Diffusion-limited extractions

Bartle et al. [286] described a simple model for diffusion-limited extractions from spherical particles (the so-called hot-ball model). The model was extended to cover polymer films and a nonuniform distribution of the extractant [287]. Also the effect of solubility on extraction was incorporated [288] and the effects of pressure and flow-rate on extraction have been rationalised [289]. In this idealised scheme the matrix is supposed to contain small quantities of extractable materials, such that the extraction is not solubility limited. The model is that of diffusion out of a homogeneous spherical particle into a medium in which the extracted species is infinitely dilute. The ratio of mass remaining (m ) in the particle of radius r at time t to the initial amount (mo) is given by ... 

The nature of the extractant affects the extraction in both the solubility and diffusion limiting cases. 

Citric acid separation from fermentation broth employs the full allotment of Sorbex beds in addition to the four basic Sorbex zones. The process utilizes a resin instead of a zeolite based adsorbent. The resin is a nonionic cross-linked polystyrene polyvinyl benzene formulation. Operating temperatures for this process are sufficient to overcome diffusion limitations with a corresponding operating pressure to maintain liquid-phase operation. The desorbent consists of water blended with acetone. Subsequent processing steps remove the desorbent from the desired extract product citric acid. 

On the basis of the Hatta number, the transformations carried out in biphasic systems can be described as slow (Ha < 0.3), intermediate (with a kinetic-diffusion regime 0.3 < Ha < 3.0), and fast (Ha > 3). These are diffusion limited and take place near the interface (within the diffusion layer). Slow transformations are under kinetic control and occur mostly in a bulk phase, so that the amount of substrate transformed in the boundary layer in negligible. When diffusion and reaction rate are of similar magnitude, the reaction takes place mostly in the diffusion layer, although extracted substrate is also present in the continuous phase, where it is transformed at a rate depending on its concentration [38, 50, 54]. 

The simulated and experimental variations of the end-of-run (i.e., 8 hr.) isomerization rates with density are compared in Figure 1. Details of the experiments are provided elsewhere [2, 3]. At subcritical densities, the extraction of coke precursors is insignificant. Hence, an increase in the concentration of the hexene and coke precursors (i.e., oligomers) leads to lower isomerization rates. At near-critical densities, the extraction of coke precursors becomes significant. Hence, the isomerization rate increases. Both the experimental and simulated rates show a decreasing trend when the density is increased from near-critical to supercritical values. This is attributed to pore-diffusion limitations as the fluid changes from gas-like to liquid-like. Above 2.0 pc, the isomerization rate increases with density as the ability of the reaction mixture to extract the coke precursors increases. 

The ratio U /Mf is inversely proportional to residence time of the eluent in the extractor tr (see equation 1), and this ratio will have to be conserved, especially for extraction limited by internal diffusion. These extractions are almost not dependent on the solvent flowrate, but the "contacting" time of the feed with the solvent is the determinant factor of plant design. Therefore, it will be necessary to use very large extractors or to use several extractors in series in order to maximize the contacting time of the solvent with the feed. On the other hand, it is possible to minimize the solvent flowrate and the energy consumption of the plant. 

Mass transfer from the water phase to the microdroplet under stationary conditions quickly reaches the steady-state because of spherical diffusion. When the extraction of X is diffusion-limited in the water phase. Equation (3) is obtained using the diffusion coefficient of X in the water phase (Z w) [18]. 

Accepting that the last traces of oil are more difficult to dissolve does not nullify the basic conclusions to be derived from the preceding theories based on simple diffusion with free miscibility of solvent and oil. If free miscibility does not exist in the latter stages of extraction, this means simply that the effective concentration of solute is not the concentration of oil in the solid seed material but a lower concentration that is limited by the solubility of the oil in the solvent. The rate of diffusion will be less than observed in the earlier stages, not because the diffusion coefficient has decreased, but because the oil content of the solid material is no longer a proper measure of its instantaneous content of diffusible material. The diffusion or extraction rate will, for example, still be inversely proportional to the square of the flake thickness. 

Third, the equations employed in a PBPK model should be consistent with the state of knowledge or biologically plausible hypotheses of the mechanisms of ADME for the particular chemical. In this regard, the uptake of chemicals in systemic circulation is described as either a diffusion-limited or perfusion-limited process (Gerlowski and Jain 1983), and metabolic clearance in individual tissues or tissue groups is described using a maximal velocity and Michaelis constant, intrinsic clearance, or hepatic extraction ratio (Krishnan and Andersen 2007). The mass balance differential equations accounting for uptake clearance, efflux clearance, and metabolic clearance are formulated as a function of identifiable input parameters (Table 21.1). 

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