# 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 |

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**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.

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**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