Reverse Osmosis Channel Flow

Due to constant permeation velocity vj), the velocity in the -direction is altered causing streamwise variation ofbulk velocity. The Reynolds number based on is small 1,  

For impermeable wall, because of the fully developed condition, Thus, we get the simplified N-S equation in the jc-direction as 

We can derive the expression for average velocity (u ) by integration, that is. 

The above derivation holds good for impermeable wall. For permeable wall, the average velocity will vary in the x-direction, that is, 

If we consider a differential element of size Ax of the channel, we can write the average 

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Figure 4.20 Similarity solution for concentration defect in developing boundary layer at inlet region of a reverse osmosis channel flow...

Figure 4.21 shows the concentration polarization of a reverse osmosis channel with complete rejection membrane. It shows how the concentration polarization increases in the streamwise direction of the reverse osmosis channel flow. Let us investigate an example to understand the effect of concentration polarization on reverse osmosis channel flow. 

To illustrate the concentration polarization problem, we again consider laminar flow in a parallel plate, reverse osmosis channel, where the channel walls are a porous support for the membrane and where the velocity profile is taken to be fully developed from the channel inlet. Let us first examine the qualitative behavior of the solute concentration distribution along the channel shown schematically in Fig. 4.4.1. The membranes are assumed to be perfectly rejecting (R = 1), that is, totally impermeable to solute. 

Figure 4.4.3 Concentration polarization in a parallel membrane reverse osmosis channel with fully developed laminar flow and complete solute rejection (after Sherwood et al. 1965).

Suppose NaCl solution of concentration 10 kg/m flow through a reverse osmosis channel having concentration in the permeate side equal to 5 kg/m. The water permeability coefficient of water through the semipermeable membrane is equal to 0.5 kg/m -MPa and AP = 10 MPa is applied from the feed side. The concentration of salt increases to 50 kg/m at a downstream location. 

Asymmetric membranes have a tight, low-permeability, retentive zone that performs the desired separation and a more open, high-permeability zone that provides mechanical strength to the overall membrane. This structure is particularly critical to the economic viability of reverse-osmosis membranes. Asymmetric membranes operated in TFF mode must have the tight side facing the feed channel so that particles are retained on its surface and can be acted upon by the tangential flow. Asymmetric membranes operated in NFF mode can... 

Figure 19.4. The spiral wound membrane module for reverse osmosis, (a) Cutaway view of a spiral wound membrane permeator, consisting of two membranes sealed at the edges and enclosing a porous structure that serves as a passage for the permeate flow, and with mesh spacers outside each membrane for passage of feed solution, then wound into a spiral. A spiral 4 in. dia by 3 ft long has about 60 sqft of membrane surface, (b) Detail, showing particularly the sealing of the permeate flow channel, (c) Thickness of membranes and depths of channels for flows of permeate and feed solutions.

Pure water can be obtainnd from brackish water by permention through a reverse osmosis membrane. Consider the stendy laminar flow of a salt solution in a thin channel between two walls composed of a membrane that rejects salt. Derive the governing equations for the salt distribution in die transverse direction for a given water peuneaiion flux (see Fig. 2.2-4),... 

Spiral-wound modules of the type used for reverse osmosis (see Fig. 26.19) are widely used for UF. They are not as prone to plugging as hollow-fiber units, since the entrance is a narrow slit about 1 mm wide, but prefiltration of the feed solution is recommended. The velocity in the feed channels corresponds to laminar flow, but the flow disturbances caused by the spacers make the pressure drop and the mass transfer greater than for true laminar flow. 

A variation of the basic plate-and-frame concept is the spiral-wound module, which is widely used today in reverse osmosis, ultrafiltration, and gas separation. Its basic design is illustrated in Figure 1.33 (c). The feed flow channel spacer, the membrane, and the porous membrane support are rolled up and inserted into an outer tubular pressure shell. The filtrate is collected in a tube in the center of the roll. 

To illustrate the behavior of the ultrafiltration flux, we here adopt Michaels model of gel layer formation. As was done for reverse osmosis, let us again consider the geometry of a two-dimensional parallel plate channel with fully developed flow. Moreover, to simplify the presentation, we examine only the limiting-flux problem. 

The mass transfer coefficient is characterised by the hydrodynamic performance of the system. It was shown in the previous section that flow conditions (velocity, viscosity, density, solute diflusion coefficient) and module geometry determine the mass transfer coefficient. So far the correlations have been used for empty flow channels or tubes. However, in man> systems mrbulence promoters are present and these affect the mass transfer coefficient. For instance, spiral wound modules are applied in reverse osmosis, nanofiltration and ulL-afiltration. These modules contain spacer materials to. separate both... 

On the basis of the above observation, Schultz and Asunmaa developed the following transport mechanism. They made an assumption that the low-density and the noncrystalUnc region of the polymer that fills the space between the circular cells is incorporated into the unit cell as its part. Those spaces (between the unit cells) were therefore assumed to be vacant. In reverse osmosis operation these vacant spaces are filled only with water, and this water is assumed to be more ordered than the ordinary water under strong influence from the polymeric material. This water flows by the viscous flow mechanism through channels that arc formed by connecting the vacant spaces. Suppose r is the effective radius of this pore (m), np is the number of the pore in a unit area (1/m ), p is the pressure drop across the membrane (Pa), 17 is the water viscosity (Pa s), L is the effective layer thickness (m), and r is the tortuosity factor (-), the volumetric... 

The concentration profiie in a laminar flow channel between two reverse osmosis membranes has been established by numerical calculation and reported [266]-[276]. The following derivation is based on the approach of Kimura and Sourirajan (51). 

Spiral-wound membrane channel in reverse osmosis" Hollow fiber membrane module Laminar tube-side flow Sh = = 0.065 Be Da Re = vdh/o) Schock and Miquel (1987) (3.1.170)... 

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