where Cb is the concentration of the solute in the bulk flow and k is the mass transfer coefficient that is the ratio of diffusivity of the solute in the solvent to the thickness of the concentration boundary layer, which can be interpreted as the mass transfer coefficient when the permeation flux approaches zero. The cause of the concentration polarization phenomenon is different in reverse osmosis, as in microfiltration or ultrafiltration. In reverse osmosis, as the low molecular weight material is retained on the membrane surface (see the following section about the characteristics of all pressure-driven membrane processes), the increase in the solute concentration causes the osmotic pressure to rise, which decreases the water flux, as illustrated in Equation 3.22. In ultrafiltration, the high concentration of larger molecules accumulated on the membrane surface does not result in significant osmotic pressure increase. However, these retained molecules may lead to precipitation and possibly formation of a gel layer on the mem
Membrane Gel Layer or rouled layer
Figure 3.13. A schematic diagram of a film model.
Membrane Gel Layer or rouled layer
brane surface. The mass transfer coefficient, k, in Equation 3.24 has to be determined experimentally because the thickness of the concentration boundary layer is usually an unknown quantity that is strongly influenced by hydrodynamics of the system. The mass transfer coefficient, k, however, can often be related to the semiempirical Sherwood number correlations with the following form of expression (Equation 3.25):
where Re is the Reynolds number, Sc the Schmidt number, and Sh is Sherwood number; and a, b, and c are all constants. In Equation 3.25, ^ and v are dynamic viscosity and kinematic viscosity, respectively while p is the density, D the diffusivity, and dh the hydrodynamic diameter. It is clear that the mass transfer coefficient k is mainly a function of the feed flow velocity, the density, the viscosity, the diffusivity of the solute, and the membrane module type. Many Sherwood relationships for different flow regimes and membrane module shape and dimensions are available in the literature (Cheryan, 1986; Rautenbach and Albreht, 1989; Mulder, 1991).
Microfiltration (MF) is a form of filtration that has two common forms. One form is cross-flow separation. In cross-flow separation, a fluid stream runs parallel to a membrane. There is a pressure differential across the membrane. This causes some of the fluid to pass through the membrane, while the remainder continues across the membrane, cleaning it. The other form of filtration is called dead-end filtration or perpendicular filtration. In dead-end filtration, all of the fluid passes through the membrane, and all of the particles that cannot fit through the pores of the membrane are stopped. Cross-flow microfiltration is used in a number of applications, as either a prefiltration step or as a process to separate a fluid from a process stream. Dead-end microfiltration is used commonly in stopping particles in either prefiltration or final filtration before a fluid is to be used. Cartridge filters are typically composed of microfiltration media. MF is a pressure-driven membrane filtration process that has a membrane with a pore size typically of 0.2-2 ^m and able to retain particles with molecular weights equal or larger than 200 kDa. MF membranes are symmetric with a characteristic spongelike network of interconnecting pores. It has been successfully used in the beer brewing industry to remove bacteria in the production of long shelf-life draft beers. The dairy industry has also found MF useful in removing bacteria or particulate substances and fractionation of milk proteins. MF is the membrane process that most closely resembles a conventional filtration unit. The transport mechanism of MF is undoubtedly sieving action. Thus, the volume flux through the MF membranes is expressed with a Hagen-Poiseuille relationship (Equation 3.26):
_ £r2 AP 8^rAz if the membrane is perceived as a bunch of straight capillaries. When a nodular structure (the space between spheres) exists, a Kozeny-Carman equation is usually applied to the following (Equation 3.27):
_ K^S2(1 -£)2 Az where S is the internal surface area, £ the volume fraction of the pores, t the pore tortuosity, ^ the viscosity, and K the Kozeny-Carman constant.
Chapter 3: Physicochemical Wastewater Treatment Processes 87 Ultrafiltration (UF)
Ultrafiltration (UF) is the most common membrane process used in the food industry and it involves the use of a membrane with a pore size ranging between 0.01-0.2 ^m. Ultrafiltration is not as fine a filtration process as nanofiltration, but it also does not require the same energy to perform the separation. Applications of ultrafiltration in food processing can most likely be found in situations that call for separating one or more desirable components from a liquid mixture. Ultrafiltration is capable of concentrating bacteria, some proteins, some fats, some colloidal minerals and constituents that have a larger molecular weight of greater than 10 kDa, but it is typically not effective at separating organic streams (Rosenberg, 1995). In UF, the chemical nature of membrane materials has only little effect upon the separation because ultrafiltration separation, like microfiltration, is based upon sieving mechanisms; thus, ultrafiltration is only somewhat dependent upon the charge of the particle and is much more concerned with the size of the particle. The mass transfer equations for UF are similar to those for MF.
Reverse Osmosis (RO)
Reverse osmosis (RO), also known as hyperfiltration, is the finest filtration known. This process will allow the removal of particles as small as ions from a solution. Reverse osmosis is used to purify water and remove salts and other impurities in order to improve the color, taste, or properties of the fluid. It can be used to purify wastewater streams that need additional treatment to remove water, which will pass through the reverse osmosis membrane, while rejecting other ions and colloids from passing. The most common use for reverse osmosis is in purifying water. It is used to produce water that meets the most demanding specifications that are currently in place. Reverse osmosis uses a membrane that is semipermeable, allowing the fluid that is being purified to pass through it, while rejecting the contaminants that remain. Most reverse osmosis technology uses a process known as cross-flow to allow the membrane to continually clean itself. As some of the fluid passes through the membrane, the rest continues downstream, sweeping the rejected species away from the membrane. The process of reverse osmosis requires a driving force to push the fluid through the membrane, and the most common force is pressure from a pump. A reverse osmosis process involves pressures 5-10 times higher than those used in ultrafiltration. As the concentration of the fluid being rejected increases, the driving force required to continue concentrating the fluid increases. Reverse osmosis is capable of rejecting bacteria, salts, sugars, proteins, particles, fats, and other constituents that have a molecular weight of greater than 0.15-0.25 kDa. The separation of ions with reverse osmosis is aided by charged particles. This means that dissolved ions that carry a charge, such as salts, are more likely to be rejected by the membrane than those that are not charged, such as organics. The larger the charge and the larger the particle, the more likely it will be rejected. The transport mechanism of RO as well as of nanofiltration is now believed to be the solution diffusion mechanism. The evaluation of RO performance can be conducted with Equations 3.21-3.24.
Nanofiltration (NF) is a form of filtration that uses membranes to preferentially separate different fluids or ions. Nanofiltration is not as fine a filtration process as reverse osmosis, but it also does not require the same energy to perform the separation. Nanofiltration also uses a membrane that is partially permeable to perform the separation, but the pores of the membrane are typically much larger than the membrane pores that are used in reverse osmosis. Nanofiltration is most commonly used to separate a solution that has a mixture of some desirable components and some that are not desirable. An example of this is the concentration of corn syrup. The nanofiltration membrane will allow the water to pass through the membrane while holding the sugar back, concentrating the solution. As the concentration of the fluid being rejected increases, the driving force required to continue concentrating the fluid increases. Nanofiltra-tion is capable of concentrating sugars, divalent salts, bacteria, proteins, particles, fats, and other constituents that have a molecular weight greater than 1 kDa. Nanofiltration, like reverse osmosis, is affected by the charge of the particles being rejected. Thus, particles with larger charges are more likely to be rejected than others. The mass transport mechanism of NF and the membrane material used are quite comparable to those of RO. In some cases, NF is grouped into reverse osmosis processes.
Membrane separations by electrical potential difference: Electrodialysis (ED)
Electrodialysis (ED) is an electrically driven membrane separation process that is capable of separating, concentrating, and purifying selected ions from aqueous solutions (as well as some organic solvents). The process is based on the property of ion exchange membranes to selectively reject anions or cations. If membranes are more permeable to cations than to anions or vice versa, the concentration of ionic solutions increases or decreases, so that concentration or depletion of electrolyte solutions is possible. Because in electrodialysis only ionic species are transferred directly, removal of ionic species from nonionic products can be accomplished so that purification is possible. Electrodialysis reversal (EDR) is an electrodialysis process in which the polarity of the electrodes is reversed on a prescribed time cycle, thus reversing the direction of ion movement in a membrane stack. The advantage of EDR is that it mitigates some of the concentration polarization and membrane fouling problems (Davis, 1990). The largest application of ED is the production of potable water from brackish water. Electrodialysis can remove salts from food, dairy, and other products, as well as concentrate salts, acids, or bases. It also finds applications in wine and juice stabilization and in removing unwanted total dissolved solids that can build up in product streams (Lopez-Leiva, 1988; Davis, 1990).
Faraday's law supplies the basis to model ion transport and affirms that the total current in an electrolytic cell is equal to the sum of the electricity conveyed by each ion species (Equation 3.28):
where I is the current density, f the Faraday's constant, Q the flow rate, AQ the concentration difference, Ji the molar flux, ei the current efficiency, and Zi the valence of ion i. Concentration polarization also severely affects the current density and the diffusive flux (the current density) through the concentration gradient over the boundary layer for a univalent ionic solution (Z = 1) (Equation 3.29):
where D is the diffusivity; Cm and Cb are concentrations at the membrane surface and in the bulk, respectively; 8c is the thickness of the concentration boundary layer; and tm and tbl are the transport numbers of the ion in j = Df (Cb - Cm ) ^c (tm - tbl )
the membrane and in the solution, respectively. The transport number of the ion is defined as the following (Equation 3.30):
Membrane separations by partial vapor pressure gradient
Pervaporation (PV) is the separation of liquid mixtures by partial vaporization through a dense permselective membrane. Unlike the other membrane processes, a phase change occurs when the permeate changes from liquid to vapor during its transport through the membrane. PV is an enrichment technique similar to distillation; however, unlike distillation, PV is not limited by the vapor-liquid equilibrium. As a matter of fact, PV has been commercially applied to the separation of azeotropic mixtures (dehydration of alcohol). The heart of the PV is a nonporous membrane, which either exhibits a high permeation rate for water but does not permeate organics, or vice versa. A gradient in the chemical potential of the substances on the feed side, and the permeate side is the driving force for the process, which can be represented by partial vapor pressures on both sides of the membrane. The driving force is kept at a maximum by applying low pressure (vacuum or sweep gas) to the permeate side of the membrane, combined with immediate condensation of permeated vapors.
Pervaporation processes have found use in the chemical industry to break azeotropic water/alcohol mixtures and to perform separations that are highly energy-intensive when distillation is used. Over the decades, a growing amount of attention has been paid to the application of pervapo-ration to environmental problem.
The performance of pervaporation is commonly evaluated by two experimental parameters: the permeation flux and the selectivity. The performance of a pervaporation process is assessed by the flux of the permeating species and the selectivity of the species (Equation 3.31):
where kiov, p', QL, Qv are the overall mass transfer rate constant, molar density of feed, bulk liquid phase concentration (mole fraction), and bulk vapor phase concentration, respectively, for component i. The most commonly used selectivity parameter is the separation factor shown in Equation 3.20. Sometimes, however, the enrichment factor, Pi, is used as an indication of the separation selectivity for component i (Equation 3.32):
As the concentration of component i is reduced, the concentration of component j will approach 1. The separation factor will therefore be close to the value of the enrichment factor, Pi, for dilute solutions (Equation 3.33):
Pervaporative transport process follows the solution-diffusion model that is also the transport mechanism of RO and NF, which consists of the following steps:
1. Diffusion through the liquid boundary layer next to the feed side of the membrane
2. Selective partitioning of molecules of components into the membrane
3. Selective transport (diffusion) through the membrane matrix
4. Desorption into vapor phase on the permeate side
5. Diffusion away from the membrane and into the vapor boundary layer on the permeate side of the membrane
Often, each step can be modeled with different approaches and fundamental assumptions; however, as with all mass transfer operations, the slowest step in this sequence will limit the overall rate of mass transfer and will be the center of research focus. Naturally, these steps are conveniently expressed in the form of the resistance-in-series model, which is expressed with mathematical symbols as the following (Equation 3.34):
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