1. Introduction

The considerable and very fast growing energy consumption consecutive to the economic development of many countries is acutely raising the question of avoiding disastrous environmental consequences which could inevitably occur if only conventional fossil sources are used. To cope with such situation, a large effort has been oriented toward other sources such as less damaging renewable ones (Lewis, et al., 2011; Kumar, et al., 2011). Aside solar, wind, geothermal and hydraulic sources, they also include other less evident ones which may however represent interesting alternatives in specific situations.

Such is the osmotic power which is the process of converting the pressure differential between water with high salinity and water with lower or no salinity into hydraulic pressure (Loeb, 1975; Mishra, 2013; Kho, 2010; The European Commission, 2004; Helfer, et al., 2013; Skilhagen, et al., 2012; Skilhagen, and Aaberg, 2012). The harnessing of this energy for conversion into power can be accomplished by means of Pressure Retarded Osmosis (PRO) (Kim, and Elimelech, 2013; Helfer, et al., 2014; Wang, et al., 2012). This technique uses a semipermeable membrane to separate a less concentrated solution, or solvent, (for example, fresh water) from a more concentrated and pressurized solution (for example sea water), allowing the solvent to pass to the concentrated solution side (Post, 2009). The additional volume increases the pressure on this side, which can be depressurized by a hydro-turbine to produce power and electricity (Kleiterp, 2012) .As seventy percent of Earth surface is covered with water, 97 percent of which is saltwater, the process created by mixing seawater with freshwater generates a resulting osmotic power which could serve as both renewable and consistent electricity source. While still in early stages, best estimates of the global production potential of osmotic power exceed 1,600 terawatt-hours, the equivalent of half of Europe entire energy demand. There are two primary sources for osmotic power: 1) natural occurrence where river water meets the sea water, and 2) merging two man-made water sources from processing plants

Both methods can be viable but in 1), seawater averages 40 grams of salt/ liter + River Water, less power is provided than in than 2), where brine (from desalination) averages 60 grams of salt/ liter + treated water. The higher the salinity, the more free energy can be extracted, and the more power can be generated. Today osmotic power is a promising renewable energy source (RES) provided conversion factor from pressure differential can be made large enough (Dinger, et al., 2012; Bræin, et al., 2010; Straub, et al., 2016). In the following, some elements on this question will be discussed in PRO case. In particular, the threshold value for permeability coefficient which characterizes membrane efficiency for viable economic application is determined.

2. Osmosis Pressure Representation

Potential osmotic pressure π, the maximum osmotic pressure in a solution separated from osmosed fluid by a selectively permeable membrane, is given by

 = CiRT (1)

with π the potential osmotic pressure (Pa), C the solute molar concentration (mol m^³), R = 8,314 , T solution temperature (°K), and I the particle number per entity. Given the solutions, D and F with respective osmotic pressures _D and _F, where D is draw solution (most concentrated), and F feed one (less concentrated), osmotic pressure difference _D_F between the two solutions is typically equal to 12 bars for water fluid, ie, 12.10⁵Pa. Let P = P_D  P_F with P_D and P_F the pressures of solutions D and F. With Spiegler-Kedem [18] modelflux equationscan be integrated and one gets volume and solute fluxes J_v and J_s flowing through the membrane with pressure retarded osmosis

J_v = A(P) ; J_s = E_m)^¹J_v C_dE_m  C_i) (2)

with J_v , J_s (m.s^¹) the volume and solute fluxes across the membrane,  the solute reflection coefficient, A(m.Pa^¹ s^¹) the membrane fluid permeability, and E_m = exp (1 ) J_vL^¹] with  the solute permeability. Similarly in the support one gets

J_s = E_s)^¹J_v C_iE_s  C_f) (3)

with C_d, C_i and C_f the solute concentration in the draw solution, at the barrier-layer/support interface, and in the feed solution respectively, and E_s = exp J_vD_e^¹] with  the support thickness. Equality of fluxes in the barrier layer and the support gives from (1,2,3) the non-dimensional equation

{(1)(1E¹^)^¹ + E^(1E^)^¹} = (1r)(1E^)^¹ 

with  = 1 C_i/C_d,  the solute reflection coefficient, r = C_f/C_d, /D_eL,  the support thickness, L the barrier layer thickness, D_e the effective diffusion coefficient of the solute, E = expX, X = F( <P>), F = ARTC_dL/,  the salt stoichiometric coefficient and finally <P> = (P_d  P_f)/RTC_d. For a given dimensionless hydrostatic-pressure difference <P>, (4) is a transcendental equation in (dimensionless concentration difference across the barrier layer) which determines the operating conditions of the osmotic plant for a given set of system parameters.

3. Osmotic Power Plant Production

The basic energy production system is composed of a compression unit which delivers pressurized salted water injected in a chamber with a filtering membrane across which a flux of salted water J_s is crossing, see Figure 1.

Figure 1. Sketch of Osmotic Plant Balance

J_S is Salt Flux across the Membrane.

The power produced per surface unit of installed membrane is given W = J_mP with W in Watts/m^², or else in non-dimensional form

<W> = W/A(RTC_d)² = <P>(<P>) (5)

<W> is maximum when <P> = 2 and is then equal to <W> = ()^But this is not necessarily possible as  and <P> are also linked by (4) which imposes a constraint on system coefficients. One effectively gets for z = the transcendental equation

z{(1)(1E(z)¹^)^¹ + E(z)^(1E(z)^)^¹} =  (1r)(1E(z)^)^¹ ²

with E(z) = expFz which relates “physical” z (concentration performance) to “technical” F (barrier quality). From this different limits can be evaluated for possible power output from the system (Yip, and Elimelech, 2012; Yip, and Elimelech, 2011; Lin et al., 2014). However simple ones are directly obtained from (6) for <W>. In the case of very large  for instance, (4) takes the very simple form

Y =  (Ln(1x))/x (7)

where x = [1r((1)^¹] and Y = (1r)F/2. In definition interval x  [0,1], Y is monotonically increasing from 1 to + , and normalized delivered power <W> = Kx², with K = ^[1r((1)^¹]², varies monotonically between 0 and K, showing that larger <W> corresponds to larger x. More generally parametric dependence of power output W can be obtained from (6) to get best parameter range. When returning to dimensional expression, W is upper bounded by W_sup = KA(RTC_d)², and a necessary condition for economic efficiency is that

W_sup = KA(RTC_d)²≥ W_crit (8)

where W_crit is the threshold value above which the osmotic plant is viable. W_sup is larger with larger draw concentration C_d, larger temperature T and larger A, which has to be determined to satisfy economic system efficiency demand, ie here A ≥ A_crit = 4W_crit^For usual figure of delivered power W_d = 5Wm^², and with typical = 12.10⁵Pa for water, one gets, for instance, A ≥ A_crit = 1.38 mPa^¹s^¹. More generally the variation of A_crit vs sees Figure 2, indicates for  [6.10⁵,20.10⁵] the “efficiency” interval A  [4.910^¹²,5.410^¹¹] which is still at technical limit today (Straub, et al., 2016; Zhang, and Chung, 2013). To cope with these economic operating constraints, extension of simple osmotic barrier effect have been recently considered (Chou, et al., 2013; Chou, et al., 2012; Yip, et al, 2010; Banchik, et al., 2014; Mc Cutcheon, and Elimelech, 2007; Cath, et al., 2013; Chen, et al., 2016; She,, et al., 2016 ; Chou, et al., 2013; Hickenbottom, et al., 2016; Dechadilok, and Deen, 2006). A coupled system with solar plant will be discussed elsewhere.

$D:\Travail ECE\ING 4\PPE\Matlab\Capture courbe 1.png$

Figure 2 . Permeability Coefficient vs Osmotic

Pressure Difference.

4. Conclusion

From analysis of equations representing osmotic physical phenomenon, it is shown that in Pressure Retarded Osmosis case, the harnessing of salinity-gradient energy taking place at the interface between waters of different salt concentration could provide an interesting and almost inexhaustible energy source if systems coefficients satisfy operating conditions which have been explicitly stated within Spiegler-Kedem model. However, even in optimum mode operating case, satisfaction of economic viability condition expressed by fixed specific membrane power output W_thres  5 Wm^²is not always met. Aside theoretical research on optimizing system operating mode, this weakness is urging further study of membrane physical properties, such as porosity and tortuosity pore length, to design most efficient hydraulic permeability of the barrier layer.

Acknowledgments

The Authors are very much indebted to ECE Paris School of Engineering for having provided the necessary setup for the development of the project, to Drs. S. Mouhali for advices during the research and Pr. M. Cotsaftis for help in the preparation of the manuscript.

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