Formulation for the Prediction of Ionic Concentration Profile in an Electrochemical System

Available online at www.rajournals.in RA JOURNAL OF APPLIED RESEARCH ISSN: 2394-6709 DOI:10.47191/rajar/v11i10.08 Volume: 11 Issue: 10 October 2025 International Open Access Impact Factor8.553 Page no.- 896-900 896 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 10 October 2025 Formulation for the Prediction of Ionic Concentration Profile in an Electrochemical System Sarwan S. Sandhu1, Nil N. Panchal2 1,2Department of Chemical and Material Engineering, University of Dayton, OH 45469, USA ARTICLE INFO ABSTRACT Published Online: 15 October 2025 Corresponding Author: Sarwan S. Sandhu An analytical expression to predict the transient concentration profile of a charge-transporting cation in the electrolyte in the vicinity of the cathode electrode of an electrochemical cell is presented in this relatively short paper. To illustrate the application of the developed formulation, the computed data are presented in the form of the Z versus Y plots for four effective diffusivity values of the cation and noting the involvement of the Z(y)-function in the prediction of the cation transient concentration profile. KEYWORDS: diffusion, ionic, electrochemical, transient, concentration I. INTRODUCTION We, at the University of Dayton, have been involved in the development of electrochemical systems, i.e. fuel cells and batteries since 1996 [1-12] to deliver electric power more efficiently and safely for space and terrestrial applications. In this pursuit, very recently, it was decided to develop the formulation to predict the transient ionic concentration profiles in the electrolyte between the electrodes of an electrochemical cell. Electrochemical systems use an electrolyte in the liquid or solid phase. When an electrochemical cell operation is initiated to deliver electric power to an external electric load at a fixed current level, the cell usually undergoes through transient behavior. That is, the ionic species local (spatial) concentrations change. Figure 1 shows the simple sketch of an electrochemical system used in the development of the formulation presented in Section 2. Figure 1: Simple sketch of an electrochemical system. II. FORMULATION It is, here, assumed that metallic ions, (e.g., or ions) are generated at the plane surface of the cell anode in contact with the electrolyte during its discharge to provide electric power to an external load. The partial differential equation [13] describing the transport of an ionic species via diffusion is: 2 2 diff cc D tx  =  (1) where c [=] 3)( mol cm is the molar concentration of an ionic species, n M+ , in the cell electrolyte separator between the electrodes, diff D [=] 21 )( m sc − is the ion effective diffusivity which is conventionally taken to be equal to the product of the ion diffusivity in the electrolyte and the separator volume void fraction divided by the tortuosity factor to account for the zigzag path along which an ion migrates, and x [=] (cm) is the spatial coordinate as shown in the sketch, and t [=] (s) is time. To solve the partial differential equation (1), the following initial and boundary conditions have been used: I.C. 0 0 ini t c c c for x    = =  (2) B.C. 1 0 0 diff ci at y for all t x nFD  = =   (3) B.C. 2 0 at x c for a tc ll  = → (4) where i [=] ( ) 2 geom A cm−  is the cell geometric current density, F is the Faraday’s constant = 96487 Coulombs of charge per g-equivalent, n is the oxidation state number of a cation (e.g., n = 1 for Na+ ions, n = 2 for Cu2+ ions). The analytical solution of the P.D.E., Equation (1), is obtained using the Laplace transform method as follows: “Formulation for the Prediction of Ionic Concentration Profile in an Electrochemical System” 897 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 10 October 2025 taking the Laplace transform of the P.D.E., Equation (1) and the conditions, Equation (2) through (4): ( ) ( ) 22 22 diff diff cc L D L D c txx      ==       L (5) ( ) ( ) ( ) 2 2 , 0, diff dc sL c t x c att x D dx − = = (6) with diff D assumed to be constant. 2 2 ini diff dc sc c D dx −= (7) where ( ) , , c c s x s== the Laplace parameter [=] ( ) 1 sec− . Rewriting Equation (7) as: 2 2 ini diff diff c d c s c DD dx     − − +=             (8) The Laplace transform of the boundary conditions, Equations (3) and (4) become, respectively: 1 0, 0; diff dc i t at x dx nFD s    = =      (9) 0, ; c t at x c s    →  =   (10) The linear differential Equation (8) has been solved using the methods provided in Chapter 3 of reference [14]. The resultant solution is: ( ) 3 2 , diff sx D ini diff cie c s x snF D s   −       =−       (11) Using the information on the inverse Laplace transforms of the functions [15, 16] ( ) ( ) 11 1 3 2 , diff ini sx D diff c c s x s ie nF D s −−   −  −  =−                    LL L (12) ( ) 1 1 3 2 1 , ini diff diff c t x c s sx iD nF D s − −  =−        −         L L (13) ( ) 2 , 2 4 2 ini diff diff diff diff i c t x c nF D tx exp tD xx erfc D tD    = −      −  −              (14) Or: ( ) 2 ,1 2 4 2 ini ini diff diff diff diff c t x i cnFc D tx exp tD xx erfc D tD     = −         −  −              (15) Equation (15) can also be expressed as: ( ) ( ) 2 ,1 2 4 ini ini diff diff diff c t x i c nFc tx exp D tD xerfc y D    = −         −  −         (16) where y is a dimensionless quantity defined as: 2diff x ytD  =   (17) Furthermore, ( ) ,1 ini ini c t x i c nFc   = −       (18) where, Z [=] (s cm-1) is a function of y and is defined below: ( ) 2 1.1284 exp 4 ) ( diff diff I diff II Ztx D tD xerfc y D y −−    = (19) Further rearrangement of the terms in Equation (18) leads to: ( ) ( ) ( ) ,, ini c c t x c c t x nF nF ii          −− = =                          (20) At any cell operational time t (s), Z can be calculated as a function of x or y. This way a Z vs. x or y profile becomes available at a time. Such profiles have been determined for some selected cell operational times for the same ion diffusivity values. By the utilization of Equation (19), (Z vs. y) data have been computed for some typical ion diffusivity values for an ion, Na+, migration through an electrolytecontaining separator between the cell electrodes. Once the “Formulation for the Prediction of Ionic Concentration Profile in an Electrochemical System” 898 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 10 October 2025 function, Z(x,t) or Z(y) for a given effective diffusivity is known, it is quite apparent from Equation (20) that ( ) ( ) ,c c t x − can be determined for an ion with its charge number (e.g. 1, Na n += ) for an electrode (or cell) current density of 2 , .i A cm−  With known c value, then ( ) ,c t x can be computed; thus, ( ) , . c t x vs x or y profile can be determined for a known or fixed value of the effective diffusivity of a cation in a solid or liquid electrolyte. Also, by the application of Equation (18), the dimensionless concentration ( ) ( ) , ini ini c t x c y cc     =             can be calculated at a desired 2diff x ytD  =    value. III. DISCUSSION OF THE CALCULATED DATA Figures 2 through 5 show the plots of the function, Z(y) vs. y, for four values of the effective diffusivity, ( ) 6 8 10 12 2 1 10 ,10 ,10 ,10 diff D cm s − − − − − = . For each diffusivity, the plots of Z vs. y are presented for three times of 4, 100 and 900 seconds as an illustration for the effect of a time period on the profile of Z vs. y The nature of the function Z vs. y plot for each diffusivity values is the same for the same time period except an increase in the value of Z by a factor of ten as the diffusivity changes from ( ) 6 8 10 12 2 1 10 ,10 ,10 ,10 cm s − − − − −  . The function Z decreases as 2diff x ytD  =    increases with an increase in the distance x (cm)for a fixed diffusivity value. It is so due to the terms I & II in Equation (19) approaching towards each other in their respective magnitudes with an increase in the distance x measured from the cell cathode-electrode, C, in the cell electrolyte in accordance with the initial and boundary conditions, Equations (2), (3), and (4). For the prediction of the dimensionless concentration profile of a cationic species near the cathode of an electrochemical cell, it is necessary to recognize that the values of the parameters, geometric current density, i; the charge number, n, on a cationic species being transported in the electrolyte separator; and the initial concentration, ini c , of the cationic species be such that the value of the lumped parameter ( / ) ini i nFc when multiplied with Z value at an x or y value is less than or equal to one. Figure 2: Plot of Z vs. y for Ddiff = 10-6 cm2 sec-1 Figure 3: Plot of Z vs. y for Ddiff = 10-8 cm2 sec-1 Figure 4: Plot of Z vs. y for Ddiff = 10-10 cm2 sec-1 Figure 5: Plot of Z vs. y for Ddiff = 10-12 cm2 sec-1 “Formulation for the Prediction of Ionic Concentration Profile in an Electrochemical System” 899 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 10 October 2025 Figures 6 and 7 are the plots of the dimensionless Na+ ion concentration versus the dimensionless distance y 2diff x tD  =    , representing the application of the developed formulation, as an illustration, to predict the concentration profile of a charge transporting ionic species in the electrolyte in the neighborhood of the cathode electrode of an electrochemical cell. The nature of these profiles are dependent on the product of the value of the group ini i nFc    and the value of the function, Z(y), at each value of y. These plots (Figures 6 & 7) are for the same diffusivity and the cell operational time of 4 second for two geometric current densities of 10 and 300 mA/cm2, respectively. Notice the two ( ) , . ini c t x vs y c     profiles approaching towards one with an increase in distance away from 0.0y= plane. Figure 6: Plot of ( ) ( ) , . ini ini c t x c y vs y cc  =    for 2 10 i mA cm− = Figure 7: Plot of ( ) , . 2diff ini x c t x vs y tD c   =       for 2 300 i mA cm− = The computed data presented here in the form of plots for some typical diffusivity values are for the illustration of the application of the developed formulation. An individual interested in the prediction of ( ) ( ) , . ini c t x Z y or vs y c         plot to gain insight of the concentration profile of a charge transporting cation near the cell cathode at any other diffusivity can use equations (18) and (19). IV. CONCLUDING REMARKS To predict the transient concentration profile of a cation in the electrolyte near the cathode of an electrochemical cell (a fuel cell or battery) used for delivery of electric power to an external load, the partial differential equation (1) was solved using the Laplace transform method with the initial and boundary conditions, equations (2), (3), and (4). The solution for the cation concentration profile in the dimensionless form is given by Equation (18). For a known value of the diffusivity of the chargetransporting cation in the cell electrolyte at any cell operational time, the function, 2diff x Zy tD  =    can be calculated and hence, the dimensionless concentration profile can be calculated from Equation (18) for a value of the lumped parameter group, ini i nFc    . One should be prudent to recognize that the product of this group value and the function ( ) Zy value must be less than one for the prediction of the cation dimensionless concentration profile. This implies that for known value of ( ) ini nand c , the cell operational geometric current density, i , should be such that the above-mentioned condition is always satisfied. REFERENCES 1. J.P. Fellner and S.S. Sandhu. Diffusion-limited SelfDischarge Reaction in the Hubble space telescope Battery. Journal of Power Sources, Vol. 58, pp. 099102 (1996). 2. J.P. Fellner and S.S. Sandhu. Diffusion-limited Model for a Lithium/Polymer Battery. Electrochimica Acta, Vol. 43, No. 10 and 11, pp. 1607-1613 (1998). 3. S.S. Sandhu and J.P. Fellner. Thermodynamic Equations for a Model Lithium-Ion cell. Journal of Electrochimica Acta, Vol. 45, No. 6, pp. 969-976 (1999). 4. S.S. Sandhu, Y.A. Saif, and J.P. Fellner. A reformer performance model for fuel cell applications. Jornal of Power Sources, Vol. 140, No. 1, pp. 88-102 (2005). 5. S.S. Sandhu, R.O. Crowther, and J.P. Fellner. Prediction of Transport Fluxes of species ( 32 , , H CH OH H O + ) through a Solid Polymer Electrolyte Membrane of a Direct Methanol Fuel “Formulation for the Prediction of Ionic Concentration Profile in an Electrochemical System” 900 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 10 October 2025 Cell. Journal of Electrochimica Acta, Vol. 50, pp. 3985-3991 (2005). 6. S.S. Sandhu and J.P. Fellner. Performance/design formulation for a solid polymer-based acid electrolyte hydrogen/air fuel cell. Journal of Power Sources, Vol. 161, No. 2, pp. 1133-1153 (2006). 7. S.S. Sandhu, J.P. Fellner and G.W. Brutchen. Diffusion-limited Model for a Lithium/Air Battery with an Organic Electrolyte. Journal of Power Sources, Vol. 164, pp. 365-371 (2007). 8. S.S. Sandhu and J.P. Fellner. Model Formulation and Simulation of a Solid-State Lithium-Based Cell. 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