Using the classical Kedem-Katchalsky' membrane transport theory, a mathematical model was developed and the original concentration volume flux (Jv), solute flux (Js) characteristics, and S-entropy production by Jv, ( ( ψ S ) J v ) and by Js ( ( ψ S ) J s ) in a double-membrane system were simulated. In this system, M1 and Mr membranes separated the l, m, and r compartments containing homogeneous solutions of one non-electrolytic substance. The compartment m consists of the infinitesimal layer of solution and its volume fulfills the condition Vm → 0. The volume of compartments l and r fulfills the condition Vl = Vr → ∞. At the initial moment, the concentrations of the solution in the cell satisfy the condition Cl < Cm < Cr. Based on this model, for fixed values of transport parameters of membranes (i.e., the reflection (σl, σr), hydraulic permeability (Lpl, Lpr), and solute permeability (ωl, ωr) coefficients), the original dependencies Cm = f(Cl - Cr), Jv = f(Cl - Cr), Js = f(Cl - Cr), ( Ψ S ) J v = f(Cl - Cr), ( Ψ S ) J s = f(Cl - Cr), Rv = f(Cl - Cr), and Rs = f(Cl - Cr) were calculated. Each of the obtained features was specially arranged as a pair of parabola, hyperbola, or other complex curves.

Department of Business Informatics University of Economics 40287 Katowice Poland

Department of Innovation and Safety Management Systems Technical University of Czestochowa 42200 Czestochowa Poland

Department of Security Services Faculty of Safety Engineering VSB Ostrava 70030 Ostrava Czech Republic

See more in PubMed

Demirel Y. Nonequilibrium Thermodynamics: Transport and Rate Processes in Physical, Chemical and Biological Systems. Elsevier; Amsterdam, The Netherlands: 2014. pp. 275–540.

Srivastava R.C., Saha S.K., Jain A.K. Thermodynamics: A Core Course. PHI Learning; New Delhi, India: 2010. pp. 214–273.

Delmotte M., Chanu J. Non-equilibrium thermodynamics and membrane potential measurement in biology. In: Millazzo G., editor. Topics Bioelectrochemistry and Bioenergetics. John Wiley Publish & Sons; Chichester, UK: 1979. pp. 307–359.

Prigogine I. Introduction to Thermodynamics of Irreversible Processes. Wiley; New York, NY, USA: 1968.

Baker R. Membrane Technology and Application. John Wiley & Sons; New York, NY, USA: 2012.

Cheng X., Pinsky P.M. The balance of fluid and osmotic pressures across active biological membranes with application to the corneal endothelium. PLoS ONE. 2015;10:e0145422. doi: 10.1371/journal.pone.0145422. PubMed DOI PMC

Jadhav S.V., Marathe K.V., Rathod V.K. A pilot scale concurrent removal of fluoride, arsenic, sulfate and nitrate by using nanofiltration: Competing ion interaction and modelling approach. J. Water. Proc. Eng. 2016;13:153–167. doi: 10.1016/j.jwpe.2016.04.008. DOI

Teugels L. Double Membrane Transducer Protector. No 7,516,665 B2. U.S. Patent. 2009 Apr 14;

Yan Y., Gu S., Gong K. Double-Membrane Triple-Electrolyte Redox Flow Battery Design. No. 9,917,323 B2. U.S. Patent. 2018 Mar 13;

Curran P.F., McIntosh J.R. A model system for biological water transport. Nature. 1962;193:347–348. doi: 10.1038/193347a0. PubMed DOI

Patlak C.S., Goldstein D., Hoffman J.F. The flow of solute and solvent across a two-membrane system. J. Theor. Biol. 1963;5:426–442. doi: 10.1016/0022-5193(63)90088-2. PubMed DOI

Richardson I.W. Multiple membrane system as biological models (current-voltage behavior) J. Membr. Biol. 1972;8:219–236. doi: 10.1007/BF01868104. PubMed DOI

Ginsburg H. Model for iso-osmotic water flow in plant roots. J. Theor. Biol. 1971;32:147–158. doi: 10.1016/0022-5193(71)90142-1. PubMed DOI

Kedem O., Katchalsky A. Permeability of composite membranes. Part 3. Series array of elements. Trans. Faraday Soc. 1963;59:1941–1953. doi: 10.1039/TF9635901941. DOI

Katchalsky A., Curran P.F. Nonequilibrium Thermodynamics in Biophysics. Harvard University Press; Cambridge, MA, USA: 1965.

Naparstek A., Caplan S.R., Katzir-Katchalsky A. Series arrays of ion-exchange membranes concentration of electric current. Israel J. Chem. 1973;11:255–270. doi: 10.1002/ijch.197300027. DOI

Ohki S. Rectification by double membranes. J. Phys. Sci. Jpn. 1965;20:1674–1685. doi: 10.1143/JPSJ.20.1674. DOI

Tsukahara S., Nanzai B., Igawa M. Selective transport of amino acids across a double membrane system composed of a cation- and an anion-exchange membrane. J. Membr. Sci. 2013;448:300–307. doi: 10.1016/j.memsci.2013.06.062. DOI

Ueno K., Doi T., Nanzai B., Igawa M. Selective transport of neutral amino acids across a double-membrane system comprising cation and anion exchange membranes. J. Membr. Sci. 2017;537:344–352. doi: 10.1016/j.memsci.2017.04.013. DOI

Kargol M., Dworecki K., Przestalski S. Graviosmotic flow amplification effects in a series membrane system. Stud. Biophys. 1979;76:137–144.

Ślęzak A., Wasik J., Ślęzak K. Effect of concentration boundary layers on mechanical pressure in a double-membrane system. Desalination. 2004;168:423–433. doi: 10.1016/j.desal.2004.07.026. DOI

Ślęzak A., Wasik J., Grzegorczyn S. Irreversible thermodynamics model equations for heterogeneous solute flux in a double-membrane system. Desalination. 2004;163:155–175. doi: 10.1016/S0011-9164(04)90187-7. DOI

Ślęzak A., Jasik-Ślęzak J., Grzegorczyn S., Ślęzak-Prochazka I. Nonlinear effects in osmotic volume flows of electrolyte solutions through double-membrane system. Transp. Porous Med. 2012;92:337–356. doi: 10.1007/s11242-011-9906-7. DOI

Ślęzak A., Ślęzak-Prochazka I., Grzegorczyn S., Jasik-Ślęzak J. Evaluation of S-Entropy production in a single-membrane system in concentration polarization conditions. Trans. Porous Med. 2017;116:941–957. doi: 10.1007/s11242-016-0807-7. DOI

Rubin A.B. Thermodynamics of Biological Processes. Moscow State University Press; Moscow, Russia: 1984.

Weber W.H., Ford G.W. Double injection in semiconductors heavily doped with deep two-level traps. Solid-State Electron. 1970;13:1333–1356. doi: 10.1016/0038-1101(70)90167-X. DOI

Anderson J.E. A model for current-controlled negative resistance in ion/membrane systems. J. Membr. Sci. 1978;4:35–40. doi: 10.1016/S0376-7388(00)83282-2. DOI

Abu-Rjal R., Prigozhin L., Rubinstein I., Zaltzman B. Teorell instability in concentration polarization. Phys. Rev. E. 2015;92:022305. doi: 10.1103/PhysRevE.92.022305. PubMed DOI

Franck U.F. Phänomene an biologischen und künstlichen membrane. Berichte Bunsengess. 1967;71:789–799. doi: 10.1002/bbpc.19670710810. DOI

Cole K.S. Membranes, Ions and Impulses. University California Press; Berkeley, CA, USA: 1968.

Artificial Intelligence and Computational Methods in the Modeling of Complex Systems