El género: desde la constitución del sujeto hasta la relación estructural

Similar to other mass transfer processes, FO is subject to concentration polar- ization (CP). CP is the formation a concentration difference between a bulk solution and a fluid boundary layer, such as a solution - membrane interface, due to unequal mass transfer rates of the different species. CP can be dilutive or concentrative, depending on whether the solute(s) concentration in the inter- face decreases or increases respectively. In pressure-driven membrane systems, such as RO and NF, CP appears externally (ECP) at the feed solution - mem- brane interface. In FO on the other hand, ECP appears twice: at both the feed and draw solution - membrane interfaces. CP also appears internally (ICP) in FO: depending on the membrane orientation, the draw solute is diluted in the support layer (dilutive ICP), or feed solutes combined with leaked draw solute accumulate in the support layer (concentrative ICP). In the former case, the membrane is oriented with the active layer facing the feed solution (AL-FS), while in the latter case, the active layer is facing the draw solution (AL-DS). The latter case is also the membrane orientation employed during PRO, which is why some texts refer to this as "PRO mode". This orientation prevents ac- tive layer delamination when the draw solution is pressurized during PRO: in AL-DS orientation, the hydrostatic pressure compresses the active layer against the backing of the support layer, while in AL-FS orientation, a pressurized draw solution would cause the active layer to tear off the support layer. The concen- tration profiles of both membrane orientations are shown in Figure 1.3. FO has 2 ECP boundaries, one concentrative and one dilutive [69, 70, 71], because FO is a membrane contactor process: the feed and draw solution are contacted by the FO membrane and both solutions are recirculated. In contrast, RO and NF depend solely on a feed solution; flux is produced due to the application

Feed

Draw

Feed

AL-FS

AL-DS

Figure 1.3: AL-FS and AL-DS membrane orientations and concentration pro- files of draw solute in both orientations. ECP boundary layers are the zones between the membrane interfaces and the dashed lines, the membrane active layer and support layer are marked as dark and light gray areas respectively. ICP is taking place in the support layer.

of pressure and the permeate is collected instead of recirculated, and all fluxes share the same direction.

In boundaries affected by CP, solvent and solute fluxes are to some extent uncoupled and convective mixing is reduced, causing the solute concentration in the boundary to deviate from the bulk solution. The solute concentration difference causes a diffusive flux counteracting the driving force of the con- centration imbalance, which leads to the establishment of a steady-state. A general 1-dimensional convection-diffusion equation linking the solute flux Js

of a solute s with the water flux Jwthrough convection and diffusion is given

below:

Js= −Ds

dc

dx+ Jwc(x) (1.9)

with x the position in the CP boundary zone and Ds the solute diffusivity.

Equation 1.9 can be integrated in either ECP or ICP zone. For FO, equation 1.9 differs depending on whether solute s is the draw solute: if so, Jsis directed

oppositely of Jw, and the relation Js = cpJwcannot be used [72]. Concentra-

tion profiles of a feed solute are given in Figure 1.4. The zones marked 1 to 5 at the bottom of the figure are the bulk feed, unmixed feed-membrane boundary, membrane active layer, membrane porous support layer and unmixed draw-

Mass transfer Jwcp Jw Jwcp F eed solute Jw 1 2 3 4 5 1 2 3 4 5

Figure 1.4: Solute concentration profile in the membrane and boundary layers in the case of partial rejection (left panel) and negative rejection (right panel). Zones 1 and 2 are the bulk feed solution and feed-membrane interface, zones 3 and 4 are the membrane active layer and support layer, zone 5 is the draw- support layer interface.

membrane boundary; the solute concentration profile is depicted as the black line. The left panel depicts a partially rejected solute, with ECP in zone 2, the unmixed feed-membrane boundary. In this zone, solute is entrained by viscous flow towards the membrane, where the solute is partially rejected. Within the boundary layer, a steady-state is established between viscous transport into the boundary layer, solute permeating through the membrane and back-diffusion towards the bulk feed. The right panel depicts negative rejection, which will be described in chapter 4. In this case, both the viscous and diffusive flux con- tribute towards solute permeating through the membrane, which causes de- pletion of the solute at the feed-membrane interface, rather than enrichment. Note that the solute is subject to dilute ICP once it has permeated through the membrane (zone 4), due to recirculation of the draw solute.

Both experimental and modeling studies have shown that ICP is the most im- portant flux limiting mechanism in FO. In fact, some of the first studies on FO have been devoted to ICP [4, 3]. This is because convective mixing is not possi- ble in the support layer, and replenishment or dilution of solute in the support layer is dependent on diffusion. The support layer is however still relatively thick, and diffusion is furthermore hindered by the limited porosity and pore size of the support layer, as well as the increased effective path length due to support layer tortuosity. This can be expressed as the structural parameter S, which has units of length and can be considered as the equivalent length over which unhindered diffusion would take place to yield a solute flux equal to the

hindered diffusion flux:

S = tsτ

 (1.10)

with ts, τ and  equaling the support layer thickness, tortuosity and porosity

respectively [3, 50]. ICP models based on equation 1.10 have come under criticism however, as Manickam and McCutcheon have shown that equation 1.10 does not incorporate all relevant diffusional resistances and leads to gross underestimation of the real hindrance against solute diffusion [73].