Theoretical trays

The relationship of this type of model to a tme differential analysis has been discussed for the case of linear equiHbrium and first-order kinetics (74,75). A minor extension of this work leads to the foUowing relations for a bed section in which dow rates of soHd and Hquid are constant. For the number of theoretical trays,... 

Fig. 7. Extractive distillation column profiles for the acetone—methanol—water separation (40). (a) Liquid composition versus theoretical tray location where...

For a general Jacobian matiix pertaining to C components and N theoretical trays, as shown by Distefano [Am. Jnst. Chem. Eng. J., 14, 946 (1968)]. Gerschgorin s circle theorem (Varga, Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, N.J., 1962) may be employed to obtain bounds on the maximum and minimum absolute eigenvalues. Accord-... 

A hydrocarbon feed gas is to he treated in an existing foiir-theoretical-tray ahsorher to remove hiitane and heavier components. The recovery specification for the key component, hiitane, is 75 percent. The composition of the exit gas from the ahsorher and the required liqiiid-to-gas ratio are to he estimated. The feed-gas composition and the eqiiilihriiim K values for each component at the temperature of the (soliite-free) lean oil are presented in the following table ... 

Colburn relationship found that the optimum number of trays varies from 2 to 3 times the number at total reflux. Gilliland [Ind. Eng. Chem, . 32, 1220 (1940)] from the establishment of an empirical relationship between reflux ratio and theoretical trays based on a study of existing columns indicated that... 

After actual theoretical trays are determined (see Actual reflux and theoretical stages) one needs to estimate the actual physical number of trays required in the distillation column. This is usually done by dividing the actual theoretical trays by the overall average fractional tray efficiency. Then a few extra trays are normally added for offload conditions, such as a change in feed composition. 

Vertical once-through One theoretical tray Simple piping and compact Not easily fouled Less cost than kettle Difficult maintenance High skirt height No control of circulation Moderate controllability... 

Horizontal natl. circ. Ease of maintenance Lower skirt height than vertical Less pressure drop than vertical Longer tubes possible Less cost than kettle No theoretical tray Extra space and piping as compared to vertical Fouls easier as compared to vertical Accumulation of higher boiling point components in feed line, i.e., temperature may be slightly higher than tower bottom... 

Forced circ. One theoretical tray Handles high viscous solids-containing liquids Circulation controlled Higher transfer coefficient Highest cost with additional piping and pumps Higher operating cost Requires additional plant area... 

Knowing required key component recovery Eg, read Ae from Figure 1 at known theoretical trays n. 

Presaturators. A presaturator to provide lean oil/gas contact prior to feeding the lean oil into the tower can be a good way of getting more out of an older tower. Absorber tray efficiencies ran notoriously low. A presaturator that achieves equilibrium can provide the equivalent of a theoretical tray. This can easily equal 3-4 actual trays. Some modem canned computer distillation/absorp-tion programs provide a presaturator option. 

TEG RATE, GAL/LB WATER REMOVED Figure 8-13. Glycol concentration vs. glycol circulation when n = 1 theoretical tray. 

Assume 8 actual trays or 2 theoretical trays. From Figure 8-17 the glycol circulation rate is 2.8 gal TEGAb H2O. Size for 3.0 galAb. 

Distillation design is based on the theoretical consideration that heat and mass transfer from stage to stage (theoretical) are in equilibrium [225-229]. Actual columns with actual trays are designed by establishing column tray efficiencies, and applying these to the theoretical trays or stages determined by the calculation methods to be presented in later sections. 

This applies to any pair of components. My experience suggests adding +1 theoretical tray for the reboiler, thus making the total theoretical trays perhaps a bit conservative. But, they must be included when converting to actual trays using the selected or calculated tray efficiency ... 

For a condition of overall total trays allowance is to be made for feed tray effect, then add one more theoretical tray to the total. As demonstrated in the tabulation to follow, allowance should be made for the reboiler and condenser. 

Because the feed tray is essentially non-effective it is suggested that an additional theoretical tray be added to allow for this. This can be conveniently solved by the nomographs [21] of Figures 8-16 and 17. If the minimum number of trays in the rectifying section are needed, the)t can be calculated by the Fenske equation substituting the limits of xpi for x jj and x i, and the stripping section can be calculated by difference. 

Sm = total number of calculated theoretical trays at total reflux, from Equation 8-30 X]k = xlk = liquid mol fraction of light key Xhk = xhk = liquid mol fraction of heavy key Ik - hk = LK - HK= average relative volatility of column (top to bottom)... 

The exact procedure is to estimate a temperature profile from top to bottom of the column and then calculate a for each theoretical tray or stage by assuming a temperature increment from tray to tray. For many systems this, or some variation, is recommended to achieve good separation calculations. 

After calculating X, and solving for Y using Equation 8-58, then solve for the theoretical trays, St, at the actual selected reflux ratio (L/D) from the equation for Y. The equation appears to represent several reliable data references. 

Example 8-12 Minimum Theoretical Trays/Plates/Stages at Total Reflux... 

Note that these values for theoretical trays do contain corrections in overall efficiency, and hence are not the actual trays for the binary distillation column. Efficiencies generally run 50-60% for systems of this type which will yield a column of actual trays almost twice the theoretical at the operating reflux. 

Figure 8-28 presents the usual determination of optimum or near optimum theoretical trays at actual reflux based on performance. It is not necessarily the point of least cost for all operating costs, febrication costs or types of trays. A cost study should be made to determine the merits of moving to one side or other of the so-called optimum point From the Figure 8-28 ...

First choice actual reflux ratio, L/D = 1.33 Corresponding theoretical trays or stages, N =... 

Figure 8-28. Relationship of reflux ratio and theoretical trays, for Example 8-12.

Fixed Number Theoretical Trays Constant Reflux Ratio and Variable Overhead Comporations... 

Point F on the figure represents conditions in the kettle or still with Xj, yj, or Xq, yo- Line DF represents slope of the operating line at minimum reflux. The step-wise development from point D cannot cross the intersection, F, where the slope intersects the equilibrium line, and leads to an infinite condition, as point F is approached. Thus, an infinite number of theoretical trays/stages is required, and... 

For a plot of xp = 0.750, slope = 0.7875, read xyy at the equilibrium line for each theoretical tray and plot similar to Figure 8-38. Then determine the area under the curve between the selected x and the product x. Then ... 

The batch distillation of a binary is somewhat simplified, as L/V values can be assumed, and since there is only enrichment of the overhead involved, only one operating line is used per operating condition. Theoretical trays can be stepped off and xib values read to correspond. The plots involved are the same as previously described. 

Figure 8-47. Short-cut solution of Fenske-Underwood-Gilliland theoretical trays for multicomponent distillation. Used by permission, Frank, O., Chem. Eng. Mar. 14 (1977), p. 109.

The minimum theoretical trays at total reflux can be determined by the Fenske relation as previously given... 

F = intermediate feed Theoretical Trays at Operating Reflux... 

The method of Gilliland [23] (Figure S-24) is also used for multicomponent mixtures to determine theoretical trays at a particular operating reflux ratio, or at various ratios. 

S = number theoretical trays in the column, including reboiler... 

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