Theoretical or equilibrium stages

A theoretical or equilibrium stage is a device or combination of devices that accomplishes the effect of intimately mixing two immiscible liquids until equilibrium concentrations are reached, then physically separating the two phases into clear layers. Crosscurrent extraction (Fig. 15-4) is a cascade, or series of stages, in which the raffinate R from one extraction stage is contacted with additional fresh solvent S in a subsequent stage. 

On X, Y coordinates this represents a straight operating line, through points of coordinates (Xq, Y and Tj) of slope —Sg/Ls. If the stage is a theoretical or equilibrium stage, the effluent streams are in equilibrium, so that the point (A", Tj) lies on the equilibrium adsorption isotherm. This is shown on the lower portion of Fig. 11.16. The equilibrium curve should be that obtaining at the final temperature of the operation. If insufficient time of contact is allowed, so that equilibrium is not reached, the final liquid and solid concentrations will correspond to some point such as A (Fig. 11.16), but ordinarily equilibrium is approached very closely. 

The countercurrent column is considered as a cascade of cells in series. The adsorption equilibrium is assumed to be reached in each cell or equilibrium stage or plate. The broadening effects, linked to the mass transfer kinetics and to the hydrodynamics, are lumped together and are quantified by the number of theoretical plates N, which can be derived from an analytical pulse injection. 

The Fenske equation estimates the minimum number of theoretical plates or equilibrium stages at total reflux. 

One way of calculating the number of equilibrium stages (or number of theoretical plates, NTP) for a mass exchanger is the graphical McCabe-Thiele method. To illustrate this procedure, let us assume that over the operating range of compositions, the equilibrium relation governing the transfer of the pollutant from the... 

Equilibrium data and material-and heat-balances to determine the number of equilibrium stages (theoretical plates or transfer units) required for the desired separation. The required height of the column can be calculated if data are available for the specific rate of transfer of material between the two phases, expressed in terms of the plate-efficiency or the height of one transfer unit. 

This is the steady-state current which is theoretically predicted if stage 1 is the rate-determining step in the sub-stages sequence represented in Equations 4.8 1.12. An important parameter to compare both in theory and experimentally is the Tafel slope or the transfer coefficient which results from it. Therefore, Equation 4.30 has to be written in a form that contains only one exponential term. Since the considered I-E curve is an oxidation wave, the effect of the reduction (second term in the right-hand part of Equation 4.30) will be negligible with potentials that are situated sufficiently far away from the equilibrium potential, and for the anodic current the following applies ... 

The variable that has the most significant impact on the economics of an extractive distillation is the solvent-to-feed flow rate ratio S/F. For close-boiling or pinched nonazeotropic mixtures, no minimum-solvent flow rate is required to effect the separation, as the separation is always theoretically possible (if not economical) in the absence of the solvent. However, the extent of enhancement of the relative volatility is largely determined by the solvent composition in the lower column sections and hence the S/F ratio. The relative volatility tends to increase as the S/F ratio increases. Thus, a given separation can be accomplished in fewer equilibrium stages. As an illustration, the total number of theoretical stages required as a function of S/F ratio is plotted in Fig. 13-95a for the separation of the nonazeotropic mixture of vinyl acetate and ethyl acetate using phenol as the solvent. 

Use of HETP Data for Absorber Design Distillation design methods (see Sec. 13) normally involve determination of the number of theoretical equilibrium stages or plates N. Thus, when packed towers are employed in distillation applications, it is common practice to rate the efficiency of tower packings in terms of the height of packing equivalent to one theoretical plate (HETP). 

The plate models assume that the column is divided into a series of an arbitrary number of identical equilibrium stages, or theoretical plates, and that the mobile and the stationary phases in each of these successive plates are in equihbrimn. The plate models are in essence approximate, empirical models because they depict a continuous column of length I by a discrete number of well-mixed cells. Although any mixing mechanism is dearly absent from the actual physical system, plate models have been used successfully to characterize the column operation physically and mathematically. Therefore, by nature, plate models are empirical ones, which cannot be related to first principles. 

---