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Monchick, L.; Mason, E. A. J. Chem. Phys. 1961, 35, 1676-1697. Moore, J. W.; Pearson, R. G. Kinetics and Mechanism, 3rd ed. Wiley: New York, 1981. O’Malley, M. M.; Bennett, M. A.; Simmons, M. B.; Thompson, E. D.; Klein, M. T. Isotopic Labelling as a Probe for Free-radical Reactions during Dibenzyl Ether Thermolysis. Fuel 1985, 64, 1027-1029. Paulaitis, M. E.; Krukonis, K. J.; Kurnik, R. T.; Reid, R. C. Supercritical Fluid Extraction. Reu. Chem. Eng. 1983a, 1 (2). Paulaitis, M. E.; Penninger, J. M. L.; Gray, R. D., Jr.; Davidson, P. Chemical Engineering a t Supercritical Fluid Conditions; AnnArbor Science: Ann Arbor, MI, 1983b; pp 179-250. Pryor, W. A., Smith, K. The Viscosity Dependence of Bond Homolysis. A Qualitative and Semiquantitative Test for Cage Return. J. Am. Chem. SOC.1970, 92 (18), 5403-5412. Rabinowitch, E. Collision, coordination, diffusion, and reaction velocity in condensed systems. Trans. Faraday SOC. 1937,80, 49. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. Rohr, D. F.; Klein, M. T. Modeling Diffusion and Reaction in Cross-Linking Epoxy-Amine Cure Kinetics: A Dynamic Percolation Approach. Ind. Eng. Chem. Res. 1990,29, 1210. Rozelius, W.; Vitzthum. 0.; Hubert, P. Method for the Production of Caffeine-free Extract. U.S. Patent 3,843,824, 1974. Sato, Y.; Yamakawa, T. Thermal Decomposition of Benzyl Phenyl Ether and Benzyl Phenyl Ketone in the Presence of Tetralin. Znd. Eng. Chem. Fundam. 1985,24 (l),12-15. Schlosberg, R. H.; Ashe, T. R.; Pancirov, R. J.; Donaldson, M. Pyrolysis of Benzyl Ether under Hydrogen Starvation Conditions. Fuel 1981a 60 (2), 155. Schlosberg, R. H.; Davis, W. H.; Ashe, T. R. Pyrolysis Studies of Organic Oxygenates. 2. Benzyl Phenyl Ether Pyrolysis under Batch Autoclave Conditions. Fuel 1981b, 60 (3), 201. Schneider, G. M. Physicochemical Aspects of Fluid Extraction. Fluid Phase Equilib. 1983, 10, 141-157. Simmons, G. M.; Mason, D. M. Pressure Dependency of Gas Phase Reaction Rate Coefficients. Chem. Eng. Sci. 1972a, 27,89-108. Simmons, G. M.; Mason, D. M. Gas-phase Kinetics Near the Critical Point. Chem. Eng. Sci. 1972b, 27, 2307-2308. Simmons, M. B.; Klein, M. T. Free-Radical and Concerted Reaction Pathways in Dibenzyl Ether Thermolysis. Ind. Eng. Chem. Fundam. 1985,24, 55. Slattery, J. C.; Bird, R. B. AIChE J. 1958, 4 (2), 137-142.
Stewart, A. T.; Dyer, G. H. Bituminous Coal Liquefaction. U S . Patent 3,850,738, 1973. Sung, W.; Stell, G. J. Chem. Phys. 1984,80,3350-3366. Swaid, I.; Schneider, G. M. Ber. Bunsen-Ges. Phys. Chem. 1979,83, 969-974. Takahashi, S. J. Chem. Eng. Jpn. 1974, 7,417-420. Towne, S. E.; Shah, Y. T.; Holder, G. D.; Deshpande, G. V.; Cronauer, D. C. Liquefaction of Coal Using Supercritical Fluid Mixture. Fuel 1985, 64,883-889. Townsend, S. H. Chemical Solvent Effects during Reactions in Supercritical Fluid Solvents. Ph.D. Thesis, University of Delaware, 1989. Townsend, S. H.; Klein, M. T. Dibenzyl Ether as a Probe into the Supercritical Fluid Solvent Extraction of Volatiles from Coal with Water. Fuel 1985, 64, 635-638. Townsend, S. H.; Abraham, M. A.; Huppert, G. L.; Klein, M. T.; Paopek, S. C. Solvent Effects during Reactions in Supercritical Water. Ind. Eng. Chem. Res. 1988, 27, 143. Wong, K. F.; Eckert, C. A. Solvent Design for Chemical Reactions. Ind. Eng. Chem. Process Des. Dev. 1969,8, 568-573. Wu, B. C. Solvent Effects on Reactions in Supercritical Fluids. Ph.D. Thesis, University of Delaware, Newark, 1990. Wu, B. C.; Klein, M. T.; Sandler, S. I. Reactions in and with Supercritical Fluids: Effect of Phase Behavior on Dibenzyl Ether Pyrolysis Kinetics. Znd. Eng. Chem. Res. 1989, 28, 255. Wu, B. C.; Klein, M. T.; Sandler, S. I. The Benzylphenylether Thermolysis Mechanism: Insights from Phase Behavior. AIChE J. 1990a, 36 (8), 1129-1136. Wu, B. C.; Klein, M. T.; Sandler, S. I. The Influence of Diffusion on Reactions in Supercritical Fluid Solvents. AIChE National Meeting, Orlando, FL, 1990b. Yang, H. H.; Eckert, C. A. Homogeneous Catalysis in the Oxidation of p-Chlorophenol in Supercritical Water. Ind. Eng. Chem. Res. 1988,27, 2009-2014. Yonker, C. R.; Wright, R. W.; Frye, S. L.; Smith R. D. Mechanism of Solute Retention in Supercritical Fluid Chromatography. Supercritical Fluids; Squires, T. G., Paulaitis, M. E., Eds.; ACS Symposium Series 329 American Chemical Society: Washington, DC, 1987; p 172. Received f o r review June 8, 1990 Revised manuscript received September 20, 1990 Accepted December 10, 1990
Dispersion and Partitioning in Short Coated Tubes Ashish Shankar and Abraham M. Lenhoff* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
The interaction of convection, diffusion, and interphase transfer is studied for a coated tube, with the fluid in the lumen in laminar flow. A solution is found directly in the time domain by series expansion, and computational results are presented for a range of parameter values and time scales to show the evolution toward classical chromatographic behavior at long times, where results are in agreement with those obtainable from the method of moments. A t shorter times multiple peaks may be seen due to the interaction of transport and retention effects. The method of moments is inadequate to characterize the more complex solutions a t short times, but a two-phase pseudocontinuum model provides a good approximation, and either this or the complete time domain solution is preferable for analysis of short-time experimental data. Introduction Analysis of the effect of partitioning between phases on axial dispersion behavior is at least about a half-century old (Westhaver, 1942). Most work over the past few decades has been based on the spirit of the Taylor (1953)-Aris (1956) result, namely, that, for sufficiently long systems, axial mass transport can be described by using a dispersive model based on the mean velocity and an effective dispersion coefficient much larger than the molecular diffusivity. Such pseudocontinuum approaches are justified by the infeasibility of obtaining a true continuum 0888-5885/91/ 2630-0828$02.50/O
description of most real process systems because of their geometric and hydrodynamic complexity, and the Taylor-Aris theory has thus been extended in various ways to account for partitioning effects (Aris, 1959a,b; Horn, 1970; Brenner and Adler, 1982; Federspiel and Fredberg, 1988). True continuum descriptions, albeit of idealized systems, are useful mainly in that the dispersion coefficients calculable in these situations can assist in estimation of suitable parameter values for use in pseudocontinuum models of real systems, such as countercurrent contacting 0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 829 operations, packed-bed catalytic reactors, and chromatography columns. There are, however, also actual processes amenable to being modeled realistically at the continuum level. This paper deals with an example of such a system, namely, the problem of chromatography in a coated tube. Westhaver’s (1942) analysis applies to this situation in the limit of a strongly retained solute, but more general formulations have been treated by Golay (1958) and Aris (1959a). These developments all apply to long systems in the Taylor-Aris sense, Le., ones in which the residence time is long relative to the characteristic time for radial transport, generally by diffusion. This situation applies in, for instance, gas chromatography, and Golay’s result has been widely used there, as well as more recently in analyzing results of the adaptation of hollow-fiber membrane devices to liquid chromatographic separations (Ding et al., 1989). Behavior in systems that are short relative to the Taylor-Aris limit has been studied for a variety of cases, including a few in which partitioning may be important (Sankarasubramanian and Gill, 1973; Reis et al., 1979; Hatton and Lightfoot, 1982; Paine et al., 1983). In most of these, the objective was, as before, to characterize the dispersion coefficient, with the results then used in a dispersion model. Such an approach may be adaptable to allow modeling of a real system, e.g., chromatography in a coated tube, but there are at least two reasons to explore alternatives. First, the validity of using generalized dispersion models has recently been shown to be questionable (Frankel and Brenner, 1989), and, second, results that would be useful for interpreting experimental data, in the form of concentration profiles as a function of time, have rarely been presented, yet they can be computed directly. The goal of this paper is thus to provide an analysis, suitable for computational implementation, of chromatography in a coated tube. The evolution of the concentration distribution is examined over a range of time scales and parameter values, including those pertinent to short systems (in the Taylor-Aris sense). Finally, the suitability of a pseudocontinuum description of the system is examined as the basis for more economical computations.
Problem Formulation and Solution The analysis presented here is for a round tube, the inner wall of which is coated with a retentive layer into which the solvent of interest can partition with a linear equilibrium relationship (partition coefficient K). The fluid is in laminar flow in the lumen, with the solute initially present as an impulse uniformly distributed across the lumen cross section. These specifications are not unduly restrictive: alternative geometries and velocity profiles can be accommodated by generalizations similar to those of DeGance and Johns (1978a,b), while distributed inputs can be accounted for by superposition. For the round tube, the mobile- and stationary-phase concentrations, c, and cg, respectively, are given as functions of radial and axial position and time, r, z , and t, by
thickness, respectively, U is the centerline velocity, and
D, and D, are the mobile- and stationary-phase diffusivities, respectively. Equations 1and 2 are to be solved subject to the radial boundary conditions acm -=0
ar
c, = Kc,
D,-
at r = O
(3)
at r = a
(4)
ac, ac, -- D , z at r = u ar
The initial condition is an impulse of mass M uniformly distributed across the lumen
M~(z)
cm = - a t t = O aa2
(7)
c s = O at t = O
(8)
Axial boundary conditions are not given explicitly, but they enter implicitly through the solution procedure used, as discussed later. The formulation is, on the whole, a specific example of a more general class of systems considered by Aris (1959a), except for a difference in the initial condition: the solute is initially in the mobile phase only in our analysis, while equilibrium partitioning is assumed in Ark’s. Our initial condition is more appropriate physically for the systems of interest here, but the effect should be small for long systems, such as those studied by Aris. Scaling is along lines commonly used for dispersion problems, the dimensionless variables being defined as
The governing equations (1-8) then become
C, = K C , at
where a and
e
are the lumen radius and the coating
(5)
=1
(13)
830 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991
The dimensionless parameters that appear in the problem statement are thus the P6clet number Pe = Ua/D,, the scaled coating thickness E* = € / a ,the diffusivity ratio D* = D,/D,, and the partition coefficient K. Solution of the governing equations and exploration of the parameter space are simplified considerably by neglecting axial diffusion in both phases; this approximation is valid for Pe >> 1, a situation generally observed in practice. For D* = 1, the high Pe solution can be used to generate solutions for arbitrary Pe by Gaussian smoothing, as can be shown by adapting the development of Wang and Stewart (1983). However, for all but very short times or very small Pe, the neglect of axial diffusion has little effect on the solution. Omission of the axial diffusion terms has the effect of keeping the axial domain of interest finite at finite times: solute initially in the input impulse (16) can be present only in z E [O,Ut],Le., l' E [-~/2,7/2],with a zero concentration outside this domain implicitly specifying axial boundary conditions. As we have shown for uncoated tubes (Shankar and Lenhoff, 1989),this allows the equations to be solved more economically than for the unbounded domain obtained when axial diffusion is retained. The solution procedure used here is a modification of the series expansion developed for uncoated tubes, which in turn is related to the methods of Yu (1976,1979,1981) and DeGance and Johns (1978b). The local solute concentration C([,[,T)is expanded in an axial Fourier series over the domain [-10/2,70/2] into which solute will be transported ; is a parameter inup to any given time of interest T ~ T~ troduced for computational purposes only, and results for a given T can be obtained by using any value of T~ > 7, although the greatest efficiency will be for T~ = 7. Thus
from which (10-17) produce a set of partial differential equations for the complex coefficients a,([,T)
to be solved subject to the radial boundary conditions (12-15) and the initial conditions
The coefficients are expanded radially in terms of the eigenfunctions of the purely diffusive radial operator, following the approach of DeGance and Johns (1978a). The eigenfunctions are Wl(5) =
where the inner product is defined as
and substitution in (19) and (20) yields a set of ordinary differential equations for the
where
The initial conditions for the an,l are
The solution thus differs from that for uncoated tubes (Shankar and Lenhoff, 1989) only in the radial eigenvalues and eigenfunctions, in the structure of the last term in (29), and in the initial condition (31). Consequently, the same considerations as in the uncoated tube case apply to obtaining the solution. In particular, the coefficientsan,lcan be solved for independently for each value of n, with a truncation scheme of the kind described for uncoated tubes required for n # 0. Numerical implementation also follows a similar course, as does obtaining the transverse averages, generally either the area average or the cup-mixing average, that are usually observable in practice. Results and Discussion With the PBclet number eliminated from the problem, the remaining parameters are a geometric one, e*, a thermodynamic one, K , and a transport one, D*. The effects of the first two of these parameters can be seen more clearly by noting that the chromatographic capacity factor, k', which is simply the effective volume of the stationary phase relative to that of the mobile phase, is given by
In most of what is presented below, the diffusivity ratio D* is taken to be unity, and the capacity factor then becomes the central parameter characterizing the response
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 831
( -vi
/;-*-.-.-.
‘_. / I I
-.-. .-.-.__
0 0
0
4
I
-0 5
03
05
10
2c/7
Figure 1. Area-average axial concentration profiles in the lumen at T = 0.15. Parameter values: k’= 0 (-); k‘= 1.025 t* = 0.05,K = 0.1 (--) and e* = 0.6,K = 1.52 (--): k’ = 10.25: e* = 0.05, K = 0.01 ( - - ) and z* = 0.6,K = 0.152 (---). All curves are for D* = 1.
of the system as a function of position and time. The physical nature of the response also influences the computational demands in that more terms in the axial and radial expansions are obviously needed to describe steep gradients. This is most noticeable a t short times, as was also the case for uncoated tubes (Shankar and Lenhoff, 1989),but in the problem being considered here, retention can introduce additional effects. In the absence of alternative solutions, several checks were performed in order to check the reliability of the computed responses. These included qualitative aspects such as smoothness (although the Gibbs phenomenon will be a factor at the ends of the domain), as well as quantitative checks such as the absence of changes with additional terms and verification of the momenta of the solution. The zero moment is simply a mass balance, while the first and second moments can be obtained analytically (Aris, 1959a). The zero moment constraint is the most reliable one since it applies at all times and is independent of initial conditions; Aris’s higher momenta are valid only at long times and are based on an initial condition in which the solute is equilibrated in both phases, rather than the single-phase input used here. Responses are presented as area-averageand cup-mixing concentrations in the form of axial profiles and elution profiles, the latter being more accessible experimentally. Being velocity-weighted, the cup-mixing concentration obviously neglects solute in the stationary phase, and most of the area averages presented below are also for the mobile phase only, although the relationship to stationary-phase concentrations is discussed. Area average axial profiles in the mobile phase at a relatively short time, 7 = 0.15, are shown in Figure 1,for several different sets of parameter values. The diffusivity ratio D* is held fixed at unity, but variations of E* and K allow capacity factors of 0 (uncoated tube), about 1, and about 10 to be examined; for each k‘ # 0, two different combinations of E* and K values are shown. The thin coatings correspond to those used in capillary gas chromatography columns, while the thick ones are typical of the wall thickness used in hollow-fiber liquid chromatography. Figure 1 shows the effects of the three key phenomena in the system: axial convection, radial diffusion, and retention. The shoulder at high 2{/7, seen in all the curves, represents solute initially near the center of the tube that is carried downstream rapidly, i.e., a convective contribution. For the uncoated tube, the major peak is that due to the interaction of convection with radial dif-
Figure 2. Area-average axial concentration profiles in the lumen at T = 1 for D* = 1. Parameter values as in Figure 1.
-1
0
-0 5
00
05
10
2C/r
Figure 3. Area-average axial concentration profiles in the lumen at T = 10. Parameter values as in Figure 1.
fusion: solute initially near the center of the tube diffuses outward into a region of lower velocity, and that initially near the wall diffuses inward into a faster moving region, giving rise to what eventually develops into the Gaussian peak described by Taylor-Aris dispersion. This dispersion peak is clearly missing from three of the curves for which k’ # 0, due to retention. This effect is to be expected, since the solute retained by the stationary phase at short times should be that which was initially near the wall. Only for the low-retention thick coating is the dispersion peak visible, because the thicker coating slows the rate of equilibration. The mobile-phase concentrations show that solute is more strongly retained at higher k’as expected, but in addition the thermodynamically dominated system (small K ) appears slightly more effective in this respect than the geometrically dominated one (large E * ) , again due to transport limitations. Axial profiles at 7 = 1 and 10 (Figures 2 and 3, respectively) show the further development of the peaks. For normal chromatographic behavior, approached a t long times, it follows from the definition of k’that the peak in the profile should be at about z = Ut/Z(l k’), which corresponds to 2 N 7 = - k ’ / ( l + k ? . This is indeed seen in the profiles, where stronger retention is manifested by a shift of the peak toward 2{/7 = -1. In addition to this information on peak position, quantitative information on the peak variance at long times is given by the results of Aris (1959a): the variance should increase linearly with time, from which it follows that the peak width in terms of 2{/7 should decrease as d2.A decrease in peak width is indeed seen, and the variances agree quite well with Aris’s at long times. The two strongly retained peaks are
+
832 Ind. Eng. Chem. Res., Vol. 30,No. 5, 1991 4
li
1
23
I 1
, 00 - 1
3
-0 5
OS
0 5
2
2{/T T
Figure 4. Area-average axial concentration profiles at T = 1 for t* = 0.6,K = 0.152 (k’= 10.25), D* 1. C, (-); KC, (- -).
Figure 5. Area-average exit concentration profiles at Parameter values as in Figure 1.
almost identical at long times, and this, too, can be seen from Aris’s results: the mobile-phase contribution to the variance is proportional to 1 6k‘+ llk’2, and for large k’ it is dominant even for the thicker coating. This broadening reflects the limitations imposed by using the low-capacity mobile phase to clear the high-capacity stationary phase, i.e., a flow limitation. Although Aris’s work is obviously very informative regarding peak width at long times, results such as those presented here provide details of peak shape not previously available. By 7 = 1 the concentration profile in the uncoated tube is essentially Gaussian, as discussed in more detail elsewhere (Shankar and Lenhoff, 1989);Aris (1956) had earlier shown that normality is approached as 7-l for a general class of single-phase dispersion problems. The retained peaks approach a symmetric form more slowly, with the rate of approach to symmetry varying inversely with k’-the strongly retained peaks are not yet symmetric at T = 10. This slower appraoch to the asymptotic situation characterized by Aris reflects the additional time required for the low-capacity mobile phase to clear the high-capacity stationary phase; this is directly related to the broadening discussed in the previous paragraph. These observations are also consistent with Paine et al.’s (1983) estimate T ~ >>/ 1~ + k’ for validity of using of a single overall dispersion coefficient. As noted previously, Aris’s results were derived for overall moments (both phases). However, experimental data are usually based on measurements in the mobile phase, so use of Aris’s results to interpret the data implies that the moments in the two individual phases behave similarly. Figure 4, which compares axial profiles of the concentrations in the two phases, corrected for the K value, for k ’ = 10.25 at T = 1, shows that the profiles are certainly different at short times (the peaks around 2{/7 = 1are due to the Gibbs phenomenon). At longer times, however, the profiles are much more similar, and their normalized moments differ little (within
