Multicomponent distillation equation

Open-loop behavior of multicomponent distillation may be studied by solving modifications of the multicomponent equations of Distefano [Am. Inst. Chem. Eng. J., 14, 190 (1968)] as presented in the subsection Batch Distillation. One frequent modification is to include an equation, such as the Francis weir formula, to relate liquid holdup on a tray to liquid flow rate leaving the tray. Applications to azeotropic-distillation towers are particularly interesting because, as discussed by and ihustrated in the Following example from Prokopalds and Seider... 

Yaws [124] et al. provide an estimating technique for recovery of each component in the distillate and bottoms from multicomponent distillation using short-cut equations and involving the specification of the recovery of each component in the distillate, the recovery of the heavy key component in the bottoms, and the relative volatility of the light key component. The results compare very well with plate-to-plate calculations. Figure 8-46, for a wide range of recoveries of 0.05 to 99.93% in the distillate. 

The complexity of multicomponent distillation calculations can be appreciated by considering a typical problem. The normal procedure is to solve the MESH equations (Section 11.3.1) stage-by-stage, from the top and bottom of the column toward the feed point. For such a calculation to be exact, the compositions obtained from both the bottom-up and top-down calculations must mesh at the feed point and match the feed composition. But the calculated compositions will depend on the compositions assumed for the top and bottom products at the commencement of the calculations. Though it is possible to... 

Colburn (1941) and Underwood (1948) have derived equations for estimating the minimum reflux ratio for multicomponent distillations. These equations are discussed in Volume 2, Chapter 11. As the Underwood equation is more widely used it is presented in this section. The equation can be stated in the form ... 

This relation enables the composition of the vapour to be calculated for any desired value of x, if a is known. For separation to be achieved, a must not equal 1 and, considering the more volatile component, as a increases above unity, y increases and the separation becomes much easier. Equation 11.14 is useful in the calculation of plate enrichment and finds wide application in multicomponent distillation. 

Even if you were only half awake when you read the preeeding chapter, you should have recognized that the equations developed in the examples eonstituted parts of mathematical models. This chapter is devoted to more complete examples. We will start with simple systems and progress to more realistic and complex processes. The most complex example will be a nonideal, nonequimolal-overflow, multicomponent distillation column with a veiy large number of equations needed for a rigorous description of the system. 

The model equations for multicomponent distillation under constant boilup rate (Robinson, 1969) are ... 

In multicomponent distillation otj components, there are j - 1 component balances and j - 1 equations describing the equilibrium relationship. 

While Eq, (2) has been found satisfactory to employ for many cases of multicomponent distillation, use this equation with caution, as it may produce high values of the minimum reflux ratio when the composition of the light key in the distillate is not predominant. 

Equations 7.3.11 and 7.3.14 are used in the development of expressions for modeling mass transfer in multicomponent distillation, a topic we consider in Chapter 12. The addition of resistances concept has seen use in distillation models by Krishna et al. (1981a), Burghardt et al. (1983, 1984) and by Gorak and Vogelpohl (1985). 

The set of differential and algebraic equations given above for modeling multicomponent distillation in a packed column must be integrated numerically in general. The complexity and nonlinearity of the above equations precludes analytical solution in most cases of practical importance. Moreover, because the vapor and liquid streams flow in opposite directions means that, in all but one circumstance—total reflux—several integrations may be required in order to properly solve the equations. An alternative method of solving approximate forms of these equations is discussed in Chapter 14. 

The next task is to set up the equations that model a complete distillation column. As noted in the introduction to this chapter, simulation of multicomponent distillation operations usually is carried out using the equilibrium stage model introduced below. 

Many equations have been proposed to describe Gilliland s curve for multicomponent distillation. However, the difficulty with some of these equations has been in meeting the end conditions of X = 0, Y = 1 and X = 1, Y = 0. A review of the many equations proposed by these authors is as follows ... 

BUBBLE-POINT AND DEW-POINT CALCULATION. Determination of the bubble point (initial boiling point of a liquid mixture) or the dew point (initial condensation temperature) is required for a flash-distillation calculation and for each stage of a multicomponent distillation. The basic equations are, for the bubble point,... 

Use the Underwood equations to determine the minimum reflux ratio for multicomponent distillation. 

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