Modeling of batch and semi-batch membrane filtration processes

Journal of Membrane Science 327 (2009) 164–173

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Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci

Modeling of batch and semi-batch membrane filtration processes Z. Kovács a,∗ , M. Discacciati b , W. Samhaber a a b

Institute of Process Engineering, Johannes Kepler University Linz, Welser St. 42, A-4060 Leonding, Austria IACS - Chair of Modeling and Scientific Computing (CMCS), Ecole Polytechinque Fédérale de Lausanne, CH-1015, Lausanne, Switzerland

a r t i c l e

i n f o

Article history: Received 3 September 2008 Received in revised form 7 November 2008 Accepted 10 November 2008 Available online 21 November 2008 Keywords: Diafiltration Nanofiltration Batch Irreversible thermodynamics Extended Nernst–Planck equation

a b s t r a c t A mathematical frame for modeling batch and semi-batch membrane filtration processes is provided. The approach followed in this work uses the feed concentrations as a basis for the calculations, rather than the concentration factor. A practical computational algorithm is proposed. Our method hands separately the design equations describing the engineering aspects of batch and semi-batch systems and the models of mass transfer through the membrane. Thus, different methods can be applied to compute the permeate flux and rejection without having to modify the general framework. In particular, we present an empirical approach to characterize the membrane separation behavior based on a minimal number of experiments. Moreover, we consider irreversible thermodynamics models and a transport model based on the extended Nernst–Planck equations. Finally, various batch and semi-batch nanofiltration operations are carried out with an organic/electrolyte binary test solution to validate the proposed algorithms. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Batch membrane filtration has a considerable industrial interest with many applications in the food and beverage, chemical, biotechnological and pharmaceutical industry. In comparison with continuous processes, batch operations are particularly suited to small-scale operations, require less expensive automatic controls, and allow to use membranes with reduced area in order to reach the target [1]. Improving batch operation performances is an active field of scientific research. Among the various batch design modes, the concentration mode and the constant-volume dilution mode (or the combination of the two which is usually referred as diafiltration) have been studied in detail by many authors [2–11]. Most recently, attention has been paid to variable volume dilution mode, as an alternative way for macrosolute concentration and simultaneous microsolute removal [12–14]. Assuming constant solute rejection coefficients, Foley developed a mathematical model and compared the variable volume dilution with the traditional twostep diafiltration process in terms of water usage and operational time [12]. Much less literature is available on semi-batch process performances. Research has been conducted on the recovery of used cleaning-in-place solutions in the dairy industry [15], immersed membrane systems for removing suspended solids [16], the treatment of waste oil/water emulsions [17], and on the performance

∗ Corresponding author. Tel.: +43 706725090. E-mail addresses: [email protected] (Z. Kovács), marco.discacciati@epfl.ch (M. Discacciati). 0376-7388/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2008.11.024

of semi-batch filtration system for treating waste metal-working fluids [18]. A very common approach in mathematical modeling is to assume constant rejection coefficients and to use the concentration factor as a basis for the calculations. The rejection coefficients are usually empirically determined from permeation experiments with the process liquor, and a mean rejection value characterizing a whole process step can be calculated. In our previous study [19], it was shown that this approach can only be applied when the rejection rates of the solutes remain constant throughout the process. In fact, when the rejections of the solutes strongly vary depending on their feed concentrations and there is a considerable interdependence in their permeation, the simulation and optimization might lead to inaccurate results. In order to express the rejections as state functions of the actual feed concentrations and incorporate them in unit operation design, numerical techniques have to be applied. Major achievements have been obtained by Cross [20] and Driscoll [21]. The approach followed in their work did not use the concentration factor as a basis for the calculations, but rather the component concentrations. They developed a simulation tool capable of determining the membrane area and the optimal area distribution required for a multi-stage system. Besides continuous operation design, their software can be used to simulate concentration mode, constant-volume dilution mode and constant-volume type semi-batch operations. Transport mechanisms through membranes have also been studied thoroughly. Bowen and Mohammad [8] developed a predictive model based on the extended Nernst–Planck equations for the performance of dilution and concentration mode nanofiltration (NF) in separating dye/salt solution. Excellent agreements between

Z. Kovács et al. / Journal of Membrane Science 327 (2009) 164–173

predicted and experimental rejections were achieved assuming a linear relationship between the charge density of the membrane and the ionic strength of the salt solution. Similar methodologies for studying the diafiltration for the removal of pyruvate from process stream were applied by Bowen et al. [22]. The simulations are developed on the basis of a transport model and, obviously, non-constant rejections are considered. In our previous study [19], numerical methods have been provided for the modeling of concentration mode and constant-volume dilution mode operations, and an empirical technique has been presented to obtain the necessary model fitting parameters. The main aim of the present paper is to provide a general mathematical framework for modeling all types of batch and semi-batch membrane filtration operations. Thus, in Section 2.2, we propose a system of ordinary differential equations, while in Section 2.3 we illustrate a possible computational algorithm. The design equations describing the engineering aspects of batch and semi-batch systems are handled separately from the models of permeation through the membrane. Therefore, our approach is valid for all pressure-driven membrane filtration processes. The estimation strategies for flux and rejection are presented in Section 2.4. We discuss the minimum experimental design necessary for an empirical approach in Section 2.4.1, while we study two theoretical NF models in Section 2.4.2. Finally, we compare the predicted values with data obtained from NF permeation experiments with an organic/electrolyte binary solution.

165

Fig. 1. Schematic representation of batch membrane filtration settings.

2. Theory

The diluant can be pure solvent, fresh process liquor, or a solution containing low concentration of solutes. The latter case is important in numerous industrial applications, where diluted process-liquors are produced and used as diluants. When fresh process water is used as diluant, the process is considered as semibatch (or fed-batch) operation. (Notice that in this case no real washing-out effect of the microsolute occurs.) Thus, all the possible operational modes can be characterized by convenient values of Q, ˛, and the concentration cd,i of component i in the diluant as illustrated in Table 1.

2.1. Batch and semi-batch operations

2.2. Mathematical modeling

A schematic flow diagram of the batch and semi-batch filtration systems considered in this study is shown in Fig. 1. In all batch and semi-batch operations, the retentate stream is recirculated to the feed tank and the permeate stream is collected separately. The main difference between the various types of operational modes is due to the quality and the quantity of the diluant stream introduced in the feed tank during the operation. In this context, the simplest operational mode is the concentration mode since no diluant is applied. The diluant can be added to the feed tank with a constant flow rate Q supplied by an external pump. Alternatively, the flow rate of the diluant can be equal or proportional to the permeate flow rate: in the former case we speak of constant-volume operational mode, while in the latter of variablevolume operational mode. In general, the diluant stream D can be described as the sum of two terms

The permeate flow rate at any time t of the operation is given as the product of the permeate flux J(t) and of the membrane area A:

D = ˛JA + Q ,

(1)

where ˛ is a proportionality factor, J is the permeate flux, A is the membrane area, and Q is the flow rate of the diluant supplied by an external pump. One of the two terms in (1) can be set to zero to represent either a constant or a flux-dependent diluant flow rate.

dVp (t) = J(t)A. dt

(2)

The volume flow entering the feed tank can be expressed as the sum of the constant and the flux-dependent term as: dVp dVd (t) = Q + ˛ (t). dt dt

(3)

The change in the volume in the feed tank during the operation is given as: dVf dt

(t) =

dVp dVd (t) − (t). dt dt

(4)

Finally, assuming to consider two solutes, the mass balance for the solute concentrations yields: dVp dVd d V (t)cf,i (t) = − (t)cp,i (t) + (t)cd,i (t), dt f dt dt

i = 1, 2

(5)

Table 1 Design of commonly used batch and semi-batch operatons Operational mode

Q

˛

cd,i

Concentration mode Constant-volume dilution mode with pure solvent as diluant Variable-volume dilution mode with pure solvent as diluant Constant-volume dilution mode with diluted process liquor as diluant Variable-volume dilution mode with diluted process liquor as diluant Constant-volume fed-batch with process liquor Variable-volume fed-batch with process liquor Constant inlet-rate dilution with pure solvent as diluant Constant inlet-rate dilution with diluted process liquor as diluant Constant inlet-rate fed-batch with process liquor

0 0 0 0 0 0 0 Q Q Q

0 1 0<˛<1 1 0<˛<1 1 0<˛<1 0 0 0

0 0 0 0 < cd,i < cf,i 0 < cd,i < cf,i cf,i cf,i 0 0 < cd,i < cf,i cf,i

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where cp,i (t) denotes the permeate concentration of solute i at time t. Substituting (2) in (3) we obtain dVd (t) = Q + ˛J(t)A dt

(6)

while using (2) and (6) in (4) we get dVf dt

(t) = Q − (1 − ˛)J(t)A.

(7)

Eq. (5) can be rewritten in the following way: dVf dt

(t)cf,i (t)+Vf (t)

dcf,i dt

(t)= −

dVp dV (t)cp,i (t)+ d (t)cd,i (t), i = 1, 2 dt dt

and using (2), (6) and (7), we obtain, for i = 1, 2, Vf (t)

dcf,i dt

(t) = J(t)A[˛cd,i (t) − cp,i (t) + (1 − ˛)cf,i (t)] + Q (cd,i (t) − cf,i (t)).

Recalling that cp,i (t) = cf,i (t)(1 − Ri (t)), where Ri (t) is the rejection of solute i at time t, we can write Vf (t)

dcf,i dt

(t) = J(t)A[˛cd,i (t) + cf,i (t)(Ri (t) − ˛)] + Q (cd,i (t) − cf,i (t)).

dt

⎩ V (0) = V 0 , f f

(8)

and, for i = 1, 2,

⎧ dcf,i ⎪ ⎪ ⎨ Vf (t) dt (t) = J(t)A[˛cd,i (t) + cf,i (t)(Ri (t) − ˛)] + Q (cd,i (t) − cf,i (t)), ⎪ ⎪ ⎩ c (0) = c0 , f,i f,i

(9)

which describe the evolution in time of the volume in the feed tank 0 denote respectively Vf and of the feed concentration cf,i . Vf0 and cf,i the initial feed volume and the initial feed concentration of the solute i. Remark that problems (8) and (9) are valid for all batch and semi batch processes described in Section 2.1. (Many numerical software packages offer built-in functions for solving Eqs. (8) and (9).) Moreover, notice that the estimation of the flux J(t) and of the rejection Ri (t) can be carried out separately using the most convenient approach for the problem at hand. Possible strategies to compute flux and rejection are presented in Section 2.4. 2.3. Computational algorithm Let us split the interval of the operation time [0, tfinal ] in subintervals [tn , tn+1 ], n = 0, . . . , N − 1, with t0 = 0 and tN = tfinal . Let tn+1 = tn+1 − tn , and denote by the upper-index n an approximan ≈ c (t ). tion of the value of any quantity at time tn : e.g., cf,i f,i n The relevant quantities for batch and semi-batch operation modes can be computed using the following algorithm. (1) At time tn , calculate the flux J n and the rejection Rni (i = 1, 2) of the two solutes. (2) The volume of permeate Vpn obtained in the time interval tn+1 is: Vpn = J n A tn+1 .

(10)

(3) The diluant volume Vdn which enters in the feed tank in the time interval tn+1 is given by Vdn = Qtn+1 + ˛Vpn

Vfn+1 = Vfn − Vpn + Vdn

(12)

(5) Finally, the concentrations at time tn+1 can be computed as follows: n+1 n n n Vfn+1 cf,i = Vfn cf,i − Vpn cp,i + Vdn cd,i , i = 1, 2.

(13)

Notice that Eqs. (10)–(13) correspond to a discretization in time of (2)–(5) using the forward Euler method. Moreover, they provide approximate solutions to problems (8) and (9). To solve (12) and (13), the following input data are required: • initial volume in feed tank V 0 , f • initial concentration of macrosolute in feed tank: c 0 , f,1 • initial concentration of microsolute in feed tank: c 0 , f,2 • constant diluant inlet-rate Q, or ratio of permeate flow to diluant flow ˛, • concentration of macrosolute in diluant: cd,1 , • concentration of microsolute in diluant: cd,2 , • exit condition (operation time, feed volume, collected permeate volume, or applied diluant volume). 2.4. Permeate flux and rejection

Thus, we have the following initial-value problems:

⎧ ⎨ dVf (t) = Q − (1 − ˛)J(t)A,

(4) At time tn+1 the volume in the tank becomes:

(11)

The flux and the rejections depend on many independent variables such as feed concentrations, temperature, applied pressure, hydrodynamic conditions, and state of fouling. In Sections 2.2 and 2.3, we have not considered any specific model to express flux and rejection. However, any suitable empirical or theoretical method can be inserted in the framework that we have presented without having to change it. We provide two examples in the following sections. 2.4.1. Empirical approach Assuming that the applied pressure, the temperature, and the hydrodynamic conditions are kept constant, flux and rejection are functions only of the feed concentrations. The objective is to determine these relationships from a minimum number of experiments. Reliable relations can be obtained when experimental data are available in the whole range of cf,1 and cf,2 , or, at least, in the extreme points of the cf,1 -by-cf,2 matrix. An effective technique for this purpose is to concentrate the initial feed solution, then to add pure water to obtain the initial volume, and to repeat this procedure several times. The operational conditions of such experimental run are described in Section 3 and the results are discussed in Section 4.1. We assume that the samples are taken from the membrane plant whose operation is to be simulated. Scale-up calculations, where an increased membrane area and a corresponding change in the hydrodynamic conditions need to be taken into account, are out of the scope of this work. Such aspects are described in [21,23–26]. 2.4.2. Irreversible thermodynamics models The fundamental models derived from irreversible thermodynamics (IT) are the Kedem–Katchalsky (KK) and the Spiegler–Kedem (SK) models [27–29]. According to Kedem and Katchalsky, the volume flux J and the molar solute flux Js for a single solute through a membrane are given by: J = Lp P − Lp RT (cf − cp ),

(14)

Js = Ps (cf − cp ) + (1 − )J c¯ ,

(15)

where Lp is the hydraulic permeability of the membrane,  is the reflection coefficient, Ps is the solute permeability of the membrane,  is the dissociation coefficient, while c¯ represents the mean concentration of the solute.

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In this model the membrane is treated as a black box: the transport mechanisms and the structure of the membrane are ignored. IT models have been applied in predicting transport through NF membranes for single and binary solute systems [30,31], for multiple systems [32,33] and also for industrial feeds [34–37]. Assuming that J > 0, we can divide (15) by J, and recalling that there holds: Js cp = J

(16)

we obtain cp =

Ps (cf − cp )

+ (1 − )¯c .

Lp P − Lp RT (cf − cp )

(17)

Recalling that cp = cf (1 − R), we can rewrite (13) as cf (1 − R) =

Ps cf R Lp P − Lp RTcf R

+ (1 − )¯c .

(18)

The mean concentration across the membrane c¯ can be estimated using the logarithmic average c¯ =

cf − cp ln cf − ln cp

=−

cf R

(19)

ln(1 − R)

so that (18) becomes cf (1 − R) −

Ps cf R Lp P − Lp RTcf R

+ (1 − )

cf R ln(1 − R)

= 0.

(20)

After assigning the transport parameters of the membrane Lp , Ps and , Eq. (20) allows us to compute the rejection R corresponding to any set of values cf and P. For more details, we refer to [38]. Generalizing (20), the rejections at time t of solutes for a binary system can be calculated solving, for i = 1, 2,

2 k=1

cf,i (t)(1 − Ri (t)) =

Lp [P − RT − (1 − i )

Ps,k cf,k (t)Rk (t)

2

k=1

J(t) = Lp P − RT

2 

Js,j = −Dj,p

ln(1 − Ri (t))

.

(21)

cjout =

−Xd +

−Xd +

Ps =

cf

cf∗

(23)

m ,

(25)



2 Xd2 + 41 2 cf,j

2



,

2 Xd2 + 41 2 cp,j

,

(26)

2

(22)

The effect of membrane charge density on the permeation of charged components is not taken into account in these models. However, many authors [39–42] established relationships between the transport parameters ( and Ps ) and the membrane electrical and structural properties and used IT models to describe permeation of electrolytes through NF membranes. Finally, let us point out that the electrolyte permeability coefficient Ps is proportional to the partition coefficient of the electrolyte [31], and its concentration dependence can be expressed as Ps∗

zj cj Dj,p d + Kj,c cj V˙ . F RT dx

2

j e−Wj /kT , j being the steric partitioning j = (1 − j ) , where j = rj /rp is the ratio between the ion and the membrane pore radii. Finally, Wj is the so-called solvation energy barrier:

Finally, let us recall that the SK model [29] states that (1 − exp(( − 1/Ps )J)) . 1 −  exp(( − 1/Ps )J)

−

where Xd is the effective charge density of the membrane. j

k=1

R=

dx

(j = 1, 2) is the dimensionless partial partition coefficient j =



k k cf,k (t)Rk (t) .

dcj

The transport equation for non-charged compounds is given as the sum of the diffusive and the convective terms due to negligible electrostatic effects. The model assumes a Hagen–Poiseuille-type solvent velocity in uniform, straight, cylindrical membrane pores. The hindered nature of diffusion and convection is accounted by incorporating steric hindrance factors (Kj,d and Kj,c ). Electrokinetic space-charge modeling systems based on the extended Nernst–Planck equation have been successfully applied by many researchers [43–47]. However, computing membrane performances using such a model can be very demanding in time [48]. Starting from the Donnan–Steric-pore-dielectric-exclusion (DSPMDE) model [49,50], a useful approach for engineering calculations has been developed by Bowen et al. [51]. The system of differential equations was reduced to a system of algebraic equations denoted as DSPM-DE model by finite difference linearization of pore concentration gradients. Let us summarize the governing equations of the linearized DSPM-DE model that we will use in the following. We consider an aqueous solution of a 1:1 type electrolyte. Supscript j = 1, 2 denotes the coion and counterion of the electrolyte, respectively. In Section 3.2 we extend the model accounting for an aqueous ternary system consisting of a neutral compound and a dissociated 1:1 type electrolyte. The concentrations within pore entrance and outlet are given by, respectively,

cjent =

The permeate flux can be then recovered as:



2.4.3. Transport models Solute transport across NF membrane pores can be described using the extended Nernst–Planck model. The molar solute flux Js,j of an ion j is given as a sum of three terms (diffusion, electromigration and convection) as follows:

k k cf,k (t)Rk (t)]

cf,i (t)Ri (t)

167

(24)

where Ps∗ is the solute permeability for a reference feed concentration cf∗ , and m is a suitable parameter.

Wj =

zj2 e2



8 0 aj

1

p

−

1



b

with the dimensionless pore dielectric constant:



p = 80 − 2(80 − ∗ )

d rp

2 + (80 − ∗ )

d rp

.

For the meaning of the other symbols we refer to Appendix A. Finally, the permeate concentration for a 1:1 type electrolyte is given by cp =

(Pe1 + Pe2 )ca2v,1 + (Pe1 + Pe2 )Xd cav,1 − (2cav,1 + Xd )c1 (Pe1 /(Kc,1 + (Pe2 /Kc,2 )))cav,1 + (Pe1 /Kc,1 )Xd

(27)

cav,i is the average concentration within the pores cav,j =

cjent + cjout 2

(28)

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while Pej is the Peclet number Pej =

Kc,j rp2 Pe Kd,j D∞,j 0

which depends on the hindrance factors Kc,j = (2 − j )(1.0 + 0.054j − 0.9882j + 0.4413j ) Kd,j = 1 − 2.30j + 1.1542j + 0.2243j

on the effective pressure Pe = P −  = P − RT

2 

(cf,j − cp,j )

(29)

j=1

on the diffusion coefficient at infinite dilution D∞,j and on the bulk dynamic viscosity 0 . To our knowledge, there is no direct method to estimate the membrane charge density Xd which depends on the membrane chemistry and on the specific adsorption of ions [52]. Therefore, estimates are based on experimental isotherm data. In this work we assume that the flux and rejections are sole functions of the actual feed concentrations and that the concentration polarization is negligible. However, the model can be extended to account for other parameters like temperature [53], pore-size distribution [54], pH or concentration polarization [8]. More recently, Geraldes and Aves [55] has developed a computer program called NanoFiltran for the simulation of mass transfer in NF based on the extended Nernst–Planck equation. The program can run in two modes: system prediction mode and membrane characterization mode. In the first mode, the rejection of solutes in single or multi-component systems can be estimated for a given permeate flux accounting for concentration polarization. In the second mode, the membrane properties can be evaluated using experimental data for the fitting. Such a program with minor changes could be applied as a subroutine for the batch system simulations described in this work. 3. Experimental The NF apparatus used in this study, the applied chemicals and the sample analysis have been described in details in the previous work [19]. In brief, technical grade sucrose and sodium chloride were used to prepare the aqueous solutions, and then cross-flow filtration experiments were carried out using a 0.45 m2 spiral-wound membrane element. The recirculation flow rate was very high (1 m3 /h) in comparison with the permeate flux, so that the effect of concentration polarization on the overall membrane separation can be neglected. During the filtration experiments, permeate and retentate samples were taken periodically always at the same time from the permeate pipe and from the feed tank, respectively. The NF membrane used in this study was Desal-DK purchased from Osmonics (today GE Water & Process Technologies). In all five experiments, the applied pressure was kept at 3 × 106 Pa and the temperature at 298 K. The first experiment was needed to determine the relations between J, R and the feed concentration. The additional four experiments were used to validate our model. These experiments are described in more details hereafter. (1) Experimental run for parameter fitting. The feed solution (0.03 m3 , 150 mol/m3 sucrose and 300 mol/m3 NaCl) was concentrated to 0.01 m3 . Then, 0.02 m3 deionized water was added into the feed tank, and it was concentrated again to 0.01 m3 . This procedure was repeated four times. The collected experimental data were used for the empirical approach.

(2) Validation run no. 1. Constant-volume dilution mode was performed with 0.015 m3 initial feed solution containing 300 mol/m3 sucrose and 300 mol/m3 NaCl. A level sensor was employed to keep constant the feed volume by continuously adding deionized water as diluant at a rate equal to the permeation rate. The process was stopped after 4 h of operation. (3) Validation run no. 2. Variable-volume dilution mode with pure water as diluant was performed. The initial feed was 0.03 m3 solution containing 150 mol/m3 sucrose and 300 mol/m3 NaCl. The ratio of diluant inlet rate to permeate flow rate was kept at ˛ = 0.75 in a quasi-continuous way: after every 0.002 m3 of collected permeate, 0.0015 m3 pure water was added into the feed tank. The operation was stopped when the feed solution was reduced to 0.01 m3 . (4) Validation run no. 3. Constant-volume dilution mode was performed using diluted process liquor as diluant. The volume in the feed tank was kept at 0.01 m3 . The initial sucrose and salt concentrations were 150 mol/m3 and 300 mol/m3 , respectively. The total volume of the diluant used during the operation was 0.06 m3 , and it contained 50 mol/m3 sucrose and 100 mol/m3 NaCl. (5) Validation run no. 4. A fed-batch process was performed. 0.01 m3 initial feed solution containing 150 mol/m3 sucrose and 300 mol/m3 salt was introduced in the feed tank. During the operation, the permeate stream was collected separately, the retentate was recycled, and fresh process liquor was continuously added to the feed tank with a constant inlet rate of Q = 2.68 × 10−6 m3 /s. 3.1. Solution procedure for the IT model The values of the transport parameters for the membrane DK and for NaCl are taken from Yaroshchuk [56]: cf∗ = 1 mol/m3 ,  =

0.9259, Ps∗ = 3.319 × 10−6 m/s, and m = 0.4569. The SK analysis of different neutral solutes is provided in [57]. The transport parameters of lactose ( = 0.9935, Ps = 3.5 × 10−9 ) were chosen for modeling the sucrose permeation, since the physiochemical nature and the size of the two neutral molecules are practically identical. We assumed constant transport parameters for sucrose over the whole range of feed concentration. Finally, the pure water permeability of the membrane was taken as Lp = 3.68 × 10−12 [m/(s Pa)]. The rejection of the solutes can be estimated for a given feed composition by solving (21) or using the SK approach. In this case, (1) (2) (3) (4) (5)

Calculate salt permeability using (24). Consider a guess for the permeate flux J. Calculate rejections for both components using (23). Calculate the permeate flux using (22). Iterate steps (3)–(4) until convergence.

3.2. Solution procedure for the DSPM-DE model The model parameters used in this study were taken from Bowen et al. [51] who applied the linearized DSPM-DE model to characterize the Desal-DK membrane. The estimated value of the dielectric constant for the oriented water layer is ∗ ≈ 31, and the predicted average pore size of the membrane is rp = 0.45 nm. The effective charge density of the membrane Xd as function of the NaCl concentration in the wall is reported in Table 2. We assume a negligible concentration polarization effect so that the feed and the wall concentrations are practically equal. The diffusivity, charge and Stokes radius of sodium and chloride ions are also taken from [51]. The value of Xd was assumed to be null for a concentration of NaCl greater than 120 mol/m3 . For less concentrated solutions,

Z. Kovács et al. / Journal of Membrane Science 327 (2009) 164–173

169

Table 2 Effective charge density of the Desal-DK membrane from analysis of NaCl permeation data with linearized model given by Bowen et al. [51] cf,2 (mol/m3 )

Xd (mol/m3 )

1.4 3.7 11.0 37.6 111.8

−0.50 −1.07 −2.05 −5.75 −1.85

we interpolated the values of Xd given in Table 2 to get a suitable function of cf,2 . Since the Stokes radius of sucrose is greater than the estimated pore radius of the membrane DK, complete sucrose rejection was assumed. However, the actual feed concentration of sucrose influences the permeate flux and the salt rejection during the process. We calculate the osmotic pressure exerted by the feed solution as follows:  = RT (cf,1 + cf,2 ),

(30)

Fig. 2. Measured data of feed concentration as a function of operation time for the experimental run.

where cf,1 is the sugar concentration, and cf,2 is the salt concentration in the feed. Thus, the effective pressure becomes Pe = P −  = P − RT

2 

(cf,i − cp,i ).

(31)

i=1

Eq. (31) is similar to (29) but it describes an aqueous sugar/salt system, while (29) accounts only for an aqueous salt solution. Since we assume a complete sugar rejection, all the other equations for the determination of the salt permeate concentration presented in Section 2.4.3 remain unchanged. The following calculation procedure was applied to determine flux and salt rejection. (1) Estimate the effective membrane charge density Xd corresponding to a given salt concentration cf,2 on the basis of the experimental data of Table 2. (2) Consider an initial guess for the permeate salt concentration cp,2 and estimate cjout using (26).

(3) Calculate cjent and cav,j using (26) and (28). (4) Calculate effective pressure driving force using Eq. (31) accounting for the osmotic pressure difference caused by both sugar and salt concentration differences across the membrane.

Fig. 3. Change in the feed composition during the experimental run.

Fig. 4. Permeate flux (left side) and rejection of salt (right side) as functions of feed composition for membrane DK. Experimental data are illustrated with closed circles and fitted empirical planes with continuous lines (298 K, 3 × 106 Pa).

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Fig. 5. Validation run no. 1: constant-volume dilution mode with pure solvent as diluant.

(5) Calculate the permeate salt concentration cp,2 using (27) and compare with the guess of point (2). (6) Iterate (2)–(5) using the Dekker-Brent method (an effective combination of the bisection and the secant methods, see, e.g., [58]) until convergence with respect to cp,2 . (7) Calculate the salt rejection as R = 1 − cp,2 /cf,2 and the permeate flux with V˙ = Lp Pe . 4. Results 4.1. Empirical (EMP) approach The feed concentrations versus the operation time obtained from the first experimental run are illustrated in Fig. 2. The change in the concentrations in the feed tank during the experimental run is shown in Fig. 3. Experimental data in the whole cf,1 -by-cf,2 matrix are available through the proposed experimental design. This allows an effective parameter estimation by data-fitting on the measured data of R and J. The measured data obtained from validation run no. 2 are also illustrated in Fig. 3. Comparing the two operational modes, one can see that the variable volume dilution mode is not sufficient to obtain the relation between the rejection and the feed composition due to the limited dates of cf,1 and cf,2 . The experimental data of permeate flux and rejection of salt obtained from the experimental run are plotted as functions of feed composition in Fig. 4. The surfaces shown in Fig. 4 were fitted to the experimental data in a least-squares sense. In particular, we have found a very good fitting using 2 2 J = (x1 cf,2 + x2 cf,2 + x3 ) · exp((x4 cf,2 + x5 cf,2 + x6 )cf,1 ),

(32)

R1 = (y1 cf,2 + y2 )cf,1 + (y3 cf,2 + y4 ),

(33)

2 2 R2 = (w1 cf,2 + w2 cf,2 + w3 ) · exp((w4 cf,2 + w5 cf,2 +w6 )cf,1 ),

(34)

where x1 , . . . , x6 , y1 , . . . , y4 , w1 , . . . , w6 are suitable coefficients. An almost total (R ≈1) and composition-independent rejection was measured for sugar. On the contrary, the rejection of salt strongly depends on the feed concentrations of both components varying between 14 and 70%. Moreover, both components have great contributions to the permeate flux. It should be pointed out that great care is needed when the empirical relations are employed for extrapolation out of the range of the experimental cf,1 -by-cf,2 matrix.

Fig. 6. Validation run no. 2: variable-volume dilution mode adjusting the proportionality factor (˛) to 0.75 and using pure solvent as diluant.

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Fig. 7. Validation run no. 3: constant-volume dilution mode with diluted process liquor as diluant.

4.2. Model validation The simulation technique discussed in Sections 2 and 3 allows a quick comparison of the performances of different batch and semi-batch operations, and also of multi-step processes involving different combinations of the basic operations. After estimating the flux and rejection as functions of concentration from the first experiment, four more experiments were carried out in order to validate the model. In each test the flux and rejection required in (10) and (13) were computed using the empirical and the theoretical approaches discussed in Sections 2.4.1–2.4.3(see also Sections 3.1 and 3.2). The predicted and the measured data of the membrane DK for the different operations are shown in Figs. 5–8. The permeate flux J of the membrane element (left-side figures) and the concentrations of the compounds (right-side figures) in the feed tank are plotted versus operation time. Concentrations of sugar and salt are illustrated with open and closed circles, respectively. In general, good predictions were obtained using the theoretical models. The permeate flux of the membrane element is under-estimated to a certain extend with both models thus giving inaccurate predictions for the validation run no. 4 shown in Fig. 8. In this case, since the change in the volume in the feed tank is given by the difference of the constant inlet and the permeate outlet, a relatively small estimation error in the permeate flux can

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Fig. 8. Validation run no. 4: constant inlet-rate (Q =2.68 × 10−6 m3 /s) fed-batch process employing process liquor as inlet.

cause inaccurate prediction for the actual volume in the feed tank, and thus for the feed concentrations. In the case of the IT model,  and Ps need to be estimated for each component either from literature data or from previous experiments with single-solute systems. It should be pointed out that these transport parameters implicitly account for the interaction between the given membrane and the specific solute, which cannot be directly inferred from the IT model on a physical basis. Moreover, their dependence on the concentration should be evaluated in order to improve the model. The DSPM-DE model requires more input parameters related to both the membrane structural and electrical properties and the characteristics of the solute. However, the increased computational cost with respect to the IT model is paid back by the good representation of the properties of the membrane. The simulations based on the empirical approach fit better the experimental data than those based on theoretical models. However, the latter are quite satisfactory considering the simplifications used to derive the theoretical models and the fact that we used parameters taken from literature without evaluating the differences in the test conditions and equipments as well as in the properties of the individual elements or membranes. Moreover, the theoretical models can be readily used for analyzing the sensitivity of the separation with respect to the applied pressure, and they can be extended on a physical basis accounting for cross-flow veloc-

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ities. On the other hand, the empirical approach would demand a rapidly increasing amount of experiments to estimate further parameters that would enrich the model, like applied pressure, cross-flow velocity, another solute or temperature. 5. Conclusion A mathematical frame is provided for modeling batch and semibatch membrane filtration processes. The approach followed in this work uses the feed concentrations as basis for the calculations, rather than the concentration factor or the dilution factor. The advantage of this approach is that it accounts for variable solute rejection coefficients. Thus, the dynamic behavior of all types of batch and semi-batch system configurations can be predicted without having to modify the given mathematical framework. A model based on a system of ordinary differential equations and a practical computational algorithm are presented. The design equations describing the engineering aspects of batch and semi-batch systems are handled separately from the estimation methods describing the mass transfer through the membrane. Thus, the permeate flux and rejection can be estimated either by empirical equations fitted to the experimental data or by equations based on theoretical models. In particular, we compared an experimental approach requiring a minimal number of experiments to the IT model and the linearized DSPM-DE model. The theoretical models might be advantageous over the empirical approach when many parameters are involved. Finally, various batch and semi-batch NF operations and simulations were carried out with an organic/electrolyte binary test solution. In all cases, good agreements were found between predicted and experimental data.

Nomenclature List of symbols aj Stokes radius of ion j (m) A membrane area (m2 ) c concentration (mol m−3 ) d thickness of oriented solvent layer (0.28−9 m) D diluant flow rate (m3 s−1 ) Di,∞ diffusion coefficient at infinite dilution (m2 s−1 ) hindered diffusion coefficient within the pore Di,p (m2 s−1 ) e electronic charge (1.602177 × 10−19 C) F Faraday constant (96487 C mol−1 ) Js molar solute flux (mol m−2 s−1 ) J permeate flux (m3 s−1 m−2 ) k Boltzmann constant (1.38066 × 10−23 J K−1 ) convective hindrance factor Kc Kd diffusive hindrance factor Lp hydraulic permeability (m s−1 Pa−1 ) Pe Peclet number solute permeability (m s−1 ) Ps Q inlet into the feed tank supplied by external pump (m3 s−1 ) ri solute radius (m) rp pore radius (m) R gas constant (8.31447 J T−1 mol−1 ) R rejection t operation time (h) T temperature (K) V volume (m3 ) ˙V solvent velocity (m s−1 ) x axial position within the pore (m)

Xd zj

effective membrane charge (mol m−3 ) valence of ion j

Greek symbols ˛ proportionality factor of diluant flow to permeate flow cj concentration difference of ion j across the pore thickness (mol m−3 ) P applied pressure (Pa)  osmotic pressure difference across the membrane (Pa) Wi Born solvation energy barrier (J) x membrane thickness (m) b bulk dielectric constant p pore dielectric constant ∗ dielectric constant of oriented water layer 0 vacuum permittivity (8.854 × 10−12 J−1 C2 m−1 ) steric partitioning coefficient i i partial partition coefficient

0 dynamic viscosity of solvent (Pa s)  ratio of solute to pore radius  reflection coefficient potential within the pore (V)  dissociation constant Subscripts d diluant f feed i component (in the experimental part: i = 1 sucrose, and i = 2 salt) j ion (in the experimental part: j = 1 Na+ , and j = 2 Cl− ) p permeate Superscripts n time discretization step Abbreviations DSPM–DE Donnan–Steric-pore-dielectric-exclusion EMP empirical approach IT irreversible thermodynamics KK Kedem–Katchalsky NF nanofiltration SK Spiegler–Kedem

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