Adsorber Dynamics in the Simulated ... - ACS Publications

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Adsorber Dynamics in the Simulated Countercurrent Moving-Bed Separator: Numerical Simulation, Detailed Comparison with Experiment, and Optimization Studies Hadrian Djohari and Robert W. Carr* Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Avenue SE, Minneapolis, Minnesota 55455

An adsorber dynamics model of a simulated moving-bed separator (SMBS) has been tested against experimentally determined concentration waveforms for the separation of propylene and dimethyl ether in a SMBS (Fish et al. AIChE J. 1993, 39, 1783). The axially dispersed, linear adsorption, linear driving force model was solved numerically by an efficient finitedifference method with adaptive gridding. The numerical simulations predict average raffinate (dimethyl ether) and extract (propylene) concentrations that are within 5% of those of the experiment, and the experimental concentration waveforms of propylene are reasonably well reproduced. However, the model gives a poor representation of the dimethyl ether waveforms and fails to accurately predict impurity levels in raffinate and extract streams. These failures may be attributed to the inadequacy of a uniform adsorption site model to describe the adsorption of dimethyl ether on Chromosorb 101 and are a reminder of the potential inadequacy of simple isotherms. Simulations were used to optimize the flow rates required to achieve the most effective separation. A study of flow rate optimization for a smaller separation factor of 1.2 shows that satisfactory separations can still be obtained. It is also shown that modifying the SMBS to a 2-1-2-1 column configuration leads to an increase of the product purity. Introduction Adsorption processes can be used to good advantage in the chemical and petrochemical industries to carry out otherwise difficult separations. Countercurrent moving beds are capable of providing high-purity adsorptive separations of continuous feed streams, but simulated moving beds (SMBs) provide advantages over continuous countercurrent operations by avoiding the problems of solids attrition, fluid flow uniformity maintenance, and backmixing.2 Countercurrent motion can be simulated by employing a fixed bed with several axially aligned inlets and outlets. The feed and product removal streams are simultaneously and sequentially moved along the direction of mobile-phase flow at fixed time intervals, as in the commercial Sorbex process.3 SMBs can also be configured as a series of packed columns organized into sections, with the number of columns as a design variable. Configurations with either three or four sections are possible.4 The periodic flow switching operations cause concentration fronts to develop and move through a SMB separator (SMBS), in contrast with the true countercurrent (TCC) moving bed, which operates in a steady state and has stationary concentration fronts. (We use SMBS here, rather than the more common SMB, to distinguish separators from SMB reactors, which we have designated as SMBR in some past work.) The SMBS is dominated by adsorption column dynamics, and the asymptotic behavior can be characterized as a periodic steady state.5 However, at the periodic steady state, the performance of the SMBS may be very similar to the performance of TCC moving-bed separators.6,7 This has led to the development of equivalent counter* To whom correspondence should be addressed. Tel.: 1-612625-2551. Fax: 1-612-626-7246. E-mail: [email protected].

current models. These give time-independent solutions and are computationally less demanding than direct, time-dependent numerical simulations, which have only comparatively recently become computationally efficient. A review that includes an account of equivalent countercurrent models has been published.4 The equivalent moving-bed approach to modeling has been developed for the optimal and robust design of SMBS units.8 A standing concentration wave approach to TCC moving beds has been developed and shown to be applicable to the SMBS design.9,10 Lu and Ching11 have concluded that while the equivalent moving-bed model is acceptable for general SMBS performance studies, models describing transient behavior are necessary for a better understanding of process dynamics, which can be simulated via differential material balances in the adsorbent beds. Examination of the transient behavior is also necessary to characterize start-up and the approach to the periodic steady state. An analytical solution has been obtained for the linear, ideal (plug-flow, adsorption equilibrium) case,12 and the influence of axial dispersion and linear mass transfer on the solution has been considered.13 Numerical simulations for single-component adsorption have been developed for a stirred tanks in series model14 and a linear, axially dispersed model with pore diffusion.15 Numerical simulations of a model incorporating axial dispersion, linear driving force (LDF) mass transfer, and extended Langmuir adsorption isotherms showed that computational efficiency can be a problem because 2-90 h of Cray J916 CPU time was required to reach the periodic steady state.11 Recent reports document computational strategies that give reasonable computational efficiency.15-17 Computationally efficient modelbased control of a SMBS for the separation of glucose and fructose has recently been reported.18

10.1021/ie0402618 CCC: $30.25 © 2005 American Chemical Society Published on Web 05/25/2005

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Ruthven19 has called for “a judicious combination of experimental measurements carried out in conjunction with modeling studies, and designed to test key elements of theory” in order to advance adsorption research. Similarly, SMBS research would be advanced by combined experimental and theoretical studies, but the extent of these studies is limited at present. In some cases, models have been used to fit experimental SMB concentration profiles at an arbitrary instant of time during a switching period, where overall profiles are determined by taking single samples of the fluid-phase composition at certain points along the SMB at a given time. See, for example, work by Strube et al.15 A more stringent model test would be obtained from a comparison of the entire model-predicted concentration waveform with an experimentally determined waveform. Fish et al.1 have experimentally determined the concentration waveforms of dimethyl ether (DME) and propylene (P) eluting from the raffinate and extract ports, respectively, of a four-section SMBS. To our knowledge, these are the only existing measurements of complete concentration waveforms eluting from an SMBS and are thus the only data for detailed testing of dynamic SMBS models. In this paper, efficient numerical simulations of transient material balance equations in an SMB are tested against these data. Adaptive gridding and finite differences are applied to solve isothermal, transient material balance equations incorporating axially dispersed plug flow, LDF mass transfer, and linear adsorption on small particles with negligible pore diffusion. The method is extendable to nonlinear adsorption and porous adsorbent pellets. Flow-rate optimization is also addressed, and the effects of decreasing the adsorption selectivity and modifying the SMBS configuration are examined.

Figure 1. Diagram of a four-section SMBS, such as the one used in the experimental separation of P and DME.

component. The feed concentration was kept very dilute, below 0.006 in mole fraction for each component, where the experimentally determined adsorption isotherms were found to closely approach the linear region. Furthermore, at low concentration, changes in the total fluid velocity due to adsorption were negligible, and other parameters such as the average molecular weight, viscosity, and diffusion coefficient, are nearly constant. Transient concentration profiles of P and DME were determined by automatically sampling the raffinate and extract streams at approximately 40-s intervals and running a gas chromatographic analysis of each sample. Solid- and fluid-phase flow speeds in TCC systems can be related to those of SMBs and are stated as simulated solid speed u and fluid speed v:

u ) L/ts, v ) F/A

Background In a multiple-column SMBS, a port that can serve as an inlet or outlet is located between each column. Feed inlet advancement in the direction of carrier fluid flow simulates countercurrent motion, although not continuously but in discrete steps at timed intervals. Either binary separation or separation of several components into two groups is possible. This is achieved by adjusting the advancement rate and the carrier flow rate, so the more strongly adsorbed component(s) travels more slowly than the feed position, while the less strongly adsorbed component(s) moves ahead of it. Just before the more strongly adsorbed component breaks through the column, the inlet/outlet ports are advanced by switching the feed stream to the next port. The time interval between switches is called the switching period. Fish et al.1 employed a four-section SMBS (Figure 1), with each section having one column, for an experimental investigation of the separation of P from DME. Section I was the extract (DME) withdrawal section, while section III was the raffinate (P) withdrawal section. Feed was injected before section III with some carrier gas to dilute the concentration, at the beginning of the first switching period. The feed concentration was a step function from zero to CF, the feed concentration. At the end of the switching period, the concentration returned to zero when the feed was advanced to the next column. The adsorbent was 60/80 mesh Chromosorb 101, and the carrier gas/desorbent was N2. The separation factor, R, for P/DME on Chromosorb 101 was approximately 2, with DME the more strongly adsorbed

(1)

where L is the length of each column, ts is the switching time,  is the bed porosity, F is the fluid flow rate, and A is the column cross-sectional area. The flow ratio, σ, is the main parameter that governs the steady-state concentration profile in countercurrent mass-transfer processes.

σ)

1- u KS ) K F  v

(

)( )

(2)

Here, K is the linear adsorption equilibrium constant for each component, while S is the flow rate of solid. For perfect separation, the mixed component should only be in sections II and III. There should only be the extract in section I and the raffinate in section IV. The necessary flow ratios, σ, to achieve this condition for P and DME in each column are listed below:

Section I Section II

σDME < 1.0, σP < 1.0 σDME > 1.0, σP < 1.0

Section III

σDME > 1.0, σP < 1.0

Section IV

σDME > 1.0, σP > 1.0

(3)

Mathematical Model The isothermal model of the SMB is given below. The list of model assumptions was made to conform with the experiments.

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1. Adsorption follows the linear isotherm at low concentration. 2. External mass transfer between the bulk phase and the adsorbed phase is assumed to follow a LDF rate:

∂q 15De (q* - q) ) ∂t R2

(4)

p

where q is the concentration of a component in the adsorbed phase and q* is that at the equilibrium. De is the effective diffusion coefficient, and Rp is the radius of the pellets. 3. The fluid velocity is constant within each column. In addition to the statements made in the previous section, the pressure drop on each column was only 100 Torr. 4. Axial dispersion is included in the partial differential equations.20 Radial effects are neglected. The axial dispersion coefficient for component k, Dk, is found by the following correlation:

1 0.5 0.3 + ) Pe (Re)(Sc) 1 + 3.8/(Re)(Sc)

(5)

where Re is the Reynolds number, Sc is the Schmidt number, and Pe is the Peclet number:

Pe ) Lv/Dk

(6)

where L is the length of the column and v is the fluid velocity. 5. Adsorbents are spherical and uniformly distributed. Also, the granules are small enough (60/80 mesh) that intraparticle diffusion and intraparticle heat transfer can be neglected. 6. External heat transfer is also negligible in isothermal adsorption. 7. Gaseous components behave as ideal gases. The governing equations are the mass balances for the gas phase in the bed void:

∂2ck 1 -  ∂qk ∂ck ∂ck +v - Dk 2 + )0 ∂t ∂z  ∂t ∂z

(7)

where c and q are the gas concentrations at the bulk and adsorbed phases, v is the fluid velocity in the column,  is the bed porosity, t is the time, z is the axial distance, Dk is the axial dispersion coefficient, and subscript k denotes component k. The initial and boundary conditions are

t < 0, ck ) ct)0,k, for 0 e z e L z ) 0, ck ) cin,k, for t g 0 z ) L,

∂ck ) 0, for t g 0 ∂z

(8)

where ct)0,k is the concentration of component k at t ) 0 and cin,k is the concentration at the inlet, varying with time. Equation 4 could be substituted into eq 7 to substitute for ∂q/∂t. The equations are solved using an explicit adaptive gridding finite-difference method.21,22 For the first derivative, we use a third-order quadratic upstream difference scheme proposed by Leonard.23 This scheme would provide better accuracy by reducing overdiffusion

Table 1. Process Parameters for Separation of Experimental P/DME (r ) 2.3) and for Simulated Separation of Binary Mixtures with r ) 1.2 run

column config

A B C D E F G H

1-1-1-1 1-1-1-1 1-1-1-1 1-1-1-1 1-1-1-1 1-1-1-1 1-1-1-1 2-1-2-1

F1 ts F2 F3 F4 (min) R1 (mL/min) (mL/min) (mL/min) (mL/min) 5 4.5 6 5 5 5 5 5

2.3 2.3 2.3 2.3 1.2 1.2 1.2 1.2

240 240 240 235 120 124 122 122

80 80 80 75 100 104 90 90

320 320 320 315 130 134 140 140

120 120 120 115 110 114 112 112

of the moving fronts compared to a regular upstream difference scheme at a reasonable additional cost.24 It also provides better stability by damping the large unphysical oscillations that are typically produced by a center-difference scheme (CDS). The second derivatives are substituted by the CDS. The time integration method selected is the implicit, second-order CrankNicholson method. We use the adaptive gridding technique to build a denser mesh around the moving fronts. This technique is very attractive because it can achieve the same accuracy as the fixed-grid scheme using only 1/4 number of nodes.22,25 Adaptive gridding methods put more grids around the large gradient and inflection regions in an optimum manner. Although some interpolation is needed to calculate the concentrations at new grids, the total number of nodes calculated is actually less than that of the fixed-grid scheme, so the calculation cost is less costly. A single adsorption column is divided into a determined number of grids, with 2 degrees of freedom per node and the concentration in the gas phase, c, and in the adsorbed phase, q. The calculation would start column-by-column from the feed column until the last column, and then it proceeds to the next switch. The parameters used in the calculation are derived from experimental parameters. The length of each column is 12-1/4 in., with 0.5-in. outside diameter. The connecting tube outside diameter is 1/8 in. The bed porosity  is 0.4, and the average particle radius is 0.101 65 mm, and for linear adsorption, KP ) 30 m3 of gas/m3 of solid and KDME ) 70 m3 of gas/m3 of solid. The effective diffusion coefficient, De, for P is 1.45 × 10-6 m2/s, and De for DME is 2.00 × 10-6 m2/s. Results and Discussion Comparison of Numerical Simulations with Experiment. Numerical simulations were compared with experimental data for the separation of P and DME.1 As stated above, all of the parameters for the simulation were chosen to match those of the experimental conditions. The experiments were done with dilute gas-phase adsorbate compositions, 0.006 mole fraction of each component in the gas flow, at most. At these concentrations, both P and DME are in the nearly linear region of the adsorption isotherm, justifying the use of a linear isotherm in the model. For higher concentrations, P and DME both display convex isotherms. However, desorption studies of DME on Chromosorb 101 gave evidence for a long desorption “tail”. This was not accounted for in the model. For experimental details, ref 1 should be consulted. Eight computational runs (Table 1) were designed to determine the effects of the switching time, ts, the separation factor, R, the flow rates F1, F2, F3, and F4 in

Ind. Eng. Chem. Res., Vol. 44, No. 13, 2005 4765 Table 2. Comparison between the Experimental Data and Numerical Results in P/DME Binary Separation average mole fraction run A: run A: run B: run B: run C: run C: run D:

experimental simulation experimental simulation experimental simulation simulation

P at port A

DME at port A

P at port B

DME at port B

4.55 × 10-3 4.574 × 10-3 4.55 × 10-3 4.469 × 10-3 4.55 × 10-3 4.533 × 10-3 4.518 × 10-3

0.32 × 10-3 0.035 × 10-3 0.404 × 10-3 0.001 × 10-3 2.65 × 10-3 1.244 × 10-3 0.012 × 10-3

0 0.002 × 10-3 0.008 × 10-3 0.073 × 10-3 0 0.002 × 10-3 0.003 × 10-3

3.35 × 10-3 3.485 × 10-3 3.7 × 10-3 3.397 × 10-3 1.45 × 10-3 2.519 × 10-3 3.390 × 10-3

each of the four sections, and the column configuration. The 1-1-1-1 column configuration means that we have one column each for sections I-IV. The feed enters column 1. Figure 1 shows the feed entering section III, which for this switching period is called column 1. P is less strongly adsorbed than DME and elutes ahead of it. This elution position is called port A, is expected to consist of high-purity P, and moves congruently as the feed position is switched. The more strongly adsorbed DME elutes from the second column behind the feed column, which is called port B. Three computational runs simulated the experimental P/DME separation. The average compositions from these numerical calculations, called runs A-C, are compared with the experimental data and are given in Table 2. The simulations predict the average concentrations of P at port A and DME at port B (Figure 2), which are in good agreement with the experimental data except for that of DME at port B in run C. The experiments show, and the simulations predict, that contamination of DME by P at port B is small. In fact, the simulations predict that DME can be recovered up to 99.9% purity. Some experimental runs showed 100% purity, but the failure to observe any P at port B (see runs A and C) can be attributed to P levels below the detection limit of the gas chromatograph. On the other hand, there is more contamination of P by DME at port A, both predicted and observed. The simulations predict that P can be recovered up to 99.2% purity, but the experiments give the purity of P as only about 93%. Table 2 shows that the observed levels of DME are significantly higher than the predicted levels. The disagreement between the simulated and observed concentrations of DME at port A can be attributed to the “tailing” of DME on the Chromosorb 101 adsorbent, and the failure of the linear isotherm adopted for the model, to account for this. A more detailed comparison between theory and experiment can be made from the time-dependent concentrations of P and DME, that is, by looking at the adsorber dynamics. The concentration profile in Figure

Figure 2. Experimental result showing the concentrations of P and DME at ports A and B for switching time ) 5 min (run A).

3 shows the comparison between the time-dependent concentration profiles for simulation and experiment at the periodic steady state. The periodic steady state is achieved when the concentration profile repeats the same pattern and concentrations during each switching period. The top graph of Figure 3 shows that the computed P concentration front is quite sharp and spread only slightly by dispersion and that the model is able to simulate the entire experimental P profile reasonably faithfully. The simulated concentration of DME at port A is significantly smaller than the concentration of DME in the experimental profile. This experimentally observed trail of DME across all of the columns is a consequence of DME “tailing”, already mentioned above. The simulations predict that DME is starting to break through the feed column. The time resolution of the experiments is not sufficiently fine to discern whether this, in fact, occurs. The bottom graph of Figure 3 shows that DME profiles at port B are not well simulated by the model. The trailing edge of the simulated DME profile falls sharply and is characteristic of desorption according to a linear isotherm with some superimposed axial dispersion. The experimental profile is considerably different. In fact, what one sees in the elution of DME is a snapshot of the DME “tail”, which can also be seen in Figure 12 of the Fish et al. work, which presents the trailing edge of DME eluting from a single packed column after injection of a rectangular wave. It is important to recognize that the adsorption isotherms were obtained from breakthrough times of step inputs and that this did not provide desorption characteristics. The observed desorption tails can be attributed to DME desorption from surface sites that are not all identical but for which desorption from some sites is slower than that from others. This can explain not only the dynamics of DME elution from the SMBS, as observed in Figure 3 but also the high average DME concentration (relative to the simulation) at port A listed in Table 2. The failure of the model with respect to DME is a failure to adequately describe DME desorption. The success of the model with respect to P can be ascribed to the adequacy of a linear isotherm for describing both adsorption and desorption, which for P on Chromosorb 101 is apparently well behaved. SMB models frequently make use of standard isotherms. While these may be satisfactory, it should be stressed that, for accurate models, adsorption and desorption characteristics should be thoroughly investigated. Figure 4 shows that it takes 12 switching periods or 3 complete cycles to reach the periodic steady state for simulation run A. The average concentrations during a switching period for P at run A and DME at run B increase from zero in the beginning to the steady-state value. The other concentrations are practically zero at all times. The number of cycles needed to reach the

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Figure 3. Comparison of the experimental simulation results for P and DME separation (run A).

Figure 5. Concentration profiles of P (at the top) and DME (at the bottom) inside the columns, with progression from the beginning of the switching period to the end (run A).

Figure 4. Transient result for run A (switching time ) 5 min) showing the average concentrations of P and DME at ports A and B. The periodic steady state is reached after 12 switching periods or 3 cycles.

steady state depends on the ratio between the feed and the recirculating flow rates. For runs A-C, the feed to recirculating flow ratio is 120 to 80 mL/min (3:2). The larger the ratio, the faster the system would reach the steady-state condition. Figure 5 shows that the simulated concentration profiles of P and DME move nearly uniformly during a switching period, leaving a little tail at the end of the column. The profile at the end of the switching period is practically the same as that at the beginning but is shifted forward one column. Although there are inflows and outflows, they do not affect this internal steadystate condition. One characteristic that determines successful separation is that column 2 (previously column 3) contains no DME at the beginning of the

switching period. This means that none of DME remains in column 3 at the end of the switching period, while none of P is in column 3 at any time. Either one would eventually bring about contamination of the product at ports A and B, respectively. Some P that gets left behind in column 3 would emerge at port B at the beginning of the switching period. DME should be behind the feed point, in columns 3 and 4 only. In run A, the front slightly crosses over into column 1. This would cause contamination of ether at port A at the end of the switching period. There is a slight discontinuity between columns 4 and 1, more evident at the halfway point in the time profile in Figure 5, because the feed is inserted at that point. This discontinuity is less obvious at the beginning and at the end of the switching period. Experimental run B has a switching time of 4.5 min compared to 5 min for run A. DME would no longer elute from the feed column before the switching period, and therefore the purity of P at port A would increase. On the other hand, P is not removed entirely from column 3 and would elute from port B at the beginning of the switching period with the rest of DME. The numerical result agrees with this trend. However, in the experiment, the left-behind DME significantly contaminates P at port A. The time required to attain the periodic steady state for runs B and C is about the same as that for run A, and that is 12 switching periods or 3 cycles. The switching time of experimental run C is 6 min. The unwanted DME shows up strongly in port A because the longer switching time allows the material

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to be desorbed in the feed column. P elutes less from port B because it has more time to be desorbed. Separation is, therefore, very low. The simulation shows a trend similar to that of the experiment. Conclusions. The model adequately predicts the average P and DME concentrations at ports A and B, respectively, and the concentration waveform of P at port A. However, it does not represent the concentration waveform of DME at port B well, and it underpredicts contamination of P by DME at port A because the linear isotherm cannot adequately describe DME desorption from Chromosorb 101. Adsorption models that assume uniformity of adsorption site characteristics are widely used in mathematical models of adsorption processes. While uniform adsorption may be an adequate approximation, as it appears to be for P on Chromosorb 101 here, caution should be exercised. The linear isotherm, or the Langmuir isotherm, or any other uniform site adsorption isotherm will not provide an adequate description of DME adsorption on Chromosorb 101. A multisite model would presumably provide a better description of this adsorbent-adsorbate pair. This serves as a reminder that mathematical modeling studies of adsorption processes employing idealized isotherms may be of limited utility. Optimization Optimization of the separation of P from DME should start with the identification of an adsorbent that is better behaved with respect to DME adsorption. With such an adsorbent, one should then ask how the separation could be improved. It seems worthwhile to explore this question, which we address by first looking at flow-rate optimization and then at the effects of the separation factor, R, and the number of columns in each section. There is a simple approach for determining the optimum flow rates. The separation depends on σ. The less strongly adsorbed component, P, should always move ahead of the feed point (σ < 1) in all sections except in section IV. P should not break through the column in section IV because DME is collected at port B in the next section. On the other hand, DME should not break through the column (σ > 1) in any section except in section I. In this section, DME is collected at port B. We can correlate this flow ratio criterion to set flow-rate limits by setting σi ) 1 and applying eq 9,

F)

(1 - )LKi tsσi

(9)

where F is a flow-rate limit,  is the bed porosity, L is the length of the column, ts is the switching time, and K is the adsorption isotherm equilibrium constant. When σ ) 1, the upper limit to the fluid flow rate is given by (1 - )LKDME/ts and the lower limit by (1 )LKP/ts. The relationships in eq 10 show the acceptable range of flow rates in each section. Table 3 shows the acceptable range of fluid flow rates in each section of the SMBS for the P/DME separation (i.e., R ) 2). Section I: Section II: lower limit < Section III: lower limit < Section IV: lower limit >

flow rate 3 > upper limit flow rate 4 < upper limit flow rate 1 < upper limit flow rate 2 (10)

Table 3. Acceptable Range of Fluid Flow Rates for Each Section of the SMBS

run A run B run C run D runs E-H

lower limit (mL/min)

upper limit (mL/min)

107.3 119.2 89.4 107.3 107.3

247.2 274.7 206 247.2 128.3

Now we can reevaluate the flow rates in runs A-C (Table 1) to see whether they fulfill the above flow requirements or not. Run A in the experiment fulfills the above requirements. The least ideal requirement fulfilled is σDME in the feed column, which is very close to 1. This means DME almost breaks through the first column for dispersionless flow. It actually has broken through in the model with axial dispersion and hence accounts for the impurity at port A. In the experiment, the impurity is caused more by tailing rather than by the early breakthrough. This brings up a question, could the ether breakthrough be postponed until the switch time has taken place? Not entirely, and minimizing the DME impurity requires optimization. Run B meets all of the requirements, but σP in section II is nearly 1, and because the simulation includes dispersion, this causes some of P to remain behind in the fourth column and exit out at port B at the next switch. Run C fails badly in section III for DME because F1 is higher than the upper limit, while it should be between the limits. The separation will be improved if F1 is, say, 180 mL/min instead. On the basis of the inequalities above, there are 4 degrees of freedom for choosing the four flow rates. Because the inflows and outflows have been determined by the experiment, the degrees of freedom are reduced by 3, leaving only 1 degree of freedom to be optimized, and the suitable variable for optimization is the recirculating flow rate (flow rate at column 2). To optimize the recirculating flow rate for a switching period of 5 min, the impurities of P and DME in ports B and A, respectively, should be kept as small as possible. The flow rate in section III is 67 mL/min for σDME ) 1. The flow rate in section II is 88 mL/min when σP ) 1. Optimization would occur at a flow rate between these two limits. To reach the optimum, σP in section II needs to be minimized while σDME in section III needs to be maximized. Figure 6 shows two lines, which are the absolute values of 1 - σP,II and 1 - σDME,III. The intersection of these two lines is the optimum point, the point where we want minimum contaminant for both the extract and the raffinate. The optimum recirculating flow rate is 74 mL/min. This optimum run, run D, yields the best result among runs A-D, as shown in Table 2. There is still some contamination of DME at port A. The purity of P in A is slightly lower, but the purity of DME in B is higher, thus reaching the optimum compromise. Considering that DME is the severely tailed component, run D represents a better option for separating both components. Separation of Binary Mixtures with Separation Factor ) 1.2 The separation of binary mixtures with separation factors larger than 2 is usually considered to be an easy separation. Adsorption processes are particularly attractive economically when the system has a large

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Figure 6. Optimizing the recirculating flow rate (flow rate in column 2) to get minimum contaminant in both ports, which is simulated in run D.

separation factor and the distillation process has a small separation factor, but this convenience does not always happen. In this section, separation of an imaginary binary mixture of L and M with a small separation factor (R ) 1.2) with a linear isotherm is examined. The more strongly adsorbed component, M, is similar to DME in the above experiment and, therefore, is susceptible to tailing. The flow-rate criteria in eq 10 are still valid for the smaller separation factor, but now the upper and lower limits are tighter. Simulation runs E-H were conducted to evaluate the SMBS performance for R ) 1.2. We have 4 degrees of freedom for choosing the flow rates. Similar to the experiment above, we can determine the feed flow rate, the makeup flow rate, and the raffinate flow rate based on the flow ratio requirement in eq 10. To determine the feed flow rate, we look at Table 3. This shows that the difference between the upper and lower limits for runs E-H is only 21 mL/ min. Because the feed enters between sections II and III and both sections should be within the flow limits, then the maximum feed rate is also 21 mL/min. When we include the dispersion effect for both fronts and tails, we need to include some spare flow rate. Therefore, in runs E-H, the feed flow rate is chosen to be 10 mL/ min. The second flow rate to be determined is the makeup flow rate. Runs E and F have a smaller makeup flow rate at 30 mL/min compared to 50 mL/min for runs G and H. Increasing the makeup flow rate improves the separation, but the outflow concentration is more diluted and more carrier gas is needed. The third flow rate is the raffinate flow rate. It is set at 20 mL/min for runs E and F and 40 mL/min for runs G and H. The final flow rate to be determined is the recirculating flow rate, which will be optimized. The results for runs F-H are shown in Table 4. In run G, σM in column 1 (1.05) is close to 1 while σL in column 4 is 0.96. The expectation is that the separation between the components would be better than that in run F, but the result is a little disappointing. Although

the purity of L is still more than 99.0%, the purity of M at port B drops to 97.4%. Purity is the percentage of a component in the binary mixture coming out from one of the ports. The overall concentrations of L and M in both ports decrease significantly because the makeup carrier flow rate in run G is much bigger than that in run F. Run G also suffers from slower start-up to the periodic steady-state condition. It takes about 14 cycles (56 switching times) compared to only 3 cycles in run A. This is caused by the much smaller ratio of the feed flow rate to the recirculating flow rate (about 0.1) in run G than that in run A (1.6). The optimal recirculating flow rate for this simulation is found by plotting the results from five runs at different flow rates. The optimal run is the one that maximizes the following parameters: (a) recovery of L at port A; (b) purity of L at port A; (c) recovery of M at port B; (d) purity of M at port B. Recovery is the fraction of a component recovered at one of the ports compared to the amount fed. The ideal separation will yield unity for all of the above parameters. In this case, L is purely recovered at port A, and M is only recovered at port B. A tradeoff between the recovery of L at port A and the purity of L at port B is observed in Figure 7. When we want to recover L more, we can set σL to be further from 1 but then σM is closer to 1. When this happens, M would contaminate port A and the purity of L would drop. A similar graph applies for component M, with the same implication. When we want to improve one of the above criteria, we need to sacrifice the others. L is recovered more at lower flow rate (120 mL/min), while M is recovered more at higher flow rate (130 mL/min). At lower feed rate, the L concentration profile is still steep and the tail does not spill over to column 3. At higher flow rate, the profile is flatter and spreads across the columns, spilling over to column 3, which causes the contamination. So even though the front of the profile is moving faster at higher flow rate, the tail is stretched and left behind in the unwanted column. The reverse is true for M. It is interesting that the curve is not monotonic but is shaped like a letter M or W with a little bump at F2 ) 125 mL/ min. If we take all of the above criteria at the same weight, then the optimum flow rate is F2 ) 124 mL/ min. Six-Column Configuration The industrial applications of SMBS usually have 1224 columns or ports to simulate the TCC process. The more columns there are, the closer the separation process would be to the TCC. The separation could still be performed with as few as four columns to represent the four sections of the SMBS, but the result would be slightly different from that of TCC. It would be interesting to compare the four-column system to systems with more columns. In the final run (run H), the SMB has six columns in a 2-1-2-1 configuration, which means two columns for section I, one for section II, two for section III, and one for section IV.

Table 4. Result from Simulated Binary Separation with r ) 1.2 average mole fraction run F: simulation run G: simulation run H: simulation

L at run A

M at run A

L at run B

M at run B

3.440 × 10-3 2.431 × 10-3 2.176 × 10-3

0.032 × 10-3 0.023 × 10-3 0.006 × 10-3

0.034 × 10-3 0.045 × 10-3 0.003 × 10-3

2.525 × 10-3 1.729 × 10-3 1.789 × 10-3

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Figure 7. Two parameters used in optimizing separation of binary materials, with separation factor ) 1.2. The optimization maximizes the total recovered fraction of L and the purity of L at port A.

The system parameters are exactly the same for run G (1-1-1-1) and run H (2-1-2-1). The flow rates and K values are identical, and so σ does not change either. The additional two columns are in sections I and III because the flow rates in these columns are larger and therefore less likely to reach steady state. The overall result is similar to the four-column configuration, but the concentrations of the impurities are much less than the ones in run G. The concentration of M in port A drops from 0.023 to 0.006 × 10-3, while the concentration of L in port B drops from 0.046 to 0.003 × 10-3. It only takes 8 cycles to reach the periodic steady state, substantially less than the 14 cycles needed in run G. This is a huge performance enhancement, which is achieved by adding two extra columns in the six-column separator. Separation is improved without additional process time because the switching time is exactly the same. Figure 8 shows the concentration profile at ports A and B for run H, which looks better than that in run G, with considerably less tailing for both components. In run H, the fourth column is empty all of the time and could be removed for this ideal separation. Summary Numerical simulation of a binary separation in a foursection SMBS has been investigated. The dynamics of the columns in the SMBS are determined by a linear uncoupled isotherm model with axial dispersion. The numerical data have been compared with the data for separation of P and DME.1 The maximum difference of the average concentrations between simulation and experiment is 5% for the raffinate (DME) and the extract (P). However, DME, the more strongly adsorbed component, tails significantly, causing reduced purity and recovery of P. The maximum concentration exiting through either port A or port B in the simulation is below the feed concentration because the interacting isotherm or physical accumulation is not accounted for. An effective separation happens when the more strongly adsorbed species, DME, almost elutes from the feed column near the end of the switching period. This can be quantified based on the flow ratio, σ, requirement. Axial dispersion broadens the concentration profiles of the components and reduces the purity of the products. In the simulation runs A-H, the dispersion is very

Figure 8. Concentration profiles inside the columns at the beginning of the 5-min switching period for 1-1-1-1 and 2-12-1 column configurations.

small, but it creates enough broadening that it would reduce the purity of the products by 0.1%. For higher dispersion, the separation would be marred by strong tailing and a dispersed front. Optimum conditions are found by determining the best flow rates that fit the flow ratio, σ, inequalities (eq 9). When the flow rates fulfill these inequalities, only axial dispersion causes the contaminations at ports A and B. Simulation runs A-C only require three cycles to attain the periodic steady state. For a smaller separation factor, near unity, as in runs F and G, it takes 14 cycles because the flow-rate restrictions are more rigid. The purity of the components can be increased dramatically by increasing the number of columns for two sections, sections I and III. The 2-1-2-1 column configuration yields a purity of more than 99% for both components, compared to 97% and 99% for the 1-11-1 column configuration. It only takes eight cycles now to reach the periodic steady state. Although the total number of columns is increased and the system is more complicated, the start-up time would be significantly shortened and the purity and recovery would justify the additions. The distinct characteristic in a four-section SMB is section IV. To ensure a proper thermodynamic mixing between the recirculating and makeup flows of the desorbent or inert gas, the exiting concentration of section IV has to contain only the desorbent or the inert gas. Even if only a small amount of one component occupies this section, the objective of a sharp separation of materials will not happen. In the 2-1-2-1 configuration, the fourth column is always empty during the

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separation process. Hence, for this particular process, it would actually need only five columns (1-1-2-1) to achieve the separation. Further possibilities could be explored with other configurations of SMBS and more columns. The benefit of adding more columns to SMBS is open to examination. Also, it would be interesting to employ a different isotherm model and compare the results with this study. Literature Cited (1) Fish, B.; Carr, R. W.; Aris, R. Design and Performance of a Simulated Counter-current Moving-bed Separator. AIChE J. 1993, 39, 1783. (2) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; John Wiley: New York, 1984. (3) Broughton, D. B.; Gembicki, S. A. Adsorptive Separations by Simulated Moving-bed Technology: The Sorbex Process. In Fundamentals of Adsorption; Myers, A. L., Belfort, G., Eds.; Engineering Foundation: New York, 1984; p 115. (4) Ruthven, D. M.; Ching, C. B. Counter-current and simulated counter-current adsorption separation processes. Chem. Eng. Sci. 1989, 44, 1011. (5) Svedberg, U. G. Numerical Solution of Multicolumn Adsorption Processes under Periodic Countercurrent Operation. Chem. Eng. Sci. 1976, 31, 345. (6) Ching, C. B.; Ruthven, D. M. An Experimental Study of a Simulated Counter-current Adsorption System. I. Isothermal Steady-State Operation. Chem. Eng. Sci. 1985, 40, 877. (7) Storti, G.; Masi, M.; Carra, S.; Morbidelli, M. Optimal Design of Multicomponent Countercurrent Adsorption Separation Processes Involving Nonlinear Equilibria. Chem. Eng. Sci. 1989, 44, 1329. (8) Storti, G.; Mazzotti, M.; Morbidelli, M.; Carra, S. Robust Design of Binary Countercurrent Adsorption Separation Processes. AIChE J. 1993, 39, 471. (9) Ma, Z.; Wang, N.-H. L. Standing Wave Analysis of SMB Chromatography: Linear Systems. AIChE J. 1997, 43, 2488. (10) Xie, Y.; Wu, D.; Ma, Z.; Wang, N.-H. L. Extended Standing Wave Design Method for Simulated Moving Bed Chromatography: Linear Systems. Ind. Eng. Chem. Res. 2000, 39, 1993. (11) Lu, Z. P.; Ching, C. B. Dynamics of Simulated Movingbed Adsorption Separation Processes. Sep. Sci. Technol. 1997, 32, 1993. (12) Zhong, G. M.; Guiochon, D. Analytical Solution for the Linear Ldeal Model of Simulated Moving Bed Chromatography. Chem. Eng. Sci. 1996, 51, 4307.

(13) Zhong, G. M.; Guiochon, G. Steady-state Analysis of Simulated Moving-bed Chromatography Using the Linear, Ideal Model. Chem. Eng. Sci. 1998, 53, 1121. (14) Chen, J. W.; Cunningham, F. L.; Buege, F. A. Computer Simulation of Plant-scale Multicolumn Adsorption Processes under Periodic Countercurrent Operation. Ind. Eng. Chem. Process Des. Dev. 1972, 11, 430. (15) Strube, J.; Altenho¨ner, U.; Meurer, M.; Schmidt-Traub, H.; Schulte, M. Dynamic Simulation of Simulated Moving-bed Chromatographic Processes for the Optimization of Chiral Separations. J. Chromatogr. 1997, A769, 81. (16) Du¨nnebier, G.; Weirich, I.; Klatt, K.-U. Computationally Efficient Dynamic Modeling and Simulation of Simulated Moving Bed Chromatographic Processes with Linear Isotherms. Chem. Eng. Sci. 1998, 53, 2537. (17) Du¨nnebier, G.; Klatt, K.-U. Modelling and Simulation of Nonlinear Chromatographic Separation Processes: A Comparison of Different Modeling Approaches. Chem. Eng. Sci. 2000, 55, 373. (18) Klatt, K.-U.; Hanisch, F.; Du¨nnebier, G. Model-based Control of a Simulated Moving Bed Chromatographic Process for the Separation of Fructose and Glucose. J. Process Control 2002, 12, 203. (19) Ruthven, D. M. Past Progress and Future Challenges in Adsorption Research. Ind. Eng. Chem. Res. 2000, 39, 2127. (20) Kubota, K.; Hata, C.; Hayashi, S. A Study of a Simulated Moving-bed Adsorber Based on the Axial Dispersion Model. Can. J. Chem. Eng. 1989, 67, 1025. (21) Finlayson, B. A. Numerical Methods for Problems with Moving Fronts; Ravenna Park Publishers: Seattle, WA, 1992. (22) Sun, L. M.; Meunier, F. An Improved Finite Difference Method for Fixed-bed Multicomponent Asorption. AIChE J. 1991, 37, 244. (23) Leonard, B. P. A Stable and Accurate Convective Modeling Procedure Based on Quadratic Upstream Interpolation. Comput. Methods Appl. Mech. Eng. 1979, 19, 59. (24) Patel, M. K.; Markatos, N. C.; Cross, M. A Critical Evaluation of Seven Discretization Schemes for ConvectionDiffusion Equations. Int. J. Numer. Methods Fluids 1985, 5, 225. (25) Sanz-Serna, J. M.; Christie, I. A Simple Adaptive Technique for Nonlinear Wave Problems. J. Comput. Phys. 1986, 67, 348.

Received for review October 12, 2004 Revised manuscript received February 28, 2005 Accepted April 8, 2005 IE0402618