Ind. Eng. Chem. Res. 1999, 38, 3519-3529
3519
Bilinear Driving Force Approximation in the Modeling of a Simulated Moving Bed Using Bidisperse Adsorbents Diana C. S. Azevedo† and Alı´rio E. Rodrigues* Laboratory of Separation and Reaction Engineering (LSRE), Faculty of Engineering, University of Porto, Rua dos Bragas, 4050-123 Porto, Portugal
The linear driving force (LDF) approximation has been extensively used to represent the intraparticle diffusion in adsorptive processes regardless of its real nature. This work introduces a bilinear driving force (bi-LDF) approximation to account for intraparticle diffusion and to obtain concentration profiles in the macropores and microparticles of the adsorbent. The bi-LDF approximation is applied to the simulation of glucose/fructose breakthrough curves in a fixed bed and to their separation in a simulated moving bed (SMB). Equivalence with the simple LDF model is well-established for both fixed-bed operation and the SMB mode. The flow rate constraints that define a region of separation (both product purities higher than 99.5%) for a SMB configuration are shown to be greatly affected by relatively small deviations in the macroand microparticle time constants, which illustrate the relevance of the proposed approximation. Introduction Most adsorbents of industrial use have a bidisperse structure composed of microparticles (crystals, for zeolites, or a polymer-based matrix, for resins) held together by means of a convenient binder. This agglomeration process to form commercial pellets generates an intricate network of pores. In general, these pores may be classified as micropores, those originated from the crystal/matrix porosity, and macropores, those originated by the agglomeration process. Several mechanisms of diffusion1 take place inside the adsorbent particles, depending on the relative sorbate/pore size and the surrounding sorbate partial pressure. In the modeling of adsorption processes, an accurate description of mass-transfer rates inside these adsorbent particles is often required. Accounting for all the mechanisms actually present in a given adsorption process requires extraordinary effort to solve the resulting equations. The simplest way to describe intraparticle mass-transfer rates is to consider the particle as a homogeneous solid in which the sorbate diffuses according to Fick’s First Law. If a parabolic concentration profile is assumed, the integration of the diffusion equation leads to the well-known linear driving force (LDF) approximation as proposed by Glueckauf.2 For linear equilibrium systems, the LDF approximation provides a good description of mass-transfer rates for the calculation of breakthrough curves in adsorption fixed beds.3 However, Nakao and Suzuki4 have shown that the conventional value of the rate constant k ()15Dpe/Rp2) is not appropriate for processes with rapid cycles, that is, those whose adsorption/desorption (halfcycle) stage is shorter than 1.5/k. They have proposed to extend the LDF approximation to include the value of the cycle time in the calculation of k. The proposed k values were calculated by comparing the solutions from * To whom correspondence should be addressed. E-mail:
[email protected]. Phone +351 2 204 1669. Fax: +351 2 2041674. † On leave from GPSA, Department of Chemical Engineering, Federal University of Ceara´ (Brazil).
the LDF approximation and from a particle diffusion model. Then, a value of k was determined which would give the same amount adsorbed for the two models. Buzanowski and Yang5 proceeded in a similar way, but they determined values of k by matching the instantaneous adsorption rates, rather than the total amounts adsorbed. A complete review on the validity of the LDF approximation in sorption/desorption cycles has been presented by Rodrigues and Dias.6 These attempts have tried to model adsorption processes using a single characteristic time constant. For bidisperse adsorbents where both resistances, macropore and microparticle, are significant, this approach may not be sufficiently accurate. Some authors have attempted to account for both mass-transfer resistances present in composite adsorbents. Ruckenstein7 analyzed the competing effects of macropore and microparticle diffusion for an isolated resin particle subject to a step change in concentration at the surface (within Henry’s Law equilibrium range). Analytical solutions were shown for different macro/ micro diffusional resistances, and a comparison with experimental data showed good agreement with the proposed model. Ma and Lee8 extended this solution to the more complicated boundary condition of an adsorbent particle immersed in a well-mixed solution of finite volume. The analytical solution was obtained from inversion of the Laplace transforms and could be reduced to the previous case studied by Ruckenstein. Haynes and Sarma9 investigated the chromatographic response of a bed packed with a bidisperse adsorbent. The proposed mathematical model was not solved but rather analyzed in terms of the moments response. The relative contribution of macro- and micropore systems in the overall diffusion process were analyzed by examining the second moment or the variance of the peaks in a chromatogram. Cen and Yang10 proposed a strategy close to a double LDF approximation to model breakthrough curves in fixed beds for gas systems. Linear equilibrium was assumed; pore accumulation and film diffusion were neglected. Analytical solutions were presented and showed good agreement with experimental data. Villermaux11 also proposed a simple
10.1021/ie990115f CCC: $18.00 © 1999 American Chemical Society Published on Web 07/30/1999
3520 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999
model based on the so-called “Chemical Engineering Approach” to represent intraparticle diffusion in multiporosity adsorbents in linear chromatography. Macropore, micropore, and external mass-transfer resistances are all grouped in a single mass-transfer time constant. For the last 3 decades, the simulated moving-bed (SMB) configuration has gained considerable interest due to its superior performance in separating low separation factor mixtures with columns of low efficiency. Using a set of fixed beds connected in series, countercurrent motion of the solid adsorbent relative to the fluid mixture is simulated by advancing inlet and outlet streams at regular time intervals. In industrial plants, this is accomplished by means of a rotary valve, which is the basis of the SORBEX technology12 by UOP. Alternatively, a system of intelligent valves may be used to synchronize the periodic shifting of outlet and inlet streams. This technology has found a wide range of applications, from hydrocarbon13,14 to fine chemicals15-17 separations. A crucial aspect of SMB operation is defining the adequate flow rates which provide separation and maximize the adsorbent use. Therefore, a precise description of mass-transfer rates should be required for the optimization of a SMB unit. The present work uses a bilinear driving force approximation to describe intraparticle mass-transfer rates in bidisperse adsorbents. Models for the adsorptive separation of fructose and glucose are presented for a fixed bed and a simulated moving bed using the equivalence with a true moving bed (TMB). Linear equilibrium is assumed. Accumulation in the pores and film diffusion is taken into account. The competing effects of macropore and micropore resistances are examined for breakthrough curves obtained in a fixed bed and internal profiles obtained in SMB units. Equivalence with the simple LDF formulation is demonstrated. Last, the regions of separation (both product purities higher than 99.5%) for SMB units are calculated for different pairs of micropore/macropore time constants so as to analyze the influence of these parameters in the performance of such units. Linear Driving Force Approximation Homogeneous Particle. Consider an isolated homogeneous particle subject to a step change in concentration of an adsorbable species in the fluid phase. The rate of adsorption, in spherical coordinates, can be expressed as follows,
( )
∂qh 1 ∂ 2 ∂qh r ) Dh 2 ∂t ∂r r ∂r
(1)
where Dh is the particle homogeneous diffusion coefficient, r is the radial coordinate, and qh is the adsorbedphase concentration in the homogeneous solid expressed in moles adsorbed per kilogram of particles. Averaging eq 1 over the particle radius rp and using the linear driving force approximation, one obtains
d〈qh〉 ) kh(q/sh - 〈qh〉) dt
(2)
where 〈qh〉 is the mean solid concentration averaged over the particle volume, kh is the homogeneous diffusion rate constant ()15Dh/rp2), and q/sh is the concentration at the surface of the particle in equilibrium with the
Figure 1. Representation of a bidisperse pellet of radius Rp containing microparticles with radius rp (a) and the equivalence between a homogeneous and a bidisperse solid (b).
external-phase concentration. This is the well-known linear driving force approximation which has been proved by Glueckauf 2 to provide an excellent description of mass-transfer rates in linear chromatography. Bidisperse Adsorbent. Let us now suppose that the particle we have just considered is a microparticle and there are thousands of other particles similar to it agglomerated so as to form a pellet. This representation is pictured in Figure 1 a. Each microparticle has its own mean concentration 〈qh〉, which depends on the concentration at the surface. For the sake of clarity, this concentration will be expressed as 〈q〉 from this point on. If there is a concentration profile throughout the pellet, each microparticle may have a different 〈q〉 from one another. An average calculated from the mean concentration 〈q〉 of the microparticles present in a pellet can be calculated from eq 2 by averaging it over the pellet volume.
d〈〈q〉〉 ) kµ(〈qs〉 - 〈〈q〉〉) dt
(3)
The rate constant kh was replaced by kµ ()15Dc/rp2) to denote that it is the microparticle (crystal) diffusion rate constant. The variables 〈〈q〉〉 and 〈qs〉 represent the mean microparticle concentration and the microparticle surface concentration, both averaged over the pellet volume, respectively. Let us now write a mass balance over a volume element of a pellet, such as the one shown in Figure 1, subject to a concentration CS at the pellet surface. Taking Cp as the pore concentration, we have
p
(
)
∂Cp ∂Cp ∂〈q〉 1 ∂ + Fp ) Dpe 2 R2 ∂t ∂t ∂R ∂R R
(4)
Averaging eq 4 over the pellet volume (radius ) Rp) and using the LDF approximation,
p
d〈Cp〉 d〈〈q〉〉 + Fp ) kp(CS - 〈Cp〉) dt dt
(5)
where CS is the concentration at the surface of the pellet, 〈〈q〉〉 and 〈Cp〉 are the mean microparticle concentration and pore concentration, respectively, averaged over the pellet volume, Fp is the pellet apparent density, and kp ()15Dpe/Rp2) is the macropore diffusion rate constant.
Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3521
The left-hand term in eq 5 is actually an overall particle concentration which takes into account not only the solute present in the pores but also that present in the microparticles (taken as a homogeneous solid). This concentration, expressed as 〈Cm〉 and defined in eq 6, will be used to establish the equivalence between the LDF approximation, as described before for a homogeneous solid, and the bi-LDF approximation, as described in this work for bidisperse solids, i.e.,
〈Cm〉 ) p〈Cp〉 + Fp〈〈q〉〉
(6)
Equations 3 and 5 are the basis of the bilinear driving force approximation which leads to the calculation of mean concentrations in the pores and microparticles independently on pellet and microparticle radial coordinates. A numerical solution of such systems is perfectly feasible and does not demand extraordinary computational effort. To solve eqs 3 and 5, the isotherm equilibrium relation must be provided. For a linear equilibrium system, such as fructose and glucose, the bi-LDF equations are eqs 5 and 7 as follows:
(
)
d〈〈q〉〉 K ) kµ 〈Cp〉 - 〈〈q〉〉 dt Fp
)
p d2〈〈q〉〉 p + K 1 d〈〈q〉〉 K + + + 〈〈q〉〉 ) CS kpkµ dt2 kp kµ dt Fp
(8)
Rewriting eq 8 in terms of a corrected kp′ ()kp/p), a dimensionless equilibrium constant K′ ()K × p) and qS ()(K/Fp) × CS):
(
)
2 1 + K′ 1 d 〈〈q〉〉 1 d〈〈q〉〉 + + + 〈〈q〉〉 ) qS 2 kp′kµ dt kp′ kµ dt
〈Cm〉 ) p〈Cp〉 + Fp 〈〈q〉〉 ) Fph〈qh〉
(12)
In equilibrium, the following is true,
〈Cp〉 )CS 〈〈q〉〉 ) 〈q〉 )
(13)
K K 〈C 〉 ) CS Fp p Fp
(14)
Fph〈qh〉 ) KhCS
(15)
Kh ) K + p
(16)
so that
(7)
since the equilibrium adsorption isotherm is Fpqs ) KCp and, therefore, Fp〈qs〉 ) K〈Cp〉. Solving eq 7 for 〈Cp〉 and substituting its value in eq 5, the bi-LDF equations are reduced to the following:
(
adsorption equilibrium occurs on the surface of the microparticles. In fact, for model equivalence purposes, we should compare 〈Cm〉, as defined for the bi-LDF approximation in eq 6, and 〈qh〉, as defined for the LDF approximation in eq 2. This can be more clearly seen in Figure 1b, where a homogeneous particle and a bidisperse particle having the same diameter are shown. For them to have the same adsorption capacity, 〈Cm〉 must be the same as Fph〈qh〉 when equilibrium is reached. Therefore,
(9)
where Kh is the equilibrium constant as measured for a homogeneous particle (simple LDF model) and K is the corrected equilibrium constant considering a bidisperse adsorbent. With equilibrium constants corrected, the equivalence between the simple LDF approximation and the bi-LDF approximation needs to be established with respect to the kinetics of mass transfer. This is more clearly visualized in the Laplace domain. The Laplace transforms of eqs 3 and 5 may be grouped into a single equation in terms of the overall particle concentration 〈Cm〉 as follows,
p s + p + K kµ 〈C h m〉 ) p + K C hS p 2 1 s + + s+1 kpkµ kµ kp
(
)
(17)
Convenient manipulation of eq 9 allows the study of the limiting cases, that is, those where diffusion is controlled either by the macropores or by the microparticles (or micropores). Macropore Controlled Diffusion. On one hand, if macropore diffusion is the limiting step in an adsorption process, kµ . 1 and eq 9 is reduced to
where the dashes over the variables 〈Cm〉 and CS denote their Laplace transforms. The transfer function for the LDF approximation as described in eq 2 is
kp′ d〈〈q〉〉 ) (q - 〈〈q〉〉) dt 1 + K′ S
Kh j h〉 Fph〈q ) C hS (1/kh)s + 1
(10)
Microparticle (or Micropore) Controlled Diffusion. On the other hand, if microparticle diffusion resistance is much stronger than macropore resistance, kp . 1 and eq 9 is reduced to
d〈〈q〉〉 ) kµ(qS - 〈〈q〉〉) dt
(11)
In both cases, an equivalent form of the simple LDF approximation is obtained. Nevertheless, the term 〈〈q〉〉 present in the bi-LDF equations is different from the term 〈qh〉 present in the simple LDF approximation (see eq 2). From the LDF approximation perspective, for homogeneous particles, adsorption equilibrum occurs on the pellet surface, whereas for the bi-LDF approach,
(18)
with Fphq/sh ) KhCS. The dashes above 〈qh〉 and CS indicate their Laplace transforms. It is clear that the simple LDF model leads to a first-order system whereas the bi-LDF model is a more complex system. If the same input CS(t) is applied to both systems, they will only respond the same way either if kµ f ∞ or for low “s” values. In these situations, eq 17 is reduced to a firstorder system. When kµf ∞, mass-transfer inside the particle is governed by diffusion in the macropores and the equivalence between the two models is verified as follows:
kh )
kp K + p
(19)
3522 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999
On the other hand, when diffusion in the microparticles is the limiting step in intraparticle mass transfer, that is, kpf ∞, eq 17 is not reduced to a first-order system, since the s-dependent term, (p/kµ)s, remains in the numerator of the transfer function. Equivalence between the two models is not always possible. The remaining equation may behave as a first-order system,11 but only for small “s” values. The steady-state gain is then Kh ) K + p and the time constant is τ ) K/[(p + K)kµ] ≡ 1/kh (see Table 1). According to Ruthven,1 the value of a constant δ, which depends on the macropore and micropore diffusion time constants, indicates the governing diffusion resistance for linear systems.
δ)
kµ(1 + K′) kµ(K + p) ) k′p kp
Figure 2. Representation of a true moving bed. Table 1. Rate Constants Equivalence between LDF and Limiting Cases of bi-LDF Approximations
(20)
If δ < 0.1, mass transfer is governed by microporecontrolled diffusion. If δ > 10, mass transfer is governed by macropore-controlled diffusion.
equivalent kh for the LDF approximation (homogeneous particles)
Rp )
1 ∂2C ∂C (1 - ) Bim ∂C ) R (C - 〈Cp〉) (21) ∂θ Pe ∂x2 ∂x 5 + Bim p
Rµ )
[
]
Rµ K ∂〈Cp〉 Rp Bim ) (C - 〈Cp〉) - Fp 〈Cp〉 - 〈〈q〉〉 ∂θ p 5 + Bim p Fp (22) Microparticle mass balance:
]
∂〈〈q〉〉 K ) Rµ 〈Cp〉 - 〈〈q〉〉 ∂θ Fp
(27)
kµLc UF
(28)
Bim ) Rp )
kfRp Dpe
(29)
Peclet number: Pe )
UFLc Dax
(30)
Modeling of a Simulated Moving Bed
(23)
Boundary conditions: at x ) 0, (24)
at x ) 1, (25)
Initial conditions: at θ ) 0, C(x,0) ) C0(x)
(26a)
〈Cp〉(x,0) ) 〈Cp〉 (x)
(26b)
〈〈q〉〉(x,0) ) 〈〈q〉〉0(x)
(26c)
0
kpLc UF
Mass Biot number:
Intraparticle fluid mass balance:
∂C (0,θ) ) 0 ∂x
kµ(K + p) K
Number of microparticle mass-transfer units:
Fluid-phase mass balance in a bed volume element:
1 ∂C Pe ∂x
kp K + p
Number of macropore mass-transfer units:
The bi-LDF approximation concept was first applied in the modeling of a fixed bed based on the following assumptions of linear equilibrium, film diffusion, axial dispersion for fluid flow, significant pore accumulation, and isothermal operation. The model equations for each of the species involved include
Cin ) C(0,θ) -
microparticle controlled diffusion
The parameters present in eqs 21-23 are
Fixed-Bed Modeling with bi-LDF Approximation
[
macropore controlled diffusion
The dimensionless variables for space and time present in the above equations are x ) Z/Lc and θ ) tUF/Lc, respectively.
A model based on the analogous true moving-bed (TMB) unit was proposed to predict the performance of a simulated moving-bed (SMB) unit. A true moving-bed unit consists of the countercurrent motion of an adsorbent solid relative to the fluid phase with fixed inlet (feed and eluent) and withdrawal (extract and raffinate) ports along the bed as shown in Figure 2. The simulated moving bed is the physical implementation of a TMB. The solid remains fixed in a finite number of beds and its countercurrent movement is simulated by advancing the inlet and outlet ports in the direction of the fluid flow at regular time intervals. The steady state is only reached in a periodic sense for the SMB, whereas there are unique fixed values for product purities for the TMB. In a SMB, product concentrations vary in a cycle, but the average concentration is approximately the same as that obtained from a TMB operating in equivalent conditions. The equivalence between SMB and TMB is summarized in Table 2. The model and remarks that follow apply to a true countercurrent bed. It has already been demonstrated19 that the average product concentrations as well as the internal profiles obtained using TMB and SMB approaches are about the same if more than one column
Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3523
To solve the model, it is necessary to provide the node balance equations which contain the information Cin present in the boundary condition given by eq 34.
Table 2. Equivalence between the SMB and TMB Systemsa SMB
a
TMB
velocity flow rate
Solid Phase 0 0
velocity flow rate
Liquid Phase U*Fj Q*j
Eluent node:
Us ) Lc/t* Qs ) Us(1- )A
Cin )
UFj ) U*Fj - Us Qj ) Q*j - Vc/t*
Q4 C (1,θ) Q4 + QE 4
(39)
Extract node:
Source: Lea˜o et al.18
Cin ) C1(1,θ)
(40)
Q2 QF C2(1,θ) + C Q3 Q3 F
(41)
Fluid-phase mass balance in a countercurrent bed volume element:
Cin ) C3(1,θ)
(42)
2 ∂Cj 1 ∂ Cj ∂Cj ) ∂θ Pej ∂x2 ∂x (1-) Bimj - Rpj(Cj - 〈Cp〉j) (31) 5 + Bimj
The design of a simulated moving bed and optimization of the operating conditions is not a straightforward task. Each of the four zones of a SMB have to perform a specific role so that good separation can be achieved. If a binary mixture of fructose (more strongly adsorbed component) and glucose (less adsorbed component) is fed into a SMB, separation will only occur if certain conditions are met. Next, these conditions will be stated, having in mind the analogy with the true moving bed (see Figure 2). Zone 1 is the adsorbent regeneration zone. The more retained component, fructose, must be displaced by the eluent to the fluid phase. Therefore, the net flow of fructose must be that of the fluid phase. In zone 2, between the extract and feed nodes, desorption of the less retained component, glucose, takes places. The net flow of glucose must be that of the fluid phase. Zone 3 is where the more retained component is adsorbed. This means that fructose must move in the same direction as the solid. In zone 4, the eluent is regenerated and the less retained component is adsorbed. The net flow of glucose is that of the solid phase in this zone. These flow constraints may be expressed mathematically as follows:
per section is used in the SMB. The TMB configuration gives rise to four distinct mass-transfer zones, each one having a very specific and important role in the intended separation. Each of these four zones is a countercurrent bed which may be modeled for each of the species involved according to the mass-balance equations:
[
]
Intraparticle fluid-phase mass balance: Bimj Rpj ∂〈Cp〉j 1 ∂〈Cp〉j ) + (C - 〈Cp〉j) ∂θ γj ∂x 5 + Bimj p j Rµj K F 〈C 〉 - 〈〈q〉〉j (32) p p Fp p j
[
]
Microparticle mass balance:
[
]
∂〈〈q〉〉j 1 ∂〈〈q〉〉j K ) + Rµj 〈C 〉 - 〈〈q〉〉j ∂θ γj ∂x Fp p j
(33)
Boundary conditions: at x ) 0, Cin j ) Cj(0,θ) -
1 ∂Cj Pej ∂x
(34)
Feed node: Cin ) Raffinate node:
at x ) 1,
Q1CFR,1
∂Cj (1,θ) ) 0 ∂x
(35)
〈Cp〉j(1,θ) ) 〈Cp〉j+1(0,θ)
(36)
Q2CGL,2
〈〈q〉〉j(1,θ) ) 〈〈q〉〉j+1(0,θ)
(37)
QS〈Cm〉GL,2
Initial conditions:
Q3CGL,3
at θ ) 0, Cj(x,0) )
C0j (x)
QS〈Cm〉FR,1
(38a)
〈Cp〉j(x,0) ) 〈Cp〉 0j (x)
(38b)
〈〈q〉〉j(x,0) ) 〈〈q〉〉 0j (x)
(38c)
The dimensionless variables are x ) z/Lj and θ ) tUF/ Lj. The parameters found in the mass-balance equations are the same as those defined in eqs 27-30 with the column length Lc being replaced by the section length Lj. Additionally, γj ) UFj/US.
QS〈Cm〉GL,3
> 1;
> 1;
>1
Q2CFR,2 QS〈Cm〉FR,2 Q3CFR,3 QS〈Cm〉FR,3
Q4CGL,4 QS〈Cm〉GL,4
