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Analytical Solution for Reactive Simulated Moving Bed in the Presence of Mass Transfer Resistance Mirjana Minceva, Viviana M. T. Silva,† 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
In this work, a new analytical solution for simulated moving bed rector (SMBR) in the presence of mass transfer is presented. A reaction of type A f B + C, where each species exhibits a linear type of adsorption isotherm, was assumed. The linear driving force model was used to describe the intraparticle mass transfer. The proposed solution is based on the steady-state equivalent true moving bed reactor (TMBR) analogy and allows the calculation of the liquid and solid concentration profiles in the SMBR. The TMBR analytical solution predicts internal concentration profiles and process performances, providing better accuracy, when compared with results of dynamic simulation of real SMBR. The main advantage of the presented solution is the fast determination of the reactive-separation regions in the presence of mass transfer limitations, according to the required product purities and reactant conversion. The analytical solution was used to study the influence of the rate of the reaction, the mass transfer rate, the number of the columns per section, and the reactant adsorption affinity on the reactive separation regions. Introduction The production-scale synthesis processes are commonly designed to operate in two sequential stages: (i) synthesis of the products by chemical reaction and (ii) separation of the reaction products. Reactive separations are integrated processes that combine chemical reaction and separation in a single unit. The main advantages of the reactive separation processes are reduction of the operating and capital cost, as well as improvement of the reaction efficiency (enhancement of conversion, yield, selectivity). Besides reactive distillation, reactive extraction, reactive crystallization, and reactive membrane separation, the combination of chemical or biochemical reaction with simulated moving bed (SMB) chromatographic separator has recently been the subject of considerable attention in scientific research. This integrated reaction-separation technology adopts the name simulated moving bed reactor (SMBR) technology. A schematic diagram of an SMBR unit and principle of its operation is presented in Figure 1, where reaction A f B + C is considered. The SMBR consists of a set of interconnected columns packed with a solid, which acts as both adsorbent and catalyst. There are two inlet streams (feed and eluent) and two outlet streams (extract and raffinate). The reactant A is used as feed, the more adsorbed product B and the less adsorbed product C are collected in the extract and the raffinate, respectively. At regular time intervals (switching time period), the inlet and outlet ports are switched for one bed distance in the direction of the fluid flow (shown with dashed lines in Figure 1). A cycle is completed when the number of switches is equal to the number of * To whom correspondence should be addressed. Phone: 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 Polo´nia, Apartado 1134, 5301-857 Braganc¸ a, Portugal.
Figure 1. Schematic diagram of SMBR.
columns. In this way, the countercurrent motion of the solid is simulated under a velocity equal to the length of a column divided by the switching time. The relative motion of the chemical species in an SMBR can be understood more visibly by considering an equivalent true moving bed reactor (TMBR). In the TMBR (see Figure 2a), the position of the inlet and outlet streams is fixed. The solid really moves in the direction opposite to the liquid flow. According to the positions of the inlet and outlet stream, the unit can be divided into four sections. In section I, placed between the eluent and extract nodes, the adsorbent is regenerated by desorption of the more strongly adsorbed product (B) from the solid. In section II (between the extract and feed node) and section III (between the feed and raffinate node), the reaction is taking place and products (B and C) are formed. The more strongly adsorbed product B is adsorbed and transported with the solid phase to the extract port. The less strongly adsorbed product (C) is desorbed and transported with
10.1021/ie049236s CCC: $30.25 © 2005 American Chemical Society Published on Web 02/17/2005
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Figure 2. Schematic diagram of equivalent TMBR (a) with concentration profiles and (b) with the notation used in the analytical solution.
the liquid in the direction of the raffinate port. In section IV, placed between the raffinate and eluent node, the eluent is regenerated before being recycled to section I. In the case of reversible reactions, the removal of products in the SMBR as they are formed allows achieving conversions well beyond equilibrium values. For reactions in series and in parallel, the desired intermediate species could be separated selectively. In reactions with inhibiting or poisoning product, its removal from the reaction medium could promote yield improvement. The main advantages of SMBR technology in view of separation/reaction are increased conversion and/or product purity, increased driving force by continuous countercurrent operation, and overcoming of the solid abrasion and handling problems by simulation of the countercurrent flow. The first application of SMBR was a zeolite-catalyzed alkylation reaction, patented by Zabrinsky and Anderson1 in 1977. The successful entrance of the SMB technology in the pharmaceutical industry has stimulated academic community interest in the SMBR process in the past decade. The potential of this technology has been investigated in the area of the following: (i) gasphase chemical reaction, as, for example, hydrogenation of 1,3,5-trimethylbenzene2 and oxidative coupling of methane,3 (ii) biochemical reactions, as inversion of sucrose by enzyme invertase4,5 and the production of lactosucrose by action of b-fructofuranosidase,6 (iii) liquid-phase chemical reactions; major examples are reversible reactions catalyzed by ion-exchange resins, as esterification of acetic acid with b-phenetyl alcohol,7 ethyl acetate synthesis,8,9 diethylacetal synthesis from ethanol and acetaldehyde,10 synthesis of bisphenol A from acetone and phenol,11 esterification of acetic acid with methanol,12 and synthesis of MTBE.13 The number of publications focusing on the design of nonreactive SMBs is quite large. Nevertheless, the same may not be said of SMBR. The design of a SMBR will define geometric and operating parameters that should lead not only to product separation but also to high reactant conversion. The design of SMB with negligible mass transfer resistance is easily performed by applications of the triangle theory,14-16 which is derived for the equilibrium theory.17 The triangular theory specifies a triangular
region, which encloses all feasible operating conditions that guarantee 100% product purities for an ideal system. The triangular region is identified in a plot of net flow ratios of liquid phase to solid phase in sections II (γII) and III (γIII). The net flow ratios in sections I and IV are fixed according to the equilibrium theory in order to ensure the complete regeneration of the adsorbent and the eluent, respectively. Most of the SMB systems operate in the presence of mass transfer effects. The triangle regions for nonideal systems are substantially smaller that the triangle regions for corresponding ideal systems; therefore, equilibrium theory could not be applied in design of the SMB. In these cases, the use of detailed mathematical models is necessary. Normally, computer simulations are used to construct the triangular region through consecutive calculations of unit performances in the (γII - γIII) plane.18-21 There are few analytical solutions for determination of net flow rates of SMB in the presence of mass transfer. Ching and Ruthven22 developed an analytical solution of the linear SMB described by dispersed plug flow model. Based on the standing concentration waves concept, Ma and Wang23 derived a set of equations to determine the zone flow rates and switching time for linear systems in the presence of mass transfer. The standing wave design was extended to nonlinear nonideal systems by Xie et al.24 In their work, the mass transfer correction terms derived for linear nonideal systems are used as approximations to counter the mass transfer effects in nonlinear systems. Recently, Silva et al.25 proposed a novel analytical solution for SMB in the presence of mass transfer resistance based on equivalent TMB concept. This solution allows calculation of the internal concentration profiles in all sections without imposing the assumption of complete regeneration of the adsorbent and eluent in sections I and IV, respectively. The analytical solution of the internal concentration profiles was used to design an SMB unit by means of the determination of separation triangles in the presence of mass transfer resistance. The first SMBR design procedure, introduced by Migliorini,9 was based on the equilibrium theory applied to the separation of the reaction products in a nonre-
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Table 1. Definition of the Coefficients MA,j, NA,j, PA,j, and QA,j for Reactant A Coefficients MA,j, NA,j, PA,j, and QA,j (Eqs 9 and 10-19) MA,j ) [(ΨA,j + ΘA,j - Rp,A,j)/(2ΘA,j)]eΨA,j+ΘA,j - [(ΨA,j - ΘA,j - Rp,A,j)/(2ΘA,j)]eΨA,j-ΘA,j NA,j ) [(νRp,A,j)/(2ΘA,jγj)](eΨA,j+ΘA,j - eΨA,j-ΘA,j) PA,j ) [(Rp,A,jKA)/(2ΘA,j)](eΨA,j-ΘA,j - eΨA,j+ΘA,j) QA,j ) eRp,A,j - [(νRp,A,j2KA)/(2ΘA,jγj)] {(2ΘA,jeRp,A,j)/[(Rp,A,j - ΨA,j)2 - (ΘA,j)2] + eΨA,j+ΘA,j/(ΨA,j + ΘA,j - Rp,A,j) - eΨA,j-ΘA,j/(ΨA,j - ΘA,j - Rp,A,j)} where ΨA,j ) [Rp,A,j(γj - νKA) + σADaj]/(2γj) ΘA,j ) x(ΨA,j)2 - (σARp,A,jDaj/γj)
active SMB. The region of complete separation of products (separation triangle) was spanned by consecutive simulation, using a detailed model that included the reaction kinetics. The algorithm proposed by Biressi et al.19 for design of a new SMB unit in the presence of mass transfer was extended by Azevedo and Rodrigues,5 by inclusion of the reaction conversion as a design constraint and an detailed model for the SMBR. Recently, two very similar works, in view of the design of the SMBR in the case of negligible mass transfer resistance, were published.26,27 These present an analytical solution for determination of the reactive-separation regions for irreversible and reversible reaction of type A f B + C where the reactant adsorption affinity is between those of the products (KC < KA < KB). In these works, expressions for determination of the reactive-separation boundaries as a function of the reactant conversion, ensuring complete separation of the products, were developed. Similarly to the SMB separation region, the SMBR reactive-separation regions in the presence of the mass transfer will be smaller, when compared to those obtained in adsorption equilibrium conditions. In this work, a new analytical solution for reactive simulated moving bed in the presence of mass transfer has been derived. The internal concentration profiles are derived based on the equivalent TMBR for the reaction of type A f B + C, where each species exhibits a linear type of isotherm and internal mass transfer described by a linear driving force (LDF) model. The proposed analytical solution is compared with the numerical solution of real SMBR and is used to study the influence of the rate of the reaction, the mass transfer rate, the number of the columns per section, and the reactant adsorption affinity on the reactive separation regions. Model of Linear SMBR in the Presence of Mass Transfer. The SMBR was modeled using the steadystate equivalent TMBR concept (Figure 2). The proposed mathematical model considers a plug flow for the liquid and solid phases, the presence of internal mass transfer described by the LDF model and linear adsorption isotherm. The reaction is irreversible and occurs in the liquid phase according to
AfB+C The following assumptions were taken in account: isothermal operation; constant bed porosity; all sections with equal length (L); different mass transfer for both components. Steady-state model equations, for section j, are
dCi,j 1 - j i,j) + σikrCA,j (1) 0 ) -νj kp,i,j(KiCi,j - q dz 0 ) us
dq j i,j + kp,i,j(KiCi,j - q j i,j) dz
(2)
where Ci,j is the liquid-phase concentration of compo-
nent i in section j, q j i,j is the average solid-phase concentration of component i in section j, νj is the interstitial liquid velocity in section j, us is the interstitial solid velocity, is the bed porosity, Ki is the adsorption equilibrium constant of component i, kp,i is the mass transfer coefficient of component i, σi is the stoichiometric coefficient of component i, and kr is the kinetic constant. Introducing the dimensionless variable for axial position x ) z/Lj and the dimensionless parameters
γj ) νj/us
fluid/solid interstitial velocity ratio (3)
Rp,i,j ) Ljkp,i/us
number of mass transfer units (4)
Daj ) Ljkr/us
Damko¨hler number
(5)
the model equations become
dCi,j - νRp,i,j(KiCi,j - q j i,j) + σiDajCA,j (6) 0 ) -γj dx 0)
dq j i,j + Rp,i,j(KiCi,j - q j i,j) dx
(7)
Model equations might be analytically solved (see Appendix 1) considering the following boundary conditions at each section:
x)0
Ci,j ) Ci,j(0) q j i,j ) q j i,j(0)
(8)
The liquid and solid concentrations at the end of each section are obtained as a function of the concentrations at the beginning of each section, according to
j A,j(0) CA,j(1) ) MA,jCA,j(0) + NA,jq
(9)
q j A,j(1) ) PA,jCA,j(0) + QA,jq j A,j(0)
(10)
The coefficients MA,j, NA,j, PA,j, and QA,j are presented in Table 1. They are defined as functions of the model parameters (ν, KA, Rp,A,j, and Daj) and the operating conditions (γj). To calculate the liquid and solid concentrations at the beginning and at the end of all sections, it is necessary to connect all sections according to the boundary conditions (see Figure 2b):
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γICi,I(0) ) γIVCi,IV(1)
(11)
j i,IV(1) q j i,I(0) ) q
(12)
Ci,II(0) ) Ci,I(1)
(13)
j i,I(1) q j i,II(0) ) q
(14)
γIIICi,III(0) ) γIICi,II(1) + (γIII - γII)Ci,F
(15)
j i,II(1) q j i,III(0) ) q
(16)
Ci,IV(0) ) Ci,III(1)
(17)
j i,III(1) q j i,IV(0) ) q
(18)
For each section, the system of algebraic equations is obtained by elimination of the liquid- and solid-phase concentrations in eqs 11-18, by using eqs 9 and 10, respectively. The system to solve is 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):
[
MA,I
-1
0
MA,II
0
0
0 γIII 0 γII MA,III -1
0 0 PA,II 0 0
γI γIV PA,I 0 0 0
0
NA,I 0
]
0
0
0
NA,II 0
0
0
0
NA,III 0
0
MA,IV 0
0
0
NA,IV ‚
0 0 PA,III 0
0 0 0 PA,IV
-1 QA,II 0 0
0 -1 QA,III 0
0 0 -1 QA,IV
QA,I 0 0 -1
[ ][ ] 0 CA,I(0) γIII CA,II(0) - CA,F -1 γII CA,III(0) 0 CA,IV(0) ) 0 q j A,I(0) 0 q j A,II(0) 0 q j A,III(0) 0 q j A,IV(0) 0
(
)
(19)
The coefficients Mi,j, Ni,j, Fi,j, Gi,j, Pi,j, Qi,j, Ri,j, and Si,j are defined in Table 2. As in the case of the reactant A, the liquid and solid concentrations of the products (B and C) at the beginning and at the end of each section were calculated by connection of all sections according to the boundary conditions (eqs 11-18). Afterward, the system of algebraic equations was written in matrix form P‚X ) Q, where X is the vector of the variables (liquid and solid concentrations of B and C at the beginning of each section):
[
Mi,I
-1
0
Mi,II
0
0
0 γIII 0 γII Mi,III -1
0 0 Pi,II 0 0
γI γIV Pi,I 0 0 0
0
Ni,I 0
]
0
0
0
Ni,II 0
0
0
0
Ni,III 0
0
Mi,IV 0
0
0
Ni,IV ‚
0 0 Pi,III 0
0 0 0 Pi,IV
-1 Qi,II 0 0
0 -1 Qi,III 0
0 0 -1 Qi,IV
QA,I 0 0 -1
[ ][ ] Ci,I(0) Φi,I Ci,II(0) Φi,II Ci,III(0) Φi,III Φ Ci,IV(0) ) Λ i,IV q j i,I(0) i,I Λi,II q j i,II(0) Λi,III q j i,III(0) Λi,IV q j i,IV(0)
(24)
The solution gives the concentrations of the products (B and C) in the liquid and solid phases at the beginning of each section, later used in the calculation of the concentration profiles (see Appendix eqs A16 and A17). The SMBR performance (see Table 3) could be also calculated from the liquid-phase concentration in the extract (Ci,X ) Ci,II(0)) and raffinate nodes (Ci,R ) Ci,IV(0)). Results and Discussion
The system of algebraic equations was solved using Microsoft Excel. The solution provides the concentrations of liquid and solid phase at the beginning of each section, which are needed for the calculation of the concentration profiles of the reactant A (see Appendix eqs A14 and A15). On the other hand, the concentration profiles of species A were used to calculate the liquid and solid concentrations of the products (B and C) at the end of each section as a function of the concentrations at the beginning of each section:
j i,j(0) - Φi,j Ci,j(1) ) Mi,jCi,j(0) + Ni,jq
(20)
q j i,j(1) ) Pi,jCi,j(0) + Qi,jq j i,j(0) - Λi,j
(21)
j A,j(0) Φi,j ) Fi,jCA,j(0) + Gi,jq
(22)
Λi,j ) Ri,jCA,j(0) + Si,jq j A,j(0)
(23)
Analysis of Concentration Profiles. The SMBR internal concentration profiles and performances calculated by the analytical solution of the linear SMBR in the presence of mass transfer for reaction type A f B + C were compared with those calculated numerically for both equivalent TMBR and real SMBR. The model parameters and the SMBR operating conditions used are presented in Table 4. The steady-state TMBR model and the corresponding transient real SMBR model were solved using gPROMSgeneral PROcess Modelling System.28 The axial domain was discretized by using third-order orthogonal collocation method in finite elements (OCFEM) over 20 elements per column. The simulation was performed on Pentium IV 3.2 GHz processor with 2 Gb RAM memory. The computational time was around 2 s for the steadystate equivalent TMBR model and 30 min for the transient SMBR model. In the case of the SMBR model, 20 cycles were needed to reach the cyclic steady state.
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Table 2. Definition of the Coefficients Mi,j, Ni,j, Fi,j, Gi,j, Pi,j, Qi,j, Ri,j, and Si,j for Products B and C Coefficients Mi,j, Ni,j, Fi,j, Gi,j, Pi,j, Qi,j, Ri,j, and Si,j (Eqs 20-24) Mi,j ) [γj/(γj - νKi)] + {1 - [γj/(γj - νKi)]}eΨi,j Ni,j ) - [ν/(γj - νKi)](1 - eΨi,j) Fi,j ) (σi/σA)[γj/(γj - νKi)] - (σiDaj/γj)[((Ψi,j - Rp,i,j)(Ψi,j - Rp,A,j)/Ψi,j((Ψi,j - ΨA,j)2 - ΘA,j2))eΨi,j + ((ΨA,j + ΘA,j - Rp,i,j)(ΨA,j + ΘA,j - Rp,A,j)/2ΘA,j(ΨA,j + ΘA,j - Ψi,j)(ΨA,j + ΘA,j))eΨA,j+ΘA,j - ((ΨA,j - ΘA,j - Rp,i,j)(ΨA,j - ΘA,j - Rp,A,j)/2ΘA,j(ΨA,j - ΘA,j Ψi,j)(ΨA,j - ΘA,j))eΨA,j-ΘA,j] Gi,j ) - (σi/σA)(ν/Rp,i,j(γj - νKi)) - (σiνRp,A,jDaj/γj2) [(eΨi,j/Ψi,j((Ψi,j - ΨA,j)2 - ΘA,j2)) + eΨA,j+ΘA,j/2ΘA,j(ΨA,j + ΘA,j - Ψi,j)(ΨA,j + ΘA,j)) (eΨA,j-ΘA,j/2ΘA,j(ΨA,j - ΘA,j - Ψi,j)(ΨA,j - ΘA,j))] Pi,j ) -[(γjKi)/(γj - νKi)](1 - eΨi,j) Qi,j ) [(νKi)/(γj - νKi)] + [1 + (γjeΨi,j)/(νKi)] Ri,j ) (σi/σA)[(γjKi)/(γj - νKi)] + [(σiRp,i,jDajKi)/γj]〈{(Ψi,j - Rp,A,j)eΨi,j/[Ψi,j((Ψi,j - ΨA,j)2 - ΘA,j2)]} + {(ΨA,j + ΘA,j Rp,A,j)eΨA,j+ΘA,j)/[2ΘA,j(ΨA,j + ΘA,j - Ψi,j)(ΨA,j + ΘA,j)]} - {(ΨA,j - ΘA,j - Rp,A,j)eΨA,j-ΘA,j)/[2ΘA,j(ΨA,j - ΘA,j - Ψi,j)(ΨA,j - ΘA,j)]}〉 Si,j ) - (σi/σA)[(νKi)/(γj - νKi)] + [(σiνRp,A,jRp,i,jKiDaj)/γj2] 〈{(eΨi,j/[Ψi,j((Ψi,j - ΨA,j)2 - ΘA,j2)) + {(eΨA,j+ΘA,j/[2ΘA,j(ΨA,j + ΘA,j - Ψi,j)(ΨA,j + ΘA,j)]} - {eΨA,j-ΘA,j/[2ΘA,j(ΨA,j - ΘA,j - Ψi,j)(ΨA,j - ΘA,j)]}〉 where Ψi,j ) [Rp,i,j(γj - νKi)]/γj Table 3. Expressions for Calculation of the SMBR Performance extract purity (%) raffinate purity (%) conversion (%)
PUX ) [CX,B/(CX,A + CX,B + CX,C)] × 100 PUR )[CX,C/(CX,A + CX,B + CX,C)] × 100 X ) [(CF,AQF - CE,AQX CR,AQR)/(CF,AQF)] × 100
Table 4. SMBR Model Parameters and Operating Conditions SMBR geometry
model parameters ) 0.4
Lc ) 29 cm dc ) 2.6 cm
KA ) 0.205 KB ) 0.795 KC ) 0.405
no. of columns: 12 configuration: 3-3-3-3
kr ) 0.61 min-1 kp,i ) 2.667 min-1 σA ) - 1 σB/C ) 1
operating conditions t* ) 3.4 min CA,F ) 80 g/dm3 γ1 ) 0.880 γ2 ) 0.493 γ3 ) 0.659 γ4 ) 0.190 ) QSMBR I 34.05 cm3/min
The comparison of the steady-state concentration profiles calculated analytically and numerically is presented in Figure 3. The cyclic steady-state SMBR profiles are presented in the middle of the switching time. The equivalent TMBR concentration profiles calculated numerically and analytically were identical as expected, since the analytical solution does not introduce any model simplification. Slight difference between the
Figure 3. Comparison of concentration profiles. The full lines present the steady-state equivalent TMBR profiles calculated numerically; the dashed lines present the cyclic steady-state SMBR concentration profiles in the middle of the switching period, and the symbols present the steady-state equivalent TMBR profiles calculated with the proposed analytical solution.
Table 5. Comparison of the SMBR Performance Calculated by TMBR Analytical Solution, TMBR, and SMBR Numerical Solution
performance extract purity (%) raffinate purity (%) conversion (%)
numerical numerical SMBR SMBR (at half (average of the over the analytical numerical switching switching TMBR TMBR time) time) 99.88 98.33 100.00
99.88 98.33 100.00
99.88 98.42 100.00
99.77 97.83 100.00
equivalent TMBR and the real SMBR concentration profiles was observed around the feed and raffinate port. The SMBR performances calculated using the relations from Table 3 are presented in Table 5. The SMBR performances were calculated in two ways: (i) using the concentrations at the middle of the switching time and (ii) using the averaged concentration over a switching time period, after reaching the cyclic steady state. There is slight difference between the SMBR performance predicted with the TMBR analytical solution and SMBR numerical solution. The proposed analytical solution provides a fast and simple method for calculation of SMBR performance (no need of calculation of the internal concentration profiles). Therefore, it could be easily applied in the prediction of reactive-separation regions for specific SMBR system (adsorption equilibrium, reaction rate, mass transfer, and SMBR unit design). Reactive-Separation Regions. Effect of the Reaction Kinetics. The effect of the reaction kinetics on the reactive-separation regions was studied by changing the Da number. The column geometry, the SMBR configuration, adsorption equilibrium parameters, mass transfer parameters, switching time, and flow rates in sections I and IV are those given in Table 4. The required extract and raffinate purities were set at 98% and the conversion that has to be reached was set at 99%. The reactive-separation regions were determined using the analytical solution for three different values of the Da number, namely, 1.40, 3.00, and 6.25 (see Figure 4). The reactive-separation region increases with increase of the Da number. Higher values of the Da number means faster reaction; the reactant A reacts completely near to the feed port, and the SMBR reactor behavior approaches to the one of the SMB fed with the reaction products, species B and C. The separation region is smaller that the one determined according to
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Figure 4. Influence of the reaction kinetics on the reactiveseparation regions for Da ) 1.40 (kr ) 0.14 min-1), Da ) 3.00 (kr ) 0.29 min-1), and Da ) 6.25 (kr ) 0.61 min-1).
the equilibrium theory as a result of the existing mass transfer limitation. The reactive-separation regions become smaller and displaced to the lower values of γIII when the Da number decrease. The reaction occurs mainly in section III (see Figure 3), since the Henry constant of reactant A is lower than the Henry constants of both products, B and C. The small quantity of reactant A in section II is due to the mass transfer limitations. For lower values of Da number, the reaction becomes slower and the products are formed near to the raffinate port, leading to impure raffinate and lower conversion. Therefore is necessary to decrease the flow rate in section III in order to guarantee the conversion of the reactant and to prevent the more adsorbed product B from reaching the raffinate port, requiring decrease of the feed flow rate. The separation regions presented in Figure 4 with gray lines correspond to the reactive-separation regions determined with the equilibrium theory. The constraint for section II (γII) was determined by the equilibrium theory for an SMB for separation of the products B and C. The reactant A is less adsorbed component and therefore imposing γII > HC it is guaranteed that A cannot reach the extract port and contained the extract. In section III, reactant A should be converted before achieving the raffinate port and product B should be adsorbent and carried out with the solid phase to the section II. The upper limit for γIII was calculated according to the equilibrium theory method proposed by Fricke and Schimdt-Traub.26
γIII e HA - DaIII/ln(1 - X)
(40)
The conversion used to determine this region was equal to the one obtained by the analytical solution of equivalent TMBR in the presence of mass transfer limitations. Effect of the Mass Transfer Rate. The effect of the mass transfer limitation on the reactive-separation region size and position was studied for different values of the number of mass transfer units (Rp,i), Rp,i ) 20.4 Rp,i ) 27.2, and Rp,i ) 30.6. The analytical solution allows a different number of mass transfer units (mass transfer coefficient) for each species in the reaction. In this study, the same values of the mass transfer parameter were used for all species in order to simplify
Figure 5. Influence of the mass transfer on the reactiveseparation regions for Rp,i ) 20.4 (kp ) 2 min-1) Rp ) 27.2 (kp ) 2.67 min-1), and Rp ) 27.2 (kp ) 3.00 min-1): (a) 95% product purities and (b) 98% product purities.
the analysis. The column geometry, SMBR configuration, adsorption equilibrium parameters, reaction kinetics parameter, switching time and flow rates in section, I and IV are given in Table 4. The required extract and raffinate purities were set on 95% and 98%. In both cases, the conversion that has to be reached was set on 99%. The reactive-separation regions for different mass transfer limitation determined by the equivalent TMBR analytical solution for 95% and 98% product purities are presented in Figure 5a and b, respectively. The reactiveseparation region increases with increase of the rate of mass transfer (the number of mass transfer units). The region is more expanded to the higher values of γIII that to the lower values of γII. This behavior is a consequence of the section in which the reaction occurs. The reaction occurs mainly in section III. In this section, the length of the reaction zone is governed by the reaction kinetics and the mass transfer rate. When the mass transfer limitation decreases the reaction zone become shorter and approaches to the feed port and consequently higher feed flow rates can be processed, for the same process performances. The reactive-separation region reduces notably when the purity requirement passes from 95% to 98%, especially for the highest mass transfer resistance (Rp,i )20.4). It is interesting to observe that in the case of 95% product purity restrictions the reactive-separation regions extend out of the separation region deter-
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Figure 7. Internal concentration profiles for 3-3-4-2 configuration, Da ) 6.25, Rp ) 27.2, γI ) 0.880, γII ) 0.397, γIII ) 0.728, and γIV ) 0.190.
Figure 6. Influence of the SMBR configuration on the reactiveseparation regions. (a) Reactive-separation regions for configurations 3-3-3-3, 3-2-4-3, and 3-4-2-3. (b) Reactive-separation regions for configurations 3-3-4-2, 4-3-4-1, and 3-3-5-1.
mined by the equilibrium theory for the case of separation of species B and C (presented with a gray line in Figure 5). Effect of the SMBR Configuration. The influence of the SMBR configuration on the reactive-separation region was also examined. The column geometry, adsorption equilibrium parameters, reaction kinetic parameter, switching time, and flow rates in sections I and IV used in this study are presented in Table 4. The mass transfer parameter kp,i was set equal to 1.8 min-1 for all species. The total number of the columns in the SMBR unit was 12. The required extract and raffinate purities were set on 95% and the conversion that has to be reached was set on 99%. The initial idea was to increase the number of the columns in section III (see Figure 6a), since reaction occurs mainly in this section. It was expected that when increasing the number of the columns in section III the flow rate of the feed that can be processed would increase. The number of the columns is in section III was increased form 3 to 4, by reducing the number of columns in section II. The size of reactive-separation region for 3-2-4-3 did not increase but was displaced up to higher values of γII and γIII comparing with the one obtained for the 3-3-3-3 configuration. The maximum feed that can be processed for both configurations was the same. When configuration 3-3-4-2 is used, the reactiveseparation region increases and has the same lower border as the one obtained with 3-3-3-3 and the same
upper border as the one obtained with 3-2-4-3 configuration. The maximum feed throughput for the configuration 3-3-4-2 is slightly higher than those allowed by the other two configurations. This shows the importance of the section II, whose role is to prevent the contamination of the extract by the less adsorbed product C. This function of section II is more pronounced when the mass transfer limitation are significant. After the analysis of the internal concentration profiles for configuration 3-3-4-2 (see Figure 7), it was verified that the eluent could be regenerated using just one column in section IV. The extra column from section IV be displaced to section III or to section I, leading to the following configurations 3-3-5-1 and 4-3-4-1, respectively. The reactive-separation regions obtained are presented in Figure 6b. For both configuration (33-5-1 and 4-3-4-1), the regions obtained are very similar and at the same time bigger than the region for the configuration 3-3-4-2. The reactive-separation regions are expanded to higher values of γIII (the border with pure raffinate region). In the case of configuration 3-3-5-1, the product B formed in the liquid has more time to be adsorbed and carried out with the solid phase to the extract port than in the configuration 3-3-4-2. On the other hand, when configuration 4-3-4-1 is used, the regeneration of the solid is improved since product B has more time to be desorbed from the solid phase in section I, compared with configuration 3-34-2. It is important to note that this analysis is valid for the values of γI ) 0.880 and γIV ) 0.190. Therefore, for different net flow rates in sections II and IV, a new study of the SMBR configuration in a view of feed throughput maximization should be performed. Effect of the Reactant Adsorption Equilibrium (Henry Constant). Finally, the effect of the reactant adsorption affinity on the reactive-separation region was studied. Three cases of reactant adsorption affinity relative to the product adsorption affinity were considered: (i) reactant as less adsorbed component (KA < KC < KB); (ii) reactant as middle adsorbed component (KC < KA < KB); (iii) reactant as most adsorbed component (KC < KB < KA). The Henry constants used for A were 0.2 (when less adsorbed), 0.6 (when middle adsorbed), and 1.0 (when most adsorbed). The column geometry, the SMBR configuration, the adsorption equilibrium parameters for B and C, mass transfer parameter, reaction kinetic parameter, switching time, and flow rates in
Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005 5253 dc ) column diameter, m K ) Henry coefficient, m3liquid/m3solid kp ) mass transfer coefficient, s-1 kr ) reaction kinetic constant, s-1 L ) TMBR section length, m Lc ) column length, m PU ) purity, % q j ) average solid-phase concentration, kg/kg Q ) liquid-phase flow rate in equivalent TMBR, m3/s QSMBR ) liquid-phase flow rate in SMBR, m3/s t* ) switching time, s us ) interstitial solid velocity, m/s ν ) interstitial liquid velocity, m/s X ) conversion, dimensionless x ) dimensionless axial position, dimensionless Figure 8. Influence of the reactant A adsorption equilibrium on the reactive-separation regions.
sections I and IV used in this study are presented in Table 4. The required extract and raffinate purities were set on 95%, and the conversion that has to be reached was set on 99%. The reactive-separation regions obtained for the reactant A as less, middle, and most adsorbed component are presented in Figure 8. The reactive-separation regions in all cases have similar size and the maximum feed throughput is almost the same. The position of the vertex moves to higher (γII, γIII) as the reactant A passes from less adsorbed to most adsorbed component. This behavior is due to the reaction zone position; in section III when A is less adsorbed component, in sections II and III when A is middle adsorbed component, and in section II when A is most adsorbed component. Conclusion A novel analytical solution for a simulated moving bed reactor in the presence of mass transfer based on steadystate equivalent TMBR concept was proposed. The solution considers reactions of type A f B + C, linear adsorption isotherms, and internal mass transfer described by the LDF model. The internal concentration profiles and process performances calculated by the proposed solution and by the dynamic simulation of real SMBR are very similar. Therefore the analytical solution provides a fast method for the design of an SMBR unit through the determination of the reactive-separation regions in the presence of mass transfer limitations, according to the required product purities and reactant conversion. The analytical solution is a useful tool for performing a study of the influence of model parameters (the rate of the reaction, the mass transfer, the number of the columns per section, the reactant adsorption affinity, etc.) on the unit design. These studies are helpful for understanding the competition between reaction and adsorption kinetics and its influence on the SMBR unit behavior. Acknowledgment This work was financially supported by the projects PCTI/EQU 40695/2001 and POCTI/EQU 44515/2002 (“Fundac¸ a˜o para a Cieˆncia e Tecnologia”).
Greek Letters Rp ) number of mass transfer units, dimensionless σ ) stoichiometric coefficient, dimensionless ) bed porosity, dimensionless ν ) solid/liquid ratio, dimensionless γ ) fluid/solid interstitial velocity ratio, dimensionless Subscripts and Superscripts i ) component, A, B, C j ) section, I, II, III, IV F ) feed R ) raffinate E ) eluent X ) extract
Appendix Applying Laplace transforms relative to dimensionless axial position, the mass balance eqs 6 and 7 become
0 ) - γj(sXi,j - Ci,j(0)) - νRp,i,j(KiXi,j - Yi,j) + σiDajXA,j (A1) 0 ) (sYi,j - q j i,j(0)) + Rp,i,j(KiXi,j - Yi,j) For the reactant, A
γjCA,j(0) + XA,j )
YA,j )
(
νRp,A,jq j A,j(0) s - Rp,A,j
νRp,A,j2KA γjs + νRp,A,jKA + - σADaj s - Rp,A,j
q j A,j(0) s - Rp,A,j
(
Rp,A,jKA γjCA,j(0) +
νRp,A,jq j A,j(0) s - Rp,A,j
)
2
)
νRp,A,j KA γjs + νRp,A,jKA + - σADaj (s - Rp,A,j) s - Rp,A,j
(A3)
(A4)
Rearranging eqs A3 and A4
Notation C ) liquid-phase concentration, kg/m3 Da ) Damko¨hler number, dimensionless
(A2)
XA,j )
/ MA,j
s - (ΨA,j + ΘA,j)
+
/ NA,j
s - (ΨA,j - ΘA,j)
(A5)
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Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005
YA,j )
/ / q j A,j(0) QA,j PA,j + + s - Rp,A,j s - Rp,A,j s - (ΨA,j + ΘA,j)
[
]
/ RA,j
s - (ΨA,j - ΘA,j)
/ PP,j )
(A6)
ΨA,j )
Rp,A,j(γj - νKA) + σADaj 2γj
ΘA,j )
x
/ ) MA,j
(A8)
νRp,A,j q j (0) γj A,j
2ΘA,j (ΨA,j - ΘA,j - Rp,A,j)CA,j(0) +
/ NA,j
)
(A9)
/ RP,j
-
/ SP,j )
νRp,A,j2KA q j A,j(0) γj2 (Ψi,j - ΨA,j)2 - ΘA,j2
2ΘA,j(ΨA,j + ΘA,j - Ψi,j) Rp,A,jKA(ΨA,j - ΘA,j - Rp,A,j)CA,j(0) +
/ RA,j )
νRp,A,j q j (0) γj A,j
- 2ΘA,j(ΨA,j - ΘA,j - Ψi,j)
(A13)
Equations A5 and A6 are easily inverted to get the liquid and solid concentration profiles of the reactant A, respectively: / / e(ΨA,j+ΘA,j)z + NA,j e(ΨA,j-ΘA,j)z (A14) CA,j(z) ) MA,j / / j A,j(0)eRP,A,jz - [PA,j eRP,A,jz + QA,j e(ΨA,j+ΘA,j)z + q j A,j(z) ) q / RA,j e(ΨA,j-ΘA,j)z] (A15)
The resolution of eqs A1 and A2 for both products is possible after substitution of eq A3 in eq A1, using a similar procedure as before. The liquid and solid concentration profiles of the products, respectively / / ΨP,jz / (ΨA,j+ΘA,j)z + LP,j e + MP,j e + CP,j(z) ) JP,j / (ΨA,j-ΘA,j)z e (A16) NP,j / Rp,P,jz / (ΨA,j+ΘA,j)z q j P,j(z) ) q j P,j(0)eRp,Pz - [PP,j e + QP,j e + / (ΨA,j-ΘA,j)z / ΨP,jz / e + SP,j e + TP,j ] (A17) RP,j
where the coefficients are
{[
(A20)
]
[
γjCA,j(0) γj((ΨP,j - ΨA,j)2 - ΘA,j2) νRp,A,j q j (0) Rp,A,j - ΨP,j A,j
{
Rp,P,jKP [γjCP,j(0) - νq j P,j(0)] γjΨP,j σPDajRp,A,j
ΨP,j )
]}
(A21)
}
(A22)
j A,j(0)] [γjCA,j(0) - νq
γj(ΨA,j2 - ΘA,j2)
(A12) νRp,A,j q j (0) γj A,j
νRp,A,j q j (0) γj A,j
2ΘA,j(ΨA,j - ΘA,j - ΨP,j)(ΨA,j - ΘA,j)
σPDaj(Rp,A,j - ΨP,j)
/ TP,j )-
(A11)
(ΨA,j - ΘA,j - Rp,A,j)CA,j(0) +
Rp,P,jKP νRp,P,j γjCP,j(0) q j (0) γjΨP,j Rp,P,j - ΨP,j P,j
(A10)
/ QA,j )
νRp,A,j q j (0) γj A,j
2ΘA,j(ΨA,j + ΘA,j - ΨP,j)(ΨA,j + ΘA,j)
σPDajRp,P,jKP γj
νRp,A,j q j (0) γj A,j
Rp,A,jKA(ΨA,j + ΘA,j - Rp,A,j)CA,j(0) +
(ΨA,j + ΘA,j - Rp,A,j)CA,j(0) +
)
- 2ΘA,j
/ ) PA,j
(A18)
(A19) (A7)
σARp,A,jDaj (ΨA,j)2 γj
(ΨA,j + ΘA,j - Rp,A,j)CA,j(0) +
q j P,j(0)
γj(Rp,P,j - ΨP,j)
/ QP,j )
σPDajRp,P,jKP γj
where
νRp,P,jKP
Rp,P,j(γj - νKP) γj
(A23)
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Received for review August 20, 2004 Revised manuscript received October 25, 2004 Accepted October 26, 2004 IE049236S
