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A simple treatment of mass transfer data in continuous dialyzer
Chemical Engineering and Processing 45 (2006) 806–811
A simple treatment of mass transfer data in continuous dialyzer ˇ akov´a, Pˇremysl Petˇr´ık Zdenˇek Palat´y ∗ , Alena Z´ ˇ Legi´ı 565, 532 10 Pardubice, Czech Republic Department of Chemical Engineering, The University of Pardubice, n´am. Cs. Received 23 May 2005; received in revised form 22 February 2006; accepted 4 March 2006 Available online 28 March 2006
Abstract The procedure, which enables to determine the basic transport properties of membrane/solution systems on the basis of the measurement in a continuous dialyzer at steady state, has been elaborated. The determination of these characteristics is based on the numerical integration of the set of differential equations describing the concentration profiles in both compartments of the dialyzer followed by the optimizing procedure. The model used takes into account changes in flow of liquid along the membrane in both compartments as a consequence of the solvent flow through the membrane. Using the procedure mentioned above, the overall dialysis coefficient, permeability of the membrane and the membrane mass transfer coefficient have been determined for the Neosepta-AFN membrane/sulfuric acid system. © 2006 Elsevier B.V. All rights reserved. Keywords: Dialysis; Continuous dialyzer; Steady state mass transfer; Sulfuric acid; Neosepta-AFN membrane
1. Introduction In the group of modern separation processes using semipermeable membranes, such as ultrafiltration, microfiltration, nanofiltration, reverse osmosis, pervaporation and electrodialysis also dialysis (or diffusion dialysis) has its own place. Low energy consumption during its application and simple construction of the apparatuses used are its considerable advantages. Transport rate of components through the membrane can be characterized by the dialysis coefficient, permeability of the membrane or membrane mass transfer coefficient. These basic characteristics are dependent not only upon the membrane properties, which are affected by the technology (chemical composition, membrane structure, concentration of fixed sites, water content, etc.), but also upon properties of mixtures to be separated. Hence, the basic information on the process and the values of the transport characteristics of membrane/solution systems can only be found by experimental procedure. For this purpose, a batch dialysis cell with two compartments separated by the membrane is the most frequently used [1–6].
∗
Corresponding author. Tel.: +420 466 037 503; fax: +420 466 037 068. E-mail address: [email protected] (Z. Palat´y).
0255-2701/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2006.03.009
Further equipment suitable to the study of dialysis is a continuous dialyzer consisted of flat membranes [4,7–14]. In the treatment of the experimental data measured in a continuous dialyzer, several approaches have been used. The overall dialysis coefficient is very often calculated by the procedure using a logarithmic mean of driving forces on both ends of the dialyzer [8,10–12]. Sato et al. [7] numerically solved ionic transport in a continuous Donnan dialyzer with a parallel-plate channel. The model used was formulated on the basis of diffusion equations in terms of diffusion, migration and convection of each ion. The validity of this model was confirmed by the comparison with experiments. A method for the calculation of the overall dialysis coefficient based on the integrated form of a differential mass balance equation over the reservoir of the feed ¨ solution was published by Cocchini et al. [13]. Lately, Ozdural and Alkan [14] proposed an original technique for the determination of the overall dialysis coefficient, which is based on monitoring the concentration in a recycle system. The explicit equation derived is restricted to the cases characterized by high volumetric flow of liquid, low membrane area and/or low permeability of the membrane. The aim of the present paper is to elaborate a simple and reliable procedure for the calculation of the basic transport characteristics of membrane/solution systems in the cases, where variable liquid flow rates along the membrane exists. This fact has not been yet taken into account.
Z. Palat´y et al. / Chemical Engineering and Processing 45 (2006) 806–811
807
Fig. 1. Balance scheme of continuous dialyzer: M, membrane; I and II, compartments.
The initial conditions for Eqs. (2) and (3) are
2. Theory Fig. 1 depicts the balance scheme of a counter-current continuous dialyzer with two identical compartments, whose thickness and width are δ and d, respectively. The height of the compartment is zT . If the concentration of the component A in the compartment I is higher than that in the compartment II, then mass transfer from the compartment I to II occurs. At the steady state, the balance of the component A over the differential volume of the compartment I is given by the following equation: I I SuI cA |z = SuI cA |z+dz + JA
A dz zT
(1)
where S is the cross-section of the compartment (S = δ × d), u the liquid flow rate, cA the molar concentration of the component A, JA the flux of the component through the membrane and A is the total membrane area. Using the definition of derivation and after arrangement, one can obtain Eq. (2) describing the concentration profile of the component A in the compartment I: I dcA 1 A cI duI = − I JA − AI dz Su zT u dz
(2)
In a similar way, it is possible to obtain Eq. (3) describing the dependence of the concentration of the component A in the compartment II upon the coordinate z: II dcA cII duII 1 A = − II JA − AII dz Su zT u dz
(3)
z = 0,
I I = cA,in , cA
II II cA = cA,out
(4)
Note: in the derivation of Eqs. (2) and (3), plug flow of liquid in both compartments of the dialyzer is assumed. As proved by earlier study [15], this assumption is fulfilled. If the concentrations of the component A in the streams at both ends of the dialyzer are known, then it is possible to numerically integrate the set of Eqs. (2) and (3), and with a suitable optimizing procedure, calculate the basic transport characteristics of the membrane/solution system, i.e. the overall dialysis coefficient, permeability of the membrane and membrane mass transfer coefficient. In order to determine the overall dialysis coefficient, the flux of the component A can be express as I II JA = KA (cA − cA )
(5)
where KA is the overall dialysis coefficient. The determination of other two characteristics is more complex than that of KA , as besides transport through the membrane, the transport of the component A through liquid films on both sides of the membrane must be solved—see Fig. 2. In this case, Eqs. (6)–(8) must be added to the basic Eqs. (2) and (3) with the initial conditions (4): I I JA = kLI (cA − cAi )
(6)
I II JA = PA (cAi − cAi )
(7)
JA =
(8)
II kLII (cAi
II − cA )
808
Z. Palat´y et al. / Chemical Engineering and Processing 45 (2006) 806–811
Fig. 2. Concentration profiles of component A in liquid films and membrane.
where kLI and kLII are the liquid mass transfer coefficients, PA is permeability of the membrane and subscript i denotes the solution/membrane interface. The mass transfer coefficients can be estimated from the following equation valid for laminar flow: Sh = CRe0.5 Sc0.33
(9)
Eqs. (6) and (8) describe the transport through the liquid films, Eq. (7) concerns the transport through the membrane. In the case of determination of the membrane mass transfer coefficient, it is necessary to consider Eqs. (6) and (8) simultaneously with the relations (10) describing the membrane/solution equilibrium: j
j j
cAM = ΨA cAi
j = I, II
(10)
The transport through the membrane is then expressed as I II JA = kAM (cAM − cAM )
(11)
Note: the concentrations of the component A in Eqs. (10) and (11) are referred to the membrane volume. 3. Experimental The experimental set-up used is shown in Fig. 3. Its main part was a laboratory dialyzer with two identical compartments separated by an anion-exchange membrane Neosepta-AFN developed and produced by Tokuyama Soda Company, Inc. (Japan). The membrane area was 3.31 × 10−2 m2 , the basic properties of the membrane are summarized in Table 1. Table 1 Properties of Neosepta-AFN membrane used Parameter
Value
Unit
Thickness Water content
160 0.418
Concentration of fixed charges
4.7
m g per g of dry membrane in Cl− form kmol m−3
Note
Considered to be monovalent
Fig. 3. Experimental set-up: 1, continuous dialyzer; 2, membrane; 3, peristaltic pump; 4, bottle with feed; 5, bottle with distilled water; 6, bottle with dialyzate; 7, bottle with diffusate; I and II, compartments.
The height of the dialyzer was 1 m, the dimensions of each compartment were 0.92 m × 0.036 m × 0.0011 m (height × width × thickness). In both the compartments there were net-type spacers made of PVC also acting as turbulence promoters. In all the experiments we used aqueous solutions of sulfuric acid. The feed of sulfuric acid entered the bottom of the compartment I (z = 0), while distilled water entered the top of the compartment II (z = zT ). The flow of both liquid streams was ensured by a peristaltic pump MC-MS/CA4 (Ismatec SA, Switzerland). In all the experiments, liquid flow rate of the feed was always equal to that of water (V˙ inI = V˙ inII ). After the steady state has been reached (a time period from 6 to 12 h in the dependence on liquid flow rate and acid concentration was necessary—as proved by preliminary experiments), three samples were taken from each stream leaving the dialyzer. At the same time, mass flow rate of both these streams was determined by weighing. Before each experiment, also mass flow rate of both feed and water was measured. Using density of liquid, mass flow rate of each stream was recalculated into volumetric liquid flow. All the experiments were carried out at a laboratory temperature of 20 ± 1 ◦ C. The acid concentration in the feed was changed in the range from 0.1 to 2.0 kmol m−3 , while the volumetric flow rate of the inlet streams was in the limits from 7.0 × 10−9 to 23.0 × 10−9 m3 s−1 (i.e. liquid flow rate was from 1.8 × 10−4 to 5.8 × 10−4 m s−1 ). The acid concentration was
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4.2. Permeability of membrane, PA In order to calculate permeability of the membrane, the procedure consisting of the following steps was used: 1. The initial estimate of PA (PA = 10−6 m s−1 ). The calcuj lation of the derivation du dz (j = I, II) from the experimental j j values of V˙ in and V˙ out (j = I, II). 2. The numerical integration of the set of Eqs. (2) and (3), where JA is expressed by Eq. (7) (at the same time, Eqs. (6) and (8) were used, too), by the Runge–Kutta fourth order method with the integration step h = 0.001 m. In each integration step, it was necessary to calculate the concentration of acid at the membrane/liquid interface, kinematic viscosity and diffusivity of sulfuric acid. As viscosity of solution and diffusivity of acid depend upon its concentration, it was necessary to determine these parameters in both liquid films by an iterative procedure using literature data [16–18]. (A) The estimate of the concentrations for the determinaI = cI II = cII tion of DA and ν—for z = 0: c¯ A ¯A A,out ; for A,in , c z > 0 these concentrations were taken to be equal to those obtained in the previous integration step. (B) The calculation of liquid mass transfer coefficients from Eq. (9). (C) The calculation of the acid concentration at the memI and cII , by solving Eqs. brane/liquid interface, i.e. cAi Ai (6)–(8). j (D) The correction of the acid concentrations c¯ A (j = I, II): (0)
Fig. 4. Dependence of overall dialysis coefficient upon volumetric flow rate of feed.
determined by titration, a standard 0.1 M NaOH solution being used. 4. Data treatment and discussion The basic transport characteristics (i.e. KA , PA and kAM ) were calculated from the data measured in continuous dialyzer at steady state by the following way. 4.1. Overall dialysis coefficient, KA
j
j
(12)
cA + cAi j = I, II (13) 2 (E) The steps from (B) to (D) were repeated until the relative j changes in the concentrations c¯ A (j = I, II) were below 0.05%. 3. The calculation of the objective function (12). 4. The realization of one step of the optimizing procedure. For that purpose, we used the Golden Section Search. This step provides the corrected value of PA . 5. The procedures from (2) to (4) were repeated until reaching the minimum of the objective function (12).
In Fig. 4 the values of KA calculated in such a way are plotted versus volumetric flow rate V˙ inI (full symbols). The parameter of the individual lines is the acid concentration in the feed. These dependences can be approximated with a sufficient accuracy by straight lines. Moreover, we calculated also the overall dialysis coefficients using the common procedure based on a logarithmic mean of driving forces at both ends of the dialyzer. Here, these are drawn by empty symbols. From the graphical presentation in Fig. 4, it can be seen that higher differences between the values of KA obtained by both procedures exist only at low acid I concentrations in the feed, i.e. at cA,in = 0.1 and 0.5 kmol m−3 . As shown in Fig. 4, the overall dialysis coefficients exhibit low sensitivity to volumetric flow rate. But on the other hand, they are negatively influenced by increasing acid concentration in the feed.
The procedure for obtaining permeability of the membrane given above needs the value of the constant C in Eq. (9). Its determination was based on the following considerations: permeability of the membrane is a membrane/solution parameter, which is not affected by flow of liquid. This means that the values of permeability obtained at various liquid flow rates and constant acid concentration in the feed must not be highly different from each other. For that reason, we defined a new criterion S¯ as a sum of I variances of PA at a constant cA,in , which is plotted versus the constant C in Fig. 5. From this dependence C was estimated to be approximately 12.5—the C at which this new criterion reaches practically a constant value. From Fig. 6, where permeability of the membrane is plotted versus the acid concentration in the feed, it is evident that
The overall dialysis coefficient was determined by the numerical integration of the basic differential Eqs. (2) and (3), where JA is expressed by Eq. (5), with the initial conditions (4). Eqs. (2) and (3) were integrated by the Runge–Kutta fourth order method with the integration step h = 0.001 m. The calculated values of I,calc II,calc and cA,in were obtained in this step. the concentrations cA,out Using a one-dimensional optimizing procedure (Golden Section Search was used) such values of KA were searched, at which the objective function (12) reached a minimum: I,exp
2
2
I,calc II,calc F = (cA,out − cA,out ) + (cA,in )
j
c¯ A =
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Z. Palat´y et al. / Chemical Engineering and Processing 45 (2006) 806–811
¯ upon constant C in Eq. (9). Fig. 5. Dependence of sum of variances of PA , S,
this transport characteristic of the membrane/solution system considered is affected by acid concentration—it decreases with increasing acid concentration. 4.3. Membrane mass transfer coefficient, kAM This transport characteristic was determined by the procedure, which was very similar to that used in the case of PA . The only difference was the calculation of the concentrations j cAM (j = I, II). For that purpose, we solved the set of Eqs. (6), (8), (10) and (11) by the Newton–Raphson procedure. The partij tion coefficients,ΨA (j = I, II), were calculated from the sorption isotherm, whose parameters are given elsewhere [19]. In the calculation of kAM , the original sorption isotherm, where the concentration in the membrane was referred to the volume of liquid inside the membrane, was recalculated using the porosity of the swollen membrane (εM = 0.30) in such a way so that the concentration would refer to the membrane volume. Fig. 7 presents the dependence of the membrane mass transfer coefficient upon the acid concentration in the feed. The course of this dependence is very similar to that published in Ref. [19], where the values of kAM were calculated on the basis of mea-
Fig. 7. Dependence of membrane mass transfer coefficient upon acid concentration in feed.
surement in a batch cell. Moreover, it can be seen that, except the lowest acid concentration used, a good agreement between the membrane mass transfer coefficients obtained by both procedures exists. (Note: a detailed inspection of the model used in Ref. [19] reveals the fact that, in reality, the values of kAM given therein must be understood as a product of the membrane mass transfer coefficients and porosity of the swollen membrane.) 5. Conclusion The procedure, which enables to calculate the basic transport characteristics from data measured in a continuous dialyzer at steady state, was suggested. It is based on the numerical integration of the set of ordinary differential equations describing the concentration dependences of the component dialyzed in both compartments upon the height coordinate. The integration is followed by the optimizing procedure. The model used takes into account a variable flow of liquid along the membrane as a consequence of the solvent transport through the membrane. For illustration, using data obtained in dialysis of sulfuric acid, the overall dialysis coefficient, permeability of the membrane and membrane mass transfer coefficient were determined by the procedure mentioned above. The results are presented graphically as dependences of these characteristics upon the volumetric flow rate of the feed and acid concentration in this stream. Acknowledgement This work was financially supported by the Ministry of Education, Youth and Sports of the Czech Republic, Project MSM 0021627502. Appendix A. Nomenclature
Fig. 6. Dependence of permeability coefficient of membrane upon acid concentration in feed.
A c c¯
membrane area (m2 ) molar concentration (kmol m−3 ) concentration for determination of diffusivity and viscosity (kmol m−3 )
Z. Palat´y et al. / Chemical Engineering and Processing 45 (2006) 806–811
C D d de F J kL kM K P Re S S¯ Sc Sh u V˙ z zT
constant in Eq. (9) diffusivity (m2 s−1 ) width of compartment (m) (=2 dδ/(d + δ)) equivalent diameter of compartment (m) objective function (kmol2 m−6 ) molar flux (kmol m−2 s−1 ) liquid mass transfer coefficient (m s−1 ) membrane mass transfer coefficient (m s−1 ) overall dialysis coefficient (m s−1 ) permeability of membrane (m s−1 ) (=ude /ν) Reynolds number cross-section of compartment (m2 ) sum of variances (m2 s−2 ) (=ν/DA ) Schmidt number (=kL DA /de ) Sherwood number liquid flow rate (m s−1 ) volumetric flow rate (m3 s−1 ) coordinate (m) height of compartment (m)
Greek symbols δ thickness of compartment (m) ε porosity ν kinematic viscosity (m2 s−1 ) Ψ partition coefficient Superscripts and subscripts A referred to component A calc calculated value exp experimental value i referred to solution/membrane interface in inlet
I, II M out (0)
811
referred to compartments I and II, respectively referred to membrane outlet initial estimate
References [1] [2] [3] [4] [5] [6] [7] [8]
[9] [10]
[11] [12] [13] [14] [15] [16] [17] [18] [19]
A. Narebska, A. Warszawski, Sep. Sci. Technol. 27 (1992) 703. Y. Wang, C. Combe, M.M. Clark, J. Membr. Sci. 183 (2001) 49. X. Tongwen, Y. Weihua, J. Membr. Sci. 183 (2001) 193. M.-S. Kang, K.-S. Yoo, S.-J. Oh, S.-H. Moon, J. Membr. Sci. 188 (2001) 61. M. Ersoz, I.H. Gugul, A. Sahin, J. Colloid Interf. Sci. 237 (2001) 130. E.G. Akgemci, M. Ersoz, T. Atalay, Sep. Sci. Technol. 39 (2004) 165. K. Sato, C. Fukuhara, T. Yonemoto, T. Tadaki, J. Chem. Eng. Jpn. 24 (1991) 81. M. Nishimura, Dialysis, in: Y. Osada, T. Nakagawa (Eds.), Membrane Science and Technology, Marcel Dekker, Inc., New York, 1992, pp. 361–376. ¨ A.R. Ozdural, Dev. Chem. Eng. Mineral Process 2 (1994) 138. S.-J. Oh, S.-H. Moon, Diffusion dialysis for recovery of acids from metal finishing wastes, in: Proceedings of the 89th Annual Meeting & Exhibition, Nashville, Tennessee, 1996. Z. Guiqing, Z. Qixiu, Z. Kanggen, J. Cent. South Univ. Technol. 6 (1999) 103. S.-J. Oh, S.-H. Moon, T. Davis, J. Membr. Sci. 169 (2000) 95. U. Cocchini, N. Cristiano, A.G. Livingston, J. Membr. Sci. 199 (2002) 85. ¨ A.R. Ozdural, A. Alkan, J. Membr. Sci. 223 (2003) 49. ˇ akov´a, J. Membr. Sci. 139 (1998) 67. Z. Palat´y, A. Z´ S. Bretsznajder, Własno´sci gaz´ov i cieczy, Wydawnictwa naukovotechniczne, Warszaw, 1966, p. 490. P. Pitter, Hydrochemick´e tabulky, SNTL, Praha, 1987, p. 119. International Critical Tables, vol. 5, McGraw-Hill, New York, 1926–1933, p. 12. ˇ akov´a, J. Membr. Sci. 119 (1996) 183. Z. Palat´y, A. Z´
A simple treatment of mass transfer data in continuous dialyzer ˇ akov´a, Pˇremysl Petˇr´ık Zdenˇek Palat´y ∗ , Alena Z´ ˇ Legi´ı 565, 532 10 Pardubice, Czech Republic Department of Chemical Engineering, The University of Pardubice, n´am. Cs. Received 23 May 2005; received in revised form 22 February 2006; accepted 4 March 2006 Available online 28 March 2006
Abstract The procedure, which enables to determine the basic transport properties of membrane/solution systems on the basis of the measurement in a continuous dialyzer at steady state, has been elaborated. The determination of these characteristics is based on the numerical integration of the set of differential equations describing the concentration profiles in both compartments of the dialyzer followed by the optimizing procedure. The model used takes into account changes in flow of liquid along the membrane in both compartments as a consequence of the solvent flow through the membrane. Using the procedure mentioned above, the overall dialysis coefficient, permeability of the membrane and the membrane mass transfer coefficient have been determined for the Neosepta-AFN membrane/sulfuric acid system. © 2006 Elsevier B.V. All rights reserved. Keywords: Dialysis; Continuous dialyzer; Steady state mass transfer; Sulfuric acid; Neosepta-AFN membrane
1. Introduction In the group of modern separation processes using semipermeable membranes, such as ultrafiltration, microfiltration, nanofiltration, reverse osmosis, pervaporation and electrodialysis also dialysis (or diffusion dialysis) has its own place. Low energy consumption during its application and simple construction of the apparatuses used are its considerable advantages. Transport rate of components through the membrane can be characterized by the dialysis coefficient, permeability of the membrane or membrane mass transfer coefficient. These basic characteristics are dependent not only upon the membrane properties, which are affected by the technology (chemical composition, membrane structure, concentration of fixed sites, water content, etc.), but also upon properties of mixtures to be separated. Hence, the basic information on the process and the values of the transport characteristics of membrane/solution systems can only be found by experimental procedure. For this purpose, a batch dialysis cell with two compartments separated by the membrane is the most frequently used [1–6].
∗
Corresponding author. Tel.: +420 466 037 503; fax: +420 466 037 068. E-mail address: [email protected] (Z. Palat´y).
0255-2701/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2006.03.009
Further equipment suitable to the study of dialysis is a continuous dialyzer consisted of flat membranes [4,7–14]. In the treatment of the experimental data measured in a continuous dialyzer, several approaches have been used. The overall dialysis coefficient is very often calculated by the procedure using a logarithmic mean of driving forces on both ends of the dialyzer [8,10–12]. Sato et al. [7] numerically solved ionic transport in a continuous Donnan dialyzer with a parallel-plate channel. The model used was formulated on the basis of diffusion equations in terms of diffusion, migration and convection of each ion. The validity of this model was confirmed by the comparison with experiments. A method for the calculation of the overall dialysis coefficient based on the integrated form of a differential mass balance equation over the reservoir of the feed ¨ solution was published by Cocchini et al. [13]. Lately, Ozdural and Alkan [14] proposed an original technique for the determination of the overall dialysis coefficient, which is based on monitoring the concentration in a recycle system. The explicit equation derived is restricted to the cases characterized by high volumetric flow of liquid, low membrane area and/or low permeability of the membrane. The aim of the present paper is to elaborate a simple and reliable procedure for the calculation of the basic transport characteristics of membrane/solution systems in the cases, where variable liquid flow rates along the membrane exists. This fact has not been yet taken into account.
Z. Palat´y et al. / Chemical Engineering and Processing 45 (2006) 806–811
807
Fig. 1. Balance scheme of continuous dialyzer: M, membrane; I and II, compartments.
The initial conditions for Eqs. (2) and (3) are
2. Theory Fig. 1 depicts the balance scheme of a counter-current continuous dialyzer with two identical compartments, whose thickness and width are δ and d, respectively. The height of the compartment is zT . If the concentration of the component A in the compartment I is higher than that in the compartment II, then mass transfer from the compartment I to II occurs. At the steady state, the balance of the component A over the differential volume of the compartment I is given by the following equation: I I SuI cA |z = SuI cA |z+dz + JA
A dz zT
(1)
where S is the cross-section of the compartment (S = δ × d), u the liquid flow rate, cA the molar concentration of the component A, JA the flux of the component through the membrane and A is the total membrane area. Using the definition of derivation and after arrangement, one can obtain Eq. (2) describing the concentration profile of the component A in the compartment I: I dcA 1 A cI duI = − I JA − AI dz Su zT u dz
(2)
In a similar way, it is possible to obtain Eq. (3) describing the dependence of the concentration of the component A in the compartment II upon the coordinate z: II dcA cII duII 1 A = − II JA − AII dz Su zT u dz
(3)
z = 0,
I I = cA,in , cA
II II cA = cA,out
(4)
Note: in the derivation of Eqs. (2) and (3), plug flow of liquid in both compartments of the dialyzer is assumed. As proved by earlier study [15], this assumption is fulfilled. If the concentrations of the component A in the streams at both ends of the dialyzer are known, then it is possible to numerically integrate the set of Eqs. (2) and (3), and with a suitable optimizing procedure, calculate the basic transport characteristics of the membrane/solution system, i.e. the overall dialysis coefficient, permeability of the membrane and membrane mass transfer coefficient. In order to determine the overall dialysis coefficient, the flux of the component A can be express as I II JA = KA (cA − cA )
(5)
where KA is the overall dialysis coefficient. The determination of other two characteristics is more complex than that of KA , as besides transport through the membrane, the transport of the component A through liquid films on both sides of the membrane must be solved—see Fig. 2. In this case, Eqs. (6)–(8) must be added to the basic Eqs. (2) and (3) with the initial conditions (4): I I JA = kLI (cA − cAi )
(6)
I II JA = PA (cAi − cAi )
(7)
JA =
(8)
II kLII (cAi
II − cA )
808
Z. Palat´y et al. / Chemical Engineering and Processing 45 (2006) 806–811
Fig. 2. Concentration profiles of component A in liquid films and membrane.
where kLI and kLII are the liquid mass transfer coefficients, PA is permeability of the membrane and subscript i denotes the solution/membrane interface. The mass transfer coefficients can be estimated from the following equation valid for laminar flow: Sh = CRe0.5 Sc0.33
(9)
Eqs. (6) and (8) describe the transport through the liquid films, Eq. (7) concerns the transport through the membrane. In the case of determination of the membrane mass transfer coefficient, it is necessary to consider Eqs. (6) and (8) simultaneously with the relations (10) describing the membrane/solution equilibrium: j
j j
cAM = ΨA cAi
j = I, II
(10)
The transport through the membrane is then expressed as I II JA = kAM (cAM − cAM )
(11)
Note: the concentrations of the component A in Eqs. (10) and (11) are referred to the membrane volume. 3. Experimental The experimental set-up used is shown in Fig. 3. Its main part was a laboratory dialyzer with two identical compartments separated by an anion-exchange membrane Neosepta-AFN developed and produced by Tokuyama Soda Company, Inc. (Japan). The membrane area was 3.31 × 10−2 m2 , the basic properties of the membrane are summarized in Table 1. Table 1 Properties of Neosepta-AFN membrane used Parameter
Value
Unit
Thickness Water content
160 0.418
Concentration of fixed charges
4.7
m g per g of dry membrane in Cl− form kmol m−3
Note
Considered to be monovalent
Fig. 3. Experimental set-up: 1, continuous dialyzer; 2, membrane; 3, peristaltic pump; 4, bottle with feed; 5, bottle with distilled water; 6, bottle with dialyzate; 7, bottle with diffusate; I and II, compartments.
The height of the dialyzer was 1 m, the dimensions of each compartment were 0.92 m × 0.036 m × 0.0011 m (height × width × thickness). In both the compartments there were net-type spacers made of PVC also acting as turbulence promoters. In all the experiments we used aqueous solutions of sulfuric acid. The feed of sulfuric acid entered the bottom of the compartment I (z = 0), while distilled water entered the top of the compartment II (z = zT ). The flow of both liquid streams was ensured by a peristaltic pump MC-MS/CA4 (Ismatec SA, Switzerland). In all the experiments, liquid flow rate of the feed was always equal to that of water (V˙ inI = V˙ inII ). After the steady state has been reached (a time period from 6 to 12 h in the dependence on liquid flow rate and acid concentration was necessary—as proved by preliminary experiments), three samples were taken from each stream leaving the dialyzer. At the same time, mass flow rate of both these streams was determined by weighing. Before each experiment, also mass flow rate of both feed and water was measured. Using density of liquid, mass flow rate of each stream was recalculated into volumetric liquid flow. All the experiments were carried out at a laboratory temperature of 20 ± 1 ◦ C. The acid concentration in the feed was changed in the range from 0.1 to 2.0 kmol m−3 , while the volumetric flow rate of the inlet streams was in the limits from 7.0 × 10−9 to 23.0 × 10−9 m3 s−1 (i.e. liquid flow rate was from 1.8 × 10−4 to 5.8 × 10−4 m s−1 ). The acid concentration was
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4.2. Permeability of membrane, PA In order to calculate permeability of the membrane, the procedure consisting of the following steps was used: 1. The initial estimate of PA (PA = 10−6 m s−1 ). The calcuj lation of the derivation du dz (j = I, II) from the experimental j j values of V˙ in and V˙ out (j = I, II). 2. The numerical integration of the set of Eqs. (2) and (3), where JA is expressed by Eq. (7) (at the same time, Eqs. (6) and (8) were used, too), by the Runge–Kutta fourth order method with the integration step h = 0.001 m. In each integration step, it was necessary to calculate the concentration of acid at the membrane/liquid interface, kinematic viscosity and diffusivity of sulfuric acid. As viscosity of solution and diffusivity of acid depend upon its concentration, it was necessary to determine these parameters in both liquid films by an iterative procedure using literature data [16–18]. (A) The estimate of the concentrations for the determinaI = cI II = cII tion of DA and ν—for z = 0: c¯ A ¯A A,out ; for A,in , c z > 0 these concentrations were taken to be equal to those obtained in the previous integration step. (B) The calculation of liquid mass transfer coefficients from Eq. (9). (C) The calculation of the acid concentration at the memI and cII , by solving Eqs. brane/liquid interface, i.e. cAi Ai (6)–(8). j (D) The correction of the acid concentrations c¯ A (j = I, II): (0)
Fig. 4. Dependence of overall dialysis coefficient upon volumetric flow rate of feed.
determined by titration, a standard 0.1 M NaOH solution being used. 4. Data treatment and discussion The basic transport characteristics (i.e. KA , PA and kAM ) were calculated from the data measured in continuous dialyzer at steady state by the following way. 4.1. Overall dialysis coefficient, KA
j
j
(12)
cA + cAi j = I, II (13) 2 (E) The steps from (B) to (D) were repeated until the relative j changes in the concentrations c¯ A (j = I, II) were below 0.05%. 3. The calculation of the objective function (12). 4. The realization of one step of the optimizing procedure. For that purpose, we used the Golden Section Search. This step provides the corrected value of PA . 5. The procedures from (2) to (4) were repeated until reaching the minimum of the objective function (12).
In Fig. 4 the values of KA calculated in such a way are plotted versus volumetric flow rate V˙ inI (full symbols). The parameter of the individual lines is the acid concentration in the feed. These dependences can be approximated with a sufficient accuracy by straight lines. Moreover, we calculated also the overall dialysis coefficients using the common procedure based on a logarithmic mean of driving forces at both ends of the dialyzer. Here, these are drawn by empty symbols. From the graphical presentation in Fig. 4, it can be seen that higher differences between the values of KA obtained by both procedures exist only at low acid I concentrations in the feed, i.e. at cA,in = 0.1 and 0.5 kmol m−3 . As shown in Fig. 4, the overall dialysis coefficients exhibit low sensitivity to volumetric flow rate. But on the other hand, they are negatively influenced by increasing acid concentration in the feed.
The procedure for obtaining permeability of the membrane given above needs the value of the constant C in Eq. (9). Its determination was based on the following considerations: permeability of the membrane is a membrane/solution parameter, which is not affected by flow of liquid. This means that the values of permeability obtained at various liquid flow rates and constant acid concentration in the feed must not be highly different from each other. For that reason, we defined a new criterion S¯ as a sum of I variances of PA at a constant cA,in , which is plotted versus the constant C in Fig. 5. From this dependence C was estimated to be approximately 12.5—the C at which this new criterion reaches practically a constant value. From Fig. 6, where permeability of the membrane is plotted versus the acid concentration in the feed, it is evident that
The overall dialysis coefficient was determined by the numerical integration of the basic differential Eqs. (2) and (3), where JA is expressed by Eq. (5), with the initial conditions (4). Eqs. (2) and (3) were integrated by the Runge–Kutta fourth order method with the integration step h = 0.001 m. The calculated values of I,calc II,calc and cA,in were obtained in this step. the concentrations cA,out Using a one-dimensional optimizing procedure (Golden Section Search was used) such values of KA were searched, at which the objective function (12) reached a minimum: I,exp
2
2
I,calc II,calc F = (cA,out − cA,out ) + (cA,in )
j
c¯ A =
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¯ upon constant C in Eq. (9). Fig. 5. Dependence of sum of variances of PA , S,
this transport characteristic of the membrane/solution system considered is affected by acid concentration—it decreases with increasing acid concentration. 4.3. Membrane mass transfer coefficient, kAM This transport characteristic was determined by the procedure, which was very similar to that used in the case of PA . The only difference was the calculation of the concentrations j cAM (j = I, II). For that purpose, we solved the set of Eqs. (6), (8), (10) and (11) by the Newton–Raphson procedure. The partij tion coefficients,ΨA (j = I, II), were calculated from the sorption isotherm, whose parameters are given elsewhere [19]. In the calculation of kAM , the original sorption isotherm, where the concentration in the membrane was referred to the volume of liquid inside the membrane, was recalculated using the porosity of the swollen membrane (εM = 0.30) in such a way so that the concentration would refer to the membrane volume. Fig. 7 presents the dependence of the membrane mass transfer coefficient upon the acid concentration in the feed. The course of this dependence is very similar to that published in Ref. [19], where the values of kAM were calculated on the basis of mea-
Fig. 7. Dependence of membrane mass transfer coefficient upon acid concentration in feed.
surement in a batch cell. Moreover, it can be seen that, except the lowest acid concentration used, a good agreement between the membrane mass transfer coefficients obtained by both procedures exists. (Note: a detailed inspection of the model used in Ref. [19] reveals the fact that, in reality, the values of kAM given therein must be understood as a product of the membrane mass transfer coefficients and porosity of the swollen membrane.) 5. Conclusion The procedure, which enables to calculate the basic transport characteristics from data measured in a continuous dialyzer at steady state, was suggested. It is based on the numerical integration of the set of ordinary differential equations describing the concentration dependences of the component dialyzed in both compartments upon the height coordinate. The integration is followed by the optimizing procedure. The model used takes into account a variable flow of liquid along the membrane as a consequence of the solvent transport through the membrane. For illustration, using data obtained in dialysis of sulfuric acid, the overall dialysis coefficient, permeability of the membrane and membrane mass transfer coefficient were determined by the procedure mentioned above. The results are presented graphically as dependences of these characteristics upon the volumetric flow rate of the feed and acid concentration in this stream. Acknowledgement This work was financially supported by the Ministry of Education, Youth and Sports of the Czech Republic, Project MSM 0021627502. Appendix A. Nomenclature
Fig. 6. Dependence of permeability coefficient of membrane upon acid concentration in feed.
A c c¯
membrane area (m2 ) molar concentration (kmol m−3 ) concentration for determination of diffusivity and viscosity (kmol m−3 )
Z. Palat´y et al. / Chemical Engineering and Processing 45 (2006) 806–811
C D d de F J kL kM K P Re S S¯ Sc Sh u V˙ z zT
constant in Eq. (9) diffusivity (m2 s−1 ) width of compartment (m) (=2 dδ/(d + δ)) equivalent diameter of compartment (m) objective function (kmol2 m−6 ) molar flux (kmol m−2 s−1 ) liquid mass transfer coefficient (m s−1 ) membrane mass transfer coefficient (m s−1 ) overall dialysis coefficient (m s−1 ) permeability of membrane (m s−1 ) (=ude /ν) Reynolds number cross-section of compartment (m2 ) sum of variances (m2 s−2 ) (=ν/DA ) Schmidt number (=kL DA /de ) Sherwood number liquid flow rate (m s−1 ) volumetric flow rate (m3 s−1 ) coordinate (m) height of compartment (m)
Greek symbols δ thickness of compartment (m) ε porosity ν kinematic viscosity (m2 s−1 ) Ψ partition coefficient Superscripts and subscripts A referred to component A calc calculated value exp experimental value i referred to solution/membrane interface in inlet
I, II M out (0)
811
referred to compartments I and II, respectively referred to membrane outlet initial estimate
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