M ViAp RTlnXi zFA j3

Concentrated Diluted Brackish solution (C) solution (D) solution (B)

Concentrated Diluted Brackish solution (C) solution (D) solution (B)

• ° •

• O ° • ° *

• #

Mixing

• ° .o °

o ° o

o

o o •

o #o • o

Figure 1 The mixing of a concentrated and a diluted solution to a brackish solution.

where is the molar free energy under standard conditions (J/mol), Dp the pressure change compared to atmospheric conditions (Pa), V the molar or specific volume of component i (m3/mol), R the universal gas constant [8.314J/(molK)], T the absolute temperature (K), xi the mol fraction of component i, z the valence of an ion (eq/mol), F the Faraday constant (96,485 C/eq), and Dj the electrical potential difference (V). Since there is no pressure change or charge transport, when the concentrated and the diluted solution are mixed, Eq. (3) reduces to mi = m0 + RT In xi (4)

When Eq. (4) is substituted in Eqs. (2) and (1), the standard chemical potential (m0) is eliminated and the final equation describes the Gibbs energy of mixing of a concentrated and a diluted salt solution:

And when n is replaced by cV, this changes into

DGmix — ^2[Ci,cVcRT ln(x,;c) + Ci;d VdRT ln(x,;d)

Because the mixing of two solutions is a spontaneous process, the Gibbs energy of mixing is negative: energy is released when two solutions are mixed. With Eq. (6), the theoretical available amount of energy available from the mixing of two salt solutions can be calculated and thus the theoretical potential of salinity gradient energy can be evaluated. This theoretically available amount of energy for an extensive range of sodium chloride concentrations is presented in Fig. 2 [3]. (Note: Because the figure shows the theoretical amount of energy available from the mixing of a diluted and a concentrated solution, the energy has a positive sign.)

Fig. 2 shows an extensive range of salt concentrations and the theoretically available amount of energy that can be obtained from the mixing of the two solutions. Values as high as ~17MJ can be obtained, depending on the concentration difference between the two solutions. Of course, this amount of energy strongly depends on the difference in

Figure 2 Theoretical available amount of energy (MJ) from mixing 1 m3 of a diluted and 1 m3 of a concentrated sodium chloride solution (T — 293 K). The shaded area is not taken into account because in this area the salt concentration of the concentrated solution is lower than that of the diluted solution [3].

Figure 2 Theoretical available amount of energy (MJ) from mixing 1 m3 of a diluted and 1 m3 of a concentrated sodium chloride solution (T — 293 K). The shaded area is not taken into account because in this area the salt concentration of the concentrated solution is lower than that of the diluted solution [3].

concentration (or chemical potential) between the concentrated and the diluted salt solution. The higher this difference, the more energy can be extracted from the system. For example, the theoretically available amount of energy from mixing 1 m3 seawater (comparable to 0.5 mol/L NaCl) and 1 m3 river water (comparable to 0.01 mol/L NaCl) both at a temperature of 293 K is 1.7 MJ, whereas the theoretically available amount of energy from mixing 1 m3 brine (5 mol/L NaCl) and 1 m3 river water (0.01 mol/L NaCl) at 293 K is more than 16.9 MJ. When mixed with a large surplus of seawater, 2.5 MJ is theoretically available from 1 m3 of river water (Table 1) [6]. Table 1 shows the amount of Gibbs energy theoretically available from the mixing of different volumes of a diluted and a concentrated salt solution [6]. This table clearly shows that when the amount of saltwater limits the process, the use of an excess of river water can be very beneficial

Table 1 Gibbs energy theoretically available from mixing different volumes of NaCl solutions at 298 K [6]

Vd (m3)

Vc (m3)

AGmix (MJ)

N

1

N

10

1

6.1

2

1

2.8

1

1

1.76

1.26

0.74

1.87

1

2

2.06

1

10

2.43

1

?

2.55

Vd is the volume of the diluted solution (0.01 M NaCl), Vc the volume of the concentrated solution (0.5 M NaCl), and AGmix the change in Gibbs energy.

Vd is the volume of the diluted solution (0.01 M NaCl), Vc the volume of the concentrated solution (0.5 M NaCl), and AGmix the change in Gibbs energy.

(compare an available amount of energy of 6.1 MJ at Vd — 10 m3 and Vc — 1m to an available amount of energy of only 1.76 MJ when both Vd and Vc are 1 m3).

Although the above-presented equations provide a good first approximation for the theoretical amount of energy obtainable from salinity gradient energy, the calculations assume that the feed solutions consist of pure sodium chloride and behave ideal (no distinction between concentrations and activities). In practice, however, sea and river water are much more complex solutions and do not behave ideal, which makes the calculations much more complex. The numbers presented here represent the theoretical, maximum amount of energy available from the mixing of fresh and saltwater. Of course, in practice, it will not be possible to harvest this total theoretically available amount of energy, due to for example, mass transfer limitations, pressure drop, nonideal behavior, and so on. In addition, depending on the location and situation, there can be also several other limitations to use the total resources available, which are related to, for example, environmental impact, shipping, recreation, and tourism. But even if only part of the available energy can be recovered, the potential of salinity gradient energy remains huge.

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