Simulated Moving Bed Cascades for Ternary Separations - American

of part of A and part of C as the product in train A, the flow rates of ... sKi. (1a) ui,j) (constanti)vj. ) Civj. (1b). uA1 e uport, uA2 g uport, uB2...
0 downloads 0 Views 86KB Size
Download PDF
Ind. Eng. Chem. Res. 2001, 40, 6185-6193

6185

Simulated Moving Bed Cascades for Ternary Separations Phillip C. Wankat† School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283

Seven cascades for simulated moving bed systems are developed for ternary separations with linear isotherms. The minimum desorbent-to-feed ratio, D/F, was determined for each cascade using the equilibrium model for 10 different separation problems. A cascade with 14 zones always had the lowest value of D/F, but other cascades could be simpler and/or have higher productivities. Rules of thumb are presented to determine reasonable cascades for more detailed simulation and detailed cost estimation. Detailed simulations showed that the cascades that the equilibrium model predicted to be best did give the best separations. Simulated moving bed (SMB) systems have been used commercially for the separation of liquid mixtures since their commercialization by Universal Oil Products in the 1960s.1-5 The method is now well-known for binary separations and has been commercialized for a large number of separations. The latest area of interest has been in bioseparations, in particular, chiral separations.6 A typical four-zone SMB for binary separations is shown in Figure 1. Binary SMBs can to some extent be considered as analogous to distillation. Zone 2 is analogous to the enriching section and zone 3 to the stripping section. Zone 4 is analogous to the reboiler, but because it is a batch total reboiler, all of the heavy component must be removed. Zone 1 is analogous to a batch total condenser; the last bit of adsorbate must be retained on the adsorbent. The four-zone system is highly efficient for binary separations and minimizes desorbent usage. The design of SMB systems for binary separations has been extensively studied by many groups and is now well understood.7-11 On the other hand, the use of SMB cascades for ternary and other multicomponent systems has not been extensively studied. The “common wisdom”11-13 is to couple two SMB cascades as either the direct connection (Figure 2a) or the indirect connection (Figure 2b). There are, of course, alternatives. Masuda et al.14 patented a method for concentration of the middle component (the component with the intermediate solute velocity) near the feed location and then elution of this component with the feed shut off. This process has been commercialized by Organo Corp., Toyko, Japan. Hashimoto et al.15 developed systems using two resins that were alternated in different sections of the SMB. A cationexchange resin and an ion retardation resin were used to separate starch, glucose, and NaCl. Balannec and Hotier16 used a system with different solvents to separate ternary mixtures. Wooley et al.17 developed a ninezone system for ternary separations and reported extensive comparisons of theory and experiments. Nicoud6 discussed five-zone systems with side streams for separation of ternary mixtures. In this paper a number of new SMB cascades for ternary separations are presented. The well-known equilibrium model is used to compare separation condi† E-mail: [email protected]. Phone: 765-494-7422. Fax: 765-494-0805.

Figure 1. Cascade for SMB for binary separation. Switching of ports is not shown but is understood to occur. There can be more than one column per zone.

tions for separation of systems with linear isotherms to determine the ideal (minimum) desorbent usage and the productivity. Then processes are compared under more realistic conditions using Aspen Chromatography simulations. Ternary SMB Designs The somewhat weak analogy between distillation and SMB systems was used to develop candidate systems for ternary SMB cascades in addition to Figure 2a,b. In distillation a variety of thermally coupled systems have been developed that do the easiest separation first.18 For a mixture of A, B, and C, where A is the most volatile and C the least, the easiest separation is the A from C separation. B is allowed to distribute between the two products of the first column. Direct application of the analogy between easy-split distillation and SMB results in the 10-zone system shown in Figure 3. In train A the ternary feed is separated with the easiest split to form the AB and BC product streams. The zones all have the same functions as those in the binary system shown in Figure 1. Zone 1 must be operated so that all A is retained and zone 4 so that all C is removed. Train B receives the two feeds and separates them into three product streams. Zones 2′ and 3′ separate A and B, while zones 4′ and 5′ separate B and C. Zone 1′ recovers A so that desorbent D can be recycled, and zone 6′ removes adsorbate C from

10.1021/ie010075r CCC: $20.00 © 2001 American Chemical Society Published on Web 11/27/2001

6186

Ind. Eng. Chem. Res., Vol. 40, No. 26, 2001

Figure 2. (a) “Direct” connection of two SMBs for ternary separation. (b) “Indirect” connection of two SMBs for ternary separation.

Figure 4. Easy-split 12-zone SMB for complete ternary separation.

Figure 3. Easy-split 10-zone SMB for complete ternary separation.

the adsorbent. Detailed conditions for successful operation of all of the new cascade designs are developed in the next section. The design in Figure 3 can be improved. By removal of part of A and part of C as the product in train A, the flow rates of products sent to train B are reduced. The resulting 12-zone easy-split cascade is shown in Figure 4. This cascade allows adjustment of the flows in each zone of train A. Train B is geometrically identical in

Figures 3 and 4, but the feeds are more concentrated in solute B in Figure 4. The theory in the next section shows that train B in Figures 3 and 4 is overconstrained. If train B is split into two binary columns, each train can be optimized. Figure 3 is then transformed into Figure 5 and Figure 4 into Figure 6. Figure 5 will have a lower value of D/F than Figure 3, and Figure 6 will have a lower value of D/F than both Figures 4 and 5. Local Equilibrium Solution The equilibrium solution has been extensively applied to analyze SMB systems with both linear5,7,10,19,20 and nonlinear8,9 isotherms. The equilibrium solution is

Ind. Eng. Chem. Res., Vol. 40, No. 26, 2001 6187

Figure 5. Ternary separation using a four-zone easy-split SMB followed by two binary-split SMBs.

Figure 6. Ternary separation done by a six-zone easy-split SMB followed by two binary-split SMBs.

easily developed for all of the schemes shown in Figures 2-6. To easily compare the cascades shown in Figures 2-6 for 10 different ternary mixtures, this paper is restricted to linear isotherms. The purpose was to find the conditions that minimized desorbent usage and then compare the values of (D/F)min and the productivity for each cascade. The equilibrium model assumes very rapid mass transfer and negligible dispersion so that the adsorbed solute is always in equilibrium with the solute in solution outside the adsorbent particles. Thus, for systems separating based on equilibrium differences, the equilibrium solution represents the best separation possible. For linear isotherms, the solute velocity is5

where the average port velocity is uport ) L/tsw. If masstransfer rates are very high and dispersion is low, the conditions in eq 2 will ensure complete separation of solutes A and B. Note that except for eq 2f these equations are identical to a binary SMB separation of A from B. Equation 2f differs because for a binary separation it involves the velocity of component B not C. This change will make velocity v4 higher in the ternary system than in the binary separation. The interstitial velocities are related by mass balances. Assuming that the densities of the liquid mixtures are identical.

vj

usolute i,zone j ) 1+

1 - p (1 - e)(1 - p) pKDi + Fs K i e e (1a)

where Ki ) qi/ci. We will find it convenient to write this as

ui,j ) (constanti)vj ) Civj

(1b)

As an illustration of the procedure, consider the indirect connection ternary SMB (Figure 2b). Train A essentially separates component A from B. We want

uA1 e uport, uA2 g uport, uB2 e uport, uA3 g uport, uB3 e uport, uC4 g uport (2a-f)

v1 ) v2 - vAprd, v2 ) v3 + vF, v3 ) v4 - vBCprd, v4 ) v1 + vD (3a-d) The velocities of products, feed, and desorbent input are related to their volumetric rates by equations of the form

vF )

F eAc

where Ac ) πdcol2/4

(4)

Because v2 is greater than v3, eqs 2b and 2e are automatically satisfied if eqs 2d and 2c are satisfied, respectively. Thus, the four eqs 2a, 2c, 2d, and 2f are sufficient. The minimum amount of desorbent under ideal conditions can be found by using the equality signs in these four relationships. For a known feed rate F, simultaneously solve eqs 1a, 2c,d, 3b, and 4. Then,

6188

Ind. Eng. Chem. Res., Vol. 40, No. 26, 2001

v2 ) CAvF/(CA - CB), uport ) CBv2 ) CBCAvF/(CA - CB), v3 ) v2 - vF (5a-c) Once the optimum value of uport has been determined, the velocities in zones 1 and 4 can be found from eq 1a,b.

v1 ) uport/CA, v4 ) uport/Cc

(6a,b)

Equations 3a, 3c, and 3d can then be used to determine vAprd,vBCprd, and vD. Equation 4 can be used to translate these into volumetric flow rates. The calculations for train B are similar. The feed to train B is the BC product from train A. Because the two trains are likely to have columns of different diameter, calculate the new volumetric feed rate as

FtrnB ) (vBCprdeAc)trnA

(7)

Then use eq 4 to find vF,trnB. The controlling velocity equations are now

uB1 e uport,trnB, uC2 e uport,trnB, uB3 g uport,trnB, uC4 g uport,trnB (8a-d) The simultaneous solution of these equations with eqs 1 and 3 for train B is very similar to the previous result. The total amount of desorbent is

Dtotal ) DtrnA + DtrnB

(9)

which allows Dtotal/F to be calculated. The productivity of the system is defined as

productivity )

volume feed/time (10a) total adsorbent volume

or

productivity ) F [(no. of columns)AcL]trnA + [(no. of columns)AcL]trnB (10b) The column length is related to the port velocity and the switching time by L ) uporttsw. Switching times and column lengths can be different in the two trains. For the local equilibrium solution, the switching time (or column length) can be arbitrarily specified. To make the productivities comparable for all systems, an arbitrarily chosen constant switching time of 7.5 min was used. This choice does not affect the calculated values of D/F. This analysis can obviously be applied to the direct connection cascade shown in Figure 2a. Equations 2a and 2f are unchanged, while eqs 2c and 2d become

uC2 e uport, uB2 g uport

(11a,b)

The feed to train B is now the AB product, and train B separates components A and B. The procedure is essentially the same as that shown previously. Application of the local equilibrium model to train A in either of the easy-split systems (Figures 3 and 4) is straightforward. For example, appropriate solute velocity equations for train A of Figure 4 are

uA1 e uport, uB2 e uport, uC3 e uport, uA4 g uport, uB5 g uport, uC6 g uport (12a-f) and the velocity relationships are

v1 ) v2 - vAprd,trnA, v2 ) v3 - vABprd, v3 ) v4 + vF, v4 ) v5 - vBCprd, v5 ) v6 - vCprd,trnA, v6 ) v1 + vD,trnA (13a-f) We can simultaneously solve eqs 1, 12c,d, and 13c at the feed location to find v3, v4, and uport. The velocities v1, v2, v5, and v6 can be found from eq 12a,b,e,f, respectively. Then the velocities vAprd, vABprd, vBCprd, vCprd, and vD are determined from eq 13. Application of the theory to train B in Figures 3 and 4 is a bit more complicated. We can easily write the appropriate set of solute velocities

uA1 e uport, uB2 e uport, uA3 g uport, uB3 e uport, uC4 e uport, uB4 g uport, uB5 g uport, uC6 g uport (14a-h) and velocity relationships

v1 ) v2 - vAprd,trnB, v2 ) v3 - vABfeed, v3 ) v4 + vBprd, v4 ) v5 - vBCfeed, v5 ) v6 - vCprd,trnB, v6 ) v1 + vD,trnB (15a-f) Unfortunately, this system is overspecified. One can calculate uport, v2, and v3 for feed AB. Superficially, it appears that the other velocities can be determined from eqs 1 and 14a,g,h using the equal signs in these expressions, and values for vAprd,trnB, vBprd, vBCfeed, vCprd,trnB, and vD,trnB can be found from eq 15a,c-f. However, the BC feed flow rate is not a variable since it must be the same as the BC product from train A. Thus, this procedure is inconsistent. A variety of schemes were devised to determine operating conditions that satisfy the inequalities in eq 14. When this is done, there is no guarantee that the best operating conditions that minimize the desorbent flow rate have been determined. The method that worked best was to guess a value for vD,trnB and, considering it a feed, to solve eq 14a,h simultaneously (using the equal signs) with eqs 1b and 15f. This gives values for uport, v1, and v6. Set uB2 and uB5 equal to uport (eq 14b,g], and find v2 and v5. Then determine vAprd,trnB, vCprd,trnB, v3, v4, and vBprd from eq 15a,c,b,d,c, respectively. Now check that eq 14c-f are satisfied. If not satisfied, increase the value of vD,trnB. If they are satisfied, decrease the value of vD,trnB. Repeat this procedure until the lowest value of vD,trnB that satisfies eqs 14 and 15 is determined. Results of Equilibrium Calculations The purpose of this research was to determine which cascade has the lowest D/F ratio and the highest productivity for a given linear separation problem. When the previous calculations are programmed in a spreadsheet, D/F and productivity can easily be determined. The difficulty of linear separations can be classified based on the selectivity

Rik ) Ki/Kk g 1.0

(16)

Somewhat arbitrarily, a separation with R ) 1.1 was

Ind. Eng. Chem. Res., Vol. 40, No. 26, 2001 6189 Table 1. Conditions for Equilibrium Calculationsa 1st hard 2nd 1st moderate 2nd 1st easy 2nd constant R′ easy-easy with R′AB ) R′BC ) 1.848 a

RBA ) 1.10 hard moderate easy RBA ) 1.50 hard moderate easy RBA ) 4.0 hard moderate easy KB ) 2.091

KB ) 0.220 KC ) 0.242 KC ) 0.330 KC ) 0.880 KB ) 0.300 KC ) 0.330 KC ) 0.450 KC ) 1.20 KB ) 0.800 KC ) 0.880 KC ) 1.20 KC ) 3.20 uB ) 0.2941v

uB ) 0.5387v uC ) 0.5334v uC ) 0.5135v uC ) 0.4165v uB ) 0.5202v uC ) 0.5135v uC ) 0.4887v uC ) 0.3752v uB ) 0.4283v uC ) 0.4165v uC ) 0.3752v uC ) 0.2318v KC ) 5.586

R′AB ) 1.009 R′BC ) 1.010 R′BC ) 1.049 R′BC ) 1.293 R′AB ) 1.045 R′BC ) 1.013 R′BC ) 1.064 R′BC ) 1.386 R′AB ) 1.269 R′BC ) 1.028 R′BC ) 1.141 R′BC ) 1.848 uC ) 0.1591v

Hard R ) 1.10, moderate R ) 1.50, easy R ) 4.00, e ) 0.40, p ) 0.45, Kd ) 1.0, KA ) 0.20, and uA ) 0.5435v.

Table 2. Results of Equilibrium Solutions cascade

separation

8-zone A + B to 2nd (Figure 2a)

8-zone B + C to 2nd (Figure 2b)

10-zone easy-split (Figure 3)

hard-hard, D/F productivity, min-1 hard-Mod, D/F productivity, min-1 hard-easy, D/F productivity, min-1 mod-hard, D/F productivity, min-1 mod-mod, D/F productivity, min-1 mod-easy, D/F productivity, min-1 easy-hard, D/F productivity, min-1 easy-mod, D/F productivity, min-1 easy-easy, D/F productivity, min-1 constant R′ easy-easy, D/F productivity, min-1

3.82 7.82 × 10-5 2.37 1.62 × 10-4 2.06 2.08 × 10-4 8.67 1.44 × 10-4 3.33 4.72 × 10-4 2.22 9.01 × 10-4 17.00 4.13 × 10-4 5.00 1.65 × 10-3 2.50 4.40 × 10-3 3.08 9.99 × 10-3

4.19 7.59 × 10-5 12.96 1.01 × 10-4 67.74 1.09 × 10-4 2.60 2.06 × 10-4 5.00 4.13 × 10-4 20.02 5.22 × 10-4 2.27 6.95 × 10-4 3.34 1.89 × 10-3 10.00 2.93 × 10-3 5.70 8.19 × 10-3

3.10a 1.12 × 10-4 a 7.48a 1.34 × 10-4 a 34.88a 1.44 × 10-4 a 5.34a 1.91 × 10-4 a 3.50a 5.79 × 10-4 a 11.01a 6.88 × 10-4 a 9.50a 5.44 × 10-4 a 3.50a 2.32 × 10-3 a 6.00a 2.32 × 10-3 a 3.85a 1.12 × 10-2 a

a

12-zone easy-split (Figure 4)

4.4.4 easy-split (Figure 5)

14-zone easy-split (Figure 6)

2.10a 1.47 × 10-4 a 6.48a 1.47 × 10-4 a 33.90a 1.47 × 10-4 a 4.34a 2.20 × 10-4 a 2.50a 7.34 × 10-4 a 10.02a 7.33 × 10-4 a 8.50a 5.87 × 10-4 a 2.50a 2.94 × 10-3 a 5.00a 4.40 × 10-3 a 2.85a 1.39 × 10-2 a

3.00 9.25 × 10-5 3.00 1.65 × 10-4 3.00 2.08 × 10-4 3.00 2.16 × 10-4 3.00 5.33 × 10-4 3.00 9.09 × 10-4 3.00 7.04 × 10-4 3.00 2.13 × 10-3 3.00 4.55 × 10-3 3.00 1.10 × 10-2

2.00 1.32 × 10-4 2.00 2.00 × 10-4 2.00 2.17 × 10-4 2.00 2.79 × 10-4 2.00 7.50 × 10-4 2.00 1.04 × 10-3 2.00 8.19 × 10-4 2.00 3.00 × 10-3 2.00 5.74 × 10-3 2.00 1.52 × 10-2

System is overconstrained. Lower values may be possible.

considered hard, that with R ) 1.5 was moderate, and that with R ) 4.0 was easy. The value of KA was arbitrarily chosen as 0.2. We then looked at nine cases where both the A-B and the B-C separations could be hard, moderate, or easy. The corresponding values of the K values and the solute velocities are given in Table 1. The last column in Table 1 lists the values of R′

R′ik ) ui/uk g 1.0

(17)

Note that this ratio varies for a second separation of the same degree of difficulty depending upon the difficulty of the first separation. For example, compare R′BC for hard-easy and easy-easy separations. This difference becomes important in interpreting the results later. The last entry in Table 1 looks at a separation where the two values of R′ are constant. The switching time was chosen to be constant at tsw ) 7.5 min. Column diameters were sized so that the maximum velocity in each train was 100 cm/min. These values are arbitrary but allow comparison of the productivities of equivalent cascades. The calculated values of the desorbent-to-feed ratio (D/F) and the productivity (eq 10b) for the equilibrium analysis of the ternary SMB systems are given in Table 2. D/F includes the most important operating cost,

desorbent usage. D/F is also important for capital costs particularly if solvent removal from the products requires distillation. The productivity includes an important capital cost, the amount of adsorbent required. Minimizing D/F and maximizing productivity will tend to minimize the cost of the SMB system. There are general trends in Table 2. The 14-zone easysplit system shown in Figure 6 always has the lowest D/F, equal to 2.00 for all cases. This is remarkable because D/F ) 2 is the thermodynamic limit without adding energy to the system. Except for the hard-hard case, this system also has the highest productivity. However, the 14-zone system is also the most complicated, and there may be reasons to select simpler systems. The second-best system follows a more complicated trend. In general, when the A-B separation is significantly more difficult than the B-C separation, Figure 2a is the second-best system. When the B-C separation is significantly more difficult than the A-B separation, Figure 2b is second-best. When the two separations are of approximately equal difficulty, the 12-zone easy-split system (Figure 4) is second-best. One exception to the trends for the second-best system appears to be the easy-easy separation. Figure 2a clearly has a lower D/F than Figure 4. However, the R′ values listed in Table 1 show that R′BC > R′AB. The A-B

6190

Ind. Eng. Chem. Res., Vol. 40, No. 26, 2001

Table 3. Dimensions for an Optimized (Minimum Desorbent) 14-Zone Easy-Split SMB System (Figure 6)a dimension column diameter, cm separation

hard-hard 10.15 hard-mod 5.89 hard-easy 2.86 mod-hard 5.89 mod-mod 4.35 mod-easy 2.48 easy-hard 2.86 easy-mod 2.48 easy-easy 1.83 constant R′ 1.64 easy-easy a

column length, cm

train 1 train 2 train 3 train 1 train 2 train 3 FAB/F 10.6 13.45 14.41 3.20 5.16 6.33 1.03 1.90 2.68 1.64

9.70 2.51 0.50 10.75 3.56 0.83 7.83 3.04 0.91 1.21

400.1 385.2 312.4 385.2 366.5 281.3 312.4 281.4 173.8 119.3

404.0 404.0 404.0 390.1 390.1 390.1 321.2 321.2 321.2 220.6

400.1 385.2 312.4 385.2 366.5 281.2 312.4 281.4 173.8 119.3

0.523 0.846 0.970 0.231 0.600 0.900 0.118 0.400 0.800 0.649

uport ) L/tsw ) L/7.5.

separation is really more difficult than the BC separation. If we look at the last row in Table 2 where the R′ values are equal for an easy-easy separation, the easy-split separation (Figure 4) is clearly preferable to Figure 2a. The 14-zone easy-split system has the minimum D/F ratio and high productivities because the easy-split configuration does not require extra desorbent and each of the three trains can be optimized. The optimized system operates with D/F ) 2, which requires FAB + FBC ) F. The results in Table 3 show that for separations between A-B and B-C of equal difficulty FAB ∼ FBC, and the designs for trains 2 and 3 will be similar. For separations of different difficulty, most of the A or C product will be withdrawn from train 1, and the ratio FAB/F will be considerably different than 0.5. Then the feed rates to trains 2 and 3 are very different, and the columns in these trains have very different diameters. For example, for the hard-easy separation, train 3 receives very little feed because the first train is able to recover most of the C product. For the hard-easy separation, Table 2 shows that there is very little penalty in choosing the considerably simpler scheme in Figure 2a instead of the 14-zone system. Changes in the column diameter in the trains (Table 3) tend to affect the productivities reported in Table 2 more than changes in the column length. The dimensions of the first train are symmetric in the sense that, for example, hard-easy and easy-hard separations have the same dimensions. Table 1 shows that KA and KC have the same values for these separations, and these two K values control the separation in train 1. Train 3 is also symmetric (equal lengths), but the diameters are different because the feed FBC varies. For train 2 all of the hard (or moderate or easy) separations are the same, and thus the columns have identical lengths. The diameters differ because the feed rates, FAB, vary. The dimensions of the columns in Table 3 can be varied by changing the arbitrary choice of maximum velocity in each train. If the maximum velocity is reduced, the columns become shorter and fatter. The productivities and D/F ratios are unchanged. The productivity and D/F ratio do not encapsulate the complete cost of an SMB system. As extra zones are added and each column needs to be different dimensions, the equipment is likely to become more expensive and the operation more complex. Thus, when the D/F ratios and productivities are similar, the simpler design may be preferable to the 14-zone system shown in Figure 6. For example, Figure 2a may be preferable for

the hard-easy separation, and Figure 4 may be preferable for the hard-hard separation. The first train in Figures 2a,b, 3, and 5 requires more desorbent than train A in Figures 4 and 6. This extra desorbent increases the feed rates to the second trains in Figures 2a,b, 3, and 5. This higher feed rate increases D/F and decreases productivity in the second train. If the desorbent is easy to remove from the solutes (e.g., by evaporation), a solvent removal system could be placed between the two trains (two solvent removal systems would be used in Figure 3). The feed(s) to the second train can then be concentrated back to its original feed concentration. At the price of solvent removal, this will decrease D/F and increase productivity. The results for intermediate desorbent removal are shown in Table 4 for Figures 2a,b and 3. An intermediate desorbent removal step was not calculated for Figure 5 because either the A or the C product will become more concentrated than the feed concentration. This may violate solubility constraints. Comparison of Tables 2 and 4 for Figures 2a,b and 3 shows that the D/F ratios are all lower and productivities are higher in Table 4. When D/F is close to 2.0 in Table 2, the improvement is very modest. If solvent removal is inexpensive and the sorbent is expensive, the systems with solvent removal between trains may be an economical alternative for the ternary separation. Rules of Thumb Based on the results in the previous section, the following tentative rules of thumb for selection of SMB systems for the ternary separation of components A-C with linear isotherms were developed. Component A is the weakest sorbed, and C is the strongest. 1. Consider the 14-zone easy-split SMB as one option. 2. The second option depends on the separations. (a) If the A-B separation is considerably more difficult than the B-C separation, consider the base case with A + B fed to the second train (Figure 2a). (b) If the A-B and B-C separations are approximately of equal difficulty, consider the 12-zone easysplit SMB (Figure 4). (c) If the B-C separation is considerably more difficult than the A-B separation, consider the base case with B + C fed to the second train (Figure 2b). 3. If removal of desorbent is inexpensive, consider removing desorbent between trains 1 and 2 using rule of thumb 2a for separation A-B more difficult than B-C, the 10-zone easy-split SMB (Figure 3) if A-B and B-C separations are of equal difficulty, and rule of thumb 2c if the B-C separation is more difficult. The rules of thumb are tentative because they are based on an equilibrium analysis of linear systems only and because new SMB processes will continue to be invented. Detailed Simulations More detailed simulations were done using the commercially available chromatography/dilute SMB software package Aspen Chromatography version 10.1. The separation of sulfuric acid, glucose, and acetic acid in water using an ion-exchange resin was modeled. Equilibrium data are given by Wooley et al.17 The compounds are sterically excluded from a portion of the resin (KDi in eq 1a). Because sulfuric acid does not sorb and is excluded from a significant fraction of the resin, it

Ind. Eng. Chem. Res., Vol. 40, No. 26, 2001 6191 Table 4. Equilibrium Results When the Intermediate Feeds Are Concentrated cascade

a

separation

8-zone A + B to 2nd (Figure 2a)

8-zone B + C to 2nd (Figure 2b)

10-zone easy-split (Figure 3)

hard-hard, D/F productivity, min-1 (solvent removal)/F hard-mod, D/F productivity, min-1 (solvent removal)/F hard-easy, D/F productivity, min-1 (solvent removal)/F mod-hard, D/F productivity, min-1 (solvent removal)/F mod-mod, D/F productivity, min-1 (solvent removal)/F mod-easy, D/F productivity, min-1 (solvent removal)/F easy-hard, D/F productivity, min-1 (solvent removal)/F easy-mod, D/F productivity, min-1 (solvent removal)/F easy-easy, D/F productivity, min-1 (solvent removal)/F constant R′ easy-easy, D/F productivity, min-1 (solvent removal)/F

2.91 1.16 × 10-4 0.912 2.183 1.87 × 10-4 0.183 2.03 2.14 × 10-4 0.0304 5.33 2.54 × 10-4 3.33 2.67 6.6 × 10-4 0.667 2.11 9.91 × 10-4 0.111 9.50 7.77 × 10-4 7.50 3.50 2.64 × 10-3 1.50 2.25 5.28 × 10-3 0.250 2.54 1.35 × 10-2 0.541

3.10 1.16 × 10-4 1.10 7.50 1.87 × 10-4 5.48 34.87 2.14 × 10-4 32.87 2.30 2.54 × 10-4 0.300 3.50 6.60 × 10-4 1.50 11.01 9.91 × 10-4 9.01 2.13 7.77 × 10-4 0.133 2.67 2.64 × 10-3 0.667 6.00 5.28 × 10-3 4.00 3.85 1.35 × 10-2 1.85

2.05a 1.80 × 10-4 a 1.00b 4.24a 2.44 × 10-4 a 1.00b 17.94a 2.83 × 10-4 a 1.00b 3.17a 3.36 × 10-4 a 1.00b 2.25a 9.57 × 10-4 a 1.00b 6.01a 1.30 × 10-3 a 1.00b 5.25a 1.01 × 10-3 a 1.00b 2.25a 3.83 × 10-3 a 1.00b 3.50a 6.95 × 10-3 a 1.00b 2.43a 1.89 × 10-2 a 1.00b

System is overconstrained. Lower values may be possible. b Total solvent removed (0.5 from both intermediate feeds).

travels at a velocity greater than the superficial velocity of the fluid. Unfortunately, Aspen Chromatography version 10.1 does not include steric exclusion. To use this software for the simulation, values were translated to a system without steric exclusion that is equivalent in the sense that the solute velocities are the same. To do this, the solute velocities were calculated from Wooley et al.17 [their formulation includes KDi in both the second and third terms of the denominator of eq 1a] and then set equal to eq 1a with p ) 0. The equivalent value of e was then determined from

e,equivalent ) T ) e + (1 - e)p ) 0.74

(18)

Solving for K in eq 1a gave the following equivalent equilibrium constants: Ksulfuric ) 1.3118, Kglucose ) 3.3118, and Kacetic ) 5.2637. Because the equivalent system has a different porosity structure, the mass transfer data17 could not be used. For linear systems zone spreading due to dispersion and mass-transfer effects can be modeled with an equivalent dispersion determined from a plate count.5 The plate count was arbitrarily set at 150 plates in each column. The results for the different ternary SMB systems are comparable but do not correspond exactly to the real system. The SMB systems were simulated with two columns per zone. A common switching time of 7.93 min was used so that productivities are directly comparable. The feed rate was 60 mL/min. The feed contained 0.5 g/L of each solute. A column diameter of 3.37 cm was used for the standard feed rate and was increased if necessary for higher feed rates in subsequent columns. Changing the column diameter changes the flow rates but has little effect on the productivities in this set of simula-

tions. All of the SMB systems were first designed to find minimum D/F values using local equilibrium theory. The resulting flow rates and column lengths were then used as the input for Aspen Chromatography. The SMB train receiving the fresh feed was simulated first. The intermediate feed streams (e.g., FBC in Figure 2b) were treated as if they were sent to a holding tank and were well-mixed. The concentrations of these feeds were obtained as the average concentrations of the intermediate products from the first train. At the minimum desorbent rate, the predicted outlet concentrations of all of the trains showed an oscillation between two cyclic steady-state results. This oscillation did not appear to have a major impact on the results reported below. When desorbent rates greater than the minimum were used, a single, stable cyclic steady state was obtained for each SMB train. The simulation results at minimum D/F for the better of the two base cases (Figure 2b), the 12-zone easy-split system (Figure 4), and the 14-zone easy-split system (Figure 6) are shown in Tables 5-7, respectively. Because the separation obtained for the 10-zone easysplit system was not as good as that for the 12-zone system and D/F was higher, these results are not shown. Comparison of the tables shows that for this particular separation the easy-split SMB systems are clearly preferable to the base case. The D/F ratio is only slightly better in the 14-zone easy-split SMB compared to the 12-zone SMB, while the productivity is better in the 12zone system. Because train 1 is identical in these two systems, the product concentrations obtained for train 1 are identical. Comparing the purity of the other products (train 2 in the 12-zone system or trains 2 and 3 in the 14-zone system) shows that the 12-zone system has higher purities. Thus, for this separation of acetic

6192

Ind. Eng. Chem. Res., Vol. 40, No. 26, 2001

Table 5. Aspen Chromatography Simulation of Sulfuric Acid, Glucose, and Acetic Acid Separation Using an 8-Zone Base Case SMB (Figure 2b) at Ideal Minimum Desorbent Flow Ratesa

Table 8. Aspen Chromatography Simulation of Sulfuric Acid, Glucose, and Acetic Acid Separation Using a 12-Zone Easy-Split SMB (Figure 4) with Increased Desorbent Flow Ratesa

Train 1 (dcol ) 3.37 cm, L ) 102.62 cm) 1.41% 2.24% 96.35% 60.00 acetic glucose sulfuric mL/min 49.13% 48.76% 2.11% 118.58

Train A (L ) 55.99 cm, D ) 70.44 mL/min, recycle ) 66.77 mL/min) 0.50% 0.62% 98.88% 32.08 acetic glucose sulfuric mL/min 0.72% 48.05% 51.23% 31.32 50.32% 49.27% 0.41% 33.72 98.91% 0.84% 0.25% 32.92

APROD BC feed BPROD CPROD

Train 2 (dcol ) 4.74 cm, L ) 105.0 cm) 7.33% 89.91% 2.76% 118.58 acetic glucose sulfuric mL/min 92.59% 5.96% 1.45% 118.58

desorbent/feed ) 3.95, feed productivity ) 0.16 (L of feed/h)/L of column a

2 columns/zone, tsw ) 7.93 min, and F ) 60 mL/min.

Table 6. Aspen Chromatography Simulation of Sulfuric Acid, Glucose, and Acetic Acid Separation Using a 12-Zone Easy-Split SMB (Figure 4) at Ideal Minimum Desorbent Flow Ratesa Train 1 (dcol ) 3.37 cm, L ) 51.92 cm, D ) 60.00 mL/min) APROD 2.56% 2.08% 95.36% 30.36 acetic glucose sulfuric mL/min AB feed 2.74% 47.80% 49.47% 29.64 BC feed 48.98% 49.23% 1.79% 30.36 CPROD 95.13% 4.07% 0.80% 29.64 Train 2 (dcol ) 3.37 cm, L ) 53.18 cm, D ) 62.92 mL/min) APROD 0.70% 2.08% 97.22% 31.09 acetic glucose sulfuric mL/min BPROD 3.95% 93.07% 2.99% 61.45 CPROD 96.02% 3.51% 0.47% 30.36 desorbent/feed ) 2.05, feed productivity ) 0.32 (L of feed/h)/L of column a

2 columns/zone, tsw)7.93 min, and F ) 60 mL/min.

Table 7. Aspen Chromatography Simulation of Sulfuric Acid, Glucose, and Acetic Acid Separation Using a 14-Zone Easy-Split SMB (Figure 6) at Ideal Minimum Desorbent Flow Ratesa Train 1 (Same as Train 1 in Table 6) APROD BPROD BPROD CPROD

Train 2 (dcol ) 3.37 cm, L ) 51.92 cm) 1.77% 5.50% 92.73% 29.64 acetic glucose sulfuric mL/min 3.73% 92.07% 4.20% 29.64 Train 3 (dcol ) 3.37 cm, L ) 51.92 cm) 6.60% 91.05% 2.35% 30.36 acetic glucose sulfuric mL/min 93.04 5.72% 1.24% 30.36

desorbent/feed ) 2.00, feed productivity ) 0.28 (L of feed/h)/L of column a

2 columns/zone, tsw ) 7.93 min, and F ) 60 mL/min.

acid, glucose, and sulfuric acid, the 12-zone easy-split system (Figure 4) is preferable. This result agrees with the rules of thumb because the sulfuric acid-glucose and the glucose-acetic acid separations are of approximately equal difficulty. As expected, the detailed simulations show that the products are not pure when the minimum desorbent flow rates are set. Because the minimum desorbent was determined using the equal signs in eqs 2, 8, 11, 12, or 14, we want to use the inequality for higher desorbent flow rates and better separations. A convenient way to solve the inequalities is to write them as equalities, but with a multiplier M times uport. For example, eqs 2a and 2d become

uA1 ) M1uport, M1 < 1.0; uA3 ) M3uport, M3 > 1.0 (19a,b)

APROD AB BC CPROD

APROD BPROD CPROD

Train B (L ) 71.83 cm, D ) 89.86 mL/min, recycle ) 85.61 mL/min) 0.48% 0.03% 99.49% 35.69 acetic glucose sulfuric mL/min 1.52% 97.56% 0.92% 76.98 98.93% 0.81% 0.26% 42.24

total desorbent/feed ) 2.67, feed productivity ) 0.26 (L of feed/h)/L of column a 2 columns/zone, t sw ) 7.93 min, dcol ) 3.37 cm, and F ) 60 mL/min.

By picking appropriate multipliers for each of the inequalities, one can easily generate the flow rates in each section of the column. The 12-zone easy-split SMB (Figure 4) was simulated with higher desorbent rates determined using multipliers and in longer columns in an attempt to improve the product purities. The results for one run with a modest increase in the desorbent rate are shown in Table 8. At the cost of more desorbent and decreased productivity, the product purities are all increased. By increasing the desorbent rate to D/F ) 4.29 (not shown), the purities of all products can be increased to greater than 99%. The productivity does drop to 0.17 (L of feed/h)/L of column. These values for the D/F ratio and productivity are similar to those reported for the base case (Table 5), but the 12-zone easy-split system has much-improved purities. Discussion These results show that the “common wisdom” using the cascades in Figure 2 will often give a cascade that is far from the optimum. The 14-zone cascade shown in Figure 6 is remarkable because it can always be optimized to operate at an ideal D/F ) 2.0. For ternary separations without the addition of energy, this represents the thermodynamic minimum amount of desorbent. The concentration of solute A in the A product streams will be equal to its concentration in the feed and likewise for solutes B and C. When energy is added to the SMB, the product streams can be concentrated.4 The equilibrium theory can be used to eliminate cascades that are poor choices for a given separation. However, it does not necessarily differentiate between systems because zone spreading is not included. For example, for binary systems the equilibrium model predicts D/F ) 1.0 for both a four-zone SMB and repeated pulses in an elution chromatography column. When zone spreading is included, the four-zone SMB can obtain the same purities as an elution chromatograph but at significantly lower D/F values. Thus, the equilibrium model is useful for initial screening, but detailed simulations are needed to check its predictions. In this study the equilibrium predictions agreed qualitatively with the detailed simulations. This agreement is expected because all comparisons are between different SMB systems.

Ind. Eng. Chem. Res., Vol. 40, No. 26, 2001 6193

A mathematical optimization algorithm could be used to determine the minimum D/F for the overconstrained systems instead of the heuristic approach used here. In practice, the current approach is probably satisfactory. If D/F is large, further optimization is unnecessary because a different design should be used. If D/F is close to 2.0, the velocities and flow rates obtained will be a satisfactory starting point for optimization using a simulator. The equilibrium model estimates identical desorbent flow rates for SMB systems with one or more columns per zone. Adding more columns per zone becomes useful as the separation deviates from the ideal equilibrium solution. The detailed separations were done with two columns per zone. The purities of the products will improve if there are more columns per zone with a constant total amount of sorbent. The ideal optimum conditions for binary separations in a four-zone SMB can be determined for nonlinear isotherms, but the algebraic manipulations are considerably more difficult than those for linear isotherms.8,9 A similar analysis can probably be developed for the ternary cascades shown in Figures 2-6. The rules of thumb developed for linear isotherms may be qualitatively correct for nonlinear systems but need to be checked. If the ideas developed in this paper are extended to SMB separations of more than three components, the number of variations increases rapidly. Acknowledgment This research was partially supported by NSF Grant CTS 9815844. The assistance of Dr. Felix Jegede was extremely helpful. Notation Ac ) cross-sectional area of the column, m2 ci ) concentration of species i in liquid, g/m3 Ci ) constant for determining the velocity of solute i, eq 1a,b dcol ) column diameter, m D ) volumetric flow rate of fresh desorbent, m3/s F ) volumetric flow rate of the feed, m3/s KDi ) steric exclusion factor (value is 1.0 if no exclusion) Ki ) qi/ci, linear equilibrium constant, g of adsorbent/m3 of solution L ) length of each column, m M1, M3 ) multipliers in eq 19a,b, respectively qi ) amount adsorbed, g of solute/g of adsorbent tsw ) switching time, s ui ) velocity of the solute, m/s uport ) port velocity ) L/tsw, m/s vj ) interstitial fluid velocity in zone j, m/s Greek Symbols Rik ) selectivity ) Ki/Kk R′ik ) ui/uk ) selectivity based on solute velocities e ) external porosity p ) particle porosity Fs ) solid density, kg/m3

Subscripts A, B, C, i, k ) solutes total ) total flow rate to all trains trnA, trnB ) conditions in train A or train B 1, 2, 3, 4, 5, 6, j ) zones in SMB

Literature Cited (1) Broughton, D. B. Molex: Case History of a Process. Chem. Eng. Prog. 1968, 64 (8), 60. (2) Ganetsos, G., Barker, P. E., Eds. Preparative and Production Scale Chromatography; Marcel Dekker: New York, 1993. (3) Ruthven, D. M. Principles of Adsorption & Adsorption Processes; Wiley-Interscience: New York, 1984. (4) Wankat, P. C. Large-Scale Adsorption and Chromatography; CRC Press: Boca Raton, FL, 1986. (5) Wankat, P. C. Rate-Controlled Separations; Kluwer: Amsterdam, The Netherlands, 1990. (6) Nicoud, R. M. Simulated Moving-Bed Chromatography for Biomolecules. In Handbook of Bioseparations; Ahuja, S., Ed.; Academic Press: San Diego, 2000; pp 475-509. (7) Ma, Z.; Wang, N.-H. L. Design of Simulated Moving Bed Chromatography Using Standing Wave Analysis: Linear Systems. AIChE J. 1997, 42, 2488. (8) Mazzotti, M.; Storti, G.; Morbidelli, M. Optimal Operation of Simulated Moving Bed Units for Nonlinear Chromatographic Separations. J. Chromatogr. 1997, A769, 3. (9) Migliorini, C.; Mazzotti, M.; Morbidelli, M. Robust Design of Countercurrent Adsorption Separation Processes: 5. Nonconstant Selectivity. AIChE J. 2000, 46, 1384. (10) Ruthven, D. M.; Ching, C. B. Countercurrent and Simulated Counter-current Adsorption Separation Processes. Chem. Eng. Sci. 1989, 44, 1011. (11) Schmidt-Traub, H.; Strube, J. Dynamic simulation of Simulated-Moving-Bed Chromatographic Processes. Comput. Chem. Eng. 1996, 20, S641. (12) Chiang, A. S. T. Continuous Chromatographic Process Based on SMB Technology. AIChE J. 1998, 44, 1930. (13) Rossiter, G. J. ISEP & CSEP: A Novel Separation Technique for Process Engineers. Israeli Mining Institution Meeting, Tel Aviv, Israel, Dec 1996. (14) Masuda, T.; Sonobe, T.; Matsuda, F.; Horie, M. Process for Fractional Separation of Multi-Component Fluid Mixture. U.S. Patent 5,198,120, 1993. (15) Hashimoto, K.; Adachi, S.; Shirai, Y.; Morishita, M. Operation and Design of Simulated Moving Bed Adsorbers. In Preparative and Production Scale Chromatography; Ganetsos, G., Barker, P. E., Eds.; Marcel Dekker: New York, 1993; pp 273-300. (16) Balannec, B.; Hotier, G. From Batch Elution to Simulated Countercurrent Chromatography. In Preparative and Production Scale Chromatography; Ganetsos, G., Barker, P. E., Eds.; Marcel Dekker: New York, 1993; pp 301-357. (17) Wooley, R.; Ma, Z.; Wang, N.-H. L. A Nine-Zone Simulating Moving Bed for the Recovery of Glucose and Xylose from Biomass Hydrolyzate. Ind. Eng. Chem. Res. 1998, 37, 3699. (18) Agrawal, R. Thermally Coupled Distillation with Reduced Number of Intercolumn Vapor Transfers. AIChE J. 2000, 46, 2198. (19) Wankat, P. C. Operational Techniques for Adsorption and Ion Exchange. Proceedings of the Corn Refiner’s Association 1982 Scientific Conference, Lincolnshire, IL, June 1982; Corn Refiner’s Association: Washington, D.C., 1982; pp 119-167. (20) Zhong, G.; Guiochon, G. Analytical Solution for the Linear Ideal Model of Simulated Moving Bed Chromatography. Chem. Eng. Sci. 1996, 51, 4307.

Received for review January 24, 2001 Accepted October 5, 2001 IE010075R