# Evaluation of Transport Properties and Energy Conversion of Bacterial Cellulose Membrane Using Peusner Network Thermodynamics

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{r}version of the Kedem–Katchalsky–Peusner formalism (KKP) for the concentration polarization (CP) conditions of solutions, the osmotic and diffusion fluxes as well as the membrane transport parameters were determined, such as the hydraulic permeability (L

_{p}), reflection (σ), and solute permeability (ω). We used these parameters and the Peusner (${R}_{ij}^{r}$) coefficients resulting from the KKP equations to assess the transport properties of the membrane based on the calculated dependence of the concentration coefficients: the resistance, coupling, and energy conversion efficiency for aqueous ethanol solutions. The transport properties of the membrane depended on the hydrodynamic conditions of the osmotic diffusion transport. The resistance coefficients ${R}_{11}^{r}$, ${R}_{22}^{r}$, and ${R}_{det}^{r}$ were positive and higher, and the ${R}_{12}^{r}$ coefficient was negative and lower under CP conditions (higher in convective than nonconvective states). The energy conversion was evaluated and fluxes were calculated for the U-, F-, and S-energy. It was found that the energy conversion was greater and the S-energy and F-energy were lower under CP conditions. The convection effect was negative, which means that convection movements were directed vertically upwards. Understanding the membrane transport properties and mechanisms could help to develop and improve the membrane technologies and techniques used in medicine and in water and wastewater treatment processes.

## 1. Introduction

^{−1})]

^{−1}is the average concentration of the solutes; and ${\zeta}_{p}^{r}$, ${\zeta}_{v}^{r}$, ${\zeta}_{s}^{r}$, and ${\zeta}_{a}^{r}$ are, respectively, the hydraulic, osmotic, diffusive, and advective coefficients of the CP [23]. For dilute nonelectrolyte solutions, ${\sigma}_{v}$ = ${\sigma}_{s}$. In contrast, for nondilute solutions, ${\sigma}_{v}$ ≠ ${\sigma}_{s}$ [16].

_{p}, σ, and ω) and the average concentration of the solutions ($\overline{C}$).

## 2. Materials and Methods

#### 2.1. Membrane System

^{3}each containing aqueous ethanol solutions, one with a concentration in the range of 1–501 mol m

^{−3}and the other with a constant concentration of 1 mol⋅m

^{−3}. The solutions in the vessels were separated by a previously described bacterial cellulose (BC) membrane called Bioprocess

^{®}(Biofill Produtos Biotechnologicos S.A., Curitiba, Brasile) [33,34,35,36] positioned in a horizontal plane with an area of A = 3.36 cm

^{2}. The BC membrane was produced in flat sheets, and its structure was made of microcellulose fibers produced by Acetobacter Xylinum [8,37].

^{−1}($\mathsf{\Delta}t$)

^{−1}. The solute flux was calculated based on the formula ${J}_{s}^{r}$ = $\left(d{C}_{s}^{r}{V}_{u}\right)A$

^{−1}($\mathsf{\Delta}t$)

^{−1}, where ${V}_{u}$ is the volume of the measuring vessel and $d{C}_{s}^{r}$ is the increase in the total concentration of the solutions. The $d{C}_{s}^{r}$ was measured by a Rayleigh interferometer based on previously calculated feature curves, i.e., the experimental dependence of the shift of the interference bars ($\mathsf{\Delta}$n) as a function of the ethanol concentration (C) [38]. The study was carried out at $T$ = 295 K. A laser interferometry method can also be used to determine $d{C}_{s}^{r}$ [39,40,41].

#### 2.2. The R^{r} Form of Kedem–Katchalsky Equations for Binary Nonelectrolyte Solutions

^{r}form of the KKP equations, which can be obtained using simple algebraic transformations presented in the paper [24,31]:

^{−1}.

^{2}s mol

^{−1}.

^{−2}.

^{−12}m

^{3}N

^{−1}s

^{−1}, $\sigma $ = (0.23 ± 0.01) × 10

^{−2}, and $\omega $ = (15.3 ± 0.5) × 10

^{−10}mol N

^{−1}s

^{−1}.

## 3. Results and Discussion

#### 3.1. The Time and Concentration Dependencies of ${J}_{v}^{r}$ and ${J}_{s}^{r}$

^{−3}and ${C}_{l}$ = 1 mol m

^{−3}are shown in Figure 3a,b. Curves 1A and 1B were obtained for mechanically stirred solutions that favored solution homogeneity. Curves 1A and 1B are symmetrical with respect to the horizontal axes passing through the points ${J}_{v}^{r}$ = 0 and ${J}_{s}^{r}$ = 0, indicating that stirring was effective. This symmetry is reflected in the linearity of the dependences ${J}_{v}^{r}=f\left(\u2206C\right)$ and ${J}_{s}^{r}=f\left(\u2206C\right)$, as illustrated by curves 1A and 1B in Figure 3c,d. In steady states, the relations $\left|{J}_{v}^{A}\right|$ = ${J}_{v}^{B}$ = ${J}_{v}$ and $\left|{J}_{s}^{A}\right|$ = ${J}_{s}^{B}$ = ${J}_{s}$ were fulfilled. In CP conditions, the time dependencies of ${J}_{v}^{r}$ and ${J}_{s}^{r},$ shown by curves 2A and 2B, are asymmetric with respect to the horizontal axes passing through the points ${J}_{v}^{r}$ = 0 and ${J}_{s}^{r}$ = 0. The consequence of this asymmetry is the nonlinear dependencies ${J}_{v}^{r}=f\left(\u2206C\right)$ and ${J}_{s}^{r}=f\left(\u2206C\right),$ illustrated by curves 2A and 2B in Figure 3c,d. The shapes of these graphs indicate that both ${J}_{v}^{r}$ and ${J}_{s}^{r}$ reached steady states relatively quickly and that in the steady states $\left|{J}_{v}^{A}\right|$ > ${J}_{v}^{B}$ and $\left|{J}_{s}^{A}\right|$ > ${J}_{s}^{B}$. This dependence was a consequence of the emergence of gravitational convection, which is destructive to CBLs. This means that, in this case, CP and gravitational convection were antagonistic processes.

#### 3.2. The Time and Concentration Dependencies of ${\zeta}_{v}^{r}$ and ${\zeta}_{s}^{r}$

^{−3}. The coefficients ${\zeta}_{v}^{r}$ and ${\zeta}_{s}^{r}$ take their values from the intervals ${({\zeta}_{v}^{r})}_{diff.}$ ≤ ${\zeta}_{v}^{r}$ ≤ 1, ${({\zeta}_{v}^{r})}_{conv.}$ ≤ ${\zeta}_{v}^{r}$ ≤ 1, ${({\zeta}_{s}^{r})}_{diff.}$ ≤ ${\zeta}_{s}^{r}$ ≤ 1, and ${({\zeta}_{s}^{r})}_{conv.}$ ≤ ${\zeta}_{s}^{r}$ ≤ 1. As shown in Figure 4a, ${\zeta}_{v}^{r}$ and ${\zeta}_{s}^{r}$ take values from the intervals 0.03 ≤ ${\zeta}_{v}^{r}$ ≤ 1, 0.26 ≤ ${\zeta}_{v}^{r}$ ≤ 1, 0.06 ≤ ${\zeta}_{s}^{r}$ ≤ 1, and 0.29 ≤ ${\zeta}_{s}^{r}$ ≤ 1. Based on these time dependencies, the concentration dependencies ${\zeta}_{v}^{r}$ = f($\u2206C$) and ${\zeta}_{s}^{r}$ = f($\u2206C$) were determined for the steady states (Figure 4b). The coefficients ${\zeta}_{v}^{r}$ and ${\zeta}_{s}^{r}$ take their values from the intervals ${({\zeta}_{v}^{r})}_{diff.}$ ≤ ${\zeta}_{v}^{r}$ ≤ ${({\zeta}_{v}^{r})}_{conv.}$ and ${({\zeta}_{s}^{r})}_{diff.}$ ≤ ${\zeta}_{s}^{r}$ ≤ ${({\zeta}_{s}^{r})}_{conv.}$. As shown in Figure 4b, ${\zeta}_{v}^{r}$ and ${\zeta}_{s}^{r}$ take their values in the range between 0.03 ≤ ${\zeta}_{v}^{r}$ ≤ 0.26 and 0.06 ≤ ${\zeta}_{s}^{r}$ ≤ 0.29. Therefore, the coefficients ${\zeta}_{v}^{r}$ and ${\zeta}_{s}^{r}$ are a measure of the CP in both convection and nonconvection states.

^{−2}, $RT$ = 24.51 × 10

^{2}J mol

^{−1}, ${\rho}_{0}$ = 998.2 kg m

^{−3}, ${\nu}_{0}$ = 1.01 × 10

^{−6}m

^{2}s

^{−1}, $\omega $ = 1.53 × 10

^{−9}mol N

^{−1}s

^{−1}, $D$ = 1.07 × 10

^{−9}m

^{2}s

^{−1}, $\partial \rho /\partial C$ = −0.009 kg mol

^{−1}, $\u2206C$ = 80 mol m

^{−3}, and $\mathsf{\zeta}$ = 0.16 (estimated based Figure 4b) in Equation (19), we obtain ${R}_{C}$ = −1155.07. The minus sign indicates that the convective currents are directed vertically upwards. In contrast, for aqueous glucose solutions, studied previously, convective currents were directed vertically downwards, and therefore ${R}_{C}$ had a positive sign [5]. The obtained critical value of ${R}_{C}$ is consistent with the values presented in the papers [44,45].

#### 3.3. Concentration Dependencies of the Resistance Coefficients ${R}_{ij}^{r}$ and ${R}_{det}^{r}$

^{−3}the condition ${R}_{11}^{A}$ = ${R}_{11}^{B}$ was fulfilled.

^{−3}the condition was ${R}_{12}^{A}$ = ${R}_{21}^{A}$ = ${R}_{12}^{B}$ = ${R}_{21}^{B}$. From Equation (6), it follows that ${R}_{12}^{r}$ ≠ ${R}_{21}^{r}$. To explain why this relation did not hold, we calculated the quotient ${R}_{12}^{r}$/${R}_{21}^{r}$ = $\left(1-{\zeta}_{v}^{r}\sigma \right)/\left(1-{\zeta}_{a}^{r}\sigma \right)$ using Equation (5), and we obtained ${R}_{12}^{r}$/${R}_{21}^{r}$ = 1.002, meaning that ${R}_{12}^{A}$ = ${R}_{21}^{A}$, with accuracy to two significant figures.

^{−3}the condition ${R}_{22}^{A}$ = ${R}_{22}^{B}$ was fulfilled.

^{−3}the condition ${R}_{det}^{A}$ = ${R}_{det}^{B}$ was fulfilled.

^{−8}m s

^{−1}, ${J}_{v}^{A}$ = ${J}_{v}^{B}$ = 0.45 × 10

^{−8}m s

^{−1}, ${J}_{s}$ = 3.01 × 10

^{−4}mol m

^{−2}s

^{−1}, and ${J}_{s}^{A}$ = ${J}_{s}^{B}$ = 0.48 × 10

^{−4}mol m

^{−2}s

^{−1}. The values of these Peclét numbers were ${\left(Pe\right)}_{v}$ = 7.5 ×10

^{−6}, ${\left(Pe\right)}_{s}$ = 7.5 × 10

^{−3}, ${\left(Pe\right)}_{v}^{r}$ = 7.99 × 10

^{−2}, and ${\left(Pe\right)}_{s}^{r}$ = 79.95, and ${\left(Pe\right)}_{s}^{r}$ > ${\left(Pe\right)}_{v}^{r}$ > ${\left(Pe\right)}_{s}$ > ${\left(Pe\right)}_{v}$.

#### 3.4. Concentration Dependencies ${({\varphi}_{ij}^{r})}_{R}$ and ${({\varphi}_{det}^{r})}_{R}$

^{−3}.

#### 3.5. Concentration Dependencies of ${({\Phi}_{S}^{r})}_{R}$, ${({e}_{ij}^{r})}_{R}$, ${({\Phi}_{F}^{r})}_{R}$, and ${({\Phi}_{U}^{r})}_{R}$

^{−3}), ${({\Phi}_{U}^{B})}_{R}$ < ${({\Phi}_{U})}_{R}$ (for $\mathsf{\Delta}$C > −500 mol m

^{−3}), ${({\Phi}_{U}^{B})}_{R}$ > ${({\Phi}_{U})}_{R}$ (for $\mathsf{\Delta}$C < −500 mol m

^{−3}), and ${({\Phi}_{U}^{B})}_{R}$ < ${({\Phi}_{U})}_{R}$.

## 4. Conclusions

- Developed within the framework of the Kedem–Katchalsky–Peusner formalism, the procedure using the Peusner coefficients ${R}_{ij}^{r}$ (i = j ∈ {1, 2}, r = A, B) and ${R}_{det}^{r}$ is suitable for evaluating the transport properties of polymer membranes and assessing the conversion of internal energy (U-energy) to useful energy (F-energy) and degraded energy (S-energy).
- Peusner coefficients ${R}_{12}^{r}$ and ${R}_{21}^{r}$ are related to the membrane Peclét coefficients ${\alpha}_{s}^{r}$ and ${\alpha}_{v}^{r}$.
- The procedure developed in this paper to evaluate the conversion of internal energy (U-energy) to useful energy (F-energy) and degraded energy (S-energy) requires the calculation of the value of the flux of S-energy ${({\Phi}_{S}^{r})}_{R}$ and efficiency factors ${({e}_{12}^{r})}_{R}$ and ${({e}_{21}^{r})}_{R}$, followed by the fluxes of F-energy $(\left({\Phi}_{F}^{r}{)}_{R}\right)$ and U-energy (${({\Phi}_{U}^{r})}_{R}$).
- The procedure proposed in the paper can be applied to membranes for which the coefficients ${L}_{P}$, ${\sigma}_{v}$, ${\sigma}_{s}$, and $\omega $ can be determined experimentally.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## List of Symbols

${L}_{P}$ | hydraulic permeability coefficient (m^{3}N^{−1}s^{−1}) |

$\overline{\mathit{C}}$ | average concentration of solutes (mol m^{−3}) |

${\mathit{\zeta}}_{\mathit{p}}^{\mathit{r}}$$,{\mathit{\zeta}}_{\mathit{v}}^{\mathit{r}}$$,{\mathit{\zeta}}_{\mathit{s}}^{\mathit{r}}$$,\mathrm{and}{\mathit{\zeta}}_{\mathit{a}}^{\mathit{r}}$ | hydraulic, osmotic, diffusive, and advective coefficients of CP |

${\mathit{R}}_{\mathit{i}\mathit{j}}^{\mathit{r}}$$\mathrm{and}{\mathit{R}}_{\mathit{d}\mathit{e}\mathit{t}}^{r};$ i, j ∈ {1, 2}, r = A, B | $\mathrm{Peusner}\mathrm{coefficients}({\mathit{R}}_{\mathbf{11}}^{\mathit{r}}$$(\mathrm{N}\mathrm{s}{\mathrm{m}}^{-3}),{\mathit{R}}_{\mathbf{12}}^{\mathit{r}}$$(\mathrm{N}\mathrm{s}{\mathrm{mol}}^{-1}),{\mathit{R}}_{\mathbf{21}}^{r}$$(\mathrm{N}\mathrm{s}{\mathrm{mol}}^{-1}),\mathrm{and}{\mathit{R}}_{\mathit{d}\mathit{e}\mathit{t}}^{r}$ (N^{2}s^{2}mol^{−2})) |

${\mathit{\delta}}_{\mathit{h}}^{\mathit{r}}$$\mathrm{and}{\mathit{\delta}}_{\mathit{l}}^{\mathit{r}}$ | thicknesses of the concentration boundary layers (CBLs) (m) |

${\mathit{e}}_{\mathit{i}\mathit{j}}^{\mathit{r}}$ | energy conversion efficiency coefficients |

${\mathit{R}}^{\mathit{r}}$ | matrix of the Peusner coefficients |

${\mathit{\Phi}}_{\mathit{S}}^{\mathit{r}}$ | flux of S-energy (W m^{−2}) |

${\mathit{\Phi}}_{\mathit{F}}^{\mathit{r}}$ | flux of F-energy (W m^{−2}) |

${\mathit{\Phi}}_{\mathit{U}}^{\mathit{r}}$ | flux of U-energy (W m^{−2}) |

${\mathit{\rho}}_{\mathit{h}}$$\mathrm{and}{\mathit{\rho}}_{\mathit{l}}$ | mass density (kg m^{−3}) |

${\mathit{r}}_{\mathit{i}\mathit{j}}^{\mathit{r}}$ | coupling coefficient |

${\mathit{R}}_{\mathit{C}}$ | concentration Rayleigh number |

${({\mathit{\varphi}}_{\mathit{i}\mathit{j}}^{\mathit{r}})}_{\mathit{R}}$$\mathrm{and}{\left({\mathit{\varphi}}_{\mathit{d}\mathit{e}\mathit{t}}^{\mathit{r}}\right)}_{\mathit{R}}$ | concentration polarization effects |

${({\mathit{\phi}}_{\mathit{i}\mathit{j}})}_{\mathit{R}}$$\mathrm{and}{({\mathit{\phi}}_{\mathit{d}\mathit{e}\mathit{t}})}_{\mathit{R}}$ | effects of gravitational convection in osmotic and diffusive transport |

${\mathit{P}}_{\mathit{e}}$ | Peclét number |

${\mathit{\alpha}}_{\mathit{s}}^{\mathit{r}}$$\mathrm{and}{\mathit{\alpha}}_{\mathit{v}}^{\mathit{r}}$ | $\mathrm{Pecl}\text{\xe9}\mathrm{t}\mathrm{coefficients}({\mathit{\alpha}}_{\mathit{s}}^{\mathit{r}}$$({\mathrm{m}}^{2}\mathrm{s}{\mathrm{mol}}^{-1})\mathrm{and}{\mathit{\alpha}}_{\mathit{v}}^{\mathit{r}}$ (s m^{−1})) |

${\mathit{\wp}}_{\mathit{v}}$$\mathrm{and}{\mathit{\wp}}_{\mathit{s}}$ | $\mathrm{solute}\mathrm{permeability}\mathrm{coefficient}({\wp}_{v}$$(\mathrm{m}{\mathrm{s}}^{-1})\mathrm{and}{\wp}_{s}$(mol m^{−2}s^{−1})) |

A and B | configurations of membrane system |

M | membrane |

CP | concentration polarization |

BC | bacterial cellulose |

${\mathit{l}}_{\mathit{l}}^{\mathit{r}}$$\mathrm{and}{\mathit{l}}_{\mathit{h}}^{\mathit{r}}$ | the concentration boundary layers (CBLs) |

${\mathit{l}}_{\mathit{h}}^{\mathit{r}}$$/\mathrm{M}/{\mathit{l}}_{\mathit{l}}^{\mathit{r}}$ | complex of CBLs and membrane |

KKP equations | Kedem–Katchalsky–Peusner equations |

## References

- Baker, R. Membrane Technology and Application; John Wiley & Sons: New York, NY, USA, 2012; ISBN 978-0-470-74372-0. [Google Scholar]
- Savencu, I.; Iurian, S.; Porfire, A.; Bogdan, C.; Tomuță, I. Review of advances in polymeric wound dressing films. React. Funct. Polym.
**2021**, 168, 105059. [Google Scholar] [CrossRef] - Demirel, Y. Nonequilibrium Thermodynamics: Transport and Rate Processes in Physical, Chemical and Biological Systems; Elsevier: Amsterdam, The Netherlands, 2014; ISBN 978-0-444-53079-0. [Google Scholar]
- Lipton, B. The Biology of Belief: Unleashing the Power of Consciousness; Hay House: Carlsbad, CA, USA, 2018; ISBN-10: 1401923127. [Google Scholar]
- Batko, K.M.; Ślęzak, A. Evaluation of the global S-entropy production in membrane transport of aqueous solutions of hydrochloric acid and ammonia. Entropy
**2020**, 22, 1021. [Google Scholar] [CrossRef] [PubMed] - Delmotte, M.; Chanu, J. Non-equilibrium Thermodynamics and Membrane Potential Measurement in Biology. In Topics Bioelectrochemistry and Bioenergetics; Millazzo, G., Ed.; John Wiley Publish & Sons: Chichester, UK, 1979; pp. 307–359. [Google Scholar]
- Gerke, K.M.; Vasilyev, R.V.; Khirevich, S.; Collins, D.; Karsanina, M.V.; Sizonenko, T.O.; Korost, D.V.; Lamontagne, S.; Mallants, D. Finite-difference method Stokes solver (FDMSS) for 3D pore geometries: Software development, validation and case studies. Comput. Geosci.
**2018**, 114, 41–58. [Google Scholar] [CrossRef][Green Version] - Blunt, M.J. Flow in porous media—Pore-network models and multiphase flow. Curr. Opin. Colloid Interface Sci.
**2001**, 6, 197–207. [Google Scholar] [CrossRef] - Koroteev, D.; Dinariev, O.; Evseev, N.; Klemin, D.; Nadeev, A.; Safonov, S.; Gurpinar, O.; Berg, S.; van Kruijsdijk, C.; Armstrong, R.; et al. Direct hydrodynamic simulation of multiphase flow in porous rock. Petrophysics
**2014**, 55, 294–303. [Google Scholar] - Veselý, M.; Bultreys, T.; Peksa, M.; Lang, J.; Cnudde, V.; Van Hoorebeke, L.; Kočiřík, M.; Hejtmánek, V.; Šolcová, O.; Soukup, K.; et al. Prediction and evaluation of time-dependent effective self-diffusivity of water and other effective transport properties associated with reconstructed porous solids. Transp. Porous Media
**2015**, 110, 81–111. [Google Scholar] [CrossRef] - Jarzyńska, M.; Pietruszka, M. The application of the Kedem-Katchalsky equations to membrane transport of ethyl alcohol and glucose. Desalination
**2011**, 280, 14–19. [Google Scholar] [CrossRef] - Jarzyńska, M.; Staryga, E.; Kluza, F.; Spiess, W.E.L.; Góral, D. Diffusion characteristics in ethyl alcohol and glucose solutions using Kedem-Katchalsky equations. Chem. Eng. Technol.
**2020**, 43, 248–252. [Google Scholar] [CrossRef] - Ślęzak, A. Irreversible thermodynamic model equations of the transport across a horizontally mounted membrane. Biophys. Chem.
**1989**, 34, 91–102. [Google Scholar] [CrossRef] - Dworecki, K.; Ślęzak, A.; Ornal-Wąsik, B.; Wąsik, S. Effect of hydrodynamic instabilities on solute transport in a membrane system. J. Membr. Sci.
**2005**, 265, 94–100. [Google Scholar] [CrossRef] - Ślęzak, A. A model equation for the gravielectric effect in electrochemical cells. Biophys. Chem.
**1990**, 38, 189–199. [Google Scholar] [CrossRef] [PubMed] - Kargol, A. Modified Kedem-Katchalsky equations and their application. J. Membr. Sci.
**2000**, 174, 43–53. [Google Scholar] [CrossRef] - Ślęzak, A.; Dworecki, K.; Anderson, J.E. Gravitational effects on transmembrane flux: The Rayleigh-Taylor convective instability. J. Membr. Sci.
**1985**, 23, 71–81. [Google Scholar] [CrossRef] - Ślęzak, A.; Dworecki, K.; Jasik-Ślęzak, J.; Wąsik, J. Method to determine the practical concentration Rayleigh number in isothermal passive membrane transport processes. Desalination
**2004**, 168, 397–412. [Google Scholar] [CrossRef] - Puthenveettil, B.A.; Arakeri, J.H. Plum structure in high-Rayleigh-Number convection. J. Fluid Mech.
**2005**, 542, 217–249. [Google Scholar] [CrossRef][Green Version] - Slezak-Prochazka, I.; Batko, K.M.; Ślęzak, A.; Bajdur, W.M.; Włodarczyk-Makuła, M. Non-linear effects in osmotic membrane transport: Evaluation of the S-entropy production by volume flux of aqueous ammonia and sulfuric acid solutions under concentration polarization conditions. Desal. Water Treat.
**2022**, 260, 23–36. [Google Scholar] [CrossRef] - Katchalsky, A.; Curran, P.F. Nonequilibrium Thermodynamics in Biophysics; Harvard: Cambridge, UK, 1965; ISBN 9780674494121. [Google Scholar]
- Friedman, M.H.; Meyer, R.A. Transport across homoporous and heteroporous membranes in nonideal nondilute solutions. I. Inequality of reflection coefficients for volume flow and solute flow. Biophys. J.
**1981**, 34, 535–544. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kargol, M.; Kargol, A. Mechanistic formalism for membrane transport generated by osmotic and mechanical pressure. Gen. Physiol. Biophys.
**2003**, 22, 51–68. [Google Scholar] - Batko, K.M.; Ślęzak, A.; Grzegorczyn, S.; Bajdur, W.M. The R
^{r}form of the Kedem–Katchalsky–Peusner model equations for description of the membrane transport in concentration polarization conditions. Entropy**2020**, 22, 857. [Google Scholar] [CrossRef] - Richter, T.; Keipert, S. In vitro permeation studies comparing bovine nasal mucosa, porcine cornea and artificial membrane: Androstendedione in microemulsions and their components. Eur. J. Pharma Biopharm.
**2004**, 58, 137–143. [Google Scholar] [CrossRef] - Twardowski, Z. Scholarly Review: History of hemodialyzers’ designs. Hemodial. Inter.
**2008**, 12, 173–210. [Google Scholar] [CrossRef] [PubMed] - Zholkovskiy, E.; Koter, I.; Koter, S.; Kujawski, W.; Yaroshchuk, A. Analysis of membrane transport equations for reverse electrodialysis (RED) using irreversible thermodynamics. Int. J. Mol. Sci.
**2020**, 21, 6325. [Google Scholar] [CrossRef] - Auclair, B.; Nikonenko, V.; Larchet, C.; Métayer, M.; Dammak, L. Correlation between transport parameters of ion-exchange membranes. J. Membr. Sci.
**2002**, 195, 89–102. [Google Scholar] [CrossRef] - Batko, K.M.; Ślęzak-Prochazka, I.; Grzegorczyn, S.M.; Pilis, A.; Dolibog, P.; Ślęzak, A. Energy conversion in Textus Bioactiv Ag membrane dressings using Peusner’s network thermodynamic descriptions. [published on line as ahead of print November 10, 2022]. Polym. Med.
**2022**, 12. [Google Scholar] [CrossRef] - Peusner, L. Studies in Network Thermodynamics; Elsevier: Amsterdam, The Netherlands, 1986; ISBN 0444425802. [Google Scholar]
- Ślęzak, A.; Grzegorczyn, S.; Batko, K.M. Resistance coefficients of polymer membrane with concentration polarization. Transp. Porous Media
**2012**, 95, 151–170. [Google Scholar] [CrossRef][Green Version] - Ślęzak, A.; Grzegorczyn, S.; Jasik-Ślęzak, J.; Michalska-Małecka, K. Natural convection as an asymmetrical factor of the transport through porous membrane. Transp. Porous. Media
**2010**, 84, 685–698. [Google Scholar] [CrossRef] - Ślęzak, A.; Kucharzewski, M.; Franek, A.; Twardokęs, W. Evaluation of the efficiency of venous leg ulcer treatment with a membrane dressing. Med. Eng. Phys.
**2004**, 26, 53–60. [Google Scholar] [CrossRef] - Batko, K.M.; Ślęzak, A.; Pilis, W. Evaluation of transport properties of biomembranes by means of Peusner network thermodynamics. Acta Bioeng. Biomech.
**2021**, 23, 63–72. [Google Scholar] [CrossRef] - Hussain, Z.; Sajjad, W.; Khan, T.; Wahid, F. Production of bacterial cellulose from industrial wastes: A review. Cellulose
**2019**, 26, 2895–2911. [Google Scholar] [CrossRef] - Tayeb, A.H.; Amini, E.; Ghasemi, M.; Tajvidi, S. Cellulose nanomaterials—Binding properties and applications: A review. Molecules
**2018**, 23, 2683. [Google Scholar] [CrossRef][Green Version] - Grzegorczyn, S.; Ślęzak, A. Kinetics of concentration boundary layers buildup in the system consisted of microbial cellulose biomembrane and electrolyte solutions. J. Membr. Sci.
**2007**, 304, 148–155. [Google Scholar] [CrossRef] - Ewing, G.W. Instrumental Methods of Chemical Analysis; McGraw-Hill Book, Co.: New York, NY, USA, 1985; ISBN 0-07-019857-8. [Google Scholar]
- Dworecki, K. Interferometric investigation of near-membrane diffusion layers. J. Biol. Phys.
**1995**, 21, 37–49. [Google Scholar] [CrossRef] - Gałczyńska, K.; Rachuna, J.; Ciepluch, K.; Kowalska, M.; Wąsik, S.; Kosztołowicz, T.; Lewandowska, K.D.; Semaniak, J.; Kurdziel, K.; Arabski, M. Experimental and theoretical analysis of metal complex diffusion through cell monolayer. Entropy
**2021**, 23, 360. [Google Scholar] [CrossRef] [PubMed] - Arabski, M.; Wąsik, S.; Dworecki, K.; Kaca, W. Laser interferometric determination of ampicillin and colistin transfer through cellulose biomembrane in the presence of Proteus vulgaris O25 lipopolysaccharide. J. Membr. Sci.
**2007**, 299, 268–275. [Google Scholar] [CrossRef] - Bason, S.; Kedem, O.; Freger, V. Determination of concentration-dependent transport coefficient s in nanofiltration: Experimental evaluation of coefficients. J. Membr. Sci.
**2008**, 310, 197–204. [Google Scholar] [CrossRef] - Kedem, O.; Caplan, S.R. Degree of coupling and its relations to efficiency of energy conversion. Trans. Faraday Soc.
**1965**, 61, 1897–1911. [Google Scholar] [CrossRef] - Lebon, G.; Jou, D.; Casas-Vasquez, J. Understanding Non-Equilibrium Thermodynamics. Foundations, Applications, Frontiers; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Batko, K.M.; Ślęzak-Prochazka, I.; Ślęzak, A.; Bajdur, W.M.; Włodarczyk-Makuła, M. Management of energy conversion processes in membrane systems. Energies
**2022**, 15, 1661. [Google Scholar] [CrossRef] - Mohammad, A.W.; Teow, Y.H.; Ang, W.L.; Chung, Y.T.; Oatley-Radcliffe, D.L.; Hilal, N. Nanofiltration membranes review: Recent advances and future prospects. Desalination
**2015**, 356, 226–354. [Google Scholar] [CrossRef] - Han, Y.; Xu, Z.; Gao, C. Ultrathin Graphene nanofiltration membrane for water purification. Adv. Funct. Mat.
**2013**, 23, 3693–3700. [Google Scholar] [CrossRef] - Weinstein, A.M. An equation for flow in the renal proximal tubule. Bull. Math. Biol.
**1986**, 48, 29–57. [Google Scholar] [CrossRef] - Huang, Y.; Chen, W.; Lei, Y. Outer membrane vesicles (OMVs) enabled bio-applications: A critical review. Biotech. Bioeng.
**2022**, 119, 34–47. [Google Scholar] [CrossRef] [PubMed] - Stamatialis, D.F.; Papenburg, B.J.; Gironés, M.; Saiful, S.; Bettahalli, S.N.M.; Schmitmeier, S.; Wessling, M. Medical applications of membranes: Drug delivery, artificial organs and tissue engineering. J. Membr. Sci.
**2008**, 308, 1–34. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) The model of a single-membrane system: M—membrane; g—gravitational acceleration; ${l}_{l}^{A}$ and ${l}_{h}^{A}$ —the concentration boundary layers (CBLs) in configuration A; ${l}_{l}^{B}$ and ${l}_{h}^{B}$ —the CBLs in configuration B; P

_{h}and P

_{l}—mechanical pressures; ${C}_{h}$ and ${C}_{l}$ —total solution concentrations (${C}_{h}$ > ${C}_{l}$ ); ${C}_{l}^{A}$, ${C}_{h}^{A}$, ${C}_{l}^{B}$, and ${C}_{h}^{B}$ —local (at boundaries between the membrane and CBLs) solution concentrations; ${J}_{v}^{A}$ —solute and volume fluxes in configuration A; ${J}_{v}^{B}$ —solute and volume fluxes in configuration B. (

**b**) Interferometric images of concentration boundary layers for a membrane system that contains ethanol solutions of concentrations ${C}_{l}$ = 1 mol⋅m

^{−3}and ${C}_{h}$ = 125 mol⋅m

^{−3}at time 80 s; M—membrane [14].

**Figure 2.**(

**a**) Measuring system (h and l—measuring vessels, N—external solution tank, s—mechanical stirrers, M—membrane, K—calibrated pipette, m—magnets, Z—plugs) [33]. (

**b**) Image of a cross section of a Bioprocess membrane obtained from a scanning electron microscope (magnification: 10,000 times) [37].

**Figure 3.**Dependences ${J}_{v}^{r}=f\left(t\right)$ (

**a**), ${J}_{s}^{r}=f\left(t\right)$ (

**b**), ${J}_{v}^{r}=f\left(\u2206C\right)$ (

**c**)

**,**and ${J}_{s}^{r}=f\left(\u2206C\right)$ (

**d**): curves 1A and 1B were obtained for homogeneous solutions (mechanical mixing), and curves 2A and 2B were obtained for concentration polarization conditions (after excluding mechanical mixing of the solutions).

**Figure 4.**Time (

**a**) and concentration (

**b**) dependences of ${\zeta}_{v}^{r}$ and ${\zeta}_{s}^{r}$ for aqueous ethanol solutions.

**Figure 5.**Concentration dependences of the resistance coefficients (

**a**) ${R}_{11}^{r},$ (

**b**) ${R}_{12}^{r}={R}_{21}^{r}$, (

**c**) ${R}_{22}^{r}$, and (

**d**) ${R}_{det}^{A}$ for aqueous ethanol solutions.

**Figure 6.**Concentration dependencies of the coefficients ${({\varphi}_{ij}^{r})}_{R}$ and (${\varphi}_{det}^{r}{)}_{R}$ (

**a**) and the coefficients ${({\phi}_{ij})}_{R}$ and (${\phi}_{det}{)}_{R}$ (

**b**) for aqueous ethanol solutions.

**Figure 7.**Concentration dependencies of ${({\Phi}_{S}^{r})}_{R}$ (

**a**) and maximum energy conversion efficiency coefficients ${({e}_{12}^{r})}_{R}={({e}_{21}^{r})}_{R}$ (

**b**) for aqueous ethanol solutions.

**Figure 8.**Concentration dependencies of the flux of F-energy ${({\Phi}_{F}^{r})}_{R}$ (

**a**) and the flux of U-energy ${({\Phi}_{U}^{r})}_{R}$ (

**b**) for aqueous ethanol solutions.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ślęzak-Prochazka, I.; Batko, K.M.; Ślęzak, A.
Evaluation of Transport Properties and Energy Conversion of Bacterial Cellulose Membrane Using Peusner Network Thermodynamics. *Entropy* **2023**, *25*, 3.
https://doi.org/10.3390/e25010003

**AMA Style**

Ślęzak-Prochazka I, Batko KM, Ślęzak A.
Evaluation of Transport Properties and Energy Conversion of Bacterial Cellulose Membrane Using Peusner Network Thermodynamics. *Entropy*. 2023; 25(1):3.
https://doi.org/10.3390/e25010003

**Chicago/Turabian Style**

Ślęzak-Prochazka, Izabella, Kornelia M. Batko, and Andrzej Ślęzak.
2023. "Evaluation of Transport Properties and Energy Conversion of Bacterial Cellulose Membrane Using Peusner Network Thermodynamics" *Entropy* 25, no. 1: 3.
https://doi.org/10.3390/e25010003