Stage theoretical

The combinations of mixing both feed and solvent until the equilibrium distribution of the solute has occurred, and the subsequent complete separation of the two phases is defined as one theoretical stage (Fig. 1). The two functions may be carried out sequentially in the same vessel, simultaneously in two different zones of the same vessel, or in separate vessels (mixers and settlers). 

Extraction may also be performed in a continuous differential fashion (Fig. 2), or in a sequential contact and separation where the solvent and feed phases flow countercurrently to each other between stages (Fig. 3). 

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Acrylic Acid Recovery. The process flow sheet (Fig. 3) shows equipment and conditions for the separations step. The acryUc acid is extracted from the absorber effluent with a solvent, such as butyl acetate, xylene, diisobutyl ketone, or mixtures, chosen for high selectivity for acryUc acid and low solubihty for water and by-products. The extraction is performed using 5—10 theoretical stages in a tower or centrifiigal extractor (46,61—65). 

Although the stagewise model is not physically reaUstic for differential contactors, it is sometimes used. The number of equivalent theoretical stages N can be determined graphically usiag the stepwise constmction illustrated ia Figure 7. For the case where both the equihbrium and operating lines are linear, it can be shown that ... 

The Alfa-Laval extractor (213) can give up to 20 theoretical stages in one unit. Depending on the system being handled, the capacity of the standard unit ranges between 5.7 and 21.2 m /h (1500—5600 gal/h). Antibiotic extractions and petrochemical processing are typical appHcations. 

Fractional vacuum distillation is the method used to separate terpene mixtures into their components. The terpene chemist usually has in the laboratory a range of columns with differing numbers of theoretical stages. Experimental distillation in the laboratory is useful in providing data for manufacturing plants that produce commercial quantities of terpene products. 

The separative capacity of the equivalent theoretical stage in the continuous process is seen to depend on the concentration difference between the countercurrent streams as well as on the concentration difference between the top and bottom of the stage. The separative capacity is zero when V is equal tojy or V is equal to x inspection shows that it attains a maximum value when V is equal to the arithmetic average of x andjy and that this maximum value... 

McCabe-Thie/e Example. Assume a binary system E—H that has ideal vapor—Hquid equiHbria and a relative volatiHty of 2.0. The feed is 100 mol of = 0.6 the required distillate is x = 0.95, and the bottoms x = 0.05, with the compositions identified and the lighter component E. The feed is at the boiling point. To calculate the minimum reflux ratio, the minimum number of theoretical stages, the operating reflux ratio, and the number of theoretical stages, assume the operating reflux ratio is 1.5 times the minimum reflux ratio and there is no subcooling of the reflux stream, then ... 

Fig. 11. Limiting conditions in binary distillation, (a) Minimum reflux and infinite number of theoretical stages (b) total reflux and minimum number of...

Fig. 12. Representative plot of theoretical stages vs reflux ratio for a given separation. Each curve is the locus of points for a given separation. Note the...

Computer solutions entail setting up component equiUbrium and component mass and enthalpy balances around each theoretical stage and specifying the required design variables as well as solving the large number of simultaneous equations required. The expHcit solution to these equations remains too complex for present methods. Studies to solve the mathematical problem by algorithm or iterational methods have been successflil and, with a few exceptions, the most complex distillation problems can be solved. 

In order to determine the packed height it is necessary to obtain a value of the overall number of transfer units methods for doing this are available for binary systems in any standard text covering distillation (73) and, in a more complex way, for multicomponent systems (81). However, it is simpler to calculate the number of required theoretical stages and make the conversion ... 

Heat Sensitivity. The heat sensitivity or polymerization tendencies of the materials being distilled influence the economics of distillation. Many materials caimot be distilled at their atmospheric boiling points because of high thermal degradation, polymerization, or other unfavorable reaction effects that are functions of temperature. These systems are distilled under vacuum in order to lower operating temperatures. For such systems, the pressure drop per theoretical stage is frequently the controlling factor in contactor selection. An exceUent discussion of equipment requirements and characteristics of vacuum distillation may be found in Reference 90. 

NTU (Number of Transfer Units) The NTU required for a given separation is closely related to the number of theoretical stages or plates required to cariy out the same separation in a stagewise or plate-type apparatus. For equimolal counterdiffusion, such as in a binary distillatiou, the number of overall gas-phase transfer units Nqg required for changing the composition of the vapor stream from yi to yo is... 

FIG. 13 50 Tridiagonal -matrix equation for a column with five theoretical stages, (a) Original equation, (h) After forward elimination. 

Example 5 Calculation of the SR Method Use the SR method with the PR equation of state for K values and enthalpy departures. The oil was taken as n-dodecane. To compute stage temperatures and interstage vapor and hquid flow rates and compositions for ahsorher-column specifications shown in Fig. 13-52. Note that a secondary ahsorher oil is used in addition to the main ahsorher oil and that heat is withdrawn from the seventh theoretical stage. 

The variable that has the most significant impact on the economics of an extractive distillation is the solvent-to-feed (S/F) ratio. For closeboiling 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 volatihty is largely determined by the solvent concentration 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 equihbrium stages. As an illustration, the total number of theoretical stages required as a function of S/F ratio is plotted in Fig. 13-75 7 for the separation of the nonazeotropic mixture of vinyl acetate and ethyl acetate using phenol as the solvent. 

FIG. 13-75 Number of theoretical stages versus solvent-to-feed ratio for extractive distillation, a) Close-boiling vinyl acetate-etbyl acetate system with phenol solvent, (h) A2eotropic acetone-methanol system with water solvent. 

Example 10 Calculation of Multicomponent Batch Distillation A charge of 45.4 kg mol (100 Ih-mol) of 25 mole percent heuzeue, 50 mole percent monochlorohenzene (MCB), and 25 mole percent orthodichloro-henzene (DCB) is to he distilled in a hatch still consisting of a rehoiler, a column containing 10 theoretical stages, a total condenser, a reflux drum, and a distillate accumulator. Condenser-reflux drum and tray holdups are 0.0056 and... 

Temperature-sensitive mixtures are to be separated. To avoid decomposition and/or polymerization, vacuum operation may then be necessary. The smaller liqmd holdup and pressure drop theoretical stage of a packed column may be particularly desirable. 

Design data for separation of the particular or similar mixture in a packea column are not available. Design procedures are better estabhshed for tray-type columns than for packed columns. This is particularly so with respect to separation efficiency since tray efficiency can be estimated more accurately than packed height equivalent to a theoretical stage (HETP). 

Computation of Tower Height The required height of a gas-absorption or stripping tower depends on (1) the phase equilibria involved, (2) the specified degree of removal of the solute from the gas, and (3) the mass-transfer efficiency of the apparatus. These same considerations apply both to plate towers and to packed towers. Items 1 and 2 dictate the required number of theoretic stages (plate tower) or transfer units (packed tower). Item 3 is derived from the tray efficiency and spacing (plate tower) or from the height of one transfer unit (packed tower). Solute-removal specifications normally are derived from economic considerations. 

Although Eq. (14-31) is convenient for computing the composition of the exit gas as a function of the number of theoretical stages, an alternative equation derived by Colburn [Tran.s. Am. Jn.st. Chem. Eng., 35, 211 (1939)] is more useful when the number of theoretical plates is the unknown ... 

In concentrated wstems the change in gas aud liquid flow rates within the tower and the heat effects accompanying the absorption of all the components must be considered. A trial-aud-error calculation from one theoretical stage to the next usually is required if accurate results are to be obtained, aud in such cases calculation procedures similar to those described in Sec. 13 normally are employed. A computer procedure for multicomponent adiabatic absorber design has been described by Feiutnch aud Treybal [Jnd. Eng. Chem. Process Des. Dev., 17, 505 (1978)]. Also see Holland, Fundamentals and Modeling of Separation Processes, Prentice Hall, Englewood Cliffs, N.J., 1975. 

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