As compared to the intersection ratio of 5 88 2.33. In the stepwise calculation, the ratio of the key components in the feed plate was 2.08 and 2.31 on the plate above. Thus the feed-plate composition utilized agrees with Eq. (9-12) satisfactorily.
In the foregoing derivations the feed could be vapor, liquid, or a mixture of the two, but it was assumed that the vapor and liquid leaving the feed plate were in equilibrium. In the case of an all-vapor feed that mixes with the vapor from the feed plate but does not react with the liquid on the plate, a similar derivation gives (Ref. 1)
This indicates the ratios of the concentrations of the key components on the plate above the feed plate, and the plate above that should straddle the intersection ratio for the key components. Or, if the nomenclature is changed such that the feed plate is the first plate with which this feed vapor reacts (i.e., the plate above) and not the plate into which it enters, then the criterion becomes the same as before.
Minimum Reflux Ratio. As in the case of binary mixtures, there is a reflux ratio below which it is not possible to obtain the desired separation of a multicomponent mixture even when an infinite number of plates is used. The calculation of this minimum for a multicomponent mixture is much more involved than for the corresponding binary mixture. The condition of the minimum reflux ratio requires that to perform the given separation an infinite number of plates must be needed, which means that there must be a pinched-in region where there are a large number of plates having the same composition, but for multicomponent mixtures of normal volatility this region usually does not occur at the feed plate as it does in the case of binary mixtures. Under this condition usually a relatively few plates above the feed serve to reduce the concentrations of the components less volatile than the heavy key component to negligible values, and then a true pinched-in condition does occur with only the heavy key and more volatile components present. Likewise, below the feed plate a relatively few plates reduce the concentrations of the components more volatile than the light key component to negligible values, and a true pinched-in condition occurs with only the light key and less volatile components present. Thus the tower operating at the minimum reflux ratio might be considered as being composed of five sections (Ref. 2): * v
(1) Starting at the still the only Aruvinnnents present in significant amounts are the light key and the less volatile components, and in proceeding up the tower in this bottom section, the concentration of the light key component increases relative to the concentrations of the heavy key and heavier components.
2. Above section 1 is a pinched-in region where the concentrations of the light key and the less volatile components are all con^iant, and an infinite number of plates is required to produce a finite change in the plate composition.
3. Next there is an intermediate region where the concentrations of the components less volatile than the heavy key component decrease to negligible values and where the concentrations of the components more volatile than the light key component increase to significant values.
4. Above section 3 is another pinched-in region where the concentrations of the heavy key and the more volatile components are all constant, and an infinite number of the plates is required to produce a finite change in plate composition.
5. A section exists where the concentration of the heavy key component decreases relative to the concentration of the more volatile components until the overhead composition is obtained.
Actually there is no sharp line of demarcation between these five sections, but this division serves as a useful picture for considering the case of the minimum reflux ratio. The feed to the fractionating column would be introduced on some plate in intermediate section 3, and the true criterion for the minimum reflux ratio should be based on matching the ratio of the concentrations of the key components above and below the feed plate under conditions such that a pinched-in section occurs both above and below the feed plate. For mixtures of normal volatility, a pinched-in region in only one section does not necessarily mean that an infinite number of plates would be required to perform the desired separation at the reflux ratio under consideration, since by relocating the feed plate, such as to shift the ratio of the concentrations of the key components at this plate, the section that was not limited could be made to do more separation and thereby relieve the load on the pinched-in section. In other words, for mixtures of normal volatilities the condition of the minimum reflux ratio is not determined by either the fractionation above or below the feed plate alone, but is determined such that the separation is limited both above and below the feed.1 The conditions in the intermediate feed section lead to
1 For mixtures with abnormal volatilities the pinched-in condition may be due to calculational difficulties because the concentrations are not constant in this region. Actually in this region the ratios of the concentrations of the key components generally change in the opposite direction from that desired for the separation; thus, proceeding up the column from the feed plate at the minimum reflux ratio, the ratio of the concentrations of the light key to the heavy key component decreases instead of increases. No satisfactory method of estimating the extent of this "retrograde" rectification has been developed. However, it is relatively simple to calculate the plate composition for a region where the concentrations are the same on successive plates, and such calculations can be made the basis for estimating the minimum reflux ratio.
Case I. It can be assumed that the concentrations of all components are constant for a number of plates above and below the feed plate. This requires that the pinched-in condition between the key components must occur with all the components present in significant amounts; actually, as pointed out before, only certain of the components are present at the pinched-in condition. The presence of these extra components makes the separation more difficult, and for that reason the minimum reflux ratio calculated on the basis of the assumptions of Case I will be equal to or greater than the true minimum reflux ratio.
Case II. Alternately it can be assumed that the ratio of the concentrations of the key components for the pinched-in region below the feed plate (section the same ratio for the pinched-in region above the feed plate (section 4), which amounts to neglecting tfieln^Brme^at^^^regon^section 3). However, for the actual operation sectiOn"Sn^usTBepresent and in this section the ratio of the concentrations of the key components changes in the opposite direction to that desired for the separation; for that reason, the assumptions of Case II correspond to an easier separation than the actual case, and the minimum reflux ratio calculated by these assumptions will be equal to or less than the true minimum reflux ratio.
Calculation of the minimum reflux ratio for these two cases will give limits for the true minimum reflux ratio. To evaluate these cases, it is necessary to calculate the concentrations of the various components for the pinch<gd-in or constant composition regions. Such compositions are easily c^cHated'By the use^of the reTaHve"volatility, a, and the fact that the ratio of the concentrations of any two components are a tangent contact between the equilibrium curve and the operating line, and the separation will not be limited both above and below the feed plate.
the same on successive plates; thus for the volatile components above the feed plate,
oihk Xhh ** Vhk (0/D)xhk + xDhk othk L (V/D)yhk — ,xDKk\ giving
and, for less volatile components below the feed plate, CXlfcSjfc ^iB ^ (°m/W)Xik — Xwlk _ ^[(Vw/WQyifc +
In these equations the last term of the denominator is usually small relative to the other factors and can be neglected for the first estimations, making only the flow quantities, relative volatilities, and terminal concentrations necessary for the calculation of the asymptotic values. Corrections can then be made for the last term in the denominators, but usually this is not necessary.
These equations can be used to calculate the concentrations of the key components for evaluating the minimum reflux ratios for Cases I and II, which would involve matching the ratio of the concentrations of the key components above the feed plate with the same ratio below the feed plate. This leads to a quadratic solution for the minimum reflux ratio. Thus for Case I the ratio of the key components above and below the feed plate are equated, and allowance is made for all components at their asymptotic values in both sections.
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