4494
Ind. Eng. Chem. Res. 2004, 43, 4494-4502
Novel Analytical Solution for a Simulated Moving Bed in the Presence of Mass-Transfer Resistance Viviana M. T. Silva,† Mirjana Minceva, and Alirio E. Rodrigues* Laboratory of Separation and Reaction Engineering (LSRE), Department of Chemical Engineering, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal
A novel analytical solution for a simulated moving bed (SMB) with a linear adsorption equilibrium isotherm in the presence of mass transfer was derived. The proposed solution is based on the equivalent steady-state true moving bed (TMB) analogy. This analytical solution allows for the calculation of the liquid and solid concentration profiles in all sections of the SMB. The analytical solution was used in the prediction of the “separation volume” for glucose/fructose separation. Emphasis was placed on the effects of the liquid/solid velocity ratios in sections I and IV (γI and γIV, respectively) on the performance of the SMB. The value of γI more significantly affects the SMB performance than the value of γIV and should be tuned carefully in a narrow zone of values. The main advantage of the presented solution is the rapid determination of the separation volume in the presence of mass-transfer limitations, according to the required purity of the extract and raffinate. Introduction The simulated moving bed (SMB) is a continuous chromatographic countercurrent process developed in the 1960s by Universal Oil Products.1 Currently, SMB technology has a wide range of applications, including large-scale separations in the sugar2 and petrochemical industries3,4 and recently developed pharmaceutical, chiral, fine chemistry, and bioseparations processes.5-10 The SMB principles of operation can best be described in reference to those of the equivalent true moving bed (TMB). In a TMB (Figure 1a), the liquid and solid phases flow in opposite directions. The inlet (feed and eluent) and outlet (extract and raffinate) ports are fixed along the unit. According to the position of the inlet and outlet streams, four different operation sections can be distinguished: section I located between the eluent and extract streams, section II located between the extract and feed streams, section III located between the feed and raffinate streams, and section IV located between the raffinate and eluent streams. The net flow rate has to be selected in each section in order to ensure the regeneration of adsorbent in section I, the desorption of the less strongly adsorbed component in section II, the adsorption of the more strongly adsorbed component in section III, and the regeneration of the eluent in section IV. These conditions will guarantee the success of the separation, as the more retained component moves to the extract port with the solid phase and the less retained component moves to the raffinate port with the liquid phase.11 The major problem in TMB operation associated with the movement of the solid phase was overcome by the introduction of SMB technology. An SMB unit consists of a set of interconnected columns in series; countercurrent flow of the solid and liquid phases is simulated by the periodic shifting of the inlets and outlets in the direction of the fluid flow. * To whom correspondence should be addressed. Tel.: 351 22 5081671. Fax: 351 22 5081674. E-mail
[email protected]. † Present address: School of Technology and Management, Braganc¸ a Polytechnic Institute, Campus de Santa Polonia, Apartado 1134, 5301-857 Braganc¸ a, Portugal.
There are two strategies for modeling SMB operation: the equivalent true moving bed concept and the simulated moving bed concept. The potential of the equivalent TMB concept in the prediction of transient behavior in an SMB process has been studied by several researchers.12-15 The main difference between these two concepts is that the steady state can be obtained only in the asymptotic sense in an SMB whereas it is a definite state in a TMB. The TMB and SMB models commonly used have different degrees of complexity (in terms of mass transfer, axial dispersion, and adsorption isotherm) and normally require numerical solution. A few analytical solutions of simple linear SMB models are available in the literature. Ching and Ruthven16 proposed an analytical solution of a linear SMB described by a dispersed plug-flow model. The solution is derived for the steady-state equivalent TMB and assumes 100% pure extract and raffinate. The concept of standing concentration waves was used by Ma and Wang17 to derive design equations for a continuous moving bed unit. The mathematical model assumes a linear adsorption isotherm, axial dispersion, and the linear driving force (LDF) pore diffusion model for mass transfer in the particle. The proposed analytical solution predicts the net flows in each section for defined products purities, recoveries, and feed flow rate. The obtained operating conditions (net flows) were validated by numerical simulations of TMB and SMB units. Zhong and Guiochon18 developed an analytical solution for a linear SMB based on an equilibrium TMB and SMB model, assuming 100% pure extract and raffinate. Both solutions led to explicit algebraic expressions for the concentration profiles and provided a good approximation only in the case of highly efficient SMB columns. The successful design of an SMB requires the proper choice of operating conditions (switching time period and flow rates in each section of the unit). Considering the dynamic complexity of SMB operation, the choice of operating conditions is not simple or straightforward. Great progress in the understanding of SMBs has been made by using an equilibrium model applied to the equivalent TMB, where axial mixing and mass-transfer resistances are neglected and adsorbent equilibrium is
10.1021/ie030610i CCC: $27.50 © 2004 American Chemical Society Published on Web 01/15/2004
Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 4495
Figure 1. (a) Schematic diagram of a true moving bed and (b) notation for the operating conditions of a true moving bed.
assumed to be established everywhere at all times in the column. Two approaches have been used, based on an equilibrium model, that could allow explicit relationships among operating parameters. One is the McCabeThiele analysis of an ideal stage-to-stage model of the unit;16 the other is an equilibrium theory based on the use of a simple concentration propagation analysis.14,19-21 With the equilibrium theory, it is possible to set the net flows in sections I and IV that ensure solid and eluent regeneration, respectively. Therefore, the design of the SMB consists of the determination of the set of pairs of net flows in sections II and III that accomplish the desired separation. These space-operating parameters lead to the separation region (triangle theory). Equilibrium models are not able to link the product purities with the zone flow rates, zone lengths, and mass-transfer parameters, so the resulting design can serve only as an initial guess for the SMB optimization.17 The optimal operating conditions can be obtained by using a complete model that includes mass-transfer effects. Pais et al.22 introduced mass-transfer effects into an SMB design described by the LDF approximation. They showed that the separation region is considerably reduced when mass-transfer effects are present. Biressi et al.23 developed an algorithm for the design and optimization of an SMB that accounts for the thermodynamic, kinetic, and hydrodynamic aspects of the system. The optimum set of the parameters found did not coincide with the predictions of the equilibrium theory. Jupke et al.24 proposed a strategy for the optimal choice of both operating and design parameters, using the separation costs as the objective function. In 1999, Azevedo and Rodrigues25 introduced the “separation volume” methodology, which accounts for the effect of the net flow in section I or IV on the separation regions, in the presence of mass-transfer resistance. The constraints on zones I, II, and III are more restrictive than those derived from the equilibrium model, whereas the constraint on zone IV was less affected.25,26 The determination of the separation vol-
umes requires the numerical solution of the model equations for each set of operating parameters. Recently, the multiobjective optimization of an SMB was performed by Zhang et al.27,28 The authors used a robust optimization technique, nondominated sorting genetic algorithm (NSGA). The multiobjective optimization allowed for the simultaneous maximization of extract and raffinate purities or for the maximization of the SMB productivity with simultaneous minimization of solvent consumption. The aim of this work is (i) to develop a novel analytical solution for a linear SMB in the presence of intraparticle mass-transfer resistances described by the LDF approach based on equivalence with a TMB, (ii) to apply this solution in the determination of the separation volumes in the case of glucose/fructose system, and (iii) to emphasize the importance of the net flows in sections I and IV for SMB performance. Analytical Solution of a Linear SMB in the Presence of Mass Transfer The simulated moving bed (SMB) was modeled using the steady-state equivalent true moving bed (TMB) concept. The proposed mathematical model considers plug flow for the liquid and solid phases, the presence of internal mass transfer describe by the LDF model, and a linear adsorption isotherm. The following assumptions were taken in account: (i) isothermal operation, (ii) constant bed porosity, (iii) equal lengths for all sections, and (iv) different mass-transfer coefficients for the two components. The steady-state model equations for a section j are as follows
0 ) γj
dci,j 1 - / -q j i,j) + Ri(qi,j dx
0)
dq j i,j / -q j i,j) + Ri(qi,j dx / ) Kici,j qi,j
(1) (2) (3)
4496 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004
where ci,j is the liquid-phase concentration of component i in section j, q j i,j is the average solid-phase concentration / of component i in section j, qi,j is the equilibrium solidphase concentration of component i in section j, is the bed porosity, and Ki is the adsorption equilibrium constant of component i. The dimensionless variable for axial position is x ) z/L, and the dimensionless parameters are
γj )
vj us
Ri )
fluid/solid interstitial velocity ratio (4) Lkh,i us
number of mass-transfer units (5)
where vj is the interstitial liquid velocity in section j; us is the interstitial solid velocity; L is the section length (the same for all sections); and kh,i is the homogeneous diffusion rate constant of component i, kh,i ) kp,i/(K′i + p), where Ki ) K′i + p, kp,i ) (15Dp,i/rp2), kp,i is the macropore diffusion rate constant of component i, and Dp,i is the pore diffusion of component i. The boundary conditions are as follows
QIci,I,0 ) QIVci,IV,1
(6)
ci,II,0 ) ci,I,1
(7)
QIIIci,III,0 ) QIIci,II,1 + QFci,F
(8)
ci,IV,0 ) ci,III,1
(9)
q j i,j,0 ) q j i,j-1,1 if j ) 1, then j - 1 ) 4
(10)
The node mass balances (Figure 1b) are
Eluent node QI ) QIV + QE
(11)
QII ) QI - QX
(12)
QIII ) QII + QF
(13)
QIV ) QIII - QR
(14)
γj dci,j + Mi,jci,j ) Ni,j Ri dx
j i,j,0 - γjci,j,0. where Mi,j ) βKi - γj and Ni,j ) βq For the boundary condition at the beginning of the section (x ) 0)
x)0
ci,j ) ci,j,0
the solution of the eq 18 is
ci,j(x) ) j i,j,0 - γjci,j,0 j i,j,0 -[Ri(βKi-γj)/γj]x βq Kici,j,0 - q e + (19) β βKi - γj βKi - γj Rearranging eq 19 and then substituting into eq 17, the liquid- and solid-phase concentration profiles are obtained as functions of the concentration at the beginning of each section according to the expressions
q j i,j,0 - Kici,j,0 (1 - e-[Ri(βKi-γj)/γj]x) (20) βKi - γj
ci,j(x) ) ci,j,0 + β
q j i,j,0 - Kici,j,0 q j i,j(x) ) q j i,j,0 + γj (1 - e-[Ri(βKi-γj)/γj]x) (21) βKi - γj To calculate the liquid and solid concentrations at the beginning of each section, it is necessary to write eqs 20 and 21 at the end of each section and connect all sections by the node mass balances. For x ) 1, by rearranging eqs 20 and 21, the relations of the solid- and liquid-phase concentrations at the end and beginning of each section can be obtained as
ci,j,1 )
Extract node
βKie-(Ri/γj)Mi,j - γj 1 - e-(Ri/γj)Mi,j ci,j,0 + β q j i,j,0 (22) Mi,j Mi,j q j i,j,1 ) q j i,j,0 +
Feed node
dq j i,j dci,j -β )0 γj dx dx
∫cc
i,j
i,j,0
∫qjqj
dci,j ) β
i,j
i,j,0
dq j i,j
(15)
(16)
the operating line relating q j i,j(x) and ci,j(x) is obtained as
j i,j,0 + q j i,j(x) ) q
γj [c (x) - ci,j,0] β i,j
(23)
Ai,j )
βKie-(Ri/γj)Mi,j - γj Mi,j
and
where β ) (1 - )/. By integrating eq 15 between the beginning of each section and some axial position along the section, i.e.
γj
γj γj ci,j,L - ci,j,0 β β
To simplify these expressions, the following parameters were introduced
Raffinate node
By substitution of eq 2 into eq 1, eq 1 becomes
(18)
(17)
By substituting eq 17 into eq 1 and rearranging, one obtains the first-order linear differential equation
1 - e-(Ri/γj)Mi,j Mi,j
Bi,j ) β
Cj )
γj β
Equations 22 and 23 now become
ci,j,1 ) Ai,jci,j,0 + Bi,jq j i,j,0
(24)
j i,j,0 + Cjci,j,1 - Cjci,j,0 q j i,j,1 ) q
(25)
Introducing the node mass balances, it is possible to eliminate the liquid and solid concentrations at the end of each section as follows
Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 4497
Section I j i,I,0 ci,II,0 ) ci,I,1 ) Ai,Ici,I,0 + Bi,Iq
(26)
q j i,II,0 ) q j i,I,1 ) q j i,I,0 + CIci,II,0 - CIci,I,0
(27)
concentration in the extract (ci,X ) ci,II,0) and raffinate (ci,R ) ci,IV,0) nodes. The internal concentration profiles and SMB performance were calculated by introducing the model parameters and operating conditions into the matrix.
Section II vF γIII ci,III,0 c ) ci,II,1 ) Ai,IIci,II,0 + Bi,IIq j i,II,0 (28) γII usγII i,F
(
j i,II,1 ) q j i,II,0 + CII q j i,III,0 ) q
)
vF γIII ci,III,0 c γII usγII i,F CIIci,II,0 (29)
Section III ci,IV,0 ) ci,III,1 ) Ai,IIIci,III,0 + Bi,IIIq j i,III,0
(30)
j i,III,1 ) q j i,III,0 + CIIIci,IV,0 - CIIIci,III,0 q j i,IV,0 ) q
(31)
Section IV γI c ) ci,IV,1 ) Ai,IVci,IV,0 + Bi,IVq j i,IV,0 γIV i,I,0
(32)
γI c - CIVci,IV,0 γIV i,I,0
(33)
j i,IV,1 ) q j i,IV,0 + CIV q j i,I,0 ) q
[
]
Equations 26-33 were then written in matrix form P‚X ) Q, where X is the vector of the variables (liquid and solid concentrations at the beginning of each section)
Ai,I
-1
0
Ai,II
0
0
0 γIII 0 γII Ai,III -1
0 CI
γI γIV -CI 0 0
0
0
0
0
Bi,II 0
0
0
0
Bi,III 0
0
Ai,IV 0
0
0
0
0
1
-1 0
0
1
-1
0
0
0
1
-1
-CIV -1 0
0
1
γIII 0 γII -CIII CIII
-CII CII
0 γI CIV 0 γIV
Bi,I 0
0
Bi,IV 0
[ ][
0
‚
vF ci,I,0 -cF ci,II,0 usγII ci,III,0 0 ci,IV,0 0 q j i,I,0 ) 0 γF q j i,II,0 cF q j i,III,0 βus q j i,IV,0 0 0
]
The system of algebraic equations (eqs 26-33) presented in matrix form was solved using Microsoft Excel. The solution provides the concentrations of the liquid and solid phases at the beginning of each section, which are necessary for the calculation of the concentration profiles (eqs 20 and 21). The SMB performance (see Table 1) can also be calculated from the liquid-phase
Results and Discussion Analysis of Concentration Profiles. The analytical solution of linear SMB in the presence of mass transfer was tested for the case of glucose/fructose separation. The model parameters and SMB operating conditions provided by Azevedo29 are presented in Table 2. The glucose/fructose steady-state concentration profiles and SMB performances were calculated according to the proposed solution. The analytical concentration profiles were compared with the concentration profiles calculated numerically using the steady-state equivalent TMB, as shown in Figure 2. The TMB steady-state model equations (eqs 1-14) were solved using gPROMS (general process modeling system).30 Orthogonal collocation on finite elements was used for the discretization of the axial domain. Each section was divided into 20 equal elements with two collocation points in each element. The simulation was performed on a Pentium III 1000 MHz computer with 785 MB of RAM memory. The computational time was around 10 s. The glucose/ fructose concentration profiles calculated numerically and analytically were identical, as expected given that the analytical solution does not consider any simplifying assumptions in the model (eqs 1-14). The SMB performances calculated using the relations from Table 1 are presented in Table 3. The proposed analytical solution provides a fast and simple way to calculate SMB performance (no need for the calculation of the internal concentration profiles). Therefore, this solution can be easily applied in the prediction of separation regions for specific systems (adsorption equilibrium, mass transfer, and SMB unit design). Effect of the Mass-Transfer and Purity Requirements on the Separation Regions. The analytical solution was used to calculate the separation regions (γII and γIII) corresponding to 90, 95, and 98% extract and raffinate purities in the case of different masstransfer limitations: (i) RFr ) 12.95 and RGl ) 18.40 (case study), Figure 3; (ii) RFr ) 25.90 and RGl ) 36.80, Figure 4; and (iii) negligible mass-transfer resistance, Figure 5. The model parameters used are those presented in Table 2. The separation region is defined by the pairs of γII and γIII values that lead to the desired extract and raffinate purities for given values of γI and γIV. These γI and γIV values are
γI > γeq I ) βKFr
(34)
γIV < γeq IV ) βKGl
(35)
The strategy used to determine the separation regions consists of successive calculations of the extract and raffinate purities within the γII × γIII triangle plane defined by the conditions γIV < γII < γIII < γI.25 In the case of negligible mass-transfer resistance (Figure 5), the separation regions for 90, 95, and 98% extract and raffinate purities are larger than the
4498 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 Table 1. Expressions for the Calculation of SMB Performance SMB performance
extract
raffinate
purity
PUX (%) )
cX,Fr × 100 cX,Fr + cX,Gl
PUR (%) )
cR,Gl × 100 cR,Fr + cR,Gl
recovery
RCX (%) )
cX,FrQX × 100 cF,FrQF
RCX (%) )
cX,FrQX × 100 cF,FrQF
productivity
PRX
solvent consumption
SCX
( ) ( )
cX,FrQX kg ) Vads m3 h
PRR
QE m3 ) kg cX,FrQX
SCR
Figure 2. Comparison of the concentration profiles: (s) calculated by the analytical solution and (- - -) simulated using the TMB steady-state model. Table 2. Glucose/Fructose SMB Model Parameters and Operating Conditions.29 SMB geometry
model parameters ) 0.4
Lc )29 cm dc )2.6 cm
khGl ) 1.86 min-1 khFr ) 1.31 min-1
number of columns ) 12 configuration: 3-3-3-3
KGl ) 0.27 m3/kg KFr ) 0.53 m3/kg
operating conditions T ) 50 °C t* ) 3.3 min QE ) 11.59 cm3/min QX ) 10.79 cm3/min QF ) 6.95 cm3/min QR ) 7.67 cm3/min QSMB ) 36.47 cm3/min 1
triangle calculated by the equilibrium theory in the absence of mass-transfer resistances. The separation regions increase with decreasing required extract and raffinate purities. Also, the separation regions decrease with increasing mass-transfer resistance. For example, the separation region in Figure 3 corresponding to 98% extract and raffinate purities is markedly smaller than that calculated from the equilibrium theory (Figure 5). For systems with high mass-transfer resistances, choosing operating conditions on the basis of equilibrium theory restrictions is not convenient, and therefore, it is necessary to solve the model equations that account for mass-transfer effects. Influence of γI and γIV on the Separation Volumes. Usually, the separation region is presented in 2D plot (γII, γIII) for fixed values of operating conditions in sections I and IV (γI and γIV) imposed by equilibrium theory restrictions. In this work, the separation volume concept was used.25 This concept considers the influence of the net flows (γI or γIV) on the separation region. It was observed in previous works25,26 that the separation volume (γII, γIII, γI) has an inverted pyramidal shape (triangular base, γII-γIII), starting from one point (γI )
( ) ( )
cR,GlQR kg ) Vads m3 h
QE m3 ) kg cR,GlQR
γImin) and going to some higher value of γI from which on the area of the triangle (γII-γIII) no longer increases. Similarly, the separation volume (γII, γIII, γIV) has a pyramidal shape (triangular base, γII-γIII), finishing in one point (γIV ) γIVmax) and having an unchanged area of the triangular base (γII and γIII) for some lower value of γIV, such that a further decrease of γIV does not change the area of separation triangle (γII-γIII). To understand better the separation volume concept, a detailed analysis of the effects of γIV on the (γII, γIII, γI) volume and of γI on the (γII, γIII, γIV) volume was undertaken. The requirement for the extract and raffinate purities was set at 95%. The separation regions were calculated for different values of γI by fixing the value of γIV. Three different values of γIV were used, namely, γIV ) 0.338, γIV ) 0.38, and γIV ) 0.393 (almost equilibrium restriction), as shown in Figure 6. For γIV ) 0.338 (Figure 6a) and values of γI lower than 0.8, separation is not possible. The separation region appears in a form of single point for γI ) 0.8. The separation regions have a triangular form and increase increasing value of γI up to 2.0. Further increasing γI influences neither the position nor the size of the separation regions. All of the separation regions have a common part of the left border that starts from the single point given by (γII ) 0.4760, γIII ) 0.4761) corresponding to γI ) 0.8. In the case of γIV ) 0.38 (Figure 6b), the separation is possible for values of γI higher than 0.828 (γII ) 0.525, γIII ) 0.526). The size of the separation region increases, and the left border of the region starts to move toward higher values of γII and γIII with increasing γI. When the value of γIV approaches the value calculated from the equilibrium theory (γIV ) 0.405), the separation is possible for values of γI between 0.87 and 4.4 (as shown in Figure 6c for γIV ) 0.393). In the case of γIV ) 0.393 (Figure 6c), the separation is possible for values of γI between 0.87 and 4.4. For γI ) 0.87 and γI ) 4.4 the separation regions are single points given by (γII ) 0.571, γIII ) 0.572) and (γII ) 0.650, γIII ) 0.659), respectively. It can be observed that the separation regions moves from lower to higher values of (γII, γIII) with increasing γI. Going from γI ) 0.87 to γI ) 1.1, the size of the separation region increases, and above this value of γI, the separation region starts to decrease, finishing in a single point for γI ) 4.4. The separation regions were also calculated for different values of γIV by fixing the value of γI. The values of γI used were γI ) 1.5, γI ) 0.954, and γI ) 0.85 (almost equilibrium restriction), as shown in Figure 7. Separa-
Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 4499 Table 3. SMB Performance Calculated by Analytical and Numerical Solutions purity (%) analytical solution numerical solution
recovery (%)
productivity, (kg/m3 h)
solvent consumption (m3/kg)
extract
raffinate
extract
raffinate
extract
raffinate
extract
raffinate
90.03 89.93
90.00 89.97
89.99 90.19
90.03 90.11
13.54 13.57
13.54 13.57
0.0460 0.0459
0.0460 0.0459
Figure 3. Separation regions for different extract and raffinate purities in the case of mass-transfer limitations: RFr ) 12.95 and RGl ) 18.40.
Figure 4. Separation regions for different extract and raffinate purities in the case of mass-transfer limitations: RFr ) 25.90 and RGl ) 36.80.
tion-region behavior similar to that observed in the case of a fixed value of γIV was found. Specifically, for γI ) 1.5 and values of γIV lower than 0.399, the separation is not possible (Figure 7a). The separation region appears as a single point for γIV ) 0.399 (γII ) 0.655, γIII ) 0.657); it increases, obtaining triangular form, with decreasing value of γIV to 0.3. Further decreasing γIV affects neither the size nor the position of the separation region. All separation regions start from the same point and share part of the right border in common. For γI ) 0.954 (Figure 7b), the separation is possible for values of γIV lower than 0.4, which corresponds to the point (γII ) 0.625, γIII ) 0.626). It was observed that, for γIV from 0.4 to 0.3, the size of the separation region increases whereas, for γIV lower than 0.3, the size of the separation region starts to decrease slightly. For values
Figure 5. Separation regions for different extract and raffinate purities in the case of negligible mass-transfer resistance.
of γIV lower than 0.1, the size and position of the separation region remain unchanged. It can be observed that the separation region moves from lower to higher values of (γII, γIII) with increasing values of γI. With decreasing γIV, the right border of the triangle moves to lower values of (γII, γIII). Near the equilibrium restriction (0.85) for section I (Figure 7c), the separation is possible only for values of γIV between 0.165 (γII ) 0.471, γIII ) 0.472) and 0.388 (γII ) 0.550, γIII ) 0.551). The separation region moves from higher to lower values of (γII, γIII) with decreasing value of γIV. Going from γIV ) 0.388 to γIV ) 0.36, the size of the separation region increases, and for lower values of γiv, the size of separation region starts to decrease, finishing in a single point for γIV ) 0.165. The separation regions for γI ) 0.85 are also presented in a 3D plot as the separation volume (γII, γIII, γIV) in Figure 8. When the parameters γI and γIV are near the equilibrium constraints, the volume is markedly reduced, as shown in Figures 7c and 6c, respectively. Operation near the equilibrium constraints implies lower eluent flow rates, but at the same time, the quantity of feed processed is smaller, reducing the overall performance of the SMB unit. Therefore, it is more convenient to use higher values of γI and lower values of γIV than those imposed by the equilibrium theory (γI ) 0.805 and γIV ) 0.405). On the other hand, the use of too high a value of γI and too low a value of γIV will lead to higher solvent consumption for the same feed processed, given that the separation region is constant (see Figures 7a and 6a). Therefore, the optimal operating point is between these two extreme conditions, and the optimization of the SMB unit must take into account the net flows in all four zones. For this purpose, the influence of γI on solvent consumption and productivityrelative to the vertex of each separation region (γII, γIII) was studied for several values of γIV. The vertex was selected as the reference for comparison, because it corresponds to the maximum feed processed with the minimum solvent consumption.
4500 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004
Figure 6. Separation regions for different values of γI, with fixed γIV: (a) γIV ) 0.338, (b) γIV )0.380, and (c) γIV )0.393 (95% extract and raffinate purities).
The results of this comparison are presented in Figure 9. Productivity increases rapidly with increasing γI until around 1.0; a further increase of γI does not significantly affect the productivity. The solvent consumption presents the minimum value around γI ) 1.0. Because the goal of the SMB operation is to reach maximum productivity with minimum solvent consumption, the optimal value for γI is around 1.0. The effect of γIV on
Figure 7. Separation regions for different values of γIV, with fixed γI: (a) γI )1.500, (b) γI )0.954, and (c) γI )0.850 (95% extract and raffinate purities).
SMB performance shows that the productivity increases with decreasing γIV from 0.393 to 0.338; a further decrease does not affect the productivity. The solvent consumption presents a minimum value for γIV ) 0.338. From this analysis, it can be concluded that the value of γI more significantly affects the SMB performance than the value of γIV. Thus, γI should be carefully tuned
Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 4501
can be used to speed up the determination of the separation volumes in the presence of mass transfer. Acknowledgment M.M. (PRAXIS XXI/BD/19503/99) and V.M.T.S. (PRAXIS XXI/BD/ 13915/97) gratefully acknowledge the Ph.D grants from the Fundac¸ a˜o para a Cieˆncia e Tecnologia. This work was financially supported by Project PCTI/EQU 32470/99 (Fundac¸ a˜o para a Cieˆncia e Tecnologia). Notation
Figure 8. Separation volume (γII, γIII, γIV) for γI ) 0.850 (95% extract and raffinate purities).
Figure 9. Influence of γI on solvent consumption and productivity for several values of γIV. Legend: solvent consumption, SC (dashed line); productivity, PR (solid line).
in a narrow zone. When the equilibrium theory is used, a “safety margin” (or safety factor, SF) is applied in the selection of the values of γI and γIV in order to guarantee the feasibility of the separation (γI ) γeq I SF and γIV ) /SF, higher safety factor is associated with safer γeq IV SMB operation). This safety margin is arbitrary chosen and could lead to SMB operation far from the optimal conditions. Conclusions A new analytical solution, based on the steady-state equivalent TMB concept, for a linear SMB in the presence of mass transfer was obtained. The proposed solution is a set of algebraic equations allows for the prediction of the liquid and solid concentration profiles in the SMB. The solution was applied in the determination of the separation volumes for the glucose/fructose SMB separation. The effect of the net flow in sections I and IV (γI and γIV) on the SMB performance was emphasized. It was found that, in addition to the net flows in sections II and III, the values of γI and γIV are also important for optimal SMB performance. The analytical solution provides an easy and rapid way of predicting SMB performance, and therefore, it
Ai,j ) parameter in eqs 24 and 25 Bi,j ) parameter in eqs 24 and 25 Cj ) parameter in eqs 24 and 25 ci,j ) liquid-phase concentration of component i in section j, kg/m3 dc ) column diameter, m Dp,i ) pore diffusion of component i, m2/s Ki ) adsorption equilibrium constant of component i kh,i ) homogeneous diffusion rate constant of component i, s-1 kp,i ) macropore diffusion rate constant of component i, s-1 L ) TMB/SMB section length, m Lc ) column length, m Mi,j ) parameter in eq 18 Ni,j ) parameter in eq 18 PR ) productivity, kg/(m3 s) PU ) purity, % SF ) safety factor q j i,j ) average solid-phase concentration of component i in section j, kg/m3 / qi,j ) equilibrium solid-phase concentration of component i in section j, kg/m3 Qj ) liquid-phase flow rate in section j of equivalent TMB, m3/s RE ) recovery, % SC ) solvent consumption, m3/kg t* ) switching time, s us ) interstitial solid velocity, m/s vj ) interstitial liquid velocity in section j, m/s x ) dimensionless axial position Greek letters Ri ) number of mass-transfer units β ) solid/liquid ratio ) bed porosity p ) particle porosity γj ) fluid/solid interstitial velocity ratio Subscripts and superscripts Fr ) fructose Gl ) glucose i ) component, i ) Gl or Fr j ) section, j ) I, II, III, IV F ) feed R ) raffinate E ) eluent X ) extract
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Received for review July 21, 2003 Revised manuscript received October 14, 2003 Accepted October 28, 2003 IE030610I
