# Aspects of Mathematical Modelling of Pressure Retarded Osmosis

## Abstract

**:**

## 1. Introduction

**Figure 1.**Schematic representation of a classic pressure retarded osmosis (PRO) energy generation plant with a compartmental geometry (adapted from [3]). The feed and draw compartments are assumed to be well stirred, so that the concentration at the membrane surface is the same as the outflowing concentration: ${C}_{f,out}={C}_{f,m}$ and ${C}_{d,out}={C}_{d,m}$.

## 2. Ideal Membrane

#### 2.1. Compartmental Configuration

_{g}) and that used by the pump (W

_{p}). Assuming that the pressure difference across the pump and the generator ($\Delta P$, also called working pressure) is the same and is equal to the hydrostatic pressure difference across the membrane, the generator and pump powers can be expressed as: ${W}_{g}=\left({S}_{m}{J}_{w}+{F}_{d,in}\right)\Delta P$ and ${W}_{p}={F}_{d,in}\Delta P$ respectively, where ${J}_{w}$ is the osmotic water flux, ${S}_{m}$ is the surface area of the membrane and ${F}_{d,in}$ is the draw solution pump flow rate or flow rate into the draw compartment. Therefore, the power generated per unit area of the membrane ($W=\left({W}_{g}-{W}_{p}\right)/{S}_{m}$) is [3]:

#### 2.2. Counterflow Configuration

**Figure 2.**(

**a**) A diagram representing tubular geometry and counterflow configuration for PRO. (

**b**) Simplified representation of the tubular geometry as a square channel. Concentrations (${C}_{d}(x)$ and ${C}_{f}(x)$) and flows (${J}_{w}(x)$, ${F}_{d}(x)$ and ${F}_{f}(x)$) are the functions of the position x along the channel for the counterflow configuration.

## 3. Concentration Polarization

**Figure 3.**A diagram representing the concentration polarization for an asymmetric membrane with the support layer due to the water flow through the membrane. Here C

_{f,m}and C

_{d,m}are concentrations of salt on the feed and draw sides of the active layer respectively and C

_{f,b}and C

_{d,b}are the bulk concentrations on the feed and draw sides of the membrane respectively.

## 4. Module-Scale Analysis of PRO

_{m}). Note that Equation (28) is similar to Equation (10), but has the positive sign in front of J

_{w}, as the direction of x selected for Equation (10) is opposite to that for s in Equation (28). The boundary conditions for Equations (27)–(30) are ${F}_{d}\left(0\right)={F}_{d,in}$, ${F}_{f}\left(1\right)={F}_{f,in}$, ${C}_{d}\left(0\right)={C}_{d,in}$ and ${C}_{f}\left(1\right)={C}_{f,in}$ [20].

^{2}, only 15% of the power density available for the small scale compartmental (coupon scale) PRO system [20] and well short of 5 W/m

^{2}necessary to produce osmotic power on commercial basis [26].

## 5. Other Aspects of Mathematical Modelling of PRO

^{+}and Cl

^{−}ions, that creates two ionic species which can differ in their permeability through the membrane. In general, solution diffusion and electro-migration have to be taken into account for multi-ionic systems [33]. Yaroshchuk et al. [33] considered mathematical modelling of transport of multiple ions where diffusion was coupled to electro-migration and concluded that such modelling was important to understand phenomenon such as negative rejection for some ions in particular that spontaneously arising electric fields may yield much higher NaCl rejection, which could be relevant for energy generating PRO systems.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- Pattle, R.E. Production of electric power by mixing fresh and salt water in the hydroelectric pile. Nature
**1954**, 174, 660. [Google Scholar] [CrossRef] - Loeb, S. Osmotic power-plants. Science
**1975**, 189, 654–655. [Google Scholar] [CrossRef] [PubMed] - Lee, K.L.; Baker, R.W.; Lonsdale, H.K. Membranes for power-generation by pressure-retarded osmosis. J. Membr. Sci.
**1981**, 8, 141–171. [Google Scholar] [CrossRef] - McGinnis, R.L.; McCutcheon, J.R.; Elimelech, M. A novel ammonia-carbon dioxide osmotic heat engine for power generation. J. Membr. Sci.
**2007**, 305, 13–19. [Google Scholar] [CrossRef] - Hon, K.C.; Zhao, C.L.; Yang, C.; Low, S.C. A method of producing electrokinetic power through forward osmosis. Appl. Phys. Lett.
**2012**, 101. [Google Scholar] [CrossRef] - Lin, S.H.; Yip, N.Y.; Cath, T.Y.; Osuji, C.O.; Elimelech, M. Hybrid pressure retarded osmosis-membrane distillation system for power generation from low-grade heat: Thermodynamic analysis and energy efficiency. Environ. Sci. Technol.
**2014**, 48, 5306–5313. [Google Scholar] [CrossRef] [PubMed] - Helfer, F.; Lemckert, C.; Anissimov, Y.G. Osmotic power with pressure retarded osmosis: Theory, performance and trends—A review. J. Membr. Sci.
**2014**, 453, 337–358. [Google Scholar] [CrossRef] - Zhao, S.F.; Zou, L.; Tang, C.Y.Y.; Mulcahy, D. Recent developments in forward osmosis: Opportunities and challenges. J. Membr. Sci.
**2012**, 396, 1–21. [Google Scholar] [CrossRef] - Altaee, A.; Zaragoza, G.; Sharif, A. Pressure retarded osmosis for power generation and seawater desalination: Performance analysis. Desalination
**2014**, 344, 108–115. [Google Scholar] [CrossRef] - Altaee, A.; Sharif, A. Pressure retarded osmosis: Advancement in the process applications for power generation and desalination. Desalination
**2015**, 356, 31–46. [Google Scholar] [CrossRef] - Straub, A.P.; Deshmukh, A.; Elimelech, M. Pressure-retarded osmosis for power generation from salinity gradients: Is it viable? Energy Environ. Sci.
**2016**, 9, 31–48. [Google Scholar] [CrossRef] - Sivertsen, E.; Holt, T.; Thelin, W.R.; Brekke, G. Iso-watt diagrams for evaluation of membrane performance in pressure retarded osmosis. J. Membr. Sci.
**2015**, 489, 299–307. [Google Scholar] [CrossRef] - Sharqawy, M.H.; Banchik, L.D.; Lienhard, J.H. Effectiveness-mass transfer units (epsilon-mtu) model of an ideal pressure retarded osmosis membrane mass exchanger. J. Membr. Sci.
**2013**, 445, 211–219. [Google Scholar] [CrossRef] - Banchik, L.D.; Sharqawy, M.H.; Lienhard, J.H. Limits of power production due to finite membrane area in pressure retarded osmosis. J. Membr. Sci.
**2014**, 468, 81–89. [Google Scholar] [CrossRef] - Mehta, G.D.; Loeb, S. Internal polarization in the porous substructure of a semipermeable membrane under pressure-retarded osmosis. J. Membr. Sci.
**1978**, 4, 261–265. [Google Scholar] [CrossRef] - Loeb, S.; Titelman, L.; Korngold, E.; Freiman, J. Effect of porous support fabric on osmosis through a loeb-sourirajan type asymmetric membrane. J. Membr. Sci.
**1997**, 129, 243–249. [Google Scholar] [CrossRef] - Yip, N.Y.; Tiraferri, A.; Phillip, W.A.; Schiffrnan, J.D.; Hoover, L.A.; Kim, Y.C.; Elimelech, M. Thin-film composite pressure retarded osmosis membranes for sustainable power generation from salinity gradients. Environ. Sci. Technol.
**2011**, 45, 4360–4369. [Google Scholar] [CrossRef] [PubMed] - McCutcheon, J.R.; Elimelech, M. Influence of concentrative and dilutive internal concentration polarization on flux behavior in forward osmosis. J. Membr. Sci.
**2006**, 284, 237–247. [Google Scholar] [CrossRef] - Chou, S.R.; Wang, R.; Shi, L.; She, Q.H.; Tang, C.Y.; Fane, A.G. Thin-film composite hollow fiber membranes for pressure retarded osmosis (pro) process with high power density. J. Membr. Sci.
**2012**, 389, 25–33. [Google Scholar] [CrossRef] - Straub, A.P.; Lin, S.H.; Elimelech, M. Module-scale analysis of pressure retarded osmosis: Performance limitations and implications for full-scale operation. Environ. Sci. Technol.
**2014**, 48, 12435–12444. [Google Scholar] [CrossRef] [PubMed] - McCutcheon, J.R.; McGinnis, R.L.; Elimelech, M. Desalination by ammonia-carbon dioxide forward osmosis: Influence of draw and feed solution concentrations on process performance. J. Membr. Sci.
**2006**, 278, 114–123. [Google Scholar] [CrossRef] - Reimund, K.K.; McCutcheon, J.R.; Wilson, A.D. Thermodynamic analysis of energy density in pressure retarded osmosis: The impact of solution volumes and costs. J. Membr. Sci.
**2015**, 487, 240–248. [Google Scholar] [CrossRef] - Yaroshchuk, A. Optimal hydrostatic counter-pressure in pressure-retarded osmosis with composite/asymmetric membranes. J. Membr. Sci.
**2015**, 477, 157–160. [Google Scholar] [CrossRef] - Thorsen, T.; Holt, T. The potential for power production from salinity gradients by pressure retarded osmosis. J. Membr. Sci.
**2009**, 335, 103–110. [Google Scholar] [CrossRef] - Trettin, D.R.; Doshi, M.R. Limiting flux in ultrafiltration of macromolecular solutions. Chem. Eng. Commun.
**1980**, 4, 507–522. [Google Scholar] [CrossRef] - Skilhagen, S.E. Osmotic power—A new, renewable energy source. Desalination Water Treat.
**2010**, 15, 271–278. [Google Scholar] [CrossRef] - Kim, Y.C.; Kim, Y.; Oh, D.; Lee, K.H. Experimental investigation of a spiral-wound pressure-retarded osmosis membrane module for osmotic power generation. Environ. Sci. Technol.
**2013**, 47, 2966–2973. [Google Scholar] [CrossRef] [PubMed] - Xu, Y.; Peng, X.Y.; Tang, C.Y.Y.; Fu, Q.S.A.; Nie, S.Z. Effect of draw solution concentration and operating conditions on forward osmosis and pressure retarded osmosis performance in a spiral wound module. J. Membr. Sci.
**2010**, 348, 298–309. [Google Scholar] [CrossRef] - Lin, S.H.; Straub, A.P.; Elimelech, M. Thermodynamic limits of extractable energy by pressure retarded osmosis. Energy Environ. Sci.
**2014**, 7, 2706–2714. [Google Scholar] [CrossRef] - Yip, N.Y.; Elimelech, M. Thermodynamic and energy efficiency analysis of power generation from natural salinity gradients by pressure retarded osmosis. Environ. Sci. Technol.
**2012**, 46, 5230–5239. [Google Scholar] [CrossRef] [PubMed] - Seppala, A.; Lampinen, M.J. Thermodynamic optimizing of pressure-retarded osmosis power generation systems. J. Membr. Sci.
**1999**, 161, 115–138. [Google Scholar] [CrossRef] - Achilli, A.; Cath, T.Y.; Childress, A.E. Power generation with pressure retarded osmosis: An experimental and theoretical investigation. J. Membr. Sci.
**2009**, 343, 42–52. [Google Scholar] [CrossRef] - Yaroshchuk, A.; Bruening, M.L.; Bernal, E.E.L. Solution-diffusion-electro-migration model and its uses for analysis of nanofiltration, pressure-retarded osmosis and forward osmosis in multi-ionic solutions. J. Membr. Sci.
**2013**, 447, 463–476. [Google Scholar] [CrossRef]

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Anissimov, Y.G.
Aspects of Mathematical Modelling of Pressure Retarded Osmosis. *Membranes* **2016**, *6*, 13.
https://doi.org/10.3390/membranes6010013

**AMA Style**

Anissimov YG.
Aspects of Mathematical Modelling of Pressure Retarded Osmosis. *Membranes*. 2016; 6(1):13.
https://doi.org/10.3390/membranes6010013

**Chicago/Turabian Style**

Anissimov, Yuri G.
2016. "Aspects of Mathematical Modelling of Pressure Retarded Osmosis" *Membranes* 6, no. 1: 13.
https://doi.org/10.3390/membranes6010013