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Ind. Eng. Chem. Res. 1996, 35, 1188-1194
Effect of Unequal Transport Rates and Intersolute Solvation on the Selective Batch Extraction of a Dilute Mixture with a Dense Polymeric Sorbent Timothy A. Barbari,* Sameer S. Kasargod, and Gregory T. Fieldson† Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218
A transport model is developed for the selective extraction of a solute from a binary solute mixture using a dense polymeric sorbent in a batch process. The effects of unequal solute transport rates and molecular solvation between solutes in the polymer phase are considered. In the absence of intersolute solvation, the model reduces to the analytical solution for mass uptake from a finite volume. The inclusion of intersolute solvation requires a numerical solution owing to the coupled nature of the species continuity equations. The model is solved numerically using the method of finite differences and compared to the analytical solution in the absence of solvation for verification. Results from the model indicate that separation factors significantly higher than the equilibrium value can be obtained at intermediate times and at high levels of extraction when the diffusion coefficient of the desired solute is greater than that of the undesired one. Intersolute solvation in the polymer can enhance this effect under certain conditions by retarding the transport of the undesired solute to a greater degree than that of the desired solute. The magnitude and range of diffusion coefficients in polymers allows for the practical application of such an approach to improve a selective extraction process based on nonporous polymeric sorbents. The measurement of solvated and unsolvated diffusion coefficients in a dense polymer film using FTIR-ATR spectroscopy is demonstrated. Introduction Liquid-liquid extraction has been studied extensively to separate organic compounds from aqueous streams using conventional or chemical complexing extractants. Examples include the extraction of chlorinated hydrocarbons and aromatics from water into undecane (Barbari and King, 1982), the extraction of polar organic pollutants from water into a variety of solvents (Joshi et al., 1984), the recovery of acetic acid from water using tertiary amines or trioctyl phosphine oxide (TOPO) in solvents such as alcohols, ketones, or chlorinated hydrocarbons (Wardell and King, 1978; Spala and Ricker, 1982; Ricker et al., 1979), the removal of phenol using amines in benzene, 2-ethylhexanol, or carbon tetrachloride (Pollio et al., 1967; Pittman, 1979; Inoue et al., 1980) or with TOPO in diisobutyl ketone (MacGlashan et al., 1985), the extraction of ethanol using phenolic extractants in toluene, chloroform, and n-butyl acetate (Arenson et al., 1990), and, more recently, the enhancement of extraction using solvation between dicarboxylic acids, water, and a ketone or ester solvent (Lee et al., 1994). These examples illustrate the ability to remove effectively a single solute from an aqueous stream but do not address the selective removal of a particular compound from a solute mixture in water. Such separations are desirable in wastewater treatment if one of the solutes is refractory or toxic to subsequent biological treatment. Another application is the selective recovery of a value-added product at dilute levels in fermentation broths or from specialty chemical processes. In the extraction of any solute from an aqueous feed with an organic solvent, true immiscibility does not exist in * To whom correspondence should be addressed. FAX: (410) 516-5510. E-mail:
[email protected]. † Present address: ALZA Corp., 950 Page Mill Road, P.O. Box 10950, Palo Alto, CA 94303-0802.
0888-5885/96/2635-1188$12.00/0
practice. A small amount of the solvent partitions into the aqueous phase, creating or adding to a waste stream. In some instances, the best extraction solvent for the selective recovery of a solute may itself be a toxic or hazardous pollutant. Dense, polymeric sorbents are one alternative to organic liquid solvents for selective extraction processes. The selectivity for the desired solute can be enhanced, as with liquid solvents, through specific groups along the backbone of or attached pendantly to the polymer chain. For example, Kawabata and Ohira (1979) and Kawabata et al. (1981) used cross-linked poly(4-vinylpyridine) to extract phenol from water and carboxylic acids from aqueous solutions, respectively. Garcia and King (1989) used a variety of basic polymeric sorbents, both nonporous gels and macroporous beads, to recover acetic acid from dilute aqueous solutions. Reversibility remains important for sorbent regeneration which can be accomplished using a leaching operation. Polymeric sorbents exhibit “one-way” immiscibility; the extractant does not partition into the aqueous phase, but water can partition into the polymer. Here, for the purposes of modeling transport processes in the dense polymeric sorbent, the aqueous and polymer phases will be taken as immiscible. In this paper, a transport model for the selective extraction of a compound from a dilute solute mixture into a dense polymeric sorbent is developed for a batch process. Owing to lower solute diffusion coefficients in polymers relative to liquids and the more dramatic effect of solute size on the magnitude of these diffusion coefficients, differences in transport rates can influence selectivity during a batch extraction in a manner similar to the effect in solution-diffusion membrane separations. In addition, intersolute solvation within the polymeric sorbent can occur, even at low concentrations, and therefore is considered here. Families of curves are generated to show the effect of various parameters on the extraction process. To fully utilize a model for © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1189
“reaction” given by
A + B h AB
(4)
The ratio of kf to kr is the equilibrium constant, Keq, which is given by
Keq ) CAB/CACB
Figure 1. Schematic of the batch extraction process using polymeric sorbent particles of uniform size.
molecular transport in a dense polymeric sorbent, the individual parameters should be experimentally measurable. The use of infrared spectroscopy to measure diffusion coefficients for both solvated and unsolvated solutes in polymers is demonstrated.
Equations 1-3 assume that the total concentration (molar density) of the polymer phase is constant and that the diffusion coefficients are independent of concentration. The latter assumption is reasonable at the low concentrations expected in polymeric sorbents used for dilute extractions. Initially, the polymer particles do not contain any solute:
at t ) 0, at all r:
CA ) 0 CB ) 0 CAB ) 0
Model Development The use of a dense polymeric sorbent in a batch extraction process is shown schematically in Figure 1. Here, nonporous polymer spheres of a given density and size are suspended in a dilute aqueous solution of two solutes, A and B, both of which can solvate with water. Solute A can also solvate with B to form AB; however, at dilute concentrations, negligible amounts of AB will exist, and A and B can be assumed to be solvated with water only. At the interface between the dense polymer sorbent and the aqueous external phase, both solutes partition into the polymer phase. The thermodynamic partitioning at the surface is taken to be selective for A. If the partition coefficients are greater than 1, the slightly higher polymer phase concentrations may allow A and B to solvate with each other to form AB. Using FTIR spectroscopy, Fieldson and Barbari (1995) showed that both unassociated and self-associated methanol exist at very low concentrations in polystyrene. In a more recent study, Plunkett (1995) demonstrated the presence of unassociated and self-associated methanol and ethanol in polybutadiene at concentrations between 0.2 and 2 wt %. On the basis of this work, intersolute solvation in hydrophobic dense polymers is expected to occur at the low concentrations considered here in the transport model. In addition to demonstrating the existence of self-associated methanol, Fieldson and Barbari (1995) showed that unassociated methanol in polystyrene interacts with the π electrons of the aromatic side groups. Similar and other stronger interactions, such as hydrogen bonding and acid-base complexation, can be incorporated into the dense polymer to improve the thermodynamic selectivity. Polymer Phase. For the polymer phase, the following continuity equations can be written for each species accounting explicitly for intersolute solvation:
(
)
(1)
(
)
(2)
∂CA DA ∂ 2 ∂CA ) 2 r - kfCACB + krCAB ∂t ∂r r ∂r ∂CB DB ∂ 2 ∂CB ) 2 r - kfCACB + krCAB ∂t ∂r r ∂r
(
(5)
)
∂CAB DAB ∂ 2 ∂CAB ) 2 r + kfCACB - krCAB ∂t ∂r r ∂r
(3)
where Ci is the concentration of component “i”, Di is its diffusion coefficient, and kf and kr are the forward and reverse rate constants, respectively, for the solvation
The boundary condition invoked at r ) 0 is the no flux condition:
at r ) 0, at all t:
∂CA/∂r ) 0 ∂CB/∂r ) 0 ∂CAB/∂r ) 0
At the interface between the particles and the external solution (r ) R), partition coefficients can be used to relate the polymer phase and aqueous phase concentrations:
at r ) R, at all t: CA + CAB ) SACAe CB + CAB ) SBCBe CAB ) KeqCACB where SA and SB are the partition coefficients for A and B, respectively, into the polymer phase. By including AB in the definition of the partition coefficient, the model can examine the effect of intersolute solvation on both selectivity and the fraction of A extracted at intermediate times without affecting their final values at equilibrium. The utility of this definition for SA and SB will become apparent when solutions of the model are examined in detail below. External Phase. If the external phase is assumed to be well mixed, then no mass-transfer gradients will be present and the concentrations of A (CAe) and B (CBe) can be taken as spatially uniform. This is reasonable given that diffusion coefficients in liquids typically are 2 or more orders of magnitude greater than those in polymers. Since the external phase is dilute and the concentrations of A and B in the polymer are low, the volumes of the external and polymer phases, Ve and Vp, respectively, will be taken as constant. As extraction proceeds, CAe and CBe are depleted according to the following material balances:
Ve[C0Ae - CAe(t)] ) Np[MA(t) + MAB(t)]
(6)
Ve[C0Be - CBe(t)] ) Np[MB(t) + MAB(t)]
(7)
where C0Ae and C0Be are the initial concentrations in the external phase, Np is the number of spherical polymer particles, and MA, MB, and MAB are the mass uptakes of A, B, and AB in the polymer phase, respectively. In
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θ2 ) 0
this paper, the external phase initially will be taken as equimolar (C0Ae ) C0Be). The mass uptake per particle for species “i” is defined as
∫0Rr2Ci(r,t) dr
Mi ) 4π
θ3 ) 0
(8)
θ4 ) 1
Numerical Solution Procedure
θ5 ) 1
The model developed above is not amenable to an analytical solution when intersolute solvation occurs owing to the coupled nature of the continuity equations. In the absence of solvation, the model reduces to the equation describing mass uptake from a finite volume (Crank, 1975) for both A and B. This analytical solution was used to verify the numerical solution. To facilitate a numerical solution, the species continuity equations were cast into dimensionless form using the following variables:
θ1 ) CA/C0Ae
∂θ2/∂ξ ) 0 ∂θ3/∂ξ ) 0 The boundary conditions at the solution-polymer interface become:
θ1 + θ3 ) SAθ4 θ2 + θ3 ) SBθ5
θ3 ) CAB/C0Ae
θ3 ) m1θ1θ2
τ ) DAt/R2 ξ ) r/R Equations 1-3 become
∂θ1 ∂2θ1 2 ∂θ1 ) 2 + - m1m2θ1θ2 + m2θ3 ∂τ ξ ∂ξ ∂ξ
]
∂θ2 ∂2θ2 2 ∂θ2 ) m4 + - m1m2θ1θ2 + m2θ3 ∂τ ξ ∂ξ ∂ξ2 2
(9)
(10)
]
∂θ3 ∂ θ3 2 ∂θ3 ) m5 + + m1m2θ1θ2 - m2θ3 (11) ∂τ ξ ∂ξ ∂ξ2 and eqs 6 and 7 become
1 - θ4(τ) ) 3m3[µA(τ) + µAB(τ)]
(12)
1 - θ5(τ) ) 3m3[µB(τ) + µAB(τ)]
(13)
where
m1 ) KeqC0Ae ) (kf/kr)C0Ae m2 ) krR2/DA
(
)
4 m3 ) Np πR3 /Ve ) Vp/Ve 3 m4 ) DB/DA m5 ) DAB/DA µi )
∫01ξ2θi dξ
The initial conditions become:
at τ ) 0, at all ξ:
∂θ1/∂ξ ) 0
θ2 ) CB/C0Ae
θ5 ) CBe/C0Ae
[
at ξ ) 0, at all τ:
at ξ ) 1, at all τ:
θ4 ) CAe/C0Ae
[
and the boundary conditions at the center of the particles become:
θ1 ) 0
This system of equations was solved using the method of finite differences. An explicit forward-in-time, centralin-space algorithm was employed. The space domain from the center of the particle to the surface was divided into 201 node points, and a dimensionless time step of 1 × 10-5 was used. Second-order accurate expressions were used for all spatial derivatives, and first-order accurate expressions were used for all time derivatives. The order of error for spatial and temporal derivatives was 2.5 × 10-5 and 1 × 10-5, respectively. The dimensionless mass uptake for each component in the polymer, µi, was calculated by integrating the dimensionless concentration profiles at each time step using Simpson’s algorithm for numerical integration. Results and Discussion The numerical procedure described above was used to generate curves for the fraction of A extracted and the selectivity as a function of time. The fraction of A extracted is equal to 1 - θ4, and the selectivity or separation factor, R, is defined as the ratio of A in the polymer (A + AB) to B in the polymer (B + AB) divided by the ratio of A in the external phase (Ae) to B in the external phase (Be):
MA + MAB µA + µAB MB + MAB µB + µAB R) ) CAe θ4 CBe θ5
(14)
An exhaustive study of the effects of all seven parameters (m1 through m5, SA, and SB) is not the intent of this paper. Rather, the purpose is to demonstrate the role that unequal transport rates and intersolute solvation play on extraction with dense polymeric sorbents by fixing some of the parameters at representative values and varying others over a limited range. The parameters fixed for all of the cases in this study were m3 ) 1.0, SA ) 5, and SB ) 1. m3 is the ratio of the total polymer particle volume to the external phase volume. The ratio of the partition coefficients, SA and SB, represents the selectivity that would be measured at equilibrium, and the individual values are consistent with the assumptions made in the development of the
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1191
Figure 2. Verification of the numerical solution (curved line) for mass uptake in the absence of intersolute solvation. The closed circles represent the analytical solution given by eq 15.
model provided CAe and CBe are dilute (
