Simulation of true moving bed adsorptive reactor: Detailed particle model and linear driving force approximations

Chemical Engineering Science 62 (2007) 1026 – 1041 www.elsevier.com/locate/ces

Simulation of true moving bed adsorptive reactor: Detailed particle model and linear driving force approximations Pedro Sá Gomes, Celina P. Leão1 , Alírio 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 Received 2 February 2006; received in revised form 21 August 2006; accepted 1 November 2006 Available online 10 November 2006

Abstract A new true moving bed (TMB) adsorptive reactor model with a detailed particle approach is presented introducing the formulation of the mass balance for the solid phase in counter-current moving systems. The system studied here is the enzymatic inversion of sucrose into fructose and glucose and subsequent separation of glucose/fructose; the reaction occurs both in the outer fluid phase and inside particles. Model equations include film mass transfer, intra-particle diffusion resistance, axial dispersion for the outer fluid phase, plug flow of the solid phase and linear adsorption equilibrium of glucose/fructose. This new model is compared with previous LDF-type approximations for reactive systems and applied to pure separative TMB process. The numerical solution of model equations is obtained for transient and steady state with commercial and public domain packages (gPROMS and COLNEW). The influence of the particle size and reaction rate constant is analyzed in the (2 × 3 ) reactive/separation region. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: True moving bed absorptive reactor; Detailed particle model; Linear driving force approximations; Dynamic simulation

1. Introduction Introduced by the universal oil products (UOP) in the early 1960s (Broughton and Gerhold, 1961), the simulated moving bed (SMB) appeared predominantly in large-scale separations in the petrochemical industry, as a practical implementation of the true moving bed (TMB) process, avoiding the problems of the solid motion by simulating the counter-current bed movement with a synchronous shift of the inlet/outlet ports, while holding the bed immobile. Recently, the SMB has found new successful applications as an attractive continuous chromatographic separation process for biotechnological, pharmaceuticals and fine chemistry areas. Examples are SMBs installed by Novasep (www.novasep.com) at UCB Pharma (www.ucbpharma.com) and Aerojet Fine Chemicals (www.aerojetfinechemicals.com). ∗ Corresponding author. Tel.: +351 225081671; fax: +351 225081674.

E-mail address: [email protected] (A.E. Rodrigues). 1 Present address: Departamento de Produção e Sistemas, Escola de

Engenharia, Universidade do Minho 4710-057 Braga, Portugal. 0009-2509/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2006.11.008

The combination of a chemical reaction and a separation process in the same unit leads, in several cases, to the improvement of reaction and separation efficiency, since in reversible reactions the conversion is increased by the products removal, the combination of the two process allows some adsorbent/catalyst savings since the same solid has two functions and also the quantity of solvent and time of operation can be reduced (Silva and Rodrigues 2004, 2005; Lode et al., 2001; Kurup et al., 2005). The SMB Adsorptive Reactor (SMBAR) is one continuous reaction/separation chromatographic process reported in literature, and the inversion of sucrose and subsequent separation of the obtained inverted sugar (glucose and fructose) is a fine example. The esterification from acetic acid and -phenethyl alcohol and subsequent separation of the product -phenethyl acetate (Kawase et al., 1996), the synthesis and separation of the methanol from syngas (Kruglov, 1994), or the diethylacetal synthesis as in Silva and Rodrigues (2004, 2005) are other examples that show the potential of this technique. The SMBAR approaches the behaviour of the true moving bed adsorptive reactor (TMBAR) model for an infinite column number, for example, a reasonable agreement of TMBAR

P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

simulation results with experimental data was observed in Azevedo and Rodrigues (2001), who considered diffusion and reaction in a porous catalyst described by a linear driving force (LDF) approximation, analogous to the Glueckauf (1955) approach. A detailed particle approach has been introduced by Dunnebier et al. (2000) for the SMBAR model; the use of TMBAR approach which provides easy computational solutions can be quite useful, but requires proper handling of the particle mass balance equation since the particle is moving relatively to a fixed reference frame. The aim of this paper, is therefore, the introduction of a TMBAR model which includes a detailed particle model for the radial concentration profile, and a first-order reaction which takes place in the fluid phase (bulk and particles pores) and at the pore walls. Moreover film mass transfer, convective fluid movement with axial dispersion, and a counter-current solid plug flow are included. The model, is applied to a first-order reaction approximation to the enzymatic Michaelis–Menten model of sucrose inversion into fructose and glucose with the subsequent product separation, and is then numerically solved using different PDE/PDAE and ordinary differential equation (ODE) solvers, namely the public domain COLNEW (Bader and Ascher, 1987), and the commercial simulation package gPROMS from PSE Ltd (www.psenterprise.com). This new TMBAR model with detailed particle description is also compared with two different LDF-type approximations, from Kim (1989) and Szukiewicz (2000). In the absence of reaction, the new detailed particle formulation of the TMB is compared to the classic TMB model based on the LDF approximation of Glueckauf (1955).

2. Model formulation This new TMBAR model considers that a first-order reaction approximation for the reaction rate of sucrose (A) inversion into fructose (B) and glucose (C) takes place in the bulk fluid phase, in particles pores and at pores walls where also products are adsorbed: kr

A → B +C invertase kr Kenz

A → B +C invertase

in the fluid phase (bulk and particle pores),

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Two sets of transient mass balances are performed to each species i in each section j, in molar basis: i. In a volume element of the bulk fluid phase, b

jCbij j2 Cbij jCbij −  b uj = b Dbj 2 jt jz jz − (1 − b )

3kf ij (Cbij − Cpij |r=RP ) Rp

+ i b R(CbAj )

(1)

with the initial and boundary conditions, Cbij (z, 0) = 0,

(2)

z = 0: Cbi(j,z=0) = Cbij (0, t) −  jCbij  =0 z = Lj : jz z=Lj

Dbj uj

 jCbij  , jz z=0

(3a)

(3b)

and the nodes mass balances, j = 1: Cbi(4,z=L4 ) =

u1 Cbi(1,z=0) , u4

j = 2, 4: Cbi(j −1,z=L(j −1) ) = Cbi(j,z=0) , j = 3: Cbi(2,z=L2 ) =

u3 uF F Cbi(3,z=0) − C , u2 u2 i

(4a) (4b) (4c)

where Cbij and Cpij are the fluid bulk and particle pores concentration for component i in section j, respectively, CiF the species i feed concentration, t, r and z the time, particle radial and column axial coordinates, b the bed porosity, Dbj the axial dispersion coefficient and uj the interstitial fluid velocity in section j, Rp the particle radius, Lj the length of section j, kf ij the film mass transfer coefficient and i the species i stoichiometric coefficient. ii. In a volume element of the spherical particle,   jCpij jCpij jCsij jCsij + p − us + p jt jt jz jz   jCpij 1 j = 2 r 2 Dpei +i [p R(CpAj )+R(CsAj )] (5) r jr jr with the initial and boundary conditions,

in the adsorbed phase,

where kr and kr Kenz represents the first-order reaction rate coefficient in the fluid and in the “adsorbed phase,” respectively. The mass transfer effects considered are intraparticle diffusion, film mass transfer, convective fluid movement with axial dispersion, and counter-current solid plug flow. The main assumptions are constant dispersion coefficients, particle diffusion and film mass transfer, constant porosities, spherical particles with homogeneous size, linear adsorption isotherms and first-order reaction in both fluid and “adsorbed phase”.

Cpij (z, r, 0) = 0,  Cpij = Cpi(j +1,z=0) for j = 1, 2, 3, z = Lj : Cpi4 = Cpi(1,z=0) ,  jCpij  = 0, r = 0: jr r=0  jCpij  kf ij r = Rp : = (Cbij − Cpij |r=RP ),  jr r=Rp Dpei

(6) (7a1 ) (7a2 ) (7b)

(7c)

where Csij is the particle adsorbed phase concentration for component i in section j , with relation to the total particle

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volume, p the particle porosity, Dpei the effective pore diffusion coefficient and us the interstitial solid velocity. The inclusion of the substantial time derivative (Bird et al., 2002; Jenson and Jeffreys, 1963) on pore and adsorbed phase concentration, jCpij jCsij D (Csij + p Cpij ) = + p Dt jt jt   jCpij jCsij − us + p jz jz

(8)

representing the particle accumulation as well as the solid counter-current movement with interstitial velocity us should be noted. This approach was taken by Leão (2003) in the modelling of TMB and also in other fields by Yao et al. (2001) and Tél et al. (2005). Considering linear adsorption isotherms for species i the “adsorbed phase” concentration is Csij = Cpij Ki , where Ki is the linear adsorption isotherm coefficient. The reaction rate in fluid phase is R(CA ) = kr [E0 ]CA ; in the adsorbed phase the enzyme concentration is [Es ] = Kenz [E0 ], and averaging the particle mass balance will lead to:   jCpij  jCpij  − us (p + Ki ) jt jz =

3kf ij (Cbij − Cpij |r=RP ) + i (p + Kenz )R(CpAj ), Rp (9)

where Cpij  =

 Rp 0

Cpij r 2 dr

 Rp 0

r 2 dr

=

3 Rp3



Rp 0

with the initial and boundary conditions, Cbij (x, 0) = 0,

(12)

 1 jCbij  x = 0: Cbi(j,x=0) = Cbij (0, ) − , Pej jx x=0  jCbij  x = 1: =0 jx x=1

(13a)

(13b)

plus the nodes mass balances, j = 1: Cbi(4,x=1) =

u1 Cbi(1,x=0) , u4

j = 2, 4: Cbi(j −1,x=1) = Cbi(j,x=0) , j = 3: Cbi(2,x=1) =

u3 uF F Cbi(3,x=0) − C . u2 u2 i

ii. Particle mass balance  jCpij 1 jCpij (p + Ki ) − j nj jx    j 1 j 2 jCpij = Npij 2  nj  j j + i (p + Kenz )Daj CpAj

(14a) (14b) (14c)

(15)

with the initial and boundary conditions, Cpij r 2 dr.

(10)

The similarity with other LDF-type models applied to the classic TMB without reaction (Leão and Rodrigues, 2004), which supports the detailed particle mass balance can be noted. Dimensionless model equations: Introducing dimensionless variables for time, axial and radial space coordinates: ⎧ t ⎧ Lc ⎪ = , ⎪ ⎪ us = , ⎪ ⎪ t c ⎪ ⎪ ⎪ ⎪ tc ⎪ ⎪ z ⎨ ⎨ x= , where Lj = Lc nj , Lj ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ t = Lj , r ⎪ ⎪ ⎪ ⎪  = , ⎪ ⎩j uj ⎩ Rp where tc is the solid space time, tj the section j bulk fluid space time, Lc the single column length, and nj the number of columns per section, the dimensionless model equations become: i. Bulk fluid mass balance  j 1 j2 Cbij jCbij jCbij = − j nj Pe jx 2 jx (1 − b ) Nf ij (Cbij − Cpij |r=RP ) b + i Daj CbAj ,

−

(11)

Cpij (x, , 0) = 0,  x = 1:

Cpij = Cpi(j +1,x=0)

(16) for j = 1, 2, 3,

Cpi4 = Cpi(1,x=0) ,  jCpij   = 0: = 0, j =0  = 1:

 jCpij  = Bi mi (Cbij − Cpij |=1 ) j =1

(17a1 ) (17a2 ) (17b)

(17c)

and  Cpij  = 3

1 0

Cpij 2 d,

where j = uj /us is the ratio between fluid and solid interstitial velocities, Pej = uj Lj /Dbj the Peclet number, Daj = kr tj the Damköhler number, Nf ij = (3kf ij /Rp )tj the number of film mass transfer units, Npij = (Dpei /Rp2 )tj the number of intraparticle mass transfer units and Bi mi = kf ij Rp /Dpei = 1 3 Nf ij /Npij the Biot number. The steady-state formulation is obtained by setting the time derivatives to zero:    jCpij jCbij =0 = 0  . j j steady state

P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

3. Numerical solution The transient model equations (11)–(17) are defined by a PDE system with two sets of mass balance equations, one for the outer fluid and other for the particle “solid phase”, for each component in each section, respectively, that are solved using the commercial package gPROMS. The steady-state formulation is solved with the public domain package COLNEW (Bader and Ascher, 1987) after discretization of the particle radial coordinate , and also with gPROMS.

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discretization the OCFE method of order 3 and 5 finite elements were used; the column axial discretization combined various methods through the different sections, order and number of elements, to improve computation, as mentioned in Table 2. After the radial and axial discretization step gPROMS integrate the system of ODEs with DASOLV and the resulting system of AE, is then solved by gPROMS BDNSOL solver (“Block Decomposition NonLinear SOLver”). 4. Operating conditions and parameter estimation

3.1. COLNEW approach

4.1. Operating conditions

For the “particle phase” balance, with initial and boundary conditions, an OCFE (orthogonal collocation in finite elements) method for the discretization of the particle radial coordinate  was applied using the Hermite polynomial as basis function. The interval 0  1 was divided into NE subintervals, with two collocation points within each subinterval; the discretization step can be found in Appendix A. The COLNEW package incorporates a new basis representation replacing B-splines, and improvements for the linear and nonlinear algebraic equation (AE) solvers and the numerical parameters used are presented in Table 1. The number and positions of steady-state radial discretization finite elements was also analyzed, and is presented in Appendix B.

The equilibrium theory applied to the separation region in a non-reactive SMB was used to define the adequate operating conditions for the SMBAR model case (Fricke et al., 1999), due to the similarities between them, namely the definition of four sections. These conditions are sufficient for the discussion of a reactive process for fast reaction kinetics as compared to the residence time, since reaction products are formed in the feed inlet neighborhood and most of the unit is utilized for their separation; in the case of slow reaction kinetics this procedure cannot be taken into account (Lode et al., 2001). Therefore, the relationships used for the SMB/TMB linear equilibrium system (Ruthven and Ching, 1989) were used to estimate the initial values of the TMBAR velocity ratios j , 1 >

3.2. gPROMS approach Due to better interface for data export/manipulation, a commercial package, gPROMS, was also used on the solution of the transient and steady-state formulations. For radial

Table 1 COLNEW numerical parameters COLNEW Axial collocation points Axial number of elements Radial collocation points Radial number of elements Error tolerance Global error

2 20 4 2a 10−8 0.0095

Global error, considering global mass balance. a (l) = 0; 0.95; 1.

Table 2 Axial gPROMS discretization method, OCFEM (orthogonal collocation on finite elements method) absolute tolerance and relative tolerance 10−5 Section (j)

Method

Order

Number of elements

1 2 3 4

OCFEM OCFEM OCFEM OCFEM

2 3 2 2

200 350 250 200

(3 (2 (5 (2

columns) columns) columns) columns)

1 − b (KB + p ), b

(18a)

1 − b 1 − b (KC + p ) < 2 < 3 < (KB + p ), b b

(18b)

1 − b (KC + p ) > 4 . b

(18c)

Notice the presence of the term p in the inequalities. The reaction takes place basically in the feed neighborhoods (sections 2 and 3) and sections 1 and 4 are regenerating sections. Consequently, if the two conditions (18a) and (18c) are fulfilled, the regeneration of the eluent and solid will be guaranteed, having no longer effect on the TMBAR performance in terms of conversion and purity, under the equilibrium theory assumptions. The values of the equilibrium constant for the more and less retained component are KB = 0.43 (fructose) and KC = 0.17 (glucose), respectively, and for the reactant KA = 0 (sucrose), not adsorbed, experimentally obtained as in Azevedo and Rodrigues (2001). The SMBAR switching time, tc , was estimated from the following expression: tc = (j + 1)

 b Vc , Q∗j

where Q∗j is the fluid flowrate in section j for the real SMB unit and assuming a flowrate of 24 ml min−1 for section 4 Q∗4 = QRec , due to the SMB unit flowrate limitation (20.120 ml min−1 ), and a fixed value of 4 equal to 0.33

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P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

(< 0.405). The value 1 was fixed on 0.95 (> 0.7959). The velocities ratios, 2 and 3 , were fixed on 0.45 (> 0.405) and 0.65 (< 0.795), respectively, and the feed flowrate of 3.62 ml min−1 . The remaining flowrates were calculated according to the previous assumptions, values from Leão (2003) work. 4.2. Parameters estimation The Wilke–Chang equation (Reid and Prausnitz, 1987) was used to predict the molecular diffusion coefficient (cm2 s−1 ), of solute M in solvent S: Dm = 7.4 × 10−8

(af Ms )1/2 T 0.6 s VM

,

where Dm the molecular diffusion coefficient, af the association factor of solvent S (af = 2.6 since the solvent is water), Ms the molecular weight of the solvent (water ≈ 18 g mol−1 ), T the absolute temperature (323 K), s the solvent viscosity ≈ 0.6 cP, VM the molar volume of solute M at normal boiling temperature. As no experimental data to obtain this value exists, the method based on the Le Bas additive volume table (Reid and Prausnitz, 1987) was used, ≈ 340.4 cm3 mol−1 for sucrose and ≈ 177.6 cm3 mol−1 for glucose and fructose. Thus DmA ≈ 8.25 × 10−6 cm2 s−1 and DmB = DmC ≈ 1.22 × 10−6 cm2 s−1 . The effective diffusivity, Dpei , is then estimated by, Dpei = Dmi p / , where p is the particle porosity (=0.1, value obtained experimentally) and the tortuosity factor (=2). As result DpeA ≈ 4.12 × 10−7 cm2 s−1 and DpeB = DpeC ≈ 6.09 × 10−7 cm2 s−1 . The film mass transfer coefficients, kf ij , for sucrose and fructose/glucose were estimated with the correlation proposed by Wilson and Geankoplis, Sh = (1.09/b )Sc0.33 Re0.33 , where Sh = 2kf Rp /Dm is the Sherwood number, kf the film mass transfer coefficient (cm s−1 ), Rp the particle radius (cm), Sc = f /f Dm the Schmidt number, and Re = u0 f dp /f the Reynolds number, with f the fluid viscosity ≈ 0.60 g(cm.s)−1 , f the fluid density ≈ 1 g cm−3 , u0 the mean TMBAR velocity ≈ 0.0924 cm s−1 and dp the particle diameter ≈ 0.0320 cm. The Sherwood number values, for sucrose, fructose and glucose, ShA = 19.0, ShB = ShC = 16.7, were obtained, and kf A = 0.29 cm min−1 , kf B = kf C = 0.38 cm min−1 for the film mass transfer coefficients. According to those values, it is possible to estimate the Biot number for mass transfer, k f Rp Bi m = ≈ Dpef



Bi mA = 188, Bi mB,C = 166.

The Peclet number Pej = uj Lj /Db was considered constant and equal to 1500, based on the average fluid velocity section (u0 the mean TMBAR fluid velocity). With the values of the effective diffusivity and the particle radius it is possible to estimate the mass transport coefficient

in the pores, kp , according to, 15Dpe Rp2  kpA = 1.45 = kpB = kpC = 2.14

kp =

min−1 for a classic separation.

For the sucrose inversion first-order reaction approximation, the Michaelis–Menten equation is considered R=

krM [E0 ]CA , Km + C A

(19)

where krM is the reaction rate constant, [E0 ] the enzyme concentration and Km the Michaelis constant, and CA the concentration of sucrose either in the bulk as in the particle pores fluid phase, in mol m−3 . If we consider Km  CA , as assumed in Leão (2003) based on Bowski et al. (1971), then the fluid-phase first-order reaction simplification can be assumed, R(CA ) = kr CA

with kr =

krM [E0 ] Km

(20a)

and for the “adsorbed phase” kr =

krM Kenz [E0 ] Km

(20b)

A summary of model parameters and operating conditions are presented in Table 3. 4.3. Performance criteria The TMBAR outlet/inlet flowrates calculated and described in Table 3, must satisfy a strict purity and reaction conversion specifications. The definitions of extract purity (PEXT , %), raffinate purity (PRFF , %), recovery of the fructose in the extract C (RecB EXT , %), recovery of glucose in the raffinate (RecRFF , %) and conversion of sucrose (X) are presented as follows: PEXT = 100(%)

CbEXT B CbEXT + CbEXT + CbEXT A B C

a similar definition to the purity of the raffinate, RecB EXT = 100(%)

QE CbEXT B QF CAF

a similar definition to the recovery of glucose in the raffinate, X=1−

QE CbEXT + QR CbRFF A A QF CAF

.

The area that defines the complete separation region was established by considering conversion of 99% or more and extract and raffinate purity higher than 95%, see Fig. 5. The safety margin parameter  (Zhong and Guichon, 1996), considered was 1.2, √ approximately the center of the  range located at 1<  < KB /KC = 1.59. At low values of , the TMBAR operating conditions are very near to the zone of no

P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

1031

Table 3 Operating conditions and model parameters Model parameters

TMBAR operating conditions

b = 0.4 p = 0.1

1 = 0.954; 2 = 0.450; 3 = 0.650; 4 = 0.325 Q1 = 17.28 ml min−1 ; Q2 = 8.15 ml min−1 Q3 = 11.77 ml min−1 ; Q4 = 5.89 ml min−1 QE = 11.39 ml min−1 ; QF = 3.62 ml min−1 QEXT = 9.13 ml min−1 ; QRFF = 5.88 ml min−1 Qs = 27.17 ml min−1 CFA = 80.0 g l−1 ; CBF = CCF = 0

Rp = 0.016 cm Pej = 1500 Bi mA = 193; Bi mB = 169; Bi mC = 169 Nf A1 = 581; Nf A2 = 822; Nf A3 = 1422; Nf A4 = 1137 Nf B1 = Nf C1 = 762; Nf B2 = Nf C2 = 1077; Nf B3 = Nf C3 = 1864 Nf B4 = Nf C4 = 1490 NpA1 = 1.00; NpA2 = 1.42; NpA3 = 2.45; NpA4 = 1.96 NpB1 = NpC1 = 1.50; NpB2 = NpC2 = 2.12; NpB3 = NpC3 = 3.68 NpB4 = NpC4 = 2.94 Da1 = 5.85; Da2 = 8.27 Da3 = 14.31; Da4 = 11.45 kr = 0.547 min−1 ; Kenz = 5 KA = 0; KB = 0.43; KC = 0.17

Columns dc = 2.6 cm Lc = 29 cm Configuration [3 2 5 2]

40

5. Simulation results

30

5.1. Steady-state solutions The model steady-state concentration profiles obtained by the COLNEW and gPROMS solvers, using the model parameters and TMBAR operation conditions as in Table 3, and the solvers numerical parameters in the Tables 1 and 2, respectively, lead to Fig. 1 where, as can be observed, both solvers give similar results. The performance parameters obtained for the case under study, for both COLNEW and gPROMS packages are purity of extract and raffinate 99.9% and recovery of B in the extract 99.9% and recovery C in the raffinate 99.8% and conversion of A, 100%, 5.1.1. Effect of particle radius In order to analyze the effect of the particle radius on the performance parameters and on the SMBAR/TMBAR concentration profiles, from the present model, three simulations were preformed using the gPROMS approach, for different values of particle radius, Rp , as presented in Table 4. The model parameters and operating conditions are constant and the same as in Table 3, with exceptions for the particle radius, Rp , the mass transfer numbers Nf ij and Npij and consequently Biot number and the Thiele modulus  calculated for the different particle radius and respective film mass transfer coefficient as in Table 4. The effects of the particle radius increase on purity and recovery of the two products are shown in Fig. 2 as in Table 4. Higher particle radius (Rp =0.020 cm) leads to lower purity and recovery of products. In all three situations, the sucrose concentration, Fig. 2(a), decreases to zero, in zone 3, very rapidly. No significant differences exist and the reaction is almost complete. The glucose and fructose concentration level, Fig. 2(b)

Concentration g.l-1

separation and at high values of  the eluent flowrate increases and consequently the products becomes more diluted.

Sucrose gPROMS Sucrose COLNEW Fructose gPROMS Fructose COLNEW Glucose gPROMS Glucose COLNEW

20

10

0 0

1

2

3

4

5

6

7

8

9

10

11

12

Columns

Fig. 1. Steady-state fluid concentration axial profiles, calculated by steady-state routines.

Table 4 Film mass transfer coefficient for the study of particle radius effect on SMBAR/TMBAR performance parameters Rp (cm)

Model parameters kf ij (cm min−1 )

0.012

0.016

0.020

B/C A

0.46 0.35

0.38 0.29

0.33 0.25

PEXT PRFF RecEXT RecRFF

100 100 100 100

99.9 99.9 99.9 99.8

99.4 99.6 99.6 99.4

Performance parameters (%)

and (c), respectively, shows that for higher particle radius the contamination of collected streams is bigger and lower values of purity in extract and raffinate are observed.

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P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

a

b 40

40

CB Rp=0.012 cm

CA Rp=0.016 cm

30

Concentration g.l-1

Concentration g.l-1

CA Rp=0.012 cm CA Rp=0.020 cm

20

10

CB Rp=0.016 cm

30

CB Rp=0.020 cm

20

10

0

0 4

5

6

0

7

1

2

3

4

5

6

7

8

9

10

11

12

Columns

Columns

c

Concentration g.l-1

40

30 CC Rp=0.012 cm CC Rp=0.016 cm

20

CC Rp=0.020 cm

10

0 0

1

2

3

4

5

6

7

8

9

10

11

12

Columns Fig. 2. TMBAR steady-state bulk profiles obtained for three different particle radius values Rp = 0.012, 0.016 and 0.020 cm, for (a) sucrose, (b) fructose and (c) glucose as in Table 4, the others model parameters and operating conditions as in Table 3.

As expected the mass transfer resistances increase with the particle radius, leading to a worse separation; it can be observed that for the higher particle radius values, the glucose contaminating front is already passing the extract collection point, and therefore decreasing the fructose purity values. 5.1.2. Complete separation region The region where both extract and raffinate purities are approximately 100% ( 99.9%) is determined and represented in Fig. 3. This prediction was based on the SMBAR/TMBAR separation assumptions. The conversion/separation region was determined by considering the assumption from Leão (2003): • the velocity ratios, 1 =0.95 and 4 =0.33, and the switching time, tc =3.4 min, after selected properly, as indicated above, were kept constant in all simulations (Leão, 2003); • the separation region is represented in the (2 , 3 ) plane with location determined by performing simulations along lines parallel to the diagonal of the (2 ,3 ) plane, with purity performance of 99.0%. The results are illustrated in Fig. 3a and b, for the model parameters described in Table 3 except for kr values and using

the steady-state TMBAR model. Four different situations are presented depending on the purity performances obtained in each simulation: (•) —complete conversion of sucrose and purity of both products 99.0%; (◦) —purity performance < 99.0% for both products; ( ) —only extract purity is 99.0%; () —only raffinate purity is 99.0%. The triangle defined by the dashed lines, given by (18b), is the region of complete separation for the case SMB/TMB separation. The region above the triangle corresponds to the region where PEXT 99.0%, whereas PRFF < 99.0% since constrained (18b) is not fulfilled. The region on the left-hand side of the triangle corresponds to PRFF 99.0% and PEXT < 99.0%. From Fig. 3(a) constructed for kr = 50.32 min−1 , it can be observed that the consideration of mass transfer resistances will result in a smaller transfer region, as expected (Azevedo and Rodrigues, 1999). The effect of the reaction rate coefficient, kr , on the reaction/separation region is shown in Fig. 3(b). The triangle-shaped separation/reaction region becomes smaller as the reaction rate

P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

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b

a 1

1

0.8

0.8

0.6

0.6

γ3

γ3

x

Pure Extract PE > 99.0% Pure Raffinate PR > 99.0%

0.4

-1

0.4

Separation PE > 99%, PR > 99%

x

purity performance < 99.05 for both products

0.2 0.2

k=50.32 min -1 k=150.96 min -1 k=1.864 min reference case conditions Table 3

0.2 0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

γ2

γ2

Fig. 3. TMBAR performance for the sucrose inversion and glucose/fructose separation in the (2 , 3 ) plane. Different regions are defined in terms of purity of the two outlet products, as in (a) The presented points define parallel lines to the diagonal at different feed flowrate, kr = 50.32 min−1 ; (b) effects of the reaction rate coefficient, kr , on the reaction/separation region.

with the area defined for higher purity specifications under the same operating conditions. The vertex of the triangle in the (2 , 3 ) plane lies outside the region defined by the equilibrium theoretical limits (Minceva and Rodrigues, 2005). From Fig. 4 it can be observed that even for 99% purity values, the consideration of a model including mass transfer resistances will result in a smaller separation region than the one established by the equilibrium theory. For lower purity values, as the 90% case presented, the separation region based in LDF model will enclose the separation region triangle from equilibrium theory.

0.9

γ3

0.7

0.5

5.2. Transient solutions

>=90% >=99% 0.3

0.3

0.5

0.7

0.9

The transient concentration profiles evolution in column bulk fluid phase is presented in Fig. 5 for the numerical solutions obtained with the gPROMS approach.

γ2

Fig. 4. Effect of the purity specifications on the separation/reaction region for the SMBAR/TMBAR for the sucrose inversion and glucose/fructose separation in the (2 , 3 ) plane, kr = 50.32 min−1 .

coefficient decreases. Three regions are defined for three different values of, kr , 1.864, 50.32 and 150.96 min−1 , with purity performance 99.0%, a similar effect on the separation region is experimented with the decrease of the LDF mass transfer coefficients as in Azevedo and Rodrigues (1999). Lower kr values lead to raffinate contamination by fructose and therefore to keep the raffinate purity >than 99% one has to reduce 3 ; the influence of kr in the contaminating front on Section 2 is very small and therefore the lower boundary of 2 is basically constant. Fig. 4 illustrates the separation/reaction region obtained for lower purity performances specifications, 90%. For lower purity specification, the separation/reaction is greater compared

5.2.1. Bulk fluid concentration profiles, gPROMS package The steady state is obtained near  = 75, about 6 complete cycles. Results obtained with gPROMS, in particular the steadystate solutions (Fig. 5c), show that the maximum concentration for fructose reaches the 25 g l−1 and below 35 g l−1 for glucose, in agreement with simulations with COLNEW (Fig. 1). As can be observed from Fig. 5(a)–(c), the dynamic bulk concentration profiles evolution, it can be concluded that the sucrose concentration profile is almost time independent, all sucrose fed to the unit is consumed in the feed neighborhoods (beginning of section 3 and end of section 2) as already mentioned, and the steady-state established in the very first unit switches. The stability of the fructose and glucose profiles is only obtained near the 75 switches, when the complete unit reaches to the steady state. 5.2.2. Intraparticle radial concentration profiles The simulation values for the internal particle pores concentration profiles, obtained by gPROMS, are presented for

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P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

a

b 40 CA CB CC

30

Concentration (g/l)

Concentration (g/l)

40

20

10

0

CA CB CC

30

20

10

0 0

1

2

3

4

5

6

7

8

9

10

11

12

0

1

2

3

4

5

Columns

6

7

8

9

10

11

12

Columns

c Concentration (g/l)

40 CA CB CC

30

20

10

0 0

1

2

3

4

5

6

7

8

9

10

11

12

Columns Fig. 5. Transient bulk fluid concentration (a)  = 4; (b)  = 8; and (c) steady state from dynamic simulation at  ± 75.

 values of 4, 8 and steady state, at feed inlet (beginning of section 3), Fig. 6. As expected the sucrose intraparticle concentration decreases, while the fructose and glucose increase to the particle interior, the higher value for glucose pore concentration can be easily explained, since it is the less retained component. The presence of a maximum value in the intraparticle fructose concentration profile can be easily understood based on the observation of Figs. 1 or 5c, which also shows a maximum in section 3 for the fructose bulk concentration characteristic of this type of reactive processes with solid motion. The intraparticle radial concentration profiles here shown represent the complete image of a particle passing the feed point, with its inherent “spatial history” of its movement throughout sections 4 and 3. During this trajectory the particle surface concentration of fructose is zero at the end of section 3, then increases and goes through a maximum higher than the “equilibrium” value obtained in section 2 as it reaches the feed point. Therefore, it is easy to understand that the fructose maximum in the intraparticle radial profile reflects, with some attenuation due to intraparticle diffusion, the surface concentration experienced by that particle when travelling from Section 4 to the feed point.

6. Linear driving force approximations 6.1. Szukiewicz approach Szukiewicz (2000) developed an approximate model for diffusion and reaction in a porous catalyst analogous to the Glueckauf’s (LDF) formula, by applying the Laplace Carson transform to Eq. (15), and after performing some mathematical manipulations; for the case under study, this model can be represented by  jCpij  1 jCpij  (p + Ki ) − j nj jx j = {Npij 3 i (Cpij |=1 − Cpij ) nj + i (p + Kenz )Daj CpAj }, where i =

2i (i cosh(i ) − sinh(i ))

2i sinh(i ) − 3(i cosh(i ) − sinh(i ))

(p + Kenz )kr , the Thiele modulus. with i = Rp Dpei

(21)

P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

a

b 40

40 CB gPROMS

CA gPROMS

CC gPROMS

Concentration (g.l-1)

CA gPROMS

Concentration (g.l-1)

1035

30

20

10

CB gPROMS

CC gPROMS

30

20

10

0

0 0

0.2

0.4

0.6

1

0.8

0

0.2

0.4

0.6

0.8

1

Radial dimensionless length

Radial dimensionless length

c 40 Concentration (g.l-1)

CA gPROMS

CB gPROMS

CC gPROMS

30

20

10

0 0

0.2

0.4

0.6

0.8

1

Radial dimensionless length

Fig. 6. Transient radial concentration profiles in particle pores at feed inlet (a)  = 4; (b)  = 8 and (c) steady state obtained by dynamic simulation at  ± 75.

For the two strategies presented here: Kim (1989) (presented after Section 6.2) and that proposed by Szukiewicz (2000), the corrective coefficient is function of the Thiele modulus, in our case constant, since it is a pseudo-first-order reaction approximation, but in the “non-reactive zone” the “pure” LDF model as stated in Glueckauf (1955) must be used. Consequently, and to avoid the reaction corrective coefficient I in pure separation zones, an “if statement” was considered: if there is a reactant, the LDF formula is the reactive corrected; if not the pure Glueckauf approximation. The steady-state results obtained with this approximation are close to the ones calculated by the particle detailed model, shown in Fig. 7. As can be observed from Fig. 7 the results obtained for both models, the detailed particle model and the LDF-type approximation are similar except for the fructose concentration profile in the feed zone, where the reaction takes place. It has to be taken into account that the approximation stated by the Szukiewicz approach is for a pure reactive system in a fixed bed; in this case the presence of a moving bed system as a reactive and separative process is a possible explanation for those types of discrepancies.

Fig. 7. Results from the Szukiewicz diffusion and reaction in a porous catalyst approximation model, and particle detailed model.

6.2. Kim’ s approach A similar approximation is introduced by Kim (1989), using the exact solution to Eq. (15) illustrated in Ramkrishna and

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P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

Table 5 Coefficients to Kim “LDF” formulas



s1 /s2

s2 /s3

1 − (s22 /s1 s3 )

0 1 2 5 10 20 50 100

15 16.88 22.70 66.56 225.0 844.4 5104.0 20202.0

10.5 11.66 15.23 42.56 145.5 553.8 3380.0 13425.0

0.3000 0.3092 0.3291 0.3606 0.3535 0.3441 0.3377 0.3355

where the coefficients, function of the Thiele modulus, were obtained by the calculation of the truncated series terms, shown in the Table 5. Considering this intraparticle reaction “LDF” approximation, only in the TMBAR reactive zones, as stated before, the results for the steady state are as shown in Fig. 9: It can be observed that with the Kim model in the reactive (plus separation) zone, the results are not so close to the detailed particle as with Szukiewicz approximation mainly in the fructose concentration profile in section 3 (Fig. 8). 7. Model without reaction The dimensionless mass balance equations to be solved for the equivalent TMB (with fructose and glucose) are for the particle:  jCpij 1 jCpij − (p + Ki ) j nj jx    j jCpij 1 j = Npij 2 2 (23a) nj  j j or, in the matrix form (radial discretization) see Appendix A,  ij  ja (Ki + p )[HMAT] j =

Or the complete formula,  jCpij  1 jCpij  (p + Ki ) − j nj jx  j s2 = Npij (Cpij |=1 − Cpij ) s3 nj + i (p + Kenz )Daj CpAj     s22 jCpij  + 1− , s1 s3 j =1

nj

{Npij ([CMAT][a ij ] + [CVEC]Cbij )}

  ij   jCbij 1 ja + (Ki + p ) [HMAT] + [CVEC] nj jx jx

Fig. 8. Results from the Kim diffusion and reaction in a porous catalyst approximation model and particle detailed model.

Amundson (1985), for the mass balance to a unsteady-state diffusion, adsorption, and first-order reaction in a porous spherical catalyst particle, using the volume-averaged fluid-phase concentration, integrating by parts, expanding the integral and then truncating the terms with higher-order derivative to obtain a LDF-type form as follows:  jCpij  1 jCpij  (p + Ki ) − j nj jx  j s1 = Npij (Cpij |=1 − Cpij ) nj s2 + i (p + Kenz )Daj CpAj  . (22a)

j

− (Ki + p )[HVEC] and j jCbij = j nj − or j jCbij = j nj −

(22b)



jCbij j

(23b)

1 j2 Cbij 1 jCbij − 2 Pe jx nj jx



(1 − b ) Nf ij (Cbij − Cpij |=1 ) b 

1 j2 Cbij 1 jCbij − 2 Pe jx nj jx

(24a)



(1 − b ) ij Nf ij (Cbij − a2NE ) b

(24b)

for the bulk liquid phase. Setting the time derivatives to zero, leads to the steady-state model. The steady-state internal concentration profiles for fructose and glucose are shown in Fig. 9, for feed concentration of 40 g l−1 in each species. A small difference in the concentration profiles for fructose in section 3 and for glucose in section 4, between the results obtained with the detailed particle model and the LDF approach, can be observed. Similar results were noticed in the TMBAR.

P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

Notation

35 30 Concentration (g.l-1)

1037

CB particle detailed

a ij Atl Btl

CC particle detailed

25

CB LDF CC LDF

20 15

Bi m Cb Cp

10 5 0 0

1

2

3

4

5

6

7

8

9

10

11

12

Columns Fig. 9. Steady-state profiles for fructose and glucose in a separative TMB: detailed model versus LDF approximation.

This is eventually related with the influence of the LDF mass transfer coefficients approximations on the net species flow, since it only occurs when the net species flow is in counter current with the fluid movement, i.e., moving downwards (with the “solid stream”), and still not in equilibrium.

Cs CF Cp  dc Db Dm Dpe Da er E , er R [Es ]

8. Conclusions A new TMBAR model was presented, with a detailed particle model, considering not only the reaction in the outer fluid phase but also inside the adsorbent particle, accounting for intraparticle radial profiles and outer fluid mass transfer resistances. The system analyzed was the sucrose inversion into glucose and fructose and subsequent separation considering linear adsorption isotherms. Model equations were numerically solved by two different packages, the COLNEW public domain for the steady-state solution and the commercial gPROMS for the transient as well as steady-state solutions. The detailed particle model was also compared with two different LDF-type approximations for reactive systems, from Kim (1989) and Szukiewicz (2000) in the reactive plus separative zones and LDF as Glueckauf (1955) in the pure separation zones, with reasonable agreement. Some of the model parameters and the operating conditions used on the TMBAR system simulation were estimated under the same constraints as for the TMB system. Is has been shown that the performance parameters, purity and recovery, decrease when particle radius increases while the sucrose inversion is slightly affected. The reaction/separation region has a triangular shape similar to those found for the TMB separation equilibrium theory. The effect of the reaction rate constant, kr , on the reaction/separation region was also studied. The triangleshaped separation/reaction becomes smaller as the reaction rate coefficient decreases. The application of the detailed particle model to a simple TMB separation of fructose and glucose was also studied and compared with the LDF approximation as stated by Glueckauf presenting again quite proximal results.

[E0 ] hk Htl kf kp kr krM K Kenz KM Lc Lj nj Nf Np NE P Pe Q r Rp Re Rec Sc Sh t tj tc

basis function coefficients cubic Hermite polynomials (first derivative) cubic Hermite polynomials (second derivative) Biot number bulk fluid phase concentration, mol m−3 pores adsorbent particle fluid-phase concen−3 tration, mol mfluid in pores particle “adsorbed phase” concentration, −3 mol mparticle feed concentration, mol m−3 average pores particle adsorbent fluid-phase −3 concentration, mol mfluid in pores Column diameter, m axial dispersion coefficient, m2 s−1 molecular diffusion coefficient m2 s−1 effective pore diffusion coefficient, m2 s−1 Damköhler number relative error of average concentration of the two components in the extract and raffinate, respectively, for two successive iterations, % adsorbed phase enzyme concentration, mol m−3 fluid-phase enzyme concentration, mol m−3 size of the finite intervalk cubic Hermite polynomials film mass transfer coefficient, m s−1 mass transport coefficient in particle pores, s−1 first-order reaction approximation constant, s−1 Michaelis–Menten reaction rate constant, s−1 constant from linear isotherm equivalence factor between liquid and “adsorbed phase” reaction Michaelis–Menten constant, mol m−3 column axial length, m section axial length, m number of columns per section number of film mass transfer units number of intraparticle mass transfer units number of radial finite elements purity % Peclet number fluid/solid flowrate, m3 s−1 radial particle coordinate, m particle radius, m Reynolds number recovery, % Schmidt number Sherwood number time variable, s bulk fluid space time, s solid space time (switch time), s

1038

T uj us u0 VM x X z

P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

temperature, K interstitial fluid velocity in a TMBAR section, m s−1 solid counter-current velocity, m s−1 mean TMBAR fluid velocity, m s−1 molar volume at normal boiling point, m3 mol−1 dimensionless axial column coordinate conversion, % axialcolumn coordinate, m

Greek letters   b p s  i i  f

 

safety margin parameter ration between fluid and solid interstitial velocities bed porosity particle porosity solvent viscosity, kg(m s)−1 dimensionless time coordinate fluid viscosity, kg(m s)−1 stoichiometric coefficient dimensionless radial particle coordinate fluid density, kg m−3 tortuosity Thiele modulus collocation point in each k

Appendix A Radial discretization step for COLNEW solution The solution Cpij in the kth subinterval and tth collocation point of  was approximated by Cpij (x, t , ) =

4 

sucrose bulk fructose glucose eluent stream extract stream feed stream chemical species TMBAR section particle raffinate stream solid

Abbreviations AE DAE LDF OCFEM ODE PDAE PDE SMB SMBAR TMB TMBAR

algebraic equations differential-algebraic equations linear driving force orthogonal collocation in finite elements method ordinary differential equations partial differential algebraic equations partial differential equations simulated moving bed simulated moving bed adsorber reactor true moving bed true moving bed adsorber reactor

(A.1)

l=1

where t are the collocation points within each subinterval of , with t = 1, 2. The cubic Hermite polynomials Htl (t ), with l = 1, . . . , 4, were defined over [k , k+1 ], as in Finlayson (1980) as Atl (t ) and Btl (t ), the first and second derivatives for each element k, obtained by differentiating (18) with reij spect to t . The al+2k−2 , with k = 1, . . . , NE are the basis function coefficients to be determined. After the discretization process, a system of (3 × 4) · (2 × NE) PDE is then obtained in the new basis function coefficients a ij ’s dependent on the onespacedimension, the dimensionless column axial coordinate x. The system resulting from the above discretization step, in conjunction with the mass balance to the bulk liquid phase, with (3 × 4(1 + 2 × NE)) equations, was then numerically integrated with COLNEW package. For the radial discretization, considering, • for k = 1 and t = 1, 2, for each species i = A, B, C and each section j = 1, 2, 3, 4

Indexes A b B C E EXT F i j p RFF s

ij

al+2k−2 (x, )Htl ,

1 − (p + Ki ) nj

 ja ja Ht1 1 + Htl l jx jx 4

ij

ij



l=3

 1 2 ij Npij Bt1 + At1 a1 = 2 2 nj h1  t h1    4  1 2 ij + B + A al 2 tl 2 tl h  h t l l l=3 j







 + i Daj (p +Kenz )

ij Ht1 a1 +

4 

 ij Htl al

(A.2a)

l=3

• for k = 2, . . . , NE − 1 and t = 1, 2, for each species i = A, B, C and each section j = 1, 2, 3, 4 − (p + Ki )

ij 4 ja 1  Htl l+2k−3 jx nj l=1

=

j nj

 Npij

4  l=1



 1 2 ij Btl + Atl al+2k−3 (t hk +k )hk h2k

+ i Daj (p + Kenz )

4  l=1

 ij Htl al+2k−3

(A.2b)

P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

1039

Fig. A1. Strategy used to obtain the numerical solutions of the PDE system in the model by using the COLNEW package.

• for k = NE and t = 1, 2, for each species i = A, B, C and each section j = 1, 2, 3, 4  2 ij  ja 1 − (p + Ki ) Htl l+2NE−3 nj jx l=1

ij

ja jCbij (x, ) + (Ht3 − Bi m Ht4 ) 2NE + Bi m Ht4 jx jx =

+

j nj



 Npij

2  l=1

2



1 h2NE



1

2 

+ Bi m

1 h2NE

Bt4 +

(At3 − Bi m At4 ) a2NE 2

(NE hNE + NE )



×

+ [CVEC]Cbij )} −

ij

(NE hNE + NE ) hNE

1 At4 Cbij (x, ) hNE

+ i Daj (p + Kenz )

ij

Htl al+2NE−3

l=1 ij

+ (Ht3 − Bi m Ht4 )a2NE  + Bi m Ht4 Cbij (x, )

,

dCbij 1 (p + Ki )[HVEC] nj dx

(A.4)

with the corresponding boundary conditions (13) and (17a), plus the nodes equations (14). The matrices HMAT and CMAT, and the vectors HVEC and CVEC have a sparse structure. Considering NE = 4 and i = 1 (A), the structure of HMAT and HVEC can be defined as in Fig. A1. The elements not shown are zero. The others matrices and vectors have similar structure. Appendix B Radial discretization element length analysis for the steady-state COLNEW solution



 2 

with

+ i Daj (p + Kenz )([HMAT][a ij ]

ij al+2NE−3



1

(A.3)

 ij  da 1 (p + Ki )[HMAT] nj dx j = − {Npij ([CMAT][a ij ] + [CVEC]Cbij ) nj

Btl

Atl (NE hNE + NE ) hNE  1 + (Bt3 − Bi m Bt4 ) 2 hNE +



applied, leading to the following ODE system:  dCbij d2 Cbij = Pe − i Daj CbAj 2 dx dx  (1 − p ) ij + Nf ij (Cbij − a2NE ) , p

(A.2c)

where hk = k+1 − k is the size of the finite element k and t = ( − k )/ hk , the two known collocation points considered in each k finite element. For the steady-state model the time derivatives on Eqs. (11) and (15) are set to zero and the radial discretization is then

The number (NE) and positions (l) of radial discretization finite elements was studied using the steady-state model with the COLNEW solver procedure. The model parameters and operating conditions used in all simulations are the same as described in Table 3 and were kept constant in the simulations, only the radial discretization was changed as presented in Table B1 . Several situations were considered to reach the TMBAR steady-state solutions, the run time was obtained for 3, 4 and 5 collocation points per finite element in the radial position, , respectively. For all the simulation performed, the global balance the global error er used was equal or less than 0.0095 (Leão and Rodrigues, 2004). This value permits the verification of the global balance (balance between the feed and each of the separation products). However, in several situations this value was not reached and then,

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P. Sá Gomes et al. / Chemical Engineering Science 62 (2007) 1026 – 1041

Table B1 Parameters used in the radial discretization analysis for the COLNEW steady-state procedure NE

(l)

No. ODE

Running time (min)

PEXT (%)

PRFF (%)

er B

er C

2 2 2 2 2 3 3 4 4 4

0; 0.50; 1 0; 0.79; 1d 0; 0.90; 1 0; 0.95; 1 0; 0.99; 1 0; 0.69; 0.87; 0; 0.79; 0.95; 0; 0.63; 0.79; 0; 0.63; 0.79; 0; 0.79; 0.87;

15 15 15 15 15 21 21 27 27 27

–, –, – –, –, – –, –, – 3, 3, 3 2,12,19 –, –, – 20,13,13 –, –, – –, –, – –, –, –

100a 100a 99.9b 99.9c 99.9c 100b 100c 99.9b 99.9b 99.9b

99.9a 100a 100b 100c 100c 100b 100c 100b 100b 100b

0.1800 0.0870 0.0310 0.0093 0.0003 0.0290 0.0086 0.0230 0.0230 0.0230

0.1300 0.0170 0.0060 0.0018 0.0002 0.0002 0.0009 0.0007 0.0007 0.0007

1d 1 0.91; 1d 0.95; 1 0.95; 1

The run time is referred for 3, 4 and 5 collocation points per finite element in radial position , (–) indicates that the stop criteria error was not verified after the maximum number of iteration, kmax , was reached. a er > 0.0095 (global balance not closed). b er ≈ 0.0095. c er < 0.0095. d Values obtained by using the isovolumetric discretization formula B1.

a

formula, used for different value of number of interior points l, and for spherical geometries, was as 

b c

(l) =

l NE

1/3 ,

l = 1, 2, . . . , (NE − 1).

(B.1)

Fig. 1 shows the steady-state concentration profiles obtained under the conditions described in Table 1, and with NE = 2 ((l) = 0; 0.95; 1) with four collocation points. References

Fig. B1. (a) Isovolumetric, (b) equidistant and (c) user-defined discretization of the radial spatial particles domain, , for NE = 3 with two collocation points in each subinterval.

the end criterion to be verified was the maximum number of iterations, 200 kmax . The symbol (–) is used to indicate these situations with the corresponding error value obtained. In some cases, the increase on the number of collocation points does not lead to better final results in terms of purities. The increase of the run time was not only due to the dimension of the ODE system to be solved but also due to the need of more iterations to reach the specified error, defined by the user. For example, for NE = 3 and with no isovolumetric discretization points (0; 0.79; 0.95; 1), with two collocation points the run time needed was higher than the one required for the case with four and five collocation points, since less iterations were needed to obtain the stop criteria error. With this table we can conclude that, for this particular case study, the isovolumetric discretization of the particle spatial radius is not a good solution. The points must be near the left boundary, x = 1, see Fig. B1 and Table B1. The best results were obtained by considering two or three finite intervals with three or four collocation points in each finite element. The isovolumetric discretization of the radial spatial domain was done considering grids with identical volumes. The

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