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Parabolic intraparticle concentration profile assumption in modeling and simulation of nonlinear simulated moving-bed separation processes
PII:
Chemical Engineering Science, Vol. 53, No. 6, pp. 1311—1315, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0009–2509(97)00442–9 0009—2509/98 $19.00#0.00
Parabolic intraparticle concentration profile assumption in modeling and simulation of nonlinear simulated moving-bed separation processes (Received 28 December 1996)
The linear driving force (LDF) model, i.e. the parabolic intraparticle concentration profile assumption, has been widely used in modeling many adsorption processes due to its remarkable simplicity and good agreement with experimental results (Liaw et al., 1979; Do and Rice, 1986; Ching and Ruthven, 1986; Ching et al., 1987). Do and Rice (1986) studied the validity of the LDF model, which was first known as the Glueckauf ’s equation (Glueckauf, 1955), and obvious deviations from the pore diffusion model were found only when the dimensionless time q is below 0.05. Modeling and simulation play a very important role in developing and studying simulated moving-bed separation processes (Ruthven and Ching, 1989; Nicoud et al., 1993), as shown schematically in Fig. 1(b). In this configuration, liquid goes up and solid goes down the different zones and countercurrent contact between solid and liquid occurs, leading to high mass transfer driving force. Provided that the adsorption affinities of species A and B on the adsorbent are different, it is possible to choose the right flow rates of the solid and the liquid to force A to move upward and B downward, thus leading to spatial separation. This system requires two inlet lines (one for the feed and one for the eluent) and two outlet lines (one for the raffinate and one for the extract). Since the movement of the solid is very difficult to handle, the above concept is normally implemented by shifting the four inlet and outlet lines between a group of fixed columns (4—24 columns) looped together, as shown in Fig. 1(a), to simulate the movement of the solid (Ruthven and Ching, 1989; Nicoud et al., 1993). Due to the complicated dynamics resulting from the process set-up of the simulated moving bed, it is advisable that simple mass transfer model be used to save computation time in solving the mathematical model equations (Lu and Ching, 1996). That may be the reason why LDF model has been widely used in mathematical modeling and simulation studies for the simulated moving bed separation processes (Hashimoto et al., 1993; Nicoud et al., 1993; Ching et al., 1987). In most of such processes the dimensionless time q is larger than 0.05, and good agreement between mathematical simulation with LDF mass transfer model and experimental results has been obtained (Ching et al., 1987; Hashimoto et al., 1993), though yet only for linear systems. This short communication will focus on the application of the LDF mass transfer model in the mathematical simulation moving-bed separation processes with nonlinear adsorp-
tion isotherms to compare with the linear pore diffusion model, in order to determine its validity as well as to establish the equivalence between the two models. The simulated moving bed separation process to be studied in this work is a binary (A#B) separation (Nicoud et al., 1993) where B is more adsorbable than A and the eluent is not adsorbable. The simulated moving bed was made of 12 columns. The sketch of the process is shown in Fig. 1 where each zone consists of three columns, and the system and operating parameters are listed in Table 1 involving three adsorbents I, II and III. The mathematical models for the separation process shown in Fig. 1 can be developed with LDF or pore diffusion models by using the following assumptions: (1) The adsorbent is homogeneous and intraparticle diffusivity is constant. (2) The flow pattern inside columns or zones can be described by axial dispersion plug flow. (3) The ‘dead’ volumes at both end of the packed adsorbent are negligible. (4) The non-linear multicomponent adsorption isotherms for species A and B follow the extended Langmuir expressions (Nicoud et al., 1993). Introducing the dimensionless axial coordinate x"z/¸ j and radial coordinate o"r/R, we can get (i) ¸inear driving force (¸DF ) model Mass balances in the fluid phase: 1 d2c dc (1!e) ij! ij! m N "0. j ij Pe dx2 dx e j
(1)
dqN ij"f (c )!qN . a i dx ij ij
(2)
LDF model:
Mass transfer between fluid and solid:
1311
dqN N " ij . ij dx
(3)
1312
Shorter communications
Fig. 1. (a) Schematic diagram of experimental system showing sequence of 12 columns with inlet and draw off points. (b) Equivalent countercurrent flow system.
Table 1. Parameters for adsorption isotherms and operating parameters for adsorbents I, II, and III Parameters
Adsorbent I
q , mg/ml m K A K B b , ml/mg A b , ml/mg B » , ml/min &% » , ml/min %» , ml/min %9 » , ml/min 3! » , ml/min 3% t , min s Elements in x coordinate
5.80 1.57 1.57 0.014 0.127 1.52 4.63 4.00 2.05 20.4 4.33
15.8 0.785 0.785 0.014 0.127 4.00 12.0 10.5 5.50 20.0 3.55
12.8 0 0 0.200 0.600 4.00 18.5 18.5 5.50 16.0 4.50
15
15
50
Boundary conditions for the four zones: 1 dc dqN ij"c , ij"0 c ! ij Pe dx ifj dx j dc ij"0; x"1, qN "qN D ij ij`1 x/0 dx
x"0,
(where if j#1'4, then j#1"1).
Adsorbent II Adsorbent III
Mass transfer between fluid and solid:
K
3 Lq ij N "! . ij a Lo j r/1
(4a)
Boundary conditions for the four zones o"0,
(4b)
(ii) Pore diffusion model Mass balances in the fluid phase: 1 L2c Lc (1!e) ij! ij! m N "0. j ij Pe Lx2 Lx e j Pore diffusion model: L2q 2 Lq Lq ij# ij"15a ij . j Lx Lo2 o Lo
(7)
o"1, (5)
(8a)
q "f (c ) ij ij
(8b)
1 Lc ij"c , x"0, c ! ij Pe Lx ifj j x"1,
(6)
Lq ij"0 Lo
Lc ij"0; Lx
Lq ij"0 Lx
(8c)
q "q D ij ij`1 x/0
(where if j#1'4, then j#1"1).
(8d)
Shorter communications (iii) Multicomponent (Nicoud et al., 1993)
¸angmuir
adsorption
isotherms
q bc m i ij f (c )"K c # . ij i ij 1#+ n b c i/1 i ij (iv) Overall mass balances at the nodes of the zones
(9)
» "» #» , » "» #» !» , 1 3% %2 #3% %9 » "» #» , » "» . 3 3% 3! 4 3% (v) Species mass balances at the nodes of zones
1313
Fig. 2. The computation times are 10 and 50 min using a Cray J916 computer for the equations with the LDF model when 15 and 50 elements were used, and 100 and 600 min, respectively, for the pore diffusion model, respectively. For simulated moving bed processes with linear adsorption isotherms the range of the operation conditions (fluid and solid flowrates in each zone) can be determined by the dimensionless parameter (Ruthven and Ching, 1989). e b " ij (1!e)m K j i
(10)
(13)
c "c D for j"2 and 4 (11) fij ij~1 x/1 » » c D #» c &% if . (12) c " 4c D , c " 2 i2 x/1 fi1 » i4 x/1 fi3 » 1 3 In the above equation, a is a dimensionless parameter, j a "(1/k)(u /¸ ), where k is the mass transfer coefficient j s j between the fluid and the adsorbent for the LDF model, and the equivalence at k"15/t between the two models is assumed d which was obtained by comparing the moments of the impulse response (Ruthven, 1984) or from the parabolic intraparticle concentration profiles assumption (Liaw et al., 1979). In the above model equations, since the boundary conditions are related between the linked zones, the equations for all zones need to be solved simultaneously. For the pore diffusion model, equations inside the particle were first reduced to ordinary differential equations in solving the variable x by applying the orthogonal collocation method in finite elements in the o coordinate and three elements were used. Then the ordinary equatins were solved by the commercial package PDECOL (Madsen and Sincovec, 1979; Lu et al., 1993). Fifteen elements were used in each zone to minimize numerical oscillations in axial concentration profiles for adsorbents I and II and 50 elements were required for adsorbent III. The nonlinear adsorption isotherms for both species A and B for the three adsorbents are shown in
and the parameter in each zones should satisfy the limiting values listed in Table 2 for good separation results. However, for nonlinear systems the operation condition estimation is extremely difficult, and it can be achieved only by using some commercial softwares for the particular separation systems (Nicoud et al., 1993). In this work, the determination of the operation conditions for the different separation systems was obtained through process dynamics analysis (Lu and Ching, 1996). The linear driving force model was used for determining the operation conditions.
Fig. 2. Adsorption isotherms for species A and B, for adsorbents I (fuel lines), II (dot-dashed lines), and III (dashed lines).
Fig. 3. On-column concentration profiles: linear driving force model (full lines) and pore diffusion model (dashed lines).
Table 2. Ranges of the values of the operating parameter for good performance Number of zone b Aj b gi
I
II
III
IV
'1 '1
(1 '1
(1 '1
(1 (1
1314
Shorter communications
Fig. 4. Intraparticle concentration profiles simulated by the pore diffusion model: Adsorbent I (full lines), II (dot-dashed lines) and III (dashed lines).
Column concentration profiles simulated by the LDF model (solid lines) and the pore diffusion model (dashed lines) for the three adsorbents are shown in Fig. 3(a)—(c) for the different adsorption isotherms shwn in Fig. 2, and operation conditions as listed in Table 2. The on-column concentration profiles by the two models are almost superimposed on each other. From these numerical simulation results, it can be concluded that the two models are essentially equivalent in modeling simulated moving bed separation processes, so long as k"15/t . The intraparticle concentration profiles d at the end of each zone (x"1) by the pore diffusion model for three adsorbents are presented in Fig. 4(h), in solid lines (adsorbent I), dot-dashed lines (adsorbent II) and dashed lines (adsorbent III). The intraparticle concentration profiles are almost parabolic in shape for all the three adsorbents except for a small regions near the particle surface at the end for zone II and zone IV where feed steam and eluent stream are introduced. The deviation from the parabolic intraparticle concentration profiles increases with the nonlinearity of the adsorption isotherms from adsorbent I, adsorbent II to adsorbent III, because bulk fluid concentration jumps take place resulting in large changes of solid surface concentration where feed and eluent are introduced.
C. B. CHING* Z. P. LU Department of Chemical Engineering ¹he National ºniversity of Singapore Kent Ridge 119260, Singapore NOTATON
b i c ij c fi D L D s k i K i ¸ j
the adsorption coefficient in the adsorption isotherm eq. (9), l/g liquid-phase concentration of species i in the zones j, g/l species i concentration in the feed or at the inlet, g/l axial dispersive coefficient in the zones j, m2/s intraparticle diffusivity, m2/s lump mass transfer coefficient of the species i between fluid and solid phases, 1/s adsorption coefficient in the adsorption isotherm eq. (9) length of the zones j, m
* Corresponding author.
Shorter communications Pe Pe j q ij qN ij q m r R t t s u s v j » %» %9 » &% » j » 3! » 3% x z
Peclet number in the bulk fluid axial Peclet number ["(v #u )/¸ )/D ] j s j Lj solid-phase concentration of species i in the zones j, g/l average solid-phase concentration of species i in the zones j, g/l adsorption coefficient in the adsorption isotherm eq. (9), g/l radius coordinate inside the adsorbent particle, m radius of the adsorbent particle, m time, min switch time, min equivalent interstitial solid velocity in the process, m/min interstitial fluid velocity in the zones j, m/min eluent flow rate, ml/min extract flow rate , ml/min feed flow rate, ml/min liquid flow rate in the zones j, ml/min raffinate flow rate, ml/min recycle flow rate, ml/min z/¸, dimensionless axial coordinate in the zones axial coordinate in the zones, m
Greek letters a reciprocal of mass transfer unit ["(1/k)(u /¸ )] j s j b dimensionless parameter, an indicator for separation ij ["(e/(1!e)m K )] j i e bed porosity m ratio of the flow rate between solid and liquid j ["(u /v )] s j q dimensionless time ("t/t ) d o diensionless radius coordinate inside the adsorbent particle ("r/R) m ratio of velocities between solid and fluid state in eq. j (13b) Subscripts i species, A, B j zones or column
1315 REFERENCES
Ching, C. B. and Ruthven, D. M. (1986) Experimental study of simulated counter-current adsorption system—IV. Non-isothermal operation. Chem. Engng Sci. 41, 3063—3071. Ching, C. B., Ho, C., Hidajat, K. and Ruthven, D. M. (1987) Experimental study of a simulated counter-current adsorption system—V. Comparison of resin and zeolite adsorbents for fructose—glucose separation at high concentration. Chem. Engng Sci. 42, 2547—2555. Do, D. D. and R. G. Rice (1986) Validity of the parabolic assumption in adsorption studies. A.I.Ch.E. J. 32, 149—154. Glueckauf, E. (1955) Theory of chromatography part 10 — formula for diffusion into spheres and their applications to chromatography. ¹rans. Faraday Soc. 51, 1540. Hashimoto, K., Adachi, S., Shirai, Y. and Morishita, M. (1993) Operation and design of simulated moving bed adsorbers. In: Preparative and Production Scale Chromatography, ed. G. Ganetsos and P. E. Barker, p. 629. Marcel Dekker Inc, New York. Liaw, C. H., Wang, J. S. P., Greekorn, R. A. and Chao, K. C. (1979) Kinetics of fixed-bed adsorption: a new solution. A.I.Ch.E. J. 25, 376—381. Lu, Z. P., Loureiro, J. M., LeVan, D. M. and Rodrigues, A. E. (1993) Simulation of a three-step one-column pressure swing adsorption process. A.I.Ch.E. J. 39, 1483. Lu, Z. P. and Ching, C. B. (1996) Dynamics of simulated moving bed adsorption separation processes. Revision, Sep. Sci. ¹echnol. (submitted). Madsen, N. K. and Sincovec, R. F. (1979) PDECOL: General collocation software for partial differential equations. ACM. ¹rans. Math. Software 5(3), 326. Nicoud, R. M., Baily, M., Kinkel, J. N., Devant, R. M., Hampe, Th R. E. and Kuters, E. (1993) Simulated moving bed applications for enantiomer separations on chiral stationary phases. In: Simulated Moving bed: Basic and Applications, ed. M. Nicoud, R., p. 65. European Meeting, Nancy, 6th December. Ruthven, D. M. and Ching, C. B. (1989) Counter-current and simulated counter-current adsorption separation processes. Chem. Engng Sci. 44, 1011—1038.
Chemical Engineering Science, Vol. 53, No. 6, pp. 1311—1315, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0009–2509(97)00442–9 0009—2509/98 $19.00#0.00
Parabolic intraparticle concentration profile assumption in modeling and simulation of nonlinear simulated moving-bed separation processes (Received 28 December 1996)
The linear driving force (LDF) model, i.e. the parabolic intraparticle concentration profile assumption, has been widely used in modeling many adsorption processes due to its remarkable simplicity and good agreement with experimental results (Liaw et al., 1979; Do and Rice, 1986; Ching and Ruthven, 1986; Ching et al., 1987). Do and Rice (1986) studied the validity of the LDF model, which was first known as the Glueckauf ’s equation (Glueckauf, 1955), and obvious deviations from the pore diffusion model were found only when the dimensionless time q is below 0.05. Modeling and simulation play a very important role in developing and studying simulated moving-bed separation processes (Ruthven and Ching, 1989; Nicoud et al., 1993), as shown schematically in Fig. 1(b). In this configuration, liquid goes up and solid goes down the different zones and countercurrent contact between solid and liquid occurs, leading to high mass transfer driving force. Provided that the adsorption affinities of species A and B on the adsorbent are different, it is possible to choose the right flow rates of the solid and the liquid to force A to move upward and B downward, thus leading to spatial separation. This system requires two inlet lines (one for the feed and one for the eluent) and two outlet lines (one for the raffinate and one for the extract). Since the movement of the solid is very difficult to handle, the above concept is normally implemented by shifting the four inlet and outlet lines between a group of fixed columns (4—24 columns) looped together, as shown in Fig. 1(a), to simulate the movement of the solid (Ruthven and Ching, 1989; Nicoud et al., 1993). Due to the complicated dynamics resulting from the process set-up of the simulated moving bed, it is advisable that simple mass transfer model be used to save computation time in solving the mathematical model equations (Lu and Ching, 1996). That may be the reason why LDF model has been widely used in mathematical modeling and simulation studies for the simulated moving bed separation processes (Hashimoto et al., 1993; Nicoud et al., 1993; Ching et al., 1987). In most of such processes the dimensionless time q is larger than 0.05, and good agreement between mathematical simulation with LDF mass transfer model and experimental results has been obtained (Ching et al., 1987; Hashimoto et al., 1993), though yet only for linear systems. This short communication will focus on the application of the LDF mass transfer model in the mathematical simulation moving-bed separation processes with nonlinear adsorp-
tion isotherms to compare with the linear pore diffusion model, in order to determine its validity as well as to establish the equivalence between the two models. The simulated moving bed separation process to be studied in this work is a binary (A#B) separation (Nicoud et al., 1993) where B is more adsorbable than A and the eluent is not adsorbable. The simulated moving bed was made of 12 columns. The sketch of the process is shown in Fig. 1 where each zone consists of three columns, and the system and operating parameters are listed in Table 1 involving three adsorbents I, II and III. The mathematical models for the separation process shown in Fig. 1 can be developed with LDF or pore diffusion models by using the following assumptions: (1) The adsorbent is homogeneous and intraparticle diffusivity is constant. (2) The flow pattern inside columns or zones can be described by axial dispersion plug flow. (3) The ‘dead’ volumes at both end of the packed adsorbent are negligible. (4) The non-linear multicomponent adsorption isotherms for species A and B follow the extended Langmuir expressions (Nicoud et al., 1993). Introducing the dimensionless axial coordinate x"z/¸ j and radial coordinate o"r/R, we can get (i) ¸inear driving force (¸DF ) model Mass balances in the fluid phase: 1 d2c dc (1!e) ij! ij! m N "0. j ij Pe dx2 dx e j
(1)
dqN ij"f (c )!qN . a i dx ij ij
(2)
LDF model:
Mass transfer between fluid and solid:
1311
dqN N " ij . ij dx
(3)
1312
Shorter communications
Fig. 1. (a) Schematic diagram of experimental system showing sequence of 12 columns with inlet and draw off points. (b) Equivalent countercurrent flow system.
Table 1. Parameters for adsorption isotherms and operating parameters for adsorbents I, II, and III Parameters
Adsorbent I
q , mg/ml m K A K B b , ml/mg A b , ml/mg B » , ml/min &% » , ml/min %» , ml/min %9 » , ml/min 3! » , ml/min 3% t , min s Elements in x coordinate
5.80 1.57 1.57 0.014 0.127 1.52 4.63 4.00 2.05 20.4 4.33
15.8 0.785 0.785 0.014 0.127 4.00 12.0 10.5 5.50 20.0 3.55
12.8 0 0 0.200 0.600 4.00 18.5 18.5 5.50 16.0 4.50
15
15
50
Boundary conditions for the four zones: 1 dc dqN ij"c , ij"0 c ! ij Pe dx ifj dx j dc ij"0; x"1, qN "qN D ij ij`1 x/0 dx
x"0,
(where if j#1'4, then j#1"1).
Adsorbent II Adsorbent III
Mass transfer between fluid and solid:
K
3 Lq ij N "! . ij a Lo j r/1
(4a)
Boundary conditions for the four zones o"0,
(4b)
(ii) Pore diffusion model Mass balances in the fluid phase: 1 L2c Lc (1!e) ij! ij! m N "0. j ij Pe Lx2 Lx e j Pore diffusion model: L2q 2 Lq Lq ij# ij"15a ij . j Lx Lo2 o Lo
(7)
o"1, (5)
(8a)
q "f (c ) ij ij
(8b)
1 Lc ij"c , x"0, c ! ij Pe Lx ifj j x"1,
(6)
Lq ij"0 Lo
Lc ij"0; Lx
Lq ij"0 Lx
(8c)
q "q D ij ij`1 x/0
(where if j#1'4, then j#1"1).
(8d)
Shorter communications (iii) Multicomponent (Nicoud et al., 1993)
¸angmuir
adsorption
isotherms
q bc m i ij f (c )"K c # . ij i ij 1#+ n b c i/1 i ij (iv) Overall mass balances at the nodes of the zones
(9)
» "» #» , » "» #» !» , 1 3% %2 #3% %9 » "» #» , » "» . 3 3% 3! 4 3% (v) Species mass balances at the nodes of zones
1313
Fig. 2. The computation times are 10 and 50 min using a Cray J916 computer for the equations with the LDF model when 15 and 50 elements were used, and 100 and 600 min, respectively, for the pore diffusion model, respectively. For simulated moving bed processes with linear adsorption isotherms the range of the operation conditions (fluid and solid flowrates in each zone) can be determined by the dimensionless parameter (Ruthven and Ching, 1989). e b " ij (1!e)m K j i
(10)
(13)
c "c D for j"2 and 4 (11) fij ij~1 x/1 » » c D #» c &% if . (12) c " 4c D , c " 2 i2 x/1 fi1 » i4 x/1 fi3 » 1 3 In the above equation, a is a dimensionless parameter, j a "(1/k)(u /¸ ), where k is the mass transfer coefficient j s j between the fluid and the adsorbent for the LDF model, and the equivalence at k"15/t between the two models is assumed d which was obtained by comparing the moments of the impulse response (Ruthven, 1984) or from the parabolic intraparticle concentration profiles assumption (Liaw et al., 1979). In the above model equations, since the boundary conditions are related between the linked zones, the equations for all zones need to be solved simultaneously. For the pore diffusion model, equations inside the particle were first reduced to ordinary differential equations in solving the variable x by applying the orthogonal collocation method in finite elements in the o coordinate and three elements were used. Then the ordinary equatins were solved by the commercial package PDECOL (Madsen and Sincovec, 1979; Lu et al., 1993). Fifteen elements were used in each zone to minimize numerical oscillations in axial concentration profiles for adsorbents I and II and 50 elements were required for adsorbent III. The nonlinear adsorption isotherms for both species A and B for the three adsorbents are shown in
and the parameter in each zones should satisfy the limiting values listed in Table 2 for good separation results. However, for nonlinear systems the operation condition estimation is extremely difficult, and it can be achieved only by using some commercial softwares for the particular separation systems (Nicoud et al., 1993). In this work, the determination of the operation conditions for the different separation systems was obtained through process dynamics analysis (Lu and Ching, 1996). The linear driving force model was used for determining the operation conditions.
Fig. 2. Adsorption isotherms for species A and B, for adsorbents I (fuel lines), II (dot-dashed lines), and III (dashed lines).
Fig. 3. On-column concentration profiles: linear driving force model (full lines) and pore diffusion model (dashed lines).
Table 2. Ranges of the values of the operating parameter for good performance Number of zone b Aj b gi
I
II
III
IV
'1 '1
(1 '1
(1 '1
(1 (1
1314
Shorter communications
Fig. 4. Intraparticle concentration profiles simulated by the pore diffusion model: Adsorbent I (full lines), II (dot-dashed lines) and III (dashed lines).
Column concentration profiles simulated by the LDF model (solid lines) and the pore diffusion model (dashed lines) for the three adsorbents are shown in Fig. 3(a)—(c) for the different adsorption isotherms shwn in Fig. 2, and operation conditions as listed in Table 2. The on-column concentration profiles by the two models are almost superimposed on each other. From these numerical simulation results, it can be concluded that the two models are essentially equivalent in modeling simulated moving bed separation processes, so long as k"15/t . The intraparticle concentration profiles d at the end of each zone (x"1) by the pore diffusion model for three adsorbents are presented in Fig. 4(h), in solid lines (adsorbent I), dot-dashed lines (adsorbent II) and dashed lines (adsorbent III). The intraparticle concentration profiles are almost parabolic in shape for all the three adsorbents except for a small regions near the particle surface at the end for zone II and zone IV where feed steam and eluent stream are introduced. The deviation from the parabolic intraparticle concentration profiles increases with the nonlinearity of the adsorption isotherms from adsorbent I, adsorbent II to adsorbent III, because bulk fluid concentration jumps take place resulting in large changes of solid surface concentration where feed and eluent are introduced.
C. B. CHING* Z. P. LU Department of Chemical Engineering ¹he National ºniversity of Singapore Kent Ridge 119260, Singapore NOTATON
b i c ij c fi D L D s k i K i ¸ j
the adsorption coefficient in the adsorption isotherm eq. (9), l/g liquid-phase concentration of species i in the zones j, g/l species i concentration in the feed or at the inlet, g/l axial dispersive coefficient in the zones j, m2/s intraparticle diffusivity, m2/s lump mass transfer coefficient of the species i between fluid and solid phases, 1/s adsorption coefficient in the adsorption isotherm eq. (9) length of the zones j, m
* Corresponding author.
Shorter communications Pe Pe j q ij qN ij q m r R t t s u s v j » %» %9 » &% » j » 3! » 3% x z
Peclet number in the bulk fluid axial Peclet number ["(v #u )/¸ )/D ] j s j Lj solid-phase concentration of species i in the zones j, g/l average solid-phase concentration of species i in the zones j, g/l adsorption coefficient in the adsorption isotherm eq. (9), g/l radius coordinate inside the adsorbent particle, m radius of the adsorbent particle, m time, min switch time, min equivalent interstitial solid velocity in the process, m/min interstitial fluid velocity in the zones j, m/min eluent flow rate, ml/min extract flow rate , ml/min feed flow rate, ml/min liquid flow rate in the zones j, ml/min raffinate flow rate, ml/min recycle flow rate, ml/min z/¸, dimensionless axial coordinate in the zones axial coordinate in the zones, m
Greek letters a reciprocal of mass transfer unit ["(1/k)(u /¸ )] j s j b dimensionless parameter, an indicator for separation ij ["(e/(1!e)m K )] j i e bed porosity m ratio of the flow rate between solid and liquid j ["(u /v )] s j q dimensionless time ("t/t ) d o diensionless radius coordinate inside the adsorbent particle ("r/R) m ratio of velocities between solid and fluid state in eq. j (13b) Subscripts i species, A, B j zones or column
1315 REFERENCES
Ching, C. B. and Ruthven, D. M. (1986) Experimental study of simulated counter-current adsorption system—IV. Non-isothermal operation. Chem. Engng Sci. 41, 3063—3071. Ching, C. B., Ho, C., Hidajat, K. and Ruthven, D. M. (1987) Experimental study of a simulated counter-current adsorption system—V. Comparison of resin and zeolite adsorbents for fructose—glucose separation at high concentration. Chem. Engng Sci. 42, 2547—2555. Do, D. D. and R. G. Rice (1986) Validity of the parabolic assumption in adsorption studies. A.I.Ch.E. J. 32, 149—154. Glueckauf, E. (1955) Theory of chromatography part 10 — formula for diffusion into spheres and their applications to chromatography. ¹rans. Faraday Soc. 51, 1540. Hashimoto, K., Adachi, S., Shirai, Y. and Morishita, M. (1993) Operation and design of simulated moving bed adsorbers. In: Preparative and Production Scale Chromatography, ed. G. Ganetsos and P. E. Barker, p. 629. Marcel Dekker Inc, New York. Liaw, C. H., Wang, J. S. P., Greekorn, R. A. and Chao, K. C. (1979) Kinetics of fixed-bed adsorption: a new solution. A.I.Ch.E. J. 25, 376—381. Lu, Z. P., Loureiro, J. M., LeVan, D. M. and Rodrigues, A. E. (1993) Simulation of a three-step one-column pressure swing adsorption process. A.I.Ch.E. J. 39, 1483. Lu, Z. P. and Ching, C. B. (1996) Dynamics of simulated moving bed adsorption separation processes. Revision, Sep. Sci. ¹echnol. (submitted). Madsen, N. K. and Sincovec, R. F. (1979) PDECOL: General collocation software for partial differential equations. ACM. ¹rans. Math. Software 5(3), 326. Nicoud, R. M., Baily, M., Kinkel, J. N., Devant, R. M., Hampe, Th R. E. and Kuters, E. (1993) Simulated moving bed applications for enantiomer separations on chiral stationary phases. In: Simulated Moving bed: Basic and Applications, ed. M. Nicoud, R., p. 65. European Meeting, Nancy, 6th December. Ruthven, D. M. and Ching, C. B. (1989) Counter-current and simulated counter-current adsorption separation processes. Chem. Engng Sci. 44, 1011—1038.