A nonlinear seawater osmotic pressure function is developed using the relation with the osmotic coefficient of seawater. The osmotic coefficient of seawater is given by a piecewise function which uses Brønsted’s equation for 0 ≤ w < 10 g/kg and an empirical correlation given by Sharqawy et al. [25] for 10 ≤ w < 120 g/kg. For details regarding the development of the nonlinear seawater osmotic coefficient function, please refer to the Appendix of Banchik et al. [19].
The nonlinear osmotic pressure of seawater as a function of salinity in g/kg is shown in Fig. (A.1). The modified van ’t Hoff coefficient (C) can be determined as the slope of the nonlinear osmotic pressure between specific ranges of interest. The osmotic pressure can be approximated as the product of the modified van ’t Hoff coefficient and the salinity of the solution using Eq. (A.1).
p »Cw (A.1)
Using a best-fit least squares method between the range of 0 and 35 g/kg, the modified van ’t Hoff coefficient is determined to be 73.07 kPa-kg/g at a temperature of 25oC. This linear model represented by Eq. (A.1) can be used for a salinity range of 0 to 35 g/kg, which is the typical range for power production at a river delta with a PRO exchanger.
For salinities between 35 and 70 g/kg, the modified van ’t Hoff coefficient is determined to be 76.76 kPa-kg/g which is the typical range for power production using seawater and disposed brine from a seawater desalination plant. For salinities between 70 and 105 g/kg, the modified van ’t Hoff coefficient is determined to be 82.65 kPa-kg/g.
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Fig. 1a Schematic drawing of a counterflow heat exchanger. A hot and cold stream enter the exchanger where a temperature difference drives heat transfer.
Fig. 1b Schematic drawing of a counterflow osmotic mass exchanger. A concentrated and dilute stream, draw and feed, enter the exchanger where a salinity difference
drives mass transfer.
Fig. 2a Schematic diagram of a PRO mass exchanger in parallel-flow configuration
Fig. 2b Schematic diagram of a PRO mass exchanger in counterflow configuration
Fig. 3 Recovery ratio vs. mass transfer units at different mass flow rate ratios for a parallel-flow configuration
Fig. 4 Concentration factor of the feed (dashed curves) and draw (solid curves) stream vs. mass transfer units at different mass flow rate ratios for a parallel-flow
configuration
Fig. 5 Recovery ratio vs. mass transfer units at different mass flow rate ratios for a counterflow configuration
Fig. 6 Concentration factor of the feed (dashed curves) and draw (solid curves) stream vs. mass transfer units at different mass flow rate ratios for a counterflow
configuration
Fig. 7 Effectiveness vs. mass transfer units at different mass flow rate ratios for a parallel-flow configuration
Fig. 8 Effectiveness vs. mass transfer units at different mass ratios for a counterflow configuration
Fig. 9 Recovery ratio vs. mass transfer units for varying mass flow rate ratios, counterflow configuration, and fixed inlet salinities representative of seawater and
river water. Lines are from analytical Eq. (34), and the points are the results of a numerical model using a nonlinear osmotic pressure function.
Fig. 10 Concentration factor vs. mass transfer unitsfor varying mass flow rate ratios, counterflow configuration, and fixed inlet salinities representative of seawater and river water. Solid and dashed lines are from analytical Eqs. (20) and
(21), respectively, and the points are the results of a numerical model using a nonlinear osmotic pressure function.
Fig. 11 Effectiveness vs. mass transfer units for varying mass flow rate ratios, counterflow configuration, and fixed inlet salinities representative of seawater and
river water. Lines are from analytical Eq. (50), and the points are the results of a numerical model using a nonlinear osmotic pressure function.
Fig. 12 Recovery ratio vs. mass transfer units for varying mass flow rate ratios, counterflow configuration, and fixed inlet salinities representative of brine and seawater. Lines are from analytical Eq. (34), and the points are the results of a
numerical model using a nonlinear osmotic pressure function.
Fig. 13 Concentration factor vs. mass transfer units for varying mass flow rate ratios, counterflow configuration, and fixed inlet salinities representative of brine
and seawater. Solid and dashed lines are from analytical Eqs. (20) and (21), respectively, and the points are the results of a numerical model using a nonlinear
osmotic pressure function.
Fig. 14 Effectiveness vs. mass transfer units for varying mass flow rate ratios, counterflow configuration, and fixed inlet salinities representative of brine and
seawater. Lines are from analytical Eq. (50), and the points are the results of a numerical model using a nonlinear osmotic pressure function.
Fig. A.1 Seawater osmotic pressure versus salinity for T = 25 °C shown as a solid curve. The points indicate the linear osmotic pressure approximation using varying
modified van ’t Hoff coefficients.
Table 1 Data input for numerical model
Input Value/Range
Ambient temperature, To 25 °C
Modified water permeability coefficient, A 3.07´10-6kg/m2-s-kPa Feed mass flow rate, mf,in 1 kg/s
Inlet draw salinity, wd,in 70 g/kg and 35 g/kg Inlet feed salinity, wf ,in 35 g/kg and 1.5 g/kg Mass flow rate ratio, MR 0.5, 0.75, 1, 2, 3 Hydraulic pressure, P (pd,in-pf ,in) / 2 Trans-membrane pressure difference, P 1.24 – 1.47 MPa
Membrane area, Am 0 – 9.45´105 m2
Table 2 Modified van ’t Hoff coefficients over three ranges for determining osmotic pressure as a function of salinity at T=25°C
C [kPa-kg/g] Range [g/kg] R2
73.07 0 - 35 0.9997
76.76 35 - 70 0.9926
82.65 70 - 105 0.9691
Table 3 Maximum error resulting from linearized osmotic pressure Case 1: Seawater + River Water RR CFd CFf ε
Parallel-flow 3.69% 1.61% 3.41% 3.36%
Counterflow 4.62% 2.12% 20.03% 4.41%
Case 2: Brine + Seawater RR CFd CFf ε
Parallel-flow 2.34% 0.43% 0.45% 1.71%
Counterflow 5.37% 1.26% 1.68% 4.95%