# Theoretical Analysis of a Mathematical Relation between Driving Pressures in Membrane-Based Desalting Processes

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Postulates for Membrane-Based Desalting Systems

**Postulate**

**1.**

^{3}$/\mathrm{min}$ of volumetric flux at 10 bar of hydraulic pressure, then the magnitude of volumetric flux at 13 bar, 16 bar, and 20 bar must be higher than 0.01 m

^{3}$/\mathrm{min}$. As the driving pressure increases, the magnitude of water transport never decreases—or, at least, remains unchanged—as long as the direction of water transport is not reversed. That is, postulate (P.1) implies that the plot of the water flux is either monotonically increasing or monotonically decreasing, according to the increase in the driving pressure. This rule also holds for a case in which the water transport results from an osmotic pressure difference. This postulate is supported by numerous studies, such as Oh et al. [23], He et al. [17], and Gaetan et al. [24]. The second postulate concerns the definition of “the ideal semi-permeable membrane.”

**Postulate**

**2.**

**Postulate**

**3.**

**Postulate**

**4.**

**Postulate**

**5.**

**Postulate**

**6.**

**Postulate**

**7.**

#### 2.2. Redefining the Model for Membrane-Based Desalting Processes

^{2}/s). This unit rule is widely accepted, even for other types of diffusivities, such as heat transfer diffusivity and momentum transfer diffusivity. Such a coincidence leads to the advent of the same type of mass transfer coefficient—a key parameter in determining the characteristics of membrane-based desalting systems.

#### 2.3. Definitions of Pseudo-Driving Pressures

#### 2.4. Similarity Coefficients and the Reflection Coefficient

#### 2.5. A Relation between Osmotic Pressure and Hydraulic Pressure

## 3. Results and Discussions

#### 3.1. A Constraint for the Monotonic Functions by the Similarity Coefficient Ratio

#### 3.2. Verification of the Relation between the Driving Pressures in FO and PRO Modes

#### 3.3. Verification of the Relation between Driving Pressures in the RO Mode

#### 3.4. Hypothesis for the Water Flux in the Transition Region between FO/PRO and RO

- (i.)
- Figure 7a. The direction of the water flux is not reversed and the absolute value of the water flux gradually increases as $\frac{\Delta P}{\Delta \pi}$ approaches one;
- (ii.)
- Figure 7b. The water flux continually decreases so that the direction of the water transport gets reversed and the absolute value of the water flux gradually increases as $\frac{\Delta P}{\Delta \pi}$ approaches one; and
- (iii.)
- Figure 7c. The water flux converges to zero and such a tendency is sustained.

#### 3.5. Practical Implications of Theoretical Analyses with Respect to Driving Pressures

## 4. Conclusions

- (I)
- $\mathsf{\Delta}\pi $ and $\mathsf{\Delta}P$ are related via the osmotic pressure difference in the boundary layer of the more concentrated side of a system, $\mathsf{\Delta}{\pi}_{\delta}$. When a given process is operated in FO/PRO modes, then $\mathsf{\Delta}\pi =\frac{{S}_{P}}{{S}_{\pi}}\mathsf{\Delta}P+\mathsf{\Delta}{\pi}_{\delta}$. On the other hand, $\mathsf{\Delta}\pi =\frac{{S}_{P}}{{S}_{\pi}}\Delta P-\mathsf{\Delta}{\pi}_{\delta}$ if the given process is RO.
- (II)
- Since $\Delta {\pi}_{\mathrm{pse}}>\Delta {P}_{\mathrm{pse}}$ in FO/PRO modes and $\Delta {\pi}_{\mathrm{pse}}<\Delta {P}_{\mathrm{pse}}$ in the RO mode, this means that $1>\frac{{S}_{P}\Delta P}{{S}_{\pi}\Delta \pi}$ for FO/PRO modes and $1<\frac{{S}_{P}\Delta P}{{S}_{\pi}\Delta \pi}$ for the RO mode. In addition, based on the postulate that specifies that the performance of actual membranes never exceeds that of the ideal membrane, ${S}_{\pi}<{S}_{P}$ in FO/PRO modes and ${S}_{\pi}>{S}_{P}$ in the RO mode. This contrast between FO/PRO and RO modes is critical for optimizing process parameters.
- (III)
- The point at which ${J}_{\mathrm{v}}=0$ always belongs to the FO/PRO region due to the reflection coefficient that states that ${S}_{\pi}<{S}_{P}$ when ${J}_{\mathrm{v}}=0$. In other words, ${J}_{\mathrm{v}}$ never becomes zero in the RO mode, theoretically.
- (IV)
- There can exist a practical water flux limit for FO and PRO processes, unless severe dilutive external concentration polarization is assumed.
- (V)
- When $\frac{\mathsf{\Delta}P}{\mathsf{\Delta}\pi}=1$, the value of the water flux made by the actual membranes cannot be defined because the value of the water flux at that point does not comply with the fundamental postulates.
- (VI)
- Given that ${J}_{\mathrm{v}}$ always monotonically increases or decreases according to $\frac{\mathsf{\Delta}P}{\mathsf{\Delta}\pi}$, the value of $\frac{\partial \left(\frac{{S}_{P}}{{S}_{\pi}}\right)}{\partial \left(\frac{\Delta P}{\Delta \pi}\right)}$ in desalting systems must be equal to or larger than a specific negative value (see Table 3).
- (VII)
- Within the range of $\frac{{S}_{\pi}}{{S}_{P}}<\frac{\mathsf{\Delta}P}{\mathsf{\Delta}\pi}<1$, in principle, the value of ${J}_{\mathrm{v}}$ is maintained as zero according to the preceding postulates. However, the practical values of ${J}_{\mathrm{v}}$ within the range fluctuate to some extent because of the presence of membrane parameters.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclatures

$B$ | The salt permeability of a semi-permeable membrane (g/mol s) |

$S$ | The structure parameter of a semi-permeable membrane (m) |

$R$ | Rejection rate of a semi-permeable membrane (-) |

$T$ | Temperature (K) |

$\Delta P$ | External hydraulic pressure (Pa) |

$D$ | The diffusivities defined in membrane-based desalting systems (m^{2}/s) |

${T}_{p}$ | The diffusivities in membrane-based desalting systems when the transported variable is the driving pressures (m^{2}/s) |

$k$ | The mass transfer coefficients in membrane-based desalting systems (m^{3}/m^{2} s) |

${S}_{\pi}$ | The similarity coefficients bridging the pseudo-osmotic pressures and the bulk osmotic pressures (-) |

${S}_{P}$ | The similarity coefficients bridging the pseudo-osmotic pressures and the bulk osmotic pressures (-) |

$a$ | Arbitrary coefficients bridging the pseudo-driving pressures and the solute concentration (N m/mol) |

$X$ | Transported variables of membrane-based desalting systems |

$C$ | Solute concentration in membrane-based desalting systems (mol/m^{3}) |

$DP$ | Driving pressures in membrane-based desalting systems (Pa) |

$y$ | Distance from the surface of a semi-permeable membrane (m) |

$Y$ | Dimensionless distance from the surface of a semi-permeable membrane to the end of a boundary layer (-) |

${J}_{v}$ | Water flux in membrane-based desalting systems (m^{3}/m^{2} s) |

${J}_{c}$ | Salt flux in membrane-based desalting systems (g/m^{2} s) |

Greek symbols | |

$\pi $ | Osmotic pressure (Pa) |

$\delta $ | Length of a boundary layer in the more concentrated side of the membrane-based desalting systems (m) |

$\alpha $ | Arbitrary pressure existing in the less concentrated side of membrane-based desalting systems (Pa) |

$\sigma $ | The reflection coefficient of membrane-based desalting systems (-)${\beta}_{ov}$ The diffusive and convective mass transfer coefficient applied to the salt flux (g/mol s) |

Subscripts and superscripts | |

$C$ | The transported variable of a system is the solute concentration |

$DP$ | The transported variable of a system is the driving pressure (specific energy) |

$h$ | A more concentrated side of membrane-based desalting systems |

$l$ | A less concentrated side of membrane-based desalting systems |

$m$ | The solute concentration at the semi-permeable membrane surface |

$b$ | The solute concentration in the bulk more concentrated region |

$pse$ | Pseudo-driving pressures |

$ideal$ | The ideal system with the ideal semi-permeable membrane |

$actual$ | The actual system with the actual semi-permeable membrane |

## Appendix A. Justification for the Equality between the Concentration-Based Differential Equation and the Pressure-Based Differential Equation

## Appendix B. Brief Derivation for the Water Flux with Respect to Driving Pressures

## Appendix C. Brief Explanation on a Notation for the Mass Transfer Coefficient, $k$

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**Figure 1.**Schematic diagrams illustrating the flow of the water flux (${J}_{\mathrm{v}}$) and the salt flux (${J}_{C}$) in (

**a**) the reverse osmosis (RO) mode and (

**b**) the forward osmosis (FO) and pressure-retarded osmosis (PRO) modes. Note that the minus sign is appended to the salt flux in FO/PRO modes due to the opposite directions of the water flux and the salt flux. The red curves in each figure represent the relative change of the solute concentration. The larger the solute concentration, the higher the red curve is placed. In the RO mode, the solute concentration at the surface of a membrane in the more concentrated side $\left({C}_{\mathrm{h},\mathrm{m}}\right)$ is higher than the bulk solute concentration in the more concentrated side $\left({C}_{\mathrm{h},\mathrm{b}}\right)$. On the other hand, ${C}_{\mathrm{h},\mathrm{b}}$ is higher than ${C}_{\mathrm{h},\mathrm{m}}$ in the FO/PRO modes. Such a difference in the distribution of the solute concentration is what distinguishes the two main types of membrane-based desalting processes.

**Figure 2.**A schematic plot with respect to the relationship between the ratio of the bulk driving pressures $\left(\frac{\Delta P}{\Delta \pi}\right)$ and the value of the water flux $\left({J}_{\mathrm{v}}\right)$ in FO and PRO modes. A presumed water flux limit in FO and PRO is marked with a dashed gray line. When $\mathsf{\Delta}P=0$ (i.e., the FO mode), the water flux must be calculated with ${J}_{\mathrm{v}}=-k\mathrm{ln}\left(\frac{\mathsf{\Delta}{\pi}_{\mathrm{m}}}{\mathsf{\Delta}\pi}\right)$. On the other hand, the water flux in the PRO mode can be determined with the equation ${J}_{\mathrm{v}}=-k\mathrm{ln}\left(\frac{\mathsf{\Delta}{P}_{\mathrm{pse}}}{\mathsf{\Delta}{\pi}_{\mathrm{pse}}}\right)$. The dashed red line indicates the change of ${J}_{\mathrm{v}}$ with the implementation of an actual membrane and the practical change of ${J}_{\mathrm{v}}$ when $\mathsf{\Delta}P\to 0$. On the other hand, the solid red curve indicates the change of ${J}_{\mathrm{v}}$ when the implementation of an actual membrane is taken into account but the practical change of ${J}_{\mathrm{v}}$ is not. That is, the solid red line does not consider the value of ${J}_{\mathrm{v}}$ when moment $\mathsf{\Delta}P=0$. The black curves indicate the changes of ${J}_{\mathrm{v}}$ when an ideal membrane is employed. The difference between the solid and dashed black lines is the same as the difference between the solid and dashed red lines.

**Figure 3.**A figure representing the values of $\frac{{J}_{\mathrm{v}}}{k}$, according to $\frac{\Delta {\pi}_{\delta}}{\Delta \pi}$, using the experimental data from a previous study [50]. In this previous study, the authors used two other FO/PRO membranes that were manufactured by HTI and Oasys. The FO experiment was conducted with the condition of ${C}_{\mathrm{h},\mathrm{b}}=1.5\mathrm{M},{C}_{\mathrm{l},\mathrm{b}}=0\mathrm{M}$, while the PRO experiment was conducted with ${C}_{\mathrm{h},\mathrm{b}}=1.5\mathrm{M},{C}_{\mathrm{l},\mathrm{b}}=0.5\mathrm{M}$. The temperature of both experiments was fixed at 20 °C. The straight red line in the figure indicates the approximation of $-\mathrm{ln}\left(1-\frac{\Delta {\pi}_{\delta}}{\Delta \pi}\right)$, which is applicable when $\frac{\Delta {\pi}_{\delta}}{\Delta \pi}$ is small enough.

**Figure 4.**The change of (

**a**) the reflection coefficient ($\sigma $) and (

**b**) the bulk driving pressure difference ($\Delta \pi -\Delta P$ ), according to the bulk osmotic pressure difference ($\mathsf{\Delta}\pi $ ). The calculated values of $\left(1-\sigma \right)\pi $ in (

**b**) are based on Equation (37) and the experimental data in (

**a**,

**b**) were adapted with permission from [57]. Copyright 2018, American Chemical Society.

**Figure 5.**Volumetric fluxes under the null-pressure condition (i.e.,$\frac{\mathsf{\Delta}P}{\mathsf{\Delta}\pi}=1$) for (

**a**) the cellulose triacetate (CTA) membrane and (

**b**) the thin-film composite (TFC) membrane with relevant standard deviations. While the water flux of CTA under the null-pressure condition is within the admittable error range of zero water flux, that of the TFC deviates far from the error range [57]. Copyright 2018, American Chemical Society.

**Figure 6.**A plot with respect to the relationship between the ratio of the bulk driving pressures $\left(\frac{\Delta P}{\Delta \pi}\right)$ and the absolute value of the water flux $\left({J}_{\mathrm{v}}\right)$ in the RO mode. Since the sign of the RO water flux is defined as a minus in the current study, the absolute value bars are appended to ${J}_{\mathrm{v}}$. The water flux is determined with the equation ${J}_{\mathrm{v}}=-k\mathrm{ln}\left(\frac{\mathsf{\Delta}{P}_{\mathrm{pse}}}{\mathsf{\Delta}{\pi}_{\mathrm{pse}}}\right)$ by setting the value of $k=5\ast {10}^{-5}\mathrm{m}/\mathrm{s}$. The solid red curve indicates the change of ${J}_{\mathrm{v}}$ when an actual membrane is employed with the condition of $\frac{\partial \left(\frac{{S}_{\pi}}{{S}_{P}}\right)}{\partial \left(\frac{\mathsf{\Delta}P}{\mathsf{\Delta}\pi}\right)}=-0.5$. The dashed red curve indicates the change of ${J}_{\mathrm{v}}$ when an actual membrane is employed with the condition of $\frac{\partial \left(\frac{{S}_{\pi}}{{S}_{P}}\right)}{\partial \left(\frac{\mathsf{\Delta}P}{\mathsf{\Delta}\pi}\right)}=0$. In both cases, the value of $\frac{{S}_{\pi}}{{S}_{P}}=0.98$ at $\frac{\mathsf{\Delta}P}{\mathsf{\Delta}\pi}=1.05$. The black curve indicates the change of ${J}_{\mathrm{v}}$ when an ideal membrane is employed. The value of the water flux is undefinable when $\frac{\mathsf{\Delta}P}{\mathsf{\Delta}\pi}=1$ due to the dilemma between postulate (P.3) and Equation (25).

**Figure 7.**Plots representing the hypothetical tendencies of the water flux $\left({J}_{\mathrm{v}}\right)$ with respect to the driving pressures within the range of $\frac{{S}_{\pi}}{{S}_{P}}<\frac{\Delta P}{\Delta \pi}<1$. In (

**a**), the value of ${J}_{\mathrm{v}}$ rebounds and increases in FO/PRO modes. In contrast, ${J}_{\mathrm{v}}$ enters the region of the RO mode in (

**b**). Lastly, (

**c**) illustrates the tendency that ${J}_{\mathrm{v}}$ converges to zero.

**Figure 8.**Summing up the overall tendency of the water flux in membrane-based desalting systems in accordance with the conditions given in Table 2. In this figure, for convenience, the value of the water flux is nondimensionalized with the mass transfer coefficient, $k$. The straight black line indicates changes in the dimensionless water flux made by the ideal membrane $\left(\frac{{J}_{\mathrm{v},\mathrm{ideal}}}{k}\right)$, the red curve indicates changes in the dimensionless water flux made by an actual membrane $\left(\frac{{J}_{\mathrm{v},\mathrm{actual}}}{k}\right)$, and the dashed red line, within $\mathrm{ln}\left(\frac{{S}_{\pi}}{{S}_{P}}\right)<\mathrm{ln}\left(\frac{\Delta P}{\Delta \pi}\right)<0$, indicates a “transition” (membrane-dominant) region.

**Figure 9.**A chart visually representing the ideal sequences for improving the performance of each membrane-based desalting process. To produce as much water as possible (i.e., making $\frac{{S}_{P}}{{S}_{\pi}}$ close to one) with the smallest energy loss (i.e., making both ${S}_{P}$ and ${S}_{\pi}$ close to one), each process must take different steps due to the second constraint in Table 3. In FO/PRO modes, (1) the operational parameters that mostly determine the value of ${S}_{P}$ should be improved before (2) the membrane parameters of. However, in the RO mode, (1) the membrane parameters that mostly determine the value of ${S}_{\pi}$ should be improved before (2) the operational parameters.

**Table 1.**This table consists of the ratios of the maximal water flux (${J}_{\mathrm{v},\mathrm{max}}$ ) from each paper to the mass transfer coefficient ($k$ ) within the boundary layer of the more concentrated side.

Value of $\frac{{J}_{v,max}}{k}$ (FO) | Value of $\frac{{J}_{v,max}}{k}$ (PRO) | Reference |
---|---|---|

0 (Assumed that $k\to \infty $) | [52] | |

0.0823 | 0.0724 | [53] |

0.3874 | 0.6854 | [50] |

0.34 | - | [54] |

0.2830 | 0.8852 | [55] |

- | 0.4329 | [56] |

**Table 2.**The conditions utilized to plot Figure 8.

Process Types | Conditions | Condition Setting |
---|---|---|

FO/PRO | ${\left(\frac{{S}_{P}}{{S}_{\pi}}\right)}_{\frac{\Delta P}{\Delta \pi}=0.05}$ | 1.2 |

$\frac{\partial \left(\frac{{S}_{P}}{{S}_{\pi}}\right)}{\partial \left(\frac{\Delta P}{\Delta \pi}\right)}$ | 1.0 | |

${\left(\frac{\Delta P}{\Delta \pi}\right)}_{{J}_{\mathrm{v}}=0}$ | 0.579 | |

${\left({J}_{\mathrm{v}}\right)}_{\frac{{S}_{\pi}}{{S}_{P}}<\frac{\Delta P}{\Delta \pi}<1}$ | 0 | |

RO | ${\left(\frac{{S}_{P}}{{S}_{\pi}}\right)}_{\frac{\Delta P}{\Delta \pi}=1.05}$ | 0.99 |

$\frac{\partial \left(\frac{{S}_{P}}{{S}_{\pi}}\right)}{\partial \left(\frac{\Delta P}{\Delta \pi}\right)}$ | −0.022 |

Constraints | FO/PRO | RO |
---|---|---|

First constraint (Pseudo-Pressure) | $1>\frac{{S}_{P}}{{S}_{\pi}}\frac{\Delta P}{\Delta \pi}$$(\Delta {\pi}_{\mathrm{pse}}>\Delta {P}_{\mathrm{pse}})$ | $1<\frac{{S}_{P}}{{S}_{\pi}}\frac{\Delta P}{\Delta \pi}$$(\Delta {\pi}_{\mathrm{pse}}<\Delta {P}_{\mathrm{pse}})$ |

Second constraint (Similarity) | ${S}_{\pi}<{S}_{P}$ | ${S}_{\pi}>{S}_{P}$ |

Third constraint (Monotonic) | $\frac{\partial \left(\frac{{S}_{P}}{{S}_{\pi}}\right)}{\partial \left(\frac{\Delta P}{\Delta \pi}\right)}\ge -\frac{\Delta {\pi}^{2}-\Delta \pi \Delta {\pi}_{\delta}}{\Delta {P}^{2}}$ | $\frac{\partial \left(\frac{{S}_{P}}{{S}_{\pi}}\right)}{\partial \left(\frac{\Delta P}{\Delta \pi}\right)}\ge -\frac{\Delta {\pi}^{2}+\Delta \pi \Delta {\pi}_{\delta}}{\Delta {P}^{2}}$ |

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## Share and Cite

**MDPI and ACS Style**

Chae, S.H.; Kim, J.H.
Theoretical Analysis of a Mathematical Relation between Driving Pressures in Membrane-Based Desalting Processes. *Membranes* **2021**, *11*, 220.
https://doi.org/10.3390/membranes11030220

**AMA Style**

Chae SH, Kim JH.
Theoretical Analysis of a Mathematical Relation between Driving Pressures in Membrane-Based Desalting Processes. *Membranes*. 2021; 11(3):220.
https://doi.org/10.3390/membranes11030220

**Chicago/Turabian Style**

Chae, Sung Ho, and Joon Ha Kim.
2021. "Theoretical Analysis of a Mathematical Relation between Driving Pressures in Membrane-Based Desalting Processes" *Membranes* 11, no. 3: 220.
https://doi.org/10.3390/membranes11030220