Fig. 4 shows the transport of water and salt through a semipermeable membrane in PRO. The membrane consists of a thin selective top layer supported by a porous support. The selective layer faces the high-pressure side in order to prevent release of the selective top layer from its support due to the pressure differences.
Because the membrane is not 100% selective, some salt will also be transported from the saltwater side to the freshwater side. In RO the transport of salt and water are in the same direction.
In PRO water is transported from the low-pressure freshwater side to the high-pressure saltwater side due to the osmotic pressure difference.
Salt water at high pressure Membrane
Selective toplayer n, n.
Fresh water at low pressure
Figure 4 Schematic representations of the osmotic profiles of a PRO membrane. P is the osmotic pressure of the bulk of the concentrated salt solution, n2 the osmotic pressure at the dense top layer of the membrane, n3 the osmotic pressure inside the membrane between the dense top-layer and the porous support, n4 the osmotic pressure at the surface of the membrane in the diluted solution, n5 the osmotic pressure in the bulk of the diluted solution.
However, salt is transported in the opposite direction due to its concentration difference. The salt transport is limited by several resistances: external concentration polarization due to stagnant layers caused by reduced mixing on the membrane surface at the saltwater side and the freshwater side; internal concentration polarization due to resistance against salt transport in the thin selective top layer of the membrane and in the porous support layer.
Intensified mixing due to high cross-flow rates at the membrane surface can lower external concentration polarization. Internal concentration polarization arises from the resistance against mass transfer that salt experiences from the dense top layer and the stagnant boundary layer in the porous support. This porous support creates a stagnant zone through which salt can only be transported by diffusion. These resistances against salt transport lower the effective osmotic pressure (n2—n3) over the selective top layer of the membrane. A good PRO membrane has a thin high selective top layer with a high resistance for salt transport and a very open (preferably thin) support layer. Loeb et al. showed that the support has a large contribution to the overall transport resistance and that the removal of the nonwoven/woven support from the membrane caused a higher osmotic water flux through the membrane . A capillary membrane (with a thin porous layer) might be very beneficial for PRO applications .
When external concentration polarization is neglected, then the water transport can be described as follows:
where Jw is the water flux through the membrane [m3/(m2 day)], A is a specific membrane transport parameter [m3/(m2 day bar)], n the osmotic pressure (bar), and AP the pressure difference between the fresh and saltwater solution (bar).
The osmotic pressure n3 is not known but can be calculated from the salt leakage through the membrane. This can be described as follows:
where Js is the salt flux through the membrane [mol/(m2 day)], B the salt permeability constant (m/day), and C the concentration (mol/m3).
This salt flux is negative since its transport is in the opposite direction of the water flow. In the porous support, the diffusion of salt is counteracted by the flow of water. The salt flux through the support can be written as follows:
where Ds is the diffusion coefficient of the salt in the membrane substrate (m2/s), e the porosity of the membrane substrate (—), and x the thickness of the porous support.
Lee et al.  solved this problem resulting in r / 1 (C4=C 2) exp(JwK) A\
For the special case of C4 — 0 with only water on the freshwater side of the membrane reduces Eq. (11) to
Both equations can be solved numerically. In these equations A and B can be obtained from RO experiment, concentrations are known and Jw is measured during osmosis experiments allowing for the determination of K. K (s/m) refers to the solute diffusion in the porous support structure and is given as tt
where t is the thickness of the membrane (m), t the tortuosity of the pores in the support (-), Ds the solute diffusion coefficient (m2/s), and e the porosity of the membrane (-).
Eq. (11) can be further simplified by assuming that p2/p4 — C2/C4  resulting in
This equation is valid when the concentrated salt solution is facing the active dense layer. If the concentrated salt solution is facing the porous support, which is sometimes applied in FO , then the following equation is valid:
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