The bond-graph description of transport processes was applied to analyse all the phenomena related to the sorption of free electrolytes into the internal solution of a polymeric cation-exchange membrane (no cation exchange reaction in the system). The kinetic constants for Nafion 120 and KNO3 solution were determined

Szczepańska, Grażyna

Szczepański, Piotr

Wódzki, Romuald

English

Repository of Nicolaus Copernicus University

 XXIII ARS SEPARATORIA – Toruń, Poland 2008  311  BOND-GRAPH DESCRIPTION AND SIMULATION  OF MEMBRANE PROCESSES. 4. NON-EXCHANGE SORPTION OF 1-1 ELECTROLYTE  INTO ION-EXCHANGE POLYMER MEMBRANE  GraŜyna SZCZEPAŃSKA, Piotr SZCZEPAŃSKI, Romuald WÓDZKI Nicolaus Copernicus University, Faculty of Chemistry,  Gagarin Str. 7, 87-100 Toruń, Poland wodzki@chem.uni.torun.pl   Abstract The bond-graph description of transport processes was applied to analyse all the phenomena related to the sorption of free electrolytes into the internal solution of a polymeric cation-exchange membrane (no cation exchange reaction in the system). The kinetic constants for Nafion 120 and KNO3 solution were determined.  1. INTRODUCTION The problem of high non-exchange sorption of electrolytes (NESE) into the internal solution ion-exchange membranes wa intensively studied in the context of its quantitative discrepancy with the classic Donnan equilibrium principle [1,2]. It was concluded that high NESE results from non-ideality of a membrane leading to the sorption of co-ions higher than that theoretically predicted. In practice, the membranes are of heterogeneous structure, i.e. they are composed of an inert backbone and an ion exchanging hydrogel dispersed therein (Fig.1).        Fig. 1. Scheme of ion-exchange polymer membrane structure   Additionally, a specific spatial distribution of the ion-exchange sites can be non-uniform which causes substantial deviation from the classic Donnan law for NESE. The equilibrium sorption was a subject of studies as () counter-ions () functional groups () co-ions  XXIII ARS SEPARATORIA – Toruń, Poland 2008  312a fundamental property of the system membrane|electro yte solution, and the corresponding sorption isotherms were proposed. On the other hand, the kinetics of this process and its detailed mechanism was not satisfactorily described. The electrolyte, when sorbed from an external medium, permeates the electrolyte|membrane interface and diffuses in the membrane with a diffusion coefficient markedly reduced by tortuosity of diffusion pathways. Thus, these two factors should be taken into account for constructing a phenomenological and mathematical description of the kinetics of the electrolyte sorption.   2. THEORY The most general theory that describes the permeation across any heterogeneous medium is the theory of percolation [3] which suggests that the dependence between the diffusion coefficient in the membrane (Deff) is related to the common diffusion coefficients in the membrane internal solution (Do) by the following equation: ( )0eff a cD D V Vτ= × −             (1) In Eq. (1), Va, and Vc denote the accessible and critical volume fraction of the internal electrolyte, respectively; τ is the universal critical exponent. Non-exchange sorption of the electrolyte into a heterogeneous membrane with non-uniform distribution of ion-exchanging sites can be described by the isotherm proposed by Glueckauf [2]: ( 2 )zA G AC k C−= ×         (2) where AC , CA are molal concentrations of co-ions (free electrolyte) in the internal membrane solution and external solution, respectively; kG – constant, z – the non-uniformity parameter characteristic for a membrane. According to Eq.(2) the distribution coefficient ofco-ions is represented by:  | (1 )( ) /f m zA w A A G AC C k Cα−= = ×    (3)  2.1. COMPARTMENTAL AND NETWORK MODEL The compartmental model of the external electrolyte (f) | cation exchange membrane (m) with a possible concentration profile of the sorbed and diffusing species S is presented in Fig.2 According to the scheme, the interfacial diffusion layer of the electrolyte of Lf thickness was divided into nf sub-layers, and the membrane of Lm thickness into nm sub-layers. The index b denotes the bulk solution, i.e. this part of the external electrolyte which is intensively agitated (convective transport only).   XXIII ARS SEPARATORIA – Toruń, Poland 2008  313     bf  1f  2f   ... 1fnf − fnf  1m  2m   ... 1mnm −   mnm                  Fig. 2. Compartmental model for external electrolyte – cation exchange membrane system  The −+ AK  species can permeate between the sub-layers and can be accumulated (or exhausted) in dependence on the local capacitance. Sorption of −+ AK  occurs at the interface, i.e. between the sub-layers marked as nf and m1. By applying the graphs of diffusion and reaction [4] one can construct the linear network of the overall process presented in Fig.3. The construction is based on the properties of “0” junction which obeys Kirchhoff Current Law.                        sorption (R1) diffusion diffusion 0 ( )( 1){KA}mi + … 0 ( )( 1){KA} mmn − mKAD  0 ( )( ){KA} mmn 0 ( )(1){KA}m  1R  mKAD  ( )(2){KA}m  … 0 ( )( ){KA}mi mKAD  0  ( )( ){KA} ffn fKAD  0 ( )( 1){KA}fi + fKAD  0 ( )( 1){KA} ffn − 0 ( )( ){KA}fi 0 ( )(2){KA}f  fKAD  0 ( )(1){KA}f  0 ( )( ){KA}fb fKAD  Fig. 3. Network model for non exchange sorption of electrolyte −+ AK  into cation-exchange membrane  XXIII ARS SEPARATORIA – Toruń, Poland 2008  3143. CALCULATIONS The network model in Fig.3 is equivalently represented by a set of ordinary differential equations describing the state of each capacitance as dependent on local flows (Tab. 1). The model can be solved numerically (Berkeley Madonna ver. 8.01) for a given set of operational, initial and boundary conditions.  Tab. 1. Mathematical descriptions of nonexchange sorption of  1-1 electrolyte into cation-exchange (CE) polymer membrane.  Local flows (N ) [mol/s]  External solution (f) ( )1 12 / [ ] [ ]bf f f f f fKA KA i bN D A L KA KA→ = × × × −( )1 1/ [ ] [ ]i if f f f f fKA KA i i iN D A L KA KA+→ += × × −  External solution | membrane interface (f | m) ( )1 1 1|, 1 ( )[ ] [ ]f f ffn m n m m nf mKA KA eff A wnN k A KA KA α→ →= × × −  CE polymer membrane (m) ( )1 1/ [ ] [ ]i im m m m m mKA KA i i iN D A L KA KA+→ += × × −   Capacitance fluxes [mol/cm3s]   External solution (f) [ ] 1/ /bf f f fKA bbd KA dt N V→= −  [ ] ( )1 1 21 / /bf f f f f fKA KA id KA dt N N V→ →= −[ ] ( )1 1/ /i i i if f f f f fKA KA iid KA dt N N V− +→ →= −[ ] ( )11/ /ff fff n mn n fKA KA ind KA dt N N V→− →= −   CE polymer membrane (m) [ ] ( )1 1 21 / /fm n m m m mKA KA id KA dt N N V→ →= −[ ] ( )1 1/ /i i i im m m m m mKA KA iid KA dt N N V− +→ →= −[ ] 1/ /m mmm n n mKA ind KA dt N V− →=   A- contact area between the membrane and electrolyte, [cm2]; 1,fn mKA effk→ - the effective permeability coefficient (mas transfer constant) of −+ AK  at the f|m interface, [cm/s]; 1/( )fm nA wα - the distribution coefficient of −A  co-ions between the membrane internal solution (sub-layer m1) and the external electrolyte solution (sub-layer nf); fKAD , mKAD - the diffusion coefficient of −+ AK  in the external electrolyte and membrane [cm2/s]; fiL , miL - the way of −+ AK  diffusion between the adjacent sub-layers in the ext rnal electrolyte and membrane, [cm]; fiV , miV - volume of i sub-layer of the external electrolyte and membrane, [cm3].  3.1. PROBLEM TO SOLVE The main goal of calculations was to calibrate the int rfacial f|m permeability coefficient 1,fn mKA effk→ . This quantity cannot be measured  XXIII ARS SEPARATORIA – Toruń, Poland 2008  315 immediately by experiments. No information is accessible on its value for KNO3 and the Nafion membrane as well. On the other hand, the value of 1,fn mKA effk→  is also necessary for constructing a reliable model f sorption. For this purpose, the dependence between the overall permeability coefficient |,f mKA effk  (which can be determined experimentally) and the local (interfacial) coefficient 1,fn mKA effk→  should be recognized. The equation relating the respective experimental variables to the time of sorpti n is:    ( )tkKAKAKAKAKAAKAKAVmfcalcKAftbftbftbfbftbftbftbfb ×−=−−×−×∞→=∞→=∞→= /,)()0()()0()()0(][][][][ln][][][      (4)  In Eq.(4) [ ] fbKA , ( 0)[ ]fb tKA = , ( )[ ]fb tKA →∞  are the concentrations of electrolyte in the bulk phase of the external solution at time t, t=0, and ∞→t , respectively.  The permeability coefficient ,f mKA calck can be calculated as a slope of the corresponding linear regression function. The value of ,f mKA calck calculated for the assumed series of 1,fn mKA effk→  can be compared thereafter with the experimental value, and the proper interfacial permeability coefficient can be estimated. The following physicochemical, structural and experimental characteristics were used in calculations: the initial concentration of the external electrolyte ftbKA )0(][ = = 1×10-4 mol/cm3; the volume of the external electrolyte, fV = 1000 cm3;  the membrane area, A= 9.2 cm2, and thickness mL = 0.0196 cm; the thickness of the diffusion layer at the f|m interface fL = 0.005 cm, [5]; the critical exponent τ = 1.5 [6]; the critical volume fraction of the internal membrane electrolyte kv = 0.10 [6]; the constant kG = 0.91 [7]; the membrane non-uniformity parameter z = 0.73 [7]; the volume fraction of the internal electrolyte in membrane mwV )( = 0.33. The diffusion coefficients of KNO3≡ −+ AK  in the aqueous solution of 1×10-4 mol/cm3 concentration was equal to fKAD =1.43×10-5 cm2/s value of which was evaluated using literature data [8]. Similarly, the value of the diffusion coefficient in the internal membrane solution was asumed as mKAD 0, =1.17×10-5 cm2/s. After the iterative procedure, the following curve calibrating the dependence of ,f mKA calck  vs. 1,fn mKA effk→  was found (Fig. 4).  XXIII ARS SEPARATORIA – Toruń, Poland 2008  316From the curve presented in Fig.4, one can conclude that the overall permeability coefficient , expf mKAk  = ,f mKA calck = 4×10-5 found experimentally corresponds with the interfacial one 1,fn mKA effk→ = 6.74×10-5 cm/s.                    Fig. 4. The relationship between the overall ,f mKA calck  and local1,fn mKA effk→ permeability coefficient of KNO3.   REFERENCES [1] E. Glueckauf, Proc. Roy. Soc. London, 1962, A 262, 350. [2] E. Glueckauf, R.E. Watts, Proc. Roy. Soc. London, 1962, A 268, 339. [3] W. Y. Hsu, T. D. Gierke, J. Membr. Sci., 1983, 13, 307. [4] J. Schnakenberg, Thermodynamic Network Analysis of Biological System, Springer Verlag, Berlin-Heidelberg-New York, 1977. [5] S. Logette, C. Eyssenie, G. Pourcelly, A. Lindheimer, C. Gavach, J. Membrane Sci., 1988, 144, 259. [6] R. Wódzki, A. Narębska, W. Kwaśnioch, J. Appl. Polym. Sci., 1985, 30, 769. [7] G. Szczepańska, Unpublished data, (PhD thesis in preparation). [8] V.M.M. Lobo, Pure & Appl. Chem., 1993, 65, 2613.   ( ) 1,ff n mKA effk→1e-6 1e-5 1e-4 1e-3 1e-21e-71e-61e-51e-4k KA, calcf / mexp. 

Bond-graph description and simulation of membrane processes. 4. Non-exchange sorption of 1-1 electrolyte into ion-exchange polymer membrane.

Thermodynamic network analysis was applied to study the electrolyte permeation across a heterogeneous cation-exchange polymer membrane. The network of local flows and the mathematical description was used to evaluate the diffusion coefficient of KNO3 in the internal solution of Nafion membran

 XXIII ARS SEPARATORIA – Toruń, Poland 2008  317  BOND-GRAPH DESCRIPTION AND SIMULATION  OF MEMBRANE PROCESSES. 5. PERMEATION OF 1-1 ELECTROLYTE  ACROSS ION-EXCHANGE POLYMER MEMBRANE  GraŜyna SZCZEPAŃSKA, Piotr SZCZEPAŃSKI, Romuald WÓDZKI  Nicolaus Copernicus University, Faculty of Chemistry Gagarin Str. 7, 87-100 Toruń, Poland wodzki@chem.uni.torun.pl   Abstract Thermodynamic network analysis was applied to study he electrolyte permeation across a heterogeneous cation-exchange polymer membrane. The network of local flows and the mathematical description was used to evaluate the diffusion coefficient of KNO3 in the internal solution of Nafion membrane   1. INTRODUCTION The permeation of free electrolytes (electrolyte leakage) across a cation-exchange polymer membrane (CEM) is associated with similar physicochemical phenomena as the sorption of electrolyte described in the preceding Part 4. Additionally, some simplifying assumption could be made to formulate a simple model in the framework of network thermodynamic analysis [1]: (i) the cation exchange sites are fully loaded with the same counter-ion as the one present in the ext rnal feed solution (f), (i.e. no ion-exchange reaction in the system), and (ii ) osmotic transport of water between the feed (f) and stripping solution (s) across the membrane (m) can be ignored. The model needs: (i) the reliable description of the membrane structure, (ii ) description of local diffusion phenomena, (iii ) description of non-exchange sorption and desorptin of co-ions (electrolyte) at the f|m an m|s interface, respectively.   2. THEORY The full description of electrolyte permeation in the system: feed solution |CEM| stripping solution can be achieved by a respective  XXIII ARS SEPARATORIA – Toruń, Poland 2008  318modification of the model presented in the Part 4. The scheme of the corresponding compartmental model is presented in Fig.1, i.e. the feed solution contains the electrolyte −+ AK , which can permeate to the stripping solution according to the following sequenc  of phenomena: diffusion at the f|m interface – sorption to the membrane internal solution – diffusion in the internal membrane solution – desorpti n from the membrane to the stripping solution – diffusion in the stripping solution at the m|s interface. To apply the network method of modeling, the diffusion layers in external solutions and the membrane were additionally divided into nf, ns and nm sub-layers.     bf  1f  ... 1fnf −  fnf  1m  ... 1mnm −  mnm  1s  ... 1sns −  sns  bs                      Fig. 1. Compartmental model for permeation of electroly e in the system: feed solution | cation exchange membrane | stripping solution  The electrolyte −+ AK  permeates between the indicated sub-layers, and at the respective interfaces is sorbed (nf|m1) or desorbed (nm|s1) into or from the membrane phase. The overall permeation process can be represented by a series of the accumulation-diffusion graphs. Also, for a bond-graph description, it was assumed that the sorption and desorption processes can be represented by the graphs of uni-molecular reversible reactions:  { } { }1 1KA KAff mnR→ →     (1)     { } { }2 1KA KAmm snR→ →     (2)  Consequently, each local flow and accumulation of the electrolyte in the sub-layers is represented by the diffusion and sorption graphs connected by “0” junctions. The corresponding linear network of the electrolyte permeation from the feed to the stripping solution with mediating non-exchange sorption/desorption of the electrolyte is presented in Fig. 2.  sorption (R1) desorption (R2) diffusion diffusion diffusion  XXIII ARS SEPARATORIA – Toruń, Poland 2008  319 3. CALCULATIONS The mathematical model for permeation of the electrolyte according to the network in Fig.2 is presented in Tab.1. The model is composed of a set of differential equations describing the capacitance fluxes as dependent on the local input and output diffusion fluxes. The model can be used for simulation of the process when all the physicochemical constants (diffusion coefficients, interfacial permeability coefficients), membrane structural parameters (z, t, kG, Vk), boundary and initial conditions for all the variables appearing in the model, and operational conditions (volume, thickness, etc.) are known. For such data, the model can be solved and it can generate the quantities which are commonly observed in this type of xperiments.                             Fig. 2. Network model for permeation of electrolyte −+ AK  across a cation-exchange membrane  0 ( )( 1){KA}mi +  … 0 ( )( 1){KA} mmn − mKAD  0 ( )( ){KA} mmn 0 ( )(1){KA}m  1R  mKAD  ( )(2){KA}m  … 0 ( )( ){KA}mi  mKAD  0 2R  ( )( ){KA} ffn fKAD  0 ( )( 1){KA}fi +  fKAD  0 ( )( 1){KA} ffn − 0 ( )( ){KA}fi  0 ( )(2){KA}f  fKAD  0 ( )(1){KA}s  sKAD  0 ( )( ){KA}si  sKAD  0 ( )(2){KA}s  0 ( )( 1){KA}si+  0 ( )( 1){KA} ssn − sKAD  0 ( )( ){KA} ssn 0 ( )( ){KA}sb  sKAD  ( )(1){KA}f  0 ( )( ){KA}fb  fKAD   XXIII ARS SEPARATORIA – Toruń, Poland 2008  320Tab. 1. Mathematical descriptions of permeation of 1-1 electrolyte across ion-exchange polymer membrane  Local flows (N ) [mol/s]  Feed solution (f) 112 / [ ] [ ] )bf f f f f fKA KA i bN D A L KA KA→ = × × × −11/ ([ ] [ ] )i if f f f f fKA KA i i iN D A L KA KA+→+= × × −  Feed solution | membrane interface (f | m) 1 1 1|, 1 ( )([ ] [ ] )f f ffn m n m m nf mKA KA eff A wnN k A KA KA α→ →= × × −  CEM (m) 11/ ([ ] [ ] )i im m m m m mKA KA i i iN D A L KA KA+→+= × × −  Membrane |strip solution interface (m | s) 1 1 1|, ( ) 1([ ] / [ ] )m m mmn s s n n sm sKA KA eff A wnN k A KA KAα→ →= × × −   Stripping solution (s) 11/ ([ ] [ ] )i is s s s s sKA KA i i iN D A L KA KA+→+= × × −2 / ([ ] [ ] )sbsn s s s s sKA KA i bnN D A L KA KA→ = × × × −   Capacitance fluxes [mol/cm3s]    Feed solution (f) [ ] 1/ /bf f f fKA bbd KA dt N V→= −[ ] 1 1 21 / ( ) /bf f f f f fKA KA id KA dt N N V→ →= −[ ] 1 1/ ( ) /i i i if f f f f fKA KA iid KA dt N N V− +→ →= −[ ] 11/ ( ) /ff fff n mn n fKA KA ind KA dt N N V→− →= −   CEM (m) [ ] 1 1 21 / ( ) /fm n m m m mKA KA id KA dt N N V→ →= −[ ] 1 1/ /i i i im m m m m mKA KA iid KA dt N N V− =→ →= −[ ] 11/ /mm mmm n sn n mKA KA ind KA dt N N V→− →= −    Stripping solution (s) [ ] 1 1 21 / ( ) /ms n s s s sKA KA id KA dt N N V→ →= −[ ] 1 1/ ( ) /i i i is s s s s sKA KA iid KA dt N N V− +→ →= −[ ] 1/ ( ) /ss s bss n sn n sKA KA ind KA dt N N V→− →= −[ ] / /s bs n s sKA bbd KA dt N V→= −   A- contact area between membrane and electrolyte, [cm2]; 1,fn mKA effk→ and 1,fn mKA effk→ - the effective permeability coefficient (mas transfer constant) of −+ AK  at the f|m and m|f interface respectively, [cm/s]; )(/1)(fnfmwAα  and ( ) / 1( )mm n sA wα - the distribution coefficient of −A  co-ions at the nf|m1 and nm|s1 interface, respectively; fKAD , sKAD , mKAD - the diffusion coefficient of −+ AK  in the feed, stripping solution and membrane [cm2/s]; fiL , siL ,miL - distance for −+ AK  diffusion between the adjacent sub-layers in the feed, stripping solution and membrane, [cm]; fiV , siV , miV - volume of i sublayer of the feed, stripping solution and membrane., [cm3].  XXIII ARS SEPARATORIA – Toruń, Poland 2008  321 3.1. PROBLEM TO SOLVE A fundamental problem in the studies of membrane permeability is the information on the exact value of the diffusion coefficients in the internal membrane solution ,0mKAD . Most frequently, this quantity is estimated as the diffusion coefficient in aqueous media additionally corrected with the use of the so called tortuosity factor. On the other hand, the experimental quantity commonly determined from the studies on “diaphragm permeability” is the permeability coefficient (effective diffusion coefficient, mass transfer coefficient) f sP . Its value can be calculated from the time dependence of the concentration of permeating species in the feed and stripping solution:   ( ) [ ] [ ][ ] ( ) [ ] ( )1,0 01/ 1/ lnf sf sf s b bb b KA calcf smb t b tKA KAAV V P tL KA KA−= = −   − × + × = ×     −            (3)  In Eq.(3): [ ] fbKA , [ ]sbKA , ( 0)[ ]fb tKA = , ( 0)[ ]sb tKA =  denote the concentrations of the electrolyte in the bulk phase of the feed (f) and stripping (s) solution at time t and t=0, respectively. The permeability coefficient ,f sKA calcP can be calculated as the slope of the corresponding linear regression function using simulated concentrations of −+ AK  - in the bulk external solutions. After a comparison of the ,f sKA calcP  coefficients, as calculated for a series of assumed ,0mKAD  values, with the experimental ,expf sKAP coefficient, the proper value of unknown ,0mKAD  can be evaluated by the standard analysis of ( ), ,0f s mKA calc KAP f D= dependence. In this way, the values of ,0mKAD  can be adjusted to the real behavior of the system. To verify the model applicability, the calculations were carried out for the Nafion-120 membrane in contact with KNO3 solution as the feed phase. All other characteristics necessary for calculations (Berkeley Madonna v. 08) correspondng with the experimental system were the same as described in Part 4. Additionally, it was assumed that the effectiv  permeability constants of KNO3 at the interfaces are 13 ,fn mKNO effk→ = 13 ,ms nKNO effk→ = 6.74×10-5 [cm/s] the value of which was evaluated after calculations f the electrolyte sorption according to the model described in Part 4.  The results of calculations are presented in Fig.3 as the plot of 3 ,f sKNO calcP  vs. 3 ,0mKNOD . As it is seen, the experimental value of the overall KNO3 permeability corresponds with ,0mKAD  value equal to 7.98×10-6 cm2/s.  XXIII ARS SEPARATORIA – Toruń, Poland 2008  322                 Fig. 3. Dependence between overall permeability coeffi ient ,f sKA calcP and the diffusion coefficient in the internal electrolyte of cation-exchange membrane  ( )3K A KNO+ − ≡ .    REFERENCES [1] J. Schnakenberg, Thermodynamic Network Analysis of Biological System, Springer Verlag, Berlin-Heidelberg-New York, 1977 1e-6 1e-5 1e-4 1e-31e-81e-71e-6,0mKADP KA, calcf /sexp. 

Bond-graph description and simulation of membrane processes. 5. Permeation of 1-1 electrolyte across ion-exchange polymer membrane.

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Bond-graph description and simulation of membrane processes. 4. Non-exchange sorption of 1-1 electrolyte into ion-exchange polymer membrane.

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