Enrichment and Separation Factor

Various phenomena that take place in a foam column are shown schematically in Figure 2. Bubbles are formed by the sparger into the liquid pool. Proteins adsorb on to the bubbles during their formation and their passage through the liquid pool. The rate of adsorption of protein depends on the rate of diffusion of protein molecules to the gas-liquid interface as well as on the adsorption activation energy at the bubble surface. The extent of the surface coverage at the gas-liquid interface is dependent on the time of formation of the bubbles and its residence time in the liquid pool (Uraizee and Narsimhan, 1995). The foam bed consists of an assemblage of gas bubbles separated by thin liquid films, creating a large gas-liquid interfacial area. The size distribution of the bubbles depends on the type of sparger employed for bubble formation. A sintered disc with fine pores where rF is the equilibrium surface concentration of the protein at the gas-liquid interface corresponding to the feed concentration. Since rF > rB, B < F, comparison of eqns [2] and [3] with eqns [6] and [7] indicates that the stripping mode yields a leaner bottom product and richer top product compared to the

Figure 2 Schematic of various phenomena that take place in a foam column.

usually results in a wide distribution of bubble sizes whereas either capillaries or orifices of uniform sizes lead to more or less uniform bubble sizes. Since the volume fraction of liquid in a foam is usually very small, the gas bubbles are distorted and are usually approximated by a dodecahedron (Narsimhan and Ruckenstein, 1986). A typical gas bubble is shown in Figure 3A. The neighbouring gas bubbles are assumed to be separated by planar films of the continuous liquid phase. Where three bubbles touch, their films drain laterally into a Plateau border. This is a channel whose length is the length of a side of the touching dodecahedral bubbles, and whose walls have a sharp concave curvature of radius Rp (Figure 3B). This lateral flow is caused by a pressure drop AP between the liquid pressure in the film, which is essentially the air pressure in the bubble, and the pressure of the liquid in the Plateau border. If a is the surface tension of the bubble-liquid interface, then:

inversely proportional to bubble size) thus leading to the growth of larger bubbles at the expense of smaller


The liquid in the Plateau border drains under gravity. Consequently, the liquid hold-up decreases with foam height. The lateral flow out of the thin films separating the gas bubbles will cause them to thin further, possibly causing them to rupture because of instability resulting from the growth of thermal and mechanical perturbations thus leading to bubble coalescence. Coalescence leads to internal reflux of the liquid from the ruptured films into the Plateau borders and a decrease in the interfacial area because of an increase in the bubble size. The former tends to enhance separation (enrichment) whereas the latter is detrimental. The former effect is usually predominant, so that coalescence leads to higher separation (enrichment). Only when coalescence is excessive, collapse of the foam bed occurs. When there is a broad distribution of bubble sizes, diffusion of gas from smaller to larger bubbles may occur because of the difference in the capillary pressure (being

In order to predict the liquid hold-up as a function of foam height, one needs to solve the balance equations for drainage of liquid from thin films into the Plateau borders. The equations describing the rate of change, with vertical position, of the volumetric hold-up of the liquid in the films, caused by their drainage into the Plateau borders and bubble coalescence is given by (Uraizee and Narsimhan, 1995):

where Xf is the film thickness, nf is the number of films per bubble, Af is the area of the film, q is the number of bubbles entrained per unit cross-section of the foam, N is the number of bubbles per unit volume of the foam, and V is the velocity of drainage of the film and ß is the coalescence frequency. q and N can be related to the superficial gas velocity G, liquid hold-up s, and the bubble volume v through:

As before, the equation describing the rate of change, with vertical position, of volumetric liquid hold-up in the Plateau borders, caused by flow from the films into the Plateau borders and bubble coalescence, and gravity drainage is given by (Uraizee and Narsimhan, 1995):

Figure 3 Schematic of a bubble in a foam column.

where np is the number of Plateau borders per bubble, ap is the area of cross-section of Plateau border, R is the radius of the bubble, l is the length of the Plateau border, and u is the velocity of gravity drainage of Plateau borders. Similarly, the protein balance in the foam can be written as:

column is much smaller than the entrainment of the liquid at the foam-liquid interface. Hence, the material balance around the foam yields:

Ggo 4

—dz ^p^p- i # dz (15 NnpuRcp, « ) # NnfAf Vfcf, '

where cp, {and c^ { are the protein concentrations in the Plateau border and film respectively. In the absence of coalescence, they would be equal. However, coalescence would enrich the liquid in the Plateau border because of reflux of adsorbed protein from the ruptured thin films. In the above equation, r,- is related to the bulk concentration c, via the Langmuir adsorption isotherm given by eqn [5]. In eqns [12] and [13], V and u are the velocities of drainage of films and Plateau borders, respectively. For an immobile gas-liquid interface, the velocity of drainage of films into the Plateau borders can be evaluated from the Reynolds equation:

The inlet bubble size R0 depends on the type of sparger and the superficial gas velocity G. The above two equations can be solved for xf0 and ap0. Also, the protein concentration in films and Plateau borders at the foam-liquid interface can be taken as equal to the pool concentration, i.e.:

The pool concentration should satisfy the overall protein balance given by:

where F, B and T refer to feed, bottom and top product flow rates expressed per unit area of cross-section of the foam column. The overall mass balance can be written as:

where Rf is the radius of the film, p is the viscosity, and AP is the pressure drop under which the film drains. The velocity of drainage of the Plateau borders for immobile gas-liquid interface is given by:

Eqns [10] to [13] can be solved with the initial conditions [16] to [20] to give the profiles of Xf, ap and cp, ,■. The liquid hold-up s at any foam height can then be calculated via:

The enrichment e, for each component is given by (Uraizee and Narsimhan, 1995):

where p is the density of the liquid.

Eqns [10], [12] and [13] are initial value problems which have to be solved with proper initial conditions at the foam-liquid interface to evaluate Xf and ap and cp, , as a function of foam height.

The liquid hold-up at the foam-liquid interface (z = 0) can be set to the void fraction of spheres (Uraizee and Narsimhan, 1995):


where (... )T refers to the evaluation of the quantity within the parenthesis at the top of the column. The separation factor S is then given by:


As the liquid hold-up at the top of the column is much smaller than 0.26, the flow rate at the top of the

The above analysis assumes adsorption equilibrium for the surface concentration of proteins at the air-water interface. Uraizee and Narsimhan (1995) have modified this analysis to account for the kinetics of adsorption of proteins on to the gas bubbles during their travel through the liquid pool before the formation of foam and demonstrated the effects of different parameters including the kinetics of adsorption and pool height on enrichment and recovery of proteins.

Solar Panel Basics

Solar Panel Basics

Global warming is a huge problem which will significantly affect every country in the world. Many people all over the world are trying to do whatever they can to help combat the effects of global warming. One of the ways that people can fight global warming is to reduce their dependence on non-renewable energy sources like oil and petroleum based products.

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