Znd. Eng. Chem. Res. 1993,32, 3123-3127
3123
Phase Equilibria in Polymer Solutions. Block-Algebra, Simultaneous Flash Algorithm Coupled with SAFT Equation of State, Applied to Single-Stage Supercritical Antisolvent Fractionation of Polyethylene Chang-keng Chen, Marco A. Duran, and Maciej R a d o d Exxon Research and Engineering Company, Annandale, New Jersey 08801
A robust but rigorous flash algorithm allows for simulating multicomponent two-phase equilibria in supercritical asymmetric solutions that contain very large and very small components, such as polymers and supercritical solvents or monomers. The approach is to solve simultaneously all the equilibrium and material balance equations using a Newton iterative procedure in the framework of block algebra. The SAFT equation of state is used to estimate the required thermodynamic properties. This approach is illustrated with an example of single-stage fractionation of polyethylene with supercritical ethylene and 1-hexene. The calculated solvent capacities and selectivities are related to ethylene concentration, pressure, and temperature.
K factors are composition independent is no longer a
Introduction Supercritical solvents are selective with respect to molecular weight, and hence can be used for polymer fractionation. For example, polymer lights can be extracted, and polymer raffinate can be enriched with heavies. This is possible because polymer solubility in the supercritical solvents rapidly increases with decreasing molecular weight. One way to screen and optimize supercritical solvents is on the basis of their equilibrium capacity and selectivity. The supercritical solvent capacity, defined as the polymer solubilityin weight percent, can be tuned by adjusting the solvent density with either temperature or pressure or solvent composition, as proposed by Radosz (1992);the higher the density the higher the solvent capacity. The selectivity, also referred to as the separation factor, is defined as the ratio of equilibrium K factors for the components of interest. The K factor is a ratio of the weight fraction in the solvent-rich phase to that in the polymer-rich phase. In general, a good solvent should have a high capacity and high selectivity, for low treat rate and good separation, respectively. In order to estimate the equilibrium capacities and selectivities,we need a thermodynamic model, usually in the form of an equation of state. In this work, we use the statistical associating fluid theory (SAFT) as our thermodynamic model. SAFT was demonstrated by Chen et al. (1992)to predict phase equilibriain solutions containing both large molecules, such as polymers, and small supercritical molecules. Also, SAFT pure component parameters are well behaved and easy to estimate for large molecules (Huang and Radosz, 1990). In order to solve all the phase equilibrium and material balance equationsthat form a system of nonlinear algebraic equations, we need an iterative algorithm referred to as the flash algorithm. Numerical procedures proposed to solve such a system of equations are reviewed by Henley and Seader (1981). An example of a conventionalapproach is the Rachford-Rice (RR) algorithm that was developed for vapor-liquid equilibria in symmetric systems, that is, systems of similarly sized molecules. The RR algorithm is relatively simple because the K factors are assumed to be independent of composition. As a result, the system of flash equations can be reduced to just one nonlinear equation that is solved by standard numerical methods. The problem with the RR approach is that its convergence becomes less and less reliable as the difference in molecular sizes increases. This is because the assumption that the 0888-588519312632-3I23$04.O0/0
reasonable approximation for strongly asymmetric systems. Therefore, the goal of this work is to develop a robust single-stage flash algorithm for multicomponent two-phase equilibriain asymmetric solutions,such as those of polyethylene in supercritical ethylene and 1-hexene.
Flash Algorithm and SAFT Equation of State An example of a typical flash problem for two-phase equilibria at constant temperature and pressure is to solve asystem of material balance, equilibrium, and consistency equations given below: firnabrid baiancs
= (1- 4 ) x i + $yi - zi = 0 for i = 1, ...,N (1)
fieqdbrium
= Kixi - y i = 0 for i = 1,...,N
(2) (3)
where 4 is the ratio of the total mass flow rate of the solventrich phase to that of the feed stream; N is the number of species in the mixture; x i , yi, and zi are the weight or mole fractions of component i (i = 1,...,N), respectively, in the polymer-rich phase, solvent-rich phase, and feed stream; and Ki is the separation factor of component i which is defined as yi/xi. In general, Ki’s are obtained from a thermodynamic model, usually in the form of an equation of state, and x i (i = 1, ..., N), yi (i = 1, ..., N), and 4 are treated as unknowns. Since there are N material balance equations (11, N equilibrium equations (2), and one consistency equation (3),we end up with a system of 2iV + 1nonlinear algebraic equations. It is easy to show (Henley and Seader, 1981) that, if the Ki’s are assumed to be independent of composition, this system of equations can be reduced to a single nonlinear algebraic equation, which can then be solved using a standard iterative method, e.g., the RR algorithm. The extent to which this approximationusually breaks down is illustrated in Figure 1for binary mixtures of ethylene and monodisperse polyethylene (PE)at 200 OC. The pressure-composition phase boundaries shown in Figure 1are calculated both ways: according to the RR approximation and rigorously (K;s are composition dependent). The rigorously calculated curves are labeled 0 1993 American Chemical Society
3124 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 800
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block linear algebra treatment and hence problem decomposition. For the specific case of the flash equations (l),(2), and (3),the incidence matrix of the Jacobian matrix (6)is a N + 1)square matrix with the followingE ucture:
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Figure 1. P-X equilibrium curves for binary mixtures of ethylene and polyethylene obtained from Rachford-Rice (RR)and blockalgebra simultaneous (BAS)flash algorithms at 200 OC for three different values of PE molecular weight (1000,3000,20OOO g/mol).
BAS, which refers to a block-algebra simultaneous approachdescribed in the next paragraph. As Figure 1shows, the RR approach converges for PE of 1000 g-mol over the whole pressure range, but becomes less reliable for PE of 3000 g-mol at lower pressures, and even less reliable for PE of 20 000 g-mol. As the size differences between the solvent and solute molecules increase, the pressurecomposition region of convergencefor RR becomes smaller and smaller. In general, it is because for systems such as that shown in Figure 1,the Rachford-Rice assumption on Ki becomes less reliable with increasing size difference and decreasing pressure. The reason for the improved convergence at high pressures is that the K dependence on composition becomes weaker as we approach the mixture critical point; dKJdxj and dKJdyj become smaller because at the critical point Ki's are equal to 1. Thus, a good initial estimate may assure smooth convergence at higher pressure. By contrast, the rigorous approach results in reliable convergence regardless of the size difference and pressure. In this approach, we solve the 2N + 1 equations (11, (21, and (3) simultaneously. These equations can be expressed as a generic system of equations: fj(0) = 0 for j = 1,...,2N + 1
(4)
where ir is the vector of 2N + 1unknowns, 8 = { ( X i : i = 1, ...,N),(yi: i = 1,...,N), 4}. Such a system of equations (4) can be solved simultaneously using a Newton iterative equation is ak+l
= irk + ~
where the first N rows represent the derivatives of the balance with respect to Xi, yi (i = equations (1)for timabrid 1,...,N),and 4 (in that order); the next N rows represent the derivatives of the equations (2) for fieqfibfium with respect to xi, yi (i = 1, ...,N), and 4 (in that order); and the last row represents the corresponding derivatives of equation with respect to Xi, yi the equation (3) for fconsisbnq (i = 1,..., N), and 4 (in that order). The matrix blocks separated by dashed lines in (7) define the following equivalent block representation of the Jacobian:
where A, B, C, and D are the matrix blocks that have the dimensions of N X N, N X (N + l),(N + 1) X N, and ( N + 1)X (N l),respectively. What is special about the Jacobian in (8)is that A is a diagonal matrix and B is an almost diagonal matrix. For efficient solution of (61, exploiting the special structure identified in (€9,let us define the following partition for ii and f vectors:
+
where iil contains the first N rows of ii vector, that is, the changes of variables x i , i = 1, ..., N,ii2 contains the remaining N + 1rows of ii vector, that is, the changes of variables, yi, i = 1,...,N, and 4. Similarly -8' contains the first N rows of f vector and -g2 contains the remaining N + 1 rows of f vector. Therefore, following Monroy and Duran (1987), we rewrite (6) as
( c k )
(5) where k is the iteration index, Bk is the current solution estimate, ii(irk)is the predicted vector of solution changes as a function of irk, and irk+' is the vector of new solution estimates. The solution step ii(8k) at iteration 12 is obtained by solving the following system of linear equations
J(Dk)ii(Dk) - f ( B k ) (6) whereT(Dk)is the constant vector of functions (4) evaluated at irk, and J(Bk)is the constant Jacobian matrix of partial derivatives (of f j with respect to ui) evaluated at irk. One specific iterative approach to solvingthe system of linear equations (6) was proposed by Monroy and Duran (1987). They realized that the Jacobian matrix in (6) has easily identifiable matrix blocks that lend themselves to
[cjD][;;] A'B =
[$]
As a result of block lower-upper (LU)factorization of the Jacobian J in (€9,and the required block linear algebra in ( l l ) , one obtains an equivalent decomposed linear system of equations: (D - C A - ' B ) ~=~&' - ~ 61 = A-'@
- BG2)
~ - 1 2 '
(12)
(13) In this decomposition, the variables ii2 are obtained independent of iil by solvingthe reduced linear subsystem of N + 1 equations (12). Once ii2 has been determined, ii1 is obtained explicitly from (13). We note that obtaining
Ind. Eng. Chem. Res., Vol. 32, No. 12,1993 3125 ErtrPct (Lights)
Table I. SAFT Molecular Parameters Used in This Work component ethylene 1-hexene PE 500 PE 1000 PE 2000 PE3000 PE8000 PE 2oooO
molarmass (g/mol) 28.05 84.16 500 lo00 2000 3000 8000 2oooO
wt % in
V""
feed 0.15 0.7 0.000105 0.001395 0.003 0.003 0.0075 0.135
uo/k
m (cm3/mol-seg) (K) 18.15 212.06 1.46 13.00 204.71 4.51 24.03 11.98 209.96 11.9 210 47.35 11.87 210 94.0 11.86 210 140.6 11.84 210 373.9 933.6 11.83 210
A-1 is simple because A is a diagonal matrix. In general, solution of (12) and (13) requires less computing and is numerically more stable than that of the original system (6) because of the reduced dimensionality and the special matrix structure of A and B. A more detailed performance comparison with straight Newton implementation is given by Monroy and Duran (1987). We refer to the algorithm described above as the block-algebra simultaneous flash algorithm, BAS for short. We customize BAS for SAFT models of polymer solutions. Specifically, we use the SAFT equation of state to calculate Ki's in (2). SAFT accounts explicitly for contributions due to the segmental size and energy in P E molecules, as well as those due to the PE chain length. SAFT is defined in terms of the residual Helmholtz energy ares
ares
= aref
+ adisp
(14) where the dispersion energy (adisp) is a mean field contribution described by Chen and Kreglewski (1977), and the reference energy (aref)is the sum of hard-sphere, chain, and association contributions: aref
= ab + achain + aassoc
(15) A more detailed description of SAFT is given by Huang and Radosz (1990, 1991). In this work, the PE model systems do not exhibit specific interactions that lead to association; thus amaoc is set equal to zero. Three pure component parameters, u o o , m, and u'lk, characterize a nonassociating SAFT molecule: uoo is the segment volume, m is the segment number, and uo/k is the segment energy. Table I lists the SAFT parameters used in this work. One binary interaction parameter, kij, associated with ad*p is adjusted to fit experimental data, when available.
Application Example: Simulating Single-Stage Fractionation of SAFT Polyethylene with 1-Hexene and Supercritical Antisolvent Ethylene We test the BAS flash on single-stage fractionation of polyethylene (PE)with supercriticalethyleneand 1-hexene and examine the effects of antisolvent, pressure, and temperature on the solvent capacity and selectivity. This is an example of a supercritical mixed-solvent separation process proposed by Radosz (1992) for polydisperse polymeric mixtures. P E is a collection of model SAFT molecules and does not mimic any real specific polyethylene. Our SAFT PE is composed of six pseudocomponents having molecular weights of 500, 1000, 2000, 3000, 8000, and 20 000 g-mol. In this case, 1-hexeneis the main solvent component whereas supercritical ethylene is an antisolvent. Therefore, we refer to such a process as supercritical &tisolvent (SAS) fractionation. A simplified SAS fractionation process concept is shown in Figure 2, where a mixture of ethylene, 1-hexene, and PE is fed into a constant-temperature and -pressure separator. The feed mixture separates into a solvent-rich
70 wt% C6 15 wt% polymer
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LIGHTS (2k-) IN SOLVENT FREE RAFFINATE, WEIGHT PERCENT Figure 3. Increasing extract yield decreases the weight percent of the 2k fraction (lights with MW less than 2000 g/mol) in raffinate. The higher the ethylene concentration the lower the extract yield. Data obtained from single-stage SAFT simulation for 15 w t 76 polymer. Symbols of the same type represent the Bame temperature and pressure but different weight percent of ethylene: 5, 15,30,45.
0.0001 0.001 0.01 CAPACITY, WEIGHT FRACTION Figure 5. Increasing antisolvent (ethylene) concentration increases selectivity (in this case along 2k/3k lines), but decreases capacity (polymer solubility). I
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CAPACITY, WEIGHT FRACTION Figure 4. Increasing capacity (polymer solubility) decreases selectivity (in this case along 2k/3k lines). The higher the ethylene concentration the lower the extract yield. Data obtained from single stage SAFT simulation for 15 wt % polymer. Symbols of the same type represent the same temperature and pressure but different weight percent of ethylene: 5, 15, 30, 45.
lower the lights weight percent in raffinate. Figure 4 illustrates the lightdheavies selectivity, for example K(2000)/K(3000), plotted against the solvent capacity. As expected, the higher the capacity the lower the selectivity. In Figures 3 and 4, all the points obtained at different temperatures, pressures, and antisolvent concentrations fall close to a single curve. One way to explain these relationships is in terms of the solvent density. For a given solvent, we find that the capacity-selectivity relationship at constant temperature can be controlled with the solvent density; qualitatively, the higher the density the higher the capacity and the lower the selectivity. Thus, one needs to understand the effects of antisolvent concentration, pressure, and temperature on the solvent density to optimize the solvent selectivity and capacity. For example, increasing antisolvent concentration in-
creases the selectivity but decreases the capacity, as predicted and shown in Figure 5. This can be explained by decreasing solvent density because increasing the antisolvent concentration decreases the solvent density. In our case, when the ethylene concentration increases from 5 to 45 wt ?6 ,the overall solvent capacity decreases by 3 orders of magnitude. Also decreasing pressure decreases density and, hence, has the same effect on the selectivity and capacity as increasing antisolvent concentration; that is, it decreases the capacity and increases the selectivity. The effect of pressure is shown in Figure 6. For example, when the pressure is increased by a factor of 2, say, from 100 to 300 bar, the capacity increases by a factor of 4 say, from 0.0008 to 0.003 weight fraction. In contrast, the capacity dependence on temperature is not monotonic; its slope depends on the antisolvent concentration and pressure. That is, at some conditions the capacity increases, whereas at other conditions it decreases, with increasing temperature. For example, as shown in Figure 7, the capacity increases with increasing temperature at 15wt 5% ethylene antisolvent. Conversely, at higher antisolvent concentrations, for example at 45 w t 5% , increasing temperature decreases the capacity while increasing the selectivity.
Ind. Eng. Chem. Res., Vol. 32,No.12,1993 3127 1000
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This block-algebra simultaneous (BAS) algorithm is demonstrated on supercritical antisolvent (SAS) fractionation of a model SAFT polyethylene with ethylene and 1-hexene. Such a simulation allows for quantifying the effects of antisolvent (ethylene) concentration, pressure, and temperature on the solvent capacity and selectivity. In general, conditions that favor high solvent densities are found to favor high capacities and low selectivities.
Acknowledgment Professor J. M,Prausnitz and Dr.A. H. Wu provided us with software for Sako et al.'s (1989)model.
-60 1
Literature Cited -90
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These effects of the solvent density on its selectivity can also be explained by analyzing the molecular weight dependence of the separation factor K. In general, the separation factor drastically decreases with increasing molecular weight at a given set of conditions, as shown in Figures 8 and 9. Increasing concentration of antisolvent, shown in Figure 8,and decreasing pressure, shown in Figure 9, make the ln(K) versus molecular weight slope more negative, which leads to higher selectivity.
Conclusions A rigorous and robust flash algorithm coupled with a SAFT model is developed for single-stage two-phase equilibria in asymmetricpolymer solutions. Conventional flash approaches, such as the Rachford-Rice algorithm, are found unreliable for such solutions. The key to improvement is an approach rooted in block algebra and simultaneous convergence etrategy.
Chen, S. S.; Kreglewski, A. Ber. Bunsen-Ges. Phys. Chem. 1977,81, 1048. Chen, S.; Economou, I. G.; Radosz, M. Macromolecules 1992, 25, 3089. Henley,E. J.; Seader,J. D. Equilibrium-StageSeparation Operatione in Chemical Engineering; Wiley: New York, 1981. Huang,S . H.;Radosz, M. Ind. Eng. Chem. Res. 1990,29, 2284. Huang, S. H.; Radosz, M. Ind. Eng. Chem. Res. 1991,30, 1994. Luft, G.; Lindler, A. Angew. Makromol. Chem. 1976,56,99. Michelsen, M. L. Fluid Phase Equilib. 1980,4, 1. Monroy, R.; Duran, M. A. A Mathematical-Structure Exploiting Strategy for Process Simulation: The Flush Problem; Internal Report, Department of Engineering, Universidad Autonoma Metropolitana-Iztapalapa: Mexico City, July 1987. Monroy, R.; Duran, M. A. The Flash Problem: Solution by a Structured Approach. In Auances en Ingenieria Quimica 1987; Academia Mexicana de Investigaciony Dodecencia en Ingenieria Quimica: San Luis Potosi, SLP, Mexico, September 1987; pp 7480. Radosz, M. Supercritical Mixed-SolventSeparation of Polydisperse Polymers. Eur. Pat. Appl. EP489574A210,June 1992,U.S. Serial No. 892,462, allowed, 1993. Sako, T.; Wu, A. H.; Prausnitz, J. M. J. Appl. Polym. Sci. 1989,38, 1839.
Received for review M a y 17, 1993 Revised manuscript receiued August 12, 1993 Accepted August 30, 1993. Abstract published in Advance ACS Abstracts, November 1, 1993.
