CHAPTER
9
Water flows through the membrane from the dilute solution to the more concentrated one. The force producing this flow is called the osmotic pressure and to stop the flow a pressure, equal to the osmotic pressure, has to be exerted externally on the more concentrated solution. Osmotic pressures in liquids arise in the same way as partial pressures in gases: using the number of moles of the solute present and the volume of the whole solution, the osmotic pressure can be estimated using the gas laws. If pressures greater than the osmotic pressure are applied to the more concentrated solution, the flow will not only stop but will reverse so that water passes out through the membrane making the concentrated solution more concentrated. 
The flow will continue until the concentration rises to the point where its osmotic pressure equals the applied pressure. Such a process is called reverse osmosis and special artificial membranes have been made with the required "tight" structure to retain all but the smallest molecules such as those of water.
Important applications for ultrafiltration are for concentrating solutions of large polymeric molecules, such as milk and blood proteins. Another significant application is to the concentration of whey proteins. Reverse osmosis, on the other hand, is concerned mainly with solutions containing smaller molecules such as simple sugars and salts at higher molar concentrations, which exert higher osmotic pressures. To overcome these osmotic pressures, high external pressures have to be exerted, up to the order of 100 atmospheres. Limitations to increased flow rates arise in this case from the mechanical weaknesses of the membrane and from concentration of solutes which causes substantial osmotic "back" pressure. Applications in the food industry are in separating water from, and thus concentrating, solutions such as fruit juices.
There are various equations to predict the osmotic pressures of solutions, perhaps the best known being the van't Hoff equation:
in which P (pi) is the osmotic pressure (kPa), M the molar concentration (moles m^{3}), T the absolute temperature (°K), and R the universal gas a constant . This equation is only strictly accurate when the dilution is infinitely great, but it can still be used as an approximation at higher concentrations. The net driving force for reverse osmosis is then the difference between the applied differential pressure DP, and the differential osmotic pressure, DP , which resists the flow in the desired "reverse" direction. Therefore it can be described by the standard rate equation, with the rate of mass transfer being equal to the driving force multiplied by the appropriate masstransfer coefficient:
where dw/dt is the rate of mass transfer, K is the mass transfer coefficient, A the area through which the transfer is taking place, DP is the net applied pressure developed across the membrane and DP is the net osmotic pressure across the membrane and resisting the flow. DP is therefore the difference in the applied pressure on the solutions at each side of the membrane and DP is the difference in the osmotic pressures of the two solutions, as in Fig. 9.9. The gas constant is 8.314 kPa m
For sucrose, the
molecular weight is 342 so for a 10% solution, molar concentration (from
tables such as Perry, 1997) is 0.304 moles m^{3} and for 20%,
0.632 moles m^{3}.
So we have for the first case, for 1 m^{2} of membrane:
Experimental values of the osmotic pressure of the sucrose solutions at 10% and 20% were measured to be 820 and 1900 kPa respectively, demonstrating the relatively small error arising from applying the van't Hoff equation to these quite highly concentrated solutions. Using these experimental values slightly reduces the predicted flow as can be seen by substituting in the equations. In ultrafiltration practice, it is found that eqn. (9.18) applies only for a limited time and over a limited range of pressures. As pressure increases further, the flow ceases to rise, or even falls. This appears to be caused, in the case of ultrafiltration, by increased mechanical resistance at the surface of the membrane due to the buildup of molecules forming a layer which is like a gel and which resists flow through it. Under these circumstances, flow is better described by diffusion equations through this resistant layer leading to equation:
where c_{i} and c_{b} are the solute concentrations at the interface and in the bulk solution respectively. The effect of the physical properties of the material can be predicted from known relationships for the mass transfer coefficient K' (m s^{1}), which can be set equal to D/d where D is the diffusivity of the solute (m^{2} s^{1}), divided by d, the thickness of the gel layer (m). This equation has been found to predict, with reasonable accuracy, the effect on the mass transfer of changes in the physicochemical properties of the solution. This is done through well established relationships between the diffusivity D, the mass transfer coefficient K', and other properties such as density (r), viscosity (m) and temperature (T) giving:
where d is the hydraulic diameter, (Sh) the Sherwood number (K'd/D); (Sc) the Schmidt number (m/rD); and a, m, n are constants. Notice the similar form of eqn. (9.20) and the equation for heat transfer in forced convection, (Nu) = a'(Re)^{m'}(Pr)^{n'}, with (Sh) replacing (Nu) and (Sc) replacing (Pr). This is another aspect of the similarity between the various transport phenomena. These ideas, and the uses that can be made of them, are discussed in various books, such as Coulson and Richardson (1977) and McCabe and Smith (1975), and more comprehensively in Bird, Stewart and Lightfoot (1960). In the case of reverse osmosis, the main resistance arises from increased concentrations and therefore increased back pressure from the osmotic forces. The flow rate cannot be increased by increasing the pressure because of the limited strength of membranes and their supports, and the difficulties of designing and operating pumps for very high pressures. The
equipment for these membrane separation processes consists of the necessary
pumps, flow systems and membranes. In the case of ultrafiltration, the
membranes are set up in a wide variety of geometrical arrangements, mostly
tubular but sometimes in plates, which can be mounted similarly to a filter
press or plate heat exchanger. Flow rates are kept high over the surfaces
and recirculation of the fluid on the high pressure, or retentate, side
is often used; the fluid passing through, called the permeate, is usually
collected in suitable troughs or tanks at atmospheric pressure.
In the case of reverse osmosis, the high pressures dictate mechanical strength, and stacks of flat disc membranes can be used one above the other. Another system uses very small diameter (around 0.04 mm) hollow filaments on plastic supports; the diameters are small to provide strength but preclude many food solutions because of this very small size. The main flow in reverse osmosis is the permeate. The systems can be designed either as continuous or as batch operations. One limitation to extended operation arises from the need to control growth of bacteria. After a time bacterial concentrations in the system, for example in the gel at the surface of the ultrafiltration membranes, can grow so high that cleaning must be provided. This can be difficult as many of the membranes are not very robust either to mechanical disturbance or to the extremes of pH which could give quicker and better cleaning. EXAMPLE 9.12. Ultrafiltration of whey It is desired to increase the protein concentration in whey, from cheese manufacture, by a factor of 12 by the use of ultrafiltration to give an enriched fraction which can subsequently be dried and used to produce a 50% protein whey powder. The whey initially contains 6% of total solids, 12% of these being protein. Pilot scale measurements on this whey show that a permeate flow of 30 kg m^{2} h^{1} can be expected, so that if the plant requirement is to handle 30,000 kg in 6 hours, estimate the area of membrane needed. Assume that the membrane rejection of the protein is over 99%, and calculate the membrane rejection of the nonprotein constituents.
The equipment has to process 30,000 kg in 6 h so the membrane has to pass the permeate at:
Membrane processes generally use only one apparent contact stage, but product accumulation with time, or with progression through a flow unit, gives situations which are equivalent to multistage units. Dialysis, which is a widely used laboratory membraneprocessing technique, with applications in industry, sometimes is operated with multiple stages.
These membrane concentration and separation processes have great potential
advantages in the simplicity of their operation and because drastic conditions,
in particular the use of heat leading to thermal degradation, are not
involved. Therefore more extensive application can be expected as membranes,
flow systems and pumps are improved. Discussion of these processes can
be found in papers by Thijssen
(1974) and in Sourirajan (1977).
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