4840
Ind. Eng. Chem. Res. 2003, 42, 4840-4848
SEPARATIONS Variable Flow Rate Operation for Simulated Moving Bed Separation Systems: Simulation and Optimization Yifei Zang and Phillip C. Wankat* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-2050
Variable flow rate operation is developed to increase simulated moving bed (SMB) separation. With this mode of operation, the average flow rates of all external and internal flows can vary with time. The separations of dextran T6 and raffinose and of 1,1′-bi-2-naphthol enantiomers, representing linear and nonlinear equilibrium systems, respectively, are tested. They are modeled with the Aspen Chromatography simulator for a four-zone SMB with one and two columns per zone. A simplex double-layer optimization method with a pulse flow design strategy is developed to maximize the SMB separation. Variable flow rate operation improves the separation for both SMBs, but improvement is more significant for the one column per zone SMB. For the SMB with one column per zone, for the linear system, the extract purity increased from 85.3 to 91.4%, and the raffinate purity increased from 89.2 to 94.9%. For the nonlinear system, the extract product purity increased from 85.0 to 94.1%, and the raffinate purity increased from 84.9 to 90.3%. The improved separation is explained using the differences between the true moving bed and the SMB. Introduction The idea of the simulated moving bed (SMB) can be tracked back to the Shanks system for leaching in 1841 in England.1 Universal Oil Products (UOP) started current industrial applications for adsorption in the petrochemical industry in the 1960s.2 Since then, SMBs have been widely used to produce petrochemicals and sugars at multi-ton scales.3,4 Recently, the principle of the SMB has been successfully adapted to enantioseparation.4 An SMB uses multiple columns (or multiple sections) on a fixed-bed system, with an appropriate sequence of column switching designed to simulate a countercurrent flow system. Figure 1 shows a series of connected fixed beds with feed and withdrawal ports between columns. The fluid flows upward, while the solid is stationary because it is packed in the columns. Within each column, the separation process is the same as in a fixed bed. As a whole, the separation profile approximates countercurrent movement of the liquid and solid phases by switching the port locations periodically. In addition to the configuration in Figure 1, there are several different types of SMBs. The second SMB studied in this paper is a four-zone SMB with two columns per zone. Because it is a closer approximation to the true moving bed (TMB) than a four-zone SMB with one column per zone, better separation results are expected. Most of the research reported in the literature1-5 has been done on the standard SMB, which uses constant flow rates and switches all ports synchronously; however, several studies have changed this operating * To whom correspondence should be addressed. Phone: (765) 494-7422. Fax: (765) 494-0805. E-mail: Wankat@ ecn.purdue.edu.
format.6-10 Kearney and Hieb invented the idea of variable flow rate operation.7 Optimization studies of variable flow rate SMBs have been tried for four subintervals and three subintervals, which change the SMB flow rates at three or four steps between ports switches, respectively.8,9 Refer to Figure 2 for the threesubinterval operation style. One particularly useful variant is the pulse mode of operation, where the flow rate is zero for two of the three subintervals in Figure 2. Partial feed operation, a subset of this mode of operation, has already proven to be an effective way of improving SMB separation.10 The “Varicol” process changes the number of columns per zone by switching the inlet/outlet valves in a nonsynchronous mode and also improves performance.6 In this paper, the variable flow rate pattern is extended to all controllable flow rates, including the recycle flow rate. To simplify simulation and optimization, the varied flow rate pattern is operated with three subintervals within each switching period (Figure 2). The subinterval step changes for all of the variable flows occur at the same times, t1 and t2. The other operating variables, such as temperature and switch time, are maintained constant. Problem Overview Variable flow rate operation introduces extra degrees of freedom that are used to specify variables, such as subinterval step-change times, shown as t1 and t2 in Figure 2, and flow rates during the subintervals. To obtain the best separation, these new operation parameters must be selected strategically. The three yardsticks for a standard SMB designsproduct purities, productivity, and D/F (desorbent consumption level)s will also be valued for variable flow rate operation. The
10.1021/ie0301785 CCC: $25.00 © 2003 American Chemical Society Published on Web 08/27/2003
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4841
Figure 1. Four-zone SMB with one column per zone under standard operation.
goal of this research is to maximize product purities at constant productivity and constant D/F value. For a binary separation, the optimization is expressed as
max purity index ) (raffinate purity) × w (%) + (extract purity) × (100 - w)% (1) subject to raff + Qfeed ) Qext Qdes i i i + Qi (i ) subintervals 1, 2, 3) (2)
Qk1t1 + Qk2(t2 - t1) + Qk3(tsw - t2) ) Qktsw
(3)
where k ) feed, extract, raffinate, or recycle
tsw > t2 > t1 g 0
(4)
Qi g 0
(5)
The purity index is defined to include both product purities. The objective is to maximize its value. The weight function w (%) is assigned to the raffinate product and can be adjusted according to the relative importance of the raffinate and extract products. For example, if the raffinate product is the only valuable product and high purity but not high recovery is required, w could be assigned as 100. If both products are equally important or one product is important but both high purity and high recovery are needed, w should be assigned as 50. Equation 2 represents the instantaneous external mass balance. Equation 3 guarantees constant productivity, and the combination of eqs 2 and 3 guarantees constant D/F. The average flow rates Qk are the flow rates for the standard SMB. Thus, the first step is to optimize the flow rates in the standard SMB
Figure 2. Variable flow rate trajectories.
(constant flow rates). For this binary separation, the use of variable flow rates adds 10 more degrees of freedom, and they need to be selected. When Qki ) Qk (i ) 1, 2, and 3) for any pair of times t1 and t2, the operation is that of a standard SMB. In the following optimization process, all control variables are normalized to T1, T2, and Fki (i ) 1-3; k ) des, feed, ext, and raff) by either tsw or the corresponding average flow rates Qk. SMB Design To achieve the desired separation in a four-zone SMB, solute A, the less strongly adsorbed component, should move faster than the ports are switched in zones 1 and 2 and slower than the ports in zone 4. Solute B, the
4842 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 1. System and Operating Parameters for Dextran T6 and Raffinose Separation (Linear) d L Rp F
System Parameters 1.4 cm 35.625 cma 17.8125 cmb 0.45 0.0011 cm 1.0 g/mL
with many columns, but it is not the best choice for studying the discrete operating conditions for this variable flow rate case, particularly with one column per zone. The column SMB model computes concentration profiles within each column individually using material balances (eq 9), mass transfer (linear lumped resistance form in this case, eq 10), and thermodynamic equilibrium (eq 11).
Operating Parameters concentration of feed, cF,i 0.05 g/mL, i ) R, D feed flow rate, Qfeed 0.016 63 mL/s extract flow rate, Qext 0.023 30 mL/s desorbent flow rate, Qdex 0.023 33 mL/s recycle flow rate, Qrec 0.0698 mL/s switching period, tsw 444 sa 222 sb
Four-zone SMB with one column per zone. b Four-zone SMB with two columns per zone a
Table 2. System and Operating Parameters for 1,1′-bi-2-naphthol Enantiomers separation (Nonlinear), four-zone SMB with One Column Per Zone System Parameters d L
2.6 cm 21 cm 0.4
Operating Parameters concentration of Feed cF,i feed flow rate, Qfeed extract flow rate, Qext raffinate flow rate, Qraff recycle flow rate, Qrec switching period, tsw
10 g/l, i ) A, B 0.0607 mL/s 0.309 mL/s 0.101 mL/s 0.590 mL/s 330 s
strongly adsorbed component, should move slower than the ports in zones 1 and 2 and faster than the ports in zone 3.5
uA,1, uA,2 > uport > uA,4
(6)
uB,3 > uport > uB,1, uB,2
(7)
The key point of these designs is to move the separating components in opposite directions at the feed port. The port velocity of the SMB is
uport ) lport/tsw
(8)
where lport is the packing height between ports. The flow rates of the standard SMB are selected as the mean value of the lower and upper limits for the linear system as the basis for variable flow rate operation (Table 1). Storti et al.11 identified a set of implicit constraints for the four-zone SMB with nonlinear isotherms from eqs 6 and 7 by defining net mass flow rate ratios in the four zones of the SMB. From these constraints, explicit bounds on the operating parameters were obtained, yielding a region in the operating parameter space that can be drawn a priori in terms of the adsorption equilibrium constants and the feed composition. The flow rates of the standard SMB are selected within the triangular area close to the apex, giving close to optimal operation for the standard SMB. These flow rates serve as the basis for variable flow rate operation (Table 2). Simulation Models There are two main approaches to SMB simulation: TMB approximation and detailed column SMB modeling. The TMB method is reasonably accurate for SMBs
∂c ∂2c ∂c ∂q + v - Ez 2 + (1 - ) )0 ∂t ∂z ∂t ∂z
(9)
∂q ) k(q* - q) ∂t
(10)
q* ) feq(c)
(11)
The ports between the columns making up the SMB unit are handled as instantaneous mixers and splitters with or without dead volume. The port switching of the SMB is interpreted as changes in the cyclic boundary conditions. Aspen Chromatography 11.1 was used to dynamically simulate the separation of dextran T6 (MW ≈ 6000) and raffinose, as well as the separation of the 1,1′-bi-2naphthol enantiomers. The isotherms for the sugar mixture are linear12
qR ) 0.56cR
(12)
qD ) 0.23cD
(13)
Their mass transfer can be represented as a linear form with lumped resistance12
∂qR ) 0.057(cR - c/R) ∂t
(14)
∂qD ) 0.0469(cD - c/D) ∂t
(15)
The axial dispersion coefficient Ez in eq 9 is assumed to vary along the length of the column. The model estimates the dispersion coefficient values for each component during the simulation using13
vrP 0.2 0.011 Re 0.48 + ) Ez
( )
(16)
The isotherms for the enantiomers have dual-site Langmuir forms14,15
qA )
2.69cA 0.1cA + (17) 1 + 0.0336cA + 0.0466cB 1 + cA + 3cB
qB )
3.73cA 0.3cB + 1 + 0.0336cA + 0.0466cB 1 + cA + 3cB
(18)
Their mass transfer also has a lumped parameter resistance form14,15
∂qi ) 0.5(q/i - qi) ∂t
i ) A, B
(19)
Axial dispersion is neglected to reduce the complexity of calculation and accelerate the optimization process. Because axial dispersion typically causes less than 5%
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4843 Table 3. Tabular Representation of the k + 1 Vertexes in a Corner Initial Simplex17 vertex
factor x1
factor x2
factor x3
1 2 3 4 ... k+1
S1 S 1 + s1 S1 S1 ... S1
S2 S2 S2 + s 2 S2 ... S2
S3 S3 S3 S 3 + s3 ... S3
... ... ... ... ... ...
factor xk
vertex
factor x1
factor x2
factor x3
Sk Sk Sk Sk ... S k + sk
1 2 3 4 ... k+1
S1 S 1 + p1 S 1 + q1 S 1 + q1 ... S 1 + q1
S2 S2 + q 2 S2 + p 2 S2 + q 2 ... S2 + q 2
S3 S3 + q 3 S3 + q 3 S3 + p3 ... S3 + q 3
Table 4. Corner Initial Vertexes Starting from Standard Operationa vertex
T1
T2
Fext I
Fext II
Ffeed I
Ffeed II
Fraff I
Fraff II
Frecy I
Frecy II
1 2 3 4 5 6 7 8 9 10 11
0.3 0.1 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
0.6 0.6 0.9 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6
c c c x c c c c c c c
c c c c x c c c c c c
c c c c c x c c c c c
c c c c c c x c c c c
c c c c c c c x c c c
c c c c c c c c x c c
c c c c c c c c c x c
c c c c c c c c c c x
a
Table 5. Tabular Representation of the k + 1 Vertexes in a Tilted Initial Simplex17 a
Case I: c ) 1, x ) 0. Case II: c ) 0 x ) 3.33
of zone spreading,16 the predicted optimum should be close to the true optimum. The operating conditions for these two systems are listed in Tables 1 and 2. Two-Layer Optimization Method A double-layer, outer and inner, optimization strategy is developed on the basis of the variable simplex method.17 The outer layer, which is the first step of optimization, explores the response surface sparsely and evenly by the corner initial simplex design. The general corner initial simplex vertexes for k-dimension space are listed in Table 3. The problem has k control variables to optimize. It needs k + 1 initial vertexes to start. The first vertex is the starting vertex with coordinates S1, S2, S3, ..., Sk. All other initial vertexes originate from this starting vertex by the diagonal addition of a variable factor s1, s2, s3, ..., sk. If the starting vertex is ignored, the remaining vertexes’ coordinates form a (k × k) square matrix. For the optimization problem treated here, two values for the starting vertex and the corresponding matrix are presented in Table 4. Case I (c ) 1, x ) 0) has two nonzero and one zero flow rate, whereas case II (c ) 0, x ) 3.33) is the pulse operation. (The first two subintervals in Figure 2 are listed in Table 4; the third subinterval is determined from the mass balance in eq 3.) The control variables are the substep switching time and the substep flow rates. They are normalized by the switching time tsw and the average flow rates, respectively. Table 4 starts from the standard operation with variation factor si ) -1. It covers the operation style in which one substep has zero flow rate while the other substeps operate with fairly high flow rates. Notice that the first three rows are identical, as changing t1 and t2 makes no difference with identical subinterval flow rates (standard operation). To include the influence of the substep time variables, t1 and t2, one other variable must be changed as well. For example, Fext 1 is assigned the value of 0 in addition to t1 being changed at vertex 2. Case II starts with all zero flow rates during the first two substeps. The third substep will then have a high pulse flow rate. The variation factors were selected to make the flow have one high pulse at any desired time,
a
factor xk ... ... ... ... ... ...
Sk S k + qk S k + qk S k + qk ... S k + pk
Si ) starting coordinate for factor i. si ) step size for factor i. pi ) si[xk + 1 + k - 1]/[kx2] qi ) si[xk + 1 - 1]/[kx2]
Table 6. Tilted Initial Vertexes Starting from Standard Operation vertex
T1
T2
Fext I
Fext II
Ffeed I
Ffeed II
Fraff I
Fraff II
Frecy I
Frecy II
1 2 3 4 5 6 7 8 9 10 11 si pi qi
0.3 0.213 0.284 0.284 0.284 0.284 0.284 0.284 0.284 0.284 0.284 -0.1 -0.087 -0.016
0.6 0.616 0.687 0.616 0.616 0.616 0.616 0.616 0.616 0.616 0.616 0.1 0.087 0.016
1 1.016 1.016 1.087 1.016 1.016 1.016 1.016 1.016 1.016 1.016 0.1 0.087 0.016
1 1.016 1.016 1.016 1.087 1.016 1.016 1.016 1.016 1.016 1.016 0.1 0.087 0.016
1 1.016 1.016 1.016 1.016 1.087 1.016 1.016 1.016 1.016 1.016 0.1 0.087 0.016
1 1.016 1.016 1.016 1.016 1.016 1.087 1.016 1.016 1.016 1.016 0.1 0.087 0.016
1 1.016 1.016 1.016 1.016 1.016 1.016 1.087 1.016 1.016 1.016 0.1 0.087 0.016
1 1.016 1.016 1.016 1.016 1.016 1.016 1.016 1.087 1.016 1.016 0.1 0.087 0.016
1 1.016 1.016 1.016 1.016 1.016 1.016 1.016 1.016 1.087 1.016 0.1 0.087 0.016
1 1.016 1.016 1.016 1.016 1.016 1.016 1.016 1.016 1.016 1.087 0.1 0.087 0.016
while the rest of the step has no flow. Take vertex 4 as an example. Its operation has a pulse extract flow for the first substep and no flow for the remaining two substeps. All other inlet and outlet flow rates follow the same style. Before performing the inner-layer optimization, one needs to identify some valuable subregions in the outer layer that might give a global optimum. Each subregion is represented by one vertex under the current condition. The next generation of initial vertexes will starts from these vertexes by the tilted initial simplex algorithm in Table 5. In this set of calculations, the step size si is chosen as very small so that the vicinity of the starting vertex can be studied. Taking the subregion around the standard operating condition as an example, the step size for t1 is chosen as -0.1. All other step sizes are chosen as 0.1 (Table 6). If all of the responses of the generated vertexes around the starting vertex are worse than the response of the starting vertex, there is no need to do the inner-layer calculation. This situation frequently occurs for some boundary vertexes, which implies that the starting vertex might be the local optimum. Although the global optimum cannot be guaranteed, the more subregions studied, the higher probability that the global optimum will be found for the set of average flow rates Fk used. A true global optimum requires optimum values of the Fk also. After several feasible subregions have been determined, the inner-layer step uses the set of available initial vertexes to start a variable simplex search, which is a form of sequential simplex method. The sequential simplex methods can be used to rapidly optimize systems containing several continuous factors. The variable-size algorithm of Nelder and Mead17 was adopted in this research. Figure 3 illustrates the possible moves in the variable-size simplex algorithm. The familiar centroid of the remaining hyperface P, the best vertex B, the next-to-worst vertex N, the worst vertex W, the reflection vertex R, the single expansion vertex
4844 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 8. Comparison between Variable Flow Rate Operation and Standard Operation for Sugar Separation in a Four-Zone SMB with Two Columns Per Zone for 50-50% Weighting for Both Productsa operating parameter T1 T2 Fext I Fext II Ffeed I Ffeed II Fraff I Fraff II Frecy I Frecy II purity index (%) dextran T6 raffinose Figure 3. Pulse flow rate at the third substep interval. Possible moves in the variable-size simplex algorithm. P ) centroid of the remaining hyperface, B ) best vertex, N ) next-to-worst vertex, W ) worst (wastebasket) vertex, R ) reflection vertex, E ) expansion vertex, CR ) contraction vertex on the R side, CW ) contraction vertex on the W side. Table 7. Comparison between Variable Flow Rate Operation and Standard Operation for Four-Zone SMB with One Column Per Zonea operating parameter T1 T2 Fext I Fext II Ffeed I Ffeed II Fraff I Fraff II Frecy I Frecy II dextran T6 raffinose dextran T6 raffinose dextran T6 raffinose dextran T6 raffinose
standard operation 1 1 1 1 1 1 1 1
optimal variable flow rate weighting factor (w, %) 50 10 90 0.356 0.545 0.015 5.051 0.031 4.958 0.056 4.881 0.056 1.113
0.279 0.615 0.006 04 0.0112 0.009 67 0.0146 0.007 68 0.0151 1.045 1.929
raffinate conc (g/L) 42.19 45.41 45.40 5.10 2.43 2.45 extract conc (g/L) 5.50 3.20 3.17 31.98 33.89 33.89 product purity (%) 89.22 94.93 94.87 85.32 91.36 91.43 recovery (%) 84.21 91.00 90.98 89.60 95.14 95.14
0.293 0.605 0.0047 0.0102 0.0096 0.026 0.0063 0.025 3.007 0.138 44.72 3.302 3.665 33.28 93.12 90.08 89.62 93.42
a System is linear (eqs 12 and 13). Operating conditions are in Table 1.
E, the contraction vertex on the reflection side CR, and the contraction vertex on the worst side CW are readily identified. The rules for the variable-size simplex algorithm are as follows:17 1. Rank the vertexes of the first simplex. 2. Calculate and evaluate R. A. If N e R e B, use simplex B‚‚‚NR and go to step 3. B. If R > B, calculate and evaluate E. i. If E g B, use simplex B‚‚‚NE and go to step 3. ii. If E < B, use simplex B‚‚‚NR and go to step 3. C. If R < N, then i. If R g W, calculate the contraction vertex on the reflection side CR, use simplex B‚‚‚NCR, and go to step 3.
dextran T6 raffinose dextran T6 raffinose dextran T6 raffinose
standard operation 1 1 1 1 1 1 1 1 92.60 raffinate conc (g/L) 44.97 2.55 extract conc (g/L) 3.52 33.80 product purity (%) 94.63 90.57 recovery (%) 90.12 94.88
optimum variable flow rate weighting factor (50%) 0.462 0.646 0.053 4.943 0.050 5.144 0.1023 5.005 0.956 0.687 93.78 45.51 2.18 2.91 34.03 95.44 92.12 91.20 95.53
a Isotherms are linear (eqs 12 and 13). Operating conditions are in Table 1.
Table 9. Comparison between Variable Flow Rate Operation and Standard Operation for Separation of Enantiomers with a four-zone SMB with One Column Per Zone for 50-50% Weighting for Both Productsa operating parameter T1 T2 Fext I Fext II Ffeed I Ffeed II Fraff I Fraff II Frecy I Frecy II purity index (%) A B A B A B A B
standard operation 1 1 1 1 1 1 1 1 84.98 raffinate conc (g/L) 5.13 0.91 extract conc (g/L) 0.29 1.67 product purity (%) 84.93 85.04 recovery (%) 85.24 84.92
optimum variable flow rate weighting factor (50%) 0.423 0.617 0.007 5.032 0.011 4.994 0.017 5.009 1.068 1.012 92.17 5.72 0.62 0.11 1.77 90.27 94.08 94.91 90.01
a Isotherms are nonlinear (eqs 17 and 18). Operating conditions are in Table 2.
ii. If R < W, calculate the contraction vertex on the worst side CW, use simplex B‚‚‚NCW, and go to step 3. 3. Never transfer the current W to the next calculation step. Always transfer the current N to W in the next calculation step. Rank the remaining retained vertexes
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4845
Figure 4. Swinging-port TMB interpretation for SMB operation (t ) 0, start of step; z ) 0, SMB ports).
Figure 5. Swinging-port TMB interpretation for a center-pulse SMB operation example (t ) 0, start of step; z ) 0, SMB ports).
in order of decreasing response in the new calculation step and go to step 2. After all of the local optima are found, they will be compared to find the best, which is usually close to the global optimum. Results For a four-zone SMB with one column per zone, variable flow rate operation shows significant improvement in product purities and recoveries for sugar fractionation compared to standard operation at the same average feed rate (Table 7). As expected, the optimal operation conditions vary significantly as the weighting factor (w) is changed. However, 50-50 weighting gives the highest purities and recoveries. This occurs because the average flow rates are also chosen to give equal weight to both products. Variable flow rate operation was also studied for a four-zone SMB with two columns per zone at w ) 50% (Table 8). Less improvement compared to standard operation is found in this case. A comparison of Tables 8 and 9 shows that raffinose and dextran T6 product purities and recoveries are highest for the variable flow rate operation for the SMB with two columns per zone; they are also higher for a variable flow rate SMB with one column per zone than in the standard operation SMB with two columns per zone. The reason for the improvements will be discussed later. Nonlinear system separation can also be improved by applying variable flow rate operation. The separation
of the enantiomers of 1,1′-bi-2-naphthol was tested on a four-zone SMB with one column per zone (Table 9). The average flow rates were first optimized to give the best purity index with 50-50 weighting for the standard SMB. Using the same 50-50 weighting, the purity index can be improved by 6.6 percentage points by applying the optimized variable flow rate operation, although the raffinate product purity improves more than extract product purity. Intelligent Design of Variable Flow Rate Operation The process of optimization including discrete dynamic SMB simulations requires considerable computational resources. An alternative requiring fewer resources is desirable. The so-called intelligent design of variable flow rate operation is inspired by the differences between SMBs and TMBs. It is well-known that the separation obtained with an SMB is not as good as that obtained with a TMB if they use the same set of operating conditions.18 The difference comes from the simulated intermittent solid particle movement of the SMB compared to the actual solid movement in a TMB. Alternatively, one can say that a TMB can simulate an SMB by swinging its feed and product ports back and forth. A TMB can simulate a four-zone SMB with one column per zone by using a cycle frequency equal to the switch frequency, 1/tsw, of the SMB (Figure 4). The movement of ports along the solid direction is continuous. Figure 4 shows five
4846 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
Figure 6. Effect of synchronized pulse duration of feed, raffinate, extract, and recycle flows for four-zone SMB with one column per zone. The system is linear (eqs 12 and 13). Operation system and average flow rates are in Table 1.
Figure 7. Effect of synchronized pulse duration of feed, raffinate, extract, and recycle flows for four-zone SMB with two columns per zone. The system is linear (eqs 12 and 13). Operation system and average flow rates are in Table 1.
snapshots at times 0, 1/4tsw, 1/2tsw, 3/4tsw, and tsw. Then, all of the ports jump back to their original positions at time 0. This is called one “swipe”. A new cycle repeats. The moving length is one SMB column with the centers at the TMB port positions. For example, the average position of a desorbent port is the intersection of zones 4 and 3. At time 0, the desorbent port is at +0.5L, corresponding to the intersection of zones 4 and 3. At the end of the switch period, the desorbent port is at -0.5L. The port traveling length is the zone length, L, which is the swinging length. Repeated execution of this procedure will give the same separation as the SMB. There is no steady state, but a cyclic steady state is achieved. Because the feed spreads around the TMB
feed position, the separation is worse than that for a standard TMB. The separation obtained with a SMB can now be explained. For a four-zone SMB with one column per zone, the swinging length is the zone length, with one swipe. For a four-zone SMB with two columns per zone, the swinging length is only one-half of the zone length, with two repeating swipes. Because the spreading caused by swinging ports with two columns per zone is much less than that for the TMB simulation of fourzone SMB with one column per zone, the separation of a four-zone SMB with two columns per zone is better than the separation of a four-zone SMB with one column per zone. By interpreting the SMB in this way, the variable flow rate operation makes more sense. The goal is to make the SMB operation closer to true TMB operation. As a start, all of the external flow rates are synchronized as center pulses (Figure 5). One trial is to shorten the swing length. Figures 6 and 7 show the improvement resulting from synchronized pulse flow of feed, raffinate, extract, and recycle, as well as the effect of pulse duration for four-zone SMBs with one column per zone and two columns per zone, respectively. The “synchronized pulse flow operation” refers to the operation of pulse flow of all external flow rates, including feed, extract, desorbent, and raffinate. The pulses of all flows des start and end synchronically; that is, tfeed ) text 1 1 ) t1 ) raff feed ext des raff k t1 , t2 ) t2 ) t2 ) t2 , and Fi ) 0 (i ) 1, 3 and k ) feed, extract, desorbent, and raffinate) (Figure 2). Because the recycle flow rate is kept constant, periods 1 and 3 are analogous to total reflux. For both SMB configurations, the synchronized pulse operation achieves better separation than the standard SMB operation corresponding to a pulse duration tsw. The pulse duration also has an optimum for both cases. These optimal pulse durations are the closest approximation of true TMB operation by the two four-zone SMBs with one column per zone and two columns per zone under synchronized pulse operation. Another interesting phenomenon is that the pulse time, defined as the time at which the center of the pulse enters the SMB, has no effect on the concentration profile for the synchronized operation. The 1/4tsw-3/4tsw pulse (Figure 5) and the 0-1/2tsw pulse (Figure 8) have the same product concentrations. This can be explained by using the swinging-port TMB to simulate an SMB. Take early pulses and center pulses as a comparison. In the SMB, the center pulses center on the solid lines in Figure 5, which are coincidently the boundaries of
Figure 8. Swinging-port TMB interpretation for an early-pulse SMB operation example (t ) 0, start of step; z ) 0, SMB ports).
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4847
the SMB zones. The early pulses center on the dashed lines in Figure 8, switched by -1/4L from the solid lines. In a synchronized swinging-port TMB, the center pulses have the zones 4, 1, 2, and 3 from the top to the bottom with zone length L; the early pulses have the zones (1/ 3 4)3, 4, 1, 2, and ( /4)3 from the top to the bottom with the same zone length L. The feed and product ports are in the center of the zones for both cases. Therefore, the early 1/2tsw pulses and the central 1/2tsw pulses defined for an SMB are identical to the central 1/2tsw pulses for the swinging-port TMB model. The separations are identical. The reasoning given above extends this phenomenon to any SMB systems with synchronized flow rates. Following the idea to make the SMB closer to true TMB operation, the pulse pattern is tested for all controllable flows in the outer-layer design (case II of Table 4), which is one pulse flow rate design. Its innerlayer calculation searches for an optimum from all of the pulse patterns. This procedure gives the global optimum for all of the systems studied (Tables 7-9). In conclusion, because it is unnecessary to explore other flow patterns (Table 4), the computation time can be cut at least by one-half. Summary and Conclusions The variable flow rate operation strategy increases SMB separation efficiency particularly for a four-zone SMB with one column per zone. It works for both linear and nonlinear systems. Under such operation, both product purity and recovery are enhanced with constant productivity. The optimal operating condition can be found by a two-layer pulse optimization method. Because of the similarities and differences between SMB and TMB systems, synchronized pulse design can be explored to reduce computation time while giving significant improvements in SMB separation. Acknowledgment This research was partially supported by NSF Grant CTS-9815844 and the Purdue Research Foundation. The optimization work started from a class project in a course taught by Prof. J. F. Pekny. The co-workers on that project, Chim Yong Chin, Gerard Rogers, and their advisor, Prof. L. N. H. Wang, were very helpful for the initial work. The simulation assistance of Dr. Andrew Stawarz of Aspen Technology is gratefully acknowledged. Notation c ) solute concentration, kg/m3 D ) column diameter, m Ez ) axial dispersion coefficient feq ) equilibrium function Fk ) normalized average flow rate for k ) feed, ext, raff, or rec Fki ) normalized flow rate in subinterval i for k ) feed, ext, raff, or rec k ) adjusted mass-transfer resistance, 1/s L ) column length, m q ) amount of solute adsorbed, kg/(m3 of adsorbent) q* ) equilibrium amount of solute adsorbed, kg/(m3 of adsorbent) Qk ) average flow rate for k ) feed, ext, raff, or rec, mL/s
Qki ) flow rate in subinterval i for k ) feed, ext, raff, or rec, mL, mL/s Re ) Reynolds number, Re ) evdpF/µ rp ) particle radius, m Si ) coordinates of the starting vertex si ) variation factor for Si t ) time, s t1, t2 ) first and second step-change times, respectively, s T1, T2 ) normalized first and second step-change times, respectively, T1 ) t1/tsw, T2 ) t2/tsw tsw ) switch time, s uport ) port velocity, m/s ui,k ) solute velocity for component i in zone k, m/s v ) interstitial fluid velocity, m/s w ) weight factor, % z ) axial coordinate, m Greek Letters ) total porosity of the bed F ) fluid density, g/cm3 Fp ) particle density of solid, including the fluid in the pores, g/cm3 µ ) viscosity of the fluid, g/(cm s) Subscripts D ) dextran T6 R ) raffinose I, II, III, IV ) zone number Definitions productivity (separation rate/efficiency) ) (feed flow rate)/ (adsorbent volume) purity ) (amount of the desired component)/(sum of all feed components in the product) recovery ) (amount of the desired component in its product)/(amount of this component in the feed)
Literature Cited (1) Wankat, P. C. Large-Scale Adsorption and Chromatography; CRC Press: Boca Raton, FL, 1986. (2) Broughton, K. B. Molex: Case History of a Process. Chem. Eng. Prog. 1968, 64, 60. (3) Rearick, E. E.; Kearney, M.; Costesso, D. D. Simulated Moving-Bed Technology in the Sweetener Industry. CHEMTECH 1997 37. (4) Schulte, M.; Strube, J. Preparative Enantioseparation by Simulated Moving Bed Chromatography. J. Chromatogr. A 2001, 906, 399. (5) Wankat, P. C. Rate-Controlled Separations; Kluwer Academic Publishers: Amsterdam, 1990. (6) Ludemann-Hombourger, O.; Nicoud, R. M.; Bailly, M. The “Varicol” Process: A New Multicolumn Continuous Chromatographic Process. Sep. Sci. Technol. 2000, 35, 1829-1862. (7) Kearney, M. M.; Hieb, K. L. Time Variable Simulated Moving Bed Process. U.S. Patent 5,102,553, 1992. (8) Kloppenburg, E.; Gilles, E. D. A New Concept for Operating Simulated Moving-Bed Processes. Chem. Eng. Technol. 1999, 22, 813-817. (9) Morbidelli, M.; Mazzotti, M. Advances in Simulated Moving Bed Chromatography. Presented at the AIChE Annual Meeting, Indianapolis, IN, Nov 3-8, 2002. (10) Zang, Y.; Wankat, P. C. SMB Operation StrategysPartial Feed. Ind. Eng. Chem. Res. 2002, 41, 2504. (11) Storti, G.; Mazzotti, M.; Morbidelli, M.; Carra, S. Robust Design of Binary Countercurrent Adsorption Separation Processes. AIChE J. 1993, 39, 471. (12) Ching, C. B.; Chu, K. H.; Hidajat, K.; Uddin, M. S. Comparative Study of Flow Schemes for a Simulated Countercurrent Adsorption Separation Process. AIChE J. 1992, 38, 1744. (13) Chung, S. F.; Wen, C. Y. Longitudinal Dispersion of Liquid Flowing through Fixed and Fluidized Beds. AIChE J. 1968, 14, 867.
4848 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 (14) Nicoud, R. M. Simulated Moving Bed: Application the Separation of Optical Isomers. Inf. Chim. 1995, 368, 113. (15) Pais, L. S.; Loureiro, J. M.; Rodrigues, A. E. Modeling, Simulation and Operation of a Simulated Moving Bed for Continuous Chromatographic Separation of 1,1′-Bi-2-Naphthol Enantiomers. J. Chromatogr. A 1997, 769, 25-35. (16) Migliorini, C.; Gentilini, A.; Mazzotti, M.; Morbidelli, M. Design of Simulated Moving Bed Units under Nonlideal Conditions. Ind. Eng. Chem. Res. 1999, 38, 2400-2410. (17) Walters, F. H.; Parker, L. R. J.; Morgan, S. L.; Deming, S. N. Sequential Simplex Optimization: A Technique for Improving
Quality and Productivity in Research, Development, and Manufacturing; CRC Press: Boca Raton, FL, 1991. (18) Liapis, A. I.; Rippin, D. W. T. Simulation of Binary Adsorption in Continuous Counter-Current Operation in Comparison with Other Operating Modes. AIChE J. 1979, 25, 455460.
Received for review February 24, 2003 Revised manuscript received July 9, 2003 Accepted July 10, 2003 IE0301785
