Optimization of the Energy Output of Osmotic Power Plants

Abstract

On the way to a completely renewable energy supply, additional alternatives to hydroelectric, wind, and solar power have to be investigated. Osmotic power is such an alternative with a theoretical global annual potential of up to 14400 TWh (70% of the global electricity consumption of 2008) per year. It utilizes the phenomenon that upon the mixing of fresh water and oceanic salt water (e.g., at a river mouth), around 2.88 MJ of energy per 1 m³ of fresh water is released. Here, we describe a new approach to derive operational parameter settings for osmotic power plants using a pressure exchanger for optimal performance, either with respect to maximum generated power or maximum extracted energy. Up to now, only power optimization is discussed in the literature, but when considering the fresh water supply as a limiting factor, the energy optimization appears as the challenging task.

1. Introduction

Due to a future lack of fossil fuels and the risk of global climate change caused by CO₂ emissions, it appears prudent to aim for an energy supply which is based on renewable energy sources. It appears to be realistic that solar and wind power plants (together with hydroelectric power) will be able to supply the future demand [1]. However, in both cases, the generation depends strongly on natural fluctuations. So additional, renewable base-load capable energy sources are desired. A possible base-load energy source is offered by the use of the mixing entropy when fresh water and salt water (e.g., sea water) are mixed, predominantly at river mouths. In fact, mixing entropy (or rather its reduction) is also derived from solar energy, which drives evaporation from the ocean leading to separation of fresh water and salt water. Its theoretical global annual potential is estimated to be up to 14400 TWh per year [2]. Including economic and ecological boundaries, Statkraft forecasts that in fact only 1600 TWh is usable [3], Zeuner [4] confirms this estimation, which is still 8% of the global demand in electricity in 2008 [5].

There are several concepts to use mixing entropy for generating electric power; a list is given in [4]. One of them is the concept of “osmotic power,” which has been introduced by Sidney Loeb in the seventies; for example, see [6]. Because several components can be adopted from “reverse osmosis” desalination plants, osmotic power plants are the most developed concept to use mixing entropy for generating electric power. In 2009, the Norwegian energy provider Statkraft started operating the first osmotic test power plant [3].

In this paper, the optimal operating salinity level and the optimal operating pressure of an actual, so-called “pressure retarded osmotic” (PRO) power plant using a pressure exchanger are derived. The pressure exchanger is crucial for the constant performance of the power plant but also leads to a significant decrease in efficiency. Its settings and losses are implemented in our model. Aiming for a well-arranged model, we ignored all minor energy losses; for example, we are assuming frictionless flow. Furthermore, the most simple membrane model is applied.

2. Theory of Osmosis

When fresh water and salt water are mixed, for example, at a river mouth, the overall entropy increases due to the larger number of possible states for the salt ions. From another point of view, this phenomenon can be explained as an effect which neutralizes the difference in chemical potential. In the case of oceanic salt water, an energy of around 2.88 MJ per 1 m³ of fresh water is released when it is mixed with a large amount of salt water. In an osmotic power plant, this energy can be used to generate electric power; in nature, this energy just results in a slight warming of the ocean at the river mouth.

We consider two reservoirs completely filled with water of different salt concentrations. They are in contact via a semipermeable membrane which is permeable for water but not for salt ions. Due to the difference in chemical potential, the system tries to neutralize the difference in the concentration. Because of the near-incompressibility of water, even a small amount of water diffusing from the lower concentration to the higher concentration will cause a large pressure increase on the higher concentration side; the so-called “osmotic pressure” occurs, which is commonly notated by Δ𝜋. Its thermodynamic formula is given by van’t Hoff’s law which reads as follows (with the ion concentration 𝑐_𝑖=𝑁_𝑖/𝑉 of the media, the Boltzmann constant 𝑘_𝐵, and the absolute temperature 𝑇): Δ𝜋=𝑐_high−𝑐_low⋅𝑘_𝐵⋅𝑇. (1) A macroscopic derivation of the osmotic pressure and van’t Hoff’s law can be looked up in most textbooks on statistical physics, for example, see [7].

On microscopic scales, the effect of osmosis is still not fully understood, but there are several explanations which work usually with diffusion models. For a list of explanations, see, for example, [8], and for a sample derivation, see [9].

2.1. Permeation

For the flow density 𝐽_𝑀 through a semipermeable membrane holds in the first approximation [10]: 𝐽_𝑀=𝐾_𝑀⋅Δ𝜋−𝑝_counter, (2) where 𝐾_𝑀 is the permeation coefficient of water, a quantity of the specific membrane, Δ𝜋 is the current osmotic pressure (only depending on the concentration difference of the solute), and 𝑝_counter is the difference in mechanical pressure.

In a closed system, 𝑝_counter=Δ𝜋 holds. If there is an outlet at the high concentration side, for example, at the turbine, 𝑝_counter is reduced. For later considerations, we define the following: 𝑝_counter∶=𝑓⋅Δ𝜋, (3) where 0≤𝑓≤1 is a dimensionless factor which can be controlled externally via, for example, the turbine flow. 𝑓 is the major parameter in this paper.

For the overall membrane flow 𝑆_𝑀 through a membrane area 𝐴_𝑀 holds: 𝑆_𝑀=𝐴_𝑀⋅𝐽_𝑀=𝐴_𝑀⋅𝐾_𝑀⋅Δ𝜋⋅1−𝑓. (4) For the power (= change of mechanical energy per unit time) generated by the flow through the total membrane area 𝐴_𝑀 against the counter pressure results: 𝑃_𝑀=𝑝_counter⋅𝑆_𝑀=𝑓⋅Δ𝜋⋅𝑆_𝑀. (5)

3. Setup of a Pressure-Retarded Osmotic Power Plant

Figure 1 shows the basic components of a pressure-retarded osmotic (PRO) power plant. The core components are the two water chambers, which are in contact via the semipermeable membrane. The pressure difference between the brackish water and the fresh water is given by 𝑝_counter, with the higher pressure on the brackish water side. In reality, very large membrane areas are required; therefore, the membranes are packed or coiled in modules in order to save space. With these devices, a packing density of 1000 m² membrane surface area per 1 m³ module is possible [11]. Membrane modules are already used in desalination plants for decades; so a well-developed membrane technology can be adopted for osmotic power plants.

Figure 1 

Setup of a PRO power plant. Variables in this figure are used later. 𝑆𝑀: flow through the membrane; 𝑆𝑇: flow through the turbine; 𝑆𝐵: flow of brackish water into the pressure exchanger; 𝑆𝑆: flow of salt water from the pressure exchanger; efficiency of the: turbine 𝜂𝑇, pump 𝜂𝑃, and pressure exchanger 𝜂PX; 𝑐fresh: salt concentration of the fresh water; 𝑐salt: salt concentration of the salt water; 𝑐∞, respectively, 𝑐opt: salt concentration of the brackish water; 𝑉0: volume of the brackish water chamber.

For generating electric power, a turbine is connected to the brackish water chamber. The water flow through the turbine has to be replenished from a fresh water reservoir, for example, a river (upper boxes in Figure 1).

Because the turbine uses brackish water but there is only a fresh water supply, the salt concentration at the brackish water side decreases and therefore the power output of the plant. Aiming for a constant performance, it is necessary to keep the salt concentration of the brackish water constant (and ideally close to the concentration of the salt water reservoir, e.g., the ocean water salinity). This is done by the “pressure exchanger.” This device is commercially available and has an efficiency of up to 0.95 [12]. It exchanges a volume of brackish water with, ideally, an equal volume of salt water, by using the mechanical energy of the brackish water volume to pump the salt water volume against the operating pressure in the brackish water chamber. Pressure exchangers are used in desalination plant for decades; up to now, they offer the most efficient way to exchange water volumes of different pressures. Nevertheless, there is an additional pump required to compensate for the pressure loss in the pressure exchanger (lower boxes in Figure 1). Furthermore, on the fresh water side, there is a small amount of flushing required, which is ensured by a fresh water outlet, to avoid buildup of salinity.

3.1. Pressure Exchanger (PX)

The efficiency 𝜂_PX of the pressure exchanger suffers two losses: a fractional loss 𝜌_𝑉 in exchanged volume (lubrication) and a fractional loss 𝜌_𝑝 (friction) in exchanged pressure [12]. Because there is no mathematical relation stated, we assume both effects to act linearly: 𝜂_PX=𝐸_mech,salt𝐸_{mech,brackish}=1−𝜌_𝑉⋅1−𝜌_𝑝, (6) where 𝐸_mech denotes the energy content of the salt, respectively, brackish water. Accordingly, the salt water volume flowing from the high pressure exit of the pressure exchanger to the brackish water volume is a little smaller than the brackish water volume entering the high pressure inlet of the pressure exchanger, 𝑆_𝑆=(1−𝜌_𝑉)⋅𝑆_𝐵, and has a little less pressure, 𝑝_salt=(1−𝜌_𝑝)⋅𝑝_counter.

The smaller pressure can be corrected by the additional pump. For the required power, 𝑃_PX holds (with the power 𝑃_{mech,brackish}=𝑝_counter⋅𝑆_𝐵 of the inflowing brackish water flow and the here defined “loss parameter” 𝜖_PX of the whole pressure exchanger system): 𝑃_PX=𝜌_𝑝⋅𝑝_counter⋅𝑆_𝑆𝜂_𝑃=𝜌_𝑝⋅1−𝜌_𝑉𝜂_𝑃⋅𝑃_{mech,brackish}=𝜖_PX⋅𝑃_{mech,brackish}. (7) The most suitable way to compensate for the reduced volume is a smaller flow through the turbine: 𝑆_𝑇=𝑆_𝑀−𝜌_𝑉⋅𝑆_𝐵. (8)

3.2. External Control and Optimizations

The flow through the membrane depends on more or less each process parameter and is therefore not directly controllable. Furthermore, the flows of the pressure exchanger depend on each other (and are ideally equal). Thus, the only ways to control the process are the flows at the brackish water inlet of the pressure exchanger and at the turbine. These two process parameters can be set externally and therefore optimized. Via the flow through the pressure exchanger, the salinity of the brackish water and, thus, the overall efficiency of the plant can be optimized (see Section 5). The setting of the turbine flow is determined by the optimal operating counterpressure which depends primarily on the economic boundaries (see Section 6).

3.3. Quantitative Values

For later considerations the following values are assumed: (i)water temperature: 𝑇=290K; (ii)mean ocean salinity: 𝑐_salt=3.5% salt (NaCl, degree of ionization = 2) = 1196 Mol/m³;(iii)efficiencies: 𝜂_𝑇=0.9, 𝜂_𝑃=0.8, and 𝜂_PX=0.94 (resp., 𝜌_𝑉=𝜌_𝑝=0.03).

4. Salinity and Operating Pressure

For the process management, it is crucial to control the salt concentration 𝑐(𝑡) of the brackish water. The brackish water chamber in a membrane module is very thin, and an operating pressure exchanger leads to turbulence. Thus, the brackish water is assumed to be well mixed and its concentration everywhere is the same. For its changes holds in general: 𝑑𝑐𝑑𝑡=lim_Δ𝑡→01[𝑐()(𝑡.Δ𝑡𝑡+Δ𝑡−𝑐)] (9) Within the period Δ𝑡, the volumes Δ𝑉_𝑀, Δ𝑉_𝑇, Δ𝑉_𝐵, and Δ𝑉_𝑆 flow in and out of, respectively, the brackish water chamber. Scaled by the constant volume 𝑉₀ of the brackish water chamber, (9) can be written as 𝑑𝑐𝑑𝑡=lim_Δ𝑡→01𝑐(𝑡)⋅𝑉Δ𝑡⁰−Δ𝑉_𝑇−Δ𝑉_𝐵𝑉₀+𝑐_salt⋅Δ𝑉_𝑆𝑉₀+𝑐_𝑚⋅Δ𝑉_𝑀𝑉₀−𝐴(𝑐(𝑡))−𝑐(𝑡), (10) where 𝑐_salt is the concentration of the salt water and 𝑐_𝑚 is the concentration of the water which is diffusing through the membrane. 𝐴(𝑐(𝑡)) describes the losses in concentration because ions diffuse through the membrane. For simplification, we assume a perfect semipermeable membrane. This leads on the one hand to 𝐴(𝑐(𝑡))=0 but also to 𝑐_𝑚=0.

Further, for the volumes hold in general Δ𝑉_𝑖=∫_𝑡^𝑡+Δ𝑡𝑆_𝑖(𝑡)𝑑𝑡. To get the desired condition of a constant performance, the easiest way is to keep also all process parameters, especially the flows 𝑆_𝑖 constant. Accordingly, we from now on set Δ𝑉_𝑖=𝑆_𝑖⋅Δ𝑡. By these assumptions, (10) reduces to 𝑑𝑐(𝑡)⋅𝑑𝑡=𝑐−𝑆_𝑇−𝑆_𝐵𝑉₀+𝑐_salt⋅𝑆_𝑆𝑉₀. (11) The general solution of this differential equation is given by 𝑐(𝑡)=𝑐(0)−𝑐_∞−𝑆⋅exp^𝐵+𝑆_𝑇𝑉₀𝑡+𝑐_∞ (12) with the steady state concentration 𝑐_∞=𝑆_𝑆𝑆_𝐵+𝑆_𝑇⋅𝑐_salt. (13) Hence, the steady state holds for the osmotic pressure: Δ𝜋=𝑐_∞−𝑐_fresh⋅𝑘_𝐵⋅𝑇. (14)

5. Pressure Exchanger Flow for Optimum Power Plant Efficiency

In the first approximation, the power plant consists of the power generating turbine and the pressure exchanger plus a pump which costs electric energy. All other technical energy losses are neglected in this paper.

For the power generated by the turbine 𝑃_𝑇=𝜂_𝑇⋅𝑝_counter⋅𝑆_𝑇 holds; for the power required by the pressure exchanger system 𝑃_PX=𝜖_PX⋅𝑝_counter⋅𝑆_𝐵 holds. Introducing these two identities as well as (3), (8), (13), and (14), one obtains for the power output 𝑃_PP=𝑃_𝑇−𝑃_PX of the plant: 𝑃_PP=𝑓⋅𝑘_𝐵⋅𝑇⋅1−𝜌_𝑉𝑆_𝐵1−𝜌_𝑉𝑆_𝐵+𝑆_𝑀⋅𝑐_salt−𝑐_fresh⋅𝜂_𝑇⋅𝑆_𝑀−𝜂_𝑇⋅𝜌_𝑉+𝜖_PX⋅𝑆_𝐵. (15) The power 𝑃_PP is a function of several constants and the independent process parameters 𝑓, 𝑆_𝑀, 𝑇, and 𝑆_𝐵. The (economic) factor 𝑓 and the membrane flow 𝑆_𝑀 will be discussed later. Manipulations of the water temperature 𝑇 are most probably not favorable; so 𝑇 is a fixed condition. For getting the maximum power, it is crucial to find the optimal setting for the pressure exchanger flow 𝑆_𝐵. (𝑑𝑃_PP/𝑑𝑆_𝐵)=0 gives (see Figure 2) 𝑆^opt_𝐵=𝑆_𝑀1−𝜌_𝑉1+𝜙^∗−1 (16) with 𝜙^∗=(((1−𝜌_𝑉)𝜂_𝑇)/(𝜌_𝑉𝜂_𝑇+𝜖_PX))⋅(𝑐_salt/Δ𝑐)+(𝑐_fresh/Δ𝑐) = ((𝜌_𝑉/(1−𝜌_𝑉))+(𝜌_𝑝/𝜂_𝑇𝜂_𝑃))⁻¹⋅(𝑐_salt/Δ𝑐) + (𝑐_fresh/Δ𝑐) and Δ𝑐=𝑐_salt−𝑐_fresh.

Figure 2 

The maximum efficiency of the power plant (divided by f) is plotted as a function of the ratio between 𝑆𝐵 and 𝑆𝑀. The optimal efficiency of 𝑓⋅0.54 is reached for 𝑆𝐵≈3⋅𝑆𝑀.

For simplification, we assume from now on 𝑐_fresh=0 and therefore 𝜙^∗ reduces to 𝜌𝜙=^𝑉1−𝜌_𝑉+𝜌_𝑝𝜂_𝑇𝜂_𝑃⁻¹≈14. (17) For the optimal salinity, the optimal operating counterpressure, and the optimal power, we obtain the following results: 𝑐_opt=11−1+𝜙⋅𝑐_salt,𝑝_opt=11−1+𝜙⋅𝑓⋅𝑐_salt⋅𝑘_𝐵𝑃⋅𝑇,_opt=𝑓⋅𝑐_salt⋅𝑘_𝐵⋅𝑇⋅𝑆_𝑀⋅11−⋅𝜂1+𝜙_𝑇−𝜂_𝑇⋅𝜌_𝑉+𝜖_PX1+𝜙−11−𝜌_𝑉. (18) Using the values introduced in Section 3, the extracted electric energy 𝐸_PP per volume of fresh water results in𝐸_PP𝑃∶=^opt𝑆_𝑀=𝑓⋅1.55MJ/m³. (19) The theoretical osmotic energy of 1 m³ of fresh water is given by 𝐸_TD=𝑐_salt⋅𝑘_𝐵⋅𝑇⋅1m³=2.88MJ. Therefore, an osmotic power plant using a pressure exchanger (but neglecting all minor losses) has an efficiency of 𝜂_PP=𝐸_PP⋅1m³𝐸_TD=𝑓⋅0.54. (20)

6. Power Optimization versus Energy Optimization

In this final chapter, the optimal setting of 𝑓 and therefore of the operating counterpressure is discussed.

From the simple formulae in Section 2, one obtains for the power (= change in mechanical energy per time) through a membrane area 𝐴_𝑀 the following: 𝑃_𝑀=𝑝_counter⋅𝑆_𝑀=𝐴_𝑀⋅𝐾_𝑀⋅(Δ𝜋)²⋅𝑓−𝑓² (21) which peaks for 𝑓=1/2.

This value is already published and well known as the setting for which the power plant obtains the maximum power output. More distinctly spoken, this factor 1/2 is throughout the literature set “axiomatically” as the optimal value for 𝑓 (see, e.g., [10, 13]). This assumption is challenged in the following.

On the other hand, a power plant is frequently not constructed for providing power but electric energy. However, the energy output of a given volume of fresh water depends linearly on 𝑓 (see (19) and (20)). So the maximum energy output would occur for 𝑓 near to 1. Unfortunately, in this limit, the flow and the power per membrane area are very low.

The missing link between these two extreme settings for 𝑓 is provided by the membrane area 𝐴_𝑀 which can be controlled externally, just by installing the optimal area. So it is possible to go for the energy optimization while still retaining a high power output by providing a larger membrane area. A higher energy output means more revenue but a larger membrane area means also higher investment cost. Hence, finally, 𝑓 depends purely on economic boundaries. A profit optimization is required for obtaining the economically optimal 𝑓. In general, Profit=Revenue−Costs,Revenue=𝑃_opt⋅𝜏⋅𝐶_𝐸 (22)holds with the power output 𝑃_opt of the power plant, the lifetime 𝜏 of the power plant, and the energy price 𝐶_𝐸 (in matters of generation costs, not corporate profit, given in €/kWh).

For our purpose, the costs might be split in membrane costs which also include all costs which are directly related to the membrane area like installation and maintenance, and other costs, which are assumed to be not affected by the extent of the overall membrane area: Costs=Membranecosts+Othercosts=𝐴_𝑀⋅𝐶_𝑀+Othercosts (23) with the membrane area 𝐴_𝑀 in m² and the membrane price 𝐶_𝑀 in €/m². The 𝑓 dependency of 𝐴_𝑀 is given by (4): 𝐴_𝑀=𝑆_𝑀𝐾_𝑀.⋅Δ𝜋⋅1−𝑓 (24) Using (19) and 𝐸_PP∶=𝑓⋅𝐸⁰_PP for the profit results inProfit=𝑆_M⋅𝑓⋅𝐸⁰_PP⋅𝜏⋅𝐶_𝐸−𝑆_𝑀𝐾_𝑀⋅Δ𝜋⋅1−𝑓⋅𝐶_𝑀−Othercosts, (25) where 𝑆_𝑀 is fixed by the limiting fresh water supply and not depending on 𝑓. The maximum profit can be reached with 𝑓_max=1−𝐶_𝑀𝐶_𝐸⋅𝜏⋅𝐸⁰_PP⋅𝐾_𝑀.⋅Δ𝜋 (26)

6.1. Visualization of the Optimal f Value

Equation (26) is visualized in Figure 3, where the dependency of 𝑓 on both economic variables is plotted. 𝑓 is shown as a function of the energy price in €cent/kWh. Varying membrane prices are indicated by the differently colored branches. Considered are 10, 20, and 40 €/m².

Figure 3 

The economically optimal value for 𝑓 as function of the energy price in €cent/kWh. The different colors indicate different values for the membrane price. Full lines indicate a membrane quality of 2 W/m2, and dashed lines indicate 5 W/m2.

By applying the values introduced above, one obtains Δ𝜋=21.6bar, 𝐸⁰_PP=1.55MJ/m³. We assume for the mean lifetime of the membrane (and therefore the whole power plant) 𝜏=10years. The test power plant operated for 2 years without damage, reported at [3].

The permeation coefficient is usually not published in m/(s⋅Pa) but in W/m² (marked with *). For our definitions, the value has to be transformed via 𝐾^∗_𝑀Wm²=𝑝_counter⋅𝑆_𝑀𝐴_𝑀=𝐾_𝑀⋅(Δ𝜋)²⋅𝑓−𝑓², (27) where 𝐾^∗_𝑀 describes the maximum power density which occurs for 𝑓=1/2: 𝐾_𝑀=2Δ𝜋²⋅𝐾^∗_𝑀. (28) The Fraunhofer IGB reported a state-of-the-art value of 𝐾^∗_𝑀=2.3W/m² for lab-intern membranes [14]. For the visualization in Figure 3, we used a value of 2 W/m² (full lines) which might be commercially available in the near future. Furthermore, [14] aims for a value of 5 W/m². This outlook is also sketched in dashed lines.

7. Discussion

The generation costs of solar and wind power are typically of the order of 10–20 €cent/kWh, for example, see [15], and the future energy price in an energy system with a higher contribution from renewable energy will probably be close to these values. For this cost range 0.5≤𝑓≤0.8 is obtained. So for the current conditions, 𝑓=1/2 appears to be the optimal value. However, on-going research on membranes and large-scale production of suitable membranes are likely to bring down membrane cost, while at the same time energy prices are probably rising. Thus, there is no justification to assume this value as the optimal one.

Please be aware that this optimization is just aiming for the maximum profit. If this profit is also positive (and therefore competitive to wind and solar power) depends also on the extent of the 𝑂𝑡ℎ𝑒𝑟𝑐𝑜𝑠𝑡𝑠 which are not the issue of this paper.

An interesting consequence of the opposing power and energy optimization should be mentioned concerning the degree of capacity utilization. So it seems reasonably to apply energy optimization in general but switching to power optimization (less counterpressure, resp., smaller 𝑓) in times of flooding, for example, in spring. Only rather small additional cost will occur, since only a larger turbine/generator will be required, which constitutes a small fraction of the total cost.

8. Conclusions

Two optimizations are performed. In the first step, the optimal salinity was derived by the correct setting of the pressure exchanger. Consequently, for the theoretical efficiency of a pressure-retarded osmotic power plant using a pressure exchanger (and neglecting minor losses), 𝜂_th=0.54 was derived. However, due to economical reasons, the actual efficiency of an osmotic power plant is reduced to 𝜂_PP=𝑓⋅𝜂_th, where 𝑓 is the ratio between the operating counterpressure and the actual osmotic pressure; it is defined 1/2≤𝑓≤1. In the second step, an estimation for the concrete value of 𝑓 was derived from an economic optimization. By considering the fresh water supply as a limiting factor and applying a realistic energy price of 10–20 €cent/kWh for the coming decades, 0.5≤𝑓≤0.8 resulted.

Please be aware that the presented model is condensed to the issues we introduced, that is, a mathematical description for an osmotic power plant using a pressure exchanger and the optimization of its energy output. Higher order membrane effects and technical losses are neglected from the model for retaining clearness. So all quantitative results extracted from this model are overestimated concerning the final energy output. However, implementing these losses would just lead to a more complicated formula for the optimal 𝑓, but the qualitative results are most likely the same.

The take-away-message of this paper is that there is no justification for the general assumption of a peaking energy output for 𝑓=1/2 which is commonly used in the literature. In fact, a final aim of membrane science appears to be an optimal value for 𝑓 as near to 1 as possible. The actual value is determined by the quality and costs of the semipermeable membrane and the energy price (of the alternative energy sources). All three quantities are likely to change in the future in favour of osmotic power which means a trend to higher settings for 𝑓 and therefore to a higher competitiveness of osmotic power.

List of Symbols

𝑓∶	Ratio of operating pressure and osmotic pressure
Δ𝜋:	Osmotic pressure, Pa
𝑝counter:	Operating counterpressure, Pa
𝑝opt:	𝑝counter for optimal settings of the pressure exchanger, Pa
𝑘𝐵:	Boltzmann constant, J/K
𝑇∶	Absolute temperature, K
𝐾𝑀:	Permeation coefficient, m/s/Pa
𝐴𝑀:	Membrane area, m2
𝐽𝑀:	Flow density through membrane, m/s
𝑆𝑀:	Flow through membrane, m3/s
𝑆𝑇:	Flow through turbine, m3/s
𝑆𝐵:	Flow of brackish water through pressure exchanger, m3/s
𝑆𝑆:	Flow of salt water through pressure exchanger, m3/s
𝑐𝑖:	Salt concentration of medium i, 1/m3
𝑐∞:	Steady state concentration of brackish water, 1/m3
𝑐opt:	𝑐∞ for optimal setting of the pressure exchanger, 1/m3
𝑉0:	Volume of the brackish water chamber, m3
𝜂𝑇:	Efficiency of the turbine
𝜂𝑃:	Efficiency of the pump
𝜂PX:	Efficiency of the pressure exchanger
𝜂PP:	Efficiency of the power plant
𝜖PX:	Loss parameter of the pressure exchanger system
𝜌𝑉:	Volume loss of the pressure exchanger
𝜌𝑝:	Pressure loss of the pressure exchanger
𝑃𝑀:	Power through membrane, J/s
𝑃opt∶	𝑃opt for optimal settings of pressure exchanger, J/s
𝑃PX:	Power required for the PX system, J/s
𝑃PP:	Power output of the overall power plant, J/s
𝐸PP:	Extracted energy per volume fresh water, J/m3
𝐶𝐸:	Energy price, €/kWh
𝐶𝑀:	Membrane Price, €/m2
𝜏:	Lifetime of power plant, s.

Acknowledgment

Florian Dinger wants to thank Werner Aeschbach-Hertig, Institute for Environmental Physics of the University of Heidelberg, for the support during his bachelor thesis.