Unsteady-State Batch Extraction

The values of C, and q which can be used for solids with varions shapes whea Eq. (10.7-3) b used in the case of betch extractions are listed in Table 10.8-1. When o is infinite Y remains constant and the dimensionless concentration ured in Eq. (LO.7-3) should be X.  

FIGURE 10.7-2 Log (X/XJ versus lime for the extraction of soybean oil from Haloed soybeans by oil-free hexane. (Reprinted from Food Technology, 36(2), 73-86 ( 962). Copyright 1982 by Institute of Food Technologists,] 

TABLE 10.8-1 Coefficients and Eigenvalues for Tick s Second Lew Solutions for Batch Extractions in Which a Is Constant and Bi Is Infinite  

The coefficients that should be used in Eq. (10.8-1), (10.8-2), and (10.8-3) for various solid shapes are listed in Table 10.8-2. While Eq. (10.7-3) is valid for al] values of r. provided enough terms ate used, Eq. (lO.S-l). (I0.S-2), and (10.8-3) ate valid only when t is small enough. Whea in doubt, the mege of validity of these equations shonld be checked by comparison with Eq. (10.7 3). 

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Mathematical models derived from mass-conservation equations under unsteady-state conditions allow the calculation of the extracted mass at different bed locations, as a function of time. Semi-batch operation for the high-pressure gas is usually employed, so a fixed bed of solids is bathed with a flow of fluid. Mass-transfer models allow one to predict the effects of the following variables fluid velocity, pressure, temperature, gravity, particle size, degree of crushing, and bed-length. Therefore, they are extremely useful in simulation and design. 

Yan [38] further simplified the equations for batch extractions by assuming an irreversible, first-order extraction reaction between the solute and the carrier, irreversible first-order stripping reaction between the complex carrier and the internal reagent and constant distribution coefficients. Weiss et al. [39] proposed an empirical model for the extraction of mercury. Recently, Baneijea et al. [40] and Chakraborty et al. [4] presented an unsteady-state mathematical models to explain type 2 facilitation. 

Pales and Stroeve [31] investigated the effect of the continuous phase mass transfer resistance on solute extraction with double emulsion in a batch reactor. They presented an extension of the perturbation analysis technique to give a solution of the model equations of Ho et al. [29] taking external phase mass transfer resistance into account. Kim et al. [5] also developed an unsteady-state advancing reaction front model considering an additional thin outer liquid membrane layer and neglecting the continuous phase resistance. 

Batchwise Extraction, Multiple Stage. This operation, usually a laboratory procedure, can be carried out in two ways. The immiscible solvents may be continuously pumped through the various stages in countercurrent flow, and the batch of mixture to be separated may be suddenly introduced near the center of the cascade. The result is an unsteady-state operation, the solutes leaving the opposite ends of the cascade in ratios different from that in the feed and in amounts varying with time. Such an operation has been considered in some detail by Martin and Synge (13) and Cornish et al. (4). 

Rate of leaching when diffusion in solid controls. In the case where unsteady-state diffusion in the solid is the controlling resistance in the leaching of the solute by an external solvent, the following approximations can be used. If the average diffusivity Da eff of the solute A is approximately constant, then for extraction in a batch process, unsteady-state mass-transfer equations can be used as discussed in Section 7.1. If the particle is approximately spherical. Fig. 5.3-13 can be used. 

It is characteristic of unsteady-state operation that concentrations at any point in the apparatus change with time. This may result from changes in concentrations of feed materials, flow rates, or conditions of temperature or pressure. In any case, batch operations are always of the unsteady-state type. In purely batch operations, all the phases are stationary from a point of view outside the apparatus, i.e., no flow in or out, even though there may be relative motion within. The familiar laboratory extraction procedure of shaking a solution with an immiscible solvent is an example. In semibatch operations, one phase is stationary while the other flows continuously in and out of the apparatus. As an example, we may cite the case of a drier where a quantity of wet solid is contacted continuously with fresh air, which carries away the vaporized moisture until the solid is dry. 

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