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Rate of Flow Through Membranes
Membrane Equipment

Membranes can be used for separating constituents of foods on a molecular basis, where the foods are in solution and where a solution is separated from one less concentrated by a semi-permeable membrane. These membranes act somewhat as membranes do in natural biological systems.

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.

Fig. 9.9 Membrane separation process
Figure 9.9 Reverse osmosis systems

Also, "looser" membranes have been developed through which not only water, but also larger solute molecules can selectively pass if driven by imposed pressure. Membranes are available which can retain large molecules, such as proteins, while allowing through smaller molecules. Because the larger molecules are normally at low molar concentrations, they exert very small osmotic pressures, which therefore enter hardly at all into the situation. Following the resemblance to conventional filtration, this process is called ultrafiltration. In general, ultrafiltration needs relatively low differential pressures, up to a few atmospheres. If higher pressures are used, a protein or solute gel appears to form on the membrane, which resists flow, and so the increased pressure may not increase the transfer rate.

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.

Rate of Flow Through Membranes

There are various equations to predict the osmotic pressures of solutions, perhaps the best known being the van't Hoff equation:

P = MRT                                                                                                               (9.17)

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 mass-transfer coefficient:

dw/dt = KA[DP - DP]                                                                                              (9.18)

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

EXAMPLE 9.11. Reverse osmosis: concentration of sucrose solution by reverse osmosis
A solution of sucrose in water at 25°C is to be concentrated by reverse osmosis. It is found that, with a differential applied pressure of 5000 kPa, the rate of movement of the water molecules through the membrane is 25 kg m-2 h-1 for a 10% solution of sucrose. Estimate the flow rate through the membrane for a differential pressure of 10,000 kPa with the 10% sucrose solution, and also estimate the flow rate for a differential pressure of 10,000 kPa but with a sucrose concentration of 20%.

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.
Applying eqn. (9.17):

For 10% solution P = 0.304 x 8.314 x 298
                              = 753 kPa
For 20% solution
P = 0.632 x 8.314 x 298
                              = 1566 kPa.

So we have for the first case, for 1 m2 of membrane:

                      dw/dt = 25 = K[5000 - 753]
                           K = 5.9 x 10-3 kg m-2 h-1 kPa-1
So for               
DP = 10,000 kPa,
                      dw/dt = 5.9 x 10-3[10,000 - 753]
                               = 55 kg m-2 h-1

And for               DP = 10,000 and 20% soln.
                      dw/dt = 5.9 x 10-3[10,000 - 1566]
                              = 50 kg m-2 h-1

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 build-up 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:

dw/dt = K’ A loge(ci /cb)                                                                                           (9.19)

where ci and cb 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 (m2 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:

(K'd/D) = a[(dvr/m)m] x (m/rD)n

or (Sh) = (K'd/D) = a(Re)m(Sc)n                                                                               (9.20)

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.

Membrane Equipment

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 non-protein constituents.

Protein in initial whey = 6 x 0.12 = 0.72%

Protein in retentate    = 12 x concentration in whey
                                = 12 x 6 x 0.12
                                = 8.6%.

Setting out a mass balance, basis 100 kg whey:
Initial whey

Water removed per 100 kg whey        = 87.3 kg

The equipment has to process 30,000 kg in 6 h so the membrane has to pass the permeate at:

                     (30,000 x 91.9)/(100 x 6) = 4595 kg h-1
and permeate filtration rate is 30 kg m-2 h-1

Therefore required area of membrane  = 4595/30
                                                       = 153 m2

Non-protein rejection rate                  = 0.7/5.3
                                                      = 13%

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 membrane-processing 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).

Contact-Equilibrium Processes - APPLICATIONS > DISTILLATION

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Unit Operations in Food Processing. Copyright © 1983, R. L. Earle. :: Published by NZIFST (Inc.)
NZIFST - The New Zealand Institute of Food Science & Technology