Stagewise Operation

Thus far we have been concerned with stagewise operations, doing just what is done in the laboratory when a solvent is mixed with an aqueous phase in a separatory funnel and allowed to settle before being separated. 

This integral clearly depends on the slope of the operating line and, as in the case of stagewise operations, if the operating line approaches the equilibrium curve too closely, then the driving force approaches zero and the inverse becomes very large. That is, when the operating line is close to... 

Adsorption can be carried out by stagewise or continuous-contacting methods. The stagewise operation of adsorption is called contact filtration because the liquid and the solid are contacted in a mixer and then the solid is separated from the solution by filtration. 

Acrivos and Amundson (SO, 51) applied matrix algebra to the unsteady-state behavior of stagewise operations in chemical processes. Instead of using a characteristic vector expansion, they emphasize the use of the Sylvester-Lagrange-Buchheim formula (52). Even though this formula is equivalent to the characteristic vector expansion, it is more difficult to manipulate and is not easily related to physical concepts such as straight line reaction paths. 

The heterogeneities of most concern to us are those that involve the presence of more than one phase. The analysis of multiphase systems can be important to the design and operation of many industrial processes, especially those in which multiple phases influence chemical reactions, heat transfer, or mixing. For example, phase-equilibrium calculations form the bases for many separation processes, including stagewise operations, such as distillation, solvent extraction, crystallization, and supercritical extraction, and rate-limited operations, such as membrane separations. 

Tiller, F. M., and R. S. Tour, Stagewise Operations— Applications of the Calculus of Finite Differences to Chemical Engineering, Trans. AIChE, 40, 317-332 (1944). 

Apparatus for the thermal separation of mixtures may be operated both discon-tinuously (intermittently, batch production, stagewise operation) and continuously (steady-state). In the following section, the operating modes are briefly illustrated. The advantages and disadvantages are listed in Table 1-3. 

This sort of richness and leanness can be demonstrated in cascade or stagewise operations, where both a permeate phase and reject phase are introduced or recycled (or refluxed) into successive membrane cells. In this respect, the use of constant molar or molal flow rates in Chapter 4 for stagewise operations may not be too far afield. 

H V/F or V/Feed Note As per Example 3.1, the symbol F can pertain to the feedstream for the entire stagewise operation (specified)... 

The Transport Phenomena approach is clearly an essential course for graduate students. However, in the undergraduate curriculum there was a definite division with many departments keeping the Unit Operations approach. Even where the Transport Phenomena was used at the undergraduate level there were segments of the Unit Operations (particularly stagewise operations) that were still used. 

In stagewise operations the time of contact is intimately connected with stage efficiency, whereas for continuous-contact equipment the time leads ultimately to the volume or length of the required device. The factors which help establish the time are several. Material balances permit calculation of the relative quantities required of the various phases. The equilibrium characteristics of the system establish the ultimate concentrations possible, and the rate of transfer of material between phases depends upon the departure from equilibrium which is maintained. The rate of transfer additionally depends upon the physical properties of the phases as well as the flow regime within the equipment. 

Chapter 11 of this undergraduate textbook devotes 77 pages to adsorption and ion exchange. Single and multicomponent equilibria, for both gaseous and liquid systems are described qualitatively. Stagewise operation, which is... 

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