The corrected distillate and bottoms component product rates are

where Pi is a corrector created by combining the overall component balance [Eq. (4.45)] with the definition of 6 [Eq. (4.46)]:

In the theta method, the total distillate rate and reflux ratio are all that can be specified and the theta function [Eq. (4.50)] is written to find a value of theta such that the sum of the corrected component distillate rates equals the total distillate rate:

For complex columns, an additional theta multiplier is defined for each side product W. The p, correction factor is expanded to include in the denominator a term for each additional product and the total flow for each side product is specified. The final values of all of the 0's will be unity.

To continue with the column trial, the p, factors are used to correct the stage-to-stage component flow rates:

and the compositions

These new compositions are used to find a new set of stage temperatures.

The Kh method. For updating the tray temperatures, the theta method relies on the Kb method. The Kb method takes advantage of the near-linear dependence of the logarithm of the if-values and the relative volatilities on temperature over short temperature spans. Relative volatilities (a^ *'s) are calculated with respect to a base component it-value, Kbhk, at the stage temperature of the current column trial, Tj k. The base component is usually a middle boiler or a hypothetical component. The ii-value of the base component for the next trial, Kbjk + is calculated using a form of the bubble-point equation unique to the Kb method:

where aSJ = Klj k!Kbj k. The stage temperature for the next trial (k + 1) is calculated from

Aj k and Bj k are the constants for a pseudo-Clausius-Clapeyron equation for the base component iT-value. These constants are unique for every stage and are updated at eveiy trial in the calculation. They are calculated as follows. Using the if-value method chosen for the simu lation, two K-values are calculated for the base component at the corrected compositions, (xy )co and (>'y)co, one at temperature TJk + ATk and the other at Tj k - ATa. Aj k and Bj k are then calculated using Eq, 4.54). The ATk should be no more than 25 to 50T for the first trial and reduced in size for succeeding trials (as the final temperature profile is approached). Billingsley (30), Boston and Sullivan (69), Lo (31), and Jelinek (73) have proposed modifications to the Kb method.

The constant-composition method. Since in the theta method temperatures are found using the equilibrium and summation equations, the next set of stage-to-stage total flow rates is calculated through the energy balances. The liquid rate leaving a stage, Lr can be found by rearranging the energy balance for that stage. The vapor entering from the stage below, Vj + 1; is still unknown but can be eliminated using the component balances for the stage. The liquid rate is found by

Because of the rearrangement, the constant-composition method requires calculating an enthalpy of a phase at the temperature of one stage but using the composition of a different phase and stage. As stated by Holland (8), this "may be represented by a thermodynamic process which occurs at constant composition." The enthalpy per mole terms above are specifically:

H(Tj y j y ¡J = vapor enthalpy at T^ a and vapor composition yir H{Tjd>ij) = vapor enthalpy at T, and vapor composition^.

HiTj+ijCy.i) - vapor enthalpy at TJ + 1 and liquid composition x^.i. h= liquid enthalpy at Tj.1 and liquid composition^.!.

H(TJ + lrx„) = vapor enthalpy at T, +1 and liquid composition jrv. h(TjrxtJ) = liquid enthalpy at 7} and liquid composition

The total vapor rate entering from the stage below is found from the total material balance:

The total vapor entering the condenser is calculated using the reflux ratio:

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