of Vaporization at Th. HVb= f(Tj,, Tc, Pc) Method: Correlation (Equation 7-11.5 in Reid et. al. [5]) tr = Tb/Tc X = 0.37691 - 0.37306*tr+ 0.15075/(Pc*tr2) Y = (0.4343*log(Pc) - 0.69431+ 0.89584*tr)/X
49 Hvb = Tb*0.008314*Y (14) Not recommended for Glycine Pitzer's Acentric Factor, co = f(Tb, Tc, Pc) Method: Lee-Kesler Correlation (2-3.4 in Reid et. al. [5])) / Constantinou & Gani [6] 0 = Tb/Tc a = -log(Pc*0.98692327) - 5.92714 + 6.09648/0 + 1.28862*log(6) - 0.169347*96 P = 15.2518 - 15.6875/9 - 13.4721 *log(9) + 0.43577*96 co = a / p (15) Glycine: 0.747 Lee-Kessler 0.673 Constantinou & Gani [6] Critical Compressibility Factor, Zc = f(Tc, Pc, Vc) Method: Theoretical (Equation) Definition ZC = (PC*VC)/(83.14*TC) (16) Glycine: 0.185 Liquid Volume at Tb, Vb =f(Vc), cm3/mol Method: Tyn and Calus Correlation (3-10.1 in Reid et. al. [5]) Vb = 0.285*Vcli)48 (17) Glycine: 86.5 cm /mol Liquid Volume at 298 K, Vm = f(Tc, Pc, co), cmVmol Method: Rackett Modified Correlation t r =1.0-298.15/T c Zra = 0.29056-0.08775*co W=l+(l-trf28571 Vm = (83.14*T0*Zrallunc)/Pc (18) Refractive Index, no = f(Soli>ar) Method: Correlation [4] nD = (0.48872*SolPar+5.55)/9.55 (19) Glycine: 1.8 Molar Refraction, Rm = f(nn, Vm) Method: Correlation [4] Rm = (((nD)2-l)*Vm*1000)/((nD)2 + 2) (20) Surface Tension at 298 K, a = f(Sol|>ar, Vm), dyne/cm Method: Correlation [4]
50 CT = 0.01707*(Sol Par ) 2 *(V m ) 0333333 (21) Entropy of Fusion. Sfus = f(HfUS, T m ), J/(mol*K) Method: Theoretical (Equation) Definition S rus =1000*H fus /T m
(22)
Glycine: 53.07 J/(mol*K) Closed Flash Temperature. Tfc = f(GCcG, Tb), K Method: Constantinou and Gani [6] Tfc = -2.03*(Sum.Groups.Ic G ) + 0.659*T b + 20.00
(23)
Open Flash Temperature, Tfo = f(GCcG, Tb), K Method: Constantinou and Gani [6] Tf0 = 3.63*(Sum.Groups.Ic G ) + 0.409*T b + 88.43
(24)
Glycine: 414 K Hansen Dispersive Solubility Parameter, 8s - f(GCcG, V m ), MPa 0 5 Method: Constantinou and Gani [6] 8s = (Sum.Groups.IcG)/Vm
(25)
Glycine: 17.74 MPa 0 5 Hansen Polar Solubility Parameter, 8 P = f(GC C o, V m ), MPa 0 ' 5 Method: Constantinou and Gani [6] 8 P = [(Sum.Groups.I C G) 05 ]/V m (26) Glycine: 12.16 MPa 0 5 Hansen Hydrogen Bonding Solubility Parameter, 8 H B = f(GCCG, V m ), MPa 0 5 Method: Constantinou and Gani [6] SUB = [(Sum.Groups.IcG)/V m ]° 5 (27) Glycine: 17.38 MPa 0 5 Dipole Moment. D m = f(8 s , V m ), debye Method: Correlation [4] D m = 0.02670*8 s *(V m )° 5 (28) Dielectric Constant, DH = F(Solpar, no, D m ) Method: Correlation [4] I f n D < 0.001. DK = (n D ) 2
51 Else, DE = (SolPar*0.48871-7.5)/0.22 (29) Henry Constant of a gas in water at 298 K, Hhenr), = (pS(298), Mw, Ws), bar*m3/mol Method: Theoretical (Equation) Definition Hhenry = ps(298)*Mw/Ws
(30)
3.3.1 Secondary Properties modeled as Primary Property For a number of secondary of secondary properties, it is sometimes possible to model them as primary properties. That is, it is possible to predict the property only as a function of the molecular structural information. Recently, Marrero and Gani [2] have developed models for Octanol-water partition coefficients, Solubility of a chemical in water at 298 K, and the Hildebrand solubility parameter. Also, the method of Martin and Young [7] for the measure of toxicity in terms of 50% mortality of Fathead Minnow after 96 hours of exposure has been adapted to the Marrero and Gani method. As in the case of primary properties listed in section 3.2, the prediction of the following properties also follow the same steps Octanol-water partition coefficient (LogKow) LogKow = A + Sum.Groups.I + Sum.Groups.II + Sum.Groups.III (31) Glycine: -3.41 (experimental:-3.21) Water Solubility, Ws, Log(mg/L) LogWs = A + Sum.Groups.I + Sum.Groups.II + Sum.Groups.III (32) Glycine: 5.41
(experimental: 5.39)
Hildebrand solubility parameter at 298 K, Solpar, M(Pa) Solpar = A + Sum.Groups.I + Sum.Groups.II + Sum.Groups.III (33) Glycine: 23.9 MPa03 Acute Toxicity (96-h LC50) to Fathead Minnow, mol/L -Log(LC50) = Sum.Groups.I (34) Glycine: 2.82
3.4 FUNCTIONAL PROPERTIES As defined in Chapter 1, pure component functional properties are those that depend on the specific value of temperature and/or pressure. Most prediction methods employ a suitable equation of state, the principle of corresponding states or a specially fitted correlation. In this
52
section, a set of functional properties and a corresponding property model is presented. Note that as in secondary properties, functional properties may also require other properties as input data. Note also that many temperature dependent functional properties are available in databases where the coefficients for the correlation of each property and chemical are stored. These correlation functions are discussed in section 3.4 of this chapter. The following steps may be employed in the estimation of functional properties. 1. For the property of interest, select an appropriate property model. 2. Verify the applicability of the model in terms of chemical species as well as the temperature (and/or pressure) limit of the method. 3. Retrieve or estimate the necessary properties to be used as input data 4. Calculate the property of interest at the condition (temperature and/or pressure) of interest using the selected method 3.4.1 Properties & Models For each property, the name of the property, the functional dependence, the units of measure, the method and the model equations are presented. Diffusion coefficient of component at infinite dilution in water, Dab = f(Vb, Tb, T) cm2/s Method: Modified Tyn & Calus Correlation (11-9.5 in Reid et. al. [5]) X = exp(-24.71 + 4209/T + 0.04527T - 0.00003376*T2) for 273.15 < T < 643.15 Dab = 0.01955/[(Vb)0433]*(T/X) (34) See also Chapter 9 for other prediction methods. Liquid Density,CTL= f(Tc, Pc, co, T), g/cm3 Method: Modified Rackett correlation (3-11.10 in Reid et. al. [5]) Zra = 0.29056-0.08775*© Tfimc = 1 + (1-T/Tcf285714 for T/Tc < 0.9 a, = (83.14*Tc*(Zra)Tfunc)/Pc (35) Thermal Conductivity, Tcon = f(Tb, Tc, T, Mw) W/m*K Method: Correlation (10-9.5 in Reid et. al. [5]) Tr = T/Tc Tbr = Tb/Tc Tcon = [l.ll/[(Mw) 05 ]*(3 + 20*(l-Tr)06666)]/[(3 + 20*(l-Tbr)06666)] (36) forT r <0.9;T b r <0.9 Vapor Pressure, pS = f(Pc, Tc, T, co) bar Method: Modified SRK EOS or any appropriate equation of state See Chapter 5 for details
53
Enthalpy of Vaporization. HV(T) = f(Tc, T, co) kJ/mol Method: Correlation (7-9.5 in Reid el. al. [5]) tr=1.0-T/Tc W = (co - 0.21 )/0.25 for (0.2 < tr < 0.9) Ri = 6.537*tr°333 - 2.467*tr°833 - 77.521 *t r ' 208 + 59.634*tr + 36.009*tr2 - 14.606*tr3 R2 = -0.133*t r ° 333 - 28.215*t r ° 833 - 82.958*t r ' 208 + 99.000*t r + 19.105*tr2 - 2.796*tr3 HV(T) = (R[ + W*R2)*Tc*0.008314 (37) Hildebrand Solubility Parameter. 8,,(T) = f(Hv(T), Vm(T), T) MPa'/2 Method: Theoretical (Equation) Definition 8h(T) = [(1000*Hv(T) - 8.314*T)/Vm(T])05 (38) 3.5 DATABASES Databases relevant to property estimation for process-product design are collections of experimental pure component and mixture properties. In many cases, these databases also include coefficients for correlations of functional properties. A large variety of databases having a wide variety of pure component and mixture property data and their correlation function coefficients can be found, among others, on the internet, as commercial database services and as part of non-commercial software (usually from academia). In this section, only a non-commercial database is discussed together with references for some of the wellknown databases. 3.5.1
The CAPEC Database8
The CAPEC database8 contains information on 13000 compounds and on 40 pure component properties of different types (primary, secondary and functional), on 9 mixture properties, classification of compounds according to molecular structure, data on molecular structural representation in terms of groups and special solvent-solubility data. Compound Classification For each compound, molecular structural description in terms of SMILES string and the UNIFAC groups (first-order and higher-orders) are available and the compounds may be identified through their chemical name, formula or CAS number. The compounds are classified in terms of nine main categories: normal fluid, polar associating, polar nonassociating, multifunctional (with respect to groups), water, polymer, electrolyte, steroid and amino acid. Each main category is further divided into sub-categories, for example, steroids are further divided in terms of adrenal corticosteroids, androgens & anabolic steroids, estrogens, progestogens, and cholesterols. Figure 2 illustrates the classification of compounds, highlighting the sub-categories under polar associating compounds.
54
1. Normal Fluids 2. Polar Non-Associating Compounds 3. Polar Associating Compounds a. Organic i. Alcohols ii. Hydroperoxides iii. Amines (& Imines) iv. Acids v. Oximes vi. Nitriles vii. Sulfonic Acids viii. Isocyanates ix. Isothiocyanates x. Oxides xi. Phosphoric Acid (including Phosphorous & Phosphoric) b. Inorganic c. Inorganic Polar Associating 4. Multifunctional Grouped Molecules 5. Water 6. Polymers 7. Electrolytes 8. Steroids 9. Aminoacids Figure 2: The 9 main categories for the classification of chemicals in the CAPEC database Pure Component Data When available, the following pure component properties are given for each compound: Primary Property
Secondary Property Functional Property
Molecular weight, critical temperature, critical pressure, critical volume, normal boiling point, normal melting point, heat of fusion at 298 K, heat of combustion at 298 K, ideal gas enthalpy at 298 K, ideal gas entropy at 298 K, ideal gas Gibbs energy at 298 K, liquid volume at normal boiling point, Hildebrand solubility parameter at 298 K, van der Waals surface area, van der Waals volume Critical compressibility factor, triple point temperature, triple point pressure, acentric factor, flash-point temperature, radius of gyration, dipole moment, refractive index, dielectric constant Vapor pressure, solid density, liquid density, solid heat capacity, liquid heat capacity, ideal gas heat capacity, liquid viscosity, vapor viscosity, liquid thermal conductivity, vapor thermal conductivity, surface tension and second virial coefficients
55 Mixture Data When available, the following mixture data can be found in the database. Binary Mixture Property (41000 data points) Ternary Mixture Property (10000 data points)
VLE, LLE, SLE, infinite dilution activity coefficients, heats of mixing, partial molar heats of mixing at infinite dilution, excess Gibbs energy, Henry's law constants, and mutual solubilities VLE, LLE, SLE, VLLE, heats of mixing
Special Data This class contains 2769 solubility data points consisting of solubility values and temperatures for 1374 binary mixtures involving 202 solutes (having molecular weights greater than 94 g/mol and having 4 < carbon atoms < 40) and 162 different types of solvents. In addition, to the solubility data, a list of 80 well-known solvents together with their solubility indicators (decomposes, miscible, insoluble, slightly soluble, soluble and very soluble) for most of the 13000 compounds in the database is also available. Search Engine An advanced search engine is available in the CAPEC database to identify compounds with a variety of search specifications. Two examples of search are given below. A. Find all compounds that are soluble in ethanol, having a boiling point > 300 K and a melting point < 250 K. B. Find all compounds having a boiling point > 300 K, a melting point < 250 K and the Hildebrand solubility parameter between 25 and 27 MPa . 3.5.2
References for Databases
In this section, a few of the well-known databases found on the internet are listed below in Table 1, while references where useful data can be found are given in Table 2. Table 1: List of well-known databases Name API TECH Database CambridgeSoft ChemFinder
Address & Comments Pure component, petroleum characterization, etc. http://www.epcon.com Searchable data and hyperlink index for thousands of compounds - the ideal starting point for internet "data-mining" http://chemfinder.cambridgesoft.com/
56 Table 1 continued CRC Handbook of Chemistry and Physics DECHEMA Chemistry Data Series DETHERM DIPPR Electrolytes GPSA Data Book IUPAC-NIST SDS Knoval Science and Engineering Resources PDB PPDS TAPP The NIST Webbook
Library Network Database (http://www.hbcpnetbase.com/) A 15 volume data collection Comprehensive collection of thermophysical and mixture properties data, includes Dortmund DDB and ELDAR DDB http://www.dechema.de/f-infsys-e.htm7englisch/dbMain.htm Critically evaluated thermophysical data http://www.aiche.org/dippr/vision.htm IVC-SEP database for properties of electrolyte systems www. i vc- sep .kt. dtu.dk/databank GPSA Engineering Data Book —section 23 (physical properties) & 24 (thermodynamic properties) http://www.gasprocessors.com Solubility Data Series http://www.unileoben.ac.at/~eschedor Library Network Database (International Critical Tables, Polymers -Property Database, Handbook of Thermodynamic and Physical Properties of Chemical Compounds, etc.) Protein Data Bank — Processing and distribution of 3-D biological and macromolecular structural data http://pdb.ccdc.cam.ac.uk/pdb/ Physical Properties Data Service http://www.tds-tds.com/fs_ppds.htm Thermochemical and Physical Properties Database http://www.chempute.com/tapp.htm An excellent source of physical and chemical data http://webbook.nist.gov
3.6 CONCLUSIONS Pure component properties are needed in the solution of various types of process-product design problems as well as input in many models for estimation of mixture properties. Usually, they are stored (experimental data) in databases, at least, the single value properties and the temperature dependent functional properties. The problem, however, is that even though the database may contain thousands of compounds, not all data is available for all the listed compounds. Also, in process-product design, new chemicals may be synthesized, which would not be present in the database. For this reason, property models for estimation of pure component properties are needed. In this respect, the chapter provides the reader a
57 quick guide in terms of the most commonly used pure component properties and a representative set of property models. Table 2: References for data Biochemistry & Biotechnology Drugs- Phase diagrams Octanol-water partition coefficients Polymer Data Solubility data Solubility data Water infinite dilution activity coefficients
Thermodynamic data for biochemistry and biotechnology, Hans-Jurgen Hinz, Editor, Springer-Verlag, 1986 J. Phys Chem Res Data, 1999, 28(4), 889-930 J. Phys Chem Res Data, 1989, 18(3) 1111-1229 Polymer DIPPR 881 Project High&Danner, 1992 Barton Handbook, CRC Press 1990 J. Marrero & J. Abildskov, Solubility and realted properties of large complex molecules. Part 1, Chemistry Data Series, Vol XV, DECHEMA, 2003 Voutsas & Tassios, Ind Eng Chem Res, 1996, 35, 1438 supporting material
REFERENCES 1. J. Marrero, R. Gani, Fluid Phase Equilibria, 183-184 (2001) 183. 2. J. Marrero, R. Gani, Industrial Engineering & Chemistry Research, 41 (2002) 6623. 3. B. E. Poling, J. M. Prausnitz, J. P. O'Connell, "The Properties of Gases and Liquids", McGraw-Hill, New York, 5th Edition, 2000. 4. A. L. Horvath, "Molecular Design", Elsevier, Amsterdam, The Netherlands, 1992. 5. R. Reid, J. M. Prausnitz, B. E. Poling, "The Properties of Gases and Liquids", McGraw-Hill, New York, 4th Edition, 1987. 6. L. C. Constantinou, R. Gani, AIChE J, 40 (1994) 1697 7. T. M. Martin, D. M. Young, Chem. Res. ToxicoL, 14 (2001), 1378 8. T. L. Nielsen, J. Abildskov, P. M. Harper, I. Papaeconomou, R. Gani, J. Chem Eng Data, 46 (2001) 1041.
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Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B.V. All rights reserved.
59
Chapter 4: Models for Liquid Phase Activity Coefficients - UNIFAC Jens Abildskov, Georgios M. Kontogeorgis and Rafiqul Gani 4.1 INTRODUCTION Computer-optimized design of the separation processes, e.g. distillation, absorption and extraction, typically encountered in the chemical industry, requires thermodynamic models, which can be applied to a variety of chemicals. The investment (capital costs) for the separation steps is often in the neighborhood of 50-70 % of the total cost, and energy costs for separations can be up to 90 % of the total cost. Measured phase-equilibrium data are typically employed at the later design stages. However, for preliminary design, at the earlier stages and for screening purposes i.e. for testing alternative separation techniques, approximate models, which can be widely applicable, are of interest. Such models can provide rapid estimation of phase equilibria over a wide range of conditions. In addition to the case of the forward property prediction problem (chemical system is known but property values need to be calculated or retrieved), the reverse property prediction problem (property values are known but the chemical system is not known) is also of interest. These are typically problems related to the design of molecules/mixtures/blends such as solvents, process fluids, etc., where the desired properties of the chemical are known but the identity of the chemicals or their mixtures are not. Finding a system with properties close to a specified set also requires rapid predictions of phase equilibria for many mixtures. In this latter case, the use of predicted data is probably the only way in which the problem becomes solvable and forms the basis for computer-aided molecular/mixture design techniques. The need of process-product design techniques for rapid estimation of phase equilibria for a wide variety of systems, conditions and problems has been the driving force behind the development of the UNIFAC and other group contribution models. Statistical thermodynamics has not as yet reached a stage where rigorous solution theories of general applicability are available, although some results of limited practical use have been reported1 (see also chapter 12). Semi-empirical methods are, thus, of interest. Of the most well-known semi-empirical methods useful for predictions of phase equilibria, the group contribution concept is possibly the most widely used. They are usually simple, easy to use and are at least qualitatively correct for many systems where no experimental data is available. Examples of such group contribution methods are the analytical solution of groups, ASOG2 and UNIFAC3, both providing the liquid phase activity coefficients for the compounds present in the solution. Despite their limitations (see section 4.3.3), such group contribution methods have found widespread application in engineering design calculations due to their relative simplicity, their predictive power and analytical form.
60 Several reviews have appeared since the 1980s summarizing the developments of these models4"8. This chapter gives a short overview of the most well-established UNIFAC methods, including the most recent developments and their efficient computer-based use. Table 1: Partial list of different versions of the UNIFAC model. UNIFAC Models UNIFAC-VLE
Special Feature 1 -parameter for group interactions
UNIFAC-LLE
1 -parameter for group interactions
Modified UNIFAC Lyngby
3-parameters for group interactions
Modified Dortmund UNIFAC
3-parameters for group interactions
Linear-UNIFAC
2-parameters for group interactions
Linear Dortmund UNIFAC
2-parameters for group interactions
KT-UNIFAC
lst-order (Linear UNIFAC) & 2ndorder group contribution terms
Remarks - Reference Original version with parameter tables [1977]3'9'" Parameters regressed from LLEdata[1981]'2 Changes in model equations & new parameter tables [1987113 Changes in model equations & new parameter tables [1987]14 Changes in model equations & new parameter tables [1992]15 Changes in model equations & new parameter tables [1992]16 Changes in model equations & new 1 st-order and 2nd-order group parameter tables [2002]10
4.2 THE UNIFAC METHOD Since its development in 19759, a number of different versions of the UNIFAC groupcontribution method for the estimation of liquid phase activity coefficients have been developed. The first UNIFAC method will be called here as the UNIFAC-VLE. One of the latest versions of UNIFAC, the KT-UNIFAC, has been developed by Kang et alw. Table 1 gives a partial list of different versions of UNIFAC that will be discussed in this chapter. The characteristic feature of all UNIFAC methods is that all versions are based on the group contribution approach where the liquid phase activity coefficient of component i in solution with one or more components is calculated as a sum of contributions of all groups representing the mixture (solution). These contributions are divided into two terms - a combinatorial term that accounts for the contributions due to differences in size and shape and are based on the group surface area and volume parameters, and a residual term accounting for the energetic differences, which includes the interaction parameters between the groups. With respect to the intensive variables, the UNIFAC method is dependent only on
61 temperature and composition but not pressure. The combinatorial term is dependent on composition only (in addition to the group surface area and volume parameters) while the residual term depends on the temperature, the composition and the group parameters (surface area, volume and interactions). With the exception of KT-UNIFAC, all UNIFAC methods employ lst-order functional groups to represent the compounds present in the liquid mixture. KT-UNIFAC employs both lst-order and 2nd-order functional groups to represent the compounds present in the liquid mixture. Therefore, the combinatorial and residual contributions have first-order and secondorder terms if 2nd-order groups are present. Two examples of representation of molecules with the UNIFAC groups are given below in Table 2. Note that each group is characterized by a main group, a sub-group, the sub-group surface area (Rf), the sub-group surface volume (Qf) and the main-group/main group interaction parameters anmo, amn,o, amn,o- Each compound therefore is represented by NSG number of sub-groups and a corresponding number of main-groups. A total of NMG maingroups are needed to describe all the compounds in the mixture. Table 2: Representation of molecular structures with different types of groups Ethanol (NSG=3, NMG=2) Sub-Groups: 1 CH3, 1 CH2, 1 OH Main-Groups: CH2, OH 2nd-Order groups: None
2-Butanol (NSG=4, NMG=2, NSOG=1) Sub-Groups: 2CH3, 1 CH2, 1 CH, 1 OH Main-groups: CH2, OH 2nd-Order groups: CHOH
4.2.1 General Model Equations (lst-order) The UNIFAC GC-method for the estimation of the liquid phase activity coefficient of component i in a solution consist of a combinatorial and a residual contribution: lny=lny\+lny":
(1)
The combinatorial term (In y/) includes entropic effects due to molecular size and shape differences while the residual term (In j,R) involves the intermolecular interactions. The different versions of the UNIFAC GC-method differ in their formulation of the combinatorial term and the expression of the temperature dependence of the residual part. Combinatorial Term - In j \ C The combinatorial term is written as17 /«r;;=/-j,+/«j,-^,(/--+/«-! * v Li Li) with
(2)
62 L> = ^ — /_, x,
; J, = ^ —
(3a)
2^ XJ r-i J
where the sums over j run over all components in the mixture while the sum over / run over all sub-groups representing component /. Residual Term - In yiR The residual term is written as17
In y? = q{l-lnl)M-
A" Gkl In ^ \
(4)
where the sum over k runs over all main groups in solution. The various variables within the summation term of Eq. 4 is defined as
Gki = Y,vl'Qi > * = Xx,G t ,
(5)
Ski = X Gm, Tmk ', r?k~^
(6)
X, Sk,
and
Tm,=exp[-^
(7)
In Eqs. 5-6, the summation with / runs over all subgroups in main group k while in Eq. 6, the summation with m runs over all main groups in solution. The group interaction between main-groups m and n, amm is calculated from the following general expression. am = am.o + alm,,(T -To) + aml,,2 (T ln^ + T-Tt>)
(8)
where To is a reference temperature while amn_o, amnj, amnj are the group interaction parameters and m and n indicate the identities of the main-groups.
63 Model Equations for Different Versions of UNIFAC The UNIFAC-VLE, UNIFAC-LLE and the Linear UNIFAC use Eqs.1-7 as they are written above. In Eq. 8, UNIFAC-VLE and UNIFAC-LLE set the values for amnj and anm2to zero. That is, these models use a temperature independent group interaction parameter. The LinearUNIFAC model sets only anm2 to zero (that is, it uses a linearly dependent group interaction parameter). The Modified-UNIFAC Lyngby and Modified-Dortmund-UNIFAC models use different expressions for the combinatorial term. Eq. 2 is replaced by Eq. 9 for the ModifiedDortmund-UNIFAC model while for the Modified-UNIFAC Lyngby model it is replaced by Eq. 10.
=1
In/,
- J,(l ) + ln J,(l)-^r('l-
— + ln — )
(9)
ln/l=l-ji(\) + lnjl(\)
(10)
j,(a)
(11)
where = —d— Z , XJ rJ j
While the Modified-UNIFAC Lyngby model employs the same residual term as given above (Eqs. 4-8), in the Modified-Dortmund-UNIFAC Eq. 8 is replaced with Eq. 12: a,,m ~ (Xmn.O + Umn. I T + Umn.2 T~
(12)
4.2.2 General Model Equations (KT-UNIFAC10) The second-order UNIFAC equation treats activity coefficients as comprised of three additive parts, a combinatorial part to account for molecular size and shape differences, a residual part to account for molecular interactions, and a second-order residual part to account for secondorder effects on molecular interactions. = \ n y c + \n y R + \ n y R 2
\ny
(13)
The combinatorial and residual terms are the same as in the Lyngby modified UNIFAC13, and given below by Eqs. 14-21. / r C
lnx =ln '
\ a
- \
CO-
' +1-' X \ 1J
X: '
(14)
64 x.r*3 m = " ,
(15)
JJ
j
(16>
>/=I>*A k
Inrf =Z^/(lnr -lnr,') i
(17)
k
k
( r
T\
<18>
lnr/t = lQk - l n f e ^ l + 1-Xv'!"* \ L ^ =
'
Jy
2
(19)
Z
rBt = e x p ( - A « M / r ) A
+
(20) 7
«^=^-' ^^- 'o) (21) The second-order residual part is derived from perturbations with respect to structural and energetic parameters by expanding the excess Gibbs energy as Taylor series around a state of zero correction37.
dAunm L
ddunm ,
"m'7 (22)
Where the sum over/ is a sum over all second-order groups identified in the mixture, the sum over m is a sum over all first-order groups in the mixture other than n, and m' is a particular first order group (type w) which is part of a second-order group j and its interactions are modified (see Abildskov37). V k^m : (Aumk := Aumk + §um-kj ; Aukm := Aukm + 8ukm-,j )
(23)
65 As pointed out by Abildskov37, only the two lower-case deltas in gR2 depend upon secondorder information. Therefore, the second-order term to be added to the total contribution of the Inyi is given by, _. lnyR2= /'
NSOG NSG NSG [ * .„J ,• f / Z Z I "' [G 3 -3 v j
RT
mejn^ml
^
m'i n
\T V U^-G
m' ni> n
'„
T G - ^
trii n't
+
s a
ni)
" m ' J \ ( G n i 0 - 9 V ) T " m - G m G m . , r'"" I RT [x ™ n m'i I ^ Sm, JJ NC
NC.
Gk, =vklQk ; &k =Y,N,Gkl ; % = £ i V A , ; Vml =9m^-Gm
(24)
NSG
; Skl = Y
This approach if unrestricted, would introduce many new parameters and the therefore, to make the model more realistic, a reduction of the number of additional parameters is introduced through the two following simple rules: • •
Second-order effects do not cause two groups to start interacting if they did not do that in the first order solution. Second-order effects on interactions apply uniformly to the 'main groups' in UNIFAC.
A list of first-order and second-order groups, along with sample assignments, group occurrences, volume parameters, surface area parameters and main-group interaction parameters (first-order and second-order) can be found in Kang10. A significant increase in the number of first-order groups has been achieved and the list of first-order groups also includes some groups that are not included in the table of the pure component method from Marrero and Gani (see chapter 3 for reference), otherwise, these two models use the same set of groups. The need for additional groups is because the pure component method is not usually used for estimation of small molecules since the needed properties are usually known. In phase equilibria predictions, however, mixtures containing small molecules could be interesting in many practical (industrial) problems. The group volume and group surface parameters are obtained from atomic and molecular structure data based on the work of Bondi17.
4.3 MODEL ANALYSIS The different versions of the models are analyzed from the point of view of numerical computation, model parameters and performance/limitations.
66 4.3.1 Numerical Analysis The analysis is highlighted in detail for only the UNIFAC-VLE model and briefly for the other versions. Consider the following problem - calculate the activity coefficient of component i in a mixture of NC components, where each compound is represented by a set of sub-groups and main-groups, and, the mixture condition is given by a pressure (P), a temperature (T) and a composition (x). This means that the following variables are known: P, T, x, NSG and NMG. In addition, retrieving the group parameters for the UNIFAC-VLE model from the model parameter tables, gives us the values for R, Q and amn,o, amnj, a,imj Now, for the UNIFAC-VLE model, we need to use Eqs. 1-8 to compute the activity coefficients. There are 9 equations (note that since Eqs. 3a, 3b, 5 and 6 have grouped, in each case, two variables, they are counted only once). The total number of variables are 20 (not counting NC, NSG and NMG). This means that there are 11 degrees of freedom, or, 11 variables that must be specified or known before the computations can be made. Since the following variables are known: v/, (number of times sub-group 1 appears in compound i) T, x R/, Qi and a,,mo (note that amnj and amnj are set to zero in the UNIFACVLE model), we have 8 variables specified. The remaining 3 variables are actually the UNIFAC VLE model constants /, z and To. So, we have 11 variables fixed and the degrees of freedom condition is satisfied. This means that the following 9 variables can be calculated: lnyh Inyt, Iny/1, (r, & q,), (.// & L,), (Gki & St), (skh r]k), rinn and amn. The model equations and
their corresponding unknown variables are shown in Figure 1 (incidence matrix), where each row represent an equation and each column represent an unknown variable. The equations are ordered to highlight the sequence in which they can be solved. Since a lower tri-diagonal form is obtained, it means there are no iterations involved for the computation of activity coefficients with the UNIFAC-VLE model. The known variables and parameters are not shown. The symbol • indicates the unknown variable in the corresponding equation. The symbol * indicates the known variables in the corresponding equation.
Figure 1: Incidence matrix for the UNIFAC-VLE model equations The same analysis can be made for all other UNIFAC models and in every case (except the KT-UNIFAC), the number of equations, the number of unknown variables and their identities are the same. The only differences are the expressions of some of the equations and
67 additional parameters, for example, a (see Eq. 12), in the case of the modified UNIFAC models. This is also a model constant that must be set. In the case of KT-UNIFAC, there will be additional rows after Eq. 4 to calculate the second-order contributions. It is interesting to note that for repetitive calculation procedures where Eqs. 1 -8 need to be solved during every trial, parts of the equations may be skipped. For example, Eqs. 7-8 does not need to be calculated if the temperature is not changing (as in a TP-flash calculation or a bubble point pressure calculation). One special feature of property models such as UNIFAC is that the variables NSG, NMG, NSOG and v/, (that is, the molecular structure related variables) may be subject to different interpretations of the representation of a component molecular structure with the available functional groups. Also, as it can be noted, the group parameters (that is, the model parameters) play an important role and may change as the computational problem is changed. 4.3.2 Model Parameters R & Q Parameters Most UNIFAC models take Rj and Qt from the tables of Bondi18, though the parameters for H2O are empirical and those for the OH and THF groups were made somewhat larger for better results. In the Modified Dortmund UNIFAC, however, all Rj and Q, are treated as adjustable parameters, regressed together with the interaction parameters from phase equilibrium data. Table 3 shows the molecular, r-, and q, for some compounds. It can be seen that the common general statement that r, is greater than qi does not apply to the Modified Dortmund UNIFAC version. Table 3: UNIFAC Parameters (r, q) as calculated from the two modified UNIFAC versions (modified Dortmund UNIFAC14, modified UNIFAC Lyngby13) Compound i n-Octane Cyclohexane Cyclooctane Anthracene Pyrene
UNIFAC14
n
5.0600 4.2816 5.7088 5.2682 6.0208
UNIFAC1J It
Tj
It
6.3702 5.1810 6.9080 5.1662 5.5888
5.8486 4.0464 5.3952 6.7738 7.5042
4.9360 3.2400 4.3200 4.4800 4.7200
These differences may have important consequences. Use of adjusted geometry parameters in the combinatorial term may sometimes lead to very large values of this term. Eq. 2 suggests that the phenomenon is particularly pronounced when qt is large and the ratio J/Lj deviates significantly from unity. This was noted in the earlier days of UNIFAC developments20. The example of phenol in water, where j2C "'' ~ 15, demonstrated that the feature is not merely of theoretical interest, but could occasionally be observed when calculating combinatorial terms of real mixtures. It can be noted that the ratio differs from unity when the ratio r\lq\ differs significantly from r-Jq-i.
68
{L2)
q,
This is unlikely when for all compounds i, rt is greater than qh which is typically the case when (r, q) are calculated from the (R,Q) values of Bondi21. On the other hand, the case of pyrene at infinite dilution in pentane calculated with the Modified Dortmund UNIFAC is illustrative of other possibilities. For pyrene, r, is greater than q,, whereas for pentane r, is less than q,. Moreover, qt of pyrene is large and, as a consequence, the combinatorial activity coefficient exceeds the value of 7.5. Large combinatorial values can also be found with the Original UNIFAC VLE, though mainly for systems involving water or alcohols, for which the R and Q are empirical. One of the largest value of y2C °° known, is for 1,1,2,2tetraphenylethane (having r2 = 12.98 and q2 = 8.936) in water (r/ = 0.92 and qi = 1.4), which is around 3575. In the Modified Dortmund UNIFAC14, most R and Q are empirical. This can lead to, perhaps, unexpected results for the combinatorial terms. Table 4 lists combinatorial terms at infinite dilution calculated from this version of UNIFAC. As can be seen, pyrene in n-pentane is not the only example of very large combinatorial contributions. In Table 4, the ratio r\lq\ generally differs from r^qi, as suggested above. Often qi has high values, but not always. Since the combinatorial of the modified UNIFAC Lyngby3 cannot yield values greater than one, such large values are not found with that model. Interaction Parameters The interaction parameters of the original, modified and linear UNIFAC are typically estimated in the range 0 - 150 °C, which thus defines approximately the range of applicability of these models. The UNIFAC-LLE parameters have been estimated in the range 10-40 °C. For most of these UNIFAC models, the interaction parameters are readily available in the literature. This is not the case for linear UNIFAC15, while some of the parameters of modified UNIFAC (Dortmund) have not been published either. 4.3.3 Performance/Limitations Since the development of the first UNIFAC model (Fredenslund el al.9) in 1975, the UNIFAC model has been extended and described in detail in a monograph . This original UNIFAC VLE model was further developed and an extensive table of group-interaction parameters was published as part of an on-going "Revision and Extension" effort. This resulted in a series of five articles19'21"22'11 starting in the late-1970s. Already during the early days in the development of the UNIFAC model, a number of problems were encountered. First, liquid-liquid equilibria could not be predicted reliably from the same parameters obtained by regression of experimental vapor-liquid equilibria data. Therefore, a separate UNIFAC model was developed exclusively based on regression of
69 experimental LLE data12. This model is referred to as the UNIFAC-LLE model and the intention has been to work with two separate tables, one for VLE ('Original UNIFAC VLE') and another for LLE ('UNIFAC LLE'). Table 4. Combinatorial Infinite Dilution Activity Coefficients calculated with the modified Dortmund UNIFAC Solvent (1) n-pentane
n
Qi
3.1625
2,3-dimethyl-2-bute ne 2,3-dimethyI-2-bute ne Formic acid
3.0529
Dimethyl ether
1.7759
hexachloro benzene
3.2190
hexachloro benzene
3.2190
Acetic acid
1.4325
4.245 9 4.701 4 4.701 4 1.274 2 2.663 0 1.906 2 1.906 2 1.982 -> 3
dichloromethane
1.8000
tetrachloro ethylene
5.6557
tetrachloro ethylene
5.6557
tetrachloro ethylene
5.6557
tetrachloro ethylene
5.6557
tetrachloro ethylene
5.6557
Dimethyl sulfoxide
3.6000
3.0529 0.8000
2.500 0 3.414 6 3.414 6 3.414 6 3.414 6 3.414 6 2.692 0
,, Coo
Solute (2) tetrachloro ethylene
r2 5.6557
3.4146
Sorbitol
10.5074
8.0884
1.36-10 5
hexachloro-1,3butadiene Pentaerythritol
8.7450
5.3510
5.25-10 A
8.0833
6.4032
1.17-10'
hexachloro benzene
3.2190
1.9062
2.79-10 2
N-butylacetate
3.8000
4.8137
2.40-10 2
ethylpropyl ether
3.6734
4.7873
3.36-10 2
Sucrose
16.4198
12.3598
4.81-10 4
dimethyl sulfoxide
3.6000
2.6920
2.10-10 '
di-n-butylether
5.5709
6.9116
1.27-10 •'
Ethylisopropyl Ketone 2,6-di-t-butyl-pcresol 2-ethylbutyric acid
4.2348
5.0920
1.27-10 2
8.5552
9.5756
1.72-10 •'
3.9625
4.8147
1.1010 2
1 -chloropentane
3.5219
4.5505
1.70-10 2
2,3-dimethyl-2butane
3.0529
4.7014
1.32-10 2
721
1.26-10 3
A second serious limitation was the temperature dependency of the original UNIFAC VLE model, which seemed not be adequate in many applications such as the prediction of heats of mixing (which requires the derivative of activity coefficients with respect to temperature). With the use of temperature dependent interaction parameters - as in Eq.(8) with at least two (sometimes three) terms, some improvements (sometimes orders of magnitude reduction of the error) have been made in reducing the uncertainty of predictions of heats of mixing in strongly temperature dependent systems. Yet, even with these
70
temperature dependent parameters, the concentration dependence of HE is still difficult to capture in many cases. Also, examinations of the combinatorial term based on hydrocarbon mixture data indicated that modifications might be required for improvements in solubility calculations21. The above reasons has led to the developments of the two (Lyngby and Dortmund) versions of the 'Modified' models1314. The two versions used fairly similar equations, but the 'Dortmund' version also treated the group surface area parameters, Q, as adjustable parameters and their values were determined, together with the interaction parameters, by fitting phase equilibrium data. This is probably the most important difference between the two modified UNIFAC versions. Also, whereas only one publication was made of the 'Lyngby' version, the parameter regression has been continued over the years for the 'Dortmund' modified UNIFAC and several publications report the updates25"29. Today the Dortmund modified UNIFAC is probably the most extensive UNIFAC parameter model. It is widely used and is now continuously revised and extended as part of an industrial/academic joint venture29. In recent years, numerous researchers30"36 have published group parameters for this model. Recently, the problem of inaccuracy of the UNIFAC methods for predicting activity coefficients of mixtures with proximity effects or isomers has been thoroughly studied37. All existing UNIFAC methods are considered as first-order methods and an extension has been undertaken using second-order interaction effects on the excess Gibbs free energy. These second-order effects arise from particular structural arrangements of groups, called secondorder groups, defined for cases when first-order groups alone do not distinguish different structures that are known to exist. For example, the conjugated pair of double bonds in isoprene is a second-order group, whereas a simple compound such as hexane has no secondorder groups. The terms added were derived from a Taylor series expansion of the UNIFAC equation around first-order interactions. This has led to the recent development of a combined first plus second-order group table, which is named 'KT-UNIFAC . It has been shown that data on alcohols, hydrocarbons and aromatics can be satisfactorily correlated and predicted for all isomers. 4.3.4 Other Applications of UNIFAC The UNIFAC method quickly became quite popular. Since activity coefficients are applied in many different thermodynamic contexts, use of UNIFAC has not been restricted to the cases treated in the foregoing sections. Many other applications have been reported. Examples are: • • • • • •
Estimation of solvent effects on chemical reaction rates " Calculation of critical micelle concentrations for surfactants solutions40'41'52 Calculation of flash points of flammable liquid mixtures42 Solvent selection for extractions43'44 Representation of petroleum fractions Viscosities of liquid mixtures46
With the introduction of some modifications, the method has also been used to describe phase equilibria of polymer solutions47 (see chapter 7) and electrolyte solutions48"51 (see chapter 8).
71 A very important application of UNIFAC (and other thermodynamic models) is in the prediction of the so-called octanol-water partition coefficients. These are of interest in environmental calculations, especially in assessing the distribution of long-lived chemicals in the various environmental ecosystems. How this is done is demonstrated in Chapter 14.
4.4 CONCLUSIONS Many semi-empirical group contribution methods have been developed over the last decades. The two most useful ones are UNIFAC and ASOG, of which only the first one has been presented in this chapter. Much progress has been made over the last decades in terms of both development and use of group contribution based excess Gibbs energy models, and the models have been improved continuously, especially new group interaction parameters have been added. This has made many types of calculations easy today, which were unthinkable a few decades ago, especially for preliminary design of separation processes. The principal users of such methods are chemical engineers who wish to predict the thermodynamic properties of a wide variety of highly non-ideal mixtures, typically for the design of suitable distillation columns and other separation units. GC methods like UNIFAC provide approximate estimations of the non-ideality of solutions but are useful when only few or no data is available and can be applied to a very wide number of non-ideal systems. Their major application is estimation of vapor-liquid equilibria of highly non-ideal mixtures at low pressures. High pressure applications require equations of state where activity coefficient models may be incorporated via the mixing rules for the energy parameter in the so-called EoS/GE models (see chapter 5), while multiphase multicomponent equilibria e.g. vaporliquid-liquid equilibria of systems containing associating fluids such as water or alcohols is best represented with more advanced equations of state such as SAFT and CPA (see chapter 6).
LIST OF SYMBOLS amn G Gki H Ji Lt qt Qk rt Rk s^ T Xj
group energy interaction parameter of UNIFAC, equation (8) Gibbs free energy defined by equation (5) enthalpy defined by equation (3) defined by equation (3) van der Waals surface area parameter of compound i van der Waals surface area parameter of group k van der Waals volume parameter of compound i van der Waals volume parameter of group k defined by equation (6) temperature in K mole fraction of compound i in the liquid phase
72
Greek
Letters
ji Qk via xmn r\k
activity coefficient of compound i defined in equation (5) number of groups of type k in compound i group parameter defined by equation (7) defined by equation (6)
Subscripts i j / m, n
compound /' compound/ sub-group / main-groups m & n
Superscripts C E R co
combinatorial property excess property residual property at infinite dilution
Abbreviations ASOG EOS GC LLE NMG NSG NSOG SLE VLE UNIFAC
Analytical Solution of Gas Equation of State Group Contribution Liquid-Liquid Equilibrium Number of main groups Number of sub-groups Number of second-order groups Solid-Liquid Equilibrium Vapor-Liquid Equilibrium Universal Function Activity Coefficient
REFERENCES 1. C.G. Gray and K.E. Gubbins, Theory of Molecular Fluids, Vol.1: Fundamentals, Oxford University Press, Oxford, 1984. 2. G.M. Wilson and C.H. Deal, Ind. Eng. Chem. Fund., 1 (1962) 20-23. 3. Aa. Fredenslund, J. Gmehling and P.Rasmussen, Vapor-Liquid Equilibria Using UNIFAC, Elsevier Scientific Publishing Company, 1977. 4. P. Rasmussen, Fluid Phase Equilibria, 13 (1983) 213-226. 5. Aa. Fredenslund and P. Rasmussen, Fluid Phase Equilibria, 24 (1985) 115-150.
73
6. R.A. Heidemann and Aa. Fredenslund, I. Chem. E. Symp. Ser., 104 (1987) A 291. 7. R.A. Heidemann and Aa. Fredenslund, Vapour-Liquid Equilibria in Complex Mixtures, Chem. Eng. Res. Des., 67 (1989) March. 8. Aa. Fredenslund, Fluid Phase Equilibria, 52 (1989) 135-150. 9. Aa. Fredenslund, R.L. Jones and J.M. Prausnitz, AIChE Journal, 21(6) (1975) 10861099. 10. J.W. Kang, J. Abildskov, R. Gani and j . Cobas, Ind. Eng. Chem. Res., 41(13) (2002) 3260. 11. H.K. Hansen, P. Rasmussen, Aa. Fredenslund, M. Schiller, M. and J. Gmehling, Ind. Eng. Chem. Res., 30 (1991) 2352. 12. T. Magnussen, P. Rasmussen and Aa. Fredenslund, Ind. Eng. Chem. Proc. Des. Dev., 20(1981)331. 13. B.L. Larsen. P. Rasmussen and Aa. Fredenslund, Ind. Eng. Chem. Res., 26 (1987) 2274. 14. U. Weidlich, and J. Gmehling, Ind. Eng. Chem. Res., 26 (1987) 1372. 15. H.K. Hansen, B. Coto and B. Kuhlmann, UNIFAC with Lineary TemperatureDependent Group-Interaction Parameters, SEP 9212 (Internal report at the Department of Chemical Engineering, The Technical University of Denmark), Lyngby, Denmark, 1992. 16. J. Gmehling, R. Wittig, J. Lohmann and R. Joh, Ind. Eng. Chem. Res., 41(6) (2002) 1678. 17. Aa. Fredenslund, P. Rasmussen, P. and M.L. Michelsen, Chem.Eng.Commun., 4 (1980) 485. 18. A. Bondi A., Physical Properties of Molecular Crystals, Liquids and Glasses, John Wiley & Sons, New York, 1968. 19. S. Skjold-Jorgensen, B. Kolbe, J. Gmehling and P. Rasmussen, Ind. Eng. Chem. Process Des. Dev., 18(4) (1979) 714. 20. Aa. Fredenslund and P. Rasmussen, Fluid Phase Equilibria, 24 (1985) 115. 21.1. Kikic, P. Alessi, P. Rasmussen and Aa. Fredenslund, Can. J. Chem. Eng., 58 (1980) 253. 22. J. Gmehling, P. Rasmussen and Aa. Fredenslund, Ind. Eng. Chem. Process Des. Dev., 21 (1982)118. 23. E.A. Macedo, U. Weidlich, J. Gmehling and P. Rasmussen, Ind. Eng. Chem. Process Des. Dev. 22 (1983) 676. 24. D. Tiegs, J. Gmehling, P. Rasmussen and Aa. Fredenslund, Ind. Eng. Chem. Process Des. Dev., 26 (1987) 159. 25. J. Gmehling, J. Li and M. Schiller, Ind. Eng. Chem. Res., 32 (1993) 178. 26. J. Gmehling, J. Lohmann, A. Jakob, J. Li and R. Joh, Ind. Eng. Chem. Res., 37 (1998), 4876. 27. J. Lohmann, R. Joh and J. Gmehling, Ind. Eng. Chem. Res., 40 (2001) 957. 28. J. Lohmann and J. Gmehling, J. Chem. Eng. Japan., 34 (2001) 43. 29. http://www.uni-oldenburg.de/tchemie/consortium/ 30. A. Jonasson, M. Savoia, O. Persson, and Aa. Fredenslund, J. Chem. Eng. Data., 39 (1994)134.' 31. J. Garcia, E.R. Lopez, J. Fernandez and J.L. Legido, Thermochim. Acta., 286 (1996)
74 321. 32. W.C. Moollan, U.M. Domanska and T.M. Letcher, Fluid Phase Equilibria, 128 (1997) 137. 33. K. Nolker and M. Roth, Chem. Eng. Sci., 53 (1998) 2395. 34. J. Garcia, E.R. Lopez, J. Fernandez and J.L. Legido, Thermochim. Acta., 317 (1998) 59. 35. H.-P. Luo, W.-D. Xiao and K.-H. Zhu, Fluid Phase Equilibria, 175 (2000) 91. 36. K. Balslev and J. Abildskov, Ind. Eng. Chem. Res., 41 (2002) 2047. 37. J. Abildskov, P. Rasmussen, R. Gani and J.P. O'Connell, Fluid Phase Equilibria, 158160(1999)349. 38. J. Gmehling and J. Fellensiek, Z. Phys. Chem. Neue Folge, 122 (1980) 251. 39. H.S. Lo and M.E. Paulaitis, AIChE J., 27 (1981) 842 40. C.-C. Chen, AIChE J., 42 (1996) 3231. 41.M.V. Flores, E.C. Voutsas, N. Spiliotis, G.M. Eccleston, G. Bell, D.P. Tassios and P.J. Hailing, J. Coll. Interfac. Sci., 240 (2001) 277.J. Gmehling and P. Rasmussen, Ind. Eng. Chem. Fundam., 21 (1982) 186. Errata, 21: 326. 42. U. Tegtmeier and K. Misselhorn, Chem. Ing. Techn., 53 (1981) 542. 43. E.A. Brignole, S. Bottini and R. Gani, Fluid Phase Equilib., 29 (1986) 125. 44. V. Ruzicka, Aa. Fredenslund and P. Rasmussen, Ind. & Eng. Chem. Process Des. Dev., 17(1983)333. 45. W. Cao, K. Knudsen, Aa. Fredenslund and P. Rasmussen, Ind. & Eng. Chem. Res., 32 (1993)2088. 46. T. Oishi and J. Prausnitz, Ind. Eng. Chem. Proc. Des. Dev., 17 (1978) 333. 47. C.-C. Chen, H.J. Britt, J.F. Boston and L.B. Evans, AIChE J., 28 (1982) 588. 48. B. Sander, Aa. Fredenslund and P. Rasmussen, Chem. Eng. Sci., 41 (1986) 11711183. 49. M. Cardoso and J.P. O'Connell, Fluid Phase Equilibria, 33 (1987) 315. 50.1. Kikic, M. Fermeglia and P. Rasmussen, Chem. Eng. Sci., 46 (1991) 2775. 51. H.Y. Cheng, G.M. Kontogeorgis and E.H. Stenby, Ind. Eng. Chem. Res., 41(5) (2002) 892.
Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B.V. All rights reserved.
75
Chapter 5: Equations of State with Emphasis on Excess Gibbs Energy Mixing Rules Epaminondas C. Voutsas, Philippos Coutsikos and Georgios M. Kontogeorgis
5.1 INTRODUCTION Accurate description of vapour-liquid equilibria (VLE) and gas-liquid equilibria (GLE) is a key to successful process design and simulation work. Equations of state (EoS) and activity coefficient (Gibbs Free-Energy) models are the two types of thermodynamic approaches that are widely used for the correlation and prediction of phase equilibrium. Activity coefficient models are successful for systems at low pressures, but cannot be extended to high temperatures and pressures. Furthermore, activity coefficient models are not applicable to mixtures containing supercritical compounds, or compounds that are not liquids when pure at the temperature and pressure of the mixture. On the other hand, the advantage of the equation of state approach is that it is applicable over wide ranges of temperature and pressure and to mixtures of various components, from light gases to heavy liquids. Since the same equation of state is applied to the various phases at equilibrium, a consistent description is obtained and questions of standard states do not arise. The most widely used are the cubic equations of state, which are often presented as corrections to the ideal gas equation, which is valid for very simple fluids and for the gas phase only:
Starting from the Nobel-Prize winner van der Waals in 1873, numerous researchers have proposed equations of state as the ones shown below. Van der Waals (1873):
Redlich-Kwong(1949): p =
RT
a
V-b
V(V + b}Jf
Soave-Redlich-Kwong (SRK)1:
76 p
_ RT V-b
a V(V + b)
Peng-Robinson (PR)2: p =
_RT
a
V-b
V(V + b)+b(V-b)
Patel andTeja3: p =
_RT V-b
a V(V + b)+c{V-b)
The SRK and PR are today the most widely employed equations of state in industry. All these vdW-type equations of state contain a positive repulsive and a negative attractive contribution to the pressure. In most cases (except for the Patel-Teja equation), they include two compound-specific parameters, (a, b). The first one (a) is related to molecular energy and the energetic interactions and the other (b), called co-volume, to the molecular volume. These parameters are fitted to experimental data for both pure compounds and mixtures. Typically, the pure fluid parameters are estimated from the critical properties (temperature and pressure) and the acentric factor. The energy parameter is usually fitted to vapor pressure data, since the reproduction of pure compound vapor pressures is crucial for accurate VLE. Phase equilibrium calculations for mixtures, however, are largely dependent on the way these parameters are extended to mixtures via the so-called mixing and combining rules. These mixing rules are so important in obtaining correct phase equilibria that it is often, not unjustifiably, stated that the proper choice of mixing rules may be more important than the actual mathematical function of the equation of state itself. Many choices have been proposed. However, two have received widespread application and acceptance. The so-called van der Waals one fluid (vdWlf), often named "classical mixing rules", and the more advanced EoS/GE mixing rules. The van der Waals one-fluid mixing rules have received widespread application especially within the oil and gas industry and are limited to non-polar fluids. They are typically not capable of representing highly non-ideal mixture behaviour of polar or associating fluids, unless large values of the interaction parameters are used and again basically only for VLE. Their application to LLE and multi-component multiphase equilibria is limited. The vdWlf mixing rules and some of their recent modifications are discussed in the next section. A number of mixing rules have been proposed to extend the applicability of equations of state to highly non-ideal mixtures with varying degrees of success. This chapter deals almost exclusively with the type of mixing rules that combine equations of state with Gibbs free energy models, resulting to the so-called EoS/GH models.
77 5.2 THE VAN DER WAALS ONE FLUID MIXING RULES The van der Waals one fluid (vdWl f) mixing rules are given by the equations: a
= E Z */xia>i '
b
= Z X x<x A
/
'
(3)
y
The summations are over all components in a mixture. For example, for a binary system, we have: a = x,2a, + x22a2 + 2x,x 2 a I2 (4) b = x*b} + x]b2 + 2x,x2ibl2
The mixing mles actually represent the interactions between the components in the mixture. As we observe from Eq. 4, they depend not only on the pure fluid parameters but also on the cross-parameters an, bnFor many years, industry (especially within oil & petroleum fields) has been using, and still use, Eqs 3 and 4 as the basic mixing rules, combined with the following rules for the cross-parameters :
<5>
bi+bj
Equations 5 which relate the cross interaction parameters with the parameters of pure fluids are called combining rules. Especially for the cross co-volume parameter, other combining rules have been also proposed:
Lorentz
rule:
bu =Ubl'3+bl2Bf
Sandier
rule:
bu =
(6)
—\
The Lorentz rule is equivalent to the arithmetic mean rule for the molecular diameter and as such has a rigorous theoretical foundation. It has found widespread application in many theoretical studies4, but for most practical purposes, the rules of equation 5 are found to be superior5. Cubic equations of state, especially SRK and PR, with the mixing and combining rules shown in Eqs 3-5 had been used for years and are still widely used in the oil & petroleum industry for modeling hydrocarbons and gas-hydrocarbon mixtures. The interaction parameter ky, is usually an adjustable parameter which must be determined from
78 experimental phase equilibrium data. Thus, some experimental data are required for calculating phase equilibria. For simple hydrocarbon mixtures, kjj can be set to zero. In practice we would optimally like to have an equation of state which can be used for the prediction of phase equilibria without the need of mixture information. Or, in other words, we would like to have methods for estimating this interaction parameter kjj as a function of pure fluid properties e.g. critical constants, molecular volume or area and of temperature. Due to the importance of having such estimation methods for ky, many efforts have been reported for different types of systems. As examples, we can mention the following investigations for different families of gases with hydrocarbons: • • • •
methane / hydrocarbons6 nitrogen / hydrocarbons7 carbon dioxide / hydrocarbons8 hydrogen sulfide / hydrocarbons9"10
Many more related investigations1 '"'4 have been reported for kjj The publications by Kordas el al.6'* and Avlonitis et alJ are from the same research group and for the same cubic EoS (a modified Peng-Robinson). These researchers examined whether the same mathematical function could be used for correlating the interaction parameter for different types of gas/hydrocarbon systems. They concluded that, when kjj is estimated based on the vdWlf mixing rules (Eqs. 3-5), no such general correlation can be established. Different mathematical functions are required. For example, for CCValkanes, Kordas et a/.8 developed the following relationship: ^ = a ( o o y ) + 6((Dy)Trt+c((oy)rrf2
(7)
where Tr; is the reduced temperature (=T/TC) of carbon dioxide and the parameters a, b and c are quadratic functions of the acentric factor of the alkane (co). The equation performs satisfactorily, but a special form was required for the CCVmethane system. Even more disturbing was the fact that different functional forms are required for other gas/alkane systems. For example, Kordas et al.6 concluded that for methane/alkanes two different correlations are required depending on the chain length of n-alkane. Thus, for methane/n-alkanes for alkane carbon number below 20: k:j =-0.13409co + 2.28543co2 -7.61455co3 + 10.46565©4 -5.2351co5
(8a)
while for methane/n-alkanes with alkane carbon number above 20: k)j = -0.04633 - 0.04367 In ©
(8b)
The same conclusion is reached by many researchers. Having different equations for kjj may be inconvenient for practical purposes. It may be considered a limitation also from a theoretical point of view. It could indicate inadequacies of cubic equations of state and/or these mixing rules for such systems.
79 The systems mentioned above, gases with hydrocarbons, are important in many industrial applications. These systems, especially when the hydrocarbon is of high molecular weight, are called (size-) asymmetric systems. Alkane solutions such as hexane-hexatriacontane and polymer solutions are also asymmetric systems. A new insight on the nature and estimation of the interaction parameter has been provided in a recent publication . These authors have combined theoretical considerations based on statistical mechanics with some empirical observations to arrive to the following general combining rule:
The parameter n is the attractive tail of the so-called Mie function for the intermolecular potential:
m — nyn)
\ r )
\ r )
where s12 is the molecular cross-energy parameter, CT12 is the molecular cross diameter and r is the distance of the molecules. A combining rule similar to Eq.9 has been previously proposed by Hudson and McCoubrey16. Upon comparing the theoretically derived Eq. 9 with Eq. 5, the following general equation is derived for estimating the interaction parameter ky:
Alternatively, the combining rule of Eq. 9 can be used with the exponent n being the adjustable parameter. Coutinho et a/.15 have performed a mathematical analysis for determining the values of the parameter n for different types of asymmetric mixtures using the PR EoS. They observed that remarkably similar trends are obtained for very different mixtures, such as CCVn-alkanes and N2/n-alkanes including polymers as well. It has been observed that although ky has different trends with the chain length of nalkanes for the two gases (decreasing for CO2 and increasing for N2), the exponent-n decreases with increasing chain length in both cases (same trend). This may indicate that the more general combining rule, Eq. 9, incorporates the size-effects and provides a physically meaningful insight into equation of state behavior. Thus, it is a useful way for estimating interaction parameters for gas/alkane systems. Despite the problems discussed e.g. different trends of the interaction parameter for different gas-hydrocarbon systems, cubic EoS with the classical vdWlf mixing and combining rules, are suitable for many practical problems, especially in the petroleum
80 industry. The satisfactory description of the vapor-phase non-ideality is the crucial factor and this is accomplished successfully. The liquid phase non-ideality (deviations from Raoult's law) does not play an important role for these non-polar systems. However, there are many cases where cubic EoS have been proven to be unsuccessful, at least when the usual mixing and combining rules are used (Eqs. 3-5) and especially for prediction purposes: i. ii. iii.
Mixtures with polar and hydrogen bonding fluids such as water, alcohols, amines, acids, etc. Situations where the 'condensed' phases (solid, liquid) dominate such as in solidliquid, vapor-liquid-liquid and solid-gas equilibria. Multicomponent, multiphase equilibria
Many of these mixtures and cases were not among the original goals of cubic EoS. At the starting time of development of cubic EoS (1950-1970's), the primary target was towards solving problems related to the oil and petroleum industry. Later, however, with the increasing growth of other industrial sectors (polymers & materials, food, biotechnology) and the expansion of oil and chemical industry in new areas, problems such as those mentioned above (i-iii) received great importance. Unfortunately, cubic EoS seemed not to be adequate for these cases. It is not entirely clear to what extent these inadequacies of cubic EoS are primarily due to the functional form of these EoS and their cubic character or the extension to mixtures i.e. the mixing and combining rules. The problems of cubic EoS for complex systems has led to the development over the last 30 years of: i. improved non-cubic EoS primarily based on Statistical Thermodynamics17 ii. activity coefficient models based on group contributions e.g. UNIFAC iii. improved mixing rules for cubic equations of state Over the period 1970-1990, the first direction (non-cubic EoS) has not led to significant advances. Recently, after 1990, a great advance has appeared with the development of the socalled SAFT and CPA equations of state (see chapter 6). UNIFAC and related models offered a solution to the thermodynamic modeling of complex systems at low pressures (see chapter 4). Still, many believe that the most important explanation for the problems is the mixing rules. The field of mixing rules is endless18. From a philosophical perspective, it reflects the efforts of science to understand and model mixture behavior having as starting point only the properties of pure fluids. Although hundreds of mixing rules have been proposed, by far the most popular advance, and one of the greatest practical achievements of applied thermodynamics over the last 15 years, is the so-called EoS/G mixing rules. These mixing rules allow cubic EoS to be used for describing highly non-ideal systems using information from well-known activity coefficient model such as UNIFAC, NRTL or UNIQUAC. Equivalently it can be stated that they extend the applicability of classical low pressure activity coefficient models like UNIFAC, NRTL and UNIQUAC to high pressure applications including gas solubilities. A discussion of these mixing rules follows in the remaining of this chapter.
81 5.3 BASIC APPROACH OF EoS/GE MODELS The idea of combining equations of state with Gibbs free energy models was originally proposed by Huron and Vidal19. The key idea of Huron and Vidal was to match the excess Gibbs free energy of an equation of state to that of an activity coefficient (AC) or excess Giggs free energy (Gh) model at a specified state and to solve for the van der Waals attractive parameter a from the equation:
GLs(x1,T,Pref)=GL(x,,T,P = low)
(12)
The van der Waals attractive parameter, a, obtained by solving Eq. 12 is the mixing rule. By using this a, the equation of state reproduces the activity coefficient model at the specified state. The G1 of the EoS changes with the volumetric properties of the EoS, thus accomplishing the objective of incorporating the activity coefficient model in the EoS. Huron and Vidal matched the GE of the activity coefficient model to that of the EoS at the state of infinite pressure (Pref= infinity), and it is referred to as the infinite pressure approach. For the EoS co-volume parameter (b) the conventional linear mixing rule, b=Lxjbj, was used. They assumed that GH at infinite pressure can be represented by the same activity coefficient models that describe low-temperature/low-pressure mixtures. The resulting model was capable of correlating high temperature/high pressure phase equilibrium but could not directly utilize existing activity coefficient model parameters correlated from lowtemperature/low-pressure data. To make possible the use of activity coefficient model parameters fitted to low-pressure data, Mollerup20 suggested matching the GH of an activity coefficient model with that of an EoS at low pressure, which may be either the zero pressure or a suitably fixed packing fraction (=b/V) where V is the molar volume. The zero-pressure approach was developed by Heidemann and Kokal21 and Michelsen22'2"3. The volume of the zero-pressure liquid is obtained by solving the EoS as part of the phase equilibrium calculations. The first disadvantage of this approach, referred to as "exact" zero-pressure approach, is that the obtained mixing rule is not explicit and the solution is found by an iterative procedure. The second disadvantage is that it cannot be applied to systems where at least one of the purecomponent and/or mixture a-values (a=a/bRT) are lower than a limiting value (anm), e.g aiim=4 for the vdW EoS, aijm=3+2°5 for the SRK, etc., since cubic EoS zero-pressure liquid exists at suitably low temperatures, but ceases to exist at higher temperatures, where an extrapolation has to be made to obtain a hypothetical zero-pressure liquid. In order to extend the applicability of this approach to a
82 Boukouvalas et al}% developed a mixing rule, resulting to the LCVM model, which is a linear combination of the mixing rules of Huron and Vidal19 and Michelsen22'23 and, as such, has no specified reference pressure. LCVM has been proved29"34 to be superior to other EoS/G1' models in predicting phase equilibrium for systems containing compounds with large differences in size (asymmetric systems). Of special interest are the models developed by matching an EoS with a predictive activity coefficient model, such as the group-contribution UNIFAC, since the resulting models become strictly predictive tools. Predictivity is an important advantage for engineers since it avoids the retrieval and correlation of experimental phase equilibrium data, which in many cases are not available. The rest of this chapter is concerned only with predictive EoS/GE models, i.e models that utilize available UNIFAC interaction parameter tables, and more specifically the MHV2, PSRK, LCVM and Wong-Sandier ones.
5.4 BRIEF PRESENTATION OF THE MODELS 5.4.1The MHV2 Model The EoS The Soave-Redlich-Kwong (SRK) EoS1 is used with this model. The mixing rule The following expression is used for parameter a: a2-Vxa2
= ^
+ V x In—
RT
(13)
b
For the co-volume parameter, the linear mixing rule is used: b= 2>,bi The UNIFAC model This model uses for G^c the modified UNIFAC model35. 5.4.2 The PSRK Model The EoS The SRK EoS is used with this model.
(14)
83 The mixing rule The following expression is used for parameter a:
a =
f^ 0.64663^ RT
+ Vxiln— l + Jx^ ^
'
b, J ^
(15)
' '
For the co-volume parameter, the linear mixing rule is used:
b = I> b .
(16)
The UNIFAC model This model uses for G^c the original UNIFAC model36. 5.4.3
The LCVM Model
The EoS The translated and modified Peng-Robinson (t-mPR) EoS37 is used with this model. The mixing rule The following expression is used for parameter a: a= ^ 0.623
\—££\ x in — -n^XiCC; 0.52 J RT 0 . 5 2 ^ [bl) ^ ' '
(17)
where A, is a constant equal to 0.36 (when the original UNIFAC model is used). For the covolume parameter, the linear mixing rule is used:
b = Ix i b, The UNIFAC model This model uses for Gj^c the original UNIFAC model36. 5.4.4 The Wong-Sandier (WS) Model The EoS The modified Peng-Robinson EoS, called as PRSV38, is used with this model.
84 The mixing rule The following expression is used for parameter a:
a = b .fy x A + -^y
(is)
Xf-1 b, 0.623J For the co-volume parameter, the following mixing rule is used: W (u a \ > > x.-xJ b b=— ^ ^ ^ RT
^
(19)
bjRT
The combining rule for the cross virial coefficient is given by: (
b
a
\
I
RTJ
( -Rf)r^
J
-T
RT J /
N
1 k
-•< - ''
(20)
5.4.5 The predictive WS model Orbey et al?9 combined the mixing rule described in 5.4.2 with UNIFAC to yield a completely predictive scheme. For the calculation of the cross virial binary interaction parameter, ky, Orbey et al.39 proposed the following procedure: First, for the binary mixture of interest, the two infinite dilution activity coefficients are predicted at 298 K from UNIFAC. Next, these infinite dilution activity coefficients are used to obtain the interaction parameters of the UNIQUAC model40. Finally, the kjj is calculated by matching the excess Gibbs free energy of the mixture calculated from UNIQUAC and from the equation of state at the midconcentration point, Xi=0.5, and T=298 K. The values of the UNIQUAC parameters and the kjj are then used at all other temperatures. Voutsas et a/.30 proposed the direct calculation of kjj by matching the excess Gibbs free energy of the mixture calculated from UNIFAC and from the EoS at Xj=0.5 and T=298 K. It must be noticed that Coutsikos el al.41 have shown that the WS mixing rule with a composition independent kjj is not able to match the EoS-obtained GH expression with that of the activity coefficient model for asymmetric systems. The UNIFAC model This model uses for G ^c the original UNIFAC model36.
85 Limitations of the WS predictive model A serious disadvantage of the WS predictive scheme is that its application is limited only to systems containing components that are condensable at the temperature of interest and, consequently, it cannot be used for phase equilibrium predictions for systems containing gases.
5.5 INPUT INFORMATION NEEDED FOR THE APPLICATION OF THE MODELS 5.5.1 EoS Pure-Compound Parameters For non-polar compounds the critical temperature (Tc), critical pressure (Pc) and the acentric factor (co) are needed for the estimation of the EoS pure-compound a and b parameters. One of the best and extensive available pure-compound property databases is that published by AIChE's Design Institute for Physical Propertied (DIPPR)42. When experimental Tc and Pc data are not available the recently developed method by Marrero and Gani43 is proposed, while for co the methods of Edmister and Lee-Kesler 5 may be used. It should be noted that for VLE/GLE calculations pure-compound vapour pressures must accurately be predicted by the EoS. To this purpose special attention should be paid to the accuracy of the Tc, Pc and co values used, since a small error in these values -especially in Tcmay have a significant impact on the accuracy of the obtained vapour pressure predictions, which is more pronounced for the large molecular weight compounds. For example, an uncertainty of 0.5% in the Tc value of n-hexane leads to tripling of the error in vapour pressure prediction obtained with the SRK EoS, while the same uncertainty in the Tc value of n-hexadecane leads to a multiplication by six of the error in vapour pressure prediction. For polar compounds the temperature dependency of the EoS a parameter is given by the expressions proposed by Mathias and Copeman46 for the PSRK, MHV2 and LCVM models, while for the WS model by the modification for the PR EoS proposed by Stryjek and Vera38. Mathias-Copeman parameters, for several substances, fitted to experimental vapour pressure data for the MHV2, PSRK and LCVM models can be found in the original publications of the models24'25'28, while PRSV pure-compound parameters fitted to experimental vapour pressure data for the WS model in Stryjek and Vera38. When Mathias-Copeman or Stryjek-Vera parameters are not available, they must be estimated by fitting pure-compound vapour pressure data. Experimental vapour pressure data for the majority of compounds are provided in the DIPPR compilation. In cases where such data are not available, vapour pressures can be predicted by the recently developed method by Voutsas et alf'', which requires the knowledge of the normal boiling point temperature (Tb) only, or by more complex methods that require additionally to Tb other pure-compound properties such as Tc and Pc, like the method of Lee-Kesler45 or the one of Gomez and Thodos4 . When Tb is not available it can be predicted by the method of Marrero and Gani43.
86 5.5.2 UNIFAC Interaction Parameters When applied to mixtures containing components that are condensable at the temperature of interest, all models use UNIFAC interaction parameters available in the existing UNIFAC tables. For LCVM, PSRK and WS models the necessary UNIFAC interaction parameters can be found in [36], while for MHV2 in [35]. Additional interaction parameters for the MHV2, PSRK and LCVM models that allow predictions in systems containing gases -not coved by UNIFAC- can be found in the original publications of the models24"25'28, while for the WS model such parameters are not available. 5.6 PERFORMANCE OF EoS/GE MODELS IN VLE/GLE PREDICTION A valuable thermodynamic model for practicing engineers is the one, which can accurately predict phase equilibrium in multicomponent mixtures. Of course, successful prediction of the constituent binary mixtures is also important. First, typical VLE or GLE predictions obtained by the various models in binary mixtures are presented, and then we proceed to ternary and multicomponent predictions. 5.6.1
Binary Mixtures
Non-polar mixtures Figures 1 to 3 present typical prediction results for non-polar binary systems. As shown in Figure 1, for relatively symmetric systems, i.e. systems where their components do not differ significantly in their size, all models give comparable good predictions. On the other hand, for asymmetric systems only the LCVM model gives good results, while all the other models give progressively poorer results (Figures 2 and 3). In order to overcome this problem with the PSRK model, Li et al.49 proposed an empirical correction to the UNIFAC van der Walls volume (Rk) and surface area parameters (Qk) for the CH3, CH2 CH and C groups based on VLE data of gases with alkanes. The PSRK model with the modification of Li et al. gives significantly better results than the original PSRK model, as shown in Figures 2 and 3. The LCVM model gives satisfactory VLE predictions even at very high pressures as indicated by the results presented in Figure 4 for three isopleths of the N2/n-tetradecane system. On the other hand, the PSRK model underestimates the experimental data, while predictions with the MHV2 model are not included due to the very high errors obtained. Figures 5 and 6 present bubble point pressure and vapour phase composition predictions for the CO2/D-Limonene system. D-Limonene is the principal compound of citrus oil and it is usually extracted from it through supercritical extraction. Also, Figure 7 presents vapour phase composition predictions for the CO2/I-Methyl Naphthalene system. The results indicate that EoS/GE models offer a valuable tool for the estimation of phase behaviour properties of supercritical fluid systems when experimental data are not available.
87
Figure 1. Prediction of the P-x-y diagram for the system Propane/n-Pentane at 344.3 K Experimental data from Knapp et al.65.
Figure 2. Prediction of infinite dilution activity coefficients of n-heptane in high molecular weight n-alkanes at 373 K. Experimental data from Pareher et al.66.
88
Figure 3. Average absolute percent deviation in bubble point pressure prediction for C2H6/11alkane systems. Sources of experimental data: Boukouvalas et al2i'.
Figure 4. Prediction of bubble point pressures of the system N2A1CI4. Experimental data from de Leeuw et al.61.
89
Figure 5. Prediction of bubble point pressures of the system CCVD-Limonene. Experimental data from Iwai et a/.68.
Figure 6. Prediction of vapor phase D-Limonene solubilities for the system CCVD-Limonene. Experimental data from Iwai et a/.68.
Polar mixtures Typical prediction results for binary polar systems are shown in Tables 1 and 2. It is again concluded that for relatively symmetric systems all models give comparable good results (Table 1) while for asymmetric systems (Table 2) the LCVM model gives the best overall predictions. Notice that in the latter case, where the large molecular weight compounds are
90 not alkanes, the empirical improvement introduced by Li et al. for the PSRK model gives worse results than the original PSRK model.
Figure 7. Prediction of vapor phase 1-Methyl Naphthalene (1MNP) solubilities for the system CO2/1- Methyl Naphthalene at 308.2 K. Experimental data from Lee and Chao69. The case of aqueous mixtures containing hydrocarbons A limitation of all EoS/UNIFAC models is that they fail to predict phase equilibria in aqueous mixtures containing hydrocarbons, which is attributed to the fact that UNIFAC does not correctly predict phase equilibria in water/hydrocarbon systems. A typical example is shown in Figure 8. All EoS/GE models overestimate n-butane solubilities, since UNIFAC underestimates the activity coefficients of alkanes in aqueous mixtures50'51.
Figure 8. Prediction of liquid-phase n-butane solubilities in the n-butane/water mixture at 298 K using various EoS/G1' models based on UNIFAC. Experimental data from Carroll et al.13.
Table 1. VLE prediction results for binary polar systems. System
Ref
NDP
T-range (K)
P-range (bar)
MIIV2 AP%
Ay
Vvs
PS RK AP%
(xlOO)
Ay
AP%
(xlOO)
LC VM Ay
AP%
(XlOO)
Ay (xlOO)
Acetone/chloroform
54
137
298-333
0.2-1.0
4.7
0.6
1.2
0.8
1.9
0.9
1.1
0.8
n-pentane-/acetone
55
27
373-423
0.2-6.7
1.7
1.0
1.9
0.9
2.3
1.4
1.5
1.0
ethanol/water
56
78
423-623
5.6-185.5
2.4
1.3
1.7
0.6
4.4
1.9
1.2
1.1
2-propanol/water
56
74
423-573
5.2-123.5
4.8
2.7
1.4
1.4
6.4
3.1
3.0
2.4
methanol/benzene
57
40
373-493
3.1-57.6
2.7
1.7
3.7
2.0
1.7
1.7
4.3
2.3
methanol/water
58
55
373-523
1.0-83.0
2.8
1.5
1.4
1.0
1.8
1.3
1.6
1.0
acclonc/watcr
58
80
373-523
1.1-67.6
2.4
1.6
3.0
1.2
2.7
1.6
2.5
1.3
acetone/methanol
58
35
373-473
3.5-39.9
1.8
1.9
2.3
2.0
2.0
1.9
2.3
1.9
3.3
1.4
1.9
1.1
3.0
1.7
2.0
1.4
Overall
526
Table 2. VLE prediction results for CCVhigh molecular weight n-alcohol systems. Experimental data from [59].
Alcohol C14OH C16OH C18OH
T-range (K) P range (bar) NDP 373.2-473.2 10.1-50.7 15 373.2-573.2 10.1-50.7 15 373.2-573.2 10.1-50.7 15
PSRK Ay x AP% 100 6.9 0.5 8.4 0.6 10.8 0.2
PSRK-Lietal. AP% 17.8 14.1 13.9
Ay x 100 0.5 0.7 0.3
MHV2 AP% 47.7 39.8 47.9
LCVM
Ay x 100 AP% Ay x 100 0.4 8.7 0.5 0.5 8.1 0.8 0.3 6.8 0.9
92 5.6.2
Ternary and Multicomponent Mixtures
Ternary mixtures Figure 9 presents bubble point pressure predictions for the system hydrogen/CO2/Methyl Naphthalene at 542.6 K. LCVM gives satisfactory predictions over the entire pressure range, while MHV2 consistently over-predicts the experimental data and PSRK gives progressively poorer results with increasing pressure.
Figure 9. Prediction of bubble point pressures of the system H2/CO2/I -Methyl Naphthalene at 542.6 K. Experimental data from Lin et al.10. Figure 10 presents experimental and predicted equilibrium ratio values of hydrogen in its mixture with n-hexadecane and benzene at 101 bar and 473 K. LCVM gives good predictions over the entire concentration range, while PSRK, due to that it over-predicts the equilibrium ratio of hydrogen in the binary H2/n-hexadecane system (Xbenzenc=0), it consistently overpredicts the experimental data in the whole concentration range. Notice, the improved results obtained by the modification of Li et al. over those obtained by the original PSRK model. The results presented with the LCVM model in Figures 9 and 10 were obtained using interaction parameters for hydrogen containing pairs developed by Polyzou5 , which are presented in Table 3. Multicomponent mixtures Bubble point pressure predictions for a fifteen-component mixture consisting of 43.8% methane 45.9% n-decane and 10.3% of a mixture of n-alkanes from nC18 up to nC30, are shown in Figure 11. The results indicate that LCVM and PSRK, as modified by Li et al., give very satisfactory results (overall absolute error in bubble point pressure of about 6%), while as temperature increases the MHV2 model gives progressively poorer results.
93
Figure 10. Prediction of hydrogen equilibrium ratios (K-values) in its mixture with nhexadecane and benzene at 101 bar and 473 K versus benzene composition in the liquid phase. Experimental data from Brunner et al.7].
Figure 11. Prediction of bubble point pressures for a fifteen-component mixture. Experimental data are from Daridon et al.72.
Table 3. UNIFAC group interaction parameters'1 for the LCVM model. Group n
Group m
E
A nm (K)
A mn (K)
H2
b
CH2
174.97
0.3811
437.53
0.2339
H2
b
ACH
160.76
-0.1155
631.81
0.3024
H2
b
ACCH 2
1275.91
4.8666
321.99
0.3099
H2
b
CH4
80.22
0.0
227.22
0.0
H2
b
C2H6
208.81
0.0
372.86
0.0
H2
b
CO
89.93
0.0
154.15
0.0
H2
b
CO 2
512.64
0.0
583.56
0.0
H2
C
H2O
343.97
6.6336
1542.16
4.9255
H2
C
MeOH
514.62
4.6127
487.37
0.7913
H2
C
OH
2829.63
0.0
1872.09
0.0
CO
C
CH4
-22.94
0.0
77.57
0.0
CO
C
C2H6
-248.95
-1.7356
424.97
2.3872
a
UNIFAC interaction parameters between pairs are estimated through the following expression: % = exJ- A - + B ^ T - 2 9 8 - 1 5 )'
b
parameters developed by Polyzou et al.52 experimental data used: H2 + H2O (323 - 423 K) from Kling and Maurer61 H2 + methanol (298 - 373 K) from Brunner et al.62 H2 + ethanol (323 K) from Vetere et al.63, +l-propanol (298 - 323 K) from Brunner64 CO + CH4 (91 - 124 K) from Knapp et al.65 CO + C2H6 (173 - 273 K) from Knapp et al.65 c
95 Figure 12a presents vapour and liquid phase composition predictions for carbon dioxide and methane in a nineteen-component mixture containing carbon dioxide, methane, paraffins (from nC5 up to nC13) and n-alkylcyclohexanes (from cyclohexane up to nheptylcyclohexane), while Figure 12b presents vapour and liquid phase composition predictions for n-hexane and n-butylcyclohexane for the same mixture. For light gases, carbon dioxide and methane, the agreement between experiment and prediction by LCVM and PSRK, for both liquid and vapor phase is satisfactory. For heavy components, n-hexane and n-butylcyclohexane, there is good agreement in the liquid phase, but the deviation in the vapor phase is not satisfactory. Notice, however, that for such low mole fractions the experimental uncertainty is comparable to the observed deviations of the predicted values53.
Figure 12a. Prediction of vapor and liquid phase compositions for CO2 and CH4 in a nineteen-component mixture. Experimental data from Angelos et al? .
Figure 12b. Prediction of vapor and liquid phase compositions for n-hexane and nbutylcyclohexane in a nineteen-component mixture. Experimental data from Angelos et al.53
96 Tables 4 and 5 present bubble point pressure and vapour phase composition predictions for a ten-component mixture containing polar, non-polar and supercritical components at two different temperatures. Missing interaction parameters for the LCVM model were estimated for the following pairs: H2/H2O, H2/Me0H, H2/OH, CO/CH4, CO/C2H6, and they are presented in Table 3. All the other parameters needed for the LCVM model were obtained from Polyzou and they are also presented in Table 3. As shown the models do not give very satisfactory results with the LCVM model to give better predictions than the other models. Table 4. VLE for a ten-component system at 313.4 K. Experimental data are from [60]. Componemt
Xi
P(bar) H2 CO CO 2 H2O CH4
C2H6 C3H8 MeOH EtOH n-PrOH
0.0071 0.0049 0.0057 0.1605 0.0170 0.0068 0.0039 0.6054 0.1359 0.0527
P (bar) and vapor phase mole fraction, y-, Exp. PSRK MHV2 LCVM 107.1 79.7 35.9 83.8 0.4502 0.3881 0.3635 0.4845 0.1701 0.2112 0.1955 0.1754 0.0192 0.0215 0.0253 0.0226 0.0002 0.0004 0.0007 0.0005 0.3039 0.3298 0.3391 0.2709 0.0384 0.0336 0.0523 0.0315 0.0144 0.0084 0.0141 0.0081 0.0082 0.0031 0.0061 0.0058 0.0005 0.0008 0.0011 0.0006 0.0002 0.0001 0.0001 0.0001
Table 5. VLE predictions for a ten-component system at 333.4 K. Experimental data are from [60]. Componemt
Xi
P(bar) H2 CO
co2
H2O CH4
C2H6 C3H8 MeOH EtOH n-PrOH
0.0073 0.0045 0.0042 0.1765 0.0152 0.0055 0.0026 0.5932 0.1366 0.0544
P (bar) and vapor phase mole fraction, y-, Exp. PSRK MHV2 LCVM 102.9 75.5 37.9 79.7 0.4495 0.3861 0.3623 0.4863 0.1872 0.1672 0.1681 0.1983 0.019 0.0216 0.0245 0.0221 0.0004 0.0012 0.0021 0.0013 0.3003 0.3364 0.3363 0.2695 0.039 0.0339 0.0535 0.0325 0.0072 0.0128 0.0137 0.0075 0.0079 0.013 0.018 0.0123 0.0018 0.0018 0.0028 0.0015 0.0004 0.0004 0.0006 0.0003
97 5.7 PERFORMANCE OF CUBIC EOS FOR SOLID-GAS EQUILIBRIA 5.7.1 General - Modelling using vdWlf mixing rules Solid-Gas equilibria (SGE) is a particular type of high pressure phase equilibria of interest to certain novel separation techniques, such as supercritical fluid extraction (SCFE) and supercritical water oxidation. SCFE is useful in cases where traditional separation methods such as distillation are expensive or for compounds such as pharmaceuticals, enzymes, food products, etc. which cannot be produced and/or separated with traditional methods, basically because they are thermally labile. SCFE can work at relatively low temperatures. Moreover, CO2 is a non-toxic "green" solvent with high solvency capacity74. Another advantages of SCFE is its selectivity, of importance in the case of very similar substances e.g. CO2 dissolves benzoic acid in concentrations 1000 times higher than p-hydroxybenzoic acid. The removal of caffeine from coffee, the extraction of nicotine from tobacoo as well as the extraction of high-value substances in the cosmetics and pharmaceutical industries and the extraction of flavours and fragrances are some of the processes for which SCFE has been used commercially. Some of the shortcomings of SCFE compared to several conventional separation methods, are the high operation pressures (up to 300 atm) and the associated high capital (equipment) cost, the need for batch operation especially for solids, the use of flammable solvents (ethane, ethylene), and the difficulty in optimised design due to, among others, the complexity of thermodynamic modelling and the often need to select useful co-solvents. The co-solvents are low molecular weight volatile compounds, which are inserted in small amounts (1-5%) in the SC solvent and modify considerably its critical properties and density. Since, in many cases, the solubilities of solids (often heavy and complex compounds) in pure SC fluids are very low, co-solvents are required to increase the solubility and make the application commercially tractable. The co-solvents may significantly increase the solubility of the solid in CO2 (up to two orders of magnitude) and often selectively. Their action is based on the strong interactions with the molecules of the solid. Since in this way less amount of solvent is required and less energy as well, the SCFE method becomes more attractive from the commercial point of view. For example, it has been reported that adding 5% methanol in CO2 increases the solubility of acridine in CO2 almost 5 times at 50 °C and 200 bar. The use of co-solvents has another role as well, to increase the selectivity, which is accomplished when the molecules of the co-solvent interact selectively with only one of the solids in the mixture. For example, adding only 1% methanol in CO2 increases the selectivity of CO2 for acridine in a solid mixture with anthracene almost 4 times. This makes apparently the separation much easier using SCFE. The phenomenon is due to the hydrogen bonds that are formed between the methanol and acridine molecules. The modeling of solid-gas equilibria (SGE) is usually based on the assumption that the solubility of the gas (SCF) in the solid phase can be neglected (x, =\,y2 = !)• Under these conditions, the solubility of a solid in a SCF {yi) is given by the equation:
^.-HT-n^r^
<2I)
98 The fugacity coefficient of the solid in the SC phase, (p? , is the key factor in these calculations and for the high pressures involved it can have very low values, of the order 1CT4 -1(T 5 . The fugacity coefficient is the property calculated by a thermodynamic model. The other properties of the solid (vapor pressure P2Sat, volume V2S) should be obtained from independent information. The typical tool used for SGE calculations is the cubic EoS using either the vdWlf mixing rules (Eqs 3-5) or the EoS/GK mixing rules described previously. In this section we report results with the former type of mixing rules and in the next one with the latter. Typically, two interaction parameters, ky and lip are required when modeling SGE using the vdWlf mixing rules, as can be seen in figure 13 for a typical system and for more CCVsolids in Table 6. The interaction parameters seem to be merely of empirical nature and can have either positive or negative values. Based on extensive investigations on the use of cubic EoS for SGE calculations75"81, the following conclusions can be drawn: i. ii.
Hi. iv. v.
ky _ ly should be regressed from experimental data and generally depend on temperature several investigators ' have noticed that the ky values obtained from regression of experimental SGE data do not seem to follow any general trends. Thus, development of generalized correlations using solid properties as adjustable parameters is difficult but still an open area for investigation the interaction parameters often include the possible errors in the thermophysical properties of the solid (vapor pressures and critical properties), which are not always known with sufficient accuracy A single interaction parameter is often not sufficient. The improvement using a second interaction parameter is impressive in some cases, esp. for complex solids such as cholesterol and carotene (see Table 6). For many (not very complex) solids, a single universal ky interaction parameter equal to 0.1 is sufficient for satisfactory predictions of SGE for many solids83. Good results (about 20% deviation in solid solubility) are obtained for aromatic hydrocarbons, phenols, acridine, aliphatic acids and ketones. Steroids and hydroxy-aromatic acids require higher values of ky (0.2-0.25).
5.7.2 Modeling with LCVM The LCVM model has been applied to SGE8j'87, for both binary and ternary systems (with cosolvents). Selected results are shown in Figures 14-20 from the work of Coutsikos et al. . The LCVM VLE-based group interaction parameter table is used. No re-estimation of parameters based on solid-gas equilibrium data has been performed and the linear mixing rule has been employed in the co-volume term (ly=0 in Eq.5). The calculations can be thus considered to be straight predictions. The basic conclusions drawn from the investigations are the following for the various categories of systems:
99 1. Poor results (very significant overestimation of the solubility) are obtained for the CCVn-alkanes systems. There is substantial uncertainty in the vapor pressures of the solid heavy alkanes. 2. Very good results are obtained for CCVaromatic hydrocarbons, with typical errors about 40-50%. LCVM performs here much better than the MHV2 and PSRK EoS/GE models. A typical result is shown in Figure 14 for hexamethyl-benzene. The mean deviations are 24% and 53% at the two temperatures.
Figure 13. Solid-gas calculations for CCVbenzoic acid at 35 °C with the PR EoS using the vdWlf mixing rules, Eqs.3-5, and zero, one or two interaction parameters83. Table 6. Correlation of solubilities of solids in CO2 with the modified Peng-Robinson EoS using the van der Waals one fluid mixing rules, Eqs. 3-5, and one or two interaction parameters. From Coutsikos et al. . The table is continued on the next page.
Fluoranthene Chrysene 1 -octadecanol 1 -eicosanol 1,10-decanediol Phenol 3,4-xylenol 2,5-xylenol 2-naphthol
T (°C) 35 35 45 35 45 36 35 35 35
P (bar) 89 - 247 84-251 152^38 89-237 205-307 79 - 249 82 - 262 74 - 267 103-362
ku
Ay2(a)
kij
lij
Ay2(b)
0.149 0.107 0.021 0.031 0.173 0.102 0.102 0.107 0.102
9.3% 10% 23.7% 11.2% 0.9% 22.3% 26.1% 34.5% 15.3%
0.113 0.081 0.075 0.062 0.179 -0.074 -0.065 -0.078 0.077
-0.087 -0.060 0.087 0.056 0.014 -0.396 -0.380 -0.403 -0.058
4.5% 9.5% 4.8% 9.2% 0.2% 6% 6.3% 7.3% 13.6%
100 Table 6 continuec p-hydroquinone p-quinone 1,4naphthoquinone 9,10anthraquinone Dodecanoic acid Tetradecanoic acid Palmitic acid Stearic acid Benzoic acid Salicylic acid Phenyl acetic acid o-methoxyphenylacetic acid m-methoxyphenylacetic acid p-methoxyphenylacetic acid Indole 5-methoxyindole Skatole Penicillin V Naproxen B-carotene Cholesterol(c) Cholesterolld) (a>
35 35 45
84-312 87-291 101-363
0.153 0.109 0.063
36% 12.4% 31.2%
-0.062 0.090 -0.010
-0.652 -0.036 -0.154
10.7% 9.5% 18%
35
84 - 298
0.108
11.2%
0.085
-0.057
8.6%
40
77-248
0.088
37.8%
0.062
-0.026
37.2%
40
82 - 249
0.069
21.8%
0.098
0.049
17%
45 35 35 40 35
142-361 90-237 120-280 100-350 83.5-184
0.059 0.095 0.043 0.090 0.028
40% 23% 11% 31% 19.5%
0.115 0.169 -0.036 -0.117 -0.051
0.090 0.147 -0.186 -0.533 -0.184
3.7% 7.6% 3% 5% 6.7%
35
122-337
0.187
40%
0.322
0.309
5%
35
117-339
0.085
20%
0.150
0.137
3.6%
35
121-339
0.145
56%
0.316
0.378
4.4%
35 35
54 - 208 80-184
0.141 0.199
33.2% 7.6%
0.024 0.212
-0.277 0.032
21% 6%
35 41 40 40
75-190 81-280 90-193 120-276
35
124-279
0.160 0.195 0.227 0.430 0.144 0.235
22% 37.9% 16.8% 106% 74.6% 87.6%
0.068 0.233 0.310 0.566 0.350 0.467
-0.218 0.084 0.206 0.349 0.363 0.447
9.1% 35% 4.6% 10% 7% 7%
correlation with one interaction parameter correlation with two interaction parameters 'c) critical constants from Singh et al85 critical constants from Kosal el al.
101
Figure 14. Solid-gas predictions with LCVM for CCh/hexamethyl benzene at two temperatures83. The points are the experimental data and the lines are the LCVM predictions. 3. Very good results are obtained for CCValiphatic carboxylic acids, as can be seen for a typical system in figure 16. For example, the deviation is only 25% in the case of the solubility of lauric and stearic acids and 40% in the case of palmitic acid. As a general rule, the LCVM model yields slightly lower solubilities than the experimental ones in these cases. From the results shown in figure 15 (for benzoic and salicylic acids), we conclude that the LCVM model yields either an underestimation or an overestimation of the solubilities in this case. The different behaviors possibly depend on whether an aromatic hydroxyl-group is contained in the molecule. When such a group is missing, e.g. benzoic acid, the model underestimates the data. Similarly, in the case of phenyl acetic acid, which does not contain an aromatic hydroxyl-group, LCVM yields an underestimation of approximately one order-of-magnitude. The percentage deviation in the solubility is 80% at 35 °C and 86% at 45 °C. On the other hand, when an aromatic hydroxyl-group is present, e.g. in salicylic (o-hydroxy benzoic acid), the model overestimates the solubility data. 4. In the case of CCValcohols, the LCVM model yields an underestimation in the solubilities of almost one order-of-magnitude (aliphatic alcohols) or overestimation (aromatic alcohols, like phenol, figure 20). For example, the deviation in the solubility is (at 35 °C) 83% in the case of 2,5 xylenol and 100% in the case of 3,4 xylenol. Rather poor results are obtained for CCVcomplex solids such as naproxen or steroids, where significant overestimation of the solubilities by often two or more orders-of-magnitude is observed. For such complex solids, of course, the thermophysical properties esp. the vapor pressures and the critical properties may be subject to significant error.
102
Figure 15. Solid-gas predictions with LCVM for benzoic acid and salicylic acid in supercritical carbon dioxide at 35°C and 40°C, respectively^.
Figure 16. Solid-gas predictions with LCVM for CCVstearic acid at two temperatures. The points are the experimental data and the lines are the LCVM predictions.
103 5. In the case of ternary systems, i.e. CCVsolid/co-solvents, different conclusions are extracted depending on the complexity of the molecules involved. i. For non-polar CCVsolid/co-solvent systems e.g. hexamethylbenzene with alkanes and some polar co-solvents, such as methanol and acetone, the predictions are at least as satisfactory as the predictions for the binary systems. Typical results can be seen in figures 17 and 18. Actually, in many cases e.g. CCVphenanthrene the predictions of LCVM using co-solvents are even better compared to the predictions in pure CO2. Generally, the LCVM model can, in these cases, satisfactorily predict both the co-solvent effect (i.e. the increase of solubility due to the presence of co-solvent), the enhancement and identify the best co-solvent. In such cases, the model can be used for co-solvent screening, i.e. for finding the most effective co-solvent for a specific solid aromatic compound. ii. In the case of polar solids e.g. 2-naphthoI, benzoic acid and hydroquinone, poor predictions are obtained with LCVM in pure CO2. It is quite remarkable that the predictions for the ternary systems are considerably improved over those of the binaries and in the case of CC>2/2-naphthol/methanol the predictions are satisfactory (figure 19). The LCVM model cannot predict the strong co-solvent effect in the case of polar solids. However, a rough distinction between the various co-solvents can be made. Thus, LCVM correctly predicts that methanol is better co-solvent than acetone and n-octane for CCh/benzoic acid. 111 In the case of strongly polar and hydrogen-bonding solids e.g. naproxen and cholesterol the capabilities of LCVM are rather limited. The predicted enhancement is, in many cases, half of the experimental value83 and furthermore the model cannot always screen between cosolvents (not even qualitatively). For hydrogen bonding solids and/or co-solvents, the qualitative method by Walsh et a/.88 is more adequate for co-solvent screening, as shown by Coutsikos et a\P
Figure 17. Solid-gas predictions with LCVM for anthracene/CC>2 with co-solvent methanol
Figure 18.Solid-gas predictions with LCVM for phenanthrene/CO2 with non-polar co-solvents83
104
Figure 19. Solid/gas predictions with LCVM for 2-naphthol for the ternary system CO2/2naphthol/methanol at 308 K.
Effect of using VLE-based parameters for making SGE predictions This is an issue of special interest, since no re-estimation of parameters has been performed. The example of phenol/CC^ system (figure 20) is particularly illustrative. At 60°C, gas-liquid equilibria (GLE) exist while at the lower temperature (36°C), the equilibria is of the solid-gas type. Excellent results are obtained in the case of GLE (19%) used in parameter estimation, while poor prediction is obtained at 36°C (187%). The poor results obtained with LCVM for this case cannot be attributed to factors such as uncertainty in critical properties and solid vapor pressure. Both the critical properties and the vapor pressure of solid phenol are available from reliable experiments. These results reveal that, in some cases, the use of VLE-based parameters for SGE-predictions may be inappropriate.
5.8 CONCLUDING REMARKS - FUTURE CHALLENGES Successful prediction of phase behaviour of complex mixtures including supercritical components with relatively simple cubic equations of state has become more and more feasible with the development, in the last two decades, of new mixing rules that combine equations of state with excess Gibbs free energy models, resulting to the so-called EoS/GE models. The design engineer, however, has to realize the various potential difficulties and shortcomings of these models before making the final selection of the model to be employed in a particular application. Following are some valuable guidelines:
105
Figure 20. Solid-gas and gas-liquid predictions with LCVM for CCVphenol at 36 °C (SGE) and 60 °C (GLE). The points are the experimental data and the lines are the LCVM predictions. From Coutsikos et a/.83.
1. Cubic Equations of State using the vdWlf mixing rules can satisfactorily correlate high pressure VLE for gas/hydrocarbons and for similar systems, using a single interaction parameter, ky, the correction to the geometric-mean cross-energy parameter. The arithmetic-mean is, for practical purposes, the best choice for the cross co-volume parameter, and using ly improved results can be obtained for sizeasymmetric systems. However, cubic EoS using the vdWlf mixing rules cannot often describe phase equilibria, VLE and LLE, for highly complex mixtures, even when two interaction parameters are employed. 2. The various EoS/Gh mixing rules are general, in the sense that they can be, in principle, combined with any (cubic) EoS and activity coefficient model. However, when interaction parameters have been estimated e.g. for gas-containing systems, the specific model combinations (EoS and activity coefficient model) proposed by the model's developers have to be used. This is the case especially for MHV2, LCVM and PSRK that have parameters for gas-containing systems.
106 3. The predictive scheme of the WS mixing rule/model cannot be used for systems containing components that are non-condensable at the temperature of interest, i.e. it can not be used in gas containing systems. 4. All EoS/GE mixing rules/models give comparable good results for both non-polar and polar systems containing components that do not significantly differ in size (symmetric systems). 5. For asymmetric systems the LCVM model is recommended since it gives reliable predictions even at very high pressures. 6. When the PSRK model is applied to asymmetric systems containing alkanes, the modification of Li et al. is recommended over the original PSRK. However, it is not recommended for non-alkane large molecular weight compounds. Ahlers and Gmehling91 have recently developed, using the conclusions of the analysis by Kontogeorgis and Vlamos34, an improved PSRK model for asymmetric systems, which does not suffer from the deficiencies of Li et al. approach. 7. For ternary and multicomponent mixtures, the LCVM and PSRK models are recommended since they give predictions that are acceptable at least for preliminary design and simulation purposes. Overall, the LCVM model gives the best predictions. 8. In the case of SGE calculations, cubic EoS with the vdWlf mixing rules are typically useful for correlating and extrapolating experimental data but not for prediction. One parameter is sufficient in some cases, but a second parameter is often needed (/,y) for correcting many deficiencies e.g. of the mixing rules, the pure solid critical properties or the solid vapor pressure. 9. The LCVM model is shown to yield quantitatively satisfactory predictions for binary solid-gas systems involving aromatic hydrocarbons, aliphatic acids and some alcohols. Poor results are obtained for complex solids e.g. naproxen and cholesterol. A number of factors, which may affect the calculations, have been illustrated. Results have been also presented for some ternary systems, with co-solvents. Prediction of the cosolvent effect using LCVM model should be restricted to relatively non-polar systems although some qualitative guidelines for polar ones can be also made. 10. For choosing the best co-solvent for hydrogen-bonding systems, the ideas by Walsh et al.88, which are supported by recent experimental data, are recommended. These ideas can be summarized in the following statement: "significant solubility enhancement in solid-SCF systems can be achieved only if solvation (cross-association) between the co-solvent and the solid takes place". The solvation can be either due to hydrogen bonding or due to charge-transfer complexes. Both types can be classified as Lewis acidLewis base (LA-LB) interactions. Thus, if the solid is a LA, a suitable co-solvent should be a strong LB, and vice versa. Use of Kamlet et alw solvatochromic parameters a, and j3 is possibly the best way to assess the acidity (LA) and basicity (LB). Coutsikos et al.1*3 showed that the use of Walsh et al. ideas can explain the significant enhancement in the solubility of hydroquinone in CO2 using TBP (tributylphosphate)90 and that of propanols for naproxen's (a pharmaceutical) solubility in CO2.
107 LIST OF SYMBOLS A a b BPP c], C2, C3 c G k K NDP n m P R T V x y Greek a an,,, y' s AP% Ay ). a co
= Helmholtz free energy (J moi"1) = attractive term parameter of the EoS (Pa m6 mol"2) = co-volume parameter of the EoS (m3 mol"1) = bubble point pressure = Mathias-Copeman pure-compound temperature-dependence parameters = parameter in the Patel-Teja equation of state = Gibbs free energy (J mol"1) = binary adjustable interaction parameter in the WS model, Eq. 20 or in equations of state cross-energy term, Eq. 5 = equilibrium ratio (=y/x) = number of data points = exponent of the attractive part of the Mie potential, appearing in Eqs.9-11 = exponent of the repulsive part of the Mie potential, Eq. 10 = pressure (Pa) = gas constant (J mol" K" ) = temperature (K) = molar volume (mJ mol"1) = mole fraction in liquid phase = mole fraction in vapour or supercritical phase
Symbols = a / (bRT) = limiting a value = infinite dilution activity coefficient = molecular energy parameter, Eq. 10 = average absolute per cent error in bubble point pressure = average deviation in vapor phase mole fraction = constant in LCVM model equal to 0.36 = molecular diameter, Eq. 10 = acentric factor
Superscripts E ref
= excess property = reference state
Subscripts i.j b c r
= compound in a mixture = boiling point = critical property = reduced property
108 LIST OF ABBREVIATIONS AC CPA EoS Exp GLE LCVM LLE MHV2 PR PRS V PSRK SAFT SC SCF SCFE SGE SRK vdW vdWl f VLE WS
activity coefficient Cubic-Plus-Association equation of state experimental (data/values) Gas-Liquid Equilibria Linear Combination Vidal and Michelsen mixing rules Liquid-Liquid Equilibria Modified Huron-Vidal mixing rule (2n order) Peng-Robinson equation of state PR Stryj ek Vera equation of state Predictive SRK equation of state/mixing rule Statistical Associating Fluid Theory Supercritical Supercritical Fluid Supercritical Fluid Extraction Solid-Gas Equilibria Soave Redlich-Kwong equation of state van der Waals van der Waals one-fluid mixing rules Vapor-Liquid Equilibria Wong-Sandier mixing rule
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110 47. E. Voutsas, M. Lampadariou, K. Magoulas, D. Tassios, Fluid Phase Equilibria, 198 (2002) 81. 48. M. Gomez-Nieto, G. Thodos, Ind. Eng. Chem. Fundam., 17 (1978) 45. 49. J. Li, K. Fischer, J. Gmehling, Fluid Phase Equilibria, 143 (1998) 71. 50. E.C. Voutsas, D.P. Tassios, Ind. Eng. Chem. Res., 35 (1996) 1438. 51. K. Kojima, S. Zhang, T. Hiaki, Fluid Phase Equilibria, 131 (1997) 145. 52. K. Polyzou, Vapor-Liquid Equilibrium in Systems Containing Hydrogen. Diploma Thesis, National Technical University of Athens, Greece, 1997. (in Greek) 53. C.P. Angelos, S.V. Bhagwat, M.A. Matthews, Fluid Phase Equilibria., 72 (1992) 189. 54. E. Hala, I. Wichterle, J. Polak, T. Boublik, Vapour-Liquid Equilibrium Data at Normal Pressures. Pergamon Press: Oxford, 1968. 55. S.W. Campbell, R.A. Wilsak, G.J. Thodos, Chem. Eng. Data, 31 (1986) 424. 56. F. Barr-David, B.F.J. Dodge, Chem. Eng. Data, 4 (1959) 107. 57. K.L. Butcher, M.S.J. Medani, Appl. Chem., 18 (1968) 100. 58. J. Griswold, S.Y. Wong, Chem. Eng. Progr. Symp. Ser., 48 (1952) 18. 59. D.-S. Jan, C.-H. Mai, F.-N. Tsai, J. Chem. Eng. Data, 39 (1994) 384. 60. K. Suzuki, H. Sue, M. Itou, R.L. Smith, H. Inomata, K. Arai, S.J. Saito, Chem. Eng. Data, 35(1990)67. 61. G. Kling, G.J. Maurer, J. Chem. Thermodynamics, 23 (1991) 531. 62. E. Brunner, W. Hultenschmidt, G.J. Schlichtharle, J. Chem. Thermodynamics, 19 (1987) 273. 63. A. Vetere, O. Sguera, A. Paggini, D. Sanfilippo, Comput. Chem. Eng., 12 (1988) 491. 64. E. Brunner, Ber. Bunsenges. Phys. Chem., 83 (1979) 715. 65. H. Knapp, R. Doring, L. Oellrich, U. Plocker, J.M. Prausnitz, Vapor-Liquid Equilibria for Mixtures of Low Boiling Substances. DECHEMA Chemistry Data Series, Vol. VI. DECHEMA: Frankfurt, 1982. 66. J.F. Parcher, P.H. Weiner, C.L Hussey, T.N. Westlake, J. Chem. Eng. Data, 20 (1975) 145. 67. V.V. deLeeuw Th. de Loos, H.A. Kooijman, J. de Swaan Arons, Fluid Phase Equilibria, 73(1992)285. 68. Y. Iwai, T. Morotomi, K. Sakamoto, Y. Koga, Y.J. Arai, J. Chem. Eng. Data, 41 (1996) 951. 69. R.J. Lee, K.C. Chao, Fluid Phase Equilibria, 43 (1988) 329. 70. H.-M. Lin, H.M. Sebastian, K.-C. Chao, Fluid Phase Equilibria, 7 (1981) 87. 71. G. Brunner, J. Teich, R. Dohrn, Fluid Phase Equilibria, 100 (1994) 253. 72. J. Daridon, P. Xans, F. Montel, Fluid Phase Equilibria, 117 (1996) 241. 73. J.J. Carroll, F.-J. Jou, A.E. Mather, Fluid Phase Equilibria, 140 (1997) 157. 74. W. Leitner, Nature, 405 (2000) 130. 75. M.E. Mackay and M.E. Paulaitis, Ind. Eng. Chem. Fundam., 18 (1979) 149. 76. R.T. Kurnik, S.J. Holla and R.C. Reid, R.C., J. Chem. Eng. Data, 26 (1981) 47. 77. R.T. Kurnik and R.C. Reid, Fluid Phase Equilibria, 8 (1982) 93. 78. W.J. Schmitt and R.C. Reid, J. Chem. Eng. Data, 31 (1986) 204. 79. A.C. Caballero, L.N. Hernandez and L. Estevez, J. Super. Fluids, 5 (1992) 283. 80. Ph. Coutsikos, K. Magoulas and D.P. Tassios, J. Chem. Eng. Data, 40 (1995) 953. 81. Ph. Coutsikos. K. Magoulas, D.P. Tassios, A. Cortesi and I. Kikic. J. Super. Fluids, 11 (1997) 21. 82. J.M. Wong, R.S. Pearlman and K.P. Johnston, J. Phys. Chem., 89 (1985) 2671.
Ill 83. Ph. Coutsikos, K. Magoulas and G.M. Kontogeorgis, J. Supercrit. Fluids, 25 (2003) 197. 84. K. Magoulas and D.P. Tassios, Fluid Phase Equilibria, 56 (1990) 119-140. 85. H. Singh, S.L.J. Yun, S.J. Macnaughton, D.L. Tomasko and N.R. Foster, Ind. Eng. Chem. Res., 32(1993)2841. 86. E. Kosal, C.H. Lee and G.D. Holder, J. Supercrit. Fluids, 5 (1992) 169. 87. N. Spiliotis, K. Magoulas and D.P. Tassios, Fluid Phase Equilibria, 102 (1994) 121. 88. J.M. Walsh, G.D. Ikonomou, G.D and M.D. Donohue, Fluid Phase Equilibria, 33 (1987) 295. 89. M. J. Kamlet, J-L.M. Abboud, M.H. Abraham and R.W. Taft, J. Org. Chem., 48 (1983) 2877. 90. R.M. Lemert and K.P. Johnston, Ind. Eng. Chem. Res., 30 (1991) 1222. 91. J. Ahlers and J. Gmehling, Ind. Eng. Chem. Res., 41 (2002) 3489.
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Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B.V. All rights reserved.
113
Chapter 6: Association Models - The CPA Equation of State Georgios M. Kontogeorgis 6.1 INTRODUCTION Associating systems are those, which contain compounds capable of hydrogen bonding e.g. alcohols, water, amines, acids, etc. Phase equilibria of complex associating systems is important in many practical cases. For example, in several hydrocabon and chemical processes as well as in order to meet environmental and low-emissions legislations, systems involving a highly polar compound, a non-polar one and a co-solvent, with both polar and non-polar functional groups, are typically present, e.g. systems involving water, alcohols or glycols and hydrocarbons. The role of the alcohol or glycol is that of a co-solvent: they improve significantly the mutual solubility of water and hydrocarbon in the hydrocarbon-rich and water-rich phases, respectively. Without such co-solvents, these solubilities are very low. Examples of cases where phase equilibria of associating systems is important: •
•
• • •
•
in the oil industry: formation of gas hydrates - calculation of the amount of hydrate inhibitors partitioning of alcohols and other chemicals between water and oil use of alcohols as additives in gasolines for 'Octane Number' improver - heterogeneous azeotropic separation in alcohol manufacturing glycol regeneration units reid vapor pressure of gasoline in the presence of additives1 for environmental calculations: octanol-water and other partition coefficients modeling of waste streams in biotechnology and biochemical engineering2'3 in the food product and process design4 in the chemical industry3: novel separation methods for Pharmaceuticals (e.g. supercritical fluid extraction) recovery of alcohols from aqueous solutions using high pressure propane (or other gases) [propane is often selected as solvent since no azeotropes are expected in mixtures of ethanol with propane and due to its high pressure extracting capability] in the polymer industry: polymer blends phase equilibria in coatings of mixed water-organic solvents
114 Typical conventional models like cubic EoS and activity coefficient models (NRTL, UNIFAC - see chapter 4) are often insufficient for such systems, especially for multicomponent and multiphase equilibria, for predictions from binary information alone, for liquid-liquid equilibria, such as prediction of liquid-liquid equilibria for ternary water alcohol- hydrocarbons. Correlation and/or prediction of binary phase equilibria of some complex mixtures are often possible with simple models, such as cubic EoS (see Figure 1). However, prediction of multicomponent phase equilibria from binary data and/or the presence of multiphase equilibria, e.g., vapor-liquid-liquid equilibria are particularly difficult to model with simple cubic equations of state and activity coefficient models.
Figure 1. Correlation of vapor-liquid equilibria for water-methanol at 1.013 bar using the SRK EoS and the van der Waals one fluid mixing rules (ki2=-0.0752) Group-contribution UNIFAC-type models and simple cubic EoS with the classical (van der Waals one fluid) mixing rules predict or correlate VLE for some binary systems including associating compounds satisfactorily (Figure 1), less so however for LLE (Figure 2 for a glycol-heptane system), especially when water is present. The limitations of conventional models become further apparent when correlation of aleohol/alkane and water/hydrocarbon VLE and LLE (or similar mixtures) are considered6'7. Over the last years, a variety of models to account for the effects of hydrogen bonding in solutions have been proposed. These are called 'association models'. The association models are designed specifically to describe phase equilibria of mixtures containing highly polar and/or strongly associating compounds such as water, alcohols, acids, amines, phenols, and others, which have the capability of forming hydrogen bonds. The hydrogen bonding formation in the same molecule is called self-association (e.g. pure water or pure methanol), while forming of hydrogen bonding complexes between two different molecules is called cross-association (e.g. water-methanol). Several reviews on association models are
115 available910. No attempt to review these theories will be made here. All theories lie in three main directions: i. ii. iii.
chemical theories lattice-fluid theories perturbation theories
Figure 2. LLE for ethylene glycol (1) - n-heptane (2) system with the Mathias Copeman (MC)—SRK with a kjj = 0.094: x, x", experimental; o, x'2, experimental; solid line, x,", MC-SRK; dashed line, x'2, MC-SRK. / = Glycol-rich phase and // = Hydrocarbon-rich phase [Derawi el a/.8]. All approaches include two (or more) contributions: a so-called physical term (phys) accounting for the deviations from ideality due to physical forces and an association term (assoc) accounting for the effect of hydrogen bonding (and other "chemical" forces). In terms of the compressibility factor, the two contributions are often (but not always) separate: z
= zphys +zassoc
^
The physical term can be either a non-cubic or a cubic equation of state. The approach used for describing the hydrogen bonding is different in the three categories of association models. Chemical theories are based on the formation of new species and the extent of association is determined by the number of oligomers formed, as a function of density, temperature and composition. Lattice (often called quasi-chemical) theories account for the number of bonds formed between segments of different molecules that occupy adjacent sites in the lattice. The number of bonds determines the extent of association. Finally, in the perturbation theories, the total energy of hydrogen bonding is calculated from statistical mechanics and the important parameter for hydrogen bonding is in this case the number of bonding sites per molecule.
116 All these theories have been developed especially in the period 1989-1999 and the engineer has now a number of alternatives to select in terms of 'closed-form' thermodynamic models (EoS), where the hydrogen bonding effect is accounted for as a separate term. An important conclusion, independently reached by a number of investigators9'11'12 is that all three theories, despite their different physical origin and formulation, yield essentially identical mathematical expressions for the hydrogen bonding contribution. This helps in e.g. the interpretation of the parameters of the popular perturbation theories (SAFT, CPA) with the help of the more familiar chemical theory, where the parameters have a well-established physical meaning (enthalpy and entropy of hydrogen bonding, equilibrium constant). In this chapter, we limit our discussion to the CPA (Chemical-Plus-Association) equation of state, which represents one of the widely applied association models. It combines a cubic equation of state (for the physical term) with the hydrogen bonding term employed in Statistical Associating Fluid Theory, SAFT13'14.
6.2 THE CPA EQUATION OF STATE - MODEL DESCRIPTION The development of the CPA equation of state started as an industrial project, with the participation of industrial and academic partners from Europe. The purpose of 'the CPA project' was to develop an equation of state suitable for complex multicomponent multiphase equilibria for systems of the type: 'water - alcohols - hydrocarbons'. Later the equation of state was extended to cover other associating compounds e.g. glycols and sulfolane. The model can be in principle extended to other types of associating molecules. It was at an early stage realised that to better achieve the above goal without sacrificing the successful performance of cubic equations of state for hydrocarbon mixtures: •
• •
the equation of state should be predictive for multicomponent systems. Prediction in this sense implies that only binary parameters should be employed, determined from binary data the cubic SRK EoS should be used as the base-model for describing the physical interactions the association term of SAFT can be adopted due to its applicability to different types of hydrogen bonding compounds.
The CPA equation of state has the following form for mixtures: Z = ZSRK + Zmx"c V-b
RT(V + b) '(_[
I
j
A
\\dxA>~
117 The association term is the one employed in SAFT (with a simplified version of the radial distribution function though). For non-associating fluids, e.g. hydrocarbons, the association term disappears and the equation reduces to SRK. In the association term, the key parameter is X A l , the mole fraction of the molecule i not bonded at site A, i.e. the monomer mole fraction. XAl is related to the association strength between two association sites belonging in two different molecules (A ' ' ) as:
V1
( XAi=
I+ P I I X ^ W
6
' ]
JBJ
1
(3)
A^=g(p)exi^j-lb/'ei The expression for zabS0C presented in equation 2 is the one originally proposed for SAFT (and CPA) by Chapman et al.u and Huang and Radosz14. However, in recent works, Yakoumis el a/.13 and Michelsen and Hendriks12 proposed a surprisingly simple general expression for the association term of CPA (which is identical to the one shown in equation 2):
Z—
=-|l+P^)lIx(-(l-^)
(4)
where Xj is the (analytical) mole fraction of component i. When comparing equations 2 and 4, it becomes apparent that using the simplified expression equation 4, there is no need for calculating the derivatives of the mole fraction of non-bonded molecules with respect to the density. This issue simplifies the calculations. For a pure associating fluid, equations 2 and 3 are simplified: 7assoc
v
f fl
l d)
)
-i A
X Jl
B
( 5 )
AB
+ pYjX A )
XA is now the fraction of a single associating molecule non-bonded at site A (=monomer fraction). The key quantity of the association term of CPA is, as in SAFT, the association strength between two sites A and B:
118 A A B = g ( p ) e x p — -lpfa
(6)
where g(p) is the so-called 'radial distribution function'. Finally, the energy parameter of the EoS is given by a Soave-type temperature dependency, while b is, in agreement with most published EoS, temperature independent:
a = ao(l + Cl (l-Vt7)) 2
(7)
The parameter ci can be, for inert compounds e.g. hydrocarbons, obtained via the acentric factor as in classical cubic equations of state. For associating substances, it has to be estimated together with the other parameters from pure component (vapour pressure and liquid density) data. 6.2.1 The radial distribution function: from the original to the simplified CPA SAFT and CPA in the publications by Kontogeorgis et al.16 and all subsequent publications up to 1999 employ the hard-sphere radial distribution function (RDF):
m
=W^f
y
<8)
^
SAFT uses also the hard-sphere radial distribution function. In the case of SAFT this is rigorously correct since the hard-sphere repulsive term is used, while for CPA only approximately correct, since CPA employs the van der Waals repulsive term (of SRK EoS). Kontogeorgis et al}6 suggested the following simplified hard-sphere radial distribution function first proposed by Elliott et al.11:
9(p)=
Y^b-y
y =
i
(9)
As shown in Figure 3, the two radial distribution functions yield very similar values, but the simplified hard-sphere radial distribution function offers some computational advantages. Thus, the simplified RDF is recommended. CPA with Eq. 9 is denoted as sCPA (simplified CPA). 6.3 THE CPA EQUATION OF STATE - PARAMETER ESTIMATION The first step in the development of the CPA EoS is to define (assign) the number and type of association sites for a specific associating substance. This is by no means a trivial issue and a carefully-designed parameter estimation procedure is required based on vapor pressures and liquid densities, taking also into account the physical significance of the parameters.
119 Hydrocarbons and other inert compounds, which do not self-associate, have zero association sites.
Figure 3. Comparison of the hard sphere RDF (employed in SAFT and CPA), Eq. 8, with the simplified radial distribution function (employed in sCPA), Eq. 9. [Kontogeorgis et a/.16] From the investigations carried so far16'15'18'19, it has been verified that: - alcohols and phenols are best modelled as two-site molecules (with equal sites) (type 2B14) - water is best modelled as a four site molecule (type 4C14) The 2B model for alcohols and the 4C one for water are in agreement with the generally accepted physical picture for these two types of associating compounds, i.e. that alcohols form linear oligomers and water forms three-dimensional structures. The notation "2B" and "4C" stems from Huang and Radosz14. 6.3.1 Pure Fluids CPA has in total five pure compound parameters for associating substances, three parameters in the physical SRK term (as in all cubic EoS) and two parameters in the association term, the association energy and the association volume. One parameter, the co-volume b, is included in both the physical and the association term, as can be seen by equations 2 and 3.
120 As observed from equation 2, the exact functional form of the association term of CPA depends on the choice of association scheme. Acids are expected to be modeled as one-site molecules (1 in Huang-Radosz notation14), alcohols as two-site (2B) and water as 3- or 4- site molecules (3B, 4C14). These schemes are in agreement with the fact that acids form dimers, alcohols linear oligomers and water three-dimensional structures. However, for other associating molecules (alkanolamines, glycols, etc.) the association schemes need to be investigated together with the parameter estimation. Other association schemes, which are not covered expcilitly in the Huang and Radosz study14, may be important for specific associating fluids e.g. inclusion of branches, closed ring structures which are known to be present for hydrogen fluoride, etc. Evidently, the need for determining the correct association scheme gives at the same time flexibility to the model and some problems, since this can be a rather time consuming process. The pure parameters are estimated from thermodynamic data for pure fluids. Vapor pressures and liquid densities are typically employed for this purpose. Multiple sets of parameters can be obtained which yield acceptable errors in both properties (within experimental uncertainty). Interestingly enough, the co-volume parameter seems to be approximately constant among the various sets, independent of the starting values of the regression18. Moreover, b seems to be related to the van der Waals volume (Vw) and, moreover, in a universal way for many compounds. For example, as shown by Zeuthen41 for alcohols, glycols and acids the b/Vw ratio of CPA is approximately constant having a value between 1.35 and 1.47. Such considerations may provide a way of reducing the number of pure parameters to be adjusted. There seems that the b-value regressed from pure data for the CPA EoS 'forces' somehow the repulsive vdW term to assume values close to those of the Carnahan-Starling expression, as shown by Kontogeorgis et al.n'. The choice of physically correct set of pure parameters is crucial as emphasized many years ago by Anderko20. Whether the regressed association parameters are physically correct may be determined upon comparing them with values for the heats and entropies of hydrogen bonding, which can in some cases be obtained from independent (spectroscopic) measurements. Table 1 gives a list of the sCPA parameters published so far and table 2 presents some values of CPA parameters compared to enthalpies of hydrogen bonding and association parameters from other models such as the SAFT, Anderko and APACT EoS. Additional sCPA parameters for other glycols are given by Derawi et a/.8. In several cases, especially for associating compounds for which it is unclear which association schemes can be used, consideration of some binary VLE/LLE data for the associating compound (under investigation) with (inert) hydrocarbons is a useful way for establishing the best set of pure parameters. This in no way implies, however, that mixture data enter directly into the regression for pure fluid parameters. Hydrocarbons and related compounds are treated as inert compounds, i.e. the association term is zero. In this case the pure parameters are three: the critical temperature, critical pressure and the acentric factor. Alternatively, the three pure fluid parameters (a0, b and cj in equation 7) can be regressed from vapor pressure and liquid densities sacrificing, thus, the description of the critical point but improving the pure compound description. When this is done, slightly better VLE19 and typically better LLE correlation is achieved32.
121 Table 1. sCPA pure-compound parameters for water, methanol, ethylene glycol (MEG), diethylene glycol (DEG) and triethylene glycol (TEG) Compound
Water Methanol MEG DEG TEG
Reference
Kontogeorgis elaL[b Kontogeorgis etal.16 Derawi et al* Derawi et al* Derawi et al.
b ao (dm3 mol"1) (bar dm6 mol"2)
P
Cl
6
0.0692
0.014515
1.2277
0.67359
(bar dm3 moF1) 166.55
0.030978
4.0531
0.43102
245.91
0.0161
0.0514 0.0921 0.1321
10.819 26.408 39.126
0.6744 0.7991 1.1692
197.52 196.84 143.37
0.0141 0.0064 0.0188
Table 2. Experimental values of enthalpies of hydrogen bonding for some alcohols and water and the corresponding association parameter (association energy) of several association models. All values are in K. Anderko2
SAFT'
SAFT2
SAFT1
3505
1809 3 site 1368 4site 2714 2759
826
1635
Compound
Expt
CPA*
APACT1
APACT2
water
1813
2003
2618
2418
methanol ethanol
2630 25263007
2957
2770 3022
2770 3021
2360 2620
3127 3127
1 propanol 2propanol lbutanol 1 pentanol lhexanol phenol
_"_ -"-"-"-
3022 3022 3022 3022 3022
3021 3021
2712 2875
3127 3127 3127 3127 3127
M
2343
Anderko1
2619 2670 2605 2587 2556 1894
2710 2803 2512 2459
Expt: Experimental values. References for the experimental values in column 2 are from : water: Koh et al}1; methanol: Nath and Bender 2 ; ethanol: Pimentel and McClellan23. Many authors accept that the 'experimental' values for the heats of hydrogen bonding of alcohols heavier than ethanol are close to that of ethanol, i.e. they do not depend considerably on chain length; phenol: Ksiazczak and Moorthi24. APACT1: Vimalchand et al.23; APACT2: Economou and Donohue26 Anderko1: Anderko27; Anderko2: Anderko28 SAFT1: original SAFT by Huang and Radosz14 and Economou and Tsonopoulos29 SAFT2: simplified SAFT by Fu and Sandier10 SAFT3: re-estimation of water parameters of original SAFT by Voutsas et al?1 *CPA values for water: sCPA: 2003 K16, CPA: 1794 K31
122 An independent test of pure parameters can be made by investigating the performance of the EoS to properties other than vapor pressures and liquid densities, which are employed in the parameter estimation. As shown by Kontogeorgis et a/.18, CPA yields very good prediction of second Virial coefficients for alcohols down to veiy low temperatures. At such extreme conditions where the second Virial coefficients achieve very low values, cubic EoS such as SRK fail completely. The prediction of second Virial coefficients through EoS is a very strict test. It is accepted that cubic EoS with parameters fitted to the critical point and the vapor pressures cannot simultaneously predict the second Virial coefficients. CPA seems to overcome this problem. However, the prediction of second Virial coefficients has not been systematically investigated with the CPA EoS, since such coefficients are not very important for practical applications. 6.3.2 Mixtures: Mixing and Combining Rules The conventional mixing rules are employed in the SRK-part for the energy and co-volume parameters. The geometric mean rule is used for the cross energy parameter and the interaction parameter ky is, in the applications reported so far, the only adjustable binary parameter of CPA, which is estimated from experimental binary phase equilibrium data:
(10)
b = JjXib, i
No mixing rules are need in the association term, since, as can be seen from equations 2-4, the association theory of Wertheim provides a theoretical extension to mixtures. However, for extending the CPA EoS to cross-associating systems e.g. water-alcohols, alcohol-alcohol, etc., combining rules are needed for estimating the cross association parameters, the energy and volume ones (s, P). An alternative simpler way, which is often adopted, is the so-called Elliott rule. Suresh and Elliott7 suggested a combining rule for the cross-association strength, which is actually the property employed in mixtures: AAB;=A/AA.B,A^B;
( n )
Elliott mentioned that equation 11 could only be applied to mixtures of alcohols or other associating fluids where compounds have the same or similar associating behavior (scheme). However, as will be shown in the next section, this mixing rule has been also employed with success for water-alcohol systems. Still, it has been verified that the results become progressively worse as the size of alcohol increases and for systems containing water and heavy glycols. Thus, results are satisfactory for systems with water and methanol, ethanol and propanol, but already for water/butanol the results are not satisfactory. As shown by Voutsas
123 et a/.33, for water/n-butanol and similar water/heavy alcohols, correlation of liquid-liquid equilibria for this system with a single ky using the Elliott rule is not successful. It is not possible to correlate at the same time the water-rich and the butanol-rich areas of the binodal curve. Similar results have been obtained for other water/heavy alcohol systems and for both vapor-liquid and liquid-liquid equilibria. We will show later that improved combining rules for the association parameters for such systems have been developped. It has been reported16 that the use of Elliott rule yields increased computational speed.
6.4 THE CPA EQUATION OF STATE - CURRENT STATUS/RESULTS At its first stages of development, CPA has been successfully applied to: pure alcohols pure water1''31 binary VLE and LLE for alcohols/hydrocarbons32'19 binary LLE for water/hydrocarbons15'31 VLE and LLE for cross-associating water/alcohols water/butanol) and water - alcohol - hydrocarbons1632 Gas-liquid equilibria for gas/alcohols and water16
(water/methanol
and
The CPA model is under continuous development. Recent results include binary and multicomponent systems (VLE, LLE, Henry's law constants) containing aromatic hydrocarbons and glycols, carboxylic acids, ketones, polymers and environmental calculations34"42. 6.4.1 Discussion of the Results Water / alkanes LLE CPA yields, for water/alkane systems, excellent correlation of the water solubility in the hydrocarbon-rich phase and very good results also for the very low hydrocarbon solubility in the water-rich phase, as shown in Figure 4. As shown in this figure and in the recent work of Voutsas et al. 3I , CPA yields much better results than both the SRK and the SAFT equations of state, especially for the hydrocarbon solubility. Nevertheless, none of the models is capable of describing the minimum, which has been experimentally confirmed for the hydrocarbon solubility in the water-rich phase at ambient temperatures. Although, for this particular case, CPA seems to perform better than SAFT, for other systems such as methanol/pentane, both CPA and SAFT perform equally well. The simplicity of the physical term of CPA makes it an attractive alternative over the SAFT EoS. VLE for mixtures with associating substances Figures 5 and 6 show two typical results for water/methane and water/methanol. Yakoumis et al}9 have showed that CPA performs considerably better than SRK in describing VLE for
124 alcohols/alkanes with a single interaction parameter. For such highly non-ideal systems, conventional models with the classical mixing rules typically fail; occasionally SRK yields a false phase split for alcohol/alkanes. The interaction parameter ky of CPA has, moreover, much lower values than in SRK and is a weak function of temperature.
Figure 4. Correlation of the water/n-hexane liquid-liquid equilibria with the sCPA EoS using either a single binary parameter or a temperature dependent interaction parameter. [Kontogeorgis et a/.16] The methanol/propane system This system is of special interest. It has been shown32 that CPA correlates very well the phase diagram of methanol/propane and predicts the azeotrope at low methanol concentrations. For the existence of this azeotrope at propane splitters there are no experimental data but there is industrial evidence43 [Meijer H., 1999; Lauerman, 1999 - personal communication]. SAFT predicts also the azeotrope at approximately the same concentration43: 0.6% methanol. Predicting this azeotrope is important for the design of propane/propene separation (C3 splitters). Comparison with other models Table 3 presents a comparison between various association models for alcohol/alkanes VLE. The models considered are: original SAFT, Simplified SAFT , Anderko's chemical EoS and CPA. All models perform satisfactorily. CPA yields very similar results to the more complicated models e.g. SAFT.
125
Figure 5. Correlation of the water/methane vapor-liquid equilibra at 75°C with the sCPA EoS using a single binary parameter16.
Figure 6. Correlation of the water/methanol vapor-liquid equilibria at 25°C with the sCPA EoS using a single binary parameter16.
Table 3. Comparison of the CPA EoS with other association models in the obtained average absolute percent error in VLE calculations (bubble-point pressures). SSAFT = Simplified SAFT, AEoS=Anderko equation of state [Yakoumis et al.19]. System Butane/octane Ethanol/butane
methanol/benzene
methanol/propane
Temperature 373.51 298.45 323.75 345.65 373.15 413.15 453.15 493.15 313.10 343.10 373.10
CPA 0.43 3.31 2.07 3.34 1.47 2.68 2.52 1.31 4.26 2.17 0.92
SAFT 2.29 1.31 2.84 2.09 3.81 3.67 1.83 5.25 2.64 1.66
SSAFT 1.75 0.35 1.20 2.50 4.47 4.76 2.73 4.92 2.33 1.45
AEoS 0.70 3.2 1.0
It may be expected that a more advanced model such as CPA would be significantly more time-consuming for performing phase equilibrium calculations compared to the simpler
126 models (SRK, NRTL). As shown in table 4, this is not entirely correct. Only for binary systems, CPA is 6 times slower than SRK and NRTL, while for multicomponent systems, such as those of interest to practical applications, CPA is only 3.6 times slower. Such a difference may be allowable considering the improvement achieved in the calculations. Table 4. Speed of various models (in hundreedths of seconds on a 200 MHz Pentium PC for 100 identical flash calculations) Model sCPA with Elliott sCPA without Elliott SRK NRTL
Test 1 166 529 28 28
Test 2 3361 4938 830 -
Test 3 3218 5405 900 -
Test 1: equimolar mixture of water and methanol at 1 bar and 350 K Test 2: mixture of 29 hydrocarbons plus methanol at 10 bar and 350 K Test 3: mixture of 29 hydrocarbons plus methanol and water at 10 bar and 350 K LLE for alcohol/alkanes As shown in Figure 7 and in the publication by Voutsas et al.i2, CPA provides much improved results for binary LLE for alcohol/hydrocarbons compared to conventional models (SRK, UNIFAC). Furthermore, a comparison with the LLE results reported for these systems in the literature with SAFT6 indicate that CPA and SAFT perform similarly. It is interesting to note that UNIFAC-LLE, although it contains special LLE-based parameters, does not yield very satisfactory results: this demonstrates the necessity of accounting explicitly for the hydrogen bonding effects dominant in these systems, especially when extended temperature ranges are involved (the UNIFAC-LLE parameter table is recommended at temperatures around 20-40 °C). In some cases, e.g. for phenol/pentane, CPA yields excellent description of the binodal curve, while UNIFAC-LLE incorrectly predicts complete miscibility. Glycol/alkane liquid-liquid equilibria The extension of the CPA (and SAFT) equations of state to new associating compounds should be done with care: establishing the correct number of sites and the exact association scheme is not a trivial task, neither is selecting the best set of pure parameters among the various sets, which optimize equally well the pure compound vapor pressures and liquid densities. To illustrate the difficulties encountered, the following example can be mentioned: for compounds, which resemble to alcohols e.g. glycols, it might be expected that the 2B or 4C association schemes could be reasonable starting points. This has been indeed verified; the 4C scheme was proved to be the most successful choice, as shown recently by Derawi et al. . The choice of the correct association scheme and parameters is based on fitting vapor pressures and liquid density data. These choices may depend on the choice of data optimized, which, as shown in Figures 8 and 9, can in some cases vary substantially. Figures 8 and 9 show the "experimental data" generated from two DIPPR correlations as well as the actual
127 raw data. Moreover, the DIPPR correlations are often extended at much greater temperature ranges than the actual raw data8. The LLE correlation for two glycol/alkane systems shown in figures 10 and 11 is very good with a single interaction parameter. Such a behavior cannot be obtained with a conventional model such as SRK (figure 2).
Figure 7. LLE for methanol/hexane with the sCPA EoS state . Methanol is described with the 2B association scheme.
Figure 8. The difference between the DIPPR correlations for vapor pressure and saturated liquid density for ethylene glycol: solid lines, DIPPR 198944; dashed lines, DIPPR 200145; x, experimental raw data [Derawi et a/.8].
128
Figure 9. The difference between the DIPPR correlations for vapor pressure and saturated liquid density for tetraethylene glycol: solid lines, DIPPR 198944; dashed lines, DIPPR 200145; x, experimental raw data [Derawi et a/.8].
Figure 10. LLE for ethylene glycol (1) + n-hexane (2) system with sCPA with a ky = 0.059: x, x" , experimental; o, x[, experimental; solid line, x" , sCPA; dashed line, x'2, sCPA. / = Glycol-rich phase and / / = Hydrocarbon-rich phase [Derawi et a/.8].
129
Figure 11. LLE for triethylene glycol (1) + n-heptane (2) system with sCPA with a kjj = 0.094: x, x'1, experimental; o, x'2, experimental; solid line, x", sCPA; dashed line, x'2, sCPA. / = Glycol-rich phase and //= Hydrocarbon-rich phase. [Derawi et a/.8].
Multicomponent Equilibria As shown in Figures 12-14, prediction of ternary liquid-liquid equilibria for water/methanol/hydrocarbons is excellent using solely one interaction parameter per binary. The results for the multicomponent systems should be considered as straight predictions, since all three interactions parameters have been obtained from the corresponding binary systems. Even, when partition coefficients of the methanol are considered (a very strict test), the prediction is very good and consistently better than conventional thermodynamic models (SRK, NRTL) available in existing commercial simulators. Figure 14 provides results of the partitioning of methanol in a quaternary system, which contains the main types of compounds of a typical petroleum mixture (water, gases, oil and a hydrate inhibitor e.g. methanol). The accurate prediction of the partitioning of methanol between the two liquid phases for such a mixture is very important for the oil industry. A difference of the order of 10 in the calculated partitioning coefficients may result up to 50% increase in the amount of methanol injected for hydrate inhibition. To realise the economical importance of such a prediction, we mention that each oil platform spends on a yearly basis approximately 4 million British pounds for the addition of methanol in the oil [Moorwood, 1998 — personal communication]. Since, as shown in Figure 14, models such as SRK overestimate almost one order of magnitude the amount of alcohol in the organic phase, this would yield to an overestimation of the amount of added methanol of almost 50%. Using CPA more reasonable estimates of the methanol required will be made. It is interesting to note that successful "predictive" results with conventional models (SRK using an NRTLbased combining rule) have been also reported for the system
130 water/methanol/methane/heptane . Future investigations may focus on further comparisons between the SRK (using advanced mixing rules) and CPA models for this type of systems.
Figure 12. Prediction with sCPA of the partitioning of methanol between the organic and the aqueous phase for the ternary system: water/methanol/propane16.
Figure 13. Prediction with various models of the partitioning of methanol between the organic and the aqueous phase of the ternary system: water/methanol/n-hexane16.
Figure 14. Prediction with sCPA and SRK of the partitioning of methanol between the organic and the aqueous phase of the quaternary system: water/methanol/methane/nheptane16.
131 6.5 USE OF THE CPA EOS The CPA EoS is being used by several researchers: At the Technical University of Berlin, CPA and other equations were tested for water / CO2 / aromatic hydrocarbons, cresol systems47. A software available on the internet also contains the CPA equation (http://vt2pc8.vt2.tu-harburg.de). as well as other association equations of state (SAFT, Anderko). At Berkeley, Prausnitz and co-workers48 developed a CPA-type EoS combining PengRobinson with the association term of Wertheim. The new feature of this work is the inclusion of an electrolyte term; thus the resulting EoS is suitable for associating mixtures in the presence of electrolytes. CPA is included in Infochem's commercial software (http://www.infochemuk.com). From the company's web-site, we read: "The CPA model reproduces the partitioning of melhanol between water and hydrocarbon vapor and liquid phases more accurately than conventional models. Thus it will predict less conservative results for the amount of methanol required for a fixed inhibition. The differences between the model predictions will be most marked for systems with low water content and/or significant amounts of C6+". CPA is included in the software SPECS of the research group IVC-SEP (Institut for Kemiteknik, Technical University of Denmark, Denmark). The simplified version is incorporated (sCPA). Parameters for water, methanol and glycols are available and new parameters for other associating compounds are being inserted as they become available. Moreover, IVC-SEP has developed CPA as a CAPE-OPEN property package (http://colan.adduce.de/portal/SoftwareCatalog/IVC-SEP.html) and thus the model can be used in commercial simulators. Recent applications of CPA include polymers38, octanol-water partition coefficient calculations39'37 (figure 19), mixtures with glycols - water - alkanes8'34 (figures 10,11,16,17), systems with chemicals such as sulfolane, gases (nitrogen, CO), water and hydrocarbons (not published in the open literature), systems with ketones36, Henry's law constants for alkane-water systems40, and finally systems containing carboxylic acids41"42 (figure 20).
6.6 CHALLENGES 6.6.1 Systems with Aromatic Hydrocarbons Olefmic and aromatic hydrocarbons in mixtures with water and other associating compounds have been considered as well. In these cases, the cross-association between aromatic (and olefinic, to a lesser degree) hydrocarbons and water or alcohols has been taken into account. Cross-association is important in these systems, since many experimental studies indicate the existence of weak complexes formed between the benzene ring (due to n -electrons) and proton donor molecules49"50. The increased solubility of water and aromatic compounds as compared to water and n-alkanes is a result of solvation between water and aromatic
132 compounds. Most results related to these systems have not as yet appeared in the open literature but further reveal the capabilities of the CPA equation of state. A sample result is given in Figure 15, obtained from a publication in a Greek journal.
Figure 15. Correlation of the water-benzene liquid-liquid equilibria with the sCPA using a single binary parameter35. The solvation between water and benzene has been accounted for. 6.6.2 Cross-Associating Systems As mentioned previously, water with heavy alcohols and glycols cannot be modelled satisfactorily using a single interaction parameter and the Elliott rule. Improved combining rules are needed for such cross-associating systems. Voutsas et al33 suggested the following set of combining rules (denoted as CR1 in figures 16 and 17) which result to much improved representation of these systems: eAiBj
eAiB>+eA& 2
(12)
A theoretical justification for this new set of combining rules can be provided. The rules of equations 12 are based on the well-known theoretically accepted rules for the cross enthalpy and entropy of hydrogen bonding:
133
AH a =
AH,- + AH J/ -
(13) AS/+AS,-
In order to show the equivalence between the combining rules in equations 12 and 13, we consider the similarities (analogy) between the chemical and the perturbation theory. The association strength (of perturbation theory) is equivalent to the equilibrium constant of chemical theory18, thus: A oc KRT =>
_6XPl RT ) ~ T Rf ^6XW J exp - J - oc exp — — \Kt J V HI J B& RT
\ R )
l^J "
=> sec -AH
(AS)
—— oc exp
eXP
(14)
(AS) => B oc exp P
\ R )
In equation 14, we have ignored the density dependence of the association strength (expressed via the radial distribution function) and of the equilibrium constant. Notice also that in the equation for the association volume parameter, the b/RT part is also ignored. If this is included, then the Elliott equation (Eq. 11) is obtained (using equations 12 and 13). Based on equations 13 and 14, the 'new combining rules', equations 12, are derived. In some few cases, e.g., for the system water/tert-butyl alcohol/i-butane, satisfactory ternary LLE results are obtained using the Elliott rule for the cross association strength of water/alcohol. Despite the fact that the correlation of water/alcohol is only fair (4% deviation), the performance of the multicomponent calculations is not greatly affected. Still, in most other cases where the Elliott rule fails for the cross-associating systems, improved combining rules need to be used. Equations 12 may be successful for some systems e.g. water with heavy alcohols or glycols. Recent results for water/heavy glycols demonstrate that equations 12 provide a better representation of cross-association effects than the Elliott rule (equation 11). Some results are shown in figures 16 and 17 (CR-1 combining rule is equation 12). We note that investigations on cross-associating systems with association EoS such as SAFT are extremely scarce. Essentially, to our knowledge, only few systematic works have been presented4849'30. Experimental investigations and prediction methods as well as ab initio (molecular orbital) calculations for the cross-enthalpy and entropy of hydrogen bonding may be very helpful in the future in defining appropriate combining rules for the association parameters51"53'21. Of
134 particular challenge, in certain future developments, may be the account of hydrogen-bond co-operativity effects in alcohol-containing systems54 as well as special associating schemes e.g. the intramolecular association in systems containing oxy-alcohols, which include both an ether and a hydroxyl group55.
Figure 16. sCPA-correlation and prediction of the MEG-water VLE at T = 363.15 K. using the CR-1 combining rule with a ky = -0.012 (solid line) and k,j = 0 (dashed line).[Derawi et a/.34. Reprinted by permission]
Figure 17. sCPA-correlation and prediction of the DEG-water VLE at T = 393.15 K using the CR-1 combining rule with a ky = -0.115 (solid line) and ky = 0 (dashed line).[Derawi et al.34. Reprinted by permission]
135 6.6.3 Critical Area of Mixtures CPA parameters for associating fluids are obtained from vapor pressure and liquid density data. The critical point constraints are not used, contrary to the classical cubic equations of state. For gas/alcohol or gas/water systems, the critical temperature and pressure are used for the gas. It may be expected that problems are encountered in the description of the critical area of associating mixtures, especially those containing gases. This is indeed, to some extent, the case and typically SAFT and CPA overestimate the critical point, less though for CPA than for SAFT31. In some cases, e.g. for methanol/methane (figure 18) and water/methane (figure 5) systems, satisfactory results are obtained. Describing satisfactorily the critical area for associating systems with equations of state such as SAFT and CPA remains an open area for investigation.
Figure 18. Correlation of the methanol/methane vapor-liquid equilibria at 25°C with the sCPA EoS using a single binary parameter16. 6.6.4 New associating compounds We have already discussed the extension of sCPA to various associating compounds such as water, alcohols and glycols. But what about 'special' chemicals such as sulfolane, which has the following structure (CH2)2-SO2-(CH2)2? A simple inspection of the structure of this compound may not be adequate for establishing whether it is or not a self-associating substance let alone to give some hints regarding the correct association scheme. Experimental activity coefficient data of sulfolane with n-alkanes have been reported3 and the values at infinite dilution of the order of 40-50 indicate a behavior similar to methanol/alkanes. at least
136 from this point of view. A similar conclusion is reached by the experimental LLE data for sulfolane/n-alkanes. Although such experimental data will not be used directly in the pure parameter estimation, they may help us identifying the correct number and nature of the association scheme of this or similar compounds. Another family of compounds, which present special interest, is the ketones. It is not clear whether ketones should be treated as associating compounds or not. Jog et al.51 state that the highly non-ideal behavior of e.g. ketone-alkane systems is due to the polarity of ketones. They have developed a version of SAFT, named polar-SAFT, and model ketones-alkanes without association (with much better results than SAFT). On the other hand, Yakoumis et al}b treated ketones as two-site associating molecules with CPA and presented very satisfactory results for the prediction of the liquid-liquid partition coefficient of methanol in the methanol/acetone/n-hexane system (using solely binary parameters). For environmentally related chemicals, Polyzou et al.i9 (figure 19) have demonstrated that the sCPA EoS can be applied to the prediction of octanol/water partition coefficients.
Figure 19. Predicted against experimental octanol-water partition coefficients with CPA39. Finally, the sCPA EoS has been recently extended to systems with carboxylic acids and a typical result is provided in figure 20. Excellent correlation is achieved at both temperatures, including the azeotrope, with a single interaction parameter. The 1A (Huang and Radosz notation14) association scheme has been used for acetic acid, in accordance to the dominant physical picture (formation of dimers).
137
Figure 20. Pressure-composition diagram for Acetic acid(l) + n-octane(2) system with sCPA at T= 323.2 K (lower curve) and T=343.2 K (upper curve) with a ktj = 0.06 : o, experimental data; solid line, CPA employing the 1A site model. From Zeuthen41. 6.7 CONCLUDING REMARKS CPA is a powerful tool for predicting multicomponent multiphase equilibria for 'water alcohol - hydrocarbons' and for other mixtures containing hydrogen bonding substances. The improved performance of CPA (and SAFT) should be attrributed to the Wertheim association term. Considering the similarity of the various association theories9'", equally good performance could be expected for the chemical and lattice theories. This is the case provided that the crucial 'details' discussed for the pertubartion theory (e.g. selection of association scheme, cross association) are carefully taken into account. The time requirement of CPA EoS does not seem to be a restrictive factor for its commercial application. Every extension of CPA (and SAFT) to new associating compounds/mixtures should be done with great care, taking into account the factors discussed in detail in this chapter. The most important of them are summarized here: careful estimation of pure parameters and selection of the appropriate association scheme the pure parameter values for the association energy and volume should be compared, when possible, with experimental (or ab initio) values for the heats and entropies of hydrogen bonding phase equilibria data for a specific associating compound with an inert non-associating alkane is a useful way in selecting the appropriate set of pure parameters as well as the most suitable association scheme the nature of cross-association should be established. It is important to choose appropriate mixing and combining rules for the association parameters as well as to investigate "hidden" (not apparent) cross-associations such as when aromatic (and to a lesser degree olefinic hydrocarbons) are present in mixtures with alcohols and/or water
138 for satisfactory multicomponent predictions, crucial binaries should be determined and described as accurately as possible tricks for faster/easier calculations are important e.g. the Elliott rule for the cross association strength, the use of homomorph method18 for reducing the number of pure association parameters, the use of estimation methods for obtaining pure vapour pressures and liquid densities when experimental data are not available, etc. Future applications of the CPA equation of state depend on specific interest from academia and industry and can involve both further theoretical and applied studies-extensions to new types of compounds. Current activities include systems containing carboxylic acids, ternary LLE for water-based systems with nitrogen-containing compounds e.g. pyridine, acrylonitrile and ethyleneamine, mixtures with aqueous alkanolamines and gases such as CO2, gas-containing systems which also include aromatic hydrocarbons and associating compounds e.g. water and alcohols or glycols, solid-liquid equilibria for glycol-waterhydrocarbons, as well as extension of CPA to mixtures with strong electrolytes.
LIST OF SYMBOLS a ao A; Bj b Cl
d g ky K Kovv x Ai
P R T x y y
v z
energy parameter of SRK and CPA EoS, defined in equation 7 parameter in the energy term, equation 7 association site A in molecule i association site B in molecule j co-volume parameter of the SRK and CPA EoS parameter in the energy term (Soave expression), equation 7 density (in figure 3) radial distribution function, defined in equations 8 or 9 binary interaction parameter, defined in equation 10 equilibrium constant octanol-water partition coefficient mole fraction of molecule i not bonded at site A pressure ideal gas constant temperature mole fraction in the liquid phase mole fraction in the vapor phase reduced co-volume, defined in Eqs. 8 or 9 volume compressibility factor
Greek Symbols association volume parameter AB
A'J
association strength
139 AH AS
enthalpy of hydrogen bonding entropy of hydrogen bonding . .
AjB;
s
'
association energy parameter
p
molar density
Superscripts A B assoc phys sat
association site A association site B association term of the EoS physical term of the EoS saturated
Subscripts i j ij r w
compound i compound j cross-parameter (interaction) between compounds i and j reduced property van der Waals
LIST OF ABBREVIATIONS AEoS CPA CR-1 DEG EoS LLE MC MEG NRTL RDF SAFT SSAFT sCPA SRK TEG vdW vdWlf VLE
Anderko Equation of State Cubic Plus Association (EoS) combining rule - 1 (defined in Eq. 12) diethylene glycol Equation of State Liquid-Liquid Equilibria Mathias-Copeman (mono)ethylene glycol Non Random Two Liquid (model) Radial Distribution Function Statistical Association Fluid Theory/EoS Simplified SAFT simplified CPA Soave Redlich Kwong EoS triethylene glycol van der Waals (EoS) vdW one fluid mixing rules Vapor-Liquid Equilibria
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141 32. E.C. Voutsas, I.V. Yakoumis, G.M. Kontogeorgis and D.P. Tassios, Fluid Phase Equilibria, 132(1997)61. 33. E.C. Voutsas, I.V. Yakoumis, and D.P. Tassios, Fluid Phase Equilibria, 158-160 (1999) 151. 34. S.O. Derawi, G.M. Kontogeorgis, M.L. Michelsen and E.H. Stenby, Ind. Eng. Chem. Res., 42(7) (2003) 1470. 35. I.V. Yakoumis, E.N. Polyzou, H. Meijer, T. Moorwood, E.M. Hendriks and G.M. Kontogeorgis, Industrial applications of the sCPA equation of state. 2nd Greek Chemical Engineering Conference, University of Thessaloniki, 16-29/5/1999, Book of Abstracts, 337-340 (in Greek) 36. I.V. Yakoumis, E.C. Voutsas and D.P. Tassios, An equation of state for systems exhibiting hydrogen bonding. 1st Greek Chemical Engineering Conference, University of Patras, 2937. -31/5/1997, Book of Abstracts, 319-324 (in Greek). 38. G.M. Kontogeorgis and I.V. Yakoumis, Novel model for phase equilibrium in the chemical and petroleum industry. Monthly Technical Review, March 1999, 39-43 (in Greek). 39. G.M. Kontogeorgis, I.V. Yakoumis and P.M. Vlamos, Computational & Theoretical Polymer. Science, 10(6) (2000) 501. 40. E.N. Polyzou, A.E. Louloudi, G.M. Kontogeorgis and I.V. Yakoumis, Prediction of octanol-water partition coefficients. 2nd Greek Chemical Engineering Conference, University of Thessaloniki, 16-29/5/1999, Book of Abstracts, 101-104 (in Greek). 41. G.C. Boulougouris, E.C. Voutsas, I.G. Economou and D.P. Tassios, J. Phys. Chem. B., 105(32) (2001) 7792. 42. J. Zeuthen, Extension of the CPA EoS to systems containing organic acids, M.Sc. Thesis, Institut for Kemiteknik, Technical University of Denmark, 2003. 43. G. M. Kontogeorgis, S. Derawi, J. Zeuthen, E.H. Stenby, N. v. Solms, Th. Lindvig, LA. Kouskoumvekaki and M. L. Michelsen, 2003. Applications of association models to problems of the oil, chemical and polymer industries, Invited (plenary) lecture to the 20th European Symposium on Applied Thermodynamics, 9-12 Ocrober 2003, Germany (article available in the proceedings of the conference). 44. P.K. Jog, A. Garcia-Cuellar and W.G. Chapman, Fluid Phase Equilibria, 158-160 (1999)321. 45. T.E. Daubert, R.P. Danner, Physical and Thermophysical Properties of Pure Compounds: Data Compilation, Hemisphere, New York, 1989. 46. T.E. Daubert, R.P. Danner, Physical and Thermophysical Properties of Pure Compounds: Data Compilation, Hemisphere, New York, 2001. 47. K.S. Pedersen, M.L. Michelsen and A.O. Fredheim, Fluid Phase Equilibria, 126 (1996) 13. 48. O. Pfohl, A. Pagel and G. Brunner, Fluid Phase Equilibria, 157 (1999) 53. 49. J. Wu and J.M. Prausnitz, Ind. Eng. Chem. Res., 37 (1998) 1634. 50. J. Suresh and E.J. Beckman, Fluid Phase Equilibria, 99 (1994) 219. 51. S. Michel, H.H. Hooper and J.M. Prausnitz, Fluid Phase Equilibria, 45 (1989) 173. 52. J.P. Wolbach and S.I. Sandier, Ind. Eng. Chem. Res., 37 (1998) 2917.
142 53.1.G. Economou, G.D. Ikonomou, P. Vimalchand and M.D. Donohue, AIChE J., 36(12) (1990) 1851. 54. J.P. Wolbach and S.I. Sandier, Ind. Eng. Chem. Res., 36(1997) 4041. 55. R.B. Gupta and R.L. Brinkley, AIChE J., 44(1) (1998) 207. 56. R.L. Brinkley and R.B. Gupta, Ind. Eng. Chem. Res., 37(1998) 4823. 57. W.C. Moollan, U.M. Domanska and T.M. Letcher, Fluid Phase Equilibria, 128 (1997) 137. 58. P.K. Jog, S.G. Sauer, J. Blaesing and W.G. Chapman, Ind. Eng. Chem. Res., 40(21) (2001)4641.
Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B.V. All rights reserved.
143
Chapter 7: Models for Polymer Solutions Georgios M. Kontogeorgis
7.1 INTRODUCTION - AREAS OF APPLICATION Knowledge of phase equilibria of polymer systems (solutions, blends, ...) is of interest to the design of a variety of processes related to polymers. Some examples are shown in the table below. Table 1. Applications of polymer thermodynamics. Properties/ Phase Equilibria Involved Polymer-solvent VLE Polymer-solvent LLE, often mixed solvents Polymer-solvent VLE, solvent activities Emissions from paint production Polymer recycling via physico-chemical Polymer-solvent LLE methods (selective dissolution) Systems with co-polymers and polymer Product design blends Design of flexible polymer pipes carrying Gas solubilities and diffusivities in polymers subsea oil Compatibility (miscibility) of PVC Finding alternative plasticizers to PVC plasticizers Deposition of polymer thin films using rapid Polymer-SCF (SGE) expansion from supercritical solution (RESS) Separation of proteins via aqueous two-phase LLE of polymer-water-protein often in presence of electrolytes systems Application Solvent devolatilization after polymerisation Selection of solvents for paints and coatings
This table shows a variety of systems and types of phase equilibria, which are of interest in the many practical situations where polymer thermodynamics plays a key role. For this reason, many different models have been developed for polymer systems and often the situation may seem rather confusing to the practising engineer. Polymer solutions and blends are complex systems: frequent existence of liquid-liquid equilibria (UCST, LCST, closed loop, etc.), the significant effect of temperature and polymer molecular weight including polydispersity in phase equilibria, free-volume effects and other factors may cause difficulties. The choice of a suitable model will depend on the actual problem and depends, specifically on:
144 -
type of mixture (solution or blend, binary or multieomponent,...) type of phase equilibria (VLE, LLE, SLLE, gas solubility,...) conditions (temperature, pressure, concentration) type of calculations (accuracy, speed, yes/no answer or complete design,...)
This chapter focuses mostly on simple activity coefficient models for polymers, which can be applied to a wide range of applications. These are group-contribution based (UNIFAC) models, which account for some special effects in polymer systems such as free-volume differences. Some few recommendations on the vast but a bit confusing literature on equations of state for polymers will be also provided at the end of the chapter. Since most models often perform better for VLE than for LLE, indirect techniques are widely applied e.g. for solvent selection. These are briefly summarized in the next section.
7.2 CHOICE OF SOLVENTS A summary of some rules of thumb for predicting polymer-solvent miscibility, with focus on the screening of solvents for polymers, is presented here. These rules are based on wellknown concepts of thermodynamics (activity coefficients, solubility parameters) and some specific ones to polymers (Flory-Huggins parameter). Then, a brief discussion of some of the concepts involved is included. It can be roughly said that a chemical (1) will be a good solvent for a specific polymer (2), or in other words the two compounds will be miscible if one (or more) of the following 'rules of thumb' are valid ' : i.
If the polymer and the solvent have 'similar hydrogen bonding degrees:
1
ii.
{cm3}
(1)
If the polymer and the solvent have very different hydrogen bonding degrees:
V 4 fo. - *J2 Y + fe, - Sp2 f + (Shl - 8h2 f
(2)
where R is the Hansen-solubility parameter sphere radius. iii.
Q" < 6 (the lower the infinite dilution activity coefficient of the solvent, the greater the solvency of a chemical). Values of the infinite dilution activity coefficient above 10 indicate non-solvency. In the intermediate region, it is difficult to conclude if the specific chemical is a solvent or a non-solvent.
145 iv.
Xn - 0-5 (the lower the Flory-Huggins parameter value, the greater the miscibility, or, in other words, the greater the solvent's capacity of a specific chemical). Values much above 0.5 indicate non-solvency.
7.2.1 The Rules of Thumb Based on Solubility Parameters They are widely used. The starting point (in their derivation / understanding) is the equation for the Gibbs Free-Energy of mixing: AGmix=AH-TAS
(3)
A negative value implies that a solvent/polymer system forms a homogeneous solution i.e. the two components are miscible. Since the contribution of the entropic term ( - TAS) is always negative, it is the heat of mixing term that determines the sign of the Gibbs energy. The heat of mixing can be estimated from various theories e.g. the Hildebrand regular solution theory for non-polars systems, which is based on the concept of the solubility parameter. For a binary solvent(l)/polymer(2) system, according to the regular solution theory: AW =
(4)
where p,is the so-called volume fraction of component i. This is defined via the mole fractions x; and the molar volumes V;, as (binary systems):
"=^vo
(5)
According to Eq. 4, the heat of mixing is always positive. For some systems with specific interactions (hydrogen bonding) the heat of mixing can be negative and Eq. 4 does not hold. Thus, the regular solution theory is strictly valid for non-polar/slightly polar systems, without any specific interactions. According to Eqs. 3 and 4, if solvent and polymer have the same solubility parameters, the heat of mixing is zero and they are thus miscible at all proportions. The lower the solubility parameter difference the larger the tendency to be miscible. Many empirical rules of thumb have been proposed based on this observation. Seymour1 suggests that if the difference of solubility parameters is below 1.8 (cal/cm3)1'2, Eq. 1, then polymer and solvent are miscible. Similar rules can be applied for mixed solvent - polymer systems, which are very important in many practical applications, e.g. in the paints and coatings industry and for the separation of biomolecules using aqueous two-phase systems. The solublity parameter of a mixed solvent is given by the equation:
8 = £
/
(6)
146 Barton3"4 provides empirical methods based on solubility parameters for ternary solvent systems. Charles Hansen introduced the concept of three-dimensional solubility parameters, which offer an extension of the regular solution theory to polar and hydrogen bonding systems. Hansen observed that when the solubility parameter increments of the solvents and polymers are plotted in three-dimensional plots, then the 'good' solvents lie approximately within a sphere of radius R (with the polymer being in the centre). This can be mathematically expressed as: ^<Sin - 8dl )2 + {Spl - 8p2 )2 + (Shi - 8h2 )2
(2)
where subscript 1 denotes the solvents and subscript 2 the polymer. The quantity under the square root is the distance between the solvent and the polymer. Hansen found empirically that a universal value 4 should be added as a factor in the dispersion term to approximately attain the shape of a sphere. This universal factor has been confirmed by many experiments. Several other two-dimensional plots have been proposed, which employ all three contributions e.g. 8p-8h,Sh-8d,5p-8d or even combined plots such as the use of 8V -8h,8v
= J8d +d2p plots suggested by van Krevelen2. With few exceptions good
solvents lie within the circle of radius R, which mathematically can be expressed as:
^8rl-8r2f+(Sh]-8h2y
(7)
The justification for this plot lies in the fact that, of the three solubility parameter increments, the dispersion one varies the least and, via this average way, it can be treated together with the polar increment. The hydrogen bonding increment is very important and it is thus accounted for separately in Eq. 7. The Hansen method is very valuable. It has found widespread use particularly in the paints and coatings industry, where the choice of solvents to meet economical, ecological and safety constraints is of critical importance5. It can explain some cases in which polymer and solvent solubility parameters are almost perfectly matched and yet the polymer won't dissolve. The Hansen method can also predict cases where two non-solvents can be mixed to form a solvent. Still, the method is approximate, it lacks the generality of a full themiodynamic model for assessing miscibility and requires some experimental measurements. The determination of R is typically based on visual observation of solubility (or not) of 0.5 g polymer in 5 cm3 solvent at room temperature. Given the concentration and the temperature dependence of phase boundaries, such determination may seem a bit arbitrary. Still the method works out pretty well in practice, probably because the liquid-liquid boundaries for most polymer/solvent systems are fairly 'flat'. A recent review of the Hansen method with extensive tables of solubility parameters is available6.
147 7.2.2 The Rule of Thumb Based on the Infinite Dilution Activity Coefficient Since in several practical cases concerning polymer/solvent systems, the 'solvent' is only present in very small (trace) amounts, the so-called infinite dilution activity coefficients are of importance. On a molar and weight basis, they are defined as follows:
r? =lim,,^o r,
nr m
=" m (xy\
(8)
The weight-based infinite dilution activity coefficient, Q", which can be determined experimentally from chromatography, is a very useful quantity for determining good solvents. Low values (typically below 6) indicate good solvents, while high values (typically above 10) indicate poor solvents according to rules of thumb discussed by several investigators7"9. The derivation of this rale of thumb is based on the Flory-Huggins model, discussed in the next section, 7.2.3. This method for solvent selection is particularly useful because it avoids the need for direct liquid-liquid measurements and it makes use of the existing databases of solvent infinite dilution activity coefficients, which is quite large (e.g. the DECHEMA and DIPPR databases). Moreover, in the absence of experimental data, existing thermodynamic models (such as the Flory-Huggins, the Entropic-FV and the UNIFAC-FV discussed later, section 7.3) can be used to predict the infinite dilution activity coefficient. Since, in the typical case today, existing models perform much better for VLE and activity coefficient calculations than directly for LLE calculations, this method is quite valuable and successful, as shown by sample results in Table 2. The thermodynamic models UNIFAC-FV (U-FV), Entropic-FV (E-FV), UNIFAC and GCLF, shown in Table 2, are used for obtaining the infinite dilution activity coefficient for several PVC systems . We can see that, with few exceptions, these models in combination with the rule of thumb mentioned above can identify which chemicals are good solvents for PVC and which are not. Similar results have been presented elsewhere ' for selecting solvents for paint polymers such as PBMA and PMMA. This rule of thumb makes use of either experimental or predicted, by a model, infinite dilution activity coefficients. However, the results depend not only on the accuracy of the model, but also on the rule of thumb, which in turns depends on the assumptions of the FloryHuggins approach. A thermodynamically more correct method is to employ the activity concentration (aw) diagram, as shown in figures 1 and 2. Results for the PVC systems are shown in Table 2. The two plots have been generated with UNIFAC-FV. The maximum indicates phase split, while a monotonic increase of activity with concentration indicates a single liquid phase (homogeneous solutions).
148 Table 2. Observed and predicted weight fraction activity coefficients at infinite dilution and ai(w), activity-weight fraction, for different PVC (Mn=50 000)-solvent systems at 298 K. Nr,
Chemicals
(E-FV)
(U-FV)
Mansen '92
Fred '75
Experimental
UNIFAC
s/ns
1
M omit; h loraben/cn c
2
Chloroform Dichlorom ethane
Uansen '(JI
Cl?
a,(iv)
(V
a,(w)
£V
a,(«)
£V
a,(w)
S
5.OX
S
3.08
S
4.54
s
2.45
S
Ns
2.98
S
4.44
S
4.54
s
2.11
S
3.79
s
1.12
s s
4.5
s s s
9.47
1.02
2.64 5.67
4
Nitrocthane
Ns
-
5
\:Ah\\ aeelale
S
6.15
S
6.46
3.78
s
6
1.4-1 )io\il!k'
S
12.2
s
19.08
1.34
5.76
s
14.5
1.22
7
Isopropanol
Ns
48.41
1.17
40.62
1.18
8.15
s
40.62
1.18
X
Meihunol
Ns
70,72
1.23
94.5
1.4
9.48
s
61.81
1.05
y
Henzj 1 alcohol
Ns
23.2
S
19.97
1.05
10.14
s
15.42
s
S
-
-
-
6.h7
-
7.32
1.01 s
10 Tetrahydrofurane
•
-
-
Toluene
S
2.54
S
19.9
1.72
9.16
s
2.87
12 Benzene
S
4.64
S
4.47
s
8.23
s
3
13 Acetone
II
Ns
6.44
S
9.5
s
6.02
S
S.M
S
6.63
s s
K.45
MI{K (a)
9,35
s
-
15 MEK(b)
s s
5.35
S
7.51
s
9.72
s
-
-
4.93
S
6
s
11.78
1.05
-
-
s
4.72
S
6.21
s
12.3
1.04
4.49
s
14
16 Methyl Isjbulyl Kelone (a) 17 Methyl Isobulyl Ketone (bl 18 Di-n-Propyl IZlher
Ns
1 3 58
1.03
17.1
1.13
23.49
1.31
9.52
1.01
19
Ns
iK.xy
1.24
16.27
1.08
29.K9
1.43
11 57
1.02
20 n-Ociane
Ns
17.72
1.2
15.72
1.06
27.34
1.36
10,72
1.0!
21 n-Pentane
Ns
15.98
1.2
15.88
1.02
22.56
1.22
8.71
s
22 n-Heptane
Ns
16.72
1.16
15.31
1.05
25.24
1.3
9.96
s
23 o-Xylene
S
11.26
1.17
5.16
s
10.2
1.02
3.85
s
S
2.06
s
1.58
s
6.22
s
2.1)1
s
24
n-Nonane
1,2-dichloroben/ene
Ns
3.67
S
8.65
1.02
12.78
1.08
4.08
s
26 F.thylcne di-chloride
S
4.1')
4.32
s
5.22
s
3.22
s
27 Amyi acetate
S
6.42
6.43
5
8.41
1.01
22
s
28 Carbon disuliide
Ns
-
-
-
4.6
s
7.14
29 Vinyl chloride
Ns
2.17
3.93
S
3.8
s
•
25 Butyl aery laic
.in
-
s s s -
-
-
-
-
Wrong answers
4
6
3
4
13
12
5
Total
26
26
26
26
28
28
26
Ns
Acctonilrile
s
10.51 26
The dark colour indicates the cases where the rule of thumb can be applied, while the light colour indicates cases where 6<XV <8. and thus no conclusion can be made. The numbers in the a-w columns indicate maximum on graphs of solvent activity vs. solvent weight fraction curves (meaning insoluble chemicals), a and b means that there are two ways of defining the compound.
149
Figure 1. Solvent-activity diagram of n- Figure 2. Solvent-activity diagram of heptane and PVAC at 300 K with UNIFAC- chloroform and PBMA at 300 K with UNIFAC-FV. Monotonically increasing line FV. The maximum indicates a phase split. indicates solubility at all concentrations. 7.2.3 The Rule of Thumb Based on the FIory-Huggins Model The Flory-Huggins (FH) model for the activity coefficient, proposed in the early 40's by Flory and Huggins12, is a famous Gibbs free energy expression for polymer solutions. For binary solvent-polymer solutions and assuming that the parameter of the model, the so-called FH interaction parameter %{1 is constant, the activity coefficient is given by the equation
x,
( n
x,
(9)
= In — + 1 YPi+XvVl x, \ r) where cp, can be volume or segment fractions and r is the ratio of the polymer volume to the solvent volume V2/V1 (approximately equal to the degree of polymerization). Using standard thermodynamics and Eq. 9, it can be found that for high molecular weight polymer-solvent systems, the polymer critical concentration is close to zero and the interaction parameter has a value equal to 0.5. Thus, a good solvent (polymer soluble in the solvent at all proportions) is obtained if Xn - 0-5, while values greater than 0.5 indicate poor solvency. Since the Flory-Huggins model is only an approximate representation of the physical picture and particularly the FH parameter is often not a constant at all, this empirical
150 rule is certainly subject to some uncertainty. Nevertheless, it has found widespread use and its conclusions are often in good agreement with experiment. This can be demonstrated by a socially important example, the choice of suitable (miscible) plasticizers with PVC10. Typical results are shown in Tables 3 and 4 and figure 3. The infinite dilution activity coefficients calculated by E-FV and U-FV are also shown in these Tables. Table 3. Classification of the solvent power of various plasticizers from different calculation (thermodynamic) methods and experiments (dilute solution viscosity, apparent melting temperature, equilibrium swelling). Plasticizer X
a
Dilute- App. £V (298K) Melting soln. (E-FV) viscosity temperature
£V (298K) (U-FV)
Equilibrium swelling (350K) at 350 K (E-FV)
n," (350K) (U-FV)
DOS
0.62 0.8
DOA
DOS
3.85
7.46
DOS
6.43
DOA
0.48 1.4
DOS
DOA
2.30
13.7
DOA
4.03
DOP
0.05 2.4
DOP
DOP
2.31
4.22
DOP
4.31
3.87
BBP
0.17 2.6
BBP
BBP
2.00
2568
BBP
3.16
19.11
DBP
0.04 3.4
DBP
DBP
2.12
3.95
DBP
3.4D
3.54
6.68 7.35
The a and x values are taken from Bigg13. The a values are simply (l-%)/MW. (MW is the molecular weight). Table 4. Classification of the solvent power of various phthalates from different calculation (thermodynamic) methods. Plasticizer
3i K (323K) (E-FV) 0.98 3.75 2.27
3.31
2.61
2.89
4.82
Dihexyl phthalate
Mw X (323K) (g/mol) DDP 446.7 0.56 DMP 194.20 0.56 DEP 222.2 0.42 DHP 334.5 -0.13
3.38
2.77
3.63
Dioctyl phthalate
DOP
390.9
2.53
3.14
3.98
Dibutyl phthalate
DBP
278.3
3.73
2.64
3.73
Didecyl phthalate Dimethyl phthalate Diethyl phthalate
0.01 -0.04
a
The a and % values are taken from Doty and Zable14.
(U-FV) 4.64 6.96
151
Figure 3. Dependency of the experimentally determined FH interaction parameter values and Bigg's alpha values on molecular weight of phthalates. The % values are taken from Barton4. There are several, still rather obscure issues about the Flory-Huggins model, which we summarize here together with some recent developments: 1. There is no single rigorous widely-accepted extension of the FH model to multicomponent systems. Several extensions have been proposed, but (at least) one Xi2-value is required per binary. 2. It has been shown that, unfortunately, the FH parameter is typically not a constant and should be estimated from experimental data. Usually it varies with both temperature and concentration, which renders the FH model useful only for describing experimental data. It cannot be used for predicting phase equilibria for systems for which no data is available. Moreover, when fitted to the critical solution temperature, the FH model cannot yield a good representation of the whole shape of the miscibility curve with a single parameter. 3. Accurate representation of miscibility curves is possible with the FH model using suitable (rather complex) equations for the temperature and the concentrationdependence of the FH-parameter15"16. 4. In some cases, a reasonable value of the FH parameter can be estimated using solubility parameters via the equation: X, 2 =Xs+X* = 0 . 3 5 + ^ ( 8 , - 8 2 ) 2
(10)
152 Eq. 10, without the empirical 0.35 factor, is derived from the regular solution theory. The constant 0.35 is added for correcting for the deficiencies of the FH combinatorial term. These deficiencies become evident when comparing experimental data for athermal polymer and other asymmetric solutions to the results obtained with the FH model. A consistent underestimation of the data is observed, as discussed extensively in the literature17, which is often attributed to the inability of the FH model in accounting for the free-volume differences between polymers and solvents or between compounds differing significantly in size such as n-alkanes with very different chain lengths. The term, which contains the "0.35 factor", corrects in an empirical way for these free-volume effects. However, and although satisfactory results are obtained in some cases, we cannot generally recommend Eq. 10 for estimating the FH parameter. Moreover, for many non-polar systems with compounds having similar solubility parameters, the empirical factor 0.35 should be dropped. 5. Recently, Lindvig el a/.18 proposed an extension of the Flory-Huggins equation using the Hansen solubility parameters for estimating activity coefficients of complex polymer solutions. (Pl
Xi
o
CPi
lny^ln^ +l - ^
+
x ^
Xi
(11)
Xl2 =0.6-^[(5 dl _ 5d2 )2 +o.25(5pl -8p2f
+0.25(8,,,
2
Sh2) ]
In order to achieve that, Lindvig et al.18, as shown in Eq. 11, have employed a universal correction parameter, which has been estimated from a large number of polymer-solvent VLE data. Very good results are obtained, especially when the volume-based combinatorial term of FH is employed, as summarized in Table 5. Table 5. Average absolute deviations between experimental and calculated activity coefficients of paint-related polymer solutions using the Flory-Huggins/Hansen method and three group contribution models. From Lindvig et a/.18. The second column presents the systems used for optimization of the universal parameter (358 points of solutions containing acrylates and acetates). The last two columns show predictions for two epoxy resins. Model FH/Hansen, Volume (Eq. 11) FH/Hansen Segment FH/Hansen Free-Volume Entropic-FV UNIFAC-FV GC-Flory
% AAD (systems in database) 22
% AAD Araldit 488
% AAD Eponol-55
31
28
25
—
—
26
—
—
35 39 18
34 119 29
30 62 37
153 6. Based on the Flory-Huggins model, several techniques have been proposed for interpreting and for correlating experimental data for polymer systems e.g. the socalled Schultz-Flory (SF) plot. Schultz and Flory19 have developed, starting from the Flory-Huggins model, the following expression, which relates the critical solution temperature (CST), with the theta temperature and the polymer molecular weight:
where \m = — is \'ls m e entropic parameter of the FH model (Eq.10) and r is the ratio of molar volumes of the polymer to the solvent. This parameter is evidently dependent on the polymer's molecular weight. The SF plot can be used for correlating data of critical solution temperatures for the same polymer/solvent system, but at different polymer molecular weights. This can be done, as anticipated from Eq.12 because the plot of 1/CST against the quantity in parentheses in Eq. 12 is linear. The SF plot can also be used for predicting CST for the same system but at different molecular weights than those used for correlation as well as for calculating the theta temperature and the entropic part of the FH parameter. It can be used for correlating CST/molecular weight data for both the UCST and LCST areas. Apparently different coefficients are needed.
7.3 THE FREE-VOLUME ACTIVITY COEFFICIENT MODELS 7.3.1 The Free-Volume Concept The Flory-Huggins model provides a first approximation for polymer solutions. Both the combinatorial and the energetic terms need substantial improvement. Many authors have replaced the random van-Laar energetic term by a non-random local-composition term such as those of the UNIQUAC, NRTL and UNIFAC models. The combinatorial term should be extended/modified to account for the free-volume differences between solvents and polymers. The improvement of the energetic term of FH equation is important. Local-composition terms like those appearing in NRTL, UNIQUAC and UNIFAC provide a flexibility, which cannot be accounted for by the single-parameter van Laar term of Flory-Huggins. However, the highly pronounced free-volume effects should always be accounted for in polymer solutions. The concept of free-volume (FV) is rather loose, but still very important. Elbro8 demonstrated, using a simple definition for the free-volume (Eq.13), that the FV percentages of solvents and polymers are different. In the typical case, the FV percentage of solvents is greater (40-50%) than that of polymers (30-40%). There are two exceptions to this rule; water and PDMS: water has lower free-volume than other solvents and closer to that of most of the polymers, while PDMS has quite a higher free-volume percentage, closer to that of most
154 solvents. LCST is, as expected, related to the free-volume differences between polymers and solvents. As shown by Elbro8, the larger the free volume differences the lower the LCST value (the larger the area of immiscibility). For this reason, PDMS solutions have a LCST, which are located at very high temperatures. Many mathematical expressions have been proposed for the FV. One of the simplest and successful equations is20'21: Vf=V-V'=V-Vw
(13)
originally proposed by Bondi20 and later adopted by Elbro et al.21 and Kontogeorgis et al.22 in the so-called Entropic-FV model (described in 7.3.3). According to this equation, FV is just the 'empty' volume available to the molecule when the molecules' own (hard-core or closed-packed V*) volume is substracted. The free-volume is not the only concept, which is loosely defined in this discussion. Even the hard-core volume is a quantity difficult to define and various approximations are available. Elbro et al.21 suggested using V*=VW, i.e. equal to the van der Waals volume (Vw), which is obtained from the group increments of Bondi and is tabulated for almost all existing groups in the UN1FAC tables. Other investigators23'24 interpreted somewhat differently the physical meaning of the hard-core volume in the development of improved free-volume expressions for polymer solutions, which employ Eq. 13 as basis, but with V values higher than Vw. Table 6 shows that, due to the closed packed structure of molecules, a higher value of the hard-core volume would have been expected e.g. around 1.2-1.3 Vw. Indeed, investigations for athermal polymer systems (without any energetic interactions) demonstrate that the optimum results with Entropic-FV (discussed below) and for both the solvent and polymer activities are obtained when V*=1.2Vw (Fig.4). This observation regarding the magnitude of hard-core volume related to Vw has helped not only in developments of the Entropic-FV model, but as shown in Figure 5, also in understanding problems of hard-core volume theories such as the one proposed by Guggenheim. This particular hard-core volume theory has been often used in models for estimating diffusion coefficients for polymeric systems. Table 6. Values of Packing Density and of the ratio V'/Vw=Vo/Vw for various packing of fluids, as well as for various fluid families23. Structure/Compound Open-packed cubic structure of spheres Closed-packed cubic structure of spheres Open-packed arays of infinite cylinders Close-packed arays of infinite cylinders Polyethylene Most organic compounds Random densely packed mixture of spheres with log normal size distribution
Packing Density (po)
v7v w =v«/v w
0.52 0.74 0.78 0.90 0.76 0.7....0.78 up to 0.8
1.92 1.35 1.27 1.11 1.31 1.43 .... 1.28 down to 1.25
155
Figure 4. Percentage deviation between experimental and calculated solvent infinite dilution activity coefficients, versus the a-parameter in the free-volume expression of the Entropic-FV model (Vf=V-aVw). From Kouskoumvekaki et al.23.
Figure 5. Plot of the ratio V /Vw calculated from Guggenheim's hard-core volume equation (V* = 0.286Vc) as a function of the van der Waals volume Vw for n-alkanes. The critical volume (Vc) is obtained from two different sets of experimental data, those by Teja and those by Steele. Only those data by Teja have been verified by independent investigations based on molecular simulation. The plot shows that, using these "correct" data, the V /Vw plot does not follow the physically expected trend. (Modified from Kontogeorgis et al.24). The original UNIFAC model does not account for the free-volume differences between solvents and polymers and, as a consequence of that, it highly underestimates the solvent activities in polymer solutions2122'25. On the other hand, the various modified UNIFAC
156 versions (Lyngby and Dortmund, see chapter 4), which use exponential segment fractions, are also inadequate for polymer solutions. Although, their combinatorial terms are more satisfactory for alkane systems, they fail completely for polymer-solvent systems and as shown26 they significantly and systematically overestimate the solvent activities. Although these UNIFAC models are not adequate for polymer solutions, the problem seems, however, to lie more in the combinatorial term rather than the residual (energetic) term. In other words, improvements are required especially for describing the free-volume effects, which are dominant in polymer solutions. 7.3.2 The UNIFAC - FV model Various modifications - extensions of the classical UNIFAC approach to polymers have been proposed. All these approaches attempt to include the FV effects, which are neglected in the UNIFAC combinatorial term. All of them employ the energetic (residual) term of UNIFAC. The most well-known is the UNIFAC-FV model by Oishi and Prausnitz25: lny, =lny™m(J + l n y r +W"
(14)
The combinatorial and residual terms are obtained from original UNIFAC. An additional term is added for the free-volume effects. An approximation but at the same time an interesting feature of UNIFAC-FV, and other models of this type, is that the same UNIFAC group-interaction parameters - i.e. those of original UNIFAC- are used. No parameter reestimation is performed. The FV term used in UNIFAC-FV has a theoretical origin and is based on the Flory equation of state:
lnyr=3c,.ln[li^)j-C,{f^-lTl-^]1
(15)
where the reduced volumes are defined as: bVhw
(16)
-
= m
wyl+w7y1 b{WlVLw+w2Vlw)
In Eq.16, the volumes Vj and the van der Waals volumes are all expressed in cnrVmol. Wj is the weight fraction. In the UNIFAC-FV model as suggested by Oishi and Prausnitz25 the parameters Cj (3CJ is the number of external degrees of freedom) and b are set to constant values for all polymers and solvents (CJ=1.1 and b=1.28). The performance is rather satisfactory, as shown by many investigators, for a large variety of polymer-solvent systems. Some researchers have suggested that, in some cases, better agreement is obtained when these parameters are fitted to experimental data27.
157 Originally, the UNIFAC-FV model was developed for solvent activities in polymers. It could be expected that the model (Eqs.14-16) is also valid for estimating polymer activities. However, such an application of UNIFAC-FV is rather problematic28. It has been shown23 that the performance of UNIFAC-FV in predicting the activities of heavy alkanes in shorter ones is not very good. Such problems limit the applicability of UNIFAC-FV to cases where the polymer activity is also of importance such as liquid-liquid equilibria for polymer solutions. Indeed, to our knowledge, UNIFAC-FV has not been applied to polymer-solvent LLE. 7.3.3 The Entropic-FV model A similar but somewhat simpler approach to UNIFAC-FV, which can be readily extended to multicomponent systems and liquid-liquid equilibria, is the so-called Entropic-FV model proposed by Elbro et al.2i and Kontogeorgis et al.22: \nli=\n1';omb-fv+\nyrjes fv
fv
lnyfo^=ln^ + */ fv _
xMjv i
l-^ */
>
(1?)
i
lny|"es -^UNIFAC
(chapA)
As can been seen from Eq. 17, the free-volume definition given by Eq.13, is employed. The combinatorial term of Eq. 17 is very similar to that of Flory-Huggins. However, instead of volume or segment fractions, free-volume fractions are used. In this way, both combinatorial and free-volume effects are combined into a single expression. The combinatorial - FV expression of the Entropic-FV model is derived from Statistical Mechanics, using a suitable form of the generalised van der Waals partition function. The residual term of Entropic-FV is taken by the so-called 'new or linear UNIFAC model, which uses a linear-dependent parameter table29: amn=amnJ +amn2(T-T0)
(18)
This parameter table has been developed using the combinatorial term of the original UNIFAC model. As with UNIFAC-FV, no parameter re-estimation has been performed. The same group parameters are used in the "linear-UNIFAC" and in the Entropic-FV models. A common feature for both UNIFAC-FV and Entropic-FV is that they require the volumes of solvents and polymers (at the different temperatures where application is required). This can be a problem in those cases where the densities are not available experimentally and have to be estimated using a predictive group-contribution or other
158 method, e.g. GCVOL30'31 or van Krevelen methods. These two estimation methods perform quite well and often similarly even for low molecular weight compounds or oligomers such as plasticizers, as shown in figure 6 for the family of phthalates. Both UNIFAC-FV and Entropic-FV, especially the former, are rather sensitive to the density values used for the calculations of solvent activities.
Figure 6. Volumes of different phthalates calculated by GC-VOL and van Krevelen, compared to the experimental volumes taken from Ellington32 at 293.15 K (From Tihic10)
7.3.4 Results and Discussion The UNIFAC-FV and Entropic-FV models have been widely applied to polymer solutions and some typical applications are shown in tables 7-9, figures 7-12 and discussed in this section, and figures 1 -2 and tables 2-4 in the previous sections. Some of the most recent applications are included, while attention is paid to the advantages and shortcomings of the models. In some cases, comparisons with two group-contribution equations of state (GCFlory and GCLF) are presented. Vapor-Liquid Equilibria Both models have been extensively applied to vapour-liquid equilibria (VLE) - solvent activities in polymers and other size-asymmetric systems, including infinite dilution conditions for binary polymer solutions22'27'33, VLE for co-polymer/solvent systems34, solvent activities in dendrimer solutions35, VLE for a large variety of polar and hydrogen-bonding systems36'37, VLE for paint-related polymer solutions including commercial epoxy resins37'11, and recently also for VLE of ternary polymer-mixed solvent systems38.
159 Table 7 shows the performance of the models for infinite dilution activity coefficients for some polyisoprene (PIP) solutions, while comparisons for complex systems (from a recent comparative study, Lindvig et al.37) are shown in Table 8. Finally, results for some ternary polymer-mixed solvent systems are shown in Table 9. Table 7. Prediction of infinite dilution activity coefficients for PIP systems with three predictive group contribution models. Experimental values and calculations are at 328.2 K
Exper. value PIP systems 68.6 +acetonitrile 37.9 +acetic acid 7.32 +cyclohexanone 17.3 +acetone 11.4 +MEK 4.37 +benzene +1,2 dichloroethane 4.25 1.77 +CC14 6.08 +1,4 dioxane 4.38 +tetrahydrofurane 7.47 +ethylacetate 6.36 +n-hexane 2.13 +chloroform ng: no groups available
UNIFAC-FV 52.3 (24%) 17.7(53%) 4.6 (38%) 13.4 (23%) 10.1 (12%) 4.4 (0 %) 6.5 (54%) 1.8(0%) 5.9 (2%) 3.9 (10%) 6.6(11%) 4.6 (27%) 2.6 (20%)
GC-Flory Ng 50.0 (32%) Ng 10.5 (39%) 7.5 (35%) 2.8 (37%) 6.6 (55%) 1.6(11% Ng Ng 4.4(41%) 3.8 (39%) 2.6 (24 %)
Entropic-FV 47.7(31%) 33.5 (12%) 5.4 (27%) 15.9(8%) 12.1 (6%) 4.5 (2.5%) 5.5 (29%) 2.1 (20%) 6.3 (4%) 4.9(14%) 7.3 (2 %) 5.1 (20%) 3.00(41%)
Table 8. Percentage deviation between calculated and experimental solvent activity coefficients from various thermodynamic models
Ntnpdarsdverts FCLMVBR
FCL FDVB FBZE
FED RVM\
FKIE FTO FVA; FWE A/.%da/.
BV 24 169 53 53 183 462 24 35 7.4
17.2
*****
25 63 183 466 23 4.0 Q8
425 4.4 25 433 1.0 63 56
7.2 131 96
*****
*****
36 366
31
137
14.1
7.2
163
222
UFV QCF 99 14.1
LFV CCF UNRflC BV ***** ***** ***** ***** ***** ***** ***** *****
UNFA:
EV
224
*****
17.8
*****
31
632
324.8
34.2
224
154 458 7.6
337 88
47.7
51.1
44.3
11.3
r-J/dqpi bcrdng schrais
Ftiar adverts
7.6
34.7 *****
LFV QCF LNFyOC ***** ***** ***** ***** ***** *****
628 358 121 450
***** ***** ***** *****
***** ***** ***** *****
***** ***** ***** *****
*****
*****
1.4 86 *****
31 240 *****
23 11.7 *****
28
7.3
4.2
20.8
11.0
134 9.6
190
28 64 *****
21.7
51.5
27.9
35
***** ***** *****
160 Table 9a: Average logarithmic deviations (xlOO) between experimental and predicted vapor phase mole fractions for some ternary polymer- mixed solvent systems. The color indications divide the deviations into the following groups: Grey: less than 20 %. Light grey: 20 - 50 %. Dark grey: above 50 %. Polymer PMMA PS PS PS PS PS. 7 = 373.15 K PS, 7 = 393.15 K PS. T = 413.15K
Solvent Butanone Toluene Benzene Toluene Toluene Ethylbenzene Toluene Cvclohexanc Chloroform Carbontetrachloride Styrcne Ethylbenzene Styrcne Ethylbenzene Styrene Ethylbenzene
SAFT FFV/UQ 49 222 23 22 40 37 6 14 18 9 9 3 33 33 37 37 23 41 17 26 9 13
FH 47 153 24 41
Pa-Vc 48 154 23 39
17 8 31 34
8 2 23 22
FFV 52 196 14 20 4 14 9 3 34 37 25 45 20 32 10 15
UFV 51 194 14 20 4 13 9 4 27 27 27 51 21 34 11 17
GCI.F FH/Ha 45 37 137 89 23 22 37 36 5 12 15 14 17 5 8 34 27 37 28 30 26 60 | 48 23 20 39 32 12 10 21 16
Table 9b: Average logarithmic deviations (xlOO) between experimental and predicted pressures for some ternary polymer-mixed solvent systems. Color indications as in Table 9a. System PMM4-butanone-tolLme PS-benzHie-toluene PS-toluene-ethylbenzene PS-toliHK-cyclohsxare PS-cHcrofaTrKarbortetrachloride PS-styrene-etlTylbenzere, T=373.15 K PS-styrene-ethylberEerE, T=393.15K PS-styrene-ethylbenzerB, r=413.15K
SAFT EFV/UC. 16 7 4
14 32 32 32
16
FH 15 12
Ri-Ve 16 7
14 11
8 4
EFV UFV GCLF FFPFfe 16 | 36 15 17 14 14 8 7 1 8 2 2 9z, 9z, 2 11 5 19 52 17 34 31 31 33 35 30 30 33 34 28 26 32
Figures 7 and 8 also show recent results 35 , demonstrating the performance of the models for dendrimer solutions, including the sensitivity of the calculations to the density value employed.
161
Figure 7. Experimental and predicted activities of methanol in the dendrimer PANAM-G2 with the UNIFAC-FV and the Entropic-FV models35. Results are shown using experimental and predicted densities (Reprinted with permission).
Figure 8. Experimental and predicted activities of acetone in the dendrimer A4 with the UNIFAC-FV and Entropic-FV models33. Results are shown using experimental and predicted densities (Reprinted with permission). Overall, we can conclude that the Entropic-FV and UNIFAC-FV models, especially the former, provide satisfactory predictions of solvent activities, even at infinite dilution, for complex polar and hydrogen bonding systems including solutions of interest to paints and coatings, and rather satisfactory predictions when mixed solvents are present. Some more specific comments can be made from comparative investigations for different types of systems.
162 Athermal systems The articles cited in this section include investigations, which compare the performance of the UNIFAC-FV, Entropic-FV and several more recent free-volume equations for athermal systems. In such systems the energetic effects are zero or very small and they can be thus used for testing the combinatorial and free-volume terms of the models. Although the database used in the various investigations is not always the same, it typically consists of solutions having components differing significantly in size but which do not exhibit energetic interactions. Examples of these nearly athermal systems are solutions of polyethylene and polyisobutylene with alkanes (only solvent activities are available), alkane solutions (where both the activity of light and heavy-chain alkanes are available), polystyrene/ethylbenzene, polyvinyl acetate/vinyl acetate as well as "pseudo" experimental data for polymer activities generated with molecular simulation techniques. In their recent review, Pappa et a/.33, considered over 200 experimental datapoints for athermal polymer solutions at intermediate concentrations and about 100 points at infinite dilution and compared the Entropic-FV and Zhong-Masuoka39 models. They found that the Entropic-FV formula yields lower error than the Zhong term, though the latter does not contain any volume terms (9% vs. 16%). Other literature comparisons40 also agree that the free-volume models with the volume-containing terms perform better than those models requiring no volume information. Thus, Entropic-FV, Flory-FV and related models provide a good basis for building a full thermodynamic model for polymers. UNIFAC-FV seems to offer no advantage over the simpler approaches and seem to be more sensitive1122'37 to the volume values employed compared to simpler free-volume equations. Despite the overall successful performance of Entropic-FV and UNIFAC-FV models for a large number of systems and types of phase equilibria, it has been shown over the last years by a number of researchers23'40"43, that the combinatorial/free-volume terms of both the Entropic-FV and UNIFAC-FV models have a number of deficiencies: i.
ii.
iii.
The solvent activities in athermal polymer solutions are systematically underestimated by, often, 10% (in the case of Entropic-FV) or more (for UNIFACFV). For athermal systems, the residual term is zero. Such an underestimation cannot be entirely attributed to the small interaction effects present in such systems. The activities of heavy alkanes in short-chain ones, available from SLE measurements, are in significant error, especially as the size difference increases. Due to the lack of experimental data on polymer activities, such SLE data can help test the models' applicability for the activities of heavy-compounds. The performance of the models is rather sensitive to the values used for the polymer density.
Numerous investigations and developments of new combinatorial/free-volume terms have been reported over the last 5 years. The general conclusions that can be drawn are: i.
The activities of alkane solvents in either alkane or athermal polymer (PE, PIB) solutions are very satisfactorily predicted (much better than with the Entropic-FV formula) by some more recent modified free-volume equations e.g. Chain-FV, pFV and R-UNIFAC. However, these models cannot be extended to multicomponent
163
ii. iii.
iv.
systems. This is a serious limitation for multicomponent systems. The Flory-FV and a recently developed model23 do not suffer from this limitation. Volume-based models perform better than those not including volume-containing terms. The UNIFAC-FV expression, the first free-volume equation proposed, which is derived from the theory of Flory, is not as successful for athermal systems compared to more recent simpler equations. This may be due to the values of the parameters b and c employed in this model. Fitting these parameters may improve the performance of the UNIFAC-FV term. The results with this model seem particularly sensitive to the density values employed. All models perform clearly less satisfactorily for the activities of heavy alkanes in short-chain ones, especially as the size-asymmetry increases. Models without freevolume corrections such as UNIFAC, and Flory-Huggins are particularly poor in these cases. Unfortunately, such activity coefficient measurements, which could bee used for testing the performance of the models for the activities of polymers, are scarce. Direct measurements for polymer activities have not been reported. Molecular simulation studies can offer help in this direction43.
Non-polar and slightly polar systems Numerous results (predictions and correlations) are available for such systems23'33'40"43. Many models perform satisfactory even when pure predictions are considered. In a recent comparison, Pappa et al.33 showed that Entropic-FV performs better than the Zhong-Masuoka model (11% vs. 20%). In the review by Lee and Danner44, GCLF (Group-Contribution Lattice Fluid Equation of State) and Entropic-FV perform similarly for non-polar systems (15%), but GCLF appears to perform better for the weakly polar ones. This is attributed to problems of the Entropic-FV model for systems containing polyacrylates and polymethacrylates with acetates. UNIFAC-FV has an average error of 23% for these types of systems and GC-Flory of 20%. Water-soluble polymers and other hydrogen-bonding systems Predictions have been provided for some hydrogen bonding systems with a number of models. Pappa et a/.33 report an average deviation of 26% (in VLE) with both Entropic-FV and Zhong-Masuoka models, which is higher than the deviations observed for non-polar and polar systems. Lee and Danner's44 comparison revealed that Entropic-FV is the best model for strongly polar solvents (23%), followed by GCLF (28%) and GC-Flory (31%). UNIFACFV does not seem to be very successful for such complex systems (mean deviation 65%). In some recent investigations ' , several of these well-known group contribution models (Entropic-FV, UNIFAC-FV, GC-Flory) have been tested for VLE of paint-related systems. These are systems of polyacetates, polyacrylates, polymethacrylates, epoxies and a variety of solvents (non-polar, polar, hydrogen bonding, water). The performance of the models is overall similar with the Entropic-FV and GC-Flory being overall better than UNIFAC-FV in most situations, in agreement to the investigations reported earlier. Some results are shown in Table 8.
164 Co-polymer solutions Comparisons for co-polymer systems are not extensive, although several VLE data for solvent/co-polymers are available. Bogdanic and Fredenslund34, Pappa et a/.33 and Lee and Danner44 have presented comparisons for such systems, using the models Entropic-FV, GCFlory, Zhong-Masuoka, UNIFAC-FV and GCLF. In their comparison, Pappa et a/.33 found that both Entropic-FV and Zhong-Masuoka models perform similarly for these systems with a deviation around 20%. Similar overall performance for the Entropic-FV, GC-Flory and UNIFAC-FV models was observed by Bogdanic and Fredenslund ', though the various models perform different for specific co-polymer systems. For example, the Entropic-FV model has problems in the presence of chloro-groups. GCLF is also shown to be quite successful for a number of co-polymer solutions in mostly non-polar/slightly polar solvents. Polymer-mixed solvent systems In a recent investigation38, a database for ternary VLE systems (polymer-mixed solvents) has been compiled and used for evaluating the performance of several group contribution models (Entropic-FV, UNIFAC-FV, and GCLF). The performance of these predictive models, though inferior to the binary systems, can be considered quite satisfactory, considering also the experimental uncertainties involved in these measurements. The experimental measurements of solvent activities in mixed solvent/polymer systems are not easy and may be often associated with significant errors (J.M.Prausnitz, 2000. Personal Communication). Liquid-Liquid and Solid-Liquid Equilibria The application of free-volume models to liquid-liquid and solid-liquid equilibria of polymer solutions is much more limited compared to VLE, and only Entropic-FV has been widely used in such cases, and for both polymer solutions and blends. Entropic-FV has been applied to SLE of asymmetric alkane solutions , LLE for binary polymer solutions and polymer blends46, SLLE for semicrystalline polymer/solvents47, as well as for LLE for ternary polymer-solvent-solvent (and solvent-antisolvent) systems48. Figures 9-12 show typical LLE results for binary and ternary polymer-solvent solutions with Entropic-FV. Entropic-FV can be readily applied to liquid-liquid and solid-liquid equilibria and can predict all types of phase diagrams present in polymeric systems (UCST, LCST, hourglass-type) e.g. the results in figure 9 for PS/acetone. However, the results are of qualitative than of quantitative value in most cases. A difference of 10-30 degrees should be expected in the predictions. The performance of Entropic-FV seems rather system-specific, e.g. for polyethylene/octylphenols the difference in UCST is 5-10 °C while for polyethylene/octanol it is approximately 40 °C. In the cases of polymer blends46, where freevolume effects are not very important, the model deviates substantially from experimental data, although it can predict the UCST-type behavior. Compared to the other free-volume models, Entropic-FV may be considered as the most successful and widely used extension of UNIFAC to polymers. Besides binary polymer-solvent LLE, Entropic-FV has been also applied to ternary polymer-solvent LLE48, and compared to the Holten-Andersen et al. equation of state (a
165 previous version of the GC-Flory EoS). The polymer/mixed solvent systems considered include both two solvent-solvent and solvent-anti-solvent systems. The comparison was limited to eight PS-two solvent systems (benzene/acetone, benzene/methanol, methylcyclohexane/acetone, toluene/acetone, MEK/acetone, ethyl acetate/acetone, NNDMF/cyclohexane) and one PMMA system (with chlorobutane/butanol-2) for which full data are available. Qualitatively good results are obtained with both models, especially Entropic-FV (despite the fact that all group interaction parameters were based on low pressure VLE of non-polymeric systems). Typical results are shown in figures 11 and 12. Finally, in one of the very few works reported on the prediction of solid-liquid-liquid equilibria47 for polymer solutions, the Entropic-FV and UNIFAC models have been shown to yield similar results for SLLE.
Figure 9. Correlation of PS/acetone LLE with Entropic/FV model (using GCVOL for the density of the polymer)45. The numbers (4800, 10300, 19800) correspond to the molecular weight of the polymer. (Reprinted with permission)
166
Figure 10. PS/cyclohexane LLE prediction with various predictive group contribution models43 (Reprinted with permission).
Figure 11. Ternary LLE for PS(300000)/benzene/methanol at T=298.15 K48. (Reprinted with permission)
167
Figure 12. Ternary LLE for PS(300000)/methyl cyclohexane/acetone, T=298.15 K48. (Reprinted with permission) 7.3.5 The Entropic-FV/UNIQUAC model Both UNIFAC-FV and Entropic-FV are group contribution models. This renders the models truly predictive, but at the same time with very little flexibility if the performance of the models for specific cases is not satisfactory. An interesting alternative approach is to employ the UNIQUAC expression for the residual term. This Entropic-FV/UNIQUAC model has been originally suggested by Elbro et a/.8'17'21 and has shown to give very good results for polymer solutions if the parameters are obtained from VLE data between the solvent and the low molecular weight monomer (or the polymer's repeating unit). The Entropic-FV/UNIQUAC model has been recently further developed and extended independently by two research groups49"51. Both VLE and LLE equilibria are considered but the emphasis is given to LLE. Very satisfactory results are obtained as can be seen for two typical systems in figures 13 and 14. It has been demonstrated that the EntropicFV/UNIQUAC approach can correlate both UCST/LCST and closed loop behavior49'50 and even show the pressure dependency of critical solution temperatures (UCST and LCST)31. 7.3.6 Extension to Semi-crystalline Polymers and Swelling When highly crystalline or cross-linked polymers are considered, e.g paints after drying, rubbers, polyolefins, the effects of cross-linking and crystallinity should be considered because they affect the solubility. Cross-linking and crystallinity are often visualized as 'similar' (in some sense) phenomena and are described with the same theories: crystalline regions are assumed to act as 'physical or giant cross-links'.
168
Figure 13. Correlation of LLE for PBMA/MEK system49. • Exp.data (Mw=200000 g/mol); correlation
Figure 14. Correlation and prediction of LLE for HDPE/1-dodecanol system49. A Exp.data (Mw=60700 g/mol), correlation o Exp.data (Mw=77800 g/mol), prediction • Exp.data (Mw=21300 g/mol), — — prediction • Exp.data (Mw=6800 g/mol), prediction
169 Crystalline and cross-linked polymers do not dissolve (with a few exceptions) in solvents but only swell. Swelling equilibria is thus important. To account for the crystalline/cross-linking effect, an additional factor (elastic term) is typically required in thermodynamic models. Two popular theories to account for this effect are the Flory-Rechner12:
lnaf=^%>'3
(19)
M
c
where: p a is the density of the amorphous polymer Vi is the molar volume of the solvent Mc is the molecular weight between cross-links and the Michaels-Hausslein53 equation:
p
, . £h' -(K)]-»» In a, =—
CT;
r-^S [3/(2ftp2)-l]
F—T
(20)
where: Tm is the melting point temperature f is the fraction of elastically effective chains in amorphous regions As observed from these equations, both theories introduce at least one extra parameter, which needs to be determined from experimental data: the molecular weight between crosslinks Mc and the fraction of elastically effective chains f. They have been combined with free-volume models ' '"" and they have been applied to semicrystalline polymer/solvent systems. The results are satisfactory but they are not predictive: the Mc and f parameters should be estimated from experimental data. However, the swelling of cross-linked polymers can be estimated with such equations. 7.3.7 Extension of Free-Volume Models to Gas Solubilities in Elastomers Thorlaksen et al.56 have recently combined the Entropic-FV term with Hildebrand's regular solution theory and developed a model for estimating gas solubilities in elastomers. The socalled Hildebrand - Entropic FV model is given by the equation: lny,=lny? + lny™'v
'* lny^'
(21)
R-T = t n
^2_ X,
+ 1_^2_
( 2 2 )
X7
170 where : 81 = solvent solubility parameter, 82= gas solubility parameter X2 = gas mole fraction in liquid/polymer. 02 is the 'apparent' volume fraction of solvent, given by: x2 -V'2
O,=
r -V1' + x • VL
and O2Ff is the 'free-volume' fraction given by: 2
~x2-(y<--vrhx]-{v<--vr)
Vj is a hypothetical liquid volume of the (gaseous) solute.
;
VJ--[Sl-5iy-Oii
f<
— = ^-exp
—
'-
(23)
A £
f2 is the fugacity of the gas and f'2 is the fugacity of the hypothetical liquid, which can be estimated from the equation: f1
4 74547
+ 1.60151-T -0.87466 -T2 +0.10971 -T3
ln^- = 3.54811—: Pc Tr
(24)
Finally, the gas solubility in the polymer is estimated from the equation:
; — r
f> =
l^.exp\ *8
(vl--{5,-S 2y-O] 2 ^ 1
I
11
R-T
, Of
L + in^^
, Of) +
x
i
i_\ x
(25) \
J 2
Calculations showed that the hypothetical gas "liquid" volumes are largely independent to the polymer used, and moreover, for many gases (H2O, O2, N2, CO2 and C2H2) these are related to the critical volume of the gas by the equation: V!,- = 1.776Vc -86.017 Very satisfactory results are obtained as shown in Table 10 and Figure 15.
(26)
171
Table 10. Summary of the performance of the models tested at T = 298 K; P = 101.3 kPa Errors associated with models for predicting gas solubilities in polymers5 .
Hildebrand/ Polymer
PIP
PIB
PBD
PDMB
PCP
Gas
Hildebrand-
Entropic FV
Entropic FV -
-1
2
N2
14.7
73
3.9
-7.9
-4.6
o2
-16.1
-4
14 -
10.8
11.8
29.4
-22.2
8.7
4.6
Ar
-32.5
-23
CO,
-3.2
4.5
N2
-2.5
13 -
6.8
3.1
5.0
o2
-6.1
-
-1.7
-8.3
1.7
Ar
-
-
-
32.8
-
-
CO2
-1.9
41.1
35.2
N2
22.3
-
8.1
8.1
12.6
o2
14.9
-
-6
8.7
10.8
Ar
12.1
-
-
111.1
24.0
CO,
-9.7
-
-4.6
0.4
-4.0
N2
-
-
-23
-7.5
-3.1
o2 Ar
-
-
-32
-
-
-
-16.8 -
-15.9 -
CO,
-
-
-24
2.3
-2.2
N2
58.1
-
49
-7.0
-4.2
43.7 -
-
60
-1.4
-1.4
-
-
-
-
8.8
-
27
-13.3
-17.1
19.8
28
18
16.8
10.6
o2 Ar CO, AAD:
Michaels/Bixler Tseng/Lloyd
Hildebrand-
Hildebrand Entropic-FV-1: The liquid volume of the gas is determined from its relationship with the critical volume, Eq. (26). Hildebrand Entropic-FV-2 : The average hypothetical liquid volume of a gas is used
172
Figure 15. The solubility of several gases in PIP as a function of temperature predicted by the Hildebrand Entropic-FV model.
7.4 EQUATIONS OF STATE FOR POLYMERS Many equations of state (EoS) have been proposed for polymers, for a recent review see Kontogeorgis . Both cubic equations of state (vdW, SRK, PR, SWP) and non-cubic ones esp. GCLF and SAFT have been applied with success. In many of the cubic EoS, mixing rules based on the EoS/GE principle have been employed using one of the previously described activity coefficient models (FH, E-FV, U-FV) in the mixing rules. In this case, the behavior of the activity coefficient model at low pressures is reproduced. In other cases (cubic EoS, SAFT) a single interaction parameter is used or group-based parameters (GCLF) are employed.
173 Numerous applications of these EoS have been reported: low and high-pressure VLE, Henry's law constants and LLE for systems including polymers, co-polymers and blends. Examples of what is possible with such EoS are shown in figures 16-20: LLE for polymer solutions with a cubic EoS58 and VLE and LLE for binary and ternary systems with a recently developed modified SAFT EoS59"61. This is still a very active open area of research and is difficult to recommend a specific approach. A serious problem with all EoS for polymers which, in our view, has not been adequately addressed as yet is the way to get the EoS parameters for polymers. Methods employed for low molecular weight compounds (see chapters 5 and 6) cannot be used since critical properties and vapour pressure data are not available (have no meaning) for polymers. Numerous indirect methods57 have been employed using volumetric data and additional information, often phase equilibria data for mixtures of polymers with low molecular weight compounds. Such methods may be necessary since use of volumetric data alone do not seem to provide polymer EoS parameters useful for phase equilibrium calculations. Use of phase equilibria data, on the other hand, may render the parameters of pure polymers sensitive to the type of information employed. A thorough investigation on methods to obtain meaningful polymer parameters for equations of state will significantly improve and enhance the applicability of equations of state for polymers. A first effort towards this direction has been recently reported for the simplified PC-SAFT equation of state62.
Figure 16. Correlated UCST phase diagrams with the van der Waals equation of state for PS/cyclohexane at various molecular weights38. A single interaction parameter is used (per molecular weight).
174
Figure 17. Predicted and correlated UCST phase diagrams with the van der Waals equation of state for PBMA(11600)/n-pentane:>8. A single interaction parameter is used. The predicted interaction parameter is based on a simple correlation with the solvent's molecular weight.
Figure 18. VLE prediction with the PC-SAFT equation of state for PVAC(170000)-acetone at three temperatures (the interaction parameter is set equal to zero). The PC-SAFT developed by von Solms et a/.59 is employed.
175
Figure 19. LLE with the PC-SAFT equation of state for PS-methylcyclohexane using a single interaction parameter.
Figure 20. Ternary LLE with the PC-SAFT equation of state for the ternary system PS(300000)-acetone-methylcyclohexane. The binary parameters are regressed from the binary systems.
176 LIST OF ABBREVIATIONS BR CST DBP DDP DEP DHP DMP DOA DOP DOS EoS Exper./exp. EFV FH FV GC GC-Flory GCF GCLF GCVOL HA Ha HDPE LCST LLE MEK MW NRTL Pa-Ve PBD PDMB PBMA PDMS PE PEO PIB PIP PMMA PR Pred. PS PVAC PVC
Butadiene Rubber Critical Solution Temperature dibutyl phthalate didecyl phthalate diethyl phthalate dihexyl phthalate dimethyl phthalate dioctyl adipate dioctyl phthalate dio(2-ethylhexyl) sebacate Equation of State Experimental Entropic-FV Flory-Huggins (model/equation/interaction parameter) free-volume Group Contribution (method/principle) Group Contribution Flory Equation of State Group Contribution Flory Equation of State Group Contribution Lattice Fluid Group Contribution Volume (method for estimating the density) Holten-Andersen equation of state Hansen solubility parameters High Density polyethylene Lower Critical Solution Temperature Liquid-Liquid Equilibria methyl ethyl ketone molecular weight non-random two liquid activity coefficient model Panayiotou-Vera equation of state Polybutadiene Polydimethylbutadiene Polybutyl methacrylate Polydimethylsiloxane Polyethylene Polyethylene oxide Polyisobutylene Polyisoprene Polymethyl methacrylate Peng-Robinsoon equation of state predicted Polystyrene Polyvinyl acetate Polyvinyl chloride
177 SCF SF SGE SRK S WP SLE SLLE UCST U-FV UNIFAC UQ vdW vdWl f VLE VOC
supercritical fluid Schultz-Flory plot solid-gas equilibria Soave Redlich Kwong equation of state Sako Wu Prausnitz equation of state Solid-liquid equilibria Solid-liquid-liquid equilibria Upper Critical Solution Temperature UNIFAC-FV Universal Functional Activity Coefficient Uniquac van der Waals equation of state van der Waals one fluid (mixing rules) Vapor-Liquid Equilibria Volatile Organic Content
REFERENCES 1. R. B. Seymour, Plastics vs. Corrosives, SPE Monograph Series, Wiley, 1982 2. Van Krevelen, Properties of polymers. Their correlation with chemical structure; their numerical estimation and prediction from additive group contributions, Elsevier, 1990. 3. A.F.M. Barton, Handbook of solubility parameters and other cohesion parameters, CRC Press, 1983. 4. A.F.M. Barton, CRC Handbook of polymer-liquid interaction parameters and solubility parameters, CRC Press, Boca Barton, FL, 1990. 5. J. Bentley and G.P.A. Turner, Introduction to Paint Chemistry and Principles of Paint Technology, 4th edition, Chapman and Hall, 1998. 6. CM. Hansen, Hansen Solubility Parameters. A User's Handbook, CRC Press, 2000. 7. J. Holten-Andersen and K. Eng, Progress in Organic Coatings, 16 (1988) 77. 8. H.S. Elbro, Phase Equilibria of polymer solutions - with special emphasis on free volumes, Ph.D Thesis. Department of Chemical Engineering, Technical University of Denmark, 1992. 9. J. Klein and H.E. Jeberien, Makromol. Chem., 181 (1980) 1237. 10. A. Tihic, Investigation of the miscibility of plasticizers in PVC, B. Sc. Thesis. Department of Chemical Engineering, Technical University of Denmark, 2003. 11. Th. Lindvig, M.L. Michelsen and G.M. Kontogeorgis, AIChE J., 47(11) (2001) 2573. 12. P.J. Flory, J. Chem. Phys, 9 (1941) 660. 13. D. C. H. Bigg, J. Appl. Polym. Sci., 19 (1975) 3119. 14. P.M. Doty and H.S. Zable, J. Polymer Sci, 1 (1946) 90. 15. C. Qian, S.J. Mumby and B.E. Eichinger, Macromolecules, 24 (1991) 1655. 16. Y.C. Bae, J.J. Shim, D.S. Soane and J.M. Prausnitz, J. Appl. Polym. Science, 47 (1993)1193. 17. J.M. Prausnitz, R.N. Lichtenthaler and E.G.D. Azevedo, Molecular thermodynamics of Fluid Phase Equilibria. Prentice-Hall International. 3rd Edition, 1999.
178 18. Th. Lindvig, M.L. Michelsen and G.M. Kontogeorgis, Fluid Phase Equilibria, 203 (2002) 247. 19. A.R. Schultz and P.J. Flory, J. Amer. Chem. Soc, 75 (1953) 496. 20. A. Bondi, Physical Properties of Molecular Crystals, Liquids and Glasses; John Wiley and Sons: New York, 1968 21. H.S. Elbro, Aa. Fredenslund and P. Rasmussen, Macromolecules, 23 (1990) 4707. 22. G.M. Kontogeorgis, Aa. Fredenslund and D.P. Tassios, Ind. Eng. Chem. Res., 32 (1993)362. 23.1. Kouskoumvekaki, M.L. Michelsen and G.M. Kontogeorgis, Fluid Phase Equilibria, 202(2) (2002) 325. 24. G.M. Kontogeorgis, LA. Kouskoumvekaki and M.L. Michelsen, Ind. Eng. Chem. Res., 41(18) (2002) 4848. 25. T. Oishi and J.M. Prausnitz, Ind. Eng. Chem. Process Des. Dev., 17(3) (1978) 333. 26. G.M. Kontogeorgis, Ph. Coutsikos, D.P. Tassios and Aa. Fredenslund, Fluid Phase Equilibria, 92(1994)35. 27. J.R. Fried, J.S. Jiang and E. Yeh, Comput. Polymer Science, 2 (1992) 95. 28. L.A. Belfiore, A.A. Patwardhan and T.G. Lenz, Ind. Eng. Chem. Res., 27 (1988) 284294. 29. H.K. Hansen, B. Coto and B. Kuhlmann, UNIFAC with lineary temperaturedependent group-interaction parameters, IVC-SEP Internal Report 9212, 1992. 30. H. S. Elbro, Aa. Fredenslund and P. Rasmussen, Ind. Eng. Chem. Res., 30 (1991) 2576. 31. E.C. Ihmels and J. Gmehling, Ind. Eng. Chem. Res., 42(2) (2003) 408-412. 32. J.J. Ellington, J. Chem. Eng. Data, 44 (1999) 1414. 33. G.D. Pappa, E.C. Voutsas and D.P. Tassios, Ind. Eng. Chem. Res., 38 (1999) 4975. 34. G. Bogdanic and Aa. Fredenslund, Ind. Eng. Chem. Res., 34 (1995) 324. 35.1. Kouksoumvekaki, R. Giesen, M.L. Michelsen and G.M. Kontogeorgis, Ind. Eng. Chem. Res., 41(19) (2002) 4848. 36. G.M. Kontogeorgis, Aa. Fredenslund, I.G. Economou and D.P. Tassios, AIChE J., 40 (1994)1711. 37. Th. Lindvig, L.L. Hestkja;r, A.F. Hansen, M.L. Michelsen and G.M. Kontogeorgis, Fluid Phase Equilibria, 663 (2002) 194. 38. Th. Lindvig, I. G. Economou, R.P. Danner, M.L. Michelsen and G. M. Kontogeorgis, Modelling of multicomponent vapor-liquid equilibria for polymer-solvent systems, Fluid Phase Equilibria (in press). 39. C. Zhong, Y. Sato, H. Masuoka and X. Chen, Fluid Phase Equilibria, 123 (1996) 97. 40. E.N. Polyzou, P.M. Vlamos, G.M. Dimakos, I.V. Yakoumis and G.M. Kontogeorgis, Ind. Eng. Chem. Res., 38 (1999) 316-323. 41. E.C.Voutsas, N. S. Kalospiros and D. P. Tassios, Fluid Phase Equilibria, 109 (1995) 1. 42. G.M. Kontogeorgis, G.I. Nikolopoulos, D.P. Tassios, D.P and Aa. Fredenslund, Fluid Phase Equilibria, 127 (1997) 103. 43. G.M. Kontogeorgis, E.C. Voutsas and D.P. Tassios, Chem. Eng. ScL, 51 (1996) 3247. 44. B.-C Lee and R.P. Danner, AIChE J., 42 (1996) 837.
179 45. G.M. Kontogeorgis, A. Saraiva, Aa. Fredenslund and D.P. Tassios, Ind. Eng. Chem. Res., 34 (1995) 1823. 46. V.I. Harismiadis, A.R.D. van Bergen, A. Saraiva, G.M. Kontogeorgis, Aa. Fredenslund, Aa. and D.P. Tassios, AIChE J., 42 (1996) 3170. 47. V.I. Harismiadis and D.P. Tassios, Ind. Eng. Chem. Res., 35 (1996) 4667. 48. G.D. Pappa, G.M. Kontogeorgis and D.P. Tassios, Ind. Eng. Chem. Res., 36 (1997) 5461. 49. G. Bogdanic and J. Vidal, Fluid Phase Equilibria, 173 (2000) 241-252. 50. G. Bogdanic, Fluid Phase Equilibria, 4791 (2001) 1-9. 51. G.D. Pappa, E.C. Voutsas and D.P. Tassios, Ind. Eng. Chem. Res., 40 (2001) 4654. 52. P. J. Flory and J. Rechner, J. Chem. Phys., 11 (1943) 521. 53. M. J. Michaels and R.W. Hausslein, J. Polym. Sci. C , 10 (1965) 61. 54. J. S. Yoo, S. J. Kim and J. S. Choi, J. Chem. Eng. Data, 44 (1999) 16. 55. S. J. Doong and W. S. W. Ho, Ind. Eng. Chem. Res., 30 (1991) 1351. 56. P. Thorlaksen, J. Abildskov and G.M. Kontogeorgis, Fluid Phase Equilibria, 211 (2003) 17. 57. G. M. Kontogeorgis, Ch. 16: Thermodynamics of polymer solutions, in Handbook of Surface and Colloid Chemistry, 2nd ed., CRC Press, 2003. 58. V.I. Harismiadis, G.M. Kontogeorgis, A. Saraiva, Aa. Fredenslund and D.P. Tassios, Fluid Phase Equilibria, 100 (1994) 63-102. 59. N. v. Solms, M. L. Michelsen and G. M. Kontogeorgis, Ind. Eng. Chem. Res., 42(5) (2003) 1098. 60.1.A.Kouskoumvekaki, N.v.Solms, M.L.Michelsen and G.M.Kontogeorgis, Fluid Phase Equilibria, 215(1) (2004) 71-78. 61. Th.Lindvig, M.L.Michelsen and G.M. Kontogeorgis, Ind. Eng. Chem. Res, 43(4) (2004) 1125-1132. 62.1. A. Kouskoumvekaki, N. v. Solms, Th. Lindvig, M. L. Michelsen and G. M. Kontogeorgis, A novel method for estimating pure-component parameters for polymers: Application to the PC-SAFT equation of state, Ind. Eng. Chem. Res. (submitted).
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Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B.V. All rights reserved.
181
Chapter 8: Property Estimation for Electrolyte Systems Michael L. Pinsky and Kiyoteru Takano 8.1 INTRODUCTION This chapter provides an overview of some of the important properties of electrolyte systems. It briefly covers experimental methods for measuring some of the properties, their analysis (visualization) though phase diagrams and a collection of the most common methods (property models) for estimation of activity coefficients of ionic species in electrolyte solutions. As supplementary material, references [1-3, 15] are recommended. The property models presented in this chapter can be considered as semi-empirical and not truly predictive. Consequently, the models require a large number of model parameters, which means that a consistent set of data (parameters) are needed for the correct application of the selected property model. In most cases, the property models require. Standard Gibbs energy and enthalpy of ions (at 298 K) in aqueous solution Standard Gibbs energy and enthalpy of each solid phase Also, the association and dissociation reactions and corresponding data need to be supplied or retrieved from a suitable database. The chapter starts with an analysis of phase diagrams of electrolyte systems, followed by a brief discussion on how to measure some of the important properties that can help to develop property models for use in phase equilibrium calculations. A few sample calculation exercises are also given. Use of the property models in phase equilibrium and other related computations is presented in chapter 15 (Computational Algorithms for Electrolyte System Properties).
8.2 PROPERTIES & DATA ANALYSIS 8.2.1 Phase Diagrams Density, viscosity, vapor pressure, and other phase properties, can be displayed on phase diagrams as continuous contours within phase regions, each contour intersecting the phase boundaries. The successful visualization of bulk properties in complex phase space goes back to 1906, when Janecke [4] introduced his method for presenting the phase diagrams for multi-component solutions using a basis of 100 for the sum of moles of the components. Whether a mass or a mole fraction basis is used to represent the mixture in two-dimensions,
182 the bulk or solution properties can be displayed as continuously varying properties within the phase diagram spaces of each phase region. Contours of the property value are easy to read and are useful because they are placed in the context of the phase diagram. It does not matter that the phase space is drawn for up to two salts with a common ion in water, or in projected coordinates with more than two salts, as Janecke pointed out. Even with the seawater components graphed as a reciprocal salt pair diagram with mole-equivalent fraction coordinates, it was possible to show the contours of water vapor pressure, (in units of mmHg) for saturated solutions as shown in his original diagram below (see Figure 1). In this example, the lowest water vapor pressure of 7.3 mmHg occurs at the invariant point with phases carnallite-bishofite-kieserite, which is also the evaporative end-point in this system. In Figure 1, the Janecke projection diagram for the symmetrical or reciprocal salt pair system K-Mg-Cl-SO4-H2O at 25°C, the salts compositions are plotted on a mole-equivalent fraction basis; the x-axis is Mg/(Mg+K2) and the y-axis is SO4/(SO4+C12).
Figure 1: Janecke projection diagram for the symmetrical or reciprocal salt pair system K-Mg-Cl-SO4-H2O at 25°C Another property of projection diagrams Janecke [4] pointed out is that their crystallization paths radiate out from the phase as indicated by the composition of that phase on the diagram. Janecke shows the same phase diagram with its crystallization paths in Figure 2. These paths logically take the composition to its evaporative end-point, which is generally the invariant point composition with the least amount of water. Note his placement of arrows on each of the radiating lines through each of the bi variant phase regions and along certain of the univariant 2-salt (4 phases in total = 2 salts + vapor + liquid) boundaries. When the crystallization paths as in the top right for K2SO4 move away from that corner and intersect the boundary line with schoenite, all of the first phase must convert to the second phase at the boundary line, before the composition enters the schoenite region and moves
183 away from its pure phase point, shown on the upper edge at an (x, y) coordinate of (0.5, 1). When the schoenite crystallization path meets the univariant (2-salt) phase boundary with epsomite, because of their mutual directions, both salts co-exist and precipitate together as the composition moves downward on the boundary line towards the invariant point with sylvite (KC1). Janecke shows the arrows along the phase boundary. It is important when using a computer model that these concepts are understood and properly represented within the context of the phase diagram.
Figure 2: Janecke phase diagrams with crystallization paths. A triangular form of a Janecke projection phase diagram is shown in Figure 3 for three salts with a common ion in water. The system is shown for Na-COrSOzi-Cl-F^O, in mole fraction units. (Any composition basis can be used providing all of the salts add to 100%). In this diagram at 20°C, there is a transition point for Na2SO4 10H2O to the anhydrous for Na2SO4 at high NaCl concentrations. A transition point between forms of hydrate phases is not an invariant point since adding heat can change the composition or phase. The crystallization paths are shown for all phases. The path for Na2SO410H2O continues into the anhydrous Na2SO4 region. Arrows also indicate the direction taken along the boundary lines, when evaporating any composition within the bivariant phase regions. The evaporative end-point is shown at the 3-salt invariant point for burkeite-Na2CO310H2Ohalite. Other solution properties for saturated solutions can be similarly presented. Typically, with projections representing phase space, water is not explicitly shown in the diagram so water contours called isohydrols, are plotted, in the same way as the vapor pressures are in Janecke's figure. , moles of water h =^ ; — ^ moles salt; i
... (1)
184 For more analysis of phase diagrams, see also Pinsky & Grubber [5] and Takano [6],
Figure 3: Ternary diagram for the system Na-CO3-SO4-Cl-H2O (water-free basis) 8.2 2 Thermodynamic Properties Thermodynamic properties such as enthalpy, heat capacity, Gibbs energy, density and vapor pressure together with properties such as viscosity and surface tension are important from a process design point of view. They are also useful for constructing other property models (used for computation of phase equilibrium for electrolyte systems) and can generally be predicted by computational models, if experimental data is not available. Measured data for heat of dilution or heat of mixing and solution heat capacity are also very useful for evaluating electrolyte property model parameters and for estimating activity coefficients over a temperature range, especially for elevated temperatures well away from 25°C. In this section some of these properties are discussed in terms of how they can be obtained and/or calculated. Density Densities and viscosities of unsaturated and saturated multicomponent electrolyte solutions can be modeled using polynomial equations with adjustable coefficients for each of the electrolyte components. The measured property is generally a linear combination of the salt components. Data regression is done, for example with density, as the difference between solution density (dsolutmn) and water (d]mler) at the temperature measured multiplied by 1000 in order to enhance the sensitivity of the regression. The fitting equation is:
185
(^,,^-^,a,e,.)-1000 = XZ«,,,^;i;2
(2)
where there are / electrolyte components, aiM are fitted coefficients and X;11/2 are concentration taken in whatever (desired) concentration units with exponents using n =1 to 3. For dVater, either experimental data or a temperature dependent correlation [Handbook of Chemistry and Physics] can be used to obtain the needed property value at the specified temperature. Liquid Volume Liquid volume of mixture without salt is calculated using Racket equation as shown in Eq.4. (Ml-*)"')
Vj^'=^^_
(4)
In Eq.4, Tc and Pc axe critical temperature and pressure of mixtures under consideration respectively. Zm is Racket parameter. When salt is dissolved in the solution, the total volume is calculated, total volume, v""al, is calculated through Eq.5. V'""" = VmIJ™" * (1 - xsall) + Vj * xsall
(5)
Therefore, if the volume of salt is known, the total liquid volume can be calculated. One way to calculate the volume of salts, when it is dissolved in a liquid phase, is to relate the volume to the concentration of salt or ionic strength. If the data of liquid volume of aqueous solution involving inorganic salts is available, it is possible to create, for example, the following relation (see Eq.6, Zemaitis et al [7]): K,S = Vmsm + AxJ"
+ Bxsal, + CxJ-5
(6)
From values of density and concentrations of the electrolyte system the molal volume change upon dissolution can be calculated. This is an important excess property as it can be used for describing the pressure dependence of solubility.
,d\nK^ (
(-AV")
^=7^T
(7)
where AV° molal volume change of the reaction (dissolution) in the standard state, which is given by (for products, P and reactants R).
AV° =IAV°P-IAV°R
186 At high pressures this relationship can mean large changes in solubility. Dissociation Constant & Solubility Product Dissociation constant and solubility product are calculated using Eqs.8-9 as function of temperature. \nKlDIS=A+ — + CAnT
(8)
\nK/s("=A+-
(9)
+ C\nT
In Eqs.8-9, tc and K are the dissociation constant for the molecular dissociation and the solubility product for the solid formation, respectively. A, B and C are parameters and T is temperature in K. When the parameters A, B and C are not available, they are determined by regressing the experimental data. Otherwise, K°ls and KSOL are calculated using Gibbs energy of each chemical species as given by Eq.10. l n ^ = ^ RT
(10)
Example 1 Determine Gibbs energy for the solid phase at standard conditions, from saturated conditions for the mixed salt glaserite, Na2SO4.3K.2SO4 at saturated conditions. Na2SO4.3K2SO4 -> 2Na+ + 6K+ + 4SO4"2 Glaserite K5p = (Na+)2(K+)6(SO4-2)4
(KS0L = KSP)
The solution for the Pitzer model is given below.
ions-
AG^^-RTkiKsp From Pitzer model on a motel basis, ln£sp = 62.0799 + .1527-T = 107.61 AGiogS;, =-6.375^104 call mole &G°GlaLer!te = 2AG°Na + 6AG°K + 4AG^ D 4 - A G i o g ^
187 AG
LogKsp = J^AGLs
~
AG
Gfaw Be
ions
AGLogRp=-RT]nKsp From Pitzer model on a molal basis, kiKsp = 62.0799+ .1527-7 = 107.61 AG
LOSRP
= -6.375;dO 4 call mole
A G ° t e r t t e = 2AG;ra + 6AG° + 4AG° O 4 - A G I o g ^
Example 2 Calculate the solubility product for a mixed salt using measured data at 25C and true mole fraction basis of LiKSO4 Solution Procedure: Convert mass% data to moles of each component and then true mole fractions of each ion. True mole fraction is based on the moles of each ion (from L^SCU and K2SO4) divided by the moles of all ions and water. Ksp = (Li)(K)(SO4). Consider the six data points given below. No activity coefficients are used in this calculation. liquid phase mass% Li2SO4 K2SO4 24.6 2.8 20.93 3.25 18.1 4.1 13.05 8.09 12.1 9.2 12 9.2 11.9 9
residue mass% Li2SO4 K2SO4 35.8 46 35.05 45.95 34 42.8 31.35 45.99 35.1 55.5 32 49 31.6 48.1
Solid Phases LiKSO4 LiKSO4 LiKSO4 LiKSO4 LiKSO4 LiKSO4 LiKSO4
From the data points, Ksp=2.989xlO"5 AG Ksp = -RTlnKsp = 6172 cal/mole Calculate AG°i.iKso4, the standard Gibbs energy of formation for LiKSC>4 using the Gibbs energy at saturation for LiKSC>4, and the standard Gibbs energy for the ions: From the dissociation of LiKSO4, LiKSO4 -» Li + + K+ +SO4"2 AG Ksp = AG°Li + AG°K + AG°S04 - AG°LiKso4 so AG°LlKso4 = AG°Li + AG°K + AG°SO4 - AGKsp AGoLlKso4 = -70100-67700-177970-6179.9 = -321950 cal/mole =-1347040 J/mole Here, the value for AGKSP is only about 2% of the total value for the Gibbs energy of formation. One temperature is not sufficient to obtain the enthalpy of formation.
188 A similar value for the standard Gibbs energy of formation is obtained from the polynomial from data at a few temperatures and regressed with the electrolyte-NRTL activity coefficient model (see section 8.3.1) which also uses true mole fraction units and calculated activity coefficients. The lnKsp equation is available for these types of calculations. When using published values for lnKsp make sure the units are known and compatible with your system. The resulting equation is: lnKsp = -48.34 + 4644.6/T K.
At 25°C, lnKsp = -32.762
AGKsp = -RTlnK5p = 19409 cal/mole AG°LiKso4 = AG°Li + AG°K + AG°So4 - AGKsp= = -70100-67700-177970 -19409 = -335179 cal/mole = -1402390 J/mole The temperature dependence of the solubility is based on the enthalpy in this example. No heat capacity data is available for LiKSC>4. The second term in the polynomial [4644.6/T] contains the enthalpy change or heat of solution of LiKSC>4 at saturation, AHKSP= R. 4664.6K. A similar procedure is used to find the standard enthalpy of formation for L1KSO4. AHLi = -66560 cal/mole AHK = -60320 cal/mole AHSO4 = -217320 cal/mole AHKsp = 9229 cal/mol = AH°Li + AH°K + AH°SO4 - AH°LlKso4 AH°i.iKso4= -66560 - 60320 -217320 - 4644.6 = -348,845 cal/mole = -1,478,746 J/mole For salts that are not the stable equilibrium phase at standard conditions, obtain as much data at different temperatures as possible. Extrapolate to 25°C to obtain an estimate for Ksp and AG°sait. Consider these constants are adjustable parameters in all cases. Use the model to predict the solubility of the salt over the temperature range of interest. Make the necessary adjustments in the values until a best fit to the data is obtained. Improving Standard Values and Property Values at Temperatures of Interest Direct measurements for the enthalpy of mixing or using standard enthalpy values for heats of formation for electrolyte solutions prepared at specific concentrations will improve the temperature dependence of the predictions for activity coefficients and the model's adjustable parameters that describe interactions between water and the electrolyte or its ions. This is so because the change in activity coefficient with temperature is related to the enthalpy of the solution through the Gibbs-Helmholz equation so enthalpies and heat capacities offer alternative accurate supporting data for the property model and its applicable temperature range. The change in the chemical potential of the electrolyte solute in the solution with temperature is given as:
189
m -^K^I and
(d\ny+) — L 3T )
= -7 m P
AH2 =r (v++v-)RT2
Here, AE^is the partial molal enthalpy of dissolving a small amount of solute in the solution. Measurements of solution heat capacity are very useful because they will allow integration to yield the enthalpy and as a consequence, improve the standard values as well. Both measurements extend the temperature range for the activity coefficient model within 25°C of the measured values. Conventional calorimetric tools are used for heats of mixing or dilution at constant temperature, while scanning calorimetry (various differential scanning calorimeters may be used) provides heat capacity measurements over a wide range of temperatures. Closed pressure vessels are required when approaching or exceeding the normal boiling point. Using tabulated standard values for enthalpy of formation for a solution with a specific concentration. Standard tables list the data as heats of formation for the various solutions. We need to convert these values to heat of mixing. They are listed as: Concentration 1 mole HC1 in lmoleH 2 O 1.5 moles H2O 2 moles H2O
Enthalpy, kJ/mole -121.55 -132.67 -140.96
The heat of formation of H2O, A//*, is-286.077 kJ/mole. For data in the first row above, the listed AH"
is -121.55 kJ/mole. The total moles of
solution, nlmal = nH1Q + nlia = 2. Of course in these tables, the solute is always 1 mole. The mole fraction of HC1, Xnn =\lnlaal =.5 The heat of mixing AH . =}±-Bl
tl
tA = -203.8\3kJ/mole
The apparent molal heat capacity of an electrolyte (2) is given by Eq. 12 and has a similar form to the above enthalpy equation. ^
=
^
^
-
(12)
190 In this equation, for 1 mole of salt (in the solution), Cp is the heat capacity of the solution of a specific concentration and is the measured quantity while the other quantities are the moles of electrolyte (112) and the heat capacity of pure water (Cpwater), and the moles of water (nwaler) in the measured solution, at that temperature and pressure. This is the solute-related property that can be regressed to assist in model development or can be output by a model to check the accuracy of the property predictions. Heat capacity measurements require conventional scanning calorimeters (DSC or C80 calorimeters, for example). These instruments have sufficient sensitivity and accuracy for modeling purposes. The measurements can be made quickly and the methods can generate a lot of data over a large range in temperatures compared with other methods such as vapor pressure, which requires time to prepare the solution under vacuum and to equilibrate at each temperature. A few solutions covering a range of solute concentrations would provide a sufficient amount of data for this purpose. Of special interest would be higher concentrations which are important for fitting activity coefficients of very soluble electrolytes. It's the high concentration range where most models have difficulties and experimental data can be insufficient. Measurements are generally made at constant volume and may have to be corrected to constant pressure values. Typically though the liquid phase remains non-compressible during the measurement and no PV work is done, so corrections would be very small and possibly insignificant, but should be considered nevertheless as part of the method. Water Activity & Osmotic Coefficient Isopiestic technique allows direct measurement for water activity but does not involve measuring vapor pressures. In this case, a reference solution of known water activity (e.g., an NaCl solution) is placed in a thermostatted closed system with only the vapor phase in contact with the unknown solution. The solutions should start out having similar concentrations and volumes. All that is needed is proper glass construction and a good vacuum pump to remove most of the air. The solutions can be frozen with dry ice or liquid nitrogen to aid the degassing procedure, in this case a sequence of freeze-pump-melt. Once most of the air has been removed the vapor pressure will equilibrate within a day or so between the sample and reference solutions. Mechanical stirring will help. Once equilibrium is established, withdraw samples and analyze both solutions. Look up the isopiestic water activity and osmotic coefficient for the reference solution at its concentration. That will also be the value for osmotic coefficient in the sample solution at its equilibrium composition. Isopiestic measurements are among the most accurate and have been extended beyond 300°C using special apparatus. Vapor pressure can increase or decrease depending on the effect of other components on water in the mixed solvent system so it is a responsive measurement for judging short-range interactions of electrolytes. Measuring vapor pressure of a solution also gives the water activity: P a
»'=-f
(12)
191
where Po is the pure water vapor pressure at that temperature, P is the measured vapor pressure of the solution, and a w is the activity of water. From these measurements the osmotic coefficient,
mr)
(13)
where, Mw is the water molecular weight, vr is moles of ion r in solution with a concentration of m,8.2.3 Experimental Methods Methods developed and published by committees of the ASTM organization are available for determining properties such as, density, viscosity, surface or interfacial tension, heat capacity, and vapor pressure of electrolyte systems. If they are not used in their precise form they are at least available as a reference or guide when adapting methods from scientific references. Other published procedures may also be important including procedures and guidelines published by the instrument manufacturer. Table 1 provides a list of some of the experimental properties. Table 1: Experimental methods for different types of properties (see ASTM at www.astm.org for all of their standards). Solution Property Density
Method - ASTM references D4502-96 (2002) el D1298-99
Viscosity D445-01 (Kinematic) D446 ISO 3105 Surface Tension or interfacial tension.
D3825-90 (2000) Other non-ASTM
Title Standard Test Method for Density and Relative Density of Liquids by Digital Density Meter. Standard Practice for Density, Relative Density (Specific Gravity), or API Gravity of Crude Petroleum and Liquid Petroleum Products by Hydrometer Method. Standard Test Method for Kinematic Viscosity of Transparent and Opaque Liquids (the Calculation of Dynamic Viscosity. Specifications and Operating Instructions for Glass Capillary Kinematic Viscometers. Glass Capillary Kinematic Viscometers-Specification and Operating Instructions Standard Test Method for Dynamic surface tension by the Fast-Bubble Technique. DeNooey ring method (static surface tension). Capillary method, calibrate capillary with known liquid, at a fixed temperature, measure unknown liquid s rise in the capillary
192 Table 1: Continued E1269-01 Heat Capacity E968-99 D2766-95 (2000) Vapor pressure
El 194-01 E1719-97
Standard Test Method for Determining Specific Heat Capacity by Differential Scanning Calorimetry. (instrument calibration) Standard Practice for Heat Flow Calibration of Differential Scanning Calorimeters. Standard Test Method for Specific heat of Liquids and Solids. Standard Test Method for vapor pressure. Standard Test Method for vapor pressure of Liquids by Ebulliometry
The procedure for heat of mixing using isothermal calorimetry is available from the manufacturers of commercial isothermal mixing calorimeters. Likewise, additional guidance is available for measuring vapor pressures at low to high pressures, from manufacturers of pressure sensors and from published articles on vapor-liquid equilibrium by T-P-x studies.
8.3 ELECTROLYTE PROPERTY MODELS In this section, a selection of property models for estimation of liquid phase activity coefficients for electrolyte systems is presented. These models are among the most widely used and are available in ICAS (see chapter 15) as well as in most commercial simulators. Because of the nature of electrolyte systems, however, the application of the models is very much dependent on the availability of model parameters and the pure species properties. The activity coefficient models covered in this section include the Pitzer model [8], the NRTLbased Chen model [9], two UNIQUAC-based models [6, 10] and an osmotic coefficient based model. All have similarities and differences, advantages and disadvantages, depending on the application and are semi-empirical in nature, requiring experimental data to evaluate the various interaction parameters in their excess Gibbs energy terms. Each model expresses the excess properties differently but each requires evaluation of their coefficients with measured data. When selecting a model, it is important to keep in mind what it is one wants to understand? If only the outcome is important, such as the stream or solution properties like vapor pressure or solubility index of the various potential salts in the system, then it does not matter how one model describes the solution species compared with another model. However, if one model uses species of interest to the user, then the model that matches the solution model best should be selected. Electrolyte solutions can have many species. It is important to note that the models become very complex as the number of species grows. For every species present, the selected property model will need the standard properties, as well as its equilibrium properties, activity coefficient and concentration (for the calculation of the solution properties or model parameter estimation). Other activity coefficient models not described here can be found in the following references [11, 12, 13].
193 8.3.1 Activity Coefficient Model Library In the model library, five activity coefficient models, which are the Pitzer-Bromley-Debye Huckel model [8, 14], the electrolyte NRTL model [9], the extended (aqueous) UNIQUAC model [10] and the extended (general) UNIQUAC model [6] and osmotic coefficient-based model are available. The Pitzer-Bromley-Debye Huckel model, the extended UNIQUAC model (aqueous) and the osmotic coefficient-based model are applicable only to aqueous systems, while the electrolyte NRTL model and the extended (general) UNIQUAC model is applicable to both aqueous systems and mixed-solvent systems. The selection of appropriate activity coefficient model is made according to the application range of each of these models. Pitzer-Bromley-Debye Huckel Model In the Pitzer-Bromley-Debye Huckel model, the activity coefficients of the ionic species are calculated as the summation of three terms: the Pitzer term to consider short range moleculemolecule and ion-molecule interactions, the Bromley-Zemaites term [14] to consider short range ion-ion interactions, and the Debye-Huckel term to consider long-range ion-ion interaction term (see Eqs.14-17). In r,FD" = In r!'z + In y>yl + In YlD" NM [ BP
In 7 / >z = X
NO i
-A\7
k
O5)
+ Z
lnr,flZ= E —^~
»r»-
1
y 2
,i+JjTBPSj
Z
(14)
P,imi
(16)
^
2
Jh'2
- ^ -
07)
In Eqs.14-17, A is Debye-Huckel constant, which is a function of density and temperature, Z is the number of charges, m is concentration in molarity. BP, BPS are interaction parameters for the Pitzer term and p is interaction term for the Bromley-Zemaites term. NM and NO are the number of molecular species except H2O and number of ionic species with opposite charges to the ionic species / under consideration respectively. I is molarity based ionic strength defined as Eq.18.
/= if;z>,
as)
The interaction parameters in Eqs.15-16, which are BP, BPS and f3, are calculated through Eqs. 19-26.
194 BP. =5^ ; c ) +5^. 1 (l + 2/ 1 / 2 )(l-exp(-2/ 1 / 2 ))/2/
(19)
BPSj = 0 . 8 6 8 5 9 ^ 5 4 ' [ l - ( l + 2/ I / 2 +2/)exp(-2/ I / 2 )]
(20)
/=i
P,i = 1
(0.06 +0.65, )|z,Z, ' ' / + Blt + C,,I + DJ2 (l + 1.5/|z,Z y |/) 2
'
"
(21)
"
+ p3T2
(22)
BP:=pA + p5T +PbT2
(23)
B,, = P, + PST + pj2
(24)
Cij=Pn + PnT + PnT2 D^Pn + PiJ + P^T2
(25) (26)
BP,;=Pl+P2T
In Eqs.19-26, NS is the number of all the chemical species, parameters, pl-pl5, are the coefficients for calculating the temperature dependency of the interaction parameters. The activity coefficients of molecular species, with the exception of H2O, are calculated through Eq.27.
Iny, = if^B^m,
(27)
7=1
In Eq.27, NS is the number of all the chemical species and BP is interaction parameters calculated through Eq.19. The activity of H2O is calculated through Eq.28 according to Gibbs-Duhem relation.
In all2O = - i - 5 £ |z,Z,|/?,m,m, - 0.01801§ [m, + 2 ^ BPW^ntj]
(28)
In Eq.28, ^Zand BPW are calculated through Eqs.29-30.
^Z = [2z>ir2'z>,l _ '=1
JL '=1
(29)
195 BPWIJ = BP:/0 + BPj exp(-2/' 2 )
(30)
In Eqs.29-30, Neat and NAUJ are the number of cation species and anion species respectively. The Debye-Huckel term is dominant only when the solution is very diluted. This model is only applicable to aqueous systems, where only water is solvent. Eq.31 calculates the derivatives of Eq.14 with respect to the model parameters. By calculating Eq.31 numerically or analytically, the sensitivity of model parameters to the activity coefficient values is decided.
d(\nr,''D") dr
z
=d(\ny>'
)
^(Iny,/*) dr
dz
[ rf(ln y,"")
dr
Since the Debye-Huckel term does not include any interaction parameters, Eq.31 is reduced to Eq.32.
d(\ny,mH) dr
=
d(\ny>'z) dr
|
d(\ny,BZ) dr
The model parameters in the Pitzer-Bromley-Debye Huckel model are/?; (i=l, 15) in Eqs. 2226, and these are adjusted to reproduce the experimental data. Electrolyte NRTL Model The electrolyte NRTL model proposed by Chen et al [8] is based on two assumptions, which are like ion repulsion (assumption 1) and electroneutrality (assumption2). In assumption 1, local composition of like-ions is zero. That is, when cation exists in the center, no other cations exist around the center cation. In assumption 2, when the solvent molecule is in the center, cation and anion exist around it so that the surrounding of the center solvent is electrically neutral. The electrolyte NRTL model consists of three terms: the Debye-Huckel term, the Born term and the NRTL term (see Eqs.33-38), and when no electrolyte exists, the electrolyte NRTL model is reduced to the well-known NRTL model for non-electrolyte systems by neglecting all the interactions involving any ionic species in the NRTL term as well as the Born term and the Debye-Huckel term. In ri'!NRTL = In y m l L + In y,B"r" + In y,DH
-"•"-&(H)T
(33)
196 For molecular species,
In ^;v"77- = -1 T,XkGk, k
k
Y
ft
/
icUiic,iAIC
k 2a *
L
Y
iC 2-i IC
, ^ t » ( , r iAiCTk<<: IAIC
kUkICMIC
M 2-1 k
v
iA
YJXkGkim k
A
IA
M2 J LA
-
]_
y
X
|VV 'T
k
+£ "" '"" Tiim <m YJXtGtim '""
C
k
2-1
kukiA,iC:A
IC 2-i k
k°'kiCMIC
kiA K
. 'iA^kiA.iCiA
kukiA,iC:A
2-t k
\_
For cations, 1
V
Y C
//I
Y
* Y
v
\_
C
, r 2-1 ^IC 2-lAkLrkiA,IOA iC k
^kimTkirn
2-1
k
k
_
k kiA.iCiATkiA.KlA
2-1
y y AK: A,/M,i,o,i ^ 2-i 2u sr Y V v r1 //I
r
* «4'M
(3J) v ;
V y r1 2-lAkKjk,A,,C,A
L
*
For anions, 1
J_in
2,
^
y -w»''. = Y
'
K
^,r
/ "
^^^M,,-^
V
y
Z'1
2-1
^,M(J»,,, x km
" 2Zxfikim
^ E ^ , - Z ^ ^ x , , i:zZ fi , IC
k
k
Y , V V
A
-^•^Vy A
A
l^,W,iA'C
1 Vyr A Lr
IC IA 2-1 IA 2-l k klCJAIC iA k
\_ / , XkGk,C
Y C
L
k
_
i,1iCTkiC i/liC
_k
T
r
k^klm^ kirn
_^s m
/TON
V
y
r
2-1A kUkiC.IAIC *
^ '
In Eqs.33-38, M is molecular weight of solvent, Z is number of chemical charges, p is the density of solvent, Ix is molefraction based ionic strength, r is Born radius. Interaction parameters G and rare calculated through Eqs.39-49. "V /
Y
r
, A iA^iCIA.M
2uA>A iA
197 YdXK:GKM,iM ( 4 0 )
GM^=^V^ iC
YaX,Aa,M,,C,A
= lA
alCA4 = aMJC
(41) LXiA iA
ZuXiCa,M,KiA
<*,A M
= a,u ,A = K' v v hXiC
(42)
IC
T
V
-
l (
-
l M /
(AT.-)
a
,C,,M
TIAAI=-
^IMIAJC.IA
~^IA,IM
fi\HCJAiC ~^C,iM
'AAI>
(44)
~^IC.IA,IM
~^~ ^M
jCiA
~^iCiAjM
^^iMjCiA
V^^j
V^®)
V«=A+y+ftl^) +^
(47)
W =Ps+f+Pi ^ - ^ + l n ^ j j
(48)
r-uv,=Ps+ — + Pw
—+ ln - —
(49)
In Eqs.39-49, iM, iA and iC represent the molecular species, anions and cations respectively. X, s, sw are concentration in mole fraction, dielectric constant of solvents and dielectric constant of water respectively. The dielectric constant of solvent is calculated through Eq.50.
198
S =A + B
(50)
» "[\:~T\
In Eq.5O, At,, Bb and Cb are parameters. For the calculation of the activity coefficients of ionic species, (ln/ )RTL ) in Eq. 33 need to be replaced with (lny*NRTL), which is unsymmetrical activity coefficient and calculated through Eq.52. In r'NRTL = In r,mTI- - In y - m T I -
(52)
In Eq.52, (ln/°NRTL) is the activity coefficients of ionic species at infinitely diluted aqueous solution. The Debye-Huckel term considers the long range interaction between ionic species and the NRTL term considers all the other interactions which are short range ion-ion, ionmolecule and molecule-molecule interactions. The NRTL term is same as in the NRTL model for non-electrolyte systems. The Born term considers the transfer energy of ionic species from infinite dilution in H2O to that in mixed solvent. This term does not require any interaction parameters and is a function of only pure properties such as dielectric constant and Born radius. When the electrolyte NRTL model is applied to aqueous systems, the contribution of the Born term is zero. When water is mixed with organic solvents with lower dielectric constant than water, the dielectric constant of the mixed solvent is lower than that of water (s<sw). Therefore, in this case, the contribution of the Born term is positive and it increases the values of activity coefficients. Eq.53 calculates the sensitivity of the model parameters to the activity coefficient values.
dQny™™) dr
=d(\ny™i')
[
dr
J(lny,B""') dr
f
djlny,"") dr
(53)
When only H2O exists as solvent, since the contribution of the second term in the right-hand said of Eq.53 is zero, it is simplified to Eq.54. dQny/*™) dr
=rf(lny,-
dz
vm
)[rf(lny/w) dr
(54)
Consider r as interaction parameters. The Debye-Huckel term does not include any interaction parameters. Therefore, Eq.54 is further simplified to Eq.55 since the second term of Eq.61 is zero.
rfflnr,"™''-) dr
=d(\ny^')
dz
The interaction parameters in the electrolyte NRTL model are p* (i=l,10) in Eqs.47-49, and these are adjusted to reproduce the experimental data.
199 Extended UNIQUAC Model (Aqueous and General) The extended (aqueous) UNIQUAC model developed by Nicolaisen et al [9] is applicable to aqueous systems. It consists of three terms: the Debye-Huckel term, then combinatorial term and the residual term (see Eqs.56-69).
\nyj-™* = \nyf +\ny? +\nr;>"
(56)
For H2O, l In yj™ =MW2£ 1 + W7 —r-2ln(l + b4l) b \_ 1+ W/
(57)
\nr
(58)
•<=in(^-L) + i_^2L_£(? / wat
Vv X wal
/
v X
»a,
fin^L +i - ^
r% J-wat \ V
2
n wal
/
v
n \ wa, )
6
J
d
\
^rJR = ^al I-MZ^J-ZY^ V
(59)
/
/
For ionic species, ln/*"H=-Z2^^^ l+W/
(60)
in r in(
^ = ^-^- in( - )+ --f4 in f-f- in( — )+ —1 X,
X,
rw
rw 2 ^
6j
0j
rwqt
In Y;R = q, - ln(X 0k¥ll) - ^ 4 ^ ~ + l n V*«i + VMl V
/
(6i)
rwqt) (62)
)
In Eqs.57-62, Mw and b are molecular weight and molar mass of water respectively, y* is unsymmetrical activity coefficient for ionic species. A is the Debye-Huckel constant calculated through Eq.63. I is the molarity based ionic strength. Z is number of charges of ionic species, z is the coordination number set to 10. r and q are volume and surface area parameters respectively, (p, 0and y/are calculated through Eqs.64-67. ^=1.131 + 1.335e-3(r-273.15) + 1.164e-5(r-273.15) 2
/
(63)
200
Ok=4^
(65)
^,= exp(-^^)
(66)
«„=«„"+««(?'-300)
(67)
In Eqs.66-67, uu and H# are temperature dependent interaction parameters between chemical species. In the model library, the activity coefficient model modified from the extended UNIQUAC model (aqueous) is called the extended (general) UNIQUAC model [6]. In the extended (general) UNIQUAC model, the Born term is added to the extended UNIQUAC (aqueous). The activity coefficients in the extended UNIQUAC model are calculated through Eqs. 34, 35, 68a, 68b and 69. In y"1"® = Inr;: + In y R + InYiK"r" + InY"H \nyc
=\-A + \ndi--q.
lnr<{=qi 1-ln '
l-^- + ln^-
(68a)
(68b)
-£ J° f"
(69)
k
J
In Eqs.68a-69,
(71)
In Eqs.70-71, (\nf'c) and (ln/"R) are the activity coefficients of ionic species at infinitely diluted aqueous solution. Eq.72 calculates the sensitivity of the model parameters to the activity coefficient values. d{\nY;^) dr
_^(ln y ; ; ) dr
[t /(ln 7 /
dz
f
)
[
d(\ny,B"r") [ d{\ny,D") dr dr
201 Since the Debye-Huckel term does not include any model parameters, Eq.72 is reduced to Eq.73. dQny™1"0) dr
d(lny,c) dr
|
dQny/1) dt
[
d(\nr,B"r") dr
(73)
Interaction parameters uu as well as r and q parameters are adjusted to reproduce the experimental data. Osmotic Coefficient-based Model From the osmotic coefficient, it is possible to calculate the electrolyte activity coefficient, using the Gibbs-Duhem relationship. That is, at equilibrium, chemical potentials (//,•) of component / in one phase equals the chemical potential of that component in all phases. For water (1) and an electrolyte (2), the chemical potential of water is given by,
(73)
Where, aw is the water activity, T is a specified temperature and R is the gas constant. Since, according to Gibbs-Duhem relationship («; is the moles of species or component i), nidpj + n2dju2 = 0
(74)
din ai = (n/ni) dIn a2
(75)
then,
The activity of the solute and solvent are therefore related as follows. (-1000 In aw) d In aw = vmd(lny±m
)
(75)
And therefore the osmotic coefficient of water and the electrolyte's mean activity coefficient are related (replacing water activity with osmotic coefficient in Eq. 75 - see Eq. 13). ( -1) (dm/m) +d(f> = dlny±
(76)
Upon integrating Eq. 76, a relation between the osmotic coefficient and electrolyte activity is obtained. = 1 + (^-))d
In
r ±
(77)
202
It follows that for v ions from the electrolyte, at m molal concentration, the excess Gibbs energy for the solution is given by, AGE = v mRT(l
-
(78)
8.3.2 Henry's Constant For system where the solubility of gaseous components can be described through Henry's law, Henry's constants are calculated through Eqs.79-80.
In (4) = ! » ( % ) Yi
is
(79) Yijs
"*=TTih*
(80)
In Eqs.79-80, Hj and Hjjs are Henry's constants in mixed solvents and in a reference solvent, respectively. y° is the activity coefficient at infinitely diluted solution and is calculated using activity coefficient models explained above. Vc is the critical volume of solvent species and x,s is mole fraction of a reference solvent. 8.3.3 Condition of Physical and Chemical Equilibrium When A, B and C parameters are available, Kms and ^ S 0 L can be calculated for the given temperature through Eqs.8-9. For the computation of solubility, the equations for physical equilibrium and chemical equilibrium relations (see Eq.81) need to be solved simultaneously. K
J =Yl(r,x,jrj
(so
In Eq.82, y; and x\ are activity coefficient and composition of chemical species i, respectively. Since K°ls and KSOL are constant for the given temperature, in the computation of solubility the compositions that satisfy Eqs.8-9 and Eq.81 at the same time are determined iteratively. When the standard Gibbs energy and standard enthalpy values from a data base are used for the calculation of K°]S and A^SOL, it is important to consider the uncertainties in these values and if necessary, it is possible to adjust them if reliable solubility data is available (the sensitivity of the parameters can be analyzed first before adjusting their values). 8.3.4 Solubility index The solubility index, SI, is a useful property that can be used for design and analysis of operations of crystallization. In an electrolyte solution, if the SI for a salt is > 1, then the salt exists as a solid. If the 57 is < 1, then it has not reached the saturation point and the salt
203
closest to unity is the one that will precipitate first (at the specified temperature). Equation 82 defines the SI for a salt /. A'C
57 = -^
(82)
Kj
8.3.4 Solubility as a function of temperature and solution pH When A, B and C parameters for Eqs.8-9 are not available and no Gibbs Energy data are available, an empirical model can be used for calculation of solubility (see Eq.83). In Eq.83, solubility is directly related to the temperature and solution pH. Therefore, within the range of experimental data, it is possible to reproduce the experimental data. However this model is system specific and cannot be applied to any other system. S=f(T,pH)
(83)
8.4 CONCLUSIONS The objective of this chapter has been to provide the reader with an overview of the important properties and parameters associated with the derivation and use of some of the commonly used models for the estimation of liquid phase activity coefficients related to electrolyte systems. The models considered in this chapter are not predictive with respect to the species present in the electrolyte solution. They may be used, however, for a range of temperature and compositions. Because of the nature of electrolyte systems, it is important to identify the system specific data that needs to be used by the property models and to check if the necessary model parameters are available. To apply the models presented in this chapter, careful check of the consistency of the model parameters, if available, needs to be made. Often, the missing parameters will need to be estimated. Normally, the list of parameters is quite large but if experimental phase equilibrium data is available, fitting only a few may be adequate for many electrolyte systems (see chapter 15 for more details).
REFERENCES 1. M. Rafal, J. W. Berthold, N. C. Scrivner, S. L. Grise, Models for Electrolyte Solutions, in Models for Thermodynamic and Phase Equilibria Calculations, Editor: Stanley I. Sandier, Marcel Dekker, Inc., New York, USA, 1993. 2. K. Thomsen, Aqueous Electrolytes: Model Parameters and Process Simulation, PhDThesis, Technical University of Denmark, Lyngby, Denmark, 1997. 3. J. R. Loehe, M. D. Donohue, AIChE J., 43 (1997) 180-195. 4. Z. Janecke, anorg. Chem, 51 (1906) 132.
204
5. M. L. Pinsky, G. Grubber, AIChE Symposium Series, 298-90 (1994) 112-126. 6. K. Takano, "Computer Aided Design and Analysis of Separation Processes with Electrolyte Systems", PhD-thesis, Technical University of Denmark, Lyngby, Denmark, 2000. 7. J. F. Zemaitis Jr., D. Clark, M. Rafal, N. C. Scrivner, AIChE J, 40 (1994) 676. 8. K. S. Pitzer, "Thermodynamics if Aqueous Systems with Industrial Applications", Ed: S.S.Newman, ACS Symposium Series, 133 (1980) 451. 9. C. C. Chen, H. I. Britt, L. B. Evans, AIChE J., 32 (1986) 1655. 10. H. Nicolaison, P. Rasmussen, J. M. Sorensen, Chem Eng Sci, 48 (1993) 3149. 11. S. G. Christensen, K. Thomsen, Ind Eng Chem Res, 42 (2003) 4260-4268. 12. C. C. Chen, C. P. Bokis, P. M. Mathias, AIChE J., 47 (2001), 2593. 13. P. M. Wang, A. Anderko, R. D. Young, Fluid Phase Equilibria, 203 (2002) 141-176. 14. Bromley, L. A., AIChE J., 19 (1973) 313. 15. W. C. Blasdale, "Equilibria in Saturated Salt Solutions", ACS Monograph Series, The Chemical Catalog Company Inc., New York, USA, 1927.
Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B.V. All rights reserved.
205
Chapter 9: Diffusion in Multicomponent Mixtures Alexander A. Shapiro, Peter K. Davis, and J.L. Duda 9.1 INTRODUCTION The term "diffusion" is used in order to describe the process of relative motion of different components in a mixture. For example, in a mixture of 10 components, 10 x 9/2 = 45 relative fluxes of the components with regard to each other may be considered. This large amount of possible interactions between fluxes of different components makes it rather difficult to provide exact description of diffusion in really multicomponent mixtures. While the theory of diffusion is well developed in the literature [1, 2, 3], the values of the diffusion coefficients in realistic multicomponent mixtures remain a problem. For binary mixtures, both large databases of experimental values and many models and correlations for diffusion coefficients are available [4, 5]. Meanwhile, only few authors tried to access the problem of determining of multicomponent diffusion coefficients. Partly this is a consequence of the fact that experimental data on multicomponent diffusion is scarce, the corresponding experiments are difficult to perform and, therefore, the models for multicomponent diffusion are difficult to verify. Since such models are normally based on a rather weak theoretical background (except the ideal gas mixtures, where the gas kinetic theory may successfully be applied), their predictive capability remains low. The first part of the present chapter reviews existing representations and models for diffusion coefficients in multicomponent mixtures. We tried to avoid consideration of the binary mixtures, since extensive literature on this subject is available; to some extent, the present chapter may be considered as a complementary to the corresponding chapter in the well-known book [6]. We considered only molecular diffusion and did not touch the convective diffusion, which requires a separate and totally different consideration. Also, we restricted ourselves with the mixtures of relative simple molecules: although the experimental data for the mixtures of the type "complex substance - solvent - co-solvent" are widely available in the literature, these systems are rather difficult for predictive thermodynamic modeling. We hope that the collected information will be of use for the specialists in many areas of engineering where diffusion is one of the determining processes. The second part of the chapter describes recent achievements in the theory of diffusion in the polymer-solvent systems. Mass transport of small molecules in macromolecular systems is of vital importance in the polymer industry. Free radial polymerization can become mass transfer dominated in the limit of high conversion due to the inability of monomer to quickly diffuse to a growing polymer chain. Ever growing governmental regulations on emissions require the plastic industry to remove impurities such as residual monomer and solvent from
206 the final product. Design and optimization of these devolatilization processes require accurate and reliable diffusion coefficients. In the coating industry, polymer solutions are dried in large ovens to make thin films. The film quality as well as proper oven operating conditions depends greatly on the mass transport of solvent in the polymeric film. Polymer diffusion is also important in medical and agricultural applications such as time released drugs and fertilizers that diffuse from a polymeric substrate into the body or soil. Most practical mass transfer problems involving polymers and solvents can be solved if the mutual binary diffusion coefficient, D, is accurately known as a function of temperature and concentration. The major exception is so called "anomalous diffusion" where polymer chain relaxation is significantly slower than the diffusion process. We will focus on determination of the mutual diffusivity for normal Fickian diffusion i.e., where the diffusive flux can be described by Fick's law and anomalous diffusion does not occur. Although the mutual diffusivity is of critical importance in the modeling of mass transfer processes, relatively few data have been collected over wide ranges of temperature and concentration. For this reason, experimentally measuring the diffusivity in polymeric systems is an area of active research. In addition, much of the work in polymer-solvent diffusion has been in the region of infinite solvent dilution while many practical problems such as film drying or devolatilization require the diffusivity at high solvent concentrations. Currently, no general theory exists that can correctly describe the mutual diffusivity for a binary polymer-solvent system over the entire concentration regime. However, the most effective theory for describing the diffusivity over large ranges of concentration is the Vrentas-Duda free volume theory. This theory can be effectively used as a correlative/predictive tool for describing the diffusivity in amorphous polymer-solvent solutions above and below the glass transition temperature. The present work focuses on various applications of the Vrentas-Duda free volume theory to correlate and predict the mutual binary diffusion coefficient in polymer-solvent systems.
9.2 MULTICOMPONENT DIFFUSION 9.2.1 General Facts About Diffusion Diffusion fluxes are determined as relative fluxes of different components in a mixture. If, for example, J, are the component molar fluxes in a n-component mixture, then one determines the molar convective flux as [7]
and the diffusive fluxes of each component as J D ,,=J,-z,J, (/ = !...«)
(2)
207
Here z, are molar fractions of different components in the mixture. Similarly, one may determine mass fluxes of each component Q,, the mass convective flux as
(3>
Q,=£Q, 1=1
The diffusion fluxes are then determined in terms of the mass fractions ci [7]: Q f l , = Q , - c , Q c (i = l,...,«)
(4)
Other definitions are also possible, and each of them has their own advantages. When thermodynamic calculations are being carried out, it is more convenient to work with molar fluxes. Meanwhile, when diffusion is coupled with hydrodynamic flow, the point of view related to mass fluxes is more appropriate, since the hydrodynamic equations (like the momentum equation) are normally formulated in terms of the mass fluxes. One fundamental property of the diffusion fluxes is valid for any their definition. It follows from Eqs. (1) to (4) that
|X,=0,
|>,,,= 0
(5)
Thus, only (w-1) diffusion fluxes are independent, which corresponds to their definitions as the relative fluxes. Sometimes diffusion fluxes are determined in a different way. If all the components but the last one are low concentrated, it is convenient to define overall fluxes of the first («-l) components as diffusion fluxes. This is especially convenient for diffusion of admixtures in solids. Thus, in this case the sum of diffusion fluxes is not equal to zero, although there are still (n-1) independent fluxes. This point of view is convenient if the flows of different lowconcentrated components do not interact, since the matrix of diffusion coefficients is diagonalized. While the total mass fluxes are simply proportional to the molar fluxes, with the proportionality coefficients equal to molar masses, Q, = M,J,,
(6)
relations between mass and molar diffusive fluxes become more complicated. Let us assume, for example, that the first («-l) diffusion flux is selected as independent, and the last flux is expressed from Eq. (5). Then the transformation matrices F, f between the two systems of fluxes are determined by formulae
Flk=M,5lk-M'Z'^-M\ IK
I
IK
-i r
M
/,=-L^-zfj ~
J IK
IK
i r
M i
I\
Ll
*f
\ f
\
{ M,
Ml
\
ii /
h
(7) • '
208
{M = Y,M,zl, i,k = \,...,n-\) It may be checked that matrices F and f are inverse to each other. The relations between the mass and the molar diffusion fluxes are given by
Q,=SX^ J,=Z.4Q,
(8)
9.2.2 Diffusion Coefficients - Different Definitions and their Connections Since diffusion is caused by spatial variation of intensive thermodynamic properties of a mixture, the diffusion fluxes are normally expressed as linear combinations of the gradients of different intensive properties. In the Fickian approach the molar diffusion fluxes are expressed in terms of the gradients of the molar fractions:
J».,="IAVz t
(9)
Here v is the molar density, Djk are Fickian molar diffusion coefficients. Similarly, the mass Fickian diffusion coefficients are introduced as proportionality coefficients between the mass diffusion fluxes and gradients of the mass fractions:
Here p is the density of the solution. Experimental conditions often lead experimentalists to expressing diffusion fluxes not in term of the fraction, but of the concentration gradients. However, to the best of our knowledge, in all such experiments the molar volume is supposed to be constant, so that the definition of the diffusion coefficients does not change. It should be stressed that equations like (9) and (10) are only valid under constant pressure and temperature. Presence of thermal gradients may lead to thermodiffusion, which may be important in some specific situations. Presence of the pressure gradients results in barodiffusion, which is often negligible, but, more importantly, under usual experimental conditions it may result in appearance of the convective fluxes, which may lead to wrong interpretation of the diffusion experiment (see discussion in [7]). The rule of transformation between mass and molar diffusion coefficients is due to the fact that the transformations between gradients of concentrations and molar fractions are given by the matrices F/M and Mi, where the coefficients of the matrices F and f and the molar mass of the mixture M are given by Eq. (7). Introducing the in -1) x (n -1) matrices D, Dm consisting of the corresponding diffusion coefficients, one obtains that
209 D = fDraF, D,,,=FDf
(11)
It follows from Eq. (11) that the mass and the molar diffusion coefficients are equal for a binary mixture (where only one diffusion coefficient is available). However, this is not the case for a larger number of components. The Fickian diffusion matrices D, Dm are not necessarily symmetric: D^D,,;
DmU*Dmj,
(12)
Therefore, in the Fickian approach diffusion is determined by {n -1)2 diffusion coefficients (which gives four coefficients for a ternary mixture). Other approaches to diffusion operate with a lower number of the coefficients than the Fickian approach. The Maxwell-Stefan formalism expresses the relations between gradients of the chemical potentials //, and the diffusion fluxes in the following form [8]: under constant pressure and temperature,
-^ v "-ti^o-i .)
(B)
Due to the Gibbs-Duhem relations, only ( « - l ) of equations (13) are independent, and the last equation should be substituted by Eq. (5). An obvious inconvenience of the MaxwellStefan relations is in the fact that for expressing the diffusion fluxes in terms of the gradients one should solve a system of linear equations. This is why the Maxwell-Stefan coefficients are not so often determined in experiments as Fickian diffusion coefficients. An advantage of the Maxwell-Stefan approach is in its generality, straightforward physical interpretation [8], symmetric form of relations (13) with regard to all fluxes, and, most important, in the fact that Maxwell-Stefan the diffusion coefficients are symmetric: DiMS) = DiMS)
{
u
j=
^
^
i
^
n
( 1 4 )
It follows from Eq. (14) that there are only n(n -1)/2 independent Maxwell-Stefan diffusion coefficients (three for a ternary mixture). The same amount of independent transport coefficients is used in the formalism of the non-equilibrium thermodynamics, which starting point is to characterize a transport process in terms of its entropy production [1]
S=£ x , i ,
(15)
Thermodynamic fluxes x,. are expressed in terms of the thermodynamic forces X, by means of the Onsager reciprocal relations:
210 *,=!«
(16) k
It was proven by Onsager [9] and extended onto continuous systems by Casimir [10] that the phenomenological coefficients Llk form a symmetric matrix: L,k=Lk,
(17)
Different choices of thermodynamic forces and fluxes are possible. One may talk about different systems of "thermodynamic coordinates" [11]. Transitions between two systems of coordinates, X = {X(}, x = {x;} and Y = {Y;j, y = {y,J are normally given in terms of a transition matrix A: Y = AX, y = (A 7 ')' 1 i, L (r) = A7L(X)A
(18)
Here L(A), L(>) are matrices of phenomenological coefficients in the two systems of coordinates. As an example, let us consider the case where the thermodynamic fluxes are the molar diffusion fluxes J ; ) / . It may be shown [1] that the conjugate thermodynamic forces may be chosen as -T^V{/ui - //„), where juk {k = 1,...,«) are chemical potentials of the mixture:
J«,=~E#"VC« i -/O (i = U.,«-l)
(19)
•* * = i
Another convenient choice of the thermodynamic system of coordinates is achieved by introducing the relative fluxes I, = zK3, - z,3H = zn3l}J - z,iDn
(i = l,...,n -1)
(20)
The reciprocal relations connect in this case thermodynamic fluxes 1} with forces X'=-V//'7znr (i = \,...,n-\): I
,=~VZ4" 2 ) V i u t (/ = !,...,»-1)
(21)
The relations between l}'kn) and L'"2) are determined by the last equation (18), by means of the following matrices A = {Ajk} and A"1 = {ajk}:
211
4 = f ( 4 " ^ X alk=znS,k+zk
(22)
If transition from L{'kn to li'k)2} is performed, the matrix A should be used. For inverse transition one should apply A"1 instead of A. Keeping the two different systems of thermodynamic coordinates (Ol) and ((92) is convenient, since the first system is closely connected to the Fickian representation of diffusion, while system (O2) may be connected with the Maxwell-Stefan representation. Indeed, for system (Ol), it follows from Eqs. (9), (19) that D = —L <0I) M. vT
(23)
where M is the matrix consisting of the elements Mlk = d(/xj - jj.n)ldzk (i,k = l,...,w —1) (all the derivatives are taken under constant pressure and temperature). The connection between the coefficients i'"2> and the Maxwell-Stefan diffusion coefficients is more cumbersome. Let the matrix I("2) = {lik} be inverse to L(02). Then it may be shown that [12]
'•>=-^r(i*jl '" = nz,znD^'s) + j^nztz]pp /*'
(24)
(25)
If the Maxwell-Stefan diffusion coefficients are known, Eqs. (24), (25) may be used in order to determine the values of //; and then, by inverting the matrix, of Z^<;2) (i,j = 1,...,«-1). Vice versa, if the coefficients li"2) are known, one may find ltJ. Then coefficients D(/JMS) for i,j = 1,...,«-1, /' ^ j are found from Eq. (24), coefficients D^us) are subsequently expressed from Eq. (25), and -£>(,f's) are found from the symmetry condition (14). We have outlined the most important representations of the diffusion formalism and the connections between them. The overall scheme of these connections is presented in Figure 1. This scheme shows that, in spite the seeming similarity between the Fick and Maxwell-Stefan pictures of diffusion, there is a long way between them, with the shortest connection passing through the area of non-equilibrium thermodynamics. Only in the case of binary diffusion there are simple proportionality relations (activity corrections) between the Fick and the Maxwell-Stefan binary diffusion coefficients [8,13].
212
Figure 1. Schema of transformations between different representations of the diffusion formalism. It was mentioned that experimental data on diffusion in multicomponent mixtures are normally expressed in terms of the Fickian diffusion coefficients. This data is normally excessive, since the Fickian coefficients are connected via symmetry relations (14), (17). Transformation rule (23) shows that, in order to relate Fickian coefficients to the symmetric matrix of Onsager coefficients, one should have either experimental data or a good model for dependencies of the chemical potentials on molar fractions under constant temperature and pressure. To the best of our knowledge, only few checks of symmetry have been presented in the literature (for example [14,15,16]). Our observations [17] show that careful application of the transformation rules described above together with a modern EoS or a Gibbs excess energy model for a mixture being studied, like UNIFAC [5] or CPA [18], normally leads to deviations from symmetry within 5-10%. 9.2.3 Diffusion Coefficients in Rarified Gases Rarified gases are the only mixtures, for which evaluation of diffusion coefficients is based on strict theoretical background [2]. In order to calculate diffusion coefficients, the Boltzmann gas kinetic theory is applied. The most convenient framework for description of gas diffusion is the Maxwell-Stefan formalism (13). For ideal gases under constant pressure and temperature, the left-hand side of Eq. (13) is reduced simply to Vz,. An advantage of the Maxwell-Stefan approach is that, in this approach, diffusion coefficients D^'IS) may be interpreted as the friction coefficients between the molecules of the two components per unit molar amounts of these components [8]. The physical picture of transport in an ideal gas mixture accepted by the gas kinetic theory assumes that only binary interactions between molecules of different components are
213 possible, and then each such interaction (collision) does not modify the state of the rest of the mixture. Therefore, the following two statements about the values of friction coefficients should be valid: 1. The values of D]JMS'1 are the same as the values of binary diffusion coefficients of the corresponding components; 2. The values of binary diffusion coefficients are independent (or only slightly dependent) on composition of the mixture. Moreover, considerations about frequency of collisions result in the conclusion that the diffusion coefficient should be inversely proportional to the molar density and proportional to the square root of the temperature (as a measure of the molecular velocity). These intuitive considerations are confirmed within the gas kinetic theory, which gives the following well-known answer for the values of the binary diffusion coefficients: 0.002667-
"
Pap
(M-+M-T
{
2
)
J
Here P is pressure, in bar, o^ is a characteristic length, in A (so that a^ is the so-called "collision cross-section"), Q is the diffusion collision integral, dimensionless, a n d / i s the second-order composition-dependent correction, normally lying between 1.0 and 1.1 [2,5]. The weak compositional dependence of the gas diffusion coefficients is reasonably confirmed by experimental data [19]. Kinetic theory of gases may be spread onto higher pressures and densities by taking into account the effect of finite molecular sizes (the so-called Enskog theory of dense gases [2]). The assumptions about impossibility of triple and multiple collisions remains in the framework of the Enskog theoiy, which should, in principle retain statement 1 unchanged. The only corrections should be additional contributions to the values of the binary diffusion coefficients and their stronger dependence on composition [20]. 9.2.4 Diffusion in Liquids There is no such a strict theory of diffusion in liquids as the Boltzmann theory for diffusion in gases. Only approximate models for liquid diffusion coefficients are available. Predictivity of these models is difficult to verify, since experimental data on diffusion coefficients in multicomponent mixtures is rather scarce. That is why the models that may, in principle work for multicomponent diffusion, are often used only for prediction of binary diffusion coefficients [21]. Among the approaches to multicomponent diffusion, the two big groups may roughly be separated: 1) Different reduction approaches, where the number of independent coefficients to be evaluated, is reduced by "lumping" of different components or by assumption about independence of certain fluxes, and 2) the approaches based on the "activation energy - free volume" concept [2]. These, as well as some other approaches are considered below.
214 Approaches based on reduction of the number of independent diffusion coefficients. Several authors have tried to reduce ternary or multicomponent diffusion to binary by considering an imaginary experiment or a process, in which the diffusion coefficient should be determined, and selecting an effective value of the binary coefficient providing the same results for a component of interest. A classical example of such an approach is the so-called Blanc law [22]. The pseudo-binary diffusion coefficient for a tracer as zth component in a homogeneous mixture is expressed as a harmonic average (
V1
D
«, = t^ki \J*i
(27) J
In [23], it is proposed to evaluate the effective Fickian diffusion coefficient for the first component in a ternary mixture as
{D ) D
" -- "[-l5$;}
(28)
The only assumption made by the author is that the concentration c2 is maintained in the point where the experimental values are taken. A more sophisticated system of assumptions has resulted in [24]
(D^=DirP^±M,
(29)
(l + q^Jp) 2
Dn D22
Dn+Dr, 2jDuD22
, Dl2D21 DUD22
This is to say that effective approximation to the value of the diffusion coefficient in a mixture with reduced number of components depends on a process being studied. A systematic approach to determination of effective diffusion coefficients for a "lumped" mixture is suggested in [25], where effective values of diffusion coefficients are found by minimization of the deviations between real and effective solutions. Treatments like (27) to (29) may simplify solution of a multicomponent diffusion problem by reducing the number of the diffusion equations. They also reduce the number of unknown diffusion coefficients. However, the values of the newly introduced effective diffusion coefficients are more empirical than the original diffusion coefficients. The problem of their determination is either reduced to the initial problem of modeling original "true" coefficients, or becomes purely experimental. It is not clear to what extent the problem of determining diffusion coefficients is simplified on this way.
215 Another way of reducing the number of independent diffusion coefficients is assuming that in a certain system of thermodynamic coordinates the fluxes become independent and, therefore, the matrix of Onsager phenomenological coefficients becomes diagonal [11]. This assumption makes it possible to transfer from «(« —1)/2 independent transport coefficients to « - 1 or n diagonal coefficients, which are normally interpreted as "friction parameters". The corresponding system of thermodynamic coordinates is normally chosen empirically, with only intuitive physical substantiation. For example Cussler [3] suggests that a good estimate for the diffusion coefficients is obtained if the Onsager coefficients are supposed to be diagonal in a system of coordinates related to the following thermodynamic forces (in our designations): X, =-Yal,VJul•> ocu=Su +^L-L (/= 1,...,«-1)
(30)
Here is a molar volume of the component;'. The thermodynamic fluxes conjugate to forces (30) are the molar fluxes with regard to the volume-average velocity. On the basis of such a representation, Cussler derives the following equation for the Fickian diffusion coefficients:
The values of Dt are assumed to be unknown constants that have to be determined experimentally. Another example of diagonalization is given by work [26], where it is assumed that in a system of spatial coordinates related to the volumetric flow of a mixture each component is driven by its own "driving force" —Vjut, which is equilibrated by the "friction force"
(32)
This assumption may be shown to straightforwardly lead to the following expressions for the phenomenological coefficients:
C^k-J^ + l-Z^lj
(33)
Thus, in this approach the Onsager coefficients are expressed in terms of only n unknown friction coefficients. In [26] the diffusion coefficients corresponding to the friction assumption (32) are found for the volumetric fluxes, provided that the total molar volume of
216 the mixture is constant. Later the authors applied these expressions for verification of the symmetry of Onsager reciprocal relations by experimental data [16]. Empirical introduction of the "friction assumption" may sometimes lead to violation of the symmetry rule for the phenomenological or Maxwell-Stefan coefficients, although the results remain acceptable from some "practical" point of view. An example is given by the recent paper [27]. The authors introduce the friction formalism in the form of V/^-^M^-v),
(34)
where v is the mass average velocity of the flow. As the authors themselves notice, this assumption is equivalent to the following assumption about the Maxwell-Stefan diffusion coefficients: D\MS)-\lfiM,
(35)
This form of the Maxwell-Stefan coefficients is clearly asymmetric. However, the authors manage to produce with this assumption a set of Fickian diffusion coefficients for a ternary mixture and compare them with experiments. In sum, the approaches described above succeed in reducing the number of unknown coefficients. However, they do not fully resolve the problem, since (n —1) or n "friction parameters", which remain after such a reduction, are still unknown, and their determination is a difficult experimental task. For a ternary mixture, the gain is very small (from 3 to, maybe, 2 independent coefficients), and the progress is achieved if these new coefficients are further treated as constants. If one deals with a diffusion theory where a diagonalization assumption is used, we would like to recommend verifying validity of the symmetry of the Onsager or Maxwell-Stefan coefficients. Approaches based on the free volume or activation energy concepts Fundamentals of the free volume and activated energy concepts and a way to introduce them properly from the basic principles of the statistical mechanics are discussed in [2]. A physical intuition behind this concept is that a molecule in the mixture occupies a certain volume where it can move in a quasi-equilibrium way and, in order to jump to a neighboring "equilibrium position" (provided that it is free), the molecule has to overcome a certain energy barrier, called the activation energy. A more detailed introduction to the free volume concept is given in the second part of this chapter, in connection with the problem of the solvent-polymer diffusion. The free volumes and the activation energies for different substances may be found by estimation of the molecular interactions [2,18]. Alternatively, they may be correlated by evaluation of the different parameters of the equations of state and/or by data on viscosities [2]. The free volume/activation energy approach to diffusion has been known for a long time, however, its first spreading onto multicomponent mixtures probably belongs to Cullinan. In the work [29] the following expression for the Maxwell-Stefan diffusion coefficients is proposed:
217
"
hNA
{
V
RT )
'
Here h is the Planck constant, NA is the Avogadro number, a is the distance between the neighboring equilibrium positions, and AG. is the activation energy for the / — j frictional interaction. If the linear mixing rule for AGlf is assumed,
AG,,=]j>,limAG,,
(37)
then the well-known empirical Vignes mixing rule for diffusion coefficients is obtained [30]:
DfX) = n(limDf s ) )'
(38)
A way to express diffusion coefficients in terms of free volumes rather than activated energies is suggested in [13]. The authors express their theory in terms of the friction coefficients C^RT/D^
(39)
In order to express the diffusion coefficients, it is suggested to work with the compressed volumes Vmin, which are determined as the minimum volumes of the substances at zero K, and with the free volumes V1 determined as V - Vmin. The values of V and Vmin for pure substances are suggested to obtain from the Guggenheim correlation [31]. Special mixing rules for the value of the free volume for a mixture are proposed. Moreover, by determining surface fractions an the authors [13] introduce individual free volumes for each component in the mixture: a,=
Z/( w
^' m : )
;
V.j =^-Vs
(i = 1,...,«)
(40)
Z
2-, k\'fr.pure) *=1
Further, the authors determine the friction coefficients £. cff for the tracer amounts of each component in the mixture as a whole: £ cff = 2Aj3hTdlPmm exp 0.7%^-
(z=l,...,«)
(41)
Vj
In the last equation, the pre-exponential multiplier comes from Einstein's theory of resistance to Brownian motion [32]. Here dj is a characteristic molecular size for the fth component, pnmx is the compressed density of a mixture, found as arithmetic average of the
218 individual compressed densities of different components. The exponential multiplier corresponds to the probability of finding a free space for a molecule of fth component to be possible to move to the neighboring position [33]. In order to obtain friction coefficient £t, the authors [13] derive the following mixing rules:
£, = crffCj'ef{ / ,zhhh A- = l
a,y=!,...,«; ^ 7 )
(42)
BIT
Other approaches to diffusion in the framework of the activation energy and free volume concepts are also possible (see,/ ex., [34]) and discussion of this paper in [35, 36]. Other approaches Apart from some useful correlations described in [5], let us mention recent approach [21], where the group contribution correlation is proposed for the Maxwell-Stefan diffusion coefficients. The final expression of the authors is rather cumbersome, and has a form similar to the UNIF AC expression for an activity coefficient. Although this expression is of a general character and looks like it may be applied to any mixture, the authors test it only on the values of the binary diffusion coefficients. An advantage of the group contribution approach is that it only requires knowledge of the properties of molecular groups, from which the necessary properties of the molecules may be calculated. Thus, the approach may be spread onto new components and their mixtures without introduction of new constants.
9.3 DIFFUSION IN POLYMER SOLVENT SYSTEMS 9.3.1 Free Volume Basics The theory of diffusion in polymers requires a more detailed analysis of the concept of the free volume than it has been done in the first part of this chapter. A volume of fluid can be partitioned into different types of volume. There is volume that is actually occupied by the molecules, and the remainder of the volume is free volume. The free volume can be further partitioned into two categories. The first is interstitial free volume, which is not available for mass transfer. The second is the hole free volume, which is available for mass transfer. A diffusive step is thought to occur when a hole of sufficient size opens into which a nearby molecule can jump. Many low molecular weight species are small enough that the entire molecule can fit into the holes, but larger species (especially polymers) cannot entirely fit into the holes. Thus, it is thought that such species move as smaller segments called jumping units. When a hole of sufficient size opens, the jumping unit will move into the hole. By a series of these jumps, large molecules can diffuse. Cohen and Turnbull [33] were the first to develop the idea that the self-diffusion coefficient of a molecule is proportional to the probability of that molecule finding a hole
219 large enough to occupy. Their expression for the self-diffusivity (Dseif) may be formulated as (c/Eq.(41)):
Dxlf=Aexp[^j
(43)
In this equation, V is the minimum hole size into which a molecule can jump, V1 is the average free volume per molecule, and y is number between 0.5 and 1 that accounts for the overlap of free volume due to different molecules sharing the same free volume. From Eq. (43), Vrentas and Duda [37,38] developed the appropriate expressions for V and ¥ for a binary polymer-solvent solution. The solvent self-diffusion coefficient based on their derivations is given as:
A = Do exp —
exp — v
7
*
^ 7
(44) )
The subscript 1 denotes the solvent, and the subscript 2 denotes the polymer. The Do parameter is a pre-exponential factor, E is the energy required for a solvent molecule to break free from its neighbors and jump into a nearby hole, and £, is the ratio of jumping unit size of the solvent and polymer. The specific volume of species at 0 K is used to represent the specific hole free volume required for a diffusive step, V*. The glass transition temperature is Tgj. The Ky parameters are free volume parameters that are related to the temperature dependency of the pure component viscosities. K\\ and Ki\ are related to the pure solvent viscosity and K\2 and K22 are related to the pure polymer viscosity. These free volume parameters are also directly related to the Williams-Landel-Ferry (WLF) equation parameters used for correlation of viscosity versus temperature data [39]. Although the solvent self-diffusion coefficient is useful for describing molecular motion in a solution with no chemical potential gradients (uniform P, T, a>,), it is the mutual diffusivity, D, that relates a diffusive flux to concentration gradients through Fick' s law. Duda et al. [40] derived an expression for the mutual diffusivity in terms of the solvent self diffusion coefficient and a gradient of solvent chemical potential: D=D££l(djA
RT (dcj,,,, The Flory-Huggins theory [41,42,43] is a popular thermodynamic theory for polymer solvent solutions, and can be used to calculate the chemical potential gradient in Eq. (45): D = Dl{\-l)2{\-2%
(46)
220
Here, % is the Flory-Huggins interaction parameter, and $ is volume fraction of species i. Thus, Eqs. (44) and (46) together represent a model for the mutual binary diffusion coefficient in polymer-solvent solution. 9.3.2 Parameter Estimation By looking at Eqs. (44) and (46), it is immediately obvious that calculation of the mutual diffusivity requires many parameters. Strength of the Vrentas-Duda model is that although there are a number of parameters, each of them has a physical significance. In addition, most of these parameters can be obtained from pure component properties. Collection of the K K variables gives the following parameter groups: E, V*, V2*, ——, ——, K2]-Tg], K22 - Tgl, %, Do, £, and the pure component densities. In Vrentas-Duda free-volume theory, the E parameter is not the activation energy for diffusion as it is in a simple Arrhenius expression. Rather, E is the energy required for a solvent jumping unit to break away from its neighbors and move into an open hole. For most hydrocarbon systems where the only intermolecular interactions are weak dispersion forces, £ is a veiy small number and is usually approximated as zero. For strongly associating systems, E may be significant and should be obtained from regression of experimental diffusivity data. The parameters V* and V* are the minimum hole specific volumes needed for the solvent and polymer respectively to make a diffusive step and are approximated as the molar volumes at 0 K. Sugden [44] and Blitz [45] developed group-contribution correlations for these molar volumes, and Zielinski and Duda [46] listed the groups for many common atoms. This list is shown in Table 1. The polymer and solvent free volume parameter groups are obtained either from pure viscosity-temperature data or from NMR relaxation data. This data can be correlated though the WLF equation and the WLF parameters have a direct relationship to the free volume parameters:
?£- = 2.303C™C™ K,, = C,f'
(47) (48)
When using the WLF equation for small molecules, low temperature viscosity data must be used since it is this low temperature regime where free volume dominates the small molecule's behavior. The free volume parameters and the values of py for some common solvents and polymers were tabulated in [46], and this list is shown in Tables 2 and 3. Also included in Table 2 is the solvent molar volume at 0 K.
221 Table 1. Group contributions for determining molar volume at 0 K. Sugden (cm3/mol) 6.7 1.1 1.1 3.6 0.9 5.9 3 10.3 19.3 22.1 28.3 12.7 14.3 13.9 8 4.5 3.2 1.8 0.6 X X
Component H C (aliphatic) C (aromatic) N N (in ammonia) 0 O (in alcohol) F Cl Br I P S Triple bond Double bond 3-membered ring 4-membered ring 5-membered ring 6-membered ring OH (alcoholic) OOH (carboxyl)
Biltz (cnrVmol) 6.45 0.77 5.1 X X X X X 16.3 19.2 24.5 X X 16 8.6 X X X X 10.5 23.2
Table 2. List of free volume parameters for some common solvents
Solvent
V,* (cm3/g)
V,(0K) (cmVmol)
K
Acetic Acid Acetone Benzene n-butylbenzene Carbon tetrachloride Chloroform Cyclohexane Cyclohexanol cis-Decalin trans-decalin n-Decane diButylphthalate
0.773 0.943 0.901 0.944
46.45 54.77 70.38 126.69
"(cm 3 /gK)x y 103 0.546 1.393 1.21 1.61
0.469
72.15
0.642
-87.93
0.51 1.008 0.882 0.928 0.928 1.082
60.89 84.8 88.34 128.3 128.3 154
0.561 2.26 0.733 0.848 0.775 0.96
-21.8 -145.3 -170.2 -86.2 -58.3 -42.8
0.737
205.2
0.777
-153.5
K 2 , - T g l (K) -12.5 -43.31 -86.4 -112.3
222 Table 2 continued di-Isobutylphthalate di-Methylphthalate n-Dodecane n-Eicosane Ethylbenzene Ethylene glycol Formic acid n-Heptadecane n-Heptane n-Hexadecane n-Hexane 2-Hexanol n-Hexylbenzene Methanol Methyl acetate Methylene chloride Methyl ethyl ketone Naphthalene n-Nonadecane n-Nonane n-Octadecane n-Octane n-Pentadecane n-Pentane n-Pentylbenzene n-Propylbenzene 1,2-Propylene glycol Styrene n-Tetradecane Tetrahydrofuran Tetralin n-Tridecane Toluene n-Undecane Water o-Xylene p-Xylene
0.737
205.2
0.737
-189.9
0.609
118.2
1.19
-193.2
1.07 1.043 0.928 0.779 0.715 1.05 1.115 1.053 1.133 0.99 0.954 0.961 0.855
182.17 294.85 98.52 48.37 32.91 252.6 117.75 238.51 97.66 101.19 154.86 30.8 63.34
0.846 0.783 1.4 0.631 0.872 0.691 1.33 0.731 1.41 1.08 2.06 0.811 1.016
-46.1 -82.2 -80 -130.9 107.7 -49 -39.9 -47.9 -26.7 -152.6 -148.6 -25.6 -33.52
0.585
49.69
1.05
-62.2
0.997
71.9
0.643
51.1
0.813 1.046 1.091 1.048 1.121 1.057 1.158 0.95 0.937
104.2 280.77 139.95 266.68 128.08 224.43 83.57 140.78 112.61
0.767 0.793 1.06 0.773 1.15 0.738 1.66 2.03 1.67
-73 -79.9 -42.9 -72.2 -37.4 -47.6 -23.6 -143.3 -109
0.815
62.05
0.58
-144.5
0.899 1.06 0.899 0.861 1.064 0.917 1.075 1.071 1.049 1.049
93.67 210.34 64.82 113.84 196.26 84.48 168.09 19.3 111.36 111.36
0.876 0.76 0.753 0.966 0.808 1.45 0.917 1.739 0.899 0.673
-48.4 -46.7 10.4 -99.2 -47 -86.3 -47.8 -144.5 -27.7 32
223 Table 3. List of free volume parameters for some common polymers
Polymer
V2* (cm3/g)
Butyl rubber Cis-1,4-poly (isoprene) Neoprene Poly(a-methylstyrene) Poly(carbonate) Poly(dimethyl siloxane) Poly(ethyl methacrylate) Poly(ethylene-propylene) Poly(ethylstyrene) Poly(isobutylene) Poly(methyl acrylate) Poly(methyl methacrylate) Poly(propylene) Poly (p-methy 1 sty rene) Poly(styrene) Poly(styrene-butadiene) Poly(vinyl-acetate)
1.004 0.963 0.708 0.859 0.732 0.905 0.915 1.005 0.956 1.004 0.748 0.788 1.005 0.860 0.850 0.789 0.728
^^(cm 3 /gK)x y 104 2.39 4.64 3.91 5.74 15.2 9.32 3.4 8.17 4.49 2.51 3.98 3.05 5.02 5.18 5.82 6.60 4.33
K 2 2 -T g 2 (K) -96.4 -146.4 -163.3 -395.7 -385.2 -81.0 -269.5 -175.3 -286.9 -100.6 -231.0 -301.0 -205.4 -330.0 -327.0 -184.4 -258.2
The E, parameter can be estimated with no diffusivity data for the system of interest if the solvent jumps as an entire molecule. Since the size of the solvent jumping unit is estimated as the molar volume at 0 K (see Table 2), only the polymer jumping unit size is unknown. Zielinski and Duda [47] have developed a correlation between the polymer jumping unit size and the polymer's glass transition temperature. The results of this correlation are shown in Figure 2. Currently, this is the only available method for estimating ^ when no diffusivity data are available. Although we have outlined a method for estimating £, in the absence of diffusivity data, it should be pointed out that for engineering design calculations, it is recommended that both £, and Do be regressed from limited experimental diffusivity data for the system of interest. 9.3.3 Applicability of the Free-Volume Theory The Vrentas-Duda free-volume theory has been shown to be most applicable in the region just above the solution's glass transition temperature since it is this region where molecular mobility is dictated by the availability of free volume. Ferry [48] has shown this region to exist up to Tg of 100 °C. At and below the Tg, the diffusivity shows a characteristic behavioral change. As the mixture goes through the glass transition, it enters into a nonequilibrium state where extra free volume gets frozen into the solution since the polymer relaxes much slower than any typical time scale. This extra free volume leads to higher
224 diffusion coefficients than would be expected if the polymer were in an equilibrium state. Eq. (44) for the solvent self diffusion coefficient cannot be used below the glass transition temperature since it does not account for the added free volume. Vrentas and Duda [38] have proposed a modified model that incorporates this frozen free volume idea to describe the solvent self diffusivity in the limit of infinite solvent dilution (ci—»()):
~^ 2
A = A, expf-|lexp {RT)
*n(K v
I r
'
(49)
+X(T_T\\ v
")
Figure 2. The polymer jumping unit size can be approximated from the glass transition temperature of the polymer. Experimental data were taken from various diffusivity measurements and NMR relaxation data.
In Eq. (49), A is a parameter describing the difference in the thermal expansion coefficients for the polymer in the glassy and rubbery states. Romdhane et al. [49] have collected solvent self-diffusion data for various solvents in polystyrene and have shown that Eq. (49) can accurately describe the self-diffusivity data as shown in Figure 3.
225
Figure 3. The solvent self-diffusion coefficient of MEK in PS around the glass transition temperature. Solid lined indicate Equation 7 for various X. Zielinski and Duda [46] have suggested that the solvent self-diffusion coefficient can be described by Eq. (44) over the entire concentration range (from c\=Q to c\=\). However, Eq. (45) typically can only describe the mutual diffusivity up to about 50 wt.% solvent because it assumes the polymer self-diffusivity to be small compared to the solvent self-diffusivity. In highly concentrated systems (ci>0.5) the polymer and solvent self-diffusivities can both significantly contribute to the mutual diffusivity. Vrentas and Vrentas [50] have developed an expression for the mutual diffusivity valid for the entire concentration range:
(l-^)(l-2^)+ te^ B-B
^
^
(50)
In this equation, the partial specific volumes of the solvent and polymer, V1 and V2 , are assumed to be independent of concentration. The
—-
parameter is the ratio of self-
diffusivities at infinite dilution. The self-diffusivity of the solvent in an infinitely dilute polymer solution!)" can be approximated by the methods for small molecule diffusion (for
226 example, Eq. (41) applied for a single component). The polymer self-diffusivity can be approximated using the Kirkwood-Riseman model [37]: _ A=
0A96kT JTJFT
( 51 )
The parameter a s in the last equation describes the deviation from a theta solvent. Parameter A is determined as:
A= ^—^-\
Here (R2\
(52)
is the mean squared end-to-end distance of the unperturbed polymer chain.
REFERENCES
1. S.R. De Groot, Thermodynamics of Irreversible Processes, North-Holland Publishing Company, Amsterdam, 1952. 2. J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954. 3. E.L. Cussler, Diffusion Mass Transfer in Fluid Systems, Cambrige University Press, Cambrige, 1997. 4. Y.D. Hsu, Y.-P. Chen, Fluid Phase Equilibria 152, 149 (1998). 5. B.E. Poling, J.M. Prausnitz, J. O' Connell, "The Properties of Gases and Liquids", McGraw-Hill Higher Education, Fifth Edition, New York, 2000. 6. R. Haase, Thermodynamics of Irreversible Processes, Addison-Wesley Publishing Company, Reading, Massachusetts, 1969. 7. E.A. Mason and A.P. Malinauskas, Gas Transport in Porous Media: The Dusty-Gas Model, Elsevier, Amsterdam, 1983. 8. R. Krishna and J.A. Wesselingh, Chemical Engineering Science, 52, 861 (1997). 9. L. Onsager, Phys. Rev. 37, 405 (1931); 38, 2265 (1931). 10. H.B.G. Casimir, Rev. Mod. Phys. 17, 343 (1945). 11. A.A. Shapiro A.A., E.H. Stenby, paper presented at the Fourth International Meeting on Thermodiffusion, IMT4, held in Bayreuth, Germany, 11-16 Sept. 2000; accepted for publication in Lecture Notes on Physics, Springer-Verlag, Heidelberg, 2002. 12. C.F. Curtiss, R.B. Bird, Ind. Eng. Chem. Research, 38, 2515 (1999). 13. J.A. Wesselingh, A.M. Bollen, Trans. IchemE, 15A, 590 (1997). 14. J.M. Zielinski, S. Alsoy, J. of Polymer Science B, 39, 1496 (2001). 15. D.G. Miller, J. of Physical Chemistry, 69, 3374 (1965). 16. T.K. Kett and D.K. Anderson, J. of Physical Chemistry, 73, 1268 (1969).
227 17. 0 . 0 . Medvedev, A.A. Shapiro, to be presented at the 191 European Conference on Applied Thermodynamics, Santorini Island, Greece 6-10 September 2002. 18.1.V. Yakoumis, G.M. Kontogeorgis, E.C. Voutsas, D.P. Tassios, Fluid Phase Equilibria, 130, 31 (1997). 19. R.D. Trengove, P.J. Dunlop, Physica 115A, 339 (1982). 20. M. Lopez de Haro, E.D.G. Cohen, J.M. Kincaid,./. Chem. Phys., 78, 2746 (1983). 21. Y.-D. Hsu, M. Tang, Y.-P. Chen, Fluid Phase Equilibria, 173, 1 (2000). 22. A. Blanc, J. Phys., 7, 825 (1908). 23. L.-O. Sundelof, J. Chem. Soc, Faraday Trans. 2, 77, 1779 (1981). 24. L.-O. Sundelof, Chemica Scripta, 18, 87 (1980). 25. S. Subramaniam,,/. Non-Equilib. Thermodyn., 24, 1 (1999). 26. T.K. Kett and D.K. Anderson, J. of Phys. Chem., 73, 1262 (1969). 27. J.M. Zielinski and B.F. Hanley, AIChE J., 45, 1 (1999). 28. A. Bondi, Physical properties of molecular crystals, liquids, and glasses, Wiley, New York, 1968. 29. H.T. Cullinan, The Canadian Journal of Chemical Engineering, 45, 377 (1967). 30. A. Vignes, lnd. Eng. Chem. Fundamentals, 5, 189 (1966). 31. E.A.J. Guggenheim,./ Chem. Phys., 13, 253 (1945). 32. A. Einstein, Ann. Physik, 17, 549 (1905). 33. M.H. Cohen and D. Turnbull, J. Chem. Phys., 31, 1164 (1959). 34. R.G. Mortimer, N.H. Clark, lnd. Eng. Chem. Fundam., 10, 604 (1971). 35. P.C. Carman, lnd. Eng. Chem. Fundam., 12, 484 (1973). 36. R.G. Mortimer, lnd. Eng. Chem. Fundam., 12, 492 (1973). 37. J.S. Vrentas, J.L. Duda ./ Appl. Polym. ScL, 20, 1125 (1976). 38. J.S. Vrentas, J.L. Duda, J. Appl. Polym. ScL, 22, 2325 (1978). 39. M.L. Williams, R. F. Landel, L. D. Ferry, J. Am. Chem. Soc, 11, 3701 (1955). 40. J.L. Duda, Y. C. Ni, J. S. Vrentas, Macromolecules, 12, 459 (1979). 41. P.J. Flory, J. Chem. Phys., 10, 51 (1942). 42. M.L. Huggins,./. Am. Chem. Soc, 64, 1712 (1942). 43. M.L. Huggins,./. Phys. Chem., 46, 151 (1942). 44. S. Sugden, J. Chem Soc, 1786 (1927). 45. W. Blitz, Rauchemie der Festen Stoffe, Voss, Leipzig, 1934. 46. J.M. Zielinski, J.L. Duda, Polymer Devolatilization, Marcel Dekker, Inc., New York (1996). 47. J.M. Zielinski, J. L. Duda, AIChE.L, 38, 405 (1992). 48. J.D. Ferry, Viscoelastic Properties of Polymers, 2" ed., Wiley, New York, 1970. 49.1.H. Romdhane, R. P. Danner, and J. L. Duda, lnd. Eng. Chem. Res., 34, 2833 (1995). 50. J.S. Vrentas, and C M . Vrentas, Macromolecules, 26, 6129 (1993).
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Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B.V. All rights reserved.
229
Chapter 10: Modelling Phase Equilibria in Systems with Organic Solid Solutions Joao A.P. Coutinho, Jerome Pauly and Jean-Luc Daridon
10.1 INTRODUCTION The formation of alloys of inorganic compounds has been known for centuries, yet it is not only on metallurgy and ceramics that metals and inorganic salts in general form mixed crystals. Reports on the formation of alloys, or solid solutions, of organic compounds, although still rare, have been appearing regularly in the last few years for aromatic compounds, such as substituted benzenes and naphtalenes [1-3], pyrene and anthracene [4-6], for carbohydrates [7], terpenes [8], neopentane derivatives [9-10] and fulerenes, besides solid solutions of C6o with C7o [11-12] solid solutions of C6o in sulphur have been reported [13]. Nevertheless it is among the molecules with long n-alkyl chains such as n-alcohols [14], fats [15], soaps [16] and alkanes [17-20] that solid solutions appear more frequently. Although most of these alloys are still just interesting academic curiosities, these new materials seem quite promising and interesting applications for some of them have been identified. N-alkane blends have been patented for energy storage and preservation of food, medicine and cell tissues [21]; solutions of aromatics based on pyrene have interesting optical properties [4-6]. Surfactants of controlled properties [16] and the manipulation of the behaviour of fats to confer them new organoleptic characteristics [15] have been surfacing showing the exciting possibilities of these new materials. Although the materials based on n-alkane alloys have been used for long as candles, waterproof coatings, pharmaceutical and cosmetics, and in spite of paraffin waxes being a valuable by-product of petroleum refining, the research on n-alkane solid solutions has been mainly prompted by its dark side: The damages and losses caused by wax precipitation from petroleum fluids. Every year wax deposition costs billions of dollars in preventive maintenance, remediation of pipeline blockages and losses of production [22,23]. Better computer models can help anticipate the problems allowing for preventive actions, optimization of actions and avoid losses by shutting down of production or by accident. The presentation of this chapter will focus on the prediction of the wax precipitation from petroleum fluids both at high pressures, in reservoir conditions, or at low pressures, as stock tank oil or fuels. In the final section the application of the model proposed to other systems of organic solid solutions will be discussed.
230 10.2 STATE OF ART The modelling of wax formation in petroleum fluids is based on the thermodynamic description of the equilibrium between the solid wax and the hydrocarbon liquid phases. The general Solid-Liquid Equilibrium equation relates, for each compound, the fugacities of both phases in the standard state,/5, with the pure component thermophysical properties [24]:
{ f"
RTJllx, { T
) RTI2, [ T
)
R { 7) a v
T
))i
All the Solid Liquid Equilibrium models available, independent of the nature of the solid phase, are based on this equation. Their differences derive from the approach to the description of the solid and liquid phases fugacities,/ the definition of the solid phase nature and composition, and consequently on the values adopted for the thermophysical properties. 10.2.1 Non-ideality of the Liquid Phase For the description of the liquid phase two approaches have been adopted. For low pressures the liquid phase is either taken as ideal [25-27] or a free energy model is used [28-32]. At high pressures the fluid phase is described by an Equation of State [33-39]. 10.2.2 Non-ideality of the Solid Phase Two opposite approaches have been adopted for the solid phase. The authors either assume that the solid phase is formed by several independent pure (pseudo) components [34-36] or that it is a solid solution, stable or not, of all the material that crystallises. Among the authors that consider the formation of a solid solution some take it as ideal [25,26,33] but most of them adopt some form of free energy model to describe the non-ideality of the solid phase [27-32]. 10.2.3 Solid Phase and Thermophysical Properties The misty region in most models is the definition of what is the solid phase forming. Early authors assume that all compounds present in the fluid eventually crystallise [28-30]. This concept can still be found in recent authors [33-36] in spite of the growing evidence that only compounds with long n-alkyl chains crystallise to form the wax. Today even the authors that consider the presence of other molecules in the solid phase agree that the main part of the wax fraction is formed by n-alkanes. These molecules have multiple solid habits (rotator, triclinic, monoclinic, orthorhombic.) depending on the temperature and parity of the alkyl chain [40]. The thermophysical properties associated to these transitions are strongly dependent on the solid phase adopted in the modelling. Most authors, behind a generic label of 'solid', fail to clearly state the solid phase they are working with. The evidence today is that the alkyl chains in the wax adopt an orthorhombic structure [41-45] even if some isoalkanes or alkyl aromatic compounds are present as inclusions [46]. The correlations for
231 thermophysical properties used are strongly associated to the solid phase adopted. When the solid phase is poorly defined the thermophysical properties used by the model are fitted to a set of wax formation data and the values adopted, typically fairly different from the pure compounds data, actually compensate for weaknesses in the non-ideality description. These correlations are of limited applicability for systems other than those used in the development of the correlation [26, 30, 35-37]. Only few authors clearly state the solid phase adopted, the compounds present in the solid phase and use, in accordance, thermophysical properties for the pure compounds [27, 31-33, 38,39]. Based on these considerations the thermodynamic wax formation models proposed in the literature can be divided into five categories: Won's, Erickson's, Ungerer's, Calange's and Predictive free energy models. These models, their differences and similarities are described below. 10.2.4 Won's Model Class Won's models such as Won [28, 47], Hansen et al. [29] and Pedersen et al. [30] are characterized by adopting the Regular Solution Theory for the description of the solid phase non-ideality and assuming that all compounds are crystallisable. The solubility parameters are usually correlated to sets of data on wax formation [29, 30] and the correlations for the thermophysical properties adopted, common to other models that often use the Won's correlations for the thermophysical properties [28] do not distinguish between the different hydrocarbon families. They are also characterised by the importance attributed to the heat capacity term on Eq. (1) being the correlation for heat capacities proposed by Pedersen et al. [30] still used by models of Ungerer's type. 10.2.5 Erickson's Model Class Erickson's models assume an ideal solid phase. On this model class can be considered the models of Erickson et al. [25, 48], Pedersen [26] and Rjanningsen et al. [37]. The predictive version of Calange's model assuming ideal solid phase [33] can also be characterized as belonging to this class. They assume the presence of non n-alkanes in the solid phase, sometimes even of very light compounds like in the Calange's model [49], and the thermophysical properties correlations used are, although with modifications, those proposed by Won [28]. 10.2.6 Ungerer's Model Class The models of this class are based on a seminal idea proposed by Leivobici to use Eq. (1) for the estimation of the solid phase fugacity [34]. This allows the description of the liquid phase by an Equation of State providing an approach that extends the description of wax formation to high pressures. This idea has made its way into all the other model classes being today generally adopted for high pressure VLSE description. What individualizes this kind of models is a solid phase as being formed by multiple pure solid phases. Ungerer et al. [34] proposed this approach for synthetic mixtures of non co-crystallisable compounds but its extension to crudes, first at low pressures by Lira-Galeana et al. [35] and then by Pan et al.
232
[36] for high pressures continues using the assumption that all pseudocomponents used in the fluid characterisation crystallise as pure independent compounds. Non-paraffins are considered to be also present in the solid phase but the thermophysical properties correlation proposed by Won [28] is used only for the n-alkanes. New correlations are proposed for the other hydrocarbon families. Like the models of Won's type these models continue to use the heat capacity term of Eq. (1) shown by some authors to be negligible [33, 50] and abandoned in most of the other model classes. As expected for these models the pseudocomponent definition and characterization plays a dramatic role in the modelling with the oil characterization acting as a tuning parameter [51]. 10.2.7 Calange's Model Class These models are characterised by using the solid phase non-ideality as a fitting parameter. Typically a cubic EOS describes the fluid phases and the activity coefficient of the solid phase is treated as a tuning parameter [33, 49,52]. Other parameters can be used to fit the experimental data such as binary interaction parameters, thermophysical properties [52, 53], and the oil characterization. Although successful in correlating experimental data given their flexibility these models have no predictive character and its use holds some danger because they reproduce the experimental data available whether it is accurate or wrong. 10.2.8 Predictive Free Energy Model Class On these models a predictive free energy model describes the non-ideality of the orthorhombic solid solution of alkyl chains. This type of models can be traced back to Won [28] and his approach to the estimation of the solid phase solubility parameter based on the heats of sublimation of pure compounds. This suggestion can be found in the Predictive Local Composition concept proposed by Coutinho and Stenby [31] where the interaction energies are estimated from the heats of sublimation of pure compounds. This approach was first applied to the Wilson Model [31, 54] and then extended to NRTL and UNIQUAC [55]. Predictive UNIQUAC [27, 32] proved to be very successful in the description of wax formation for a broad range of fluids [56,57] considering that the wax phase was formed by only n-alkane molecules and adopting correlations for the thermophysical properties based on the orthorhombic solid phase of the pure n-alkanes. 10.2.9 Modelling High Pressure Wax Formation Although the initial models proposed for each of these approaches focused in the low pressure fluids [26-33, 35, 47] the need for a model that could describe the wax formation in live oil at reservoir conditions promoted the development of the models for high pressures [34, 36-39, 48-49, 53]. The approached followed in all cases is based on the suggestion by Leivobici [34] for the estimation of the solid fugacities, with the Poyinting correction to take into account the pressure dependence, and to use a cubic Equation of State for the description of the fluid phases. The dependence of the solid phase non ideality with pressure was taken into account, deliberately or not, by underestimating the volume decrease in the phase transition as demonstrated by Pauly et al. [38].
233
A systematic effort of comparison of these models performance for crudes has not been reported yet. Unfortunately the authors usually do not perform comparisons between the models proposed and other models available in the literature. The limitations of the Won's type models are evident from the results reported by Pedersen [26] and Calange [33]. Erickson's type model if not carefully used can lead to the problems reported for Calange's model [49] where due to the ideal nature of the solid phase even methane can be found on it. Ungerers's type models in their approach to the description of the solid phase lack a sound thermodynamical basis as discussed by Pedersen and Michelsen [51] and physics as the solid phase used does not agree with the reality. Fitting models, if carefully tuned, can describe about any set of experimental data. These models are of limited utility as, lacking predictive capacities, experimental data must be available and they reproduce the experimental data even if this is of poor quality. The soundest thermodynamical approach and that has been most widely applied in the open literature to quite a number of different systems from complex synthetic mixtures to fuels and crudes are the Predictive free volume models. This approach was also found to be the best in a comparison with other wax formation models on a complex synthetic mixture [54] reported by Pauly and Daridon. The multicomponent SLE data for hydrocarbon systems produced at University of Pau became a reference in the field and today all the new models reported are being tested against them.
10.3 WAX FORMATION MODEL As discussed above and according to the experimental evidence [41-45] the model hereafter presented considers the wax formed by n-alkanes crystallising from a fluid as orthorhombic solid solutions [31-32]. The condition of equilibrium between the Liquid, Solid and Vapour phases is given by the equality of the fugacities in all phases for each individual component, i: f:(T,P,xJ)
= f!(T,P,x',)
= f;(T,P,x;)
(2)
Taking the liquid phase as the reference phase, the equilibrium ratios, KJ and K*, are defined as:
K J = ^ =^ 1
(3)
x', tf(P)
*;=4=4^
(4)
where <j>;, are the fugacity coefficients and the x> the mol fractions for the different phases .
234
10.3.1 Low Pressure At low pressure the vapour phase is neglected and the solid phase equilibrium constants are obtained directly from Eq. (1) as:
K; = ZU^f^-,]+ ^f^-,]l s
\
DT
\
T
T>T
(5)
T
The heat capacities term of this equation was found negligible [33, 50] and is not used. The correlations used for the thermophysical properties are presented on the Fluid Characterization section below. The liquid phase activity coefficient Since the solubility of n-alkanes is similar in different non-polar solvents as shown by Coutinho and Darridon [27] the liquid phase may be assumed ideal and the activity coefficient taken as unit. This approach simplifies considerably the calculation and the Fluid Characterization. Alternatively a more rigorous approach may be used with liquid phase non-ideality described by a modified UNIFAC model: lny, =lnf, ro +\nr"""l"/r
(6)
The residual term, In fes, describing the energetic interactions between unlike molecules, such as aromatics and aliphatics, is given by the modified UNIFAC [59-61], while the size difference effects and free volume contributions, In fomb-fv^ a r e described by the Flory-free volume equation [27]:
l n y r - ' = ln4"Hl-^ x x
'
'
with +,= 'y-
-V" 3
T4V -K Y
(7)
j
where V{ is the molar volume obtained from the literature or estimated by Elbro group contribution method [62]. The Vwj is the van der Waals volume of component i estimated using the Bondi method [63] or alternatively using the UNIFAC volume parameters r [59-61] Fw,= 15.17 xr, The solid phase activity coefficient - Predictive UNIQUAC For the solid phase non-ideality the Predictive UNIQUAC model is used [32, 55]
(8)
235
Here TJ; is T, =exp
'-
(10)
and the X$ are the interaction energies. The area fraction, 9j, and the segment fraction,
(ID
J
(12)
j
using a new definition for the structural parameters r and q [32]. The correlations for the r and q values with the n-alkane chain length are: rn=0.1Cni+0.0672
(13)
qn=0.1Cni+0.1141
(14)
The predictive local composition concept [31 ] allows for the estimation of the interaction energies. The interaction energies, Xa, are estimated from the heat of sublimation of a pure orthorhombic crystal, ^=~(^bH,-RT)
(15)
with Z being the coordination number. It has a value of 6 for orthorhombic crystals. Correlations for the heat of sublimation are provided at the end of the Chapter. The interaction energy between two non-identical molecules is given by ^=Xii=Ait where./ is the n-alkane with the shorter chain of the pair ij.
(16)
236 This model is purely predictive, requiring for the calculation of the phase behaviour nothing but the pure component thermophysical properties. 10.3.2 High Pressure At high pressure a gamma-phi approach will be used to describe the VLSE. The fugacity coefficients for the equilibrium ratio of the fluid phases, Eq. (3), will be estimated by a cubic equation of state while the fugacity of the solid phase required for the equilibrium ratio of the solid phase, Eq. (4), will be obtained from Eq. (1) corrected for the pressure by the Poynting correction.
Fluid phase fugacities The evaluation of liquid and vapour fugacity coefficients is performed with the SoaveRedlich-Kwong Equation of State [64]: a
p = -™
(V-b)
—
(17)
V(V + b)
For mixtures, the linear mixing rule is kept for the parameter b
(18)
& = X*A whereas for the a parameter the LCVM mixing rule is used: ( a)
(X
]-X](G'-:)
\-X^
(b)
^
Here Am, Av and X are constant. The excess Gibbs free energy GE of the liquid mixture is calculated using the modified UNIFAC group contribution method [61] with interaction parameters on the following form:
%, = exp\- -^
'±j
-J
(20)
The parameters were estimated by Boukouvalas et al. [65, 66]. The critical properties for light gases are widely available in the literature (e.g. Poling et al. [67]). For the heavy components the Twu correlations [68], described in the next section, are used.
237
Alternatively to the LCVM mixing rule the quadratic mixing rule for the a parameter can also be applied
however due to a mismatch of the fugacities of the fluid and solid phases in the gamma-phi approach the predictive character of the model is lost. With the quadratic mixing rule the interaction energy mismatch for the solid phase, ^, must be fitted to the WAT of the fluid as described below [69]. The loss in predictive character is compensated by a gain in flexibility. Solid phase fugacity As discussed above the fugacity of the solid phase is estimated from Eq. (1). The equilibrium constant for the solid phase is given by [38]
r?[P.] UJ exp{ (1 -^ (P -^ + ^f^-lV^f^-llI
(22)
where p=0.9 is the ratio between the solid and liquid molar volumes of n-alkanes [38], and the Peneloux volume corrections, Q, are introduced to improve the description of the liquid densities [38]. The fugacities of the liquid phase are obtained by the SRK-EOS. Po is the reference pressure taken as atmospheric pressure. The activity coefficients can be modelled by the Predictive UNIQUAC model described above. Often this model will entail the presence of multiple solid phases requiring multiple solid equilibrium constants to be used and making the problem more complex from a numerical point of view. The use of the Predictive Wilson for the solid phase activity coefficients allows for a good description of the WAT's with a better numerical stability since a single solid phase will be involved. The solid phase activity coefficient - Predictive Wilson For multicomponent mixtures the activity coefficients may be alternatively evaluated by Predictive Wilson [31]:
iny,=i-in5>A-lY7r j
with:
(23)
238
A
-"4r^ir)
<24)
The interaction energies are estimated as discussed above. Although this solid phase model allows for a good description of the WAT's it underestimates the non-ideality of the solid phase [32] and consequently overestimates the solid fraction formed below the WAT. The use of the van de Waals one fluid mixing rule for the Equation of State instead of LCVM is possible and simplifies the calculations. Due to the mismatch of the fugacities between the solid and fluid phases caused by the gamma-phi approach used a correction to the interaction energies in the solid phase must be used A,J=1J,=1JJ(\+^
(25)
the interaction energy mismatch for the solid phase, £, is fitted to the WAT at atmospheric pressure or other experimental data available. The model loses some predictive character since at least one data point must be available for the fitting. However this loss is compensated by a greater flexibility of the model and an improved description of the phase equilibrium. Moreover some guidelines for the estimation of the interaction energy mismatch can provide an approximation to the phase equilibrium even in absence of experimental data [69]. 10.3.3 Multiphase Flash Calculations The presence of at least three phases (gas, liquid and solid) with multiple solid phases being possible requires the use of a multiphase flash in the calculation of wax formation in hydrocarbon fluids. The algorithm of resolution of the Rachford-Rice equations applied to multiphase systems proposed by Leibovici and Neoschil [70] can be used in the phase equilibrium calculations. It is a very stable algorithm because of its continuity across phase boundaries leading to non-physical phase amounts for the unstable phases. The number of solid phases is a product of the multiphase flash calculation resulting from the instability caused by the non-ideality of the solid solutions. With this approach if p phases are present and the liquid phase is used as reference phase the phase fractions, y/h are obtained from the solution of a systems of(p-l) equations [70, 71]. Given a fluid with a composition zh for a generic phase k
'-'i+2>/-iV, ./=2
239 The Kjk are the equilibrium constants for compound / in phase k. Using this algorithm it is possible to easily estimate the phase fractions and the compositions of each phase present at the equilibrium. Other algorithms more or less time consuming and with various degrees of complexity are also possible [71, 72].
10.4 FLUID CHARACTERIZATION For crude oils and other real fluids a detailed composition is generally not available and a fluid characterisation is essential. The characterization used with this model considers that the fluid consists of two types of compounds: the solute, comprising the n-alkanes that crystallise to form a solid phase, and the solvent, formed by all the non-crystallisable compounds. A different approach is used for the characterization of the two types of compounds. 10.4.1 Characterization of n-Alkanes The work developed by our team indicates that an accurate description of n-alkanes SolidLiquid Equilibrium cannot be achieved unless each n-alkane is considered individually [43, 54, 56, 73]. The n-alkanes characterization aims at develop a description of the n-alkanes distribution as close as possible to the actual fluid. This distribution is described by two parameters: the total n-alkane content and their decay. These can be obtained in a number of different ways: Total n-alkane content 1 - Chromatographic measurements. Gas Chromatography (GC) or High Temperature Gas Chromatography (HTGC) allows for the direct detection and quantification of n-alkanes up to very high molecular weights. It can provide both the total n-alkane content and its decay. 2- Solvent precipitation. The UOP 46-85 essay [74], or any of its variants [75-77], are standard procedures that provide a good estimate of the total wax content in a crude oil. 3- In the absence of further analytical information the wax content can be obtained from a correlation involving some known property of the oil. A correlation for paraffmic waxy crudes of the total wax content with the oil average molecular weight, Mw, was proposed by Coutinho and Daridon [27] Wax content (C20+)[wt%]= 0.070 * Mw - 8.3
(27)
The n-alkane decay The exponential decay of the n-alkanes in oils is well documented in the literature [78-79]. It is defined as the ratio between the mass fractions of two successive n-alkanes
240 a =^ -
(28)
Its value can be estimated by: 1- Chromatography: Again GC or HTGC can provide information about the paraffin distribution with even better accuracy than for the wax content. 2- Simulated distillation: The n-alkane decay can be related to the decay of True Boiling Point or Simulated Distillation fractions [27] by «„-*,»,=«„,„* "0.03 (29) 3- Average value: If no information at all is available the average decay value of a=0.88 (
(30)
Tt2,i [K] = 420.42 - 134784 exp( -4.344 ( Cni+ 6.592 ) ° I4627 )
(31)
And A,otHi [kJ/mol] = 3.7791Cni - 12.654
(32)
241 AfusH, [kJ/mol] = 0.00355C,,,3 - 0.2376C,,,2 + 7.400C nl -34.814
(33)
with Al2H = AlolH - AfusH
(34)
where Cnj is the number of carbon atoms in n-alkane i. These equations are valid from pentane to n-alkanes larger than n-CiooH2O2 for the melting temperatures, TfUSj, and total heats of melting, AlotH. The solid phase transitions occur for n-alkanes between n-C9H2o and nC 4 iH 84 inclusive. The heat of sublimation, AsubH=AvapH+AfusH+At2H
(35)
is calculated at the melting temperature of the pure component, and the heat of vaporisation, AvapH, is assessed using the PERT2 correlation by Morgan and Kobayashi [82] A vap H/RT c = AHV(O) + co AHV(' > + co2AHv<2)
(36)
with AHV(O>= 5.2804 x°' 3 3 3 3 +l 2.865 x a 8 3 3 3 + 1.171 x 1 2 0 8 3 - 13.116 x +0.4858 x 2 - 1 . 0 8 8 x 3
(37)
AHV(1)= 0.80022x O3333 + 273.23 x O8333 + 465.08x 1 2 0 8 3 - 638.51 x - 145.12x 2 + 74.049x 3 (38) AHV(2>= 7.2543 x 0 3 3 3 3 - 346.45 x 0 8 3 3 3 - 610.48 x 1 2 0 8 3 + 839.89 x + 160.05 x 2 - 50.711 x 3 (39) where x=(l-T/T c ). The acentric factor is a quadratic function of the carbon number, Cnj, co = - 0.000185397 C ni 2 + 0.0448946 C n i - 0.0520750
(40)
and the critical properties are obtained from the correlations of Twu [68] 6=ln(Mw)
(41)
T b [K]=[exp(5.71419+2.715796-O.286590 2 -39.8544/8-0.122488/9 2 )-24.75220+ 35.31559 2 ]/1.8
(42)
T c [K]=T b (0.533272+0.343831 x 10"3Tb+2.52617x 10"7Tb2-1.65848x 10"I0Tb3 +4.60774xl0 24 /T b 13 )" 1
(43)
242 Pc[atm]=(1.0+0.312(l-Tb/Tc)1/2+9.1(l-Tb/Tc)+9.4417(l-Tb/Tc)2+27.1793(l-Tb/Tc)3)2
(44)
10.5 RESULTS A comparison of the performance of the model described above with predictions by models of the Won, Erickson and Ungerer classes is presented in Figure 1 for a mixture of n-alkanes in the C20-C30 range in decane [54, 83]. The superiority of the Predictive UNIQUAC model is clear. The correlations for the thermophysical properties used are those of Eqs. (30-34) without which the results would be much worse. It is interesting to notice that although the models generally over-predict the solid fraction but are qualitatively correct the multiple solids model (Ungerer) shows an odd behaviour.
Figure 1: Comparison of the performance of Predictive UNIQUAC model [32, 55] with other literature models for the solid phase non-ideality. Reprinted from [83]. To illustrate the performance of the model at low pressures results for a crude oil and diesel are presented. Figure 2 shows the fraction of wax formed in crude 16 [76] below the WAT [83] and Figure 3 the change in liquid phase composition with the crystallisation of nalkanes for a diesel below the cloud point [56, 83]. In both cases the solvent has been described by a single pseudo-component chosen to match the average molecular weight of the fluid. The n-alkanes for the diesel were obtained from a GC analysis and for the crude the wax content was measured by ketone precipitation [74, 76] and the distribution obtained from the simulated distillation data using Eq. (29).
243
Figure 2: Comparison between experimental data and model predictions for the fraction of wax forming in Oil 16 [76]. Reprinted from [83].
Figure 3: Experimental data and model predictions for the temperature dependence of the paraffins composition in the liquid phase for Diesel S [56]. Reprinted from [83].
244
The description of the phase envelope of a North Sea gas condensate obtained using the LCVM mixing rule is reported in Figure 4 [39, 83]. For the characterization of the fluid no lumping was attempted for the C7- fraction. For the C7+ the n-alkanes were obtained by a GC analysis and a single pseudo-component for the non-crystallisable was used. The critical properties for this pseudo-component were fitted to one data point on the VLE boundary line. A comparison between the performance of the high pressure modelling using the LCVM and the quadratic mixing rules is reported on Figure 5 for a diesel [69]. The characterization was done as previously [56]. The interaction energy mismatch, t,, was fitted to the atmospheric cloud point. It can be seen that the use of the quadratic mixing rule not only simplifies the calculation procedure allowing for the use of a well established mixing rule but also provides a better description of the wax formation below the cloud point. The values for the parameter £, correlate well with the number of n-alkanes present in the distribution as shown in Figure 6. For distributions with a number of alkanes above 15 the value of the interaction energy mismatch becomes approximately constant with ^=0.06. This is a consequence of the maximum number of n-alkanes that can be present in a single solid phase.
Figure 4: Measured and calculated phase envelope for a live oil from North Sea [37]. Reprinted from [83].
10.6 OTHER SYSTEMS The applicability of the model described above is not limited to petroleum fluids. The thermodynamic approach used is very general and can be applied to any organic alloy as long as the estimation of interaction energies described by Eq. (16) is relaxed. This relationship was derived for n-alkanes and it is not applicable to other compounds. For solutions of nalkyl chains the Eq. (25) can still be used since the interaction energy mismatch will take into
245 account the deviations to the n-alkane behaviour presented by other systems. This approach, while never attempted before, should work for iso-alkanes, n-alcohols, fats, a-olefins and other sort of waxes. Of particular interest would be the description of the behaviour of Fischer-Tropsch waxes, which involve several of these compounds with long n-alkyl chains, and the dewaxing of base-oils by solvent. For a generic system the interactions between two unlike molecules cannot be associated to the interactions between identical molecules, as suggested by Eqs. (16) and (26). For these systems the interaction energies, Xy, have to be fitted to experimental data. Approaches to the description of solid solutions using other excess Gibbs energy models are also possible. Oonk and co-workers have studied multiple solid solutions [2-3, 8-9, 14] successfully using Redlich-Kister polynomials to correlate the excess properties of the solid phases. For waxes in the high temperature rotator phase Coutinho el al. [83] proposed CDLP, a model based on a Margules equation for the description of the excess properties of the solid phase.
Figure 5: Comparison between the performances of the LCVM and quadratic mixing rules for the high pressure behaviour of a diesel [69].
246
Figure 6: Interaction energy mismatch for various fluids as function of the number of nalkanes present [69].
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248 43. M. Dirand, V. Chevallier, E. Provost, M. Bouroukba, D. Petitjean, Fuel, 77 (1998) 1253. 44. V. Chevallier, E. Provost, J.B. Bourdet, M. Bouroukba, D. Petitjean, M. Dirand, Polymer, 40(1999)2121. 45. V. Chevallier, A.J. Briard, D. Petitjean, N. Hubert, M. Bouroukba, M. Dirand, Mol Cryst Liq Cryst, 350 (2001) 273. 46. D.L. Dorset, Energy and Fuels, 14 (2000) 685. 47. K.W. Won, Fluid Phase Equilibria, 53 (1989) 377. 48. T.S. Brown, V.G. Niesen, D.D. Erickson, SEP 28505 (1994). 49. K. Delmas, Etude du comportement des bruts paraffmiques dans les conditions de Poffshore profond. PhD thesis, IFP, France, 2001. 50. J.A.P.Coutinho, S.I. Andersen, E.H. Stenby, Fluid Phase Equilibria, 103 (1995) 23. 51. K.S. Pedersen, M.L. Michelsen, AIChE J., 43 (1997) 1372. 52. J.Y. Zuo, D.D. Zhang, Chemical Engineering Science, 56 (2001) 6941. 53. N. Lindeloff, R. Heidemann, S.I. Andersen, E.H. Stenby, Ind. Eng. Chem. Res., 38 (1999)1107. 54. J. Pauly, C. Dauphin, J.L. Daridon, Fluid Phase Equilibria, 149 (1998) 173. 55. J.A.P. Coutinho, Fluid Phase Equilibria, 158-160 (1999) 447. 56. J.A.P. Coutinho, C. Dauphin, J.L. Daridon, Fuel, 79 (2000) 607. 57. J.A.P. Coutinho, Energy and Fuels, 14 (2000) 625. 58. T.S. Brown, V.G. Niesen, D.D. Erickson, SEP 28505 (1994) 59. Aa. Fredenslund, R.L. Jones, J.M. Prausnitz, AIChE J., 21 (1975) 1086. 60. Aa. Fredenslund, J. Gmehling, P. Rasmussen, Vapor-Liquid Equilibria using UNIFAC, Elsevier, Amsterdam, 1977. 61. B.L. Larsen, P. Rasmussen, Aa. Fredenslund, Ind. Eng. Chem. Res. 26 (1987) 2274. 62. H.S. Elbro, Aa. Fredenslund, P. Rasmussen, Ind. Eng. Chem. Res., 30 (1991) 2576. 63. A. Bondi, Physical Properties of Molecular Crystals, Liquids and Glasses, Wiley, New York, 1968 64. G. Soave, Chem. Eng. Sci., 27 (1972) 1197. 65. C.J. Boukouvalas, N. Spiliotis, P. Coutsikos, N. Tzouvaras, D.P. Tassios, Fluid Phase Equilibria, 92 (1994) 75. 66. C.J. Boukouvalas, K.G. Magoulas, S.K. Stamataki, D. P. Tassios, Ind. Eng. Chem. Res., 36, (1997) 5454. 67. B.E. Poling, J.M. Prausnitz, J.P. O'Connell, The Properties of Gases and Liquids, 5th Ed. McGraw-Hill, NY, 2000. 68. C.H. Twu, Fluid Phase Equilibria, 16 (1984) 137. 69. J.A.P. Coutinho, J.M. Sansot, J. Pauly, J.L. Daridon, Proceedings of 19th ESAT, Santorini, Greece, September 2002. 70. C. F. Leibovici, J. Neoschil, Fluid Phase Equilibria, 112, (1995) 217. 71. M.L. Michelsen, Fluid Phase Equilibria, 9 (1982) 21. 72. R.M. Abdel-Ghani, A.V. Phoenix, R.A. Heidemann, Presented at the 1994 annual AIChE meeting San Fransisco, Ca. 73. C. Dauphin, J.L. Daridon, J.A.P. Coutinho, P. Baylere, M. Potin-Gautier. Fluid Phase Equilibria, 161 (1999) 135.
249 74. Laboratory test methods for petroleum oils, Universal Oil Products, Des Plaines, IL (1969). 75. E.D. Burger, T.K. Perkins, J.H. Striegeler, J. Pet. Tech., 33 (1981) 1075. 76. H.P. R.0nningsen, B. Bjorndal, A.B. Hansen, W.B Pedersen, Energy and Fuels, 5 (1991)895. 77. B.J. Musser, P.K. Kilpatrick, Energy and Fuels, 12 (1998) 715. 78. T.G. Monger-McClure, J.E. Tackett, L.S. Merrill, SPE 38774 (1997). 79. A. Hammami, M. Raines, SPE 38776 (1997). 80. M.G.J. Broadhurst, Res. Nat. Bur. Stand., 66A (1962) 241. 81. J.J. Marano, G.D.Holder, Ind. Eng. Chem Res., 36 (1997) 1895. 82. D.L. Morgan, R. Kobayashi, Fluid Phase Equilibria, 94 (1994) 51. 83. J.A.P. Coutinho, J. Pauly, J.L. Daridon, J. Braz. Chem Eng., 18 (2001) 411. 84. J.A.P. Coutinho S.I. Andersen E.H. Stenby , Fluid Phase Equilibria, 117 (1996) 138.
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Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B.V. All rights reserved.
251
Chapter 11: An Introduction to Modeling of Gas Hydrates Eric Hendriks and Henk Meijer
11.1 INTRODUCTION 11.1.1 What are the Gas Hydrates? Gas hydrate is shorthand for: clathrate hydrate(s) of natural gas. This is a subset of a group of solid, crystalline structures - inclusion compounds - and may be characterized as follows [Zett5] Gas hydrates are crystalline molecular complexes formed from mixtures of water and low molecular weight compounds. The water molecules form a lattice structure, based on hydrogen bonding. The 'gas' molecules reside in interstitial vacancies (cages) of the lattice without occupying a position in the water lattice. A gas hydrate is thus a solid, not a gas. The difference with ordinary (stochiometric) hydrates is that the composition is not fixed, and cannot be expressed in small integers. The reason for this can be understood by comparing the type and strength of the interactions involved. Each oxygen atom is connected to two hydrogens by covalent bonds and to two hydrogens by hydrogen bonds. The energy stored in a covalent bond is ca. 430 kJ/mol, in a hydrogen bond it is ca. 20 kJ/mol, whereas the lattice-hydrate Van der Waals interaction represents energy of ca. 1.3 kJ/mol. So the binding of the guest to the lattice is comparatively weak. However, the lattice cannot exist without the guest molecule: it is thermodynamically unstable (or meta-stable). In fact, a substantial fraction (in the order of 95%) of the cages is always occupied. Complete occupation - which would correspond to the ideal composition, seems impossible, but the fact that it is always close to it made early investigators believe gas hydrates to be stochiometric. In Appendix B (nomenclature), there is list of terms that are useful to know when reading literature on gas hydrates. 11.1.2 Physical Properties of the Hydrates When being formed, hydrates look and behave like wet snow. In the vast majority of cases, the higher the pressure, the higher the temperature at which hydrates melt. Wet hydrate crystals stick to each other and to metal surfaces. The physical properties of gas hydrates depend on the composition of the hydrate, which in turn is a consequence of the pressure, the temperature and the composition of the gas and/or liquid phase(s) in equilibrium with it. Some typical ranges of physical properties can be given. The density of gas hydrates
252
typically ranges between 900 and 950 kg/m3. One cubic meter of liquid water will, when combining with gas to form hydrates, result in the formation of 1.22 m3 of hydrates. Characteristic properties of hydrates are their low thermal conductivities and large heats of dissociation. The thermal conductivity of gas hydrates is approximately 0.5 W/(m.K) which is one quarter of the thermal conductivity of ice. Approximately 450-500 kJ of heat is needed to dissociate 1 kg of hydrates into liquid water and gas at atmospheric pressure and zero degrees centigrade. This is approximately 1.5 times the heat required to convert ice into liquid water. One of the factors responsible for this large heat of dissociation is the gas expansion during the dissociation process. 11.1.3 History Clathrate compounds were first discovered in the early 1800s in experiments with chlorinewater mixtures, in which Davy and Faraday observed the formation of a solid above the normal freezing point. In thel930s, Hammerschmidt discovered that gas hydrate was responsible for plugging natural gas pipelines. After that, a number of researchers investigated the physics of various clathrates. Predictive models of their formation were formulated, and methods to inhibit hydrate formation were and still are developed. In the late 1960s methane hydrate was observed as a naturally-occurring constituent of subsurface sediments in Western Siberia and in Alaska. 11.1.4 Importance of Gas Hydrates The oil industry is interested in gas hydrates mainly because they are nuisance. Hydrates may form from natural gases at elevated pressures at temperatures well above 0°C. In the 1930s it was discovered that they could block pipelines, e.g. gas transmission lines at temperatures above the melting point of ice, and deepwater pipelines transporting water and condensed hydrocarbons. They can also reduce the efficiency of oil recovery, by decreasing the reservoir permeability. In principle, positive use could be made of hydrates, too. Only molecules of the right size and shape enter the hydrates, and the composition in the hydrate is different from the composition of the phases in equilibrium with it. Potential applications include separation of light gas components and desalination of water. Other applications include storage and transportation of natural gas, including underground storage of radioactive gases. Last but not least, hydrates are a potential (unconventional) energy resource. The quantity of natural gas stored as hydrates present on earth has been (optimistically) estimated to be, in order of magnitude, twice the amount of combined fossil fuel. A large research effort is now devoted (mainly in the US, Japan and India) to study the possible exploitation of these resources. 11.1.5 Methods to Prevent Hydrates The methods to prevent hydrate formation may be divided into thermodynamic1 and 'nonthermodynamic'. The former include all methods that can be rationalized on the basis of equilibrium measurements and calculations. They are generally well established and accepted by the industry. They fall in one of the following categories
253 • • •
Removal of water from the mixture below the level of saturation with respect to hydrate formation. Control of temperature and pressure Addition of an inhibitor to the mixture, shifting the equilibrium
The 'non-thermodynamic' methods include application of growth modifiers, transportation of hydrate crystals, kinetics, and so on. This is much more difficult to model, and there are no generally accepted theories or prediction methods. 11.1.6 Types of Gas Hydrates Clathrathes come in different crystal structures, depending on host molecule, guest molecule and thermodynamic conditions. For gas hydrates, in the oil industry, two structures are of practical relevance: structure I (S-I) and structure II (S-II). A third structure, structure H was discovered in the 1980s. Its relevance to the oil industry is still being debated. The structures differ in the exact arrangement of water molecules in the lattice but they have also characteristics in common. Both structures I and structure II are of high, cubic symmetry and both contain two types of cages. These cages are comparatively large cavities, where guests may 'sit'. The unit cell in structure H has hexagonal symmetry (hence the name). The small cage in each structure is a dodecahedron (12-hedron), made up of 12 pentagons (5-gons). In the hydrates, they are distorted somewhat, and in fact the average radii differ a little bit between the two structures. In S-I, the large cage is a tetrakaidecahedron (14-hedron), made up of twelve 5-gons and two hexagons (6-gons); in S-II it is a hexakaidecahedron (16hedron) made up of four 6-gons and twelve 5-gons (this is written as 5 12 6 2 ). In structure H there are three cage types: 5 12 , 4 13 5 6 6 3 and 5 12 6 8 .
Table 1. Number of water molecules, types and number of cages in the unit cells of hydrate structure I, structure II and structure H.
254 11.1.7 Which Components Form Hydrates? A large number of substances may form (simple) hydrates, when brought into contact with water. The following heuristic rules apply. The molecules may not be too small or too large. They should fit into at least the largest cavity. The range is roughly 3.8A(argon) to 7.lA(normal butane). The latter component forms a hydrate only in the presence of a second guest molecule "hilfsgas". The smallest molecules (3.8A to 4.4A) form structure II with both cages occupied. Structure II has a relative large number of small cages. These are most suited to containing the small molecules. Guests with intermediate sizes form hydrate structure I, the smaller ones (4.5A to 4.9A) entering both cages, the larger ones (5.lA to 5.8A) entering only the large cages. The largest guests (5.8A to 7.1 A) form structure II, where the small cage is left empty. Some components, if the size is near the transition, may form either structure, depending on the conditions. Examples are C-C3H6 (cyclo-propane) and (CFb^O. The chemical nature of the component has an influence as well. The guest must not contain a strong hydrogen bonding group, or a number of moderately strong functional groups. Most natural gas hydrate formers are hydrophobic, with the notable exceptions of H2S and CO2. The shape of the guest molecule seems to play a minor role in hydrate formation. For example, n-butane is enclathrated (as a mixed hydrate) as the gauche isomer, rather than the trans isomer, which is preferred in the gas phase Some larger molecules, such as benzene, can form structure II if a second component such as methane is present. Or, phrased differently, if enough benzene is present, methane will no longer form structure I, but structure II, at a lower pressure. Such heavy structure II hydrate formers include benzene, cyclo-hexane, cyclo-pentane and neo-pentane. There are no known simple structure H hydrates, i.e. always at least two different hydrate formers are needed: a small one, such as methane, to occupy the smaller cages, and a larger component to occupy the large cages. Many components of structure H have been identified, including iso-alkanes such as 2,3-dimethylbutane, cyclo-alkanes such as cyclopentane, and alkenes. 11.1.8 Reviews There is a large literature on gas hydrates, or more generally, clathrates. For an overview - of both the subject and the literature - we refer to the review papers and books listed at the end of this paper. Extensive information can also be found on the internet. An absolute must is the 1959 paper of Van der Waals and Platteeuw3, which not only contains the description of the famous solid solution model, but also serves as a splendid review, still in many respects up-to-date.
255 11.2. THERMODYNAMIC BEHAVIOUR OF HYDRATES 11. 2.1 Hydrate Equilibrium Curve Figure 1 shows a typical phase diagram of multi-component mixture of light hydrocarbons (methane to n-butane) and water. The pressure on the vertical axis has been plotted on a logarithmic scale to be able to better see details. Various regions in which different type of phases coexist are shown. The following notation is used: V=vapor phase: H=hydrate phase; I.ii=hydrocarbon liquid phase: I,n=aqueous phase; lee=lce phase. The banana shaped curve is the familiar oil-gas phase envelope. It is intersected twice by the hydrate dissociation curve. On the low temperature side of the hydrate equilibrium curve, hydrates are stable and will form in thermodynamic equilibrium, provided enough water is present. In general, as is the case in this example, the hydrate dissociation temperature increases with increasing pressure. The hydrate equilibrium curve is dependent on the composition of the fluid in question, and in general thermodynamic models must be applied to predict it. but some simple rules exist that provide understanding and rough estimates.
Figure 1. Calculated multi-component phase diagram showing various regions of coexisting phases. The composition of the simple natural mixture is: methane (70 moles), ethane (20 moles), propane (10 moles), n-butane (10 moles). 11.2.2 Engineering Rules There arc some simple approximate engineering rules which are based on sound physics and which provide insight in the thermodynamic behavior of hydrates. Some of these generalities are presented below. The hydrate dissociation temperature is determined by the partial pressures of the hydrate forming compounds (including water) in the gas phase.
256 This is an important rule by which one can predict the effect of changes in the composition of simple (only hydrocarbons and water) systems on the hydrate dissociation temperature. Some examples are given below: The removal or the addition of liquid water from a system containing hydrocarbons and water hardly affects the hydrate equilibrium curve of the system (as long as liquid water and a vapor phase will be present). This is because the hydrocarbons hardly dissolve in the water and therefore the partial pressures of these compounds in the vapor phase are very weakly dependent on the absolute quantity of liquid water. Vice versa, the small amounts of hydrocarbons dissolved in the water phase will hardly affect the partial pressure of the water (which will therefore be about equal to its vapor pressure). As long as both liquid water and a vapor phase are present, the hydrate equilibrium curve is weakly dependent on the absolute and relative amounts of heavy non-hydrate forming hydrocarbons in the system ("the amount and composition of the oil"). This is because the partial pressures of the heavier (non-hydrate forming) hydrocarbons are usually much lower than the partial pressures of the hydrate forming compounds and therefore do not significantly contribute to the total pressure of the system. However, large changes in the position of the hydrate equilibrium curve in a pressure-temperature diagram are expected if a change in the amount of oil, or in the composition of the oil, would lead to a substantial change in the relative abundances of the hydrate forming compounds in the gas phase (i.e. when some hydrate forming compounds are much more soluble in the oil phase than others). In practice, this is not often the case and the position of the hydrate equilibrium curve is relatively insensitive to the amount and composition of the oil. If non-hydrate forming compounds are dissolved in the liquid water phase the position of the hydrate equilibrium curve in the pressure-temperature diagram is shifted towards lower temperatures. This is because, as a first approximation ("Raoult's law"), the partial pressure of water in the vapor phase is proportional to the mole fraction of water molecules in the aqueous phase. This mole fraction is decreased if other species are dissolved in the water and hence the hydrate equilibrium (for a given pressure) shifted to lower temperatures. This explains the action of anti-freezes such as methanol, glycol and sodium chloride. Obviously, also during gas drying the partial pressure of water in the vapor phase is reduced thereby lowering the hydrate dissociation temperature at a given pressure. The hydrate dissociation temperature decreases by about 0.8 K per mole % of inhibitor added to the aqueous phase, if the quantity added is not too large. Provided that no phase transition occurs in the P-T region under consideration, a plot of the logarithm of the pressure versus the reciprocal (absolute) hydrate dissociation temperature will be nearly linear. In fact this rule is an application of the Clausius-Clapeyron relation for the thermodynamic equilibrium of a multi-component system involving a hydrate phase. The slope of the line on an inP-1/7" plot is a measure for the heat of formation (or dissociation of the hydrate phase). In practice it is found that often between 450 and 500 Joules are required to convert one gram of hydrates into liquid water and gas and between 80 and 160 Joules to convert hydrates into gas and ice.
257 11.2.3 Binary Systems The empty hydrate lattice is always unstable thermodynamically, so the simplest phase diagram involving hydrates is for a binary: water + hydrate former. This system is called a simple hydrate. There are various possibilities, depending on the relative location of critical points, the solubility of the guest in water, and so on. In Figure 2 the hydrate equilibrium curve is shown for pure propane, along with other phase boundaries. As this is a binary system, all lines shown are three-phase lines, which are univariant according to the Gibbs phase rule. This means that the specification of a single quantity, such as temperature, is enough to fix all other variables (pressure and composition). The following phase types exist in this temperature-pressure region: hydrate (structure II) H, vapor V, propane rich liquid Ln, aqueous phase Lw and Ice.
Figure 2. Phase diagram for water-propane (water in excess) showing different regions of coexisting phases. Most important to the petro-chemical industry is the absolute hydrate boundary, formed by the three phase lines Ice-H-V, Lw-H-V and LW-H-LH- These lines divide the plane into two areas. In the high temperature range, hydrates will never form, irrespective of the overall composition. In the low temperature area, hydrates may form, depending on the overall composition. The three three-phase lines, meet in a lower quadruple point Qi (I-V-Lw-H) and an upper quadruple point Ch (I-LH-LW-H), from which, of course, additional three phase lines emerge. Note that the upper quadruple point, Q2, exists because the critical temperature of propane is high enough so that the pressure-pressure curve of propane (more precisely, the HV-LH three phase line) intersects the V-Lw-H three phase line. For methane, which has a much lower critical temperature, this upper critical point does not exist. As a consequence,
258 whereas the absolute hydrate range for propane is bounded in temperature, it seems that that methane hydrates can form at any temperature, if the pressure is high enough. The slope of each three-phase line in the diagram satisfies dp_=AS_ dT AV
^
This formula is rigorously valid for any uni-variant curve, where AV and AS must be interpreted as changes accompanying the phase transition where, say, n moles of one of the phases is converted into the other phases, for the uni-variant equilibrium in question. Note that the LH-LW-H three-phase line is very steep due to the small values of AV. Here hydrate is formed at the expense of liquid propane and liquid water. The V-Lw-H is not steep, because here the gas entering the hydrate undergoes a comparatively large decrease in volume 11.2.4 Multi-component Systems We already showed a typical phase diagram for a multi-component system involving hydrocarbons and water in Figure 1. A simple but non-rigorous way to think of this diagram is as a superposition of the binary phase diagram, discussed before, and the familiar oil gasphase envelope with its dew and bubble point branches connected at the mixture critical point. For a pure component, the phase-envelope reduces to a sharp line (the vapor pressure curve), which ends at the critical point. So, conversely, the upper quadruple point becomes a four-phase line in the multi-component diagram. This is strictly true only for a ternary system (water and two hydrocarbons, one of which is a hydrate former). In reality, the hydrate curves shown in Figure 1 are boundaries of three and four phase areas. These areas are usually very narrow, i.e. the temperature-range over which all the liquid water is converted to hydrate is very small (typically less than 0.1 K). In Figure 1 the hydrate curve and the phase envelope cross. Another possibility is that the phase envelope lies completely inside the hydrate region, which will happen for methane rich mixtures. The oil-gas phase envelope itself is normally quite insensitive to the amount of water, due to the low water-hydrocarbon mutual solubilities. Effect of drying Not shown in Figure 1 is the water-dew point curve, which if the mixture contains enough water, will lie completely to the right of the hydrate formation curve. However, if the mixture contains less water, the water dew point curve will shift to lower temperatures, and may then intersect the phase envelope. If the mixture contains even less water, the water dew point curve will meet the hydrate curve and actually (at first partly) disappear as if the hydrate region were a carpet. At yet lower water contents, the hydrate formation curve - now corresponding to hydrate formation without free water present - will move to lower temperatures. This effect of drying the mixture is shown in Figures 3a and 3b.
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Figure 3a. Effect of drying on the hydrate equilibrium curve. If enough water is present to form a free water phase, then — inside the phase envelope — the hydrate equilibrium curve corresponds to a four-phase situation (V-Lw- LH-H) and is not sensitively dependent on the amount of water present. If the mixture is very dry, then the hydrate equilibrium curve corresponds to a three-phase situation (V- Ln -H) and is very sensitive to the amount of water present (yw corresponds to the mole fraction of water in the mixture). Depressurization To understands what happens when a mixture is depressurized, for example in an gas production system, on may compute curves of constant enthalpy — indicative of a JouleThomson valve — or of constant entropy — indicative of a turbo expander. Such lines are shown in Figure 4. As one can see, depressurization may cause the system to enter the hydrate region, due to the accompanying cooling. Retrograde behavior An interesting phenomenon appears for mixtures rich in carbon dioxide. Water is well soluble in liquid carbon dioxide, so inside the phase boundary, the stability of hydrates relative to the liquid phase decrease, i.e. the chemical potential of water in the liquid phase is lowered by the presence of liquid carbon dioxide. The effect of this on the hydrate curve is a drastic change in slope, which may cause a portion of curve where
260
Figure 3b. Effect of drying on the hydrate equilibrium curve. This diagram corresponds to an isobaric cross-section in Figure 3a, for a pressure of 50 bar. The vertical lines are tie-lines connecting compositions (water mole fractions) of phases in equilibrium.
Figure 4. Isenthalpic and isentropic lines superposed on the phase diagram. hydrate forms in increasing the temperature! So a pressure may exist for which hydrates form at a certain temperature, disappears at a lower temperature and re-appears at a yet lower temperature. This is shown in Figure 5. This figure is based on calculations, but they
261 reproduce the data (Sloan ) quite well. Note that in this diagram, S represents a solid carbon dioxide phase. Azeotropy At a given temperature, a given component may form a (simple) hydrate at a certain pressure. If two simple hydrate formers are combined to form a mixture, it is not necessary that the hydrate equilibrium pressure lies between the hydrate equilibrium pressure of the two separate hydrate formers. By occupying different cages, the hydrate formers may collaborate to increase the stability of the hydrate. This is analogous to the phenomenon of azeotropy found in binary vapor-liquid equilibria for close-boiling components. Azeotropes are also found in hydrate equilibria, such as shown in Figure 6. This system shows both a homogeneous (minimum pressure) azeotrope, where vapor, ice and hydrate structure II are in equilibrium, around a water free mole fraction of hydrogen sulphide of 0.6. Below this mole fraction, hydrogen sulphide has a preference for the hydrate phase over the vapor phase, above it is the other way around. If the feed stream gets richer in hydrogen sulphide, at some point hydrate structure I may become more stable than hydrate structure II. As a simple hydrate former, hydrogen sulphide forms hydrate structure I. This gives rise to a heteroazeotrope, a four-phase equilibrium point where vapor, ice and the two different hydrate structures are in equilibrium.
Figure 5. Diagram for a mixture rich in carbon dioxide.
262
Figure 6. Azeotropes in the hydrogen sulphide-propane-water system at 270 K.
Inhibition Substances dissolved in the aqueous phase will lower the activity of water, which will cause a depression of the hydrate equilibrium temperature. This is shown in Figure 7, based on a calculation for the same mixture as in Figure 1, by adding 0.2 moles of methanol to each mole of water (to produce the shifted hydrate equilibrium curve). In the same figure, also the freezing point of water is plotted for both pure water and the water-methanol mixture. Clearly, the two temperature shifts are related, and their ratio can be expressed in terms of the slopes of the V-H-Lw and the V-H-Ice equilibrium curves. These slopes are related to the enthalpy and volume changes associated with the phase transition — which are mainly determined by the melting of ice (hydrate) and the expansion of gas — and turn out to be fairly independent of the gas mixture. This observation led to the following engineering rule of the thumb
!- = /--—]
J T
T
1
J
hydr
hydr.O
T V,
ice
W
T •" icc.O J
where subscript '0' refers to the situation without inhibitor. Here, c is a constant, which is for most natural gas mixtures around 0.7 for structure I and around 0.78 for structure II. (The inverse temperatures appear because when plotted against 1/7/the curves appear to be more linear). So, from measurements of the freezing point depression of water, the decrease of the
263 hydrate equilibrium temperature can be estimated, which is especially useful for complicated electrolyte solutions.
Figure 7. Shifting of the hydrate equilibrium curve caused by the presence of inhibitor (feed mole ratio of methanol/water equal to 1:5) in the aqueous phase. Note that this rule will break down if a substantial amount of inhibitor will dissolve in other phases than the aqueous phase. So since methanol is more volatile than glycol, it will work better for glycol, and better still for sodium chloride. Substances when dissolved in water lower the water activity primarily via the dilution effect. The chemical of water in the free water phase may be written as n[:=^+RT\n(jvxv)
(3)
Here, xw is the mole fraction of water and yw is its activity coefficient. So by reducing xv, the free water phase becomes more stable with respect to the hydrate (and ice) phases. Cheap and light materials that dissolve well in water (such as methanol) are good thermodynamic inhibitors. On a weight basis they dilute the water phase effectively. An activity coefficient (well) below unity —as with glycols— will enhance this effect. So the amount of inhibitor in the aqueous phase is the primary variable influencing the temperature depression. However, to accurately compute the total amount of inhibitor required — in order to determine injection rates — also inhibitor losses the vapor and hydrocarbon liquid phases become very important. In gas production systems, the vapor stream is in general much larger than that of the free water stream, so it may contain a relatively large amount of methanol. The
264 partitioning of inhibitors such as methanol and glycol over aqueous, hydrocarbon and vapor phases is one of the more difficult aspects of hydrate modeling. Structure transition The prediction of which structure will form is a subtle matter. The empty lattice is always only meta-stable and has never been observed. Based on the model, comparing structures I and structure II, it is found that the empty lattice of structure II is more stable, but the difference is small. Structure I has three large cavities for each small cage, whereas in structure II there are two small cages for each large cage (which is in absolute terms, larger than the large cage in structure II). So if a hydrate former is not too large, it may be easier to stabilize structure I than is to stabilize structure II. This happens for methane. If, on the other hand, a hydrate former is too large to fit into even the large cages of structure I, then it can only form structure II. This holds for propane. This implies that for the mixture methanepropane, there must be a composition at which the structure changes, and this has indeed been observed to happen for quite small amounts of propane. There are also systems for which the mixture may form a hydrate structure different from the one formed by each hydrate former separately. This happens for the very familiar small hydrocarbons methane and ethane, both of which form structure I. It has been predicted on the basis of the solid solution model and later confirmed experimentally that for certain compositions, a mixture of methane and ethane forms structure II. This may be explained crudely by saying that, in structure I, the components compete for the cages, whereas in structure II, each prefers its own cage type, and they collaborate.
11.3 PHASE EQUILIBRIUM CALCULATIONS & THERMODYNAMIC MODELS 11.3.1 Equilibrium Conditions In natural gas production and processing, one wants to answer questions such as: do hydrates form at the given composition and conditions? At which conditions do hydrates form for the first time? Under adiabatic depressurization, will the hydrate disappear or not? How much water must be removed from the mixture to avoid hydrates? How much inhibitor is required to avoid hydrate? These questions may be answered by applying thermodynamics, by performing phase equilibrium calculations, for which the Gibbs free energy of the system is required
G = YLpax'?M»{pJX) /
(4)
a
Here J3a are the phase mole fractions and //;a is the chemical potential of species i in phase a, which is a function of pressure, temperature and composition. For rigorous multiphase calculations, a model is required for all phases. Stability of a mixture at a given composition
265 implies that the Gibbs energy must be at an absolute minimum, under the following constraints a) Phase fractions should be between zero and unity 0<^
(5)
b) The condition of material balance, i.e. component amounts in the phases should add up to reproduce the overall amount zj of the component i present a
c) Normalization of phase fractions
Z/?a=l
(7)
At the minimum, the familiar phase equilibrium equations hold
rf=rf
(8)
for each pair of phases present. Note that the number of phases is not a priori given. It cannot, however, exceed the value 2+ Nc, where Nc is the number of components (Gibbs phase rule). 11.3.2 Thermodynamic Model In order to compute hydrate equilibria, a thermodynamic model is required for each of the phases that may potentially form in the region of interest. That is, for given pressure, temperature and composition, it should be possible to assert that the phase in question can be formed, and if so, what the value of the chemical potentials (or for this purpose equivalently, fugacities) of all the components are. The following phase types need to be considered in practice: vapor (V), aqueous liquid phase (L w ), liquid hydrocarbon phase (L), hydrate structure I (Hi), hydrate structure II (Hn), hydrate structure H, ice (I), other solids (S), such as carbon dioxide, wax and asphaltenes. Combinations of all these phases may be simultaneously present. Most hydrate computational programs are based on (some modification of) the solid solution theory of Van der Waals and Platteeuw3. The fluid phases are modeled via an equation of state or a liquid activity coefficient model. Parameters have to be based on various types of experimental data, such as hydrate equilibria, solubilities and freezing point depression, spanning a wide range of conditions. For a thermodynamically consistent description, all phases should be defined with respect to the same reference phase. This may be conveniently chosen as the vapor phase, which is usually represented by an equation of state. If a liquid phase is described using the same equation of state, then the description is automatically consistent. The properties of a hydrate phase may be computed by adding a difference term to the properties of pure liquid water.
266 The same holds for the ice phase. Since hydrates often and ice always forms below 0°C this means that the properties of meta-stable liquid water must be represented. The fugacity of water in the ice phase can be computed via such a difference term involving the heat of fusion A///us and the difference of heat capacity AcJp"s between solid and liquid6.
RTm \Tm
)
RTn\
Tm
\T))
Here, Tm is the melting point of ice (273.15 K). The parameters in the equation of state for water must be derived for a wide range of temperatures. At the lower temperatures, this should be done such that the sublimation curve of water is reproduced. If, beside water, other components are present in the aqueous phase, the chemical potential of water will decrease. This causes a lowering of the freezing point of water and, for the same reason, a decrease of the hydrate equilibrium temperature. Consequently, components such as methanol, glycol and salts, act as inhibitors. The effect they have on the chemical potential of water may be described through an activity coefficient model or through an equation of state. This is quite a challenging task when for example both methanol and salts are present simultaneously, as they have a mutual effect on their activities. Equally challenging is the prediction of the partitioning of inhibitors like methanol between the aqueous, the vapor and the condensate phase, which is important in the calculation of loss of inhibitor and amount of inhibitor required. Traditional equations of state fail. Modern equations of state capable of describing hydrogen bonding in the liquid phase, such as Cubic Plus Association (CPA) — discussed in chapter 6 — have shown some promises. Hydrate phase Concerning the mathematical description of the hydrate phase, we go into more detail here. Consider a clathrate crystal consisting of water and a number of encaged compounds, the 'guests', m. For each hydrate structure, there are different cage types /, with vt denoting the number of cages per water molecule. The following assumptions are made in the solid solution theory a. The empty lattice (E) is a meta-stable modification of water (which is stabilized through the inclusion of guest molecule). b. The contribution of the water molecules to the free energy is independent of the mode of occupation of the cages. c. The encaged molecules are localized in the cages and a cage can never hold more than one guest molecule d. The mutual interaction of guest molecules can be neglected
267 e. Classical statistics (as opposed to quantum statistics) are valid Under these assumptions, the thermodynamic functions for the hydrate phase are embodied in the following relations (a derivation is presented in Appendices Al and A2). Chemical potential of water in the hydrate phase In the solid solution model, the chemical potential of water in the hydrate emerges as a sum of three terms: the chemical potential of pure liquid water, the change corresponding to converting liquid water into a empty hydrate lattice, and the contribution arising from inclusion of the guest molecules.
^=/^'+A^w*rXl/iogfi- £©J V
guests m
0°)
)
Here, ©
s
-^i-
(11)
is the degree of occupation of the cages of type i by guest molecules m. The label Lwo refers to pure water, and ELwo to the difference between empty hydrate lattice and pure water. Note that the two first terms on the right hand side represent properties of 'pure water alone'. Nim is the number of guests m in cages of type i, and Nw is the number of water molecules. Composition of the hydrate The fraction of the cages i occupied by hydrate formers m is given by ©
=
9JEISI
(12)
guests*
This relates the fugacities fm (or, equivalently, the chemical potentials) of all the guests to the total composition, since Y.N»i=N»
(13)
268 The Langmuir coefficients Cm, which depend only on temperature, are related to the partition function of guest m (isolated) in its cage /. The values of these parameters must be known in order to apply the theory. They may be determined directly from experiments, but according to the theory they are related to the guest-host interactions, via the following expression 1
1/
C =— f e - w (' 1 )- kT jV~_^l e - w » ;kT "" kT J kT
(14)
cage
where W(f) is the potential the guest experiences in the cage. The Langmuir coefficient is an inverse pressure or fugacity. At the pressure where the product of the fugacity of a hydrate former and the Langmuir constant is unity, the hydrate former starts to occupy (stabilize) the cage in question. Two approaches exist concerning the computation of the Langmuir coefficient. The first approach is to fit the parameters occurring in an expression for the Langmuir coefficient itself. Most used is the following expression, based on . C = — exp —
I1-5)
Parameter values are required for each hydrate former in each cage. The second approach is to introduce a model potential experienced by the guest molecule in the cage, based on waterguest interactions. This potential is then used in Zetts5. The Kihara potential — which has three parameters — is commonly used. To carry out the integration, the size and shape of each cage is required, but this is approximated as spherical with radius Rj. In this more physical approach, the Langmuir constants in different cages are not independent and this may have an influence on the predictive power of the model. Thermodynamic properties of the empty lattice In addition to the Langmuir coefficients, the thermodynamic properties of the empty lattice must be computed. This problem is tackled (Zetts ) by considering the difference between the empty lattice and pure liquid water. By integrating the relation Td(/.i/T) = -(H/T) + VdP
„„
— typically valid for a pure substance — from a reference point '0' to the conditions of interest, P and T
269
AAf- Arf? ^ f ' V . W f -
(17)
In practice Av"'" is assumed constant, AH"'" is expanded in T about the reference point Ta (using Acp, the difference in heat capacity between pure liquid water and empty lattice). The reference point is usually chosen as 273.15 K and 1 bar. The properties [i'^°, A//*7'"', Acp and Av^''"° required are referred to as 'empty lattice properties'. As the meta-stable empty lattice does not exist by itself, these properties have to be obtained indirectly. Status and developments of the solid solution model The solid solution model of Van der Waals and Platteeuw3 is able to correlate and 'predict' data for simple hydrates and natural gas mixtures quite well. For more complicated mixtures it is more difficult. The difficulties lie mostly in the description of the liquid phase(s), and extrapolation to lower temperatures, higher pressures and high concentration of inhibitors. Of course, for these more complicated systems, data are always scarce. The fundamentals of the solid solution theory do not go unchallenged. For example, molecular dynamics studies indicate that the empty lattice is unstable, rather than metastable. Various attempts have been made to improve upon the assumptions, keeping the basic idea. To list a few of them: generalization of the Lennard-Jones potential initially used in the calculation of the Langmuir coefficient to a Kihara potential, to account for guest molecules with a hard core; additional interactions ('guest'-'guest', 'guest'-'next shell of water molecules'); the influence of the size of the guest on the effective empty lattice parameters; influence of steric hindrance on the rotational motion of the guest; introduction of binary parameters; quantum-mechanical model. What makes it difficult to evaluate all this is the presence of the hypothetical empty lattice. Values for parameters such AH"'" (enthalpy difference between liquid water and hydrate) scatter a lot in the literature. They cannot be measured directly. However, the empty lattice parameters have a large influence on the predictions, and errors will be compensated for by 'improvements' elsewhere. Another complicating factor is that mixtures may contain molecules that form a different structure as a simple hydrate former than the structure stabilized by the mixture. For methane and ethane, for example, this holds for both molecules in the mixture! So to obtain the model parameters for these components in the structure not formed by them as pure hydrate formers, one has to rely on mixture data. On the whole, though, the solid-solution model has been very successful in applications. The most challenging areas are high-pressure condensate systems in the presence of inhibitors including electrolytes. For these systems, the progress has to come mainly from the modeling of the non-hydrate phases.
270 ACKNOWLEDGEMENT The authors thank Dr. Ulfert Klomp for useful suggestions and material. REFERENCES 1. Davidson, D. W., Clathrate Hydrate. Ch.3 In Franks: Water, a comprehensive treatise, Vol.2, Plenum Press, NY, 1973. 2. Sloan E. D. Clathrate hydrates of natural gases. Marcel Dekker, New York and Basel, 1990. 3. Van der Waals J. H. and Platteeuw J. C, Clathrate solutions. Adv. Chem. Phys., 2(1959)1-57. 4. Yamamuro, O. and Suga, H., Thermodynamic studies of clathrates hydrates. J. Therm. An, 35(1989)2025-2064 5. Zetts S. P., Holder G. S. and Pradhan N. Phase behavior in systems containing clathrates hydrates. Reviews in Chem. Eng., 5(1988)1-70 6. Prausnitz, J.M., Lichtenthaler, R. N. and de Azvedo, E. G., Molecular thermodynamics of fluid-phase equilibria, Prentice Hall Inc. APPENDIX A: Derivation of the hydrate phase equations Here we give a derivation of the thermodynamic relations describing the hydrate phase for the solid solution model, first using a simple analogy in terms of a set of chemical reactions, and second, more rigorously, using statistical mechanics. The original formulation of Van der Waals and Platteeuw3 is largely followed. A.I Simple derivation based on chemical reaction theory One may consider the formation of hydrates as an inhomogeneous reaction in which a guest molecule Gm binds to an empty cage r,0 to form a filled cavity G m +r,o^r,,,,
(A.1}
This can be described by an equilibrium equation of the form JESL
=C
(A-2)
I1 fOj/m
in which Clm is a chemical reaction constant related to the free energy of enclathration ('adsorption'), fm is the fugacity of the guest component, and the notation[x] denotes
271 concentration. The ratio of the concentrations is equal to a ratio of corresponding occupation probabilities
[r, 0 ] ®/o where ®im denotes the probability that the cage is occupied by a guest of type m, and ©/0 the probability that it is empty. These probabilities add up to unity Z0™+0-o=l
(A-4)
guests 111
Combining A-3 and A-4 we obtain Clmfm=—^
(A-5) guests k
or inversely
e
"=^Hi
'
guests k
This is a relation between the composition and the fugacities of the guest molecules. In order to complete the description, the fugacity (or chemical potential) of water must be derived, which may done through application of the Gibbs-Duhem relation (at constant pressure and temperature)
* » = - ^ E 7fJln/<» guests m
(A 7)
"
w
where JVmand Nw denote the number or moles of guest m and of water present in the hydrate, respectively. Introducing vj as the number of z'-cages in the hydrate per water molecule, and Nmj as the number of guest molecules in them, we may write
^,,,= 2 X , = A',, Iv,©,,,, cages/
cages/
Combining A-5, A-7 and A-8 we obtain
(A-8)
272
& cages?
guestsm
^
/ fe'ut:sls*
V
(A 9)
)
"
,™ki 7
On the right hand side of this equation, the occupation probabilities should be considered as the set of independent variables defining the composition. Working out the derivative of the logarithm and collecting the coefficient of each differential, with some algebra it is found that this equation can be rewritten in a simple form that permits direct integration, with the result
(A-10)
^,=^-^X^lnfl-X©J cages/
\
guests m
J
where the integration constant //,'': corresponds to the situation of zero occupation probabilities, i.e. to the chemical potential of the empty lattice. This depends on pressure and temperature alone, an expression for it is derived in the main text. The chemical reaction constant may be identified with the Langmuir coefficient. It is related to the partition function of a guest molecule m in cage i, and depends only on temperature. A.2 Mixed grand partition function and thermodynamic functions Given are: • •
Nw host (water) molecules in a volume F, at temperature T M different guest species m at fixed chemical potentials, jum .
•
The independent variables used are £ ^{T.V.N^.,^,^,...,^}.
The absolute activity
of component m is defined in terms of chemical potential as Xm = e''°''RI . The corresponding (mixed) grand partition function is defined as
q(g) = Yl~2dZ{T,V,Nw,Ni,N2,...,NM}fl(Amy;V,
A',,
(A-l 1)
m=l
where Z is the canonical partition function. The relation between the associated thermodynamic potentials takes the form of a Legendre transformation, which follows from taking the maximum term in the summation in A-11 logq = logZ + YjNM logA,,, The canonical partition function Z is related to the Helmholtz free energy
(A-12)
273 A = -RT\ogZ
(A _ 13)
Using the latter two equations, the following relations between q and the familiar thermodynamic functions can be derived
(A"14)
d(RTlogq) = SdT + pdV-nJNw +YjNk4d/um
RTdQog q) = —dT + pdV - /.iJNv + RT^ Nm =^-
( A "! 5 )
A.3 Evaluation of mixed grand partition function q We think of the clathrate as built in two steps: first the creation of the empty lattice, then the filling of the cages by guest molecules. As a consequence of assumption b (see main text), the partition function can be written as a product of two factors, one for the empty lattice, and one for the distribution of guests over the cages. Introducing Nmi as the number of guests of type m in cages of type i — for which Nm = ^ q
= Z>;(T,V,Ntr)ZZ(T,V,{NIIN},Nw)Yl(Aj->
Nlm holds — we can rewrite A-l 1 (A-l 6)
where Zh =exp[-Ah I RTj is the partition function of the empty lattice, and Z(T,V,{Nlm},Nn)the constrained canonical partition function, for fixed ({Nmi},Nw) which can be evaluated by counting all possible ways in which the M kinds of indistinguishable mguests can be distributed over v,A^1(, distinguishable /-cages (v,- is the number of/-cages per water molecule in the lattice) (vN) 2y
i
V
oi' \ \ m
( A " 1? )
) iy
nn '
k )
274
Here zml is the partition function of an w-guest in an z'-cage and NOI is the number of empty /-cages. We define z0/ = 1 for convenience. By the multinomial theorem, this leads to the following result for the grand partition function
««wn(>* Z'-<.r cages \
guests m
"'
)
A.4 Derivation of thermodynamic functions Combining A-15 with A-18, we can now compute the (guest) composition through differentiation of log q with respect to the absolute activities Xm
TV =N \
—JSL-rn—
v
V
*
J
It is often more convenient to use fugacities instead of absolute activities or chemical potentials. The fugacity of guest species m may be defined through the formula Am=e""IST=^^
(A-20) kTQ>m(T)
where Om (T) = ZU!tmZiM IV (which depends only on temperature) is the product of translational and internal partition functions for guest molecule m, with the volume factor removed. Introducing the Langmuir constant <:„,,=—^— kTOm(T)
(A-2i)
and realizing that Nm must be a linear and homogeneous function of the number of cages of different types, we obtain from A-19
©
s
J^. =
C
^J'«
(A-22)
guests*
The chemical potential of water in the hydrate phase follows from the mixed grand partition function by differentiation with respect to Nw
275
cages/
^
guests m
)
where ^ is the contribution from the empty lattice partition function Zk , i.e. the chemical potential of the empty lattice. This can be expressed in a different and useful form, by observing from A-22 that
1- 2 > m , = gueslsw
=j-^ ^ '
(A-24)
/ , ^miJm gncsts in
so that
^=^+RT^Vl\ollcages/
(A 25)
£©J ^
guests m
"
J
The internal energy of the hydrate can be obtained from the partition function by differentiation
=U":+RT'N YV®
U = RT4^M]
alOgZ
""
(A-26)
8
(A"27)
and, from this, the enthalpy of the hydrate,
H
= H':- + p(V-V-) + RT'N^v,®,,, mi
-^-
01
From A-21, the definition of 0 m/ (see A-22) and A-27 we see then that the inclusion ('adsorption') enthalpy per mole of hydrate former present in cages of type / is related to the temperature derivative of the logarithm of the Langmuir constant HH " g u e s t mm I
_Hv " guest in
=RT2d\og(TCmi) l l
'
~j,
(A_2g)
v
'
APPENDIX B: Glossary of Terms The following terms are useful to know when reading literature on gas hydrates Simple hydrate. A hydrate made of water and a single guest molecule (binary system).
276 Mixed hydrate. A hydrate in which a cage of the same kind is occupied by various guest molecules. Double hydrate. A hydrate in which each cage is primarily occupied by a different guest molecule. Hilfsgas. (Help gas.) A component like oxygen or nitrogen, that increases the hydrate stability of a second component. Hydrate equilibrium curve. (Also dissociation or melting curve). Locus of thermodynamic conditions of pressure and temperature corresponding to incipient formation of hydrates from a free water phase (analogous to dew point). For a binary system (water + guest), if the feed is within certain boundaries, this is a three-phase line. Hydrate number. Number of water molecules per guest molecule in the hydrate (an indirect way of specifying the composition). One writes XnH20. Ideal composition. (Equivalently, ideal hydrate number). Composition corresponding to complete occupancy of the cages (available to the guest molecule in question). Cavities or cages. These are the 'voids' in the lattice providing room for the hydrate former molecules. As an example, a cage bounded by 12 pentagons and 2 hexagons is denoted by 51262. The make-up of empty lattice structure I is denoted concisely as 2(5I2)6(5I262)46H2O Inhibition. Lowering of hydrate formation temperature by a special chemical agent. Methanol is the best known example, which acts as a hydrate point depressant by lowering the activity of water.
Part III: Applications of property models and databases
12. Molecular simulation of phase equilibria for industrial applications Ioannis G. Economou 13. Property models in computation of phase equilibria Rafiqul Gani & Georgios M. Kontogeorgis 14. Application of property models in chemical product design Georgios M. Kontogeorgis, Jens Abildskov & Rafiqul Gani 15. Computational algorithms for electrolyte system properties Rafiqul Gani & Kiyoteru Takano
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Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B.V. All rights reserved.
279
Chapterl2: Molecular Simulation of Phase Equilibria for Industrial Applications Ioannis G. Economou 12.1 INTRODUCTION The unprecedented increase of computing power at relatively low cost in recent years and the development of novel methods and algorithms has made mathematical modeling of industrial processes possible at all levels of characteristic length- and time-scales of the corresponding system (material system such as fluids, polymers, solids, process system, or enterprise system). In Figure 1, the various levels of modeling are presented for the case of a large corporation with several production facilities spread throughout the globe [1]. Today's tools enable mathematical modeling at the submolecular level of electrons and atoms using quantum chemistry methods, at the molecular level using molecular simulation techniques and all the way to the modeling of the entire enterprise using linear and non-linear programming methods. This book is devoted to the various methods available for fluid property estimation for process and product design. Most of the chapters in this book deal with macroscopic thermodynamic models for different systems of industrial interest. In this chapter, an overview of the state-of-the art methods that can be used efficiently for industrial applications is presented. Emphasis is given to Monte Carlo simulation methods, specifically for phase equilibrium and other thermodynamic property calculations. Novel elementary moves that allow efficient sampling of highly dense fluids and/or systems of long chain molecules are discussed. Realistic force-fields for hydrocarbons and for water are elaborated in detail and representative calculations for highly non-ideal fluid mixture phase equilibria are presented. Comparison with experimental data is made in all cases. For most of these mixtures, molecular simulation predictions are more accurate than predictions using state-ofthe art macroscopic equations of state. Molecular simulation allows the fundamental understanding of the molecular mechanisms (atomic and molecular motion, intra- and inter-molecular interactions, etc.) and phenomena that control macroscopic thermodynamic, transport, mechanical and other properties. The characteristic time- and length-scales of these various molecular processes span several orders of magnitude and, thus, hierarchical modeling is necessary in order to obtain physically realistic results, even with the more powerful computers available today [2]. A number of different atomistic simulation methods have been proposed for different systems and properties that includes molecular mechanics (MM), molecular dynamics (MD), Monte Carlo (MC) simulation, Brownian dynamics and transition state theory [3,4]. At the mesoscale level, simulation methods include kinetic MC, dissipative particle dynamics.
280 dynamic density functional theory, etc [2]. The focus of this chapter is on the molecular simulation methods.
Figure 1. Schematic representation of multi-scale mathematical modeling of chemical process industries in terms of characteristic length- and time-scale of the system. A necessary requirement in order that simulation results are accurate and in agreement with the experimental data (when available) is that the force field used represents accurately the inter- and intramolecular interactions. Considerable work has been devoted by different research groups to develop realistic detailed and more coarse-grained molecular models [5]. Some of these models are presented later in this chapter. Force-fields for hydrocarbons and for water are discussed in great detail and comparisons between the different models and with experimental data are made. Finally, representative mixture phase equilibrium calculations are presented and conclusions regarding the future impact of molecular simulation to engineering problems are made.
281 12.2 MOLECULAR SIMULATION METHODS A brief introduction to the most commonly used molecular simulation methods is initially given. MM is a relatively fast simulation method based on the minimization of the potential energy of the system with respect to the coordinates of atoms and molecules [5]. As a result, MM allows calculation of the system configuration at the local minimum energy. In this respect, MM is a method suitable for the calculation of single molecule structure but not for equilibrium thermodynamic property calculations. MD simulation consists of solving numerically Newton's equations of motion for all the particles (atoms, molecules, etc.) of the system of interest [3]. MD allows monitoring of the evolution of the system with time, and thus, time-dependent properties (such as viscosity, diffusion coefficient, etc.) can be calculated. MD is usually performed in the microcanonical (NVE) statistical ensemble; however, the method has been extended to canonical (NVT), NPT and other statistical ensembles [3]. An important parameter concerning the robustness of the MD simulation is the time step used for the numerical integration of the equations of motion. For systems characterized by a relatively stiff potential (as for example the case of chain molecules), a typical time step is in the order of 0.1 - 1 fs. In this way, a typical MD calculation is able to simulate a real system for no more than a few hundreds of ns, even with the use of the most powerful supercomputers available today. This time scale is considerably insufficient to calculate phenomena such as the relaxation of long polymer chains, the diffusion of large molecules etc. posing a considerable limitation to the method. Metropolis MC simulation is a stochastic method that allows efficient sampling of the multidimensional phase space of the system. In other words, this method allows "jumps" in the phase space and so, no real time monitoring of the system is possible [3,4]. In Metropolis MC, the different states of the system are visited with a probability proportional to the Boltzmann factor of the energy of the system. The system goes from one configuration (state) to the next configuration (state) based on different types of moves that satisfy microscopic reversibility and preserve the macroscopic properties of the system that are set constant. In this way, MC simulations are performed in the canonical (NVT), grand canonical (fiVT), NPT and many other statistical ensembles, depending on the system (pure fluid or mixture) and conditions (one phase, two or more phases, etc.) examined. In a typical NVT Metropolis MC simulation particles are displaced randomly one at a time within the simulation box and the new configuration is accepted or rejected according to the Boltzmann factor of the energy difference between the two states, that is: /7,VIT = min[l,exp(-/?At/)]
(1)
where [i = 1/kT, with k being the Boltzmann's constant, and AU = jJ"ew> - lf°'d) is the energy difference between the old and the new configuration. Additional moves in the NPT, fxVT and other ensembles include volume fluctuation, random insertion and deletion of particles and so on, and acceptance criteria are modified accordingly [3,4].
282 12.2.1 Monte Carlo Simulation Methods for Phase Equilibria The accurate knowledge of phase equilibria of pure fluids and mixtures is of great importance for the optimum design of processes in the oil, chemical and petrochemical industry. As a result, considerable emphasis has been given by the research community for the development of efficient molecular simulation (mostly MC) methods for such calculations. A concise review of these methods is given here. Unequivocally, the development of Gibbs Ensemble MC (GEMC) method by Panagiotopoulos [6,7] revolutionized research in this area and initiated further developments. According to GEMC, coupled simultaneous simulation of the two phases of interest (for example, vapor and liquid) is performed through random particle displacement, volume fluctuation and random particle transfer from one phase to the other. The temperature is selected (same value in both phases) and set constant in the beginning of the simulation. For the case of a single-component vapor-liquid equilibrium, the total number of particles (molecules) and total volume remain constant to satisfy Gibbs phase rule. For the case of a random particle displacement within one of the phases, the acceptance probability is the same as in conventional NVT MC simulation, given in Eq. (1). For the case of volume fluctuation, where the volume of phase / increases by AV and of phase // decreases by the same amount to preserve constant total volume, the acceptance probability is:
p,,.^^;r ) '' fc -r ) "' e *p(-/^, -*
<2)
where N/ and NJJ are the total number of particles in phase / and //, respectively. Finally, for the case of a particle transfer (let's say from phase / to phase II), the acceptance probability is:
/ W , = min l,,NN'^"y
exp(-jSAU, -J3AU,,)j
(3)
Using these moves, the chemical potential and the pressure in each phase are equalized and phase equilibrium is attained. The resulting pressure is the pure component vapor pressure at the given temperature. For the case of mixtures (two or more components), an extra macroscopic degree of freedom is available and so pressure can be also specified. In this way, GEMC simulation can be performed in the N1N2PT ensemble, for a binary mixture. In this case, independent volume fluctuation in each phase is possible and the corresponding acceptance probability is (for the case of volume fluctuation in phase I): / W . = ™n l , ^ L ± ^ l _ e x p ( - pAU, - J3PAV)
(4)
283 Over the last 15 years, GEMC has gained wide acceptance by the research and industrial community for fluid phase equilibrium calculations for pure components and mixtures. Recently, the method was incorporated into a popular commercial software product of the leading company worldwide in the field of scientific software for materials design [8]. Despite its success, GEMC has several limitations especially for highly dense systems and systems containing large molecules (heavy hydrocarbons, polymers etc.), where the successful particle transfer from one phase to the other becomes highly improbable, and for near-critical calculations where the system becomes unstable. To improve acceptance of particle transfers for the case of large molecules, a number of methods have been proposed using also novel elementary moves for chain molecules (Section 12.2.3). For example, configurational-bias (CB) sampling methods improve considerably the acceptance of a move (displacement, insertion, or deletion) of a large chain molecule [9-11]. CB consists of a segment-by-segment move of the molecule and for each segment move several trial directions are attempted. For each segment growth, a preferential direction is selected according to an expression containing the interaction energy of the segment at each trial position (the so-called Rosenbluth weight). Even using CB techniques, GEMC becomes impractical for components larger than approximately n-eicosane. Lisal el al. [12] proposed an interesting variation of the GEMC method for multicomponent mixtures where the phase equilibrium conditions are treated as a special type of chemical reaction. In the so-called reaction GEMC (RGEMC), the experimental and simulated (calculated by GEMC) pure component vapor pressures are incorporated into an "ideal-gas" driving term Fj for the reaction, given by: psal '
psal I,GEMC
^
>
1
In this way, the acceptance probability for particle transfer from phase / (liquid) to phase // (gas) becomes (compare to Eq. 3):
/ W , =min 1.r, tNN'lly
ex
P(" PAU' ~ PAU»)
(6)
RGEMC was shown to provide accurate prediction of vapor - liquid equilibria of non-ideal mixtures, including mixtures forming an azeotrope [12]. A powerful method for phase equilibrium is the so-called Gibbs-Duhem integration method of Koike [13] where the coexistence curve of a pure component is constructed through the numerical integration of the Clausius-Clapeyron equation:
_^J -__^_ d(\/T)lon
(7)
AV/T
where the differences in the right hand side correspond to the two coexisting phases. A single point in the coexistence curve should be known (possibly through GEMC at high
284 temperature, for example) in order that Eq. 7 can be integrated numerically. In the course of two single-phase NPT MC (or even MD) simulations, AH, and AV can be measured. Furthermore, the right-hand side of Eq. 7 is fairly constant, away from the critical point. As a result, a numerical integration scheme converges fast to the correct pressure value that corresponds to the temperature increment. In this way, particle transfer between phases is avoided. Caution should be taken with error propagation related to uncertainties at the initial point, especially when approaching the critical point. The method has been extended to mixtures [14]. In this case, however, particle transfer for one of the components for a binary mixture is necessary. For a light component - heavy component binary mixture, the light component can be chosen as the transfer component. The Gibbs-Duhem integration method is applicable also to solid - fluid phase equilibria [14]. An iterative MC simulation scheme was proposed for the simulation of phase equilibria of long chain systems (SPECS), which involves the simultaneous simulation of a two-phase multicomponent mixture [15]. For the case of a binary mixture of a light component 1 (i.e., solvent) and a heavy component 2 (i.e., heavy hydrocarbon), the two phases (/ and II) are simulated under the same fugacity of the light component, temperature and pressure while the number of heavy component molecules in each phase remains constant (/A^/T ensemble). To attain the imposed value o f / insertions and deletions of the solvent molecules are performed in both phases. An insertion attempt is accepted with probability: p,m = min 1, , ' v"vyf'
exp(- pAU)
|_ (JV,+i)z;'"
ra
v H
;
(8)
J
whereas for the case of attempted deletion, the move is accepted with probability: \T
pM = min 1,
yimra
' '
exp(- pAU)
(9)
where JiJm and .//,<&/ are the corresponding Jacobians of transformation from Cartesian to generalized coordinates and Z""m is a dimensionless configurational integral over all the orientations and conformations of a single solvent molecule. Equilibrium is achieved by equalizing the fugacities of the heavy component in each of the two phases, using an iterative scheme based on the Gibbs-Duhem equation that revises the common value of/: A, f A / I , / / I n / / - I n / / , „ ln/;f-ln/; A In/, = (l-x, )x2 n- =x2x2 n 1
Xj
X2
X2
(10)
X2
Iteration proceeds until Aln/} reaches a sufficiently small value, x/ corresponds to the mole fraction of component i in phase ,/. SPECS requires calculation of the fugacity (or equivalently the chemical potential) of both components in all phases. Simulation methods for chemical potential calculation are discussed in the following section. In the case of a
285 heavy component, SPECS makes use of the chain increment ansatz to evaluate the chemical potential [16]. A powerful approach in molecular simulation is to combine information from simulations at different conditions using histogram-reweighting methods. The method is based on constructing individual histograms for different sets of conditions and then combines the results to obtain a global picture [17-19]. Care should be taken so that partial overlap exists between histograms that are to be combined. Although such calculation is more CPU time consuming than a conventional single-state simulation, a wealth of data can be obtained over a wide range of conditions. Typically, a range of temperature and chemical potential values can be spanned or, equivalently, different Hamiltonians [20]. In this way, histogramreweighting grand canonical MC (HRGCMC) can be used to calculate the vapor-liquid equilibrium of pure fluids over a wide temperature range and furthermore, to optimize forcefield parameters [20, 21]. HRGCMC method has been extended also to multicomponent mixtures [22, 23]. Escobedo [24] presented a general formulation for the development of pseudoensembles for simulation of multicomponent phase equilibria and showed the relationships between some of the methods presented above as well as several other methods. 12.2.2 Simulation Methods for Chemical Potential Calculation The chemical potential is the thermodynamic property that controls phase equilibrium for pure components and mixtures. Consequently, methods that allow its accurate calculation are of immense importance. The most widely used and efficient method is based on Widom's approach [25]. According to this method, the chemical potential is calculated from the ratio of the partition function of a system containing N+l molecules divided by the partition function of a system containing N molecules, where both systems are under the same conditions of temperature and volume or pressure. In practice, a "ghost" molecule of the component whose chemical potential is evaluated is inserted into the simulated system. The interaction energy of the "ghost" molecule with the remaining molecules (UghOsi) is evaluated and used for the calculation of the excess chemical potential (juex) of the component: Mp, T)- ft* (A Thju" (A T)=~
ln(exp(- jBURhml ) m , r
(11)
The brackets in Eq. 11 denote ensemble averaging over all configurations and spatial averaging over all "ghost" molecule positions. Eq. 11 holds for the canonical (NVT) statistical ensemble; similar expressions exist for other ensembles too. Widom's approach is applicable to MC simulation as well as to MD. Furthermore, the "ghost" molecule insertion can be performed during the simulation or as a post-processing calculation. In the latter case, a number of system configurations are stored every few thousand MC moves or MD timesteps and in each configuration many (from few thousand to several hundred thousand) "ghost" molecule insertions are attempted. This post-processing calculation is relatively fast compared to the MC or MD simulation of the system. Widom's method is applicable both to pure components and mixtures.
286 Widom's method becomes impractical for large chain molecules and/or highly dense fluids. In both cases, significant overlap occurs between the "ghost" molecule and the system molecules, so that Ughost assumes very high values resulting in very poor statistics. An alternative method was proposed based on the argument that the excess chemical potential of a long chain molecule varies linearly with molecule size above a characteristic length, at the same conditions [16]. The slope of this curve is the segmental excess chemical potential (/i"'eg). According to the chain increment ansatz, the excess chemical potential of an «-mer can be written as: H" (n)= f
{nWKlll)+ (n - nsmoll ) M ^
(12)
Consequently, Widom's insertions are performed for the «OTO/r-mer (an "oligomer" of the long species) to evaluate //*(nsman) and for a single "ghost" segment which is appended to the "ghost" molecule or to a system molecule of the same identity. This method can be used for very long molecules, including polymers, and can be combined with MC schemes for systems of long chain molecules, such as SPECS [15]. A powerful approach to overcome limitations related to "ghost" molecule overlap with system molecules is based on a test particle removal (inverse Widom) scheme [26]. The method is implemented in two stages: An intermediate system is introduced that consists of (N-l) molecules and a hard core corresponding to the removed molecule. The free energy differences between the JV-molecule system and the (AL7)-molecule + hard core system, and between the (N-l )-mo\ecu\e system and the (N-/)-molecule + hard core system are calculated. In summary, ff is calculated from the expression:
(n/4JexP(/?t/<4 *-(*r)=-^'"
I"" \ '=1
/
(13)
N-U'J
where lfN) is the intermolecular energy felt by the JVth molecule due to its interactions with the remaining N-l molecules of the system, and H(riN) is a Heaviside step function: /
N
0 if 1 if
r, - rv < d (p) r.,-^ >dCOK{fi)
where dcore is the diameter of the hard core of the deleted molecule. The inverse Widom method was extended to chain molecules and to mixtures [27], and it was shown that computational time for the evaluation of //"* can be decreased by as much as 5 times compared to the Widom insertion method [26, 27]. An alternative efficient approach for chemical potential calculation is based on the combination of Widom insertions of a small weakly interacting "ghost" molecule and expanded ensemble simulations [28-30]. A scaling parameter y e [0,1] is used to scale the
287 full Hamiltonian of the "ghost" molecule (parameters e and a for the case of Lennard-Jones potential). Initially, /f of the "weak" molecule (y < 1) is calculated using Widom insertions. Followed this initial calculation, a number of sub-ensembles at the same macroscopic conditions (temperature and pressure for the case of NPT simulation) are sampled. Each of these sub-ensembles consists of the iV molecules and the (N+l)th molecule with a different y value (different Hamiltonian). In the course of the simulation, attempts are made to move from one sub-ensemble, /, to another sub-ensemble, /. The frequency with which each subensemble is visited is dictated by the free energy of the Hamiltonian. To sample each Hamiltonian with equal frequency, weighting functions are introduced. The acceptance probability for the attempted move from sub-ensemble i to / is:
/7,^=min lAexp{-^(r 7 )-t/( r ,))}
(15)
where U(yt) and co; are the energy and weight of Hamiltonian i, respectively. The weights are adjusted prior to the production phase of the simulation. The difference in pf* for two Hamiltonians, i.e. i and/ is evaluated by the expression:
^"(rj)-^c1r,)=-ln[^j+ ln^j
(16)
where /?, is the probability of visiting sub-ensemble i. The total if is the sum of the differences between adjacent sub-ensembles and of jfx of the "weak" molecule from the Widom insertions. 12.2.3 Monte Carlo Moves for Long Chain Molecules In order that a MC simulation provides statistically reliable data, it is necessary that important configurations are sampled adequately in the course of the simulation. It was made clear thus far that for systems of long chain molecules, typical moves such as particle displacement and insertion result in overlaps with other molecules of the system and are rejected. Furthermore, long chain molecules are entangled above a characteristic size value and the characteristic time for relaxation can be very high. Considerable effort has been devoted over the last decade towards the development of efficient MC moves for long chain molecules. Although such moves are physically unrealistic, nevertheless, they allow efficient importance sampling of the system. In this section, the most successful and widely used moves are briefly discussed. In the reptation move (Figure 2) [31], an end segment of a randomly selected chain is cut off and appended to the other end with a randomly selected torsion angle. In the CB method (Figure 3) [9-11], an end part (in the order of three to six segments) of a randomly selected chain is cut and removed. Subsequently, the removed part is regrown segment by segment in a biased way, by accounting for the energetic interactions with neighboring segments in the
288 same or different chains. The bias introduced in this case is removed from the system using appropriate acceptance criteria.
Figure 2. Schematic representation of the reptation move [31]. Both reptation and CB moves concern the end part of chain molecules. As the chain size increases, efficient relaxation of the internal part of the molecule becomes an important issue. A revolutionary solution to the problem was proposed by Theodorou and co-workers based on the concerted rotation (CONROT) move [32, 33]. CONROT addresses the following topological problem: Given two dimers in space, connect them with a trimer, such that the resulting heptamer has prescribed bond lengths and angles. In the general formulation of CONROT (Figure 4), a randomly selected internal trimer of skeletal atoms of the chain is excised. The two atoms neighboring the excised trimer are displaced through randomly selected changes to both torsion angles neighboring the trimer on its two sides. Finally, the trimer is reconstructed. In total, five skeletal segments are relocated and eight torsion angles are altered.
Figure 3. Schematic representation of the CB move [9-11]. From the initial configuration (left), an end trimer (shown in black) is cut and is regrown segment by segment (right). In each step of the regrowth process, five randomly selected positions are considered.
Figure 4. Schematic representation of the double-driven CONROT [33].
289 The original configuration is shown in green and the final configuration is shown in red. The trimer 2-3-4 is cut and regrown (2'-3'-4') while atoms 1 and 5 are displaced due to changes in torsion angles (fi and
Figure 5. Schematic representation of the EB move [33, 34].
Figure 6. Schematic representation of the DB move [35].
290
Figure 7. End-to-end vector autocorrelation function at T = 450 K and P = 0.1 MPa for the monodisperse C5oo and Ciooo linear polyethylene melts, consisting of the same number of interacting sites. For comparison, results are shown for a monodisperse Ciooo polyethylene melt with the NVE MD method. All CPU times are on a DEC Alpha single-processor system at 667 MHz. All calculations are from [35].
For the same initial configuration, two different final configurations are shown. If the two chain segments shown on the left belong to different chains then the move is considered as intermolecular DB, otherwise it is intramolecular DB. A number of improved methods for CB and CONROT have been proposed that result in efficient relaxation of long chain molecules [36-42]. A recently proposed method for the efficient simulation of long chain molecules is based on the parallel tempering concept where a number of replicas of the system are simulated at different conditions (typically different temperatures and in some cases different chemical potentials) and configurations are swapped between replicas at adjacent temperatures so that equilibration of the lowest temperature replica is accelerated. Parallel tempering has been applied to simple Lennard-Jones fluid [43] as well as to complex molecular architectures such as cis-1,4 polyisoprene [44]. Parallel tempering can be used both in MC simulation and MD.
3. FORCE-FIELDS FOR COMPLEX FLUIDS The representation of atoms and molecules through a potential model is of extreme importance for a simulation both in terms of accuracy of the calculations with respect to reality and of computation time. Given the fact that the majority of CPU time in a typical simulation is spent in the calculation of interactions between segments in the same or different molecules, optimization of the force field is of immense importance. In general, force fields are separated into lattice models and into continuum models. In the former case,
291 the lattice can be cubic, hexagonal or more complex and a molecule occupies one (in the case of spherical molecules) or more lattice sites, depending on its size [3]. Lattice simulations are considerably faster than continuum model-based simulations (by more than an order of magnitude) and were used widely in the early stages of the simulation era. Lattice simulations provide results that require further processing in order to be comparable directly to experimental data (as for example structural data, volumetric properties, phase equilibria, etc.). However, they provide important results in terms of scaling relations [45], testing theoretical models etc. Detailed atomistic models take explicitly into account the complex molecular architecture and the interactions between the different functional groups in the same or different molecules. All of the force fields developed for non-polar fluids and many of the force-fields developed for polar fluids are pair wise potentials. Three-body, four-body and higher order interactions are extremely time consuming and are accounted implicitly by appropriate tuning of the pair-wise potential. As a result, for a fluid of chain molecules the potential energy function can be written as the sum of the contributions due to bond stretching, bond angle bending, dihedral angle torsion and non-bonded intra- and inter-molecular interactions, according to the general expression [5]:
F(r-,...,FV)= XKO+ !>(*,)+ I > W + I > U all bonds
all bond angles
all lorsional angles
(17)
all pairs
where 4 6it and (pt denote bond length, bond angle and torsional angle, respectively. Bond stretching and bond angle bending potentials are typically calculated through a harmonic-type potential, in the form:
V{l,)=k-f(l,-O
(18)
and
v&h^f^-ej
(19)
respectively, where subscript o denotes the equilibrium value. For the torsional potential almost always a cosine series expansion is used. A typical functional form is [5]:
^ , ) = £ ^ [ l + cos(«^-r)]
(20)
where N assumes a small integer value (in the order of 4) and y is the phase factor determining where the torsion angle passes through its minimum value. Cross terms can be also considered for the bonded interactions reflecting coupling between the internal
292 coordinates of the molecule [5, 46]. Such terms are important for the prediction of vibrational spectra but do not affect structural and thermodynamic properties. Non-bonded intra- and inter-molecular non-polar interactions are usually calculated through a 6-12 Lennard-Jones potential:
fc> 4
<2)i
" = 4&H^)]
where e and a are the energy and size Lennard-Jones parameters. Other potentials have been also proposed to describe the weak van der Waals interactions {i.e. Buckingham exp-6 potential etc.). Interactions between unlike segments (/ ^ J) in the same or different molecules are calculated using appropriate combining rules. The most commonly used are the LorentzBerthelot combining rules: / £
£
°V/+Cr/,/
, £
iJ=yj >J j.j
a n d
a
u,=
Y~^
zoo-, (
^
Potential parameters for the bonded interactions are evaluated based on ab initio quantum mechanics calculations, experimental measurements (such as spectroscopic studies), or fitted to experimental thermodynamic and other property data for the species of interest. Nonbonded interaction parameters are most of the time fitted to experimental data (i.e., second virial coefficient, liquid density, vapor pressure, enthalpy of vaporization, etc.). Polar (such as dipolar, quadrupolar etc.), hydrogen bonding and ionic interactions are usually taken into account by assuming partial charges on the species of interest. These types of interactions are of long-range nature and special techniques (such as Ewald summation method, reaction field method, etc.) are needed in order to get converging results within the finite simulation box considered [3]. These techniques increase computational time considerably, compared to the non-polar force fields. In such complex fluids, many body interactions play a significant role in the bulk macroscopic properties. Explicit inclusion of these effects into the force field increases further (by an order of magnitude or more) the computation time needed. As the CPU time is analogous to TV", where N is the number of interacting sites in the same or different molecules, an effort is made to account for light atom interactions implicitly. In the so-called united atom (UA) approach, hydrogen atoms are ignored. In this way, the methyl group, for example, is considered as a single interacting site rather than one carbon and three hydrogen atoms. In this case, an appropriate parameterization is needed of course. Such approach does not affect the model accuracy for phase equilibrium predictions. Recently, Panagiotopoulos showed that finely discretized lattice models provide practically identical structural and thermodynamic results as the corresponding continuum models [47]. The critical parameter for such agreement is the discretization factor C = a/I where a is the characteristic monomer size and / is the lattice grid spacing. Depending on the molecular architecture, a Rvalue in the order of 10 (for the case of n-alkanes) to 20 (for the
293 case of branched polyolefins) is required so that lattice models and UA continuum models are equivalent [48]. In Sections 12.4.1 and 12.4.2 below, a detailed analysis is given for state-of-the art forcefields for hydrocarbons and water and their prediction capability for phase equilibrium calculations. A summary on force-fields for other compounds can be found in the recent review of Panagiotopoulos [49].
12.4 APPLICATIONS 12.4.1 Force-fields for hydrocarbons Normal and branched alkanes are some of the most commonly encountered components in the oil, chemical and petrochemical industry and, at the same time, are used as prototypes for several other homologous series of more chemically complex species (i.e., alcohols, ketones, etc.), including biomolecules. Subsequently, significant work has been done towards the development of force fields that describe accurately a series of properties over a wide range of conditions. Early models (as for example the OPLS model [50]) focused mainly on the representation of structure and energetic properties of these compounds, especially at ambient conditions. As a result, they are inaccurate for phase equilibrium calculations over a wide temperature range, even for pure fluids [51]. Over the last five years, a number of UA force-fields were developed for the accurate representation of phase equilibrium and thermodynamic properties of n-alkanes, branched alkanes, cyclo-alkanes, olefins and aromatics over a wide temperature and pressure range, including the critical region. In all of these models, an effort is made to develop transferable potential parameters, so that the properties of components consisted of the same pseudoatoms as the components used in the fitting procedure can be predicted accurately. In the TraPPE-UA force-field [52], the bond length is kept constant, whereas bond angle bending is governed by a harmonic potential (Eq. 19) and dihedral angle torsion by a cosine series expansion (Eq. 20). Non-bonded intra- and intermolecular interactions are calculated by a Lennard-Jones 12-6 potential (Eq. 21). Coexisting densities for methane through ndodecane are predicted very accurately [52]. Vapor pressure values are overpredicted by the model for all n-alkanes examined. The deviation is large for the light n-alkanes, especially for low temperature (for ethane, the simulation value at 178 K is more than double the experimental value), and decreases as the hydrocarbon size increases (for n-dodecane the simulation value is only 9 % larger than the experimental value). TraPPE-UA was further extended to branched alkanes (i-butane, neo-pentane, dimethyl-butane and dimethyl-hexane) [53], to linear (a-olefins) and branched alkenes, and alkylbenzenes [54]. In all cases, deviations between simulation predictions and experimental data for saturated densities and vapor pressure are similar to the deviations for n-alkanes: good agreement is obtained for the saturated density and over-prediction of the vapor pressure (deviation increases at low temperature). Significant improvement is obtained when hydrogens are taken into account explicitly in the TraPPE-EH model [55]. The deviation between simulation and experiments for the vapor
294 pressure is within a few percent, for methane through n-dodecane. However, the price that is paid for this improved accuracy is the considerable increase in CPU time. The NERD force-field is based on a similar functional form as TraPPE-UA with the exception that bonds are flexible (Eq. 18) [56]. NERD was applied to a wide range of nalkanes, from ethane up to n-octatetracontane, and to branched alkanes [57]. Simulation results for the coexisting densities are in good agreement with the experimental data, when available. Furthermore, vapor pressure predictions for n-pentane, n-octane, i-butane and 2methyl-pentane are slightly above the experimental data at low temperatures and below the experimental data at higher temperatures. A transferable force-field for n-alkanes was proposed using the Buckingham exponential6 (exp-6) potential for the non-bonded intra- and intermolecular interactions instead of the Lennard-Jones potential [21]:
r '-^
V(rJ= l-6/«M[«,,; .°°
r exp a
P
r 1
~\\ ( Y'~ —^
for r >r
"alX
[ '•'[ c J J U J J f°r
r
(23)
max
where e/j , rmjj and atj are model parameters: rmjj is the radial distance at which the exp-6 potential is a minimum. Parameter a, corresponding to the distance where V(rt<j) = 0, is calculated from Eq. 23. The exp-6 potential has an additional parameter {atJ) over the Lennard-Jones potential, that is also fitted to experimental data. In this force-field, bond lengths are kept constant. Exp-6 provides very accurate representation of liquid density (within less than 1 %) and vapor pressure (within approx. 2 %) values of ethane through ndodecane. This force-field was further extended to cyclohexane and to benzene with similarly accurate results [58]. In all the force-fields presented above, the force center coincides with the carbon atom of the UA. Ungerer and co-workers proposed recently an anisotropic UA (AUA) potential, where the force center is shifted between the carbon and hydrogen atoms of the related group [59]. In the AUA, bond lengths are kept constant, while the Lennard-Jones potential is used for the non-bonded interactions. The AUA potential provides good representation of the vapor pressure and very accurate representation of the liquid density over a wide temperature range for n-alkanes (ethane through n-eicosane), branched alkanes (heptane isomers) [60] and cycloalkanes (cyclopentane through cyclooctane) [61]. Interestingly, while in all UA discussed here the Lennard-Jones sphere for d-L- is larger than that of CH3, in the AUA the opposite is true, in agreement with intuitive expectations. In addition, CH size parameter is smaller than CH2. Furthermore, AUA is able to predict with reasonable agreement transport properties of n-alkanes, such as the self-diffusion coefficient [59]. In general, UA forcefields are considered incapable for simultaneous prediction of phase equilibrium and transport properties. Olefins are used widely as reactants, intermediates or end products by the chemical process industry and so accurate knowledge of their phase equilibrium properties is highly desirable. Spyriouni et al. [62] proposed the first transferable UA force-field for the phase equilibrium of a-olefms (oleUA) that represents accurately the liquid density and vapor
295 pressure of 1-butene through 1-octene. More recently, TraPPE-UA, NERD and AUA were extended to include a-olefins [54, 63, 64]. Critical parameters are used extensively in various engineering correlations for thermophysical properties required for chemical process design, etc. Measurement of such parameters, especially for heavy components, is far for trivial and thus accurate predictive methods are of high importance. Molecular simulation using an accurate force-field can be used in this respect. The seminal work of Smit el al. [65] resolved the open question, at that time, concerning the variation of critical density of heavy n-alkanes as a function of carbon number. In Table 1, experimental values and predictions from the various force fields for the critical parameters of representative n-alkanes, branched alkanes, cyclic hydrocarbons and aolefins are shown. Results for additional components can be found in the original publications. Errington and Panagiotopoulos [21] calculated the critical parameters of nalkanes from the different models using a mixed-field analysis on histograms collected during grand canonical MC simulations at near critical conditions that provides more accurate results than other studies where subcritical simulation data were fitted to scaling relationships using a critical exponent of 0.32. These parameters are reported here for TraPPE-UA and NERD models over the parameters quoted originally [52, 56]. The agreement between experimental data and simulation predictions from the different force fields is, in general, good. 4.2. Force-fields for water Water is the most widely used solvent in industrial processes due to its unique physical properties, non-toxicity, non-flammability and low cost and at the same time it plays a vital role in biological processes that are of interest to biotechnology and other life-sciences industries. The strong directional forces between water molecules and the small molecular size result in 3-dimensonial hydrogen bonding clusters that affect macroscopic properties considerably. Some of the unique thermodynamic properties of water are the maximum density of liquid water at 4 °C, its liquid nature at ambient conditions (whereas all other molecules of similar molecular weight are gaseous), its very high critical temperature, etc. A complete understanding and modeling of such highly non-ideal behavior is still an open problem both from the purely scientific as well as from the engineering point of view, as will be made clear later on. Electrostatic interactions in water result in a very high dipole moment, equal to 1.85 D in the gas phase and approximately 2.5 D in liquid and 2.6 D in solid phase, although recent studies indicate a dipole moment of 3.0 D for both condensed phases [70, 71]. For comparison, the dipole moment of n-alkanes in the gas phase is less than 0.1 D, in general. The increased dipole moment in the liquid phase results from the polarization of water molecules due to the electric field generated by the permanent dipoles. As a result, forcefields for water can be divided into two-body pair potentials that account implicitly for the polarizability effects and many body potentials that account explicitly for the induced field.
296 Table 1: Critical parameters of representative hydrocarbons from experiments and simulation (TraPPE = TraPPE-UA). Pc (MPa) p c (kg/m 3 ) Ref Compound Model Tc(K) Zc [66] 0.279 Ethane expt. 4.87 ±0.01 206.6 ±3.0 305.32 ±0.04 [21] TraPPE 4.98 + 0.05 208.8 ±0.8 301.4 + 0.5 0.286 + 0.002 [21] NERD 4.95 ± 0.04 197.6 ±1.0 313.0±0.4 0.290 ± 0.002 [21] exp-6 4.90 ± 0.06 206.2 ± 0.8 305.3 ±0.6 0.282 ± 0.002 [66] expt. 0.268 n-Pentane 3.37 ±0.02 232 ±3 469.7 ±0.2 [21] TraPPE 3.59 ±0.03 238.9 ±1.8 464.2 ± 0.4 0.282 ± 0.003 [21] NERD 3.49 ± 0.04 227.4 ±0.8 476.9 ±0.8 0.278 ± 0.003 [21] 3.37 232 469.7 exp-6 0.268 AUA [59] 468.9 218+11 [66] expt. 0.259 n-Octane 2.49 ± 0.03 232 ±3 568.7 ±0.3 [21] TraPPE 2.74 + 0.05 240.3 + 3.1 564.3 + 1.6 0.278 + 0.006 [21] NERD 2.67 ± 0.04 231.2 ±2.9 569.2 ±1.2 0.278 ± 0.004 [21] 232 2.49 568.7 exp-6 0.259 [66] expt. 0.251 n-Dodecane 1.82 + 0.1 226+10 658+ 1 [52] TraPPE 2.3 ±0.2 235 ±6 667 ±5 0.30 ±0.03 [21] exp-6 1.98 ±0.03 225.2 ±1.4 657.1 ±1.0 0.274 ± 0.003 AUA [59] 651.6 217 + 11 [67] expt. 0.278 Isobutane 407.8 ± 0.5 3.64 ± 0.05 224 ±3 [53] TraPPE 408.9 ± 2.3 4.4 ±0.9 228 ±3 0.33 ±0.07 [67] 0.272 expt. Neopentane 433.8 ±0.1 3.196 ±0.01 235 ±3 [53] TraPPE 432.3 ±2.2 233 ±4 3.3 ±0.7 0.29 ± 0.06 [67] 0.273 expt. Cyclohexane 553.8 + 0.2 273 + 2 4.08 + 0.03 [58] exp-6 553.8 ±0.4 4.21 ±0.03 272.8 ±1.1 0.282 ± 0.003 AUA [61] 559 ±5 271 ±10 [68] 0.268 expt. Benzene 562.05 ±0.07 4.895± 0.006 305 ±4 [54] TraPPE 565 + 3 304 + 5 [58] exp-6 561.5 ±0.9 5.08 ±0.05 305.5 ±1.6 0.279 ± 0.003 [69] 0.2775 expt. 1 -Butene 4.02 + 0.05 233 + 12 419.5 + 0.5 [62] 0.289 OleUA 424 ±2 4.0 ±0.1 220 ±10 [54] TraPPE 414±2 241 ±2 AUA [64] 418 + 8 240 + 8 [69] 0.266 expt. 1 -Octene 567.0 ±0.8 2.68 ± 0.08 240 ±10 [62] 0.273 OleUA 2.7 ±0.2 581 ±7 230 ±10 [54] TraPPE 567 ±4 638 ±4 AUA [64] 554 + 10 237 + 8
The development of force-fields for water goes back in the 1930s when Bernal and Fowler proposed the first two-body potential [72]. Since then, tens of force-fields have been proposed to describe very many different properties of aqueous systems. We will concentrate here on the most widely used for thermodynamic property and phase equilibrium
297 calculations. In all cases, water is modeled as a Lennard-Jones sphere and a number of partial charges whose interactions are calculated using a Coulombic term, so that [5]:
K<,M)=4jf^'-(2*TUi-i-Mi [{r',J J
I ' M J J t^ip47r£<> V ,
(24)
where n is the total number of charges per water molecule, q, are the charges and so is the permittivity of vacuum. The number and the position of charges vary: In the simplest case, the negative oxygen charge coincides with the Lennard-Jones center and the two hydrogens are located at a certain distance from the oxygen (three-site model). In the four-site models (including the Bernal-Fowler model), the negative charge is shifted from the Lennard-Jones center to a point along the bisector of the H-O-H angle towards the hydrogens. Furthermore, in the five-site models charges are put on the lone pairs of oxygen and on hydrogens. The increase in the number of sites makes the models more "expensive" in CPU time, since more site-site distances are needed per water dimer. Parameters for the three site water force-fields include Lennard-Jones parameters (e and a), partial charges, O-H distance and H-O-H angle. For the case of four and five site models, the position of negative charge(s) is an additional parameter. These parameters are calculated based on ab initio calculations or, in most cases, by fitting experimental structural and thermodynamic properties. Typical properties used to adjust the model parameters include the radial distribution function, liquid density, vapor pressure, enthalpy of vaporization, second virial coefficient, dielectric constant, and diffusion coefficient. For most of the recently developed models, parameter estimation is based on experimental data over a wide temperature range from the triple point to the critical point. Many-body polarizable force-fields account explicitly for the total electric field on the center of mass of molecule i through the expression [73]: pl=aLlEi = aiJ(E?+E!')
(25)
where pl is the induced dipole moment, a / ; is the dipolar polarizability tensor (in most cases polarizability is isotropic so a, ; is a scalar parameter) and Et is the electric field that includes contributions both from the permanent charges ( E'1) as well as from the polarization of the molecules (Ef). Eq. 25 is usually solved iteratively, thus the computation time increases significantly over the two-body potentials. A polarization term is added to the expression given by Eq. 24, so that the total energy function becomes [73]:
298 Another category of polarizable force-fields are the fluetuating-charge force-fields which are based on the electronegativity equalization method [70]. Furthermore, force-fields for water are developed by fitting appropriate expressions to ab initio data for small water clusters (in the order of 2 to 6 molecules) [74]. In both cases, the functional forms for the force-field are rather complex requiring iterative procedures. Saturated liquid density and vapor pressure from low temperature up to the critical region of pure water as predicted from four two-body force-fields (namely SPC [75], SPC/E [76]), MSPC/E [77] and exp-6 [78]), an isotropic polarizable force-field (SPC-pol-1) [70], an anisotropic polarizable force-field (PPC) [79], and an ab initio polarizable model (MCHO) [80], and from experiments [81] are shown in Figures 8 and 9, respectively. In Table 2, the critical properties from the same force-fields are presented. A similar comparison for other polarizable water models can be found elsewhere [82].
Figure 8. Saturated liquid density for pure water. Experimental data (solid line) [81], and predictions from SPC (A), SPC/E (o), MSPC/E (•) (all from [77]), exp-6 (+) [78], SPC-pol-1 (x) [70], PPC (•) [79] and MCHC
Figure 9. Vapor pressure for pure water. Symbols are the same as in Figure 8.
SPC and SPC/E model parameters were fitted to structure and thermodynamic properties of water at ambient conditions, and they are less accurate for phase equilibria at elevated temperatures. MSPC/E is a re-parameterization of the SPC/E model to improve water phase equilibrium prediction (especially vapor pressure) at sub-critical conditions, whereas exp-6 (here the exp-6 potential is used instead of the Lennard-Jones potential in Eq. 26) parameters were fitted to critical and to sub-critical phase equilibrium properties. SPC-pol-1 is definitely an improvement over SPC and of the same accuracy as MSPC/E and exp-6 for liquid density
299 and vapor pressure predictions. However, SPC-pol-1 is more accurate than the two-body force-fields for the second virial coefficient, due to the more realistic dipole moment of SPCpol-1 compared to the other models. PPC is of the same accuracy as the best two-body force fields. Finally, MCHO severely underestimates the critical temperature of pure water and, consequently, is very inaccurate over the entire temperature range, both for saturated densities and vapor pressure predictions. A clear message from this failure is that ab initio models developed from water clusters are not appropriate for the prediction of bulk thermodynamic and other properties. It should be pointed out, that none of these models is able to predict the maximum saturated water density at 4 °C [70]. Finally, the water structure in terms of the radial distribution function is better predicted by the polarizable force-fields rather than the simple two-body force fields [71]. Table 2. Critical properties of pure water from experiments and simulation using various force fields. Model Expt. SPC SPC/E MSPC/E exp-6 SPC-pol-1 PPC MCHO
TC(K) 647.1 596 593.8 630 638.6 602 609.8 645.9 650 606 495
Pc (MPa) 22.064 12.6 12.9 14.8 13.9 14.8 13.9 18.3 19.5 6.8
Pc(kg/m3)
322 289 271 295 273 310 287 297 300 265
zc 0.229 0.158 0.173 0.172 0.173 0.172 0.172 0.207 0.231 0.112
Reference [81] [77] [78] [77] [78] [77] [78] [78] [70] [84] [80]
Significant work has been done recently on developing smart methods for reducing the computation time associated with the iterative procedure required by polarizability and other many-body interactions [83-85]. Despite this, a simulation with a many-body potential is still at least 50 % more expensive than an equivalent simulation with a two-body potential. Given the fact that the accuracy of the best two-body force-fields is similar to the accuracy of the best polarizable force-fields, there is no clear advantage of using the latter type of force fields for thermodynamic and phase equilibrium calculations of real fluid systems. 4.3. Mixture Phase Equilibria For engineering applications, the ultimate goal in terms of phase equilibrium calculations is to develop a method to predict accurately multicomponent phase equilibria based on information from the corresponding pure components. Most of the methods and elementary moves described in Sections 2.1 - 2.3 above have been applied in recent years to different types of mostly binary mixtures of complex fluid phase equilibria. The vapor-liquid equilibrium of n-alkane binary mixtures is generally predicted with good accuracy over a wide temperature range, including mixtures of very asymmetric n-alkanes [86-89]. Similarly
300
good results were obtained for CO2 - n-alkane and N2 - n-alkane binary mixtures [90]. For the CO2 — n-alkane mixtures, it was shown that Kong combining rules provide better agreement with experimental data than the widely used Lorentz-Berthelot rules [89]. For mixtures where one of the components is polar or hydrogen bonding (i.e., water, methanol), agreement between MC simulation and experiments is less satisfactory [89,91]. Nevertheless, recently GEMC simulation results were reported for the ternary mixture water - methanol - carbon dioxide in the vicinity of carbon dioxide critical point [92]. Liquid liquid and vapor - liquid - liquid equilibrium predictions for such highly non-ideal mixture were in reasonably good agreement with experimental data. In the remaining of this section, mixture phase equilibrium results obtained at NRCPS "Demokritos" in recent years are reviewed in more detail. High pressure light solvent - heavy hydrocarbon phase equilibria are very important for the chemical and the polymer industry. For example, the accurate knowledge of phase boundaries allows the optimum design of chemical reaction and separation processes in polymer production industry. Spyriouni el al. [15] predicted the high pressure phase equilibria of ethane and of ethylene with heavy hydrocarbons using SPECS method. In Figures 10 and 11, experimental data [93,94] and SPECS results are shown for ethane - n-eicosane at 402.8 K and ethylene - n-tetracontane at 398.15 K fluid phase equilibria. At pressures above 18 MPa for the former and 40 MPa for the latter, mixtures are fully miscible. As the pressure lowers, immiscibility occurs resulting in a solvent-rich phase (with very low heavy hydrocarbon composition) and a heavy hydrocarbon-rich phase. In all cases, simulation results are in excellent agreement with experimental data.
Figure 10. Ethane - n-eicosane phase equilibria at 402.8 K. Experimental data (A) [93] and MC simulation using SPECS method (O) [15].
Figure 11. Ethylene - n-tetracontane phase equilibria at 398.15 K. Experimental data (A) [94] and MC simulation using SPECS method (O)[15].
301 In Figure 12, representative snapshots of the two coexisting phases for the ethylene - ntetracontane mixture at 33.2 MPa are presented. For such mixtures, it was shown that macroscopic cubic and even higher order equations of state designed for polymer mixtures fail to predict accurately the phase behavior, and an adjustable parameter is thus needed to improve the agreement [15].
Figure 12. Representative snapshots of the ethylene - n-tetracontane coexisting phases at 398.15 K and 33.2 MPa simulated with the SPECS method: Ethylene-rich phase (left) and ntetracontane-rich phase (right) [95]. Water - hydrocarbon mixtures are highly non-ideal due to the very different nature of intermolecular forces between their molecules. As a result, they are immiscible and, in fact, the typical hydrocarbon (i.e., methane, n-hexane etc.) solubility in water is several orders of magnitude lower than the corresponding water solubility in the hydrocarbon. The accurate description of the phase equilibria of these mixtures is very important for a number of applications in the petroleum and petrochemical industry as well as for environmental control. Solubilities are conveniently described through the Henry's law constant, which is inversely proportional to the solubility. In Figure 13, the methane Henry's constant in water is shown along the saturation line of pure water. Predictions using SPC/E and MSPC/E for water and TraPPE for methane are in very good agreement with experimental data [96]. In the high temperature range, deviations from experiments are due to the lower critical temperature predicted by the two force fields (Table 2). Calculations were performed using the Widom's insertion methodology (Eq. 11). Similar calculations for the case of ethane in water are also in good agreement with experimental data [97]. In Figure 14, the n-butane Henry's constant in water is shown, again along the pure water saturation line. In this case, the exp-6 force-field was used for both compounds and the Henry's constant was calculated using the expanded ensemble methodology (Eqs. 15 - 16) [30]. Simulations are in reasonable agreement with experiments. In addition, calculations for n-hexane, cyclohexane and benzene in water have been performed with same order of accuracy as for n-butane [30, 58]. Solubilities of n-butane and of n-octane in water were also calculated recently through MD simulation using a fluctuating point-charge model for water,
302
in reasonable agreement with experiments [98]. It should be pointed out, that state-of-the art macroscopic equations of state are less accurate for this type of calculations, requiring temperature-dependent binary adjustable parameters in order to describe the entire temperature range [99].
Figure 13. Henry's constant of methane in Figure 14. Henry's constant of n-butane in water. Experimental data (solid line) [96] water. Experimental data (solid line) [100] and MC simulation using the SPC/E and MC simulation (points) [30]. (diamonds), and MSPC/E (squares) models for water [97].
High pressure water - hydrocarbon phase equilibria are important for chemical engineering as well as geochemical applications. As the pressure increases, mutual solubilities increase and pressure - composition diagrams are preferred. In Figure 15, water - ethane phase equilibria at 523, 573 and 623 K and up to 350 MPa is shown [97]. GEMC simulation results using the exp-6 force-field for both compounds are in excellent agreement with experimental data, even in the vicinity of the water critical point. Such a success clearly reflects the fact that accurate force-fields for pure compounds can be used reliably for mixture calculations even at extreme conditions without the need for adjustable parameters. Aqueous mixture simulations presented in Figures 1 2 - 1 4 were based on effective twobody force fields. To the best of our knowledge, none of the more elaborate force fields presented in Section 4.2 has been used to date beyond pure water properties (with the exception of n-butane and n-octane solubility in water based on molecular dynamics calculations using a fluctuating point-charge model in ref. [98]). As polarizable force-fields become progressively more accurate and as computing power increases, we expect that these more complex force-fields will be used in order to eliminate deviations between experiments and simulations (as the ones shown in Figure 14, for example).
303
Figure 15. Water - ethane phase equilibria at 523, 573 and 623 K. Experimental data (lines) [101] and GEMC simulation (points) [97].
Figure 16. Performance of top supercomputers since mid-1950s. Today (March 2004), the most powerful supercomputer is in Kanazawa, Japan with 35.9 Tflops. As it has been announced, in 2005 IBM will put in operation Blue Gene with computing power of approx. 400 Tflops.
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12.5 CONCLUSIONS This book is devoted to state-of-the art thermodynamic methods directly applicable to engineering problems. Consequently, the most important question that this chapter tries to address is: Can molecular simulation be used to solve industrial problems? If the question was asked twenty years ago the answer would be a diplomatic "maybe". However, today we have available in our disposal an arsenal of methods and algorithms developed specifically for systems of industrial interest. All of the methods, force-fields and systems discussed in this chapter concern real fluids and simulation results are in good agreement with experiments. Of course, by no means molecular simulation will ever replace macroscopic engineering models as some people are afraid of. The two methods are not in competition with each other but they are rather complimentary. On the one hand, molecular simulation has the unique capability of linking basic chemical structures and molecular mechanisms to macroscopic properties. In this way, it can assist in the development of better macroscopic models with a stronger theoretical basis and, hopefully, higher accuracy. On the other hand, molecular simulation cannot be routinely used to generate data for a process simulation, as is the case with an equation of state. Of course, simulation results based on reliable force-fields can be considered as pseudo-experimental data so that simulation becomes an invaluable tool for systems and conditions where experimental measurements are not possible (highly toxic compounds, extreme conditions, etc.). For this reason, additional work is needed to develop accurate force-fields for a wide range of complex compounds of interest to industrial applications. Almost all of the applications presented here were related to thermodynamic properties and in particular to phase equilibria. Molecular simulation plays an important role also in other areas of chemical process industry, as for example in the design of novel materials (polymers, catalysts, microelectronic materials etc.) with tailor-made properties, design of new pharmaceuticals and other biomolecules, etc. [102]. Commercial molecular simulation products are also available today and have become an invaluable tool for researchers in the chemical, petrochemical, polymer and life-sciences industry [8]. As computing power continues to rise (Figure 16), molecular simulation and other computing intensive computational chemistry tools are able to provide reliable solutions to a wide range of problems that are of interest to industry and to economy, in general.
ACKNOWLEDGMENT I am grateful to Dr. Philippe Ungerer for providing his manuscripts prior to publication, to Nikos Karayiannis for providing the raw data in Figure 7, to Professor Allan Mackie for providing the raw data for the MCHO force-field for water (Figures 8 and 9), to Dr. Loukas Peristeras for drawing Figures 2 through 6, and to Professor Doros Theodorou for his very thorough comments on the manuscript.
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Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B. V. All rights reserved.
309
Chapter 13: Property Models in Computation of Phase Equilibria Rafiqul Gani and Georgios M. Kontogeorgis 13.1 INTRODUCTION The objective of this chapter is to highlight how the selected property models are used in various types of phase equilibrium calculations, using simple illustrative examples. The chemical systems covered in this chapter do not include electrolyte systems (see chapter 15 for applications covering electrolyte systems). The model equations (including the property model) are analyzed and the calculation procedure is discussed. The aim here is not to discuss the most efficient and robust algorithms for phase equilibrium calculations but to point out the relationships between the property models and the process-product model from which they are called from during the solution of the model equations. Numerical analysis is also provided of the model equations in terms of incidence matrix and a sample calculation order for the equations for various types of phase equilibrium computations. For a thorough discussion on efficient algorithms for phase equilibrium calculations, the reader is referred to a recent book by Michelsen and Mollerupfl]. 13.2 DIFFERENT TYPES OF PHASE EQUILIBRIUM PROBLEMS Consider a closed two-phase system maintained at constant temperature and pressure. The equilibrium state of a closed system is that state for which the total Gibbs energy is a minimum with respect to all possible changes at the given temperature (T) and pressure (P). G = f(P,T,n 1 ,n 2 ,
nM)
(1)
The chemical potential, u.j is defined as, Pi=(SG/8ni)p^n>.
(2)
For a system as a whole comprising phases I and II in equilibrium, the condition for equilibrium can then be written as, (dG)p5 T = (dG!)p, T = (dG n ) P , T = 0
(3)
310 or, in terms of the chemical potentials, Z(jiiI-HiII)dniI = Z(jiiI-niII)dniII = O
(4)
Since the quantities dnj and dnj are independent & arbitrary, the condition of equilibrium becomes, Hj^uj11 or fj^fi11
(5)
where, the relation between the chemical potential and fugacity is given by, Hi = RTIn(fi)
(6)
and the fugacity fj is defined as, f' = cp' * ' P
(7)
where 9/ is the fugacity coefficient of component i present in phase I while x/ is the mole fraction of component i present in phase I and is calculated through an appropriate model. In generic form, 9/ can be represented as a function of temperature, pressure and composition, 9/ = f(T, P, x1)
(8a)
Therefore, to compute any phase equilibrium condition, an appropriate property model for the fugacity coefficients for each component in each coexisting phase is necessary. For a given chemical system and the type of the phase equilibrium model, the choice and the dependence of the intensive variables on the fugacity coefficient can vary. Among the various forms, the three most common are the following (in addition to Eq. 8a), (Pi1 = f(T) = P; sat (T)
(8b)
(pi^fitT.x^YiCr^Pi^Cr)
(8c)
In Eqs. 8a and 8b, Pj
(T) is the vapor pressure of component i at a specified temperature,
T while yi (T, x) is the activity coefficient of component i in the liquid phase (usually calculated as a function of temperature and composition). Usually, for the vapor phase, Eqs. 8a or 8b (that is, a suitable model representing these equations) are used, while for the liquid phase, Eqs. 8a-8c are used. Note, that Eq. 8c is only used for the liquid phase (for a discussion on the choice of models, see chapter 1) and is a simplified form, which neglects for example, the Poynting correction term. Usually, Eq. 8a represents an equation of state (EOS), Eq. 8b represents an ideal system where a vapor pressure correlation is used, while
311 Eq. 8c represents a GE-based activity coefficient model. If Eq. 8a is used for all the coexisting phases in equilibrium, the phase equilibrium calculation procedure is usually referred to as a phi-phi approach. If Eqs. 8a or 8b is used for the vapor phase and Eq. 8c for the liquid phase, the phase equilibrium calculation procedure is usually referred to as a gamma-phi approach. For a two-phase system, there can be different combinations of liquid, vapor and solid phases. Most typical are the vapor-liquid (VLE), liquid-liquid (LLE) and solid-liquid (SLE). For three phase systems, there may be one vapor phase and two liquid phases (VLLE) or a vapor, a liquid and a solid phase (SLV). More details of different types of phase equilibrium and different combinations of phases can be found in [2, 3]. 13.2.1 Computation of VLE Considering again a closed two-phase system and Eq. 5 (which is also known as the isofugacity criteria), the following equilibrium conditions are obtained, At low pressure, non-ideal liquid and non-ideal vapor (gamma-phi approach) ^.satp.saU^Vp
( 9 )
At low pressure, non-ideal liquid and ideal vapor (gamma-phi approach) xj Yi Pi s a t = y;P
(10)
At any temperature and pressure, the same fugacity coefficient model for both phases *i
(11)
From the above equations, it becomes clear that in order to predict VLE, we need to predict first, Yj, 9j , cpj (mixture properties), cpj , Pj s (pure component properties), for which appropriate models are needed. We will now analyze the phase equilibrium model equations by inserting the property model (only for the fugacities) and observe the form of the ordered set in an incidence matrix. Model Analysis Consider the following VLE calculation - Given the pressure P, and the liquid phase composition x, calculate the saturation temperature T and the vapor composition y, using the phi-phi approach with Eq. 8b used for both phases. The model equations for the above problem are the following (Eq. 13 is obtained by assuming liquid to be ideal, i.e., yj = 1) Pj s a t =A; -B;/(T + Ci)
(12)
312 yi
= x{Pfat(T)/P
(13)
Residy = Z; y; - 1 = 0
(14)
In the above model, Eq. 12 is a constitutive equation (property model) and Eq. 14 is a constraint equation (a necessary condition for VLE). Equation 13 is an explicit algebraic equation, that is, given the values of xj, Pj s a t (T), and P, it is possible to calculate explicitly, yj. Note that if Eqs. 12-14 are combined together, that is, Eqs. 12 and 13 are inserted into Eq. 14, we will have one implicit equation with one unknown variable, T. Instead, we will add one more equation relating the sum in Eq. 14 to T. T = f(T, 2j Yi , Residy)
(15)
For a binary system, we can say that there are 4 equations with 4 unknown variables (yj, Pj , Residy, and T). Since P and xj are known (input information), together with model parameters A;, B; and Q, the ordered incidence matrix (including Eq. 15) is shown below in Figure 1.
Eq. 12 Eq. 13 Eq.14 Eq. 15
Ai
Bi
Ci
*
*
*
i
P
*
*
x
P .sat (T)
y;
Residy
• * *
•
T •
*
•
Figure 1: Incidence matrix for VLE calculation model (ideal system) In Figure 1, * in the incidence matrix signifies a known variable where as • indicates an unknown variable. Since only T has • in the upper triangular part of the incidence matrix, an iterative solution technique can be generated by starting with an assumed value of T in Eq. 12 and ending (checking) the new value of T in Eq. 5. Note also that the incidence matrix tells us that we have a system of equations that may be solved simultaneously as a system (if we do not want to use the iterative approach). In either case, iterations are necessary since there is one unknown variable (that is, T) in the upper tridiagonal part of the incidence matrix. Note that a similar analysis can also be made when the temperature is specified instead of the pressure (and also the vapor composition instead of the liquid composition). Note also that the property model (Eq. 12) belongs to the equations within the repetitive cycle of equations (or the set of simultaneous equations). Now let us consider a liquid that is non-ideal, that is, yj ^ 1. So, we now need to introduce a model for the liquid phase activity coefficients (yj). Taking the UNIFAC VLE [4] model considered and analyzed in chapter 4, in addition to Eqs. 10,12, 14-15, we have Eqs. 3-1, 3-2,
313 3-3a, 3-3b, 3-4, 3-5, 3-6, 3-7 & 3-8 from chapter 4. All these equations are now shown as an ordered set in the incidence matrix of Figure 2 (note that the known variables Aj, Bj, Q, x and P are not listed). Not counting the GE-model equations, there are still 4 equations and 4 unknown variables, as in the previous example.
Figure 2: Incidence matrix for VLE calculation model (gamma-phi approach with ideal vapor phase) It can be seen that a change in the constitutive model in this case did not change the structure of the incidence matrix and the iterative procedure. The main reason for this is that the fugacity coefficient model for the vapor phase is composition independent. Again, the property model belongs to the equations within the repetitive cycle of equations (or the set of simultaneous equations). Let us now consider the use of the phi-phi approach with Eq. 8a inserted into Eq. 11. Instead of Eq. 12, we now have Eq. 8a (used twice - for liquid phase with x and for vapor phase with y) together with Eqs. 11, 14 & 15. Again, x and P are specified. It is assumed that all the EOS model parameters are available. Now, the incidence matrix shows additional unknown variables (the vapor compositions) on the upper tri-diagonal part of the incidence matrix (see Figure 3). This means that in addition to T, the values of the vapor compositions also needs to be iterated upon, when a numerical solution technique is employed. As shown in Figure 3, we have 5 equations with 5 unknown variables (for a binary mixture), which are listed in the last 5 columns of row 1 in the incidence matrix. For a detailed model analysis of an EOS in terms of calculation of the fugacity coefficient, given the intensive variables, see chapter 1 where the SRK EOS [5] has been used as an example. Here also, the property model belongs to the equations within the repetitive cycle of equations (or the set of simultaneous equations).
314
Eq. 8a-Liquid Eq. 8a-Vapor Eq.ll Eq.14 Eq.15
x, *
P * *
Residy
*
• *
• • *
T •
• *
•
Figure 3: Incidence matrix for VLE calculation model (phi-phi approach with an EOS) Now let us consider a typical two-phase PT-flash calculation involving the vapor and liquid phases. We can analyze the incidence matrices for the gamma-phi and phi-phi approaches. In either case, considering only a mass balance model, we need to introduce the following mass balance equation. F z ^ V y j + Lxj
i=l,2,....N
(16a)
Equation 16a can be rearranged to the following form, where p = V/F. zi = pyi + (l-P)x i
i=l,2,...-N
(16b)
We also need a constraint on the liquid composition (x), that is, Residx = Zj xj - 1
(17)
Since the Residx and Residy both must be zero at solution of the model equations, this means that inserting Eqs. 14 and 17 into Eq. 16a, results into the total mass balance equation, F=V+L
(18)
For the PT-flash problem described above, the model equations are those listed in Figure 2 plus Eqs. 16-17. The variables known are F (feed flowrate), z (feed composition), T (temperature of flash) and P (pressure of flash). We have 2 additional variables, L (liquid flowrate) and V(vapor flowrate). Instead of writing all the equations for the constitutive property model (Eqs. 3-1, 3-2, 3-3a, 3-3b, 3-4, 3-5, 3-6, 3-7 & 3-8 from chapter 4), we represent them as one equation in a generic expression since they have a tridiagonal form. Yi = f(T,x)
(19)
Again, for a binary mixture, the unknown variables are - P (=V/F), L, V, x, y, yj and Pj s (7 variables) and the TP-flash model is represented by Eqs. 10, 12, 14, 16b, 18, P=V/F & 19 (7 equations). These equations and variables together with the specified variables are shown in the incidence matrix of Figure 4. From the above analysis, it is clear that a sub-set of equations (Eqs. 19, 16b, 10 & 14) need to be solved simultaneously for yj, Xj, yi and p. Also, by assuming values for the 2 variables lying in the upper tridiagonal (XJ and p.), an iterative solution scheme can be
315 generated. Note that in this case not all the property models belong to the equations within the repetitive cycle of equations (or the set of simultaneous equations).
Figure 4: Incidence matrix for a TP-flash (mass balance) model with gamma-phi approach The same analysis as above can also be shown for the phi-phi approach using an EOS. Here, the model equations are Eqs. 8a (twice), 10, 14, 16, 18 and P=V/F. The unknown variables are, cpj ,
Figure 5: Incidence matrix for a TP-flash (mass balance) model with phi-phi approach 13.2.2 Computation of LLE & VLLE LLE-phase Diagrams Consider a two-liquid phase system with N components. The criteria for LLE equilibrium is given by Eq. 20 in terms of liquid activities of each component in each liquid phase together with the constraint equation (Eq. 21), while Eq. 19 represents the property model.
316 *iV = X i V 1
i=l,2,....N
%xil = Zixi11 = 1
(20) (21)
In order to generate LLE phase diagrams (for example, for a ternary system), the degrees of freedom is N (for a 2 phase system with N components). Therefore, N variables from Eqs. 19-21, need to be specified. A typical scenario (considering a ternary system) for these types of calculations is to start with N-l compositions from one phase and the temperature as the N=3 specified variables. Figure 6 shows the incidence matrix for this model and it can be seen that an iterative procedure is necessary, even though the calculation of the activity coefficients is lower tridiagonal (see chapter 4). Clearly, Eqs. 19 (for phase II), 20 and 21 (for phase II) need to be solved simultaneously, or an iterative scheme needs to be derived for updating the guessed x". For generating ternary LLE phase diagrams, it is recommended to start with the binary pair (out of ternary mixture) that will split into 2-liquid phases for specified N-l compositions of one liquid phase at a specified temperature. Calculations are then repeated by adding small amounts of the third component. This produces a binodal curve at a constant temperature. Note that pressure has not been used as a variable because the property model (Eq. 19) does not have pressure dependence. If the dependence of pressure needs to be considered, then a mixed GE-EOS model [6] needs to be selected (see property models in chapter 6 models).
Figure 6: Incidence matrix for LLE phase diagram calculations. LLE Phase Separation Calculations This calculation is similar to a VLE PT-flash calculation, except that the two coexisting phases are both liquids. The model equations are the following: Eqs. 19 (for both phases), 20, 21 and the following mass balance equation, Z z — L . x j 1 +L2XJ11
i=l,2,....N
(22)
For a 2-phase system, the degree of freedom is N + 2. That is, N + 2 variables need to be specified (for example, T, Z, z; the temperature, the feed flowrate and the feed compositions, respectively). The incidence matrix for this model is shown in Figure 7.
317
Figure 7: Incidence matrix for LLE phase diagram calculations (superscript + indicates that a total mass balance Z = Li + L2 is obtained when Eq. 21 is satisfied) Again, we see the need for an iterative solution technique or for solving a sub-set of equations simultaneously (Eqs. 19-1, 19-11, 22, 20 & 21-1). Note that if G -models are used, the pressure effect cannot be considered for reasons listed above. To obtain the pressure p effect, combined G -EOS models need to be used. VLLE Phase Diagrams (3-phases) The case of one vapor phase and 2 liquid phases is considered here. Eqs. 20 and 21 are modified by adding another phase, as shown below. x
i V
i = 1,2,....N
EixiI = SixiII = E i y i = 1
(23) (24)
Note that in addition to Eqs. 23-24, property models for the various property terms are also needed (such as Eq. 8a, 12, 19, etc.). The calculations can be started with the binary pair (from ternary mixtures) that will show a liquid-liquid split. That is, given x & P; calculate T and corresponding y and x . Analyzing the incidence matrix (not shown) will indicate again the need for an iterative (or simultaneous solution of a sub-set of equations) solution technique. PT-Flash for VLLE The model in this case, includes the model equations for VLLE phase diagram plus the following mass balance equation (for a mass balance model). Z z j ^ V y j + Lixj 1 +L2XJ11
i=l,2,....N
(25)
318 Even though there are now 3 phases in coexistence, the degree of freedom is still N+2, which means that the following variables are specified - Z, T, P and (N-l) compositions of z. Although, not shown, the incidence matrix again will show unknown variables in the upper tridiagonal, indicating a need for iterative solution techniques. 13.2.3 Computation of SLE Consider a liquid solution from which a single component may precipitate out as a solid. One scenario for the saturation point calculation is to specify the temperature and the compound that is to precipitate as a solid (from a binary mixture of a solid and a solvent), and calculate the saturation liquid phase composition. The solid-liquid equilibrium condition is given by Eqs. 26-27, where AHj is the heat of fusion of component i, R is the universal gas constant and T m j is the melting point of component i. Note that sj is the composition of component i in the solid phase and since this is the only component in the solid phase, it is equal to unity. In addition to Eqs. 26-27, Eq. 19 is also needed as the model for the constitutive equation. Note that this model is neglecting the entropic effect and that the pressure effect is not considered in this model. 1 = s; y;S = XJ Yi L exp[AHiF/(RTmi){(T - Tmi)/T}] i = s
l - x s = xN
(26)
(27)
In Eq. 27, for a binary system, xN is the composition of the second component. For a ternary mixture, x^ represents the sum of N-l compositions (of the solvents). The solvent distributions remain constant for a set of data-points. The incidence matrix for the above SLE model is shown in Figure 8. The unknown variables are xs and yj (liquid) and xN. It can be seen that an iterative solution technique where Eq. 19 belongs to the repetitive cycle is obtained. For higher dimension problems, for example, two solids and one solvent or two solvents and one solid, the saturation curve for each solid can be calculated for a fixed amount of the third compound. The calculations may be started for each binary pair that forms a SLE system (there will be 3 for a ternary system). Then from each of the binary points, constant temperature saturation curves can be generated for a fixed amount of the third component. If instead of temperature, the saturation composition of the solid in the liquid phase is specified, the incidence matrix would also look very similar.
Figure 8: Incidence matrix for SLE calculations using a G -model
319 13.3 APPLICATION EXAMPLES: SPECS The calculations shown below have been performed with the software SPECS developed by the IVC-SEP research center (www.ivc-sep.kt.dtu.dk) at the Technical University of Denmark. The standard opening page is shown below. The software SPECS can perform a variety of phase equilibrium calculations (VLE, LLE, low & high pressure) for ordinary systems (standard database, shown below) as well as for mixtures containing electrolytes, polymers, oil, etc. A variety of models can be chosen, including various cubic equations of state, possibility of different mixing rules, various UNIFAC variants, the CPA and PC-SAFT equations of state, various free-volume models for polymers and the extended UNIQUAC for electrolytes. For polymers, electrolytes, PC-SAFT and oil, different choices should be made in "the component database", shown below in Figure 9. A number of examples illustrating the capabilities of the software are shown below, together with the corresponding screen shots from SPECS.
Figure 9: Screen shot from SPECS
320 13.3.1 Example 1 - Evaporation of isopropanol from a PVAC film This example has been modified from Prausnitz et al. [7]. A film of poly(vinyl acetate) [PVAC] contains traces of isopropanol. For health reasons, the alcohol content of the film must be reduced to a very low value: government regulations require that that the volume fraction of isopropanol should be: (p, <1(T 4 . To remove the alcohol, it is proposed to evaporate it at 125 °C. At this temperature, chromatographic experiments, which are carried at infinite dilution of alcohol, give a value of the Flory-Huggins parameter equal to 0.44. The vapor pressure of alcohol is 4.49 bar. Assuming that the PVAC is of very high molecular weight, calculate the low pressure, which must be maintained in the evaporator to achieve the required purity of the film using: i. The above information and the Flory-Huggins model ii. The software SPECS and one of the available suitable models for polymer solutions Compare the results obtained from the two methods. What do you observe? Under the conditions prevailed here, the activity of the solvent is simply given as the ratio of the partial pressure to the solvent vapor pressure and the polymer is assumed to be completely non-volatile. Solution At low pressures, the vapor-liquid equilibria for a binary solvent (1) - polymer (2) solution is given by (see Chapter 7): P = xxYxPr'=aAsc"
(28)
Since the solvent vapor pressure is known, the problem is reduced in estimating the solvent activity. For this, we can employ either the Flory-Huggins (FH) activity coefficient model or other models, using the software SPECS. Using the FH model The value of the FH interaction parameter is known (=0.44). The FH equation can be expressed in terms of the activity, which is the property actually required in Eq. 28, as (see also Chapter 7): l n ^ = l n ^ + 1—U,+2- p ^ 2 -=> X|
l
r)
( n In a, = l n ^ + 1—Wi+XnVl
(29)
V r) -
Since the polymer is of very high molecular weight, the r-parameter (ratio of volumes) is very high, thus 1/r is approximately zero and, using the volume fraction value provided in the problem (10~4), we get the value of the activity as well as the pressure of the evaporator:
321
lnct, = l n c p , + 1 \?2+ln'?22 => V r) - a, = 0.000422 => P = 0.00189
(30) bar
Using SPECS We can select different activity coefficient models e.g. Entropic-FV, UNIFAC-FV and UNIFAC and with various choices for the UNIFAC parameter tables (see Chapter 7 for references to these models). The Models window, with the Entropic-FV chosen, is shown below (Figure 10). The vapor phase is assumed ideal (as previously).
Figure 10: Screen shot from SPECS highlighting the choice of the property model Assuming that volume and weight fractions are close to each other, the activity can be calculated at 398 K for the isopropanol/PVAC system. Then, the pressure can be calculated using Eq. 28. The default molecular weight of the polymer in SPECS is 50000 g/mol (but can be modified, if this required). We have selected the linearly temperature dependent parameters for all models. Entropic-FV yields an activity equal to 0.0006, UNIFAC-FV equal to 0.001 and UNIFAC equal to 0.0009. [Entropic-FV is closest to FH in this case and results to a pressure equal to 0.002694 bar]. All three models are UNIFAC based and since they are based on group contributions, all calculations can be considered to be straight predictions.
322
13.3.2 Example 2 - The effect of polymer molecular weight on activity coefficients for polymer solutions Using the software SPECS and a suitable model for polymer solutions, investigate the effect of polymer molecular weight on VLE calculations and especially on the values of activity coefficients for polymer solutions. Consider e.g. the system hexane/PVAC and calculate the molar and weight-based activity coefficients at infinite dilution at three different polymer molecular weights (e.g. 50000, 500000, 2000000 g/mol). What do we observe? Solution The Tasks and Output windows are shown below in Figures 11 and 12, respectively when the Entropic-FV model is used and the molecular weight is equal to 50000 g/mol. The concentration of n-hexane must be set equal to zero.
Figure 11: The task window in SPECS Using SPECS and Entropic-FV at 300 K (any other model or temperature can be used), we obtain the results shown below.
323
Molecular weight 50000 500000 2000000
Infinite Dilution Activity Coefficient (molar based) 0.0611 0.0061 0.0015
Weight-based activity coefficient at infinite dilution 35.47 35.63 35.63
Figure 12: The output window in SPECS We observe that, while the molar-based activity coefficient depends a lot (decreases) with increasing molecular weight, the activity coefficient based on weight fractions is essentially independent of the molecular weight. Thus, it is preferred in many calculations involving polymers and in rules for thumb for selecting solvents for polymers (chapter 7). We recall that the weight-based activity coefficient is given by the relation:
and at infinite dilution, the molar-based and weight-based activity coefficients are related by the equation:
324
where Mj is the molecular weight of the solvent and M2 is the molecular weight of the polymer. 13.3.3 Example 3 - Comparison of models for polymer-solvent VLE with SPECS Using SPECS, various models can be chosen for polymer solutions. Thus, their performances can be compared, as shown in the table below (Table 1), where the predictions are compared also to the experimental data. For all three models, the UNIFAC linearly temperature dependent parameter table is used (but other choices are possible). Table 1: Comparison of calculated values from SPECS with experimental data Entropic-FV
UNIFAC-FV
UNIFAC
423.2
Experimental value 5.69
5.09
4.37
2.89
400.2
4.27
4.55
3.94
2.91
393.2
11.8
9.33
9.52
5.24
383.2
45.0
14.85
14.76
8.67
323.2
5.30
5.32
6.08
3.50
298.2
2.53
2.74
3.32
1.54
System
T(K)
benzene/PS (86700) benzene/PS (3524) acetone/PVC (41000) heptane/PVC (41000) toluene/PIB (53000) CCL4/PIB (40000)
Alternatively, plots of activity against concentration can be performed, as shown for three systems in Figures 13a-13c. The clear maximum of PVC/heptane indicates immiscibility (verified by the activity coefficient values of the table), while CC14 is clearly a good solvent for PIB. Less clear is the situation for toluene/PIB (incipient instability). 13.3.4 Example 4 - SRK with various mixing rules using SPECS Examples are shown below with two VLE systems, the low pressure acetone/hexane at 328.15 K and the high pressure acetone/water at 473 K using SRK with two different mixing rules. The standard vdWlf mixing rules (without interaction parameters) and the MHV2 mixing rule (using the UNIFAC as the activity coefficient model). Both systems are azeotropic. The Models, Tasks and Output Windows are shown (see Figures 14a-14d). Notice that various mixing rules can be chosen. Bubble point temperature (P-xy) calculations are performed. It can be concluded that only MHV2 can successfully predict the phase equilibria for these polar systems e.g. as shown in Figure 14d for acetone/water. SRK using the standard vdWlf mixing rules cannot predict the azeotrope for acetone/hexane (Figure 14c), while very bad results are obtained for acetone/water (not shown here).
325
Figure 13b: Plotted results from SPECS (Example 3)
326
Figure 13c: Plotted results from SPECS (Example 3)
Figure 14a: Models window from SPECS (Example 4)
327
Figure 14b: The tasks window from SPECS (Example 4)
Figure 14c: The output window from SPECS for example 4 (xy-diagram)
328
Figure 14d: The output window from SPECS for example 4 (P-xy diagram) 13.3.5 Example 5 - Complex systems with SRK & CPA using SPECS The final example is a T-xy plot at 1 atm for the thermodynamically complex system methanol-benzene, which is experimentally shown to have an azeotrope at 0.6 mole fraction of methanol. Results are shown below (see Figures 15a-15b), as output files from SPECS, using either the SRK or the CPA equations of state. In both cases, no interaction parameters are used, thus the calculations are straight predictions. CPA can predict very correct the phase diagram, including the azeotrope, while SRK fails.
Figure 15a: Output from SPECS for example 5 (T-xy diagram)
329
Figure 15b: Output from SPECS for example 5 (T-xy diagram)
13.4 APPLICATION EXAMPLES: ICAS The thermodynamic properties options in ICAS [8] includes a database (CAPEC Database [9]), a pure component properties package (ProPred [10]), a thermodynamic model library (TML [11]), a property model creation package (SoluCalc [12]) and a utility toolbox (for phase equilibrium calculations). ICAS has been developed at CAPEC (Computer Aided Process-Product Engineering Center), which is a center for research (www.capec.kt.dtu.dk) at the Chemical Engineering Department of the Technical University of Denmark. A selection of options from the ICAS-utility toolbox is shown in the examples below. An example from SoluCalc is shown in chapter 14, the property model parameter estimation feature of TML has been highlighted in chapter 1, use of ProPred has also been highlighted in chapters 1 and 14. Details of the CAPEC database can be found in Chapter 3. 13.4.1 VLE Calculation Examples Problem 1 - VLE phase diagram calculation VLE calculations are needed for the synthesis and design of the separation of a binary azeotrope of acetone-chloroform. Step 1: Verify binary azeotrope and pressure dependence of azeotrope. Select the gamma-phi two-model approach with KT-UNIFAC model for the liquid phase activity coefficient and ideal gas law for vapour phase. This property model configuration
330
needs a model for vapour pressure as a function of temperature (select the DIPPR correlation 101). Figure 16a shows the property model selection. Figures 16b-16d highlight the KTUNIFAC model parameters and the DIPPR 101 vapour pressure correlation parameters. Figure 17 shows the calculated T-xy diagram for acetone chloroform at latm pressure. Step 2: Select a solvent and study the ternary mixture VLE Benzene is a well-known solvent and can be used to "break" the azeotrope. For product (solvent) design problems, the calculated properties for this ternary mixture will help to establish the target (desirable) properties of the replacement solvent. The property models are the same as in step 1. The only additional calculations now are the identification of binary azeotropes on the ternary diagram, checking if the solvent causes a phase split of the liquid phase and the prediction of distillation boundaries. Figure 18 shows the distillation boundary on a ternary diagram together with residue curves and the binary acetone-chloroform azeotrope. This system does not show a liquid-phase split. That is, it is a homogeneous azeotropic system. ICAS also has tools for solvent selection and design, called ProCamd [13], which is based on the computer aided molecular design (CAMD) technique. Use of the CAMD technique for solvent selection related to acetone-chloroform separation has been reported by Hostrup et al. [14]. The ternary diagram in Figure 18 shows the calculated binary azeotropes as well as a distillation boundary, which helps to identify the feasible products for a specific feed mixture.
Figure 16a: Selection of property models for VLE computation.
331 KT-UNIFAC group representation Acetone: 1 CH3; 1 CH3C0 Chloroform: 1 CHC13 Benzene: 6 ACH
Note that it is possible to define new groups and estimate their parameters as well as fine-tune existing group parameters. Figure 16b: KT-UNIFAC group interaction parameters.
Figure 16c: R & Q values for the KT-UNIFAC groups.
Figure 16d: DIPPR correlation parameters for the vapour pressure as a function of temperature.
332
T-xy diagram
Driving force (y-axis) as a function of composition of acetone in the liquid phase
Figure 17: Calculated T-xy and driving force diagram for acetone-chloroform mixture from ICAS (see chapter 2 for definition of driving force)
Figure 18: Ternary diagram showing the binary acetone-chloroform azeotrope and the distillation boundary starting from the binary azeotrope and ending on the chloroformbenzene axis. The two other curves are residue curves where the light products are acetone and chloroform respectively and the heavy product in both cases is benzene. The KT-UNIFAC model has been used for this calculation.
333 13.4.2 LLE & VLLE Calculation Examples Problem 2 - Investigate the ternary system ethanol-water-benzene at a pressure of 1 atm Step 1: Property model selection For LLE calculations, only a liquid phase activity coefficient model is needed. The UNIFACLLE model is selected, as all the necessary parameters are available. For VLLE, the gammaphi two-model approach is selected with various versions of UNIFAC as the liquid phase activity coefficient model and the SRK equation of state (without binary interaction parameters) as the vapour phase fugacity coefficient model. For this property model selection, the following pure component properties are also needed: critical properties of each component and the vapour pressure as a function of temperature. For high pressure calculations, the combined GH-EOS models, such as, the MHV2 model can also be selected. Step 2: Calculate the LLE and VLLE diagrams The utility toolbox in ICAS is used. For the LLE phase diagram, calculations are started on the binary water-benzene axis and then the calculations are repeated for a step increase in the amount of the third component (ethanol) until only a one phase liquid system results. This is repeated for a range of temperatures. Figure 19a shows the calculated LLE phase diagram through the property utility option in ICAS. For VLLE, the calculations also start with the binary water-benzene mixture and the binary azeotrope together with the liquid-liquid split of the azeotropic mixture is calculated. This procedure is repeated for each step increase in the amount of ethanol. Note that unlike the LLE curve, the VLLE curve represents a heterogeneous liquid surface (that is, each tie-line represents a different temperature). In these calculations, first a VLE problem is solved, followed by a calculation of the liquid split of the condensed vapour followed by a final correction of the VLLE compositions and temperature. Figure 19b shows the calculated heterogeneous surfaces together with the binary and ternary azeotropes. 13.4.3 SLE Calculation Examples Problem 3 - Investigate the solubility of morphine in single and mixed solvent systems. Step 1: Property model selection For calculation of solubility of morphine in non-electrolyte solvents, only the normal melting point, the heat of fusion and the liquid phase activity coefficients are needed. For a specified temperature, the composition of the solid in the liquid is calculated. The solid is assumed to be pure morphine. Solvents for morphine are identified through its Hildebrand solubility parameter value by searching in the CAPEC database. The ProPred toolbox in ICAS is used to estimate the needed pure component properties of morphine (see Figure 20a). For the SLE calculations, the utility toolbox in ICAS is used. Step 2: Pure component property prediction with ProPred in ICAS From the calculated Hildebrand solubility parameter of morphine, it is clear that good solvents for morphine may be hydrocarbons (or non-hydrogen bonding solvents). First
334 hexane is considered as a solvent and then mixtures of hexane with cyclohexane and benzene are also considered analysis of solubility as a function of temperature and solvent mixtures. Note that since the database does not have the needed pure component properties for morphine, these are passed from ProPred into the database.
Figure 19a: Calculated ternary phase diagrams as a function of temperature for ethanol-waterbenzene system at 1 atm pressure
Figure 19b: Two calculated VLLE diagrams with the UNIFAC-VLE model (left) and the Modified UNIFAC-Lyngby model (right)
335
Step 3: SLE calculation with the utility toolbox in ICAS Morphine and the solvents are selected as compounds present in the mixture. The UNIFACVLE model is selected as the model for the liquid phase activity coefficient. First a binary phase diagram with morphine and hexane is computed, followed by a ternary mixture, where cyclohexane is added to the binary mixture of morphine and hexane (see figures 20b-c). Finally, addition of benzene is also investigated (not shown in the figures). Note that the estimated values have not been compared with any experimental data and therefore, should not be used without any prior verification. Note also that the pressure is not specified.
3D-structure of Morphine
Figure 20a: Pure component properties estimated through ProPred The estimated heat of fusion and the UNIFAC group parameters have been used together with the available experimental melting point temperature to calculate its solubility in hexane (solvent). From the total solubility parameter, it can be seen that hexane may be a good solvent (since the solubility parameter is close to that of morphine). At the same time, the solubility of morphine in water is calculated to be 2.41 (log mg/L).
336
Figure 20b: Binary SLE phase diagram for solubility of morphine in hexane
Figure 20c: Ternary SLE phase diagram for solubility of morphine (at 504 K) Note that the specified temperature is quite high (in fact, higher than the boiling points of both solvents). This means that either, the heavier solvents should be used to operate at the high temperature or, a higher pressure where the solvents will be liquid would be needed.
337 13.4 CONCLUSIONS The calculation orders for the property model equations as part of models for various types of phase equilibrium computations have been analyzed and highlighted through representation of the model equations and variables in incidence matrices. It can be clearly seen that while on their own, the property models analyzed are tridiagonal, when they are used from within phase equilibrium and two-phase separator models, there are unknown variables appearing in the upper tridiagonal part. This means that the property model equations need to be solved for every repeat of the calculation cycle. Therefore, from a computational time point of view, they need to be fast. Also, from a numerical stability point of view, they need to have correct derivatives with respect to the unknown variables (usually the intensive variables). The generation of various types of phase diagrams has been highlighted through two property prediction tools, SPECS and ICAS.
REFERENCES 1. M. L. Michelsen, J. Mollerup, 2004. Thermodynamic Models: Fundamentals & Computational Aspects, Tie-Line Publications, Denmark (www.tie-tech.net). 2. J. de Swaan Aarons, Th. W. de Loos, Phase Behavior: Phenomena, Significance and Models, in Models for Thermodynamic and Phase Equilibria Calculations, Editor: Stanley I. Sandier, Marcel Dekker, Inc., New York, USA, 1993. 3. J. S. Rowlinson, F. L. Swinton, Liquids and Liquid Mixtures, 3rd Edition, Butterworths, London, 1982. 4. Aa. Fredenslund, J. Gmehling and P.Rasmussen, Vapor-Liquid Equilibria Using UNIFAC, Elsevier Scientific Publishing Company, 1977. 5. G. Soave, Chem. Eng. Sci., 27 (1972) 1197. 6. M. L. Michelsen, Fluid Phase Equilibria, 60 (1990) 47. 7. J. M. Prausnitz, R. N. Lichtenthaler, and E. G. d. Azevedo, 1986. Molecular Thermodynamics of Fluid-Phase Equilibria, Prentice-Hall Inc., Engewood Cliffs, N.J. 07632. 8. ICAS Documentations, PEC02-14, CAPEC Internal Report, Technical University of Denmark, 2002. 9. T. L. Nielsen, J. Abildskov, P. M. harper, I. Papaeconomou, R. Gani, J. Chem Eng Data, 46(2001)1041-1044. 10. J. Marrero, ProPred manual, PEC02-15, CAPEC Internal Report, Technical University of Denmark, 2002. 11. T. L. Nielsen, R. Gani, Fluid Phase Equlibria, 185 (2001) 13-20. 12. S. Christensen, SoluCalc Manual, PEC02-17, CAPEC Internal Report, Technical University of Denmark, 2002. 13. R. Gani, ProCamd Manual, PEC03-25, CAPEC Internal Report, Technical University of Denmark, 2002. 14. M. Hostrup, P. M. Harper, R. Gani, Computers & Chemical Engineering, 23 (1999) 13941405.
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Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B. V. All rights reserved.
339
Chapter 14: Application of Property Models in Chemical Product Design Rafiqul Gani, Jens Abildskov and Georgios M. Kontogeorgis 14.1 INTRODUCTION Two types of chemical product design problems are treated in this chapter - pure component process fluids such as solvents, refrigerants, drugs, etc., and formulated products such as solutions, multiphase mixtures and emulsions. In the case of pure component process fluids, the designed chemical product has a particular role in the process such as extract, cool, heat, etc., while in the case of formulated products, additional chemicals are added to achieve (in most cases) a significantly improved product and/or enhanced behaviour that otherwise could not be obtained. In this sense, process fluids such as solvents and refrigerants could also be formulated products. For both types of chemical products, their desired behaviour can be described through a set of properties, classified as essential, desirable and regulatory. Essential properties are related to specific function of the chemical product - for example, as a solvent the chemical product must be able to dissolve while as a refrigerant, it must be able to cool. On the other hand, as part of a formulated (paint) product, a solvent mixture must able to mix with the paint and must be able to evaporate at a desired rate. As part of a drug/pesticide product, the chemicals or solutions must be able to increase the solubility of the product in the carrier. The desirable properties are usually related to the cost of the process and/or operation. For example, in order to reduce the cost of compression, in addition to its essential property (cooling) it needs to have desirable vaporization-condensation curves so that at low pressures and with small amounts of refrigerant, the essential cooling can be achieved. Similarly, in the case of solvent-based extraction, the desirable property would be that it is easy to recover the solvent and that the solvent is able to dissolve selectively if more than one solute is present. The regulatory properties are related to environment, safety and hazards and chemical products must not exceed the threshold values for these properties. While properties help to define the desired (target) behaviour of the chemical product, property models help to design the chemical product by predicting the target properties of the chemical product alternatives. That is, for a generated (design) alternative, the three types of properties are estimated and if they satisfy the specified target values, then the alternative is a (feasible) candidate product. Repeating the procedure generates a list of (feasible) candidate products from which the optimal can be identified by evaluating performance criteria such as cost. Computer-aided techniques for simultaneous generation and test of design alternatives are popularly known as Computer Aided Molecular/Mixture Design, or, CAMD. Important issues related to chemical product design has been addressed recently by Achenie, Gani & Venkatasubramanian , who also provide a collection of CAMD techniques together with their application.
340
The objective of this chapter is to highlight the use of property models in solving various types of chemical product design problems. Emphasis is given on the estimation of properties and their corresponding model rather than the solution of the design problem, examples of which can be found in Achenie, Gani & Venkatasubramanian1. The chapter is divided into three main sections - first a general definition of the property model needs for typical chemical product design problems is discussed. Then two examples of how special property models can be generated/adapted for the estimation of the needed (target) properties are given. The final section highlights the application of the property models for specific chemical product design problems.
14.2 PROPERTY MODEL NEEDS IN CHEMICAL PRODUCT DESIGN Contrary to process design where a set of properties need to be estimated for a fixed number of identified components and their mixtures (solutions), in chemical product design, a similar set of properties may need to be estimated for a wide range of chemicals (species) and their mixtures. In many cases, however, identities of some of the chemicals will be known. Therefore, in terms of components and their mixtures, while in process design a fine-tuned (to the component system of the process) parameter set of a property model needs to safely interpolate, in product design a general parameter set (of wide application range) of the same property model needs to safely extrapolate. With respect to the intensive variables (temperature, pressure and/or composition), however, property models need to safely extrapolate as well as interpolate for both process and product design. Therefore, with respect to the application range, a property model needs to be truly predictive for the needs of chemical product design, at least in the design phase related to the generation of alternatives. In the verification (test) phase, a property model with finely tuned parameters will be more appropriate depending on the availability of experimental data. Consider the following typical solvent design problem - identify a set of solvents that can selectively dissolve phenol from an aqueous mixture that is environmentally acceptable and is easier to recover from the solvent. Note that the identities of two of the components (water and phenol) of the final ternary mixture are known. However, since the third component is unknown, for every candidate, the property model will need to establish a new set of model parameters — since the identity (molecular structure) of the candidate third compound is not known in advance, use of property models that need regressed or pre-calculated molecular parameters is not feasible. Here, uses of predictive models that calculate the molecular parameters from group or atom or connectivity index are more practical. The properties needed to solve this problem are the following: • •
Phase split calculation (must split the feed liquid into two liquid phases - activity coefficient calculation within LLE computation plus phase stability check) - essential Normal boiling point and melting point of the solvent (needed to ensure that the solvent is a liquid at the condition of operation - pure component primary properties) - essential
341 •
• •
• •
Toxicity and other environmental properties of the solvent (must be lower than threshold values - predicted with GC-property models, QSAR based models, retrieved from databases) - regulatory Solubility of phenol in solvent (high solubility - infinite dilution activity coefficient calculation in binary mixtures) - essential Selective solubility of phenol in solvent compared to water (high selectivity is desirable — infinite dilution activity coefficient calculation in binary mixtures) desirable Heat of vaporization of solvent (low value desirable as this would reduce the cost of solvent recovery by distillation - pure component primary property) - desirable Binary azeotrope calculation (azeotrope between solvent-phenol will make the solvent recovery difficult - VLE calculations) - desirable
Alternatively, consider the following formulation design problem: find a single component or a mixture that when added to an aqueous solution of a solid product (drug, pesticide, etc.) in water, increases the solubility of the solid product and thereby, its uptake into the system. The properties needed in this case are the following: •
• •
Miscibility (formulation of dissolved solid, water and added chemical must form a single liquid phase - check first chemicals that are totally miscible in water) essential Solubility (solubility of solid increases as amount of third component is increased SLE calculations) - essential Diffusion (needed for drug/pesticide uptake calculation - diffusion coefficients needed or correlations with respect to octanol-water partition coefficients and distribution coefficients may be applicable) - essential for uptake calculation and increased diffusion coefficient values are desirable
From the above discussion it becomes clear that a large range of property models or a single model with a wide application range (in terms of intensive variables and chemical systems) are needed for reliable solution of product design problems. Also, in the case of relatively small to medium sized molecules, the needed property models may already be available or can be adapted. The ideal type of property models are those that are predictive in nature, such as additive group contribution and topology index based models, since the same groups or indices can be used to generate a very large of molecules whose properties can be predicted with at least qualitatively correct values. The predicted properties are more accurate and the application range is wider for the pure component property models than the mixture property models (because binary group interaction parameters are more likely to be missing than the pure component group parameter). Also, for a number of product design problems, the needed target properties can be calculated from specially developed secondary property models based on known values of solubility, distribution coefficients and/or octanol-water partition coefficients. Therefore, predictive property models for these base properties are desirable and treated in the next section.
342
14.3 PREDICTIVE PROPERTY MODELS FOR BASE PROPERTIES 14.3.1 Distribution Coefficients and Octanol-Water partition coefficients with UNIFAC Introduction Numerous articles have been published on the calculation of distribution coefficients using UNIFAC. Usually applications are based on the idealized ternary LLE situation where the solute 1 is distributed between two liquid phases a and ft containing completely immiscible solvents 2 and 3. The partition coefficient for the distribution of the solute 1 is then expressed as:
K.23.1
7
~
xi
r,
\l)
If only a very small amount of solute is added, so that the solute concentration is very small in both phases, then the activity coefficients are essentially the infinite dilution activity coefficients, so that Eq.(l) can be written as:
&23.I
a
»
«~
»
XI
VZJ
7,2
In many environmentally-related systems, the pollutant species 1 is generally present at very low concentrations, so that Eq.(2) is applicable. This approach simplifies substantially the calculations. A well-established partition coefficient employed in environmental studies is the octanolwater partition coefficient of species i, Kon.j, which is related to the ability of chemicals to bio-accumulate and absorb on soils and sediments (see section 14.5). Kow is defined as: K
_ u / _ xi u _ Y; u _ Y< u o,x, o y,u y, o
^
where C" and C" a r e the concentrations of the chemical i in the octanol-rich phase (o) and in the water-rich phase (w), respectively and C° and Cvv are the total molar concentrations of the octanol-rich and water-rich phases, respectively. The mole fractions can be substituted by the activity coefficients based on the assumption of liquid-liquid equilibrium in the system. Finally, /•'•'" and y,°'"are the infinite dilution activity coefficients in the water-rich and octanol-rich phases, respectively. Use of the infinite dilution activity coefficients is justified by the fact that the distribution chemical is inserted only in very small amounts.
343
As can be seen in Eq. 3, the concentration variables can be readily converted to activity coefficients (at infinite dilution) so that use of a thermodynamic model is feasible for estimating K,,,,,. Equation 3 can therefore be rewritten as1"2:
Kow = 0.151 *4i=-
W
Equation 4 has been derived considering that the water-rich phase is essentially pure water (99.99% water) while in the octanol-rich phase the solubility of water in octanol is 27.5 mol %, which is a result confirmed by most experimental measurements reported for the binary octanol-water system. [Generally, there is some disagreement on this issue, and water solubilities in octanol between 20.7 and 27.5% have been reported]. Before proceeding with the estimation methods, a small parenthesis here is required in order to elucidate why studies of models for estimation of octanol-water partition coefficients (Kow), which will be shown later, are of major social and environmental importance. Kow is related to both the toxicity (via QSPR type of methods) and accumulation of organic compounds in the various ecosystems (soil, sediment, biota) via other partition factors, e.g. soil-water or sediment-water, as discussed in section 14.5. For the living organisms (biota), this partition coefficient is called bio-concentration factor (BCF) or bio-magnification if the increase in the concentration of the pollutant via the food chain should be accounted for as well. Health protection via proper assessment of the effect of existing and new chemicals, design and manufacturing of 'better' (in the sense of environmentally more suitable) products, substitution of existing hazardous chemicals and designing wastewater treatment processes for complex / dangerous / slowly-biodegrading chemicals are some of the cases where partition coefficients, such as the octanol-water one, are of interest. The policies of the European Union and the USA emphasize environmental protection and more specifically, the water quality standards and the determination of the effect (fate, transport, distribution) of toxic and hazardous pollutants in aqueous ecosystems (both water and water-living organisms) - Official Journal of the EU [No L 129/23, 18-5-1976; 1984; 1986; 1988]. The US-EPA (Environmental Protection Agency of USA) has established criteria for maximum permissible concentration of pollutants in the various ecosystems based on octanol-water and other partition coefficients. Due to lack of experimental data and/or reliable models, these criteria cover for the time being a small part of potential pollutants. For example, only a small part of the 219 different chemical compounds in the class of PCBs are included, but in time, criteria for the most important pollutants can be expected to be established. Estimation Methods Up to 15000 experimental Koll, data are available3. The reason for such an abundance of KOT, data is due to its extensive use and its adoption by many international and governmental agencies as a physical property of organic pollutants, which is directly related to the uptake in tissue and fat of living species and thus their potential toxicity. Numerous empirical direct correlation methods specifically designed for Kolr have been proposed. The most widely used, which are based on the group contribution concept, are the methods of Leo & Hansch4"5 (abbreviated as ClogP) and the AFC correlation model6,
344
sometimes also abbreviated as KOWWIN in its computerized form. These methods are based on the fragment or group contribution concept. They have been developed using large databases of KoW data, and they are often (in their most general formulations) very complex. These specific correlations are often capable of distinguishing between isomers and also can take into account, to some extent, the proximity and the intermolecular effects. They are described in an excellent textbook7 and many of these methods exist also in computer programs readily available from the internet. For example, Environmental Science Centre (ESC) of Syracuse Research Corporation (SCR) has a computerized form of the AFC method available on its homepage, (http://esc.syrres.com/interkow/estsoft.htm). Similarly, Daylight Chemical Information Systems, Inc (http://www.daylight.com/release) provides computerized versions of the Leo-Hansch (ClogP), while for the ACD method (which has not been described in the scientific literature) the reader is referred to http://www.acdlabs.com. Recently, Marrero & Gani21, have proposed a rather simple yet accurate higher-order groupcontribution method for the prediction of log Kow. Alternatively, UNIFAC has been widely applied to Kow and the major contributions and conclusions are discussed here. There is a need for a comparative discussion, due to the large number of UNIFAC variants, including additional ones developed specifically for Kow calculations. It should be noted, however, that the applicability of group-contribution models such as UNIFAC for highly polar and asymmetric systems, in particular aqueous solutions, can be rather problematic for several reasons: 1. The UNIFAC expression for the activity coefficient does not explicitly take into account the association effect (hydrogen-bonding) often dominant in such systems 2. Without extra correction factors, UNIFAC may fail in complex systems due to the presence of proximity effects (caused by the presence of many polar groups in a single molecule). 3. The UNIFAC table covers only a small part of the important chemical families to environmental and pollution studies. 4. The group interaction parameters are, in most cases, based on phase equilibrium (VLE & LLE) data at finite concentrations whereas infinite dilution condition conditions are of interest to environmental studies. However, the UNIFAC concept is under continuous development. Several new versions of UNIFAC have been proposed the last years for complex phase equilibria, including versions specifically designed for estimating ^"and Kow8'9'1'15'16. Some important reports related to the applicability of UNIFAC-type models in environmental thermodynamics are listed below: i. Critical evaluations of several UNIFAC-type models in predicting the activity coefficient at infinite dilution in aqueous solutions have been recently reported by Voutsas and Tassios10 and Zhang et al.n. ii. An example of early UNIFAC based octanol-water partition coefficient calculations using Eq.4 is given by Arbuckle12.
345
iii.
Campbell and Luthy13 evaluated UNIFAC for prediction of partition coefficients, KSK?h for aromatic solutes in various organic solvent-water systems. They used the expression:
K
"'~c»-r:.v.
iv. v. vi.
vii.
for systems comprised of water and an immiscible organic solvent. This is useful for solvent screening purposes as well as for assessing the relative distribution of various compounds between water and a given solvent, Kuramochi el al.u recently presented a comparison between four UNIFAC-type models for Kmr Special versions of the UNIFAC model are suitable for estimating water solubilities as well as octanol-water partition coefficients ' . Chen et al} developed a special UNIFAC method (often called water-UNIFAC) for Kow calculations. The only difference from the original UNIFAC-VLE is that special group parameters have been introduced for water systems, i.e. the group interaction parameters containing water have been re-estimated, Another interesting recent development is the so-called KOW-UNIFAC (Wienke and Gmehling16), which was targeted specifically for the estimation of octanol-water partition coefficients.
The KOW-UNIFAC is distinguished from other group contribution models in the sense that the group interaction parameter table is entirely based on a large database of KOT1, and y™ data. Moreover, Wienke and Gmehling16 have used the following equation: ,r — I m /v aw /
fio
In the case of the octanol-water binary, the assumption of completely immiscible solvent species is not valid. As mentioned previously, at T = 298.15 K, octanol and water split into two liquid phases, one which is almost pure water and one octanol-rich phase containing about 27.5 mole-% water. Yet, the calculation procedure of Wienke and Gmehling16 evaluates the octanol-water coefficients simply as the ratio of infinite dilution activity coefficients of solute (1) in the water-rich and octanol-rich phases. Their method is tailored for octanol-water partition coefficients, but the resulting group table is rather limited. Thus, until now this KOW-UNIFAC has had limited application and to only few environmentally significant compounds. A systematic comparison between various UNIFAC models and two specific models for Kow calculations (the AFC model mentioned earlier and a recent method by Lin and Sandier17) has been recently presented18. The following UNIFAC models were considered in this work (see also Chapter 4, Table 1):
346 1. the original UNIFAC VLE-1 (temperature independent parameters obtained from VLE data) 2. the original UNIFAC LLE (temperature independent parameters obtained from LLE data) 3. the new UNIFAC VLE-2 (linear temperature dependent parameters obtained from VLE data - Linear UNIFAC) 4. the modified UNIFAC VLE-3 (logarithmic temperature dependent parameters obtained from VLE and excess enthalpies data - Modified UNIFAC Lyngby) 5. the water-UNIFAC1 (temperature independent parameters estimated from water systems) Using an extensive database, retrieved from the Sangster compilation, the following compound families have been investigated: alkanes, aromatics, cycloalkanes, ethers, alcohols, aldehydes, ketones, acids, esters, amines, and some poly-functional compounds (production chemicals). An overview of the results for complex compounds is given in Figure 1.
Figure 1. Average absolute deviation between experimental and predicted from various models log(KoW) values for complex chemicals (glycols and alkanolamines). Modified from Derawie/a/.18. The following conclusions summarize the basic findings: 1. The AFC correlation performs best for both the mono-functional and the multi-functional compounds 2. The recent model by Lin and Sandier fails for multi-functional compounds, although it performs very well for mono-functional compounds18 3. The performance of the various UNIFAC versions varies a bit, but the water-UNIFAC version by Chen et at., which has special water-based parameters, performs best,
347 especially for multi-functional compounds. Among the other UNIFAC methods, the linear UNIFAC is a bit worse than the others, while the VLE-1, LLE, VLE-3 seems to perform similarly. These conclusions seem to be common for mono- and multi-functional compounds. 4. Typically problems occur for environmentally interesting compounds (potential pollutants), as demonstrated by the results shown in Tables 1 and 2. Again the AFC model performs best. The UNIFAC-LLE is the second best model for these complex chemicals, while the linear UNIFAC is again the worse among these models. Table 1. Experimental and calculated values of log KoW at 298.15 K from the various UNIFAC models and the AFC correlation model19 Solute
N-octane N-tetradecane Pyrene Benzo(a)pyrene Octachloronaphthalene DDT Pentachlorobenzene Decachlorobiphenyl Lindane
log Kow Expt. value 5.15 8.00 5.10 6.06 8.40 6.19 5.17 8.26 3.85
UNIFAC VLE-1 3.59 6.20 5.48 6.84 1.56 5.95 0.93 1.81 4.35
UNIFAC LLE 3.95 6.80 5.88 7.33 4.68 7.20 2.91 5.72 4.51
UNIFAC VLE-2 3.81 6.56 1.47 1.95 4.28 2.32 2.42 5.26 4.17
UNIFAC VLE-3 3.13 5.64 4.60 5.77 3.88 6.97 2.47 4.99 3.63
AFC Model 4.27 7.22 4.93 6.11 8.33 6.79 5.22 10.20 4.26
Table 2. Average absolute deviation between experimental and predicted log Kow values at 298.15 K from the various UNIFAC Models and the AFC Correlation Model19 Solute
AAD (%) UNIFAC VLE-1 N-octane 1.56 N-tetradecane 1.80 Pyrene 0.38 0.78 Benzo(a)pyrene Octachloronaphthalene 6.84 DDT 0.24 Pentachlorobenzene 4.24 Decachlorobiphenyl 6.45 Lindane 0.50 2.53 AAD(%) =
±ho$Kow^'-l°gK 0W ca/J
UNIFAC LLE 1.20 1.20 0.78 1.27 3.72 1.01 2.26 2.54 0.66 1.63
UNIFAC VLE-2 1.34 1.44 3.63 4.11 4.12 3.87 2.75 3.00 0.32 2.73
UNIFAC VLE-3 2.02 2.36 0.50 0.29 4.52 0.78 2.70 3.27 0.22 1.85
AFC Model 0.88 0.78 0.17 0.05 0.08 0.60 0.05 1.94 0.41 0.55
348 Tables 3-5 present Kow results from another recent investigation20 for complex chemicals and phthalates. This investigation has considered all three "direct" computerized methods discussed above (AFC, ClogP, ACD), the UNIFAC VLE-1 and the method of Sandier. Figure 2 shows a plot of experimental and estimated K<,w values for a series of alcohol ethoxylates with 4 alkane groups and varying number of hydrophilic groups. Only two experimental points are available. An interesting conclusion is that, while UNIFAC VLE-1 and AFC follow the correct trend with respect to experimental data, the two other commercial software packages do not, thus being in qualitative disagreement with experimental observations. Similar results have been observed for other families of surfactants as well. There is no problem in the case of Kow plots with respect to increasing hydrophobic (alkane) part. All models exhibit the expected increase of Kow with the alkane chain length. From this analysis, it can be concluded that: 1. The three commercial programs, "direct methods" for octanol-water estimations yield reliable results, similar to each other. 2. The GCS does not perform very well, especially, for alcohol ethoxylates 3. Of the various UNIFAC models, the UNIFAC-VLE 1 performs best 4. The UNIFAC- VLE1 performs similar to the "direct" methods for both families of compounds considered (phthalates and alcohol ethoxylates) 5. Only the UNIFAC-VLE 1 and the AFC methods predict the correct trend of Kow with increasing hydrophilic part for the alcohol ethoxylate surfactants Table 3. logAbw prediction results for phthalates with various estimation methods. GCS is the "solvation" method by Lin and Sandier I7(* all names for phthalates are given in substitute group). Phthalates dimethyl Diethyl dipropyl di-iso-propyl Dibutyl di-iso-butyl dipentyl Dihexyl Dioctyl Didecyl di-sec-octyl ditridecyl Diallyl dutylbenzyl dicyclohexyl Average deviation%
Exp.* 1.6
2.42 3.64 2.83 4.50 4.48 5.62 6.82 8.18 8.83 7.06 8.4
2.98 4.73 4.9
UNIFAC-VLE1 1.74 2.64 3.53 3.53 4.43 4.43 5.32 6.21 8.00 9.79 8.00 12.47 3.39 4.79 6.02
ACD
ClogP 1.56 2.62 3.68 3.24 4.73 4.47 5.79 6.85 8.97 11.08 8.71 14.26 3.11 4.98 5.62
AFC
GCS
1.62 2.69 3.75 3.38 4.81 4.44 5.87 6.94 9.06 11.19 8.69 14.38 3.28 4.99 5.74
1.66 2.65 3.63 3.48 4.61 4.46 5.59 6.57 8.54 10.5 8.39 13.45 3.36 4.84 6.20
-0.08 0.97 2.03 1.73 3.09 2.79 4.15 5.21 7.34 9.46 7.03 12.6 0.93 2.37
12
14
13
13
37
5.0
349 Table 4. logiTow prediction results for alcohol ethoxylates with various estimation methods. Compounds
Abbr.
Exp.*
ClogP
AFC
ACD
GCS
2-methoxethanol 2-ethoxyethanol 3,6-dioxa-1 -octanol Iso-propoxyethanol 2-butoxyethanol 3,6-dioxadecanol 2-(hexyloxy) ethanol 3,6-dioxa-l-dodecanol Average deviation
C1E1 C2E1 C2E2 C3E1 C4E1 C4E2 C6E1 C6E2
-0.77 -0.28 -0.54 0.05 0.8 0.56 1.86 1.7
-0.75 -0.22 -0.15 0.09 0.84 0.91 1.90 1.96 33
-0.91 -0.42 -0.69 0.00 0.57 0.29 1.55 1.28 39
-0.80 -0.27 -0.26 0.08 0.80 0.81 1.86 1.87 22
-1.56 -1.02 -1.89 -0.65 0.03 -0.83 1.09 0.22 311
UNIFACVLE1 -0.83 -0.38 -0.75 0.07 0.51 0.15 1.41 1.04 36
Table 5. log^Tow prediction results for alcohol ethoxylates with various UNIFAC methods Compounds
UNIFAC VLE1 -0.77 -0.83 -0.28 -0.38 -0.54 -0.75 0.05 0.07 0.51 0.8 0.56 0.15 1.86 1.41 1.04 1.7
Exp.
UNIFAC LLE1 -1.51 -1.0 -1.93 -0.53 -0.04 -0.95 0.94 0.03
H2OUNIFAC -0.96 -0.52 -0.88 -0.07 0.38 0.01 1.27 0.90
UNIFACVLE2 -0.78 -0.31 -0.72 0.15 0.62 1.55 0.21 1.14
C1E1 C2E1 C2E2 C3E1 C4E1 C4E2 C6E1 C6E2 Average 287 48 80 36 deviation % CxEy means x hydrocarbon groups and y oxyethylene groups - see Table 4.
UNIFACVLE3 -0.60 -0.06 -0.30 0.47 1.01 2.08 0.77 1.84 135
Figure 2. Experimental and predicted ATow values for C4En (Cheng el al.2<X)
350
14.3.2 Local UNIFAC Model for Estimation of Solubility of Complex Molecules This section highlights the generation of local property models for specific property estimation problems. The problem highlighted in this chapter is the estimation of solubility of a complex solute, carbazole. Since group contribution models such as UNIFAC (see chapter 4) are not expected to handle these problems, the objective is to fit the group interaction parameters so that a "local" UNIFAC model with parameters only for the specific system under study is obtained. Note that, as defined in chapter 13, the calculation of solubility of a solid in a liquid solvent needs the normal melting point and heat of fusion of the solute as well as the activity coefficient of the solute in a liquid solution with the solvent or solvent mixture. The problem solution is highlighted through a sequence of steps, each providing the data needed for the next steps. We start with the drawing of the solute molecule and the representation of the molecule with the appropriate functional groups so that the needed pure component and mixture properties can be estimated. Then the pure component properties are estimated, if experimental values for them are not available. The next step is to compute the activity coefficient with the chosen G1'-model (in this case, UNIFAC) even though all group interaction parameters are available. In the case of a solute that cannot be represented by the presently available functional groups, new groups may be defined. In the case of carbazole, this is not necessary. The next step is to analyze the solubility data and evaluate all the groups and their interaction parameters needed to estimate the activity coefficients of the solute in different binary mixtures with solvents. If all group interaction parameters are available, the activity coefficients and the corresponding solubilities in different solvents can be estimated to make a preliminary estimate of the accuracy of prediction. If the accuracy of prediction and/or some group interaction parameters is not available, a sensitivity analysis is performed on all group interaction parameters to identify the most sensitive pair of group interaction parameters. In the next step, the identified most sensitive group interaction parameters are regressed to match the available experimental data. If all the data cannot be matched adequately, the next most sensitive group interaction parameter pair is selected and the regression step is repeated. The final results of the regression step are a "local" model that has been specially fitted to the available solubility data for the solute. Because of the predictive nature of the group contribution method, many other solvent-solute systems can also be handled through the specially regressed parameters as long as the same groups can represent them and the most sensitive group interaction parameters are the same. The step-bystep algorithm is listed below. 1. Representation of carbazole with the appropriate functional groups 2. Estimation of pure component properties (normal melting point and heat of fusion) needed to estimate the solubility of carbazole in a specified solvent. 3. Solubility data analysis and setting up of the group interaction parameters 4. Estimation of liquid phase activity coefficients with the UNIFAC method 5. Sensitivity analysis of the group interaction parameters 6. Estimation of the most sensitive group interaction parameter that fits the experimental data
351 7. Estimation of solubility of Carbazole in different solvents covered by the regressed parameters from step 6. Step 1 - Represent the solute with the appropriate groups
Figure 3: Molecular structure of Carbazole The Carbazole molecule can be represented by 8 ACH groups, 3 AC groups and 1 ACNH group (which is a new group since it is not available in the original UNIFAC model). The addition of the new ACNH group requires the R- and £>-group parameters (the size and volume parameters, respectively - see chapter 4). This is calculated as follows RACNII
=
RACNII2
+
(RCIB -
Ran) = 0.8333
QACNH
=
QACNHI
+ (QcHi-Qcm) = 0.5080
Step 2 - Estimate the needed pure component properties This is calculated by the Marerro & Gani2' method. Use of a computer program, ProPred (ProPred manual22) that allows the prediction of a wide range of pure component properties through different group-contribution is another option. In this case, ProPred has been used. More details of the Marerro & Gani method can be found in chapter 3. Since experimental values for both melting point and heat of fusion are available, these are used in the solubility calculations. Figure 4 shows some of the calculated values from ProPred with the Marerro & Gani methodStep 3 - Solubility data analysis and setting up of the group interaction parameters The solvents, for which carbazole solubility data are available, are listed in Table 6, which also lists the UNIFAC main-groups representing the solvents and the Carbazole solubility values. Step 4 - Estimation of liquid phase activity coefficients with the UNIFAC method Without any adjustment of the group interaction parameters, the solubility values have been calculated. This involves an iterative calculation where the Carbazole activity coefficient in the liquid phase is calculated for an assumed liquid composition of Carbazole at a fixed temperature until the condition of equilibrium has been satisfied for the assumed liquid composition.
352
Figure 4: Calculated pure component property values for Carbazole with ProPred Step 5 - Sensitivity analysis of the group interaction parameters The most sensitive group interaction parameters are found through a sensitivity analysis. This is done by perturbing the parameter value by ± 5 % and evaluating the resulting activity coefficient value. Alternatively, if analytical derivatives are available (for derivative of activity coefficient as a function of group interaction parameter), this can be very easily estimated. For carbazole-solvents. the most sensitive group interaction parameter pair has been found to be the pair involving the main-groups ACNH-CH2 (see Figure 5). Step 6 - Estimation of the most sensitive group interaction parameter that fits the experimental data The most sensitive main-group parameter has been fitted and the resulting estimation of solubility parameters as a function of solubility parameter is illustrated in Figure 6. It can be noted that except for a small number of solvents (solubility parameter values in the range of 15-16), most of the other solubility data have been fitted quite accurately. This indicates that even if a model may not be suitable for a generic application over a wide range of compounds, for a specific problem with well defined application range, it may be possible to adapt the model for a "local" application.
353 Step 7 - Estimation of solubility ofCarbazole in different solvents covered by the regressed parameters from step 6. Final verification of the model performance is carried out by comparing the performance of the fitted model against solubility data not used in the regression as well as qualitative evaluation of the SLE phase diagrams.
Table 6. Solubility data for Carbazole no. 1 2
C;LS no.
111-66-9 124-18-5 3 67-W5-3 4 540-S4-1 5 64-17-5 6 U1-78-G 7 342-18-7 344-10-5 9 111-83-3 1Q 2425-543 11 109-99-9 12 2<)0(i3-28-3* 13 112-40-3 14 123-i)l-l 15 71-43-2 1(> 110-82-7 17 61)3-65-2 18 123-86-4 19 71-3(5-3 21) 3178-22-1 21 G7-W-1 22 71-23-8 23 142-82-5 24 344-76-3 25 110-34-3 26 10046-3 27 7732-18-5 28 142-1)6-1 l 29 2 J2-l54-8 31) 78-93-3 31 108-87-2 32 (iO-29-7 33 111-01-3
Solvent iiiiiiu1
Solubility (mot/atol) n-Octiuic 0.000198 li-Dcciiuc 0.000898 Oh [uniform 0.1)037 2.2.4- trit't h y t] >< «rti u« • 0.000126 Eihitiuil aOO358 Ethyt-ncftiitf 0.02 ryelolicxiinc. <:l»l
(].1)(M)183
<)i-ll-JH'lltyl CtlllT
0.1)1)364
u-Bntyl iu-rt.it(> 1-lmtuiml t - B l ity Iry cl< >h roculie
1). Ill 63
aeetone 1-Proj>nn<jl li-UcptulK' li-Hi1Xii(l(>c;ili<1 ii-H(>j;au(*
Alii sole Water Butyl ctlitT Cyelooctaue N[<-tijyl cHiyl ki-touo Mftliyl i yclolirxiiiic Dit'tliyl i>th(T T<'trnc*)s;iu*"
(],()(U44
0.1)1)025 0.(1469 0.00365 (].()(M)173 ().1)(H)414 0.1100139 0.1)13 9.SE-S Q.00601 a000309 0.0424 0.000196 0,1)129 0.1)(M)tl71
L'NIFAC main groups CII2 CII2 CC13 CH2 CH2 Oil CCOO CH2 CH2 CCl CH2 CCl CH2 CCl C3I2 CCl CH2O CH2 CH2 Oil CH2 CH2O CH2 ACH CH2 CH2C) CH2 CCOO CH2 CH2 OH CHJ CH2CO CK2 CH2 OH CH2 CIIJ CH2 ACH CH2O H2O CH2O CH2 CH2 CH2CO CH2 CH2 CII2O CH2 CH2
354
Figure 5: Sensitivity analysis of the UNIFAC group interaction parameters
Figure 6: Plot of solubility versus solubility parameter after UNIFAC model parameter estimation
355
14.4 PRODUCT DESIGN EXAMPLES 14.4.1 Example 1: Assessment of the miscibility of plasticizers in PVC Dioctyl phthalate (DOP) is widely used as plasticizer in PVC, one of the most important plastics. The role of plasticizers is to make the material "soft", by decreasing the glass transition temperature of PVC. However, suitable plasticizers should be completely miscible in PVC, otherwise migration of the plasticizer to the surface of the material can occur. Such problems have been observed in practical situations and have initiated discussions on the possible dangers from the plasticizers. Step 1 - Problem definition To determine whether DOP is miscible with PVC at different conditions (temperature, DOP concentration). Step 2 - Property model selection The various techniques, discussed in chapter 7, can be used for "screening of suitable solvents for polymers". These are based on solubility parameters, the Hildebrand and Hansen ones, the Flory-Huggins parameters and the infinite dilution activity coefficients. Another possibility is to perform full phase diagram (LLE or SLE) calculations for PVCDOP with a suitable predictive model such as Entropic-FV or UNIFAC-FV (Chapter 7), but such an approach should be first tested for similar systems, since no experimental thermodynamic data for PVC-DOP are available. The following tables (Tables 7-8) provide values for solubility and FH parameters for the PVC-DOP systems as extracted from various sources, mostly handbooks. Different values have been reported for the FH and the solubility parameters. Su el al2^ have reported the concentration dependence of the FH parameter for the PVC-DOP system. Table 7. Solubility parameters for PVC and DOP (all values are in Hildebrand units = (cal/cm3)172) Compound
dispersion polar
hydrogen bonding
DOP-1 DOP-2 PVC-1
Total Value 7.94 8.92 9.314
8.14
3.2
1.52
PVC-2 PVC-3
11.03 10.49
9.4 8.91
4.5 3.68
3.5 4.08
R-Hansen
Source
3.2 1.7
Barton.83 Barton,83 Barton,90 preferred Hansen-1 Hansen-2
Step 3 - Model validation and parameter estimation It is difficult to validate the model since there are no direct thermodynamic data for PVCDOP (The FH parameters can be considered as indirect thermodynamic data). However, one way to validate the results will be by comparing with each other the various procedures -
356 "rules of thumb", based on the different type of information (solubility parameters, FH parameters, activity coefficients). Table 8. Flory-Huggins parameters for the PVC/DOP system from different sources Reference T(°C) Doty, P. M., Zable, H. S., 1946. J. Polym. 53 Sci., 1:90 76 Anagnostopoulos, C. E, Coran, A. Y., 116-118 Gamrath, H. R., 1960. J. Appl. Polym. Sci., 4: 181 Bigg, D.C.H, 1975. J. Appl. Polym. Sci., 19:3119 Patel, S. V., Gilbert, M., 1986. Plast. 113-114 Rubber Process Appl., 6: 321.
FH parameter 0.01 0.03 -0.03 0.06 - extrapolated from swelling data 0.05 -0.13
Step 4 - Property estimation Table 9 shows the difference in the solubility parameters, both in terms of the Hildebrand and Hansen ones. Table 9. Difference of the solubility parameters between PVC and DOP PVC
DOP
PVC-1 (preferred) PVC-1 (preferred)
DOP-1 DOP-2
Difference in solubility parameters 1.4 0.4
PVC-2 PVC-2 PVC-3 PVC-3
DOP-1 DOP-2 DOP-1 DOP-2
3.1 2.1 2.6 1.6
(Hansen-1) (Hansen-1) (Hansen-2) (Hansen-2)
According to the two "solubility parameters-based" rules of thumb, miscibility is expected if:
h-*,isi-i.(^r
r4)
I4)
\cm~ J
V 4 fo. -S d 2 f + {Spl -Sp2 f + (Shl -Sk2f
(5)
where R is the Hansen-solubility parameter sphere radius. Eq. 4 is strictly valid for nonpolar/slightly polar compounds, while the second is more general.
357
We first observe that there are many "experimental" values on solubility parameters for both PVC and DOP and both for the total (Hildebrand) values as well as for the Hansen increments (dispersion, polar and hydrogen bonding). Apparently, there is some confusion on the correct solubility parameter values that should be used and this should be taken into account in our evaluation. Using the different combinations in Table 7 we can construct the results as shown in the Table 8 above. It is clear that in most combinations, according to rule of equation 1, complete miscibility of DOP and PVC should be expected although maybe on the border. This is further confirmed using the Hansen parameters together with equation 2, where the Hansen distance is approximately equal to 3 (3-3.4), which is actually a bit higher than the solubility radius in one case (R=3.2) and much higher in the other case (R=1.7). Thus, in conclusion, the solubility parameter methods indicate in most cases miscibility (at 25 °C) will exist between PVC and DOP. On the other side, it could be that miscibility between PVC and DOP is only in the borderline and when some factors are changed (amount of DOP, temperature, etc.), the system may enter the immiscibility region. Thus, use of additional "rules of thumb" would be useful. Miscibility based on Flory-Huggins parameters According to this rule, miscibility is expected for low values of the FH parameters, typically whenj ] 2 <0.5. Generally, the lower the Flory-Huggins parameter value, the greater the miscibility, or in other words the greater the solvency of the chemical. Values much greater than 0.5 indicate non-solvency. We are given two sources of the Flory-Huggins parameter for the PVC/DOP system, Table 8 and the plot presented by Su et alP showing the dependency of the FH parameter for the PVC-DOP system with respect to concentration. This is done in one of the few such investigations reported. The different FH parameter values shown in Table 2 have been obtained from different experimental methods e.g. swelling studies and melting point depression studies and possibly at different plasticizer concentrations. The conclusion is that, at these conditions and at the rather high temperatures involved, the reported FH parameters are rather low, often negative, and always far below the critical value of 0.5. This indicates, according to the FH-based rule of thumb, complete miscibility of PVC in DOP. It should be reminded that this conclusion is valid only if the assumptions led to the FH-based rule of thumb are also valid. The most important such assumption is that the FH parameter is independent of concentration, which may not be true especially for high plasticizer content. This is verified by the work by Su et al.2* who have presented a plot of the FH parameter as a function of the concentration. These investigators showed that the FH parameter increases dramatically as the plasticiser content increases, especially above concentrations of 40-60%. It could be the case that this indicates dangers of thermodynamic instability in the PVC/DOP system at high DOP concentrations. The investigation was made at rather high temperatures (110-130 °C). In Chapter7, results with the last rule of thumb, based on infinite dilution activity coefficients (miscibility when Q™ < 6 ) are presented.
358 Conclusions & Perspectives All the above "rules of thumb" have some theoretical foundation. The limiting values, however, (1.8, R, 0.5, 6, see Chapter 7), should be at best considered as "semi-empirical" verified by many investigations. Thus, when the parameters involved in these rules for a chemical and a polymer i.e. the solubility parameter differences, the FH parameter and the activity coefficient at infinite dilution are rather close to these "limiting values", it is rather difficult to conclude on miscibility. For the specific case of PVC-DOP we conclude that: 1. Using the information given, it may be difficult to arrive to definite conclusions due to lack of DIRECT thermodynamic (phase equilibria) data for PVC/DOP or similar systems of any type (activities, activity coefficients, LLE, etc.). 2. Based on the available data (FH and solubility parameters), it may be with some reservation concluded that: i. PVC and DOP are miscible at low (ambient) temperatures and low DOP concentrations. However, the miscibility seems not be as 'obvious' as could be anticipated from the extensive use of DOP ii. Miscibility problems may occur at higher temperatures and especially at higher DOP concentrations It is worth noticing that many commercial PVC-based products with DOP concentrations up to 60% are available in the market. 14.4.2 Example 2: Choice of Suitable Mixed Solvents in the Paint Industry In the paints and coatings industry, mixed solvents are often employed, while many polymers are not well-characterized in the sense that their structure and molecular weight may not be known. The purpose of this example is to study the miscibility of two commercial resins (Desmophen, Epikote 1001) in a hydrocarbon and an alcohol solvent as well as in their mixtures with the purpose of selecting a suitable mixed solvent that can dissolve these commercial paints. Step 1 - Problem definition To determine whether the two commercial resins are miscibile in two typical solvents such as xylene and n-butanol as well as their mixtures. Step 2 - Property model selection Fast calculations for polymers which are not well-characterized can be performed if their solubility parameters, esp. the Hansen ones, are known together with the solubility radius. In addition the solubility parameter of a mixed solvent is given by the equation:
S = 2>,o, i
where
359 For the two commercial resins stated, the polyester resin Desmophen and the epoxy resin Epikote 1001 and for two potential solvents e.g. n-butanol and xylene, the Hansen solubility parameters are available from the literature and they are given in Table 10. Table 10. Hansen solubility parameters for the compounds of interest Compound n-butanol Xylene epoxy resin polyester resin
8
7.8 8.7 9.95 10.53
P
2.8 0.5 5.88 7.30
8, 7.7 1.5 5.61 6.0
The radius of solubility of the polyester resin has been experimentally determined to be equal to R=8.2 and that of epoxy resin is R=6.20. All solubility parameter values are in Hilderbrand units. Step 3 - Model validation and parameter estimation It is difficult to validate the model unless direct solubility experiments will be carried out. Step 4 - Property estimation The basic equation which determines the solubility range according to Hansen method is:
V 4 ^ * " s d2Y + foi "S p 2 ) 2 + (Shl -Sh2f
s = I>,s, for estimating each one of the solubility parameter increments of the mixed solvent. Thus, we have:
360 g d =£cp,.8 d/ =0.2x7.8 + 0.8x8.7 = 8.52 / 8 p =X ( P, 8 p, =0.2x2.8 + 0.8x0.5 = 0.96 8A = £9,8,,,. =0.2x7.7 + 0.8x1.5 = 2.74 Using again the Hansen equation we have for the system Desmophen-mixed solvent 80% xylene / 20% butanol: ^4(8.52 -10.53) 2 +(0.96-7.30) 2 +(2.74-6.00) 2 = 8.18 < 8.2 thus the polyester resin is (on the limit) soluble in the mixed 80% xylene - 20% butanol solvent. Similar calculations can be made e.g. for the system Epikote-mixed solvent 55% xylene / 45% butanol We can use for the liquid solvent mixture of 55% xylene / 45% butanol the same equation:
S = l9,S, for estimating each one of the solubility parameter increments of the mixed solvent: Stf =^cp,8 t f =0.45x7.8 + 0.55x8.7 = 8.295 8 p =YJ(?i8pi
=0.45x2.8 + 0.55x0.5 = 1.535
8h = ^ ( p . 8 w =0.45x7.7 + 0.55x1.5 = 4.29 Using the Hansen equation we obtain for the system Epikote-mixed solvent 55% xylene / 45% butanol: ^4(8.295-9.95) 2 +(l.535-5.88) 2 +(4.29-5.6l) 2 = 5.62 < 6.2 thus the epoxy resin is soluble in the mixed 55% xylene - 45% butanol solvent. Discussion The two resins are quite polar and have also a large value of hydrogen bonding solubility parameter. Thus, as expected, they are not soluble in the non-polar xylene, which has a value of hydrogen bonding solubility parameter much lower than that of any of the two resins. However, the mixed solvent xylene/butanol has much higher polar and especially hydrogen bonding solubility parameter increments than xylene. Thus, the resins are soluble in suitable mixed solvents containing sufficiently high percentage of n-butanol. The system 80% xylene/20% butanol provides possibly only limiting/limited solubility for the polyester resin since the value of the Hansen difference is nearly equal to that of the radius of solubility.
361 However, as shown by the calculations in the third case, as we increase the amount of the polar butanol in the mixed solvent, the solubility tendency is clearly increased. 14.4.3 Example 3: Choosing Alternative Surfactants for Stabilizing an Emulsion Emulsions are typically stabilized with mixed surfactants, having both hydrophilic and hydrophobic characteristics. One of the key concepts in surfactant science and very important in choosing suitable emulsifiers is the so-called HLB (Hydrophilic - Lipophilic Balance), an empirical parameter introduced by Griffin, which however is directly related to thermodynamic parameters such as the solubility parameter, the octanol-water partition coefficient or the critical micelle concentration (CMC), all of which can be estimated from thermodynamic models. Both HLB and Kow are both measures of the amphiphilic (hydrophilic/hydrophobic) characteristics of a compound. Interestingly enough, Kow is mostly used in environmental science and drug design, while HLB is used in the emulsion field. Actually, in the surfactant science alone (irrespectively of emulsions) a third "geometric" measure of the hydrophilic/lipophilic character is often used, the so-called CPP (critical packing parameter). An example of the HLB-CMC relationship is given in Figure 7 for polyoxyethylene alkyl sulfates (Cheng20), but similar relationships have been proposed / can be easily developed for other surfactant families. Relationships of Kow with HLB are also very common.
Figure 7: CMC of the polyoxyethylene sodium alkyl sulfates as a function of the corresponding HLB values How can the HLB be used in emulsion stability analysis and for choosing suitable surfactants as emulsifiers? As example, we take a particular product that is an o/w (oil-inwater) emulsion and is based on equal amounts (on weight percent) of two surfactants, which are both hydrophilic. The first surfactant has a HLB=13 and the other has a HLB=11. The company, which manufactures the particular emulsion product wishes, for some reasons, to change the formulation of the surfactants used for the stabilisation of this o/w
362 emulsion product. Of course the product must have the same o/w emulsion characteristics. Different reasons could exist for such a decision e.g. one of the surfactants turned out to be hazardous or suddenly became very expensive or is simply out of production and cannot be supplied any more. One idea was to use this time a mixture of hydrophobic and hydrophilic surfactants, both of which are non-ionic and belong in the family of polyoxyethylene alcohols (polyethylene oxides). Two possibilities are the surfactants Q2E30 and C16E6, both of which seem to be widely available and not very expensive. This family of non-ionic surfactants has generally the formula:
CxEy
=CH3(CH2)x_l(OCH2CH2)yOH
Before starting detailed experiments, it would be desired to know how much (weight) percentage of the new surfactants is needed for this specific o/w emulsion. Design The design will be based on the HLB parameter. The HLB of the (original) o/w emulsion product is: HLB = £w;HLB; =0.5*13+0.5*11=12 / We want that our new product has the same o/w emulsion characteristics, which among other factors, it also implies that it should also have the same HLB number. Actually, this argument is crucial in emulsion design. In this case, for the polyoxyethylene alcohols (polyethylene oxides), which are wellknown non-ionic surfactants, a thermodynamic method e.g. via using a relationship of HLB with CMC or Kow may not be necessary. The group-contribution method of Davies-Rideal, as explained in many textbooks (Shaw) can be used.
HLB = 7 + ZniHLBfYdroPhlllc + i
^HLB^^0^
i
where the sum in the first term is over all hydrophilic groups (positive group parameters) and the sum in the second term is over all hydrophobic groups (negative group parameters), and n; is, respectively, the number of hydrophilic and hydrophobic groups. The necessary group parameters are provided in Table 11:
363 Table 11. Group parameters for calculation of HLB Group -OH -CH2-,CH3-, etc. -(CH2-CH2-O)-
HLB value 1.9 -0.475 0.33
For Ci2E30 we have: HLB = 7 + 12*(-0.475) + 0.33*30+1.9=13.1 For C]6E6 we have: HLB = 7 + 16*(-0.475) + 0.33*6+1.9=3.28 Since HLB=wiHLB]+w2HLB2=12 and 12=Wi*13.1+(l-w,)*3.28 we get that wi=0.8879, that is, amount of the new hydrophilic surfactant should be 88.79 %.
14.5 BASIC RELATIONSHIPS IN ENVIRONMENTAL ENGINEERING Many thermodynamic calculations in environmental engineering can be performed using a few basic thermodynamic quantities, which are the vapor pressures of chemicals, their water solubility (or infinite dilution activity coefficients), the Henry's law constant and the octanolwater partition coefficient Kow. The latter is related to several other important partition coefficients, which determine the concentration of long-lived chemicals in the various environmental compartments, air (A), water (W), soil and sediment (S) and biota (B), the latter including also the living organisms and humans. These calculations are possible for long-lived chemicals, where the assumption of phase equilibria is possible. The basic relationships of environmental engineering giving the fugacities of the chemicals in the various ecosystems and the role of octanol-water partition coefficients are provided here. 14.5.1 Fugacity of a Chemical in Air This is the most well-established fugacity in environmental engineering. At environmental conditions, air is an ideal gas, thus the fugacity of a chemical is equal to the partial pressure:
f,A=x?P = \^y
= C?RT
(6)
where XjA is the mole fraction of chemical i in the air, Q A is the molar concentration of the chemical and CA is the molar concentration of the air. R is the ideal gas constant and T is the absolute temperature. Thus, the distribution of a chemical between air and water can be obtained starting from the equilibrium equation:
364 f,w = f* =s> C?H° = C*RT => _ C , A _ H , C _ 0.2165/
(7) ,
a
where T is in K and the vapor pressure in atm. coefficient.
KAW.J
is the most well-defined partition
14.5.2 Fugacity of a Chemical in Water The fugacity of a chemical in water is:
= xf/1JPisat
=xf/Hi=CfHf
(8)
where: x;w is the mole fraction of the chemical i in water; fj° is the pure component fugacity, which can be replaced by the pure vapor pressure PjSal; y, is the activity coefficient, which is the key property in this calculation. Since in most cases of environmental interest, the concentration of the chemical (pollutant) in the aqueous phase is quite low, the activity coefficient can be replaced by its value at infinite dilution y™ as shown in Eq. 8. As can be seen by Eq. 8, if the infinite dilution activity coefficient of a species in water is known as well as its vapor pressure, its fugacity in rivers, lakes and the oceans can be estimated. For many chemicals, the Henry's law constant (Hj or H\~) is known instead of the activity coefficient at infinite dilution and the vapor pressure. The infinite dilution activity coefficients of many chemicals in water can be very high, which indicates that they are almost insoluble in water; or in other words that they are highly hydrophobia Other chemicals with lower values of water activity coefficients are less hydrophobic (more hydrophilic). Vapor pressures values of several chemicals are found in several handbooks. For environmental chemicals, which are often heavy complex compounds, experimental data are not available and need to be estimated with predictive methods. A recent compilation of vapor pressures for environmentally important compounds is provided by Site . 14.5.3 Fugacity of a Chemical in Soil and Sediment The fugacity of a chemical in soil/sediment as well as in biota (see later) cannot be estimated directly. They are instead calculated via the knowledge of the distribution of the chemical between soil/water and biota/water respectively. We first define the distribution (ratio) of a chemical between soil/sediment and water:
365 K
Cs d = sw =7^r K
'
(9)
Ps
where CjVV and Q s are the concentrations of the chemical in water and soil/sediment, and p s is the density of soil/sediment (in e.g, kg/dm ). From Eq. 9 it can be seen that the distribution coefficient Kd is dimensionless since both concentrations are expressed in the same units (e.g. mg/dm3), while K
f* =C?H< J-tlOc?
(11)
[KdPs) 14.4.4 Fugacity of a Chemical in Biota Similarly as in the case of the fugacity of a chemical in soil/sediment, we first define the distribution (ratio) of a chemical between biota and water. This defines the important concept of Biocencentration Factor (BCF). BCF =
Kb=Kb!w=^ (12)
Kb=^ where Q w and Q b are the concentrations of the chemical in water and biota, and pb is the density of biota (in e.g, kg/dm ). From equation (12) can be seen that the distribution coefficient Kb is dimensionless since both concentrations are expressed in the same units (e.g. mg/dm3), while Kb' has dimensions e.g. dm /kg. All these are in equivalence to what was stated for the soil/sediment-water partition coefficients. Assuming that equilibria is achieved in the distribution of the chemical between biota and water, we have:
366
(13)
C?H?=x?1>! Combining equations (12) and (13) we have:
/f =CrH, c =f-^Ucf
(14)
{K'bPbJ 14.4.5 The Octanol-water and other Partition Coefficients The fugacities of chemicals in soil/sediment and biota can be estimated via equations 11 & 14 provided that the distribution coefficients Kd and BCF (=Kb) are known. These distribution coefficients are difficult to assess quantitatively and to be measured but to a first approximation they are related to the octanol-water partition coefficient Kow, which, as stated, is a measure of the hydrophobicity of a chemical. The reason that a correlation between Kd and BCF with Kow exists is due to the fact that noctanol is considered to be a good measure for the lipid-organic part (phase) of living organisms (biota) and of the organic part of soil and sediment. It is assumed that the organic pollutants (chemicals) preferentially partition only into the organic matter of soil/sediment and the lipid part (fatty tissue) of the biota. We assume that these chemicals enter mainly the fatty tissues of fish and people rather than muscles, other tissues or skeletal structures. Octanol-water partition coefficients (Kow) are widely considered today to be a useful measure of the accumulation of pollutants / chemicals by aquatic organisms. Kow-values range from below unity, in the case of hydrophillic compounds (e.g. triethylene glycol) up to several million, in the case of very dangerous compounds. For chemicals with log(Kow) > 4, it is believed that they constitute dangerous pollutants which can be transferred through the food chain to higher organisms22. In this case, it is said that these chemicals "biomagnify" or "bioaccumulate", i.e. their concentration is increased as they are transferred via the food chain. For example, the well-known insecticide DDT with log(Kow) almost 6.2 has been found in fish, higher birds such as eagles and even human beings. While the mechanism for food bioaccumulation is not completely understood, the important point is that the extent can be directly correlated with the value of the octanolwater partition coefficient. Examples of Kow values for typical commercial chemicals are: KoW(pyrene) = 135000, Kow (Aldrin) = 3160000, Kow (biphenyl) = 12300 and KoW (ppDDT) = 2344000. The important chemicals-pollutants of interest to environmental science are PAHs (polynuclear aromatic hydrocarbons), PCBs (polychlorinated biphenyls), heavy metals (Cr, Ni, Fe, Cu, Zn, Cd,..) and inorganic salts. Due to this significantly large range of values (over ten orders of magnitude: 10"3- 107 !) and other factors as well (e.g. the important influence of isomerism on Kovv, etc.), knowledge of KQW is a non-trivial problem especially for new structurally complex chemicals and either experimental measurements or reliable estimation methods are required.
367 Returning to the relationship between KoW and the various distribution coefficients, the following empirical equations have been proposed: BCF =
fbKow=0.05Kow
Ksw=fOoKoc
(15)
K-sdw = 'oc^-oc
where fj, is the lipid fraction in biota. For aquatic organisms e.g. fish it is often assumed that ft is approximately 5%, but other correlations have been proposed of the type: \ogBCF = a + blog(Kow)
(16)
where a and b are constants depending on the specific biota system considered. The Kb distribution ratios (for soil and sediment) are functions of KoC, the organic carbon partition coefficient which is also considered to be a function of KoW e.g. Koc = 0.63 Kow (for soil) and KoC = 0.53 Kow (for sediment). [Other correlations using slightly different constants have been proposed] The fractions of organic carbon in soil and sediment are also typically different e.g. foc is assumed to vary between 0.2-3% with most typical value equal to 0.02 and foc typically around 5% (=0.05). Using these assumptions, the distribution ratios involving soil and sediment can be given also as functions of Kow: Ksw
=foc0.63Kow=0.02x0.63xKow (17)
Ksdw =foc0.53Kow
= 0.05x0.53xK o w
Similarly relationships have been proposed for the distribution coefficients of chemicals between water and suspended matter (Kmw) and water and animal biota (Kvw): Kvw =/v0.41K ow = 0.2x0.63xKOM, (18)
Kmw
=fm0A\Kow=0A7x0.4lxKow
From equations 15-18, we understand that all partition coefficients can be expressed -to some approximation- as a function of the octanol-water coefficient. Thus, K<,w is the key property in environmental engineering. Several approximate relationships between Koc and Kow have been reported in the literature. Sandier • indicates that all these simple relationships should be used with care since soil is an extremely complex heterogeneous material and under the single heading of soils one finds materials ranging from clays to sand. Moreover, some soils e.g. clays have quite different physical and chemical properties depending on whether they are wet or dry. Apparently, all these important complexities are ignored when the soils are characterized simply by their organic carbon content.
368 LIST OF SYMBOLS C Kjj# Kow TV T Vj Xj yt
concentration (mol/1) distribution coefficient of compound a among solvents i and / octanol-water partition coefficient (total) number of compounds temperature in K liquid molar volume of compound /' in a solution liquid phase mole fraction of compound i gas/vapour phase mole fraction of compound i
Greek Letters a, p
fluid phases
y,
activity coefficient of compound i
Subscripts i
compound i
ow
octanol-water
Superscripts cal expt QO o w Abbreviations AAD AFC BCF EPA KOW LLE VLE QSPR
calculated experimental infinite dilution octanol water Average absolute deviation Atom Fragment Contribution Bioconcentration factor Environmental Protection Agency Octanol-water (method / partition coefficient) Liquid-liquid equilibria Vapour-liquid equilibria Quantitative structure property relationships
REFERENCES 1. F. Chen, J. Holten-Andersen and H. Tyle, Chemosphere, 26 (1993) 325.
369 2. W.J. Lyman, W.F. Reeh and D.H. Rosenblatt, Handbook of Chemical Property Estimation Methods. Environmental Behavior of Organic Compounds, American Chemical Society; Washington, 1990 3. J. Sangster, J. Phys. Chem. Ref. Data, 18(3) (1989) 1111. 4. A. Leo, C. Hansch and D. Elkins, Chem. Reviews, 71 (1971) 525. 5. C. Hansch and A. Leo, Substituent constants for correlation analysis in chemistry and biology. Wiley, New York, 1979. 6. W.M. Meylan and P.H. Howard, J. Pharm. Sci., 84 (1995) 83. 7. J. Sangster, Octanol-Water Partition Coefficients: Fundamentals and Physical Chemistry, Wiley, New York, 1997. 8. S. Banerjee and P.H. Howard, Environ. Sci. Techn., 22 (1988) 839. 9. A. Li, W.J. Doucette and A.W. Andren, Chemosphere, 29 (1994) 657. 10. E.C. Voutsas and D.P. Tassios, Ind. Eng. Chem. Res, 35 (1996) 1438. 11. S. Zhang, T. Hiaki, M. Hongo and K. Kojima, Fluid Phase Equilibria, 144 (1998) 97. 12. W.B. Arbuckle, Environ. Sci. Technol., 17 (1983) 537. 13. J.R. Campbell and R.G. Luthy, Environ. Sci. Technol., 19 (1985) 980. 14. H. Kuramochi, H. Noritomi, D. Hoshino, S. Kato and K. Nagahama, Fluid Phase Equilibria, 144(1998)87. 15. A.T. Kan and M.B. Tomson, Environ. Sci. Technol., 30(4) (1996) 1369. 16. G. Wienke and J. Gmehling, Toxicological and Environmental Chemistry, 65 (1998) 57. 17. S-T. Lin and S.I., Sandier, Ind. Eng. Chem. Res., 38 (1999) 4081. 18. S.O. Derawi, G.M. Kontogeorgis and E.H. Stenby, Ind. Eng. Chem. Res., 40 (2001) 434. 19. J.-S Cheng, Internal report, Institut for Kemiteknik, Technical University of Denmark, 2001. 20. H.Y. Cheng, G.M. Kontogeorgis and E.H. Stenby, Internal report, Institut for Kemiteknik, Technical University of Denmark, 2003. 21. J. Marrero, R.Gani, Fluid Phase Equilibria, 183-184 (2001) 183 22. J. Marrero, ProPred Manual, PEC02-15, CAPEC Internal Report, Tech Univ Denmark, 2002. 23. Su, Patterson and Schreiber, J. Appl. Polymer Sci, 20 (1976) 1025. 24. A. Site, J. Phys. Chem. Ref. Data, 26(1) (1997) 157. 25. S.I. Sandier and H. Orbey, Fluid Phase Equilibria, 82 (1993) 63. 26. S.I. Sandier, J. Chem. Thermodynamics, 31 (1999) 3.
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Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B. V. All rights reserved.
371
Chapter 15: Computational Algorithms for Electrolyte System Properties Rafiqul Gani and Kiyoteru Takano
15.1 INTRODUCTION In this chapter, computational algorithms for calculation of phase diagrams and for the creation of problem specific property models, especially suitable for electrolyte systems, are presented. In section 15.2, a collection of algorithms for analysis of solubility and various phase equilibriumbased computations are presented together with illustrative examples. The phase equilibrium computations cover vapor-liquid, liquid-liquid and solid-liquid equilibrium (electrolyte) systems as well as solid-liquid-vapor and solid-liquid-liquid systems. In section 15.3, an algorithm for the creation of "problem specific" property model package is presented together with illustrative examples. The property models are activity coefficient models, which also require additional properties and their models when used in various types phase equilibrium computations. In section 15.4, a model for a multiphase separation is presented. All the algorithms and models in this chapter build on the background theory and the property models described in chapter 8 for electrolyte systems.
15.2 COMPUTATIONAL ALGORITHMS This section presents a collection of algorithms for solubility analysis, and various types of phase equilibrium computations. All the computations in the illustrative examples have been performed through ELEC-ICAS, which is a toolbox in ICAS [1], for calculations involving electrolyte systems. The appendix A of this chapter shows some of the main features of ELEC-ICAS. More details of ICAS can also be found in chapter 13. 15.2.1 Algorithm for Solubility Analysis The effect of temperature and solution pH on the solubility of an organic/inorganic salt can be calculated using the solubility analysis algorithm. This algorithm is only used for analyzing the solubility of the salt in one solvent (i.e. a binary system). When multiple solvents need to be considered, the algorithm for ternary phase diagram computation involving two (or more)
372 solvents and one salt is used. For multiple salts systems, the analysis is repeated for each salt. The main steps in this algorithm are given below. Step 1: Specification of system and type of calculation Specify the electrolyte system and type of calculation (calculate the effect of temperature or of pH). Determine the range of either temperature or pH and the number of data points to be calculated. For calculation of the effect of pH, HC1 and NaOH are selected as acid and base, respectively, to change solution pH (other components can also be used). Step 2: System Classification Classify the electrolyte system, in terms of the number of salts present, into two groups: a single salt system or multiple salt systems. Create multiple-salt compounds or complexes from information on the identified salts. Step 3: Generation/solution of the solubility model equations Generate the equations for electroneutrality, molecular dissociation, solid-liquid equilibrium and atomic balances (see section 15.2.2 "solid-liquid equilibrium phase diagrams" for the model equations). If multiple solids need to be considered, first, the temperature and type of solids coexisting at the invariant point (where one liquid phase and two solid phases coexist) need to be specified. According to this information, the calculation range specified in step 1 may be divided into regions, because the solids that will precipitate below and above the temperature of the invariant point are different. The algorithm identifies the type of solid that will precipitate in each region. To calculate the effect of solution pH on the solubility of salt, first, the solubility of the salt at natural pH is calculated, and it is used for subsequent calculations as the initial value. When pH needs to be lowered (stronger acidity), HC1 is added to get the desired pH, while when pH needs to be increased (stronger base), NaOH is added. In this algorithm, how much HC1 or NaOH needs to be added for one mole of solvent is also calculated together with the solubility. This algorithm can be combined with a data regression algorithm so that when solubility data as a function of temperature change and/or pH change are available, the interaction parameters for the activity coefficient models (see section 15.3) or the parameters for solubility product can be regressed to reproduce the experimental data. Example 1: The Effect of pH on the Solubility of DUP860 in H2O In Figure 1, the effect of solution pH on the solubility of DUP860, which is Alpha Styryl Carbinol Antifungal agents (Maurin et al. [2]), in H2O is calculated for 0.01 < pH < 10 (step 1). The chemical structure of DUP860 is shown in Figure 1. The algorithm finds that this system has a single solid (step 2) and the calculation is performed according to step 3. The algorithm, first calculates the solubility at the isoelectric point, where pH is around 5. HC1 and NaOH are then added to go to acid and base sides from the isoelectric point, respectively. As the acidity of the
373 electrolyte solution increases, the solubility of DUP860 in H2O decreases and in the region of pH > 3, it approaches a constant value. This is due to the fact that only one molecular dissociation occurs for DUP860 and this takes place in the low pH region. Example 2: The effect of temperature on the solubility of Na2SO4 in H2O In Figure 2, the effect of temperature on the solubility of Na2SC>4 in H2O is calculated between 270 K and 370 K (step 1). This system has multiple solids, which are Na2SO4-10H2O and Na2SC>4 (step 2). These solids are precipitated at different condition. According to step 3, it is necessary to calculate the invariant point, where Na2SO4-10H2O and Na2SC>4 coexist with one liquid phase, and the temperature of invariant point is found to be around 308 K. Therefore below it, Na2SC>4-10H2O is found to precipitate and above it Na2SC>4is found to precipitate. The algorithm divides the whole range of temperature into two regions: one is below the invariant point and the other is above it. Calculation is continued from within a specified temperature range.
Figure 1: (a) Chemical structure of DUP860 (b) Effect of pH on the solubility of DUP860 in H2O
15.2.2 Algorithms for Phase Diagram Generation The algorithm for computation of phase diagrams for electrolyte systems generates three types of phase diagrams, which are phase diagrams for solid-liquid equilibrium, vapor-liquid equilibrium and liquid-liquid equilibrium. The details of the computational algorithm are given below.
374
Vapor-liquid Equilibrium Phase Diagram In this section, the algorithm for the calculation of the effect of salt (called salting in/salting out effect) on the vapor-liquid phase equilibrium of binary systems (components 1 and 2) is presented. For computation of vapor-liquid phase diagram, it is necessary to define the electrolyte system and to identify a salt. The computational problem is formulated as, Given, temperature (or pressure) and the compositions of components 1 and 2 in the liquid phase Calculate, compositions of components 1 and 2 in the vapor phase by solving Eqs.l- 2 simultaneously The vapor liquid equilibrium can be computed by solving Eqs. 1-7, where Eq. 1 is related to the physical equilibrium, Eq. 2 is related to the chemical equilibrium constants, Eq. 3 represents the charge balance while Eqs. 4-5 are constraints on the component compositions (mole fractions) for the two phases. It can be noted that an activity coefficient model (constitutive equations) is needed for Eq. 1 and indirectly for Eqs. 2-5.
fja" = ,p(T, P, y)ymp = y(T, P, x)xlmplms = fj"
im = \ NM
(1)
375 N('
Kl = \ \ a,A vH
i = \,NC, j = 1, NR
(2)
i«=0
(3)
i>,=i
(4)
f>,=l
(5)
xM,, = const.
(6)
xIor2 = const.
(7)
In Eqs.1-7, NM is number of molecular species, NC is number of chemical species, and NR is number of chemical reactions to be considered. P, T, ps and a are total pressure, temperature in K, vapor pressure of pure component and activity, respectively. Zj is the number of chemical charges of component i. Vj, is stoichiometric coefficient of component / in reaction/. Eqs.5-7 are used to specify the concentration of the salt and concentration of either component 1 or component 2. It is important to note that mass balance equations are not required when only computations of phase equilibrium are to be performed. This algorithm also calculates the depression of vapor pressure for pure components by the addition of a salt. In this case, it is necessary to define the system and to specify the salt together with its composition. For a specified temperature, the saturation vapor pressure is calculated by solving Eq. 8. S = f(T,pH)
(8)
Example 1: The salt effect ofCaCL on the vapor-liquid equilibrium of Ethanol-H2O system Consider the Ethanol-H2O-CaCl2 ternary system as an example. In this example, the aim is to calculate the effect of CaCl2 on the vapor-liquid equilibrium of Ethanol-H2O (see Figure 3) at 298.15 K. There are 6 (=NC) chemical species, which are H2O, H+, OH-, Ca2+, Cl" and Ethanol. There are 2 (=NR) chemical reactions, which are complete dissociation of CaCh, molecular dissociation of H2O. The unknown variables are 9, which are the concentrations of 6 chemical species in liquid phase, the compositions of 2 molecular species in vapor phase, and either the T or P. To solve for these unknown variables, the generated model equations are the following: 2 from Eq.8, 2 from Eq. 1, 1 from Eq. 2, 1 from Eq. 3, 1 from Eq. 4, 1 from Eq. 5 and 1 from Eq. 6. Since there are 9 unknown variables and 9 equations, this problem can be solved. The calculated phase diagrams (see Figure 3) have been obtained with the electrolyte NRTL model (see chapter 8 for more details on the model).
376 Example 2: The salt effect ofCaCh on the vapor-liquid equilibrium of Methanol-Ethanol system In this example, the effect of CaCb on the vapor-liquid equilibrium of Methanol-Ethanol system is calculated. The result is shown in Figure 4. The calculation is made with the electrolyte NRTL model (see chapter 8). The calculated results confirm that the electrolyte NRTL is also applicable to organic systems without H2O (see Figure 4).
Figure 3: Effect of CaCl2 on the vapor-liquid equilibrium of H2O-Ethanol system [3]
Figure 4: Effect of CaCi2 on the vapor-liquid equilibrium of Methanol-Ethanol system [3]
Solid-liquid Equilibrium Phase Diagram In this section, the algorithm for the calculation of the ternary and quaternary solid-liquid phase equilibrium diagrams, which can be used for the design of separation processes with crystallization, is presented. Case studies involving the following three types of systems have been tested. Type 1) 1 solvent (H2O) with two salts (inorganic and organic ternary system) Type 2) 1 solvent (H2O) with three salts (inorganic and organic quaternary system) Type 3) 2 solvent (H2O and organic solvent) with one salt (ternary system) When the electrolyte system is defined and temperature is given, the algorithm calculates the solubility of solids by simultaneously solving Eqs. 9-12, where Eq. 9 represents the condition of physical equilibrium while Eq. 10 represents the condition for chemical equilibrium. fj"> = y(T, P, x)xlmpims = U""'
im = \, NM
(9)
377
Kj =Y\<*?
' ='»NC> J = ^NR
00)
%x,z,=0
(11)
§>,=1
(12)
In Eqs. 9-12, NM is the number of molecular species, A/C is the number of chemical species, and NR is the number of chemical reactions to be considered. P is the total pressure, T is the system temperature and ps is the vapor pressure of pure component, z, is the number of chemical charges of component i. K is the solubility product. The solid-liquid equilibrium phase diagram is generated according to the following steps 1 - 4. Step 1: System classification Classify the system as single salt system or multiple salts from the data of the types of salts. Step 2: Invariant point calculation Calculate the invariant point(s) of the given system, where multiple solid phases may coexist with the liquid phase. The identification of the invariant point requires the calculation of the solubility index SI, which is defined by Eq.13.
S/ = -s! K,
(13)
In Eq.13, Kj, a; and Vj, are the solubility product of solid j , the activity of component i, and the stoichiometric coefficient of component / in the reaction for the solid formation of solid j , respectively. When the solution is saturated with respect to solid /, 5/ for solid/ is equal to one and for other solids, it is less than one. Therefore, for all the combinations of solid, it is necessary to calculate SI and check if above condition is satisfied. Only the combinations of the solids, which satisfy above condition, will have invariant points. Step 3: Start- and end-points of saturation curves Given the locations of invariant points, identify the location of intersection (also called edge point) of saturation curve and the binary-axis of the triangular diagram. Determine the starting point and end point for all the saturation curves.
378 Step 4: Intermediate points for the saturation curves Using the information from step 3, Solve Eqs.9-12 are solved to identify the intermediate points on all the saturation curves. Example 1: SLE phase diagram for H2O-Glycine-(l)Asparagine-Fumaric acid quaternary system Consider the solid-liquid equilibrium for the H2O-Glycine-(l)Asparagine-Fumaric acid quarternary system (type 2). There are 12 chemical species, which are H 2 O, Gly, Asparagine, Fumaric acid, H+, OH", Gly , Gly", ASP+, ASP", FA" and FA ". There are 10 chemical reactions: which are 7 molecular dissociations and 3 solid formations. Unknown variables are concentrations of all the chemical species, xim (im=\, 12) in the liquid phase. In the step 1, the algorithm finds that the solids to be considered are three, which are Fumaric acid, L-Asparagine and Glycine. This means that no compound or complex is created from these components. In this system, there is only one invariant point, where three solids and one liquid coexist. Furthermore, there are three edge points (see step 2). There are three saturation curves and the starting point of each of these three curves is the invariant point (identified in step 2), while the end points are the edge points (identified in step 2) - this is step 3. Finally the intermediate points on the three saturation curves are calculated in step 4. Figure 5 shows the calculated phase diagram. Note that once the start- and end-points have been identified, they can also be joined by straight lines as a first approximation. Example 2: SLE phase diagram for H2O-Na7SO4-K2SO4 ternary system Consider the solid-liquid equilibrium for the Na2SO4-K.2SC>4 ternary system (type 1). There are 7 chemical species, which are H 2 O, H+, OH", Na+, SO42", K+ and HSO4". In the step 1, the algorithm finds that the solids to be considered are three, which are Na2SO4, K2SO4, Na2SO4-10H2O and NaK3SO42. In step 2, the algorithm classifies this system as a multiple-solid electrolyte system. From Gibbs phase rule, at invariant point, only two solids can coexist. Therefore, which of two salts will be present at the invariant points need to be identified. The algorithm finds that there are two invariant points: at one of them, NaK3(SO4)2 and Na2SO4 coexist with a liquid phase, while at the other, K2SO4 and NaK.3SO4 coexist with a liquid phase (see step 3). The algorithm finds three saturation curves and calculates the intermediate points on these three curves (see step 4). Figure 6 shows the calculated phase diagram. Example 3: SLE phase diagram for H2O-Butanol-Benzilpenicillin potassium salt ternary system Consider the solid-liquid equilibrium for the H2O-Butanol-BenzilpeniciIIin potassium salt (also called KBP) ternary system (type 3). There are eight chemical species, which are H2O, Butanol, KBP, K , BP", BP, H and OH". In step 1, the algorithm classifies this system as a single salt system, because only KBP will precipitate at the temperatures of interest. Furthermore, this system does not have any invariant point (see step 2). In step 3, the composition of edge points, that is, the solubility of KBP in H2O and Butanol, are calculated. Finally the intermediate points on the saturation curves are calculated in step 4. Figure 7 shows the calculated phase diagram on a salt-free basis.
379
Figure 5: Solid-liquid phase diagram for H2OGlycine-Fumaricacid-Lasparagine quaternary system
Figure 6: Solid-liquid phase diagram for H2O-Na2SO4-K2SO4 ternary system
Figure 7: Solid-liquid phase diagram for F^O-KBP-Butanol ternary system [4]
Liquid-liquid Equilibrium Phase Diagram In this section, the algorithm for the calculation of the effect of salts on the liquid-liquid equilibrium of binary systems (components 1 and 2) is presented. For the computations, it is
380 necessary to define the system and to identify a salt. The concentration of the salt also needs to be specified. For a specified temperature, the compositions of components 1 and 2 in the two liquid phases are calculated by solving Eqs. 14-20 simultaneously. Again, Eq. 14 represents the condition for physical equilibrium while Eq. 15 represents the condition for chemical equilibrium. fJ=rl(T,P,x)xllmpJ
Kj=f\<>,"°
J = UNR
f>,z,=0
I>,=1 X
x
=r\T,P,x)x2lmPn:=fJ
mh=COmt-
\,,r2 = const.
ix^ m =fiix F /
im = l,NM
(14)
(15)
(16)
(17) (18)
(19)
(2°)
In Eqs. 14-20, NM is number of molecular species, NC is number of chemical species, NP is number of phases (equal to 2) and NR is number of chemical reactions to be considered. P is total pressure, T is system temperature, Fu is flowrate of component k and p s is saturation vapor pressure of pure component, z, is the number of chemical charges of component i. The algorithm tests the existence of miscibility gaps using phase stability analysis. This algorithm first identifies whether the binary system consisting of the two solvents has a miscibility gap or not. Although a ternary system consisting of two solvents and one salt has 3 binary pairs, any of the binary pair involving the salt are not tested, since the salt is not likely to create a miscibility gap with the solvent. If the binary system consisting of two solvents has miscibility gap, the liquid-liquid equilibrium phase diagram algorithm continues from the phase split calculation by adding the salt until it comes close to the plait point. In the vicinity of the plait point, linear extrapolation method is used to find the approximate location of the plait point. In this case, the conjugate line may also be used to decide the location of the approximate plait point. At the plait point, the conjugate line and the boundary of LLE region intersect.
381 Example 1: Liquid-liquid phase diagram for HiO-Butanol-NaCl ternary system Consider the liquid-liquid equilibrium for the I-bO-Butanol-NaCl ternary system (see Figure 8). There are 6 chemical species, which are H2O, Butanol, H+, OH-, Na+ and Cl". There are 2 chemical reactions: which are two molecular dissociations. The unknown variables are the concentrations of all the chemical species (6*2=12) in the two liquid phases. The algorithm, first, checks if two solvents, which are H2O and Butanol, have miscibility gaps. Since the H2OButanol binary system has a miscibility gap, the algorithm continues with the liquid-liquid equilibrium calculation by increasing the concentration of the salt. Calculation is continued until it reaches close to plait point. In this electrolyte system, however, the boundary of the liquidliquid phase equilibrium (denoted by the solid line in Figure 8) intersects with the boundary of solid-liquid phase equilibrium (denoted by dashed line in Figure 8) below plait point. This indicates that a solid-liquid-liquid phase equilibrium region exists in this system. Therefore, the calculation is continued until two the boundaries, the boundary of liquid-liquid phase equilibrium and the boundary of solid-liquid phase equilibrium intersect. Example 2: LLE phase diagram for H2O-Butanol-Benzilpenicillin potassium salt ternary system Consider the liquid-liquid equilibrium for the F^O-Butanol-Benzilpenicillin potassium salt (KBP) ternary system (see Figure 9). The algorithm, finds that the F^O-Butanol binary system has a miscibility gap. Therefore by increasing the salt, KBP, liquid-liquid equilibrium calculation is continued until it comes close to plait point. The approximate plait point is determined by extrapolating the phase boundary of liquid-liquid phase equilibrium. In this system, miscibility gap is in the unsaturated region with respect to KBP.
Figure 8: Liquid-liquid equilibrium for H2OButanol-NaCl ternary system [5]
Figure 9: Liquid-liquid equilibrium for H2OButanol-NaCl ternary system [4]
382 15.2.3 Algorithms for Data Regression In this algorithm, a least square minimization method is used for regressing the experimental data. The objective function is given by Eqs.21-22. Min{Objective function) Objective function = Y
(21) Yl
—
-JJ—
a
.'=i ' I
>
(22)
)
In Eqs.21-22, p is a particular physical property and a is the relative tolerance of the experimental data of the particular physical properties. Ndata and Npro are the numbers of experimental data and physica properties to be regressed, respectively. Superscript cal and exp represent the calculated value and the experimental value, respectively. For example, when the experimental data for vapor-liquid equilibrium is regressed, the objective function is written in Eq.23.
Ndata ( J'exp _ j,cal \
Objective function = V <\
' [{
a
f pexp _ peal V
+
i
)
I
a
2
)
In Eq.23 T, P, x, and y are the temperature, the pressure, the liquid composition and the vapor composition, respectively. Eq.23 minimizes the error of temperature, pressure and compositions. If the errors of the distribution coefficient (physical equilibrium constant) need to be minimized, Eq. 22 may be written in Eq.24. Ndala / T e x P
Tcal \
( D ex P
PCL)I V
\:C'ompl f r^ exp
is cal \
\\
OV0.fi***- ? l ^ - j + ( ^ - j + ? [ ( ^ - j jj (24, In Eq.24, K is distribution coefficient defined by Eq.25. K,=%-
(25)
383 When the experimental data for liquid-liquid equilibrium is regressed, before minimizing Eq.22, Eq.26 may be minimized to obtain a good initial guess [6].
Objective function = ^
\^
(y'x^p
- y,2x,2™ f \
(26)
In Eq.26, x is a vector of compositions from experimental data and % is a vector of calculated activity coefficient values using experimental data.
3.3 ALGORITHM FOR CREATION OF PROPERTY MODEL PACKAGE In this section, an algorithm for creating problem and system specific property model packages in the sense of adapting the selected (appropriate) property models for the purposes of the specified calculation problem is presented. The property model package includes pure component physical property (density, dielectric constant, etc.) models, mixture physical property (activity coefficient, solubility etc.) models and parameters for them. The main steps for the creation of the property model package are shown in Figure 10. An explanation for each steps are given below. Step 1: Problem definition Creation of a property model package starts from the definition of the given problem. In this algorithm, mainly three types of problems, which are multi-phase separation calculations (flowsheet simulation), computation of phase diagrams (solid-liquid/liquidliquid/vapor-liquid phase diagrams), and solubility analysis, are considered. These problems are also classified in terms four types of phase equilibrium problems, which are VLSE problem, SLE problem, LLE problem and VLE problem, based on the phases to be considered (see Table 1). The flowsheet simulation is classified as a VLSE problem. The computation of solid-liquid phase diagrams, liquid-liquid phase diagrams and vaporliquid phase diagrams are classified as SLE problem, LLE problem, and VLE problem, respectively. Solubility analysis calculates the temperature and pH dependencies of the solubility of the salt under consideration and is classified as a SLE problem. Table 1: List of problem types together with corresponding problems Problem type VLSE (or VLLSE) SLE LLE VLE
Corresponding problems Multiphase separation Solubility analysis, SLE phase diagram LLE phase diagram VLE phase diagram
384 The list of necessary physical properties is made based on the problem specified. The four types of phase equilibrium problems (types 1 -4) require different sets of physical properties. The selection of the property model package (step 3) and the creation of a list of necessary physical properties (step 4) are made based on the problem type specified at this step. If the created property model package is to be used only for phase diagram computations, properties such as enthalpy, heat capacity etc., may not be needed, while for design and simulation of crystallization processes, they will be necessary.
Figure 10: The algorithm for creation of property model package
Step 2: Electrolyte system definition In this step, the electrolyte system is defined from the information on the presence of solvents, salts and ionic species. This includes identification of the ionic species (cation/anion/dipole ion), types of salts (organic and/or inorganic), and solvents (water and/or organic). Based on the presence of organic solvent(s), the system is classified as
385 "mixed-solvent system" or "aqueous system". In addition, the phase(s), where each of the chemical species may exist, is identified. For example, "liquid phase" for ionic species, since they exist only in liquid phase(s). This means that in vapor-liquid equilibrium calculations, equality of fugacity between both phases need not be considered for ionic species. The salts, for which complete dissociation in the liquid phase can be assumed, are classified in the "solid" group, because it does not exist in the liquid phase. Gases, for which Henry's law is valid for solubility calculations, are identified together with their corresponding Henry's constant in water. Example In Table 2, the system definition for the H2O-Glycine-Potassium Chloride (KCl) ternary system is highlighted as an application example. Since H2O is the only solvent, this system is classified as an "aqueous system". There is one inorganic salt, KCl. Since KCl is assumed to be dissociated completely and does not exist in the liquid phase, the identified phase for KCl is "Solid". All the information needed in this step is stored in the electrolyte database (see the CAPEC database in chapter 3). Glycine is a dipole ion, which has both negative and positive charges and the total charge is zero. HC1 is a gas component, for which Henry's law is valid. Therefore, Henry's constant for HC1 in H2O is required. Table 2: Definition of H2O-Glycine-KCL ternary system Component No. 1 2 3 4 5 6 7 8 9 10
Name
Phase(s)
Ion type
Salt type
H2O Glycine KCL H+ OHGlycine+ GlycineK+ ClHCL
Vapor/solvent Liquid/solid Solid Liquid Liquid Liquid Liquid Liquid Liquid Vapor/liquid
Dipole Cation Anion Cation Anion Cation Anion -
Organic Inorganic -
Other spec.
Henry
Step 3: Property model selection Four property models for calculation of activity coefficients of electrolyte systems have been implemented in the model library of ELEC-ICAS and their creation/adaptation is discussed below. The four activity coefficient models (to be referred to as G -models) are, the Pitzer-Bromley-Debye Huckel model [7], the electrolyte NRTL model [8], the
386 extended UNIQUAC model [9], and the extended (general) UNIQUAC model [10]. The extended UNIQUAC model was developed for aqueous systems and many interaction parameters for it have been estimated by Thomsen et al. [11]. In the developed model library, the extended (general) UNIQUAC model has been developed by modifying the extended UNIQUAC model by adding the Born term. Together with these GE-models, a number of pure component physical property models {e.g., models for vapor pressure and dielectric constant) and mixture property models (e.g., models for fugacity coefficients and density) are also needed to perform the necessary calculations. Excess properties are calculated directly from the selected G -models. In this step, the appropriate property model is selected based on the problem type (which is specified in step 1), the application region and the application range with respect to the ions and solvent(s) present in the system. Tables 3a and 3b help to identify the feasible property models with respect to the problem type and the application range. For example, for mixed-solvent systems, only the electrolyte NRTL model or the extended UNIQUAC model is feasible. Model selection is based on "pattern matching". That is, a model is selected if it's known pattern (see Tables 3c and 3d) matches with the generated pattern for a specified problem. In Tables 3c and 3d, index equal to 1 indicates the model is feasible, while index equal to zero indicates model is infeasible. Step 3-a: Determine pattern index For the given problem k and system /, the pattern (product of index) for model i is calculated through Eq.27. Index of model i = index(i, /) x index(i, k)
(27)
Step 3-b: Compare pattern For a specified problem and system, GE-models are identified as appropriate models, if their corresponding index in Tables 3c and 3d are also 1. Example Consider the problem to generate the VLE phase diagram for a mixed solvent system. The VLE phase diagram belongs to problem type 4 according to Table 1. According to Table 3a, three GKmodels, which are the Pitzer model, the electrolyte NRTL model, and the extended (general) UNIQUAC model, are suitable, as the index for these three models for the specified problem is equal to one. From Table 3b, the electrolyte system is identified as "mixed". When the pattern is calculated for these three GK-models (see step 3-a), only that of the electrolyte NRTL model, and the extended (general) UNIQUAC model are found to be one. Therefore, these two GE-models are selected as the appropriate property models (see step 3-b). The final selection regarding
387 which G -model to use will depend on, for example, the availability of interaction parameters, the model complexity and other factors. In this example, the electrolyte NRTL model is selected. Table 3a-d: List of thermodynamic models with their application range represented by index. Superscripts A and G indicates "aqueous" and "general", respectively. Model name Pitzer-Bromley-DH
VLSE Possible
SLE Possible
LLE
VLE Possible
Extended-UNIQUAC0 Extended-UNIQUACA Electrolyte-NRTL
Possible
Possible Possible Possible
Possible
Possible
Possible
Possible
Possible
(a) Selection rule based on problem type Model name
Ionic strength limit.
Pitzer-Bromley-DH ExtendedUNIQUAC 0 ExtendedUNIQUACA Electrolyte-NRTL
6 mole/kg No limitation
Aqueous system Aqueous 1 Possible Possible
No limitation No limitation
Aqueous 2 Possible Possible
Non-aqueous system Mixed. Organic Possible
Possible
Possible
Possible
Possible Possible
Possible
(b) Selection rule based on the application range (system type and ionic strength limit.) VLSE 1
Model name Pitzer-Bromley-DH
SLE 1
LLE 0
VLE 1
1 1 1 1 Extended-UNIQUAC 0 0 0 0 1 Extended-UNIQUACA 1 1 1 1 Electrolyte-NRTL (c) Selection rule based on problem type represented by index Model name Pitzer-Bromley-DH ExtendedUNIQUAC 0 ExtendedUNIQUAC^ Electrolyte-NRTL
SI > 6 SI < 6 Aqueous system mole/kg mole/kg Aqueous 1 Aqueous 2 1 1 1 0 1 1 1 1
Non-aqueous system Mixed. Organic 0 0 1 1
1
1
0
1
0
0
1
1
1
1
1
1
(d) Selection rule based on the application range represented by index
388 Finally the notable point of current model selection algorithm is that the addition of new selection rule is easy. When new selection rules (assuming the number of the selection rules to be added is N) are added, the number of columns in the index matrix (see Tables 3b and 3d) is increased by N. When a new GE-model is added, the number of rows in the index matrix (see Tables 3a-3d) is increased by 1. Step 4: List of necessary properties and constants Once the problem has been specified and an appropriate G1 -model selected, the necessary physical properties are identified, based on the nature (type) of specified problem and a list of necessary physical properties is created according to the following three steps. In the first step, the list of problem specific properties, which do not depend on the selected GE-model, is created (see Table 4a). In the next step, properties specific for the selected GH-model, such as the dielectric constant and the Born radius, are added to the list (Table 4b). For example, when the Pitzer model is selected, the dielectric constant or the Born radius are not necessary although they are needed for the electrolyte NRTL model and the extended (general) UNIQUAC model for the same problem. Finally in the case where the given system includes gaseous components for which Henry's law is valid and the problem type is either 1 (VLSE) or 4 (VLE), Henry's constant is added to the list of the needed properties. In the case where equilibrium constants are calculated by thermodynamic properties, such as Gibbs energy and enthalpy, they are also added to the list of properties (Table 4c). The complete list of the necessary physical properties is created by combining Table 4a-4c. After the list of necessary physical properties has been created, the corresponding property models for the identified properties, are selected based on the availability of model parameters and the application range of models. Table 4a: List of problem specific physical properties
Pure properties
Mixture Properties
Properties name Density Vapor pressure Enthalpy of formation Heat capacity Gibbs energy of formation Fugacity coefficient Activity coefficient Dissociation constant Solubility Henry's constant
VLSE
SLE
LLE
* * * * * * *
VLE *
* * *
* *
* *
389 Table 4b: List of model specific physical properties. Pitzer: Pitzer-Bromley-Debye Huckel model; NRTL: electrolyte NRTL model; UNIQ(aq): extended UNIQUAC model (aqueous); UNIQ: extended (general) UNIQUAC model Properties name Pure properties
Mixture Properties
Born radium Van der Waals volume Van der Waals surface area Density Dielectric constant Density
Pitzer
NRTL
UNIQ (aq)
* * * *
*
* *
*
UNIQ * * * * * *
Table 4c: List of additional physical properties. Case 1: When the system includes gas component, for which Henry's law is applied; Case 2: When the equilibrium constants are calculated by thermodynamic properties
Pure properties
Properties name Enthalpy of formation Heat capacity Gibbs energy of formation Henry's constant
Casel * * *
Case 2
*
Two calculation approaches are available for solubility analysis - a solubility product approach and an empirical model approach. In the solubility product approach, solubility is calculated as a function of temperature through the calculated activity coefficients and the equilibrium constants. A selected G'-models is evaluated through experimental data sets if they are available, and the GE-model does not describe the experimental data sets, another GE-model needs to be selected or model parameters have to be adjusted (data regression) to describe the experimental data sets. In the case where the equilibrium constant is not known, it is calculated from the Gibbs energy at the standard state. In the empirical model approach, an empirical solubility function of temperature and pH is determined through regression of solubility data. Example Consider the computation of VLE phase diagram for the same mixed-solvent electrolyte system, which was used in step 3. Since this problem is classified as a VLE problem, the list of necessary properties for the corresponding problem is shown in Table 5a. According to step 3,
390 both the electrolyte NRTL model and the extended (general) UNIQUAC model (general) are applicable to this problem. Assuming that the extended (general) UNIQUAC model (general) is selected, according to Table 4b, the properties shown in Table 5b are also required for the calculations related to this example. Since no gaseous component is present, no additional properties need to be added to the list of the properties in Table 5b. Finally, the complete list of the necessary properties is shown in Table 5c. Table 5a-5d: List of the necessary properties for the example Properties name VLE Pure Vapor pressure * Properties Mixture Fugacity coefficient * Properties Activity coefficient * (5a) List of the necessary properties from Table 4a
Pure properties
Properties name Born radium van der Waals volume van der Waals surface area Density Dielectric constant Density
UNIQ * * * * * *
Mixture properties (5b) List of the necessary properties from Table 4b Properties name UNIQ Vapor pressure * Born radium * van der Waals volume * van der Waals surface * area Density * Dielectric constant * Mixture Fugacity coefficient * properties Activity coefficient * Density * (5c) Complete list of the necessary properties
Pure properties
391 Step 5: Parameter tables The next step is to determine which parameters are needed by the selected G -model and associated property models and how many of these are available in the database. Note that model parameters here also include the dissociation constants, solubility products, GE-model interaction parameters (e.g., ion-ion, ion-solvent and solvent-solvent interactions) and pure component properties (densities, melting points, etc.). A parameter estimation procedure, which generates a list of the necessary parameters and the corresponding data that are needed to estimate them, for which a data regression algorithm will be considered next. Data Regression Algorithm In the data regression algorithm, the following types of experimental data sets for mixture properties can be regressed, in addition to pure component properties, such as density and dielectric constant. I. Vapor pressure depression data (the effect of salts on the vapor pressure of pure component) II. Activity coefficient data (include osmotic coefficient data, mean activity coefficient) III. Solubility data IV. Vapor-liquid equilibrium data V. Solid-liquid equilibrium data VI. Liquid-liquid equilibrium data Most of the experimental data sets for electrolyte systems found in the open literature may be classified into one of the six types of data sets listed above. These data sets are stored in the wellstructured electrolyte component database and used not only for data regression but also for the validation of models and is a part of the CAPEC database implemented in ICAS [1]. To generate a list of candidate parameters for estimation, information from the system definition step is used. For example, for ion-ion interaction parameter, two different ionic species, one from the list of cation and the other from the list of anion are selected, based on the assumption that interactions between cation-cation and anion-anion are not important. The most difficult problem here is to address the question of how to reconcile the lack of data to the large number of parameters that need to be regressed? For multi component systems, the number of interaction parameters may be quite large - just the ion-ion and ion-solvent interaction parameters alone may involve a parameter matrix of dimension 15 by 15. The objective of a sensitivity analysis is to help to regress the available experimental data sets with only a minimum number of parameters. In Figure 10, the link of sensitivity analysis to data regression algorithm is highlighted. The sensitivity analysis is made within the composition/temperature/pressure range of the experimental data sets. It identifies the order of the most sensitive parameters based on the derivatives of calculated physical property values with respect to each of the model parameters (see Eq. 28).
392
dF
— =Y dz Y
d(z\{Prl-pr)ipr
dF
^ dz
=V Y
D,
•-
i = \,NC,j = \,NE
(28)
dz
The derivatives are calculated for the entire range of conditions given by the experimental data sets. This sensitivity analysis is used mainly for the regression of activity coefficient data, where the matrix of model parameters is large. Model parameters for four G -models are listed in Tables 6a-6d. Based on the definition of the system and the information in Tables 6a-6c, the list of interaction parameters to be used for sensitivity analysis is created. Table 6a-6d: The list of model parameters for activity coefficient models
Pitzer-Bromley-DH
Debye-Huckel term No parameter
Bromley term Ion - Ion
Pitzer term Ion-molecule Molecule-molecule
(6a) Pitzer-Bromley-Debye Huckel model
electrolyte-NRTL
extendedUNIQUAC (general)
ExtendedUNIQUAC (general)
Debye-Huckel term No parameter
NRTL term
Debye-Huckel term No parameter
UNIQUAC term
Molecule-Molecule Molecule-(cation, anion) (cationl, anionl)-(cationl,anion2) (cation 1, anion 1 )-(cation2,anion 1) (6b) Electrolyte NRTL model
Molecule-Molecule Molecule-ion; Cation-anion van der Waals volume van der Waals surface area (6c) Extended UNIQUAC (general) model Debye-Huckel term No parameter
Born term Born radius
Born term Born radius
UNIQUAC term Molecule-Molecule Molecule-ion; Cation-anion van der Waals volume van der Waals surface area (6d) Extended UNIQUAC (aqueous) model
393 Once the order of the most sensitive parameters has been specified, data regression is started for the minimum number of parameters. In the case where the selected parameters do not describe the entire set of experimental data, the next most sensitive parameter pair is added for data regression. This procedure is repeated until all experimental data sets are reproduced with acceptable accuracy. This interaction is shown by the bi-directional arrow in Figure 11. Experimental data
Sensitivity analysis
Identify the order of the most sensitive parameters
Data regression
Figure 11: The relation between sensitivity analysis and data regression The identification of the sensitive interaction parameters is highlighted for solubility and activity coefficient as the property in Eq.28. Consider property as the solubility of a salt, which can be calculated using Eq.29. Kk=(r,x,)
i = \,NC
(29)
In Eq.29, x-, is the mole fraction of component i, y, is the activity coefficient of component i, Kt is the solubility product for solid k. The derivatives of Eq.29 with respect to model parameters (r) for activity coefficient models is given by Eq.30.
dz
dz dyt
The first term on the right side of Eq.30 can be calculated from the G1 -models and the second term is calculated from the models for yand K. Therefore it is possible to see how the solubility is changed for the small changes of the model parameters. Considering property of Eq.28 as the activity coefficient, the derivatives of logarithm of activity coefficient of chemical species / {In y) for the electrolyte NRTL model with respect to interaction parameter ris calculated through Eq.31 or Eq.32 (see chapter 8 for GH-model details). dQny,) _d(\nr;'DH) dz dz djlny,) _d(lRy,k) dt dz
^ d(\nrik) dz
[
d{\nYlBor") dz
From a study of various systems, it has been found that one of the easiest ways to identify the sensitive parameters is to find the pairs of chemical species with high concentration. Normally,
394 there are only a few pairs of high concentration chemical species in the electrolyte system. The interaction parameters for these high concentration pairs are significantly more sensitive to the activity coefficient value than the interaction parameters for all others. Changes in the interaction parameters for the other pairs do not change the activity coefficient value significantly. Alternatively, these sensitive interaction parameters for the high concentration pairs can also be identified through solving Eq.28. Therefore, even with a few data points, it is possible to create an appropriate thermodynamic package with all the needed parameters. The sensitivity analysis can also be used for other purposes. In data regression, it is used to determine the initial estimates for parameters to be adjusted as well as for identifying the sensitive parameters. This is useful when the system behavior is very sensitive to small changes of parameters. It may also be used to evaluate how the uncertainties of interaction parameters affect the computation of thermodynamic properties. Example 1 This example illustrates how to identify the order of the most sensitive parameters from the whole interaction parameter matrix. Consider the regression of the activity coefficient data of H2O-KCI binary system. In Figure 12a, the matrix of all the interaction parameters is shown. In Figure 16b, only the interaction parameter for the H2O - (K+,C1~) pair (shown by *) has been found to be sensitive to the activity coefficient value according to the calculated derivative values of Eq.28 (see Table 7).
Figure 12. Interaction parameter matrix of electrolyte NRTL model for H2O-KC1 system (a) before sensitivity analysis (b) after sensitivity analysis - only the shaded interaction parameter is needed
395 Table 7: The derivatives with respect to interaction parameter calculated by Eq.28, where p is activity coefficient Component 1 H2O H2O H2O H2O (H+, OH-) (H+, C1-) (K+, OH-) (K+, C1-)
Component 2 (H+, OH-) (H+, C1-) (K+, OH-) (K+, C1-) H2O H2O H2O H2O
dF/dx 0.126192995508617 -6.003976149582257E-005 0.197063115632209 2.71500842362036 0.000E+0 0.000E+0 0.000E+0 6.59206906490653
Example2 The purpose of this example is to highlight that the sensitivity analysis is also useful in the determination of good initial values for the parameters to be regressed. In this example, the solidliquid phase diagram for FkO-butanol-Benzilpenicillin potassium salt (KBP) ternary system (phase diagram is shown in Figure 7) is considered. This system generates 8 chemical species (H2O, Butanol, KBP, H+, OH', K+, BP", BP) , and KBP is assumed to be dissociated completely. For the computation of activity coefficients, the extended (general) UNIQUAC model is selected. According to the sensitivity calculation results of Eq.28, the H 2 O, Butanol, K+, and BP" are four most abundant species and 8 binary interaction parameters for sensitivity analysis are selected (shown in Table 8). Table 8: The behavior of KBP solubility against UNIQUAC interaction parameter Interaction parameter H2O - BPBP- - H2O
Positive value Increase Increase
Negative value Decrease decrease
H2O - K+ K+ - H2O Butanol - BPBP- - Butanol Butanol - K+ K+ - Butanol
Increase Increase Increase Decrease Decrease Decrease
Decrease Decrease Decrease increase to decrease increase to decrease increase to decrease
In Table 8, the effect of the change of the interaction parameter on the phase equilibrium behavior of the given system is shown. In this study, the parameter values used are within the limits of+1000 and -1000. Based on the fact that solubility curve has maximum point, (Butanol-
396 K+) interaction parameter is selected for regression and a negative value is used as an initial estimate. In addition, (H2O-BP'), (Butanol-BP'), and (H2O-K+) interaction parameters together with their R and Q parameter values for BP" are regressed based on the result of sensitivity analysis. This analysis is useful to decide what is a good initial value for each interaction parameter? For the (Butanol-K+) interaction, a negative value is needed to get the peak in the solubility curve and the final regressed parameter value has also found to be negative. When a positive value is given for this parameter as a starting point, the regressed results has been found to be not very good (as shown by dashed line in Figure 7) and a positive value for this parameter contributes only to a decrease of the solubility. By changing starting point to a negative value, the calculated solubility has been found to match the experimental data as shown by the solid line in Figure 7. Note that the solubility of KBP in pure Butanol calculated by Extended (general) UNIQUAC would not match with experimental data as this data has not been included in the data regression. Final values for binary interaction parameters are shown in Table 9. Table 9: The interaction parameters regressed in this study Interaction parameter H2O(l)-BP-(2) Butanol (l)-BP-(2) Butanol ( 1 ) - K + (2) H2O(1)- K+(2)
xl2 -160.5085 1146.33 0 (fixed) 39.1926
T21
0 (fixed) 0 (fixed) -771.0212 0 (fixed)
9.4 COMPUTATIONAL ALGORITHMS FOR PROCESS SIMULATION This section presents an equilibrium-based crystallizer model representing a multiphase separator (MPS) with and without a decanter (see Figure 13). The mass balance equations and equilibrium constrains in terms of fugacity for the MPS models are given below.
Figure 13: Crystallizer model consisting of MPS model and decanter
397 15.4.1 The MPS Model Equations The MPS (Multi Phase Separation model) model is applicable for the simulation of vapor-liquid, liquid-solid and vapor-liquid-solid separations. This general model of phase separation calculation for electrolyte systems includes mass balance equations (Eqs.33, 34 and 37), equations for interand intra-phase equilibrium (see Eqs.35 and 36), and equation for electroneutrality (chemical charge balance, Eq.38). MWmokcllFlm,eJ"
= £MW mdeciX F molec J
MWnmkcp_Fm>hcJ' =YXMWmokcaFmokcJ
j \ = 1, NM(NE)
+ MWlm)2Fuml2")
j2 = 1, NM(E)
(33)
(34)
p.i
(Fmkc//F")
= K{Fmok,cJ/F")
(F,,J"^F"") = a(Fmokvi'"'/Fhq) FP="L{F,,,^/ + Fml/) Y;ZkFk=0
k = l,NC
j = l,NM, p,q = l,NP {p*q)
(35)
j = \,NM,i = \,NI
(36)
k = \,NC
(37) (38)
In Eqs.33-38, F is molar flow rate, MW is molecular weight of each chemical species, K is equilibrium constant which is a function of temperature and composition, a is the degree of dissociation which is function of temperature and composition, z is the number of charges of ionic species, NI is number of ionic species, NM is number of molecules, NM(NE) is number of nonelectrolytes, NM(E) is number of electrolytes, NC (=NM+NI) is number of chemical species, NP is number of phases. Applying the elemental balance equations [12, 13], Eqs.33-34 and 37 can be combined into Eq.39.
4=1
p=\ k=\
398 In Eq.39, Nt is the number of particular balancing element present in chemical species k. To generate the correct number of elemental balance equations, Gram-Schmidt orthogonalization [14] algorithm has been incorporated together with knowledge of chemistry [12] in the MPS model. The complete MPS model is given by Eqs.35, 36, 38 and 39. The solution algorithm of the MPS model together with a detailed analysis of the model equations and validation has been reported in [10, 15].
9.5 CONCLUSIONS A collection of algorithms for various types of phase equilibrium based calculations for electrolyte systems has been presented together with an algorithm for the creation of the appropriate problem specific property model. The basis for these algorithms is a systematic approach that has analyzed the model equations, the model parameters, the needs of the calculation problems and the nature of the electrolyte systems. The algorithms can be converted into efficient and flexible computer programs. Because of the nature of the electrolyte systems and the need for data regression for most new electrolyte systems, a highly flexible computeraided system that can adapt a basic set of property models according to the needs of the problem is required. The algorithms presented in this chapter, at least partially fulfills this objective.
REFERENCES 1. ICAS Documentations, PEC02-15, CAPEC Internal Report, Technical University of Denmark, 2002. 2. M. B. Maurin, W. J. Addicks, S. M. Rowe, R. Hogan, Pharmaceut Res, 10 (1993) 309312. 3. Y. Kumagae, K. Mishima, M. Hongo, M. Kusunoki, Y. Arai, Can J. Chem Eng, 70 (1992) 1180-1185. 4. M. C. Annesini, F. Gironi, L. Marinelli, J. Chem Eng Data, 39 (1994) 502-505. 5. R. deSantis 1974 6. Zerres H., and M. Prausnitz (1994): Thermodynamics of Phase Equilibrium in AqueousOrganic Systems with Salt, AIChE./., 40(4), 676-691. 7. Pitzer, K. S., Thermodynamics if Aqueous Systems with Industrial Applications, S.S.Newman, Ed., ACS Symposium Series, 133, 451, 1980. 8. Chen, C. C , Evans, L. B., AIChE J., 32, 1655, 1986. 9. Nicolaison, H., Rasmussen, P., Sorensen, J. M., Chemical Engineering Science, 48(18), 3149, 1993. 10. K. Takano, "Computer Aided Design and Analysis of Separation Processes with Electrolyte Systems", PhD-thesis, Technical University of Denmark, Lyngby, Denmark, 2000. U . K . Thomsen, P. Rasmussen, Chem Eng Sci, 54 (1999) 3575.
399 12. J. F. Zemaitis Jr., D. M. Clark., M. Rafal, N. C. Scrivner, Handbook of Aqueous Electrolyte Thermodynamics, American Institute of Chemical Engineering, New York, 1986. 13. Zemaitis Jr., J. F., Clark, D., Marshall, R., Scrivner, N. C. AIChE J, 40(4), 676, 1994. 14. D. C. Lay, "Linear Algebra And Its Applications (2nd edition)", Addison Wesley Longman, Inc., 1977. 15. K. Takano, R. Gani, T. Ishikawa, P. Kolar, Trans IChemE, 78-A (2000) 763-772.
Appendix A This appendix highlights some of the features of ELEC-ICAS. In particular, the steps of the property model creation (without data regression) and problem solution is highlighted for a sample problem. Step a: Draw a stream and select the compounds from the electrolytes database
Figure Al: Compound selection from the electrolyte database
400 Step b: The next step is to launch the ELEC-ICAS from the ICAS-utility toolbox as shown in the next screen shot (figure A2).
Figure A2: Lunching of ELECICAS from the utility toolbox
Step c: From ELEC-ICAS, follow the ELEC-T wizard (Steps 1-5) to solve a problem — see Figure A3.
Figure A3: Problem set-up in ELEC-ICAS and ELEC-T wizard
401 Step 1: The compounds and ions are shown. The association and dissociation reactions can also be viewed (see Figure A3). Steps 2, 3a-3c: Click on next to go to step 2 where the GE-model and the type of problem need to be selected (as shown in Figure A4). Click on next to move from steps 3a-3c (the corresponding screens are shown in Figures A5a-A5c).
402 Step 4: Click on next to view the list of properties and their corresponding model and/or data source (see Figure A6) Step 5: This step involves data regression and parameter sensitivity analysis. It has not yet been implemented in the current version of the ELEC-T wizard, although a stand-alone program exists. Step 6: Click on next to view a summary of the problem definition (see Figure A7)
Figure A6: List of selected properties from Step 4.
Figure A7: Problem definition summary from 5.
403 Step 7: Add any other problem information not covered in the earlier steps (see Figure A8) Step 8: Start the calculation by clicking on perform calculations. Note that compound 7 has been selected as the solid salt (see Figure A9)
Figure A8: Additional problem details (step 7)
Figure A9: Perform calculations (step 8)
404 The calculated results are shown below in Figure AlO.
Figure AlO: The calculated solubility diagram for the two salts. Note that the interaction parameters for the extended UNIQUAC model used in this calculation have not been tuned to the experimental data. The black point indicates the feed mixture, which means that either water needs to be evaporated or the temperature needs to be reduced in order to precipitate the Na2SO4-10H2O salt.
Part IV: Challenges and Opportunities
16. Challenges and opportunities for property modeling Georgios M. Kontogeorgis & Rafiqul Gani
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Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B. V. All rights reserved.
407
Chapter 16: Challenges and Opportunities for Property Modeling Georgios M. Kontogeorgis and Rafiqul Gani 16.1 INTRODUCTION It is often stated that chemical engineering is in a state of rapid transition in different directions and that thermodynamics (property modeling) should follow theses changes. Currently, many of the new directions are related to life sciences & biotechnology [1-3], soft knowledge-based materials & nanotechnology [4-7] as well as adapting to the needs for today's postmodern world [8], which requires increased attention to human needs and the environment, among others. Also, chemical product design, where properties have a very important role, is receiving increased attention. Future developments in thermodynamics are expected to follow these trends in addition to addressing the problems related to the design of traditional petrochemical systems and conventional separations (distillation, absorption, extraction). It is, however, worth mentioning that even in these "traditional" areas there are still numerous "thermodynamic" challenges [9], especially in relation to condensed phases (multiple liquids and solids) and complex molecules, such as polymers [10-13]. The directions of biotechnology, materials/complex products and sustainable processes that can produce them are quite complex because they often require a highly multidisciplinary approach where thermodynamics is coupled with other fields, especially transport phenomena, mathematical modeling and kinetics. As examples can be mentioned the design of biodegradable polymers such as hydrogels for use in controlled drug release devices [2, 4]. Multiscale modeling and integration across size and time (microscopic properties versus end-use properties) is another example. The objective of this chapter is to briefly outline some of the challenges and opportunities for property modeling in the areas outlined above: Classical Separations, Materials & Chemical Products, Biotechnology and Post-Modernism.
16.2 CLASSICAL SEPARATIONS Despite the availability of cubic equations of state, tuned often to specific applications using local-composition based mixing rules, group-contribution methods such as UNIFAC and association based advanced equations of state (CPA, SAFT), there are still challenges to be met related to design of "conventional" separations (distillation, extraction, absorption, adsorption) and flow assurance. Some of these are:
408 •
•
• •
•
The prediction of multicomponent, multiphase equilibria, based exclusively on binary parameters, for example, for immiscible systems such as water-alcohols (glycols)hydrocarbons. The simultaneous prediction of vapor-liquid, liquid-liquid and solid-liquid equilibria over extended temperature ranges with a common set of model parameters, for example, for alcohol-alkanes. The prediction of thermal properties such as heats of mixing and heat capacities at different conditions based on parameters estimated from phase equilibrium data [14]. The calculation of simultaneous physical and chemical equilibria, for example, for absorption of CO2 or H2S in aqueous alkanolamine solutions using a minimum amount of data/interaction parameters. The calculations in the presence of multiple solid phases (such as waxes, asphaltenes, and gas hydrates).
In addition to the above, property models of wider application range than traditionally available, are needed for process-product design. A well-known bottleneck is what can one do when the needed model parameters (such as the UN IF AC groups and their parameters) are not available? Rather than wait for experimental data to regress the parameters, the challenge is to predict them. Marrero and Gani [15] have shown how this can be done for pure component properties. It is quite clear that, despite the now famous statement of Tsonopoulos and Heidman [16] about the adequacy of cubic EoS for the needs of the oil & gas industry, such simple models are often not adequate for several of the above "classical" challenges. Cubic EoS are reliable high-pressure models for mixtures containing gases and hydrocarbons, but even advanced mixing rules often cannot provide good description of, for example, multicomponent LLE of water-alcohol-hydrocarbon systems [17] or other liquid phase properties such as enthalpies and heat capacities [11]. Difficult separations, such as those involving azeotropic mixtures or close boiling mixtures require additional accuracy, which may not be available [18]. In this respect, it is worth mentioning the statement of Zeck [9] - "There is still considerable potential for improvement for phase equilibrium thermodynamics even in long-established areas of the chemical industry. With all the enthusiasm about the possibilities of thermodynamics in new areas, it is necessary first to concentrate on performing the basic tasks". The "basic tasks" according to Zeck are those mentioned in points highlighted above, which are important for solvent selection for extractive distillation, design of liquid-liquid extraction, and absorption with chemical reactions as well as multicomponent adsorption. Zeck also pointed out the many discrepancies between measurable data for even simple systems such as the binary chlorobenzene-ethylbenzene system, the big gaps in the parameter tables of the group contribution based methods and their inability to describe VLB & LLE with a common set of parameters for the same set of compounds, or provide reliable excess enthalpies. While separation of mixtures commonly found in petrochemical industries can be routinely tackled through the property models available in most commercial simulation packages, many of the classical separations involving mixtures in the specialty chemicals, fine chemicals, food, aroma and other industries are quite difficult to handle. The predictions in most cases are unreliable unless model parameters are specially fitted to a set of system
409 specific data. Another difficulty is the lack of data - for many of the chemicals of interest, neither pure component data nor mixture data are available. Often when data is available, especially fitted model parameters are used, making them system-process specific. An important question here is the following - have the traditional models reached their application limits? Can their application range (in terms of chemical systems as well as conditions of state defined by temperature, pressure and composition) be increased even if more experimental data were made available? Also, since in many industrial separation problems, quantitative accuracy is very important, would it be better to generate processsystem specific model parameters? A principal deficiency of most traditional models is that application range can only be increased at the expense of accuracy or vice versa.
16.2 MATERIALS AND CHEMICAL PRODUCTS Chemical products involving specialty chemicals, micro-structured materials and formulated products are often multicomponent systems where the presence of 5-20 chemicals (species) is not unusual. Often, these chemical products comprise a range of different chemical compounds such as polymers, surfactants, solid particles, and water. Milk is an example of such a product involving both solid/liquid phases and (non-equilibrium) metastable states. Thus, many of these products are colloidal systems of different types - liquid-liquid emulsions, suspensions, powders, solid and liquid dispersions, aerosols and sprays. Many of these are often also classified as "soft materials". Such materials have found use in diverse industries and applications ranging from pharmaceuticals and paints to food, detergents and chemical devices. Therefore, the needs for thermodynamic modeling are very diverse The physical chemistry (thermodynamics, stability) of chemical products is often also as important as their manufacture and a number of non-traditional manufacturing/separation processes such as emulsification, foaming, gelation, granulation and crystallization have become relevant. Today, serious gaps exist in our property (thermodynamic) modeling abilities when we try to describe and understand chemical products with traditional thermodynamic models, typically applicable to problems of petrochemical industries. The chemicals that need to be handled often have fairly large molecular size and complex structures. This also makes the traditional models less accurate unless system specific special model parameters are regressed. Even in this case, it may be difficult to obtain acceptable qualitative and quantitative accuracy. It is clear that development of thermodynamic tools to quantitatively and qualitatively describe structured materials and formulated chemical products may require much more advanced techniques than those used for calculation of vapor-liquid and liquid-liquid equilibria for "ordinary" (low molecular weight) mixtures, of interest to petrochemical industries [6]. Even if quantitative accuracy can be achieved through specially fitted parameters or advanced techniques, unless they are also predictive in terms of chemical systems, they can only be used for the verification (screening) step of design of chemical products. For the generation of design alternatives, qualitatively accurate and truly predictive models having very large application range with respect to chemical systems are necessary. Note that an important issue in chemical product design is to define the product needs and functions through a set of properties, which are a combination of microscopic (atomic
410 structure based) and macroscopic (molecular structure based) properties. Therefore, in the generation step of the chemical product design problem, we may even need to use more than one property model for the same property type (for example, one at the microscopic level and another at the macroscopic level). The difficulties with the traditional property models (which are all modeling properties at the macroscopic level), however, is that they are not able to capture the molecular structural differences of isomers that the atomic level models are able to. More work is needed in this area as most chemical products of interest are isomers of a certain molecular type. We outline below some areas where serious gaps in current thermodynamic knowledge exist and where, based on the trends of current research activities, significant future developments appears to be very likely. The topics covered represent only a small portion of phenomena related to chemical product design where thermodynamic developments are both desired and hopefully also feasible. Many more areas within materials/chemical products require thermodynamic knowledge, which is either more or less mature (metal alloys) or extremely complex such as food [27]. However, even in the area of food products some first steps have been reported [28] including success stories e.g. the phase behavior of margarine and solidification of chocolate [27]. 13.3.1 Structured Polymers and Complex Interactions An area where much research is currently being done involves the new polymeric stmctures such as dendrimers, hyperbranched polymers, inorganic polymers used in antifouling paints & new complex blends and co-polymers. Thermodynamic and transport models should include the effect of structure in the properties especially swelling, diffusivity and permeability of gases. Moreover, in most cases, models have been independently developed for polymer-solvent systems and electrolytes. Such division is done for simplicity but should be abandoned [12] and new tools should focus on materials where both charged molecules (interactions) and polymeric structures are present such as poly electrolyte hydrogels [2]. An interesting question here would be to investigate if with the current property models for polymers, structured products can be designed. That is, can we design, for example, polyethylenes with specific branch parameters (number of branches, branch length and branch size) having desired end-use properties such as density, solubility of gases and kinematic viscosity? If such property models were available, computer-aided polymer (structure) design would also become feasible. 13.3.2 Surface Phenomena Surface phenomena are very important in the design (and function/operation) of many materials and products (surfactants, emulsions, adhesion). In this case, joint developments of advanced models like SAFT (Statistical Association Fluid Theory) with frameworks such as the DFT (Density Functional Theory) in order to predict surface and interfacial tensions are highly desirable. Such work is in progress, which is in the early stages of development [19].
411 13.3.3 Multicomponent, Multiphase Mixtures and Complex Phase Diagrams This is a general need that spans from the description of water-oil-surfactant-co surfactant mixtures of importance to the design of microemulsions up to the complex paint formulations containing mixed solvents. Existing models are typically not adequate and advanced theories need to be developed that can, at least qualitatively, predict the multiple liquid or solid phases present. These models should be formulated in a way so that they can be easily extended to multicomponent, multiphase systems and much focus should be given on distribution coefficients (rather than absolute values). Models expected to treat both polymers and electrolytes should be extended to cover surfactants as well. Polar interactions, hydrogen bonding, and association in all forms (including intramolecular association) are all expected to play an important role in these materials. When even ternary LLE for a system as simple as water-methanol-benzene is a challenge today for advanced thermodynamic models, we understand that even the "simple microemulsion" ternary system water-alkane-nonionic surfactant might be a serious challenge to meet in future developments, but could at least serve as the first milestone. 16.3.4 Emulsions Metastability is a challenge which, of course, is not solely within thermodynamics regime. However, the description of spinodal phenomena would be highly useful. 16.3.5 Pharmaceuticals and Agrochemicals The presence of solvents is essential in all steps of pharmaceutical processes (reaction, separation, and formulation). For toxicological reasons, drug manufacturers are increasingly required to minimize the amounts and number of solvents employed in processing. The selection of solvents is an important area of applied thermodynamics at industrial companies. Increasingly, solvents in current use need to be replaced with more "green"' or environmentally friendly substitutes. It has been estimated [20] that 30 % of the work of an industrial thermodynamic group can be related to various aspects of solvent selection. The molecules dealt with in the pharmaceutical area are substantially different from common petrochemicals. The high polarizability, the presence of heteroatoms and the existence of multiple functional groups in pharmaceuticals make the molecules liable to a wide variety of specific interactions with polar solvents, such as, protonation, hydrogen bonding and solvation. Solvent selection methods based on CAMD (computer aided molecular design) techniques are receiving increasing attention. A new solvent selection method for organic synthesis [21], related to pharmaceutical and agrochemical industries, is an interesting new development in this area. The problem, however, is that the application range of CAMD techniques is limited by the application range of property models being used. Extension of these techniques to solvent selection in the area of synthetic pharmaceuticals or at least prototypes of moderately flexible, highly functionalized molecules could serve as targets for future development. The needs for property models in the agrochemical sector, where also formulated products, mixed electrolyte systems, solubility of complex chemicals, solvent selection, etc.,
412 are encountered, have been recently highlighted by Cordiner [22, 23]. The use of property models in the study of pesticide uptake in plants and in the controlled delivery of pesticides have recently been reported [24]. 16.3.6 Polymorphism The ability to predict what polymorph precipitates under a given set of conditions is highly relevant in the food and pharmaceutical industries ([25], [26]). Crystal morphology is an important factor that determines a product's quality (e.g., dissolution rate), processing characteristics in downstream units (filtration, washing, drying) as well as formulation possibilities. For example, monoglycerides are often crystallized in an a-form and upon storage a recrystallization is initiated resulting in a change in crystal form (a—»P). The energy release connected with this transition is approximately equal to 50 kcal/kg. If this (exothermic) process happens in the final packaging, the result is the formation of one big lump of material rather than a free flowing powder, because heat released could not escape. There are currently no model to predict this behavior, which is highly dependent upon the solvent medium, the influence of which does not reduce to a simple set of rules. Most synthetic Pharmaceuticals are medium sized molecules containing 10-50 non-hydrogen atoms. The molecules are typically composed of several interlinked aromatic cores and multiple substituents containing heteroatoms N, P, O, S, and X = F, Cl. Due to the presence of the aromatic delocalized Ti-electrons and the electronegative heteroatoms, the molecules are highly polarizable so there is a strong interplay between the solute and solvent properties. Furthermore, in most cases, the solutes are conformationally flexible. By the possibility of exposing certain parts of the molecules, the conformational flexibility may affect both the reactivity and solvation behavior of the molecules, and also the formation of polymorphs. The identification of conformers has been intensively studied using molecular simulation and new methods for conformer searches in flexible molecules are currently being developed.
16.4 LIFE SCIENCES AND BIOTECHNOLOGY This is a vast area and much has been written on topics associated with thermodynamics. It is worth mentioning that several editorials of the AIChE J in the period 1999-2003 have been devoted to various aspects of biotechnology (biophysics, biochemistry, biomedical engineering, functional genomics, bioinformatics, metabolic engineering, and Pharmaceuticals). Some chemical industries are focusing more on biotechnology and biochemicals than traditional chemicals [29]. The role of thermodynamics has been also extensively discussed [1-4] and special courses on biothermodynamics have been organized [30]. The challenges are numerous and involve many fields: • •
Estimation methods for pure complex molecules, e.g. penicillin, for use in thermodynamic tools. Extension to charged macromolecules.
413 •
• •
Combination of the knowledge from the fields of polymers and electrolytes in the modeling of the thermodynamics of the so-called '"aqueous two-phase systems" of importance to some bioseparations (water-polymers-electrolytes-proteins). Phase separation / agglomeration phenomena in protein systems, of interest in the understanding of certain diseases such as Cataract and Alzheimer [3]. Computer aided drug design, where the uses of molecular modelling techniques, quantum mechanics and QSAR-type property (activity) models have been reported.
16.5 COMPUTER-AIDED SYSTEMS Recent advances in computer science and increased use of computers means that many of the simulation and design related calculations are being done through commercial software available through software vendors. Also, increasingly more and more software are developed by academic/research institutions, based on their research results. The transfer of property models from the model developers to the model users (that is, users of software), has been, however, very slow. Since industries use mainly commercial software and some "academic" special purpose software, the use of traditional property models is very high, compared to the new advanced property estimation techniques. Also, as shown in chapters 1 and 13, since property model equations need to be in the inner-loops of repetitive calculation cycles, unless the property model is computationally fast, their application becomes very limited. Another difficulty has been that the new advanced models also involve more complex calculations and therefore, their calculation efficiencies need further improvements to compete with the traditional models. While for the petrochemical industry, most of the chemicals of interest have been identified and well documented (in terms of available properties), the same is not true for the pharmaceutical, biochemical, agrochemical, etc., industries where new chemicals are continuously being synthesized. Special models need to be introduced in order to design or study processes handling these chemical systems. Consequently, for many current and future process-product design problems property models will need tuning or adaptation and the software will need to be flexible with features for property model "creation" according to the needs of the process-product design problem. This will require a thorough analysis of the property model equations and a verification of their derivative properties, including analytical expressions for the derivatives (at least, the first derivatives) with respect to the intensive variables. Use of computer-based property model analysis and creation will speed-up the transfer of the property model from the developer to the user. Finally, computer-aided techniques can also play an important role as the "integrator" or "glue" in multiscale modeling or integration of property models across different scales of size and time.
16.6 THERMODYNAMICS IN THE POST-MODERN WORLD Prausnitz [8] writes,
414 "For about 100 years, chemical engineering has had the loyal support and applause of the public. As the public becomes increasingly critical, that support is eroding. Encouraged by the ideas ofpostmodernism, the applause is fading, replaced by mistrust and accusations". Indeed, postmodernism, as a tendency/philosophical movement, does not reject science, but denies that science offers the only path to the truth. Postmodernism stresses the multidimensionality of phenomena and their interaction with society but, in addition, demands that those who practice science and engineering accept the responsibilities that such interaction implies. As a part of the society it serves, chemical engineering curriculum needs in the future to be "more human" and more "society/human/environment-oriented". In brief, the essential messages of postmodernism to chemical engineering (and consequently also to thermodynamics / property modeling, as part of the latter) are: • • •
Better products, not just in terms of properties, but also healthier and with respect to environment. Cleaner, safe processes, with respect to environment and the employees. Clean environment, minimum pollution, sustainable development.
Which challenges can be expected from thermodynamics in the future in order to meet these needs, especially if they continue to exist with the same persistence as today? In brief, more emphasis to thermodynamics / property modeling: •
•
•
For environment - Estimations of the distribution of long-lived chemicals in the environmental ecosystems [31], or more emphasis to environmentally-friendly separations such as membranes and supercritical fluid extraction [32-33]. At least approximately, correlations exist between the bioconcentration / biomagnification factors and the toxicity limits with the octanol-water partition coefficient [34-35]; the latter can be estimated from thermodynamics. For safety - Estimations of flammability limits, flash points and other safety parameters [11], assessment of the emissions from paint production and generally the development of environmentally-friendly coatings (with low VOCs and possibly largely water-based). The design of such paints is complex and largely empirical, but the multicomponent liquid-liquid equilibria (most paints employ mixed solvents) may provide some guidance of miscibility as a function of environmental factors (for example, temperature) [36, 37]. Another emerging field, where the role of thermodynamics seems important, is the simulation of corrosion in industrial systems. For health - Thermodynamics of Pharmaceuticals & drugs [38] and the other challenges from biotechnology discussed previously.
16.7 CONCLUSIONS Several future challenges and opportunities for property modeling have been outlined. The cases vary in complexity, and in the degree of multidisciplinarity. Several of the tools presented in this book may provide a starting point for addressing some of these challenges.
415 Solutions to these challenging problems may, in some cases, be obtained from extensions of group-contribution/solubility parameter techniques but are most likely to come from the developments in statistical theories such as SAFT-type approaches, in molecular simulation and other computational methods and tools, not covered here, such as quantum mechanics [39] and methods deriving from it [40]. Much can also is expected to be gained by the integration of property models and estimation approaches, across scales of size and time, across disciplines, across chemical systems and many more. It is necessary to have a solid foundation (root or basis or framework for the estimation technique) that will generate many new products (strong branches or safe methods or reliable tools for accurate and wide ranging property estimation).
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J. M. Prausnitz, Fluid Phase Equilibria, 104 (1995) 1-20. J. M. Prausnitz, Fluid Phase Equilibria, 53 (1989) 439-451. J. M. Prausnitz, J. Chem. Thermodynamics, 35 (2003) 21-39. J. M. Prausnitz, Fluid Phase Equilibria, 158-160 (1999) 95-111. E. Favre and M. Kind, "Formulation engineering: towards a multidisciplinary and integrated approach of the training of chemical engineers", Paper presented in the 2nd European Congress of Chemical Engineering (Montpellier), 1999 6. J. Villadsen, Chem. Eng. ScL, 52 (1997) 2857. 7. J.C. Rainwater, J. Chem. Eng. Data, 46 (2001) 1002-1006. 8. J. M. Prausnitz, Chem. Eng. Sci., 56 (2001) 3627-3639. 9. S. Zeck, Fluid Phase Equilibria, 70 (1991) 125-140. 10. W. Arlt O. Spuhl, A. Klamt, Chem. Eng. & Proa, 43 (2004) 221-238. U . S . Gupta and J. D. Olson, JD, Ind. Eng. Chem. Res., 42 (2004), 6359-6374. 12. C. C. Chen and P. M. Mathias, AIChE J., 48 (2002) 194-200. 13. E. C. Carlson, Chem. Eng. Progr., (1996) 35-46. 14. R. Agarwal, Chem. Eng. Progress, (2001) 64-72. 15. J. Marrero, R. Gani, "Method for prediction of unavailable group contributions: large and complex multifunctional molecules", paper presented at 20th ESAT, Lahnstein, Germany, October 2003. 16. C. Tsonopoulos and J.L. Heidman, Fluid Phase Equilibria, (1986) 391-414. 17. P. M. Mathias and H. C. Klotz, Chem, Eng. Progr., (1994) 67-75. 18. R. Dohrn and O. Pfohl, Fluid Phase Equilibria, 194-197 (2002) 15-29 19. G. J. Gloor, F. J. Bias, E. M. de Rio, E. de Migeul and G. Jackson, Fluid Phase Equilibria, 194-197(2002)521-530. 20. P. Kolar, J-W. Shen, A. Tsuboi, and T. Ishikawa, Fluid Phase Equilibria, 194-197 (2002) 771-782. 21. R. Gani, S. Christensen, C. Gonzalez-Jimenez, "A new solvent selection method for organic syntheis", paper presented at AIChE Annual Meeting, San Francisco, November 2003. 22. J. L. Cordiner, Computer Aided Chemical Engineering, 9 (2001) 27-40. 23. J. L. Cordiner, Computer Aided Chemical Engineering, 15A (2003) 125-140.
416 24. N. M. Sune, R. Gani, G. Bell, I. Shirley, Predictive property models for use in design of controlled release of pesticides", Fluid Phase Equilibria (submitted) 2004. 25. S. Karaborni, "Computer Simulations of Nanostructures in the Petrochemical and Pharmaceutical Industries", Plenary paper presented at FOMMS, USA, 2003. 26. P. G. Karamertzanis, and C. C. Pantelides, "Optimal Site Charge Models for Molecular Electrostatic Potentials", Paper A37 presented at FOMMS, USA, 2003. 27. T. Moorwood, "CAPE tools and techniques for the 21 st century. Properties of Materials and Mixtures - Where do we need to be 10 years from now ?", Report for Eureka Project 3211,2001. 28. S. Bruin, Fluid Phase Equilibria, 158-160 (1999) 657-671. 29. U. von Stockar, and L. A. M. van der Wielen, L.A.M., Journal of Biotechnology, 59 (1997) 25-37. 30. S. I. Sandier, and H. Orbey, Fluid Phase Equilibria, 82 (1993) 63-69. 31. S. I. Sandier, J. Chem. Thermodynamics, 31 (1999) 3-25. 32. M. Perrut, Ind. Eng. Chem. Res., 39 (2000) 4531-4535. 33. K. A. Larson and M. L. King, Biotechnology Progress, 2 (1986) 73. 34. H. Y. Cheng, G. M. Kontogeorgis and E. H. Stenby, Ind. Eng. Chem. Res., 41 (2002) 892. 35. H. Y. Cheng, H.Y., PhD Thesis. Institut for Kemiteknik, Technical University of Lyngby, Denmark, 2003. 36. T. Lindvig, M. L. Michelsen and G.M. Kontogeorgis, Ind. Eng. Chem. Res, 43 (2004) 11251132. 37.1. A. Kouskoumvekaki, G. Krooshof, M.L. Michelsen, G. M. Kontogeorgis, Ind. Eng. Chem. Res, 43 (2004) 826-834. 38. J. Sangster, J. Phys. Chem. Ref. Data, 28 (1999) 889-930 39. S. I. Sandier, "Quantum mechanics: a new tool for engineering thermodynamics", Fluid Phase Equilibria, (in press) 2003. 40. A. Klamt, A., F. Eckert, Fluid Phase Equilibria, 172 (2000) 43-72.
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Subject Index Subject
Page
Activity coefficient model library Algorithm for solubility analysis - electrolyte systems Algorithms for data regression - electrolyte systems Algorithms for phase diagram generation - electrolyte systems Application of property models in chemical product design Applications - UNIFAC Aqueous mixtures with hydrocarbons - Model performance - EoS/GE models ASOG Assessment of miscibility of plasticizers in PVC Azeotropy - thermodynamic behavior of hydrates Basic relationships in environmental engineering Binary non-polar mixtures - model performance - EoS/GE models Binary polar mixtures - model performance — EoS/GE models Binary systems - thermodynamic behavior of hydrates Calange's model class Carbazole Challenges — critical area of mixtures — CP A EoS Challenges - cross associating systems - CPA EoS Challenges — new associating compounds — CPA EoS Challenges (modeling) - systems with aromatic hydrocarbons Characterization of n-Alkanes Characterization of solvent Chemical potential of water in the hydrate phase Chemical reaction theory — hydrate phase equations Choice of solvents Choice of suitable mixed solvents in the paint industry Choice/needs of property models Choosing alternative solvents for stabilizing an emulsion Classical separations Classification of properties & models Combinatorial term - UNIFAC Comparison of models for polymers - application of SPECS Comparison of the performance of various UNIFAC models Complex phase diagrams Complex systems with SRK and CPA EoS - application of SPECS Composition of the hydrate
193 371 382 373 339 70 90 59 355 261 363 86 89 257 232 351 135 132 135 131 239 240 267 270 144 358 35 361 407 4 61 324 346,347 411 328 267
418 Compound classification Computation of LLE & VLLE Computation of SLE Computation of VLE Computational algorithms - electrolyte systems Computer-aided systems Condition of physical and chemical equilibrium Corticosterone CPA EoS - extension to free-volume models for gas solubility in elastomers CPA EoS - extension to semi-crystalline polymers and swelling CPA Equation of State Current status/results with CPA EoS Data - Mixture Data - Pure compound Data - Special Databases Definitions - diffusion coefficients Density - electrolyte systems Depressurization - thermodynamic behavior of hydrates Derivation of the hydrate phase compositions Derivation of thermodynamic functions - hydrate phase equations Derivative analysis Diffusion (solution approaches ) based on free volume or activation energy Diffusion in liquids Diffusion in multicomponent systems Dissociation constant & solubility product Driving force Effect of drying - thermodynamic behavior of hydrates Effect of pH on the solubility of DUP860 in water Effect of temperature on the solubility of Na2SO4 in H2O Effect of VLE-based parameters - in SGE predictions ELEC-ICAS Electrolyte NRTL model Emulsions Engineering rules for insight to thermodynamic behavior of hydrates Entropic-FV model Entropic-FV/UNIQUAC model EoS pure-compound parameters EoS/GE models - basic approach Equations of state - polymers Equilibrium conditions - hydrates Erickson's model class Estimation method for LogKow Evaluation of mixed grand partition function - hydrate phase equations Evaporation of isopropanol from a PVAC film - application of SPECS
53 315 318 311 371 413 202 13 169 165 113 123 55 54 55 53 208 184 259 270 274 39 216 213 205 186 35 258 372 373 104 399 195 411 255 157 165 85 81 172 264 231 343 273 320
419 Example of property model use - density estimation of polymers Example of property model use - estimation of pure component properties Example of property model use - extraction of heavy compounds using supercritical gas Example of property model use - heat pump fluids Example of property model use - parameter estimation Example of property model use - selective separation with SCFE Experimental methods for properties of electrolyte systems Extended UN1QUAC model (electrolyte system) Fluid characterization Force fields for complex fluids Force fields for hydrocarbons - application of molecular simulation Force fields for water - application of molecular simulation Free volume concept - polymer thermodynamics Free volume GE-models for polymers Free volume theory - applicability with respect to diffusion Fugacity of a chemical in air Fugacity of a chemical in Biota Fugacity of a chemical in solid and sediment Fugacity of a chemical in water Functional properties Functional properties Gas hydrates - importance Gas hydrates - methods to prevent Gas hydrates - reviews Gas hydrates - types Gas hydrates - which components form gas hydrates? General facts about diffusion General model equations - UNIFAC lst-order Henry's constant - for gas solubility in electrolyte systems High pressure - wax formation model Hydrate equilibrium curve Hydrate phase Hydrates - history ICAS: Integrated Computer Aided System Improving standard values for property variables Incidence matrix - LLE phase diagram model Incidence matrix - PT-flash mass balance model (gamma-phi approach) Incidence matrix - PT-flash mass balance model (phi-phi approach) Incidence matrix - SLE phase diagram model Incidence matrix — SRK EoS Incidence matrix - UNIFAC Incidence matrix - VLE computation model (gamma-phi approach) Incidence matrix - VLE computation model (ideal system) Incidence matrix - VLE computation model (phi-phi approach)
17 13 15 23 14 21 191 199 239 290 293 295 153 153 223 363 365 364 364 4 51 252 252 254 253 254 206 61 202 236 255 266 252 329 188 316 315 315 318 9 66 313 312 314
420
Inhibition - thermodynamic behavior of hydrates Integrator role versus choice/needs of property models Investigation of SRK EoS with various mixing rules - with SPECS Janecke phase diagrams KT-UNIFAC LCVM model Life sciences & biotechnology Liquid volume -electrolyte systems LLE & VLLE calculation examples - application of ICAS LLE and SLE - discussion of results - CPA EoS LLE for alcohol/alkanes - CPA EoS LLE phase diagrams LLE phase diagrams - electrolyte systems LLE phase separation calculations Local UNIFAC model for estimation of solubility of complex molecules Low pressure - wax formation model Material and chemical products Methanol/propane system - CPA EoS results MHV2 model Miscibility based on Flory-Huggins parameters Mixed grand partition functions - hydrate phase equations Mixture: mixing and combining rules for CPA EoS Model analysis - UNIFAC Model analysis - VLE computation Model description - CPA equation of state Model parameters - group interaction parameters for UNIFAC Model parameters - R & Q for UNIFAC Model performance - EoS/GE models in VLE/GLE predictions Model performance/limitations - UNIFAC Modeling high pressure was formation Modeling using vdW If mixing rules - performance evaluation Modeling with LCVM - performance evaluation Models for primary properties Molecular simulation methods Monte Carlo moves for long chain molecules Monte Carlo simulation methods for phase equilibria Multicomponent equilibria - CPA EoS Multicomponent mixtures - Model performance - EoS/GE models Multicomponent systems - thermodynamic behavior of hydrates Multicomponent, multiphase mixtures Multiphase flash calculations Multiphase separation model - electrolyte system Multi-scale mathematical modeling of chemical processes Non-ideality of liquid phase Non-ideality of solid phase
262 38 324 182 63 83 412 185 333 164 126 315 379 316 350 234 409 124 82 357 272 122 65 311 116 68 67 86 68 232 97 98 45 281 287 282 129 92 258 411 238 397 280 230 230
421 Octanol-water partition coefficient with UNIFAC Osmotic coefficient-based model for activity coefficient Parameter estimation - CPA EoS for pure fluids Parameter estimation - CPA EoS Parameter estimation - models for diffusion coefficient Partition coefficients - environmental engineering Performance of cubic EoS for SGE Pharmaceuticals and agrochemicals Phase diagrams (electrolyte systems9 Phase equilibrium calculations & thermodynamic models - hydrates Physical properties of the hydrates Pitzer-Bromley-Debye-Huckel model Polymer solutions - models Polymer/solvent systems - Diffusion -free volume basics Polymorphism Predictive free energy model class Predictive property models for base properties Predictive UNIQUAC Predictive Wilson Predictive Wong-Sandier model Primary properties Primary property models Properties & data analysis (electrolytes) Property model - use - examples Property model creation - electrolyte systems Property model development Property model needs in chemical product design Property modeling — challenges and opportunities Property models -electrolytes Property models - selection Property models - selection - decision tree Property models - use PSRK model PT-flash for VLLE Radial distribution function — CPA EoS Rarified gases — diffusion coefficients References for databases Residual term-UNIFAC Retrograde behavior - thermodynamic behavior of hydrates Role of properties Role of properties - integrator role Role of properties — service role Role of properties — service/advice role Rule of thumb — Flory —Huggins model Rules of thumb - infinite dilution activity coefficients
342 201 119 118 220 366 97 411 181 264 251 193 143 218 412 232 342 234 237 84 4 46 181 13 383 5 340 407 192 10 11 8 82 317 118 212 55 62 259 27,28 33 30 31 149 147
422
Rules of thumb - solubility parameters - polymer models Saturated polymers and complex interactions Search engine - database Secondary properties Secondary property models Service role versus choice/needs of property models Service/advice role versus choice/needs of property models Simulation methods for chemical potential calculation SLE phase diagram - electrolyte systems Solid phase and thermophysical properties Solid solution model development - hydrates Solubility as a function of temperature and pH Solubility index Solubility of morphine - Application of ICAS SPECS Structure transition - thermodynamic behavior of hydrates Study of activity coefficients for polymer solutions - application of SPECS Surface phenomena Ternary mixtures - Model performance - EoS/GE models The CAPEC database The UNIFAC method Thermodynamic behavior of hydrates Thermodynamic functions - hydrate phase equations Thermodynamic model - hydrates Thermodynamic properties Thermodynamic properties of the empty lattice Thermodynamics in the post-modern world Thermophysical properties for the n-alkanes Types of phase equilibrium problems Ungerer's model class UNIFAC UNIFAC - different versions UNIFAC interaction parameters - for EoS/GE models UNIFAC-FV model (polymers) UseofCPAEoS Van Der Waals one fluid mixing rules VLE for polymers - discussion of results - CPA - EoS VLE mixtures with associating substances - CPA EoS results VLE phase diagram - electrolyte systems VLE phase diagram calculations - application of ICAS VLLE phase diagrams Water activity & osmotic coefficient Water-alkanes (LLE) - CPA EoS results Water-butanol-benzilpenicillin Water-butanol-sodium chloride system
145 410 55 4 48 36 37 285 376 230 269 203 202 333 319 264 322 410 92 53 60 255 272 265 185 268 413 240 309 231 59 63 86 156 131 77 158 123 374 329 317 190 123 378 381
423
Water-calcium chloride-ethanol system Water-glycine-(l)asparagines-fumaric acid system Water-glycine-potassium chloride system Wax formation model What are gas hydrates? Won's model class Wong-Sandier model
376 378 385 233 251 231 83
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425
Author Index Author
Page
Jens Abildskov Joao A. P. Coutinho Philippos Cousikos Jean-Luc Daridon Peter K. Davis J. L. Duda Ioannis G. Economou Rafqul Gani Eric Hendriks Georgios M. Kontogeorgis Jorge Marrero HenkMeijer John P. O'Connell Jerome Pauly Michael L. Pinsky Alexander Shapiro Kiy oteru Takano Epaminondas C. Voutsas
59, 339 229 75 229 205 205 279 3, 27, 45, 59, 309, 339, 371, 407 251 3, 59, 75, 113, 143, 309, 339, 407 45 251 27 229 181 205 181,371 75
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