Stage feed

Thus far, the McCabe-Thiele construction has not considered the feed to the column. In determining the operating lines for the rectifying and stripping sections, it is very important to note that although xD and x can be selected independently, R and VB are related by the feed phase condition. 

Consider the section of the column at the tray where the feed is introduced. The quantities of the liquid and vapor streams change abruptly at this tray since the feed may consist of liquid, vapor, or a mixture of both (fraction vaporized = VjJF). If, for example, the feed is a saturated liquid, Lst will exceed L by the amount of the added feed liquid. To establish a general relationship, an overall material balance around the feed plate is 

The vapors and liquids inside the tower are all saturated, and the molal enthalpies of all saturated vapors at this section are esentially identical since the temperature and composition changes over one tray are small. Therefore, HG f+x = Hc j. The same is 

Since there are only small composition changes across the feed plate, Hc j. is basically the molal enthalpy that the feed would have if it were a saturated vapor while HUf is basically the molal enthalpy that the feed would have if it were a saturated 

The feed may be introduced under five different thermal conditions, ranging from a liquid well below its bubble point to a superheated vapor. For each condition, the value of q will be different, as the next table shows  

A component material balance on component 1 on any part of the stripping section that includes stages j through N (for j greater than f, the feed stage) gives 

The feed stream, F, with composition Xp enters the column on stage/ mixes with the vapor and liquid entering that stage from adjacent stages, and reaches equilibrium. A material balance on component 1 on the stage/gives 

An energy balance on the feed stage determines the relative flow rates of liquid 

Equation 5.16 is combined with an overall material balance on the feed stage, 

If the feed is a saturated vapor at 7 and Fy, then ATI = 0 and q = 0. The vapor rate leaving the feed tray is simply the sum of vapor from the bottom column section and the feed, 1/ = 1/ -i- F. Since no external liquid is involved, the liquid rate entering the tray from the upper section equals the liquid rate leaving it L,. = L,.  

OLEj Tangential operating line, enriching section, from A out through the equilibrium curve OLS Operating line, stripping section 

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Milk consists of 85—89% water and 11—15% total soflds (Table 1) the latter comprises soflds-not-fat (SNF) and fat. Milk having a higher fat content also has higher SNF, with an increase of 0.4% SNF for each 1% fat increase. The principal components of SNF are protein, lactose, and minerals (ash). The fat content and other constituents of the milk vary with the animal species, and the composition of milk varies with feed, stage of lactation, health of the animal, location of withdrawal from the udder, and seasonal and environmental conditions. The nonfat soflds, fat soflds, and moisture relationships are well estabhshed and can be used as a basis for detecting adulteration with water (qv). Physical properties of milk are given in Table 2. 

The primes denote the streams below the stage to which the feed is introduced. The is a measure of the thermal condition of the feed and represents the moles of saturated hquid formed in the feed stage... 

FIG. 13-31 Typical intersection of the two operating lines at the q line for a feed stage. The q line shown is for a partially flashed feed. 

Pressures can be specified at any level below the safe working pressure of the column. The condenser pressure will be set at 275.8 kPa (40 psia), and all pressure drops within the column will be neglected. The eqnihbrinm curve in Fig. 13-35 represents data at that pressure. AU heat leaks will be assumed to be zero. The feed composition is 40 mole percent of the more volatile component 1, and the feed rate is 0.126 (kg-mol)/s [1000 (lb-mol)/h] of saturated liquid (q = 1). The feed-stage location is fixed at stage 4 and the total number of stages at eight. 

The solution starts with an assumed arbitrary split of all the components to give estimates of top and bottom compositions that can be used to get initial end temperatures. The (Xj s evaluated at these temperatures are averaged with an assumed feed-stage temperature (assumed to be the bubble point of the feed) by using Eq. (13-36). The initial assumption for the split on i-Cs will be Dxp/Bxb = 3.15/16.85. As mentioned earlier, N usually ranges from 0.4N to 0.6N, and the initial value assumed here will be (0.6)(10) = 6.0. Equation (13-32) can be rewritten as... 

All stage-to-stage methods that work from both ends of the column toward the middle suffer from two other disadvantages. First, the top-down and the bottom-up calculations must me somewhere in the column. Usually the mesh is made at a feed stage, and if more than one feed stage exists, a choice of mesh point must be made for each component. When the components vary widely in volatility, the same mesh point cannot be used for all components if serious numerical difficulties are to be avoided. Second, arbitrary procedures must be set up to handle nondlstrihuted components. (A nondistributed component is one whose concentration in one of the end-product streams is smaller than the smallest number carried by the computer.) In the LM and TG equations, the concentrations for these components do not natur ly take on nonzero values at the proper point as the calculations proceed through the column. 

Stage compositions in the TG method are obtained by stage-to-stage calculations from both ends toward the feed stage. With reference to Fig. 13-1, the calculations work with the ratios v /d, Jd, v /h, and instead of v or f directly. 

A knowledge of the reflux ratio (obtained from the specified distillate and top vapor rates) permits the calculation of (f d) from which ( i/d) is obtained, etc. Equation (13-50) is applied to each stage in succession until the ratio 2/d in the overflow from the stage above the feed stage is obtained. The calculations are then switched to the stripping section. 

Tbe values all are fixed by assumed temperature and pbase-rate profiles. Equation (13-55) is applied to eacb of tbe stripping stages in sequence until tbe ratio in tbe liquid entering tbe feed stage is obtained. 

FIG. 13-45 Effect of feed on stream rates just above feed stage M + 1. (a) Subcooled or bubble-point feed, (h) Superheated or dew-point feed, (c) Partially flashed feed. 

It should be noted in Table 13-13 that it is not necessary to list two values of V, L, and T for the feed stage (stage 6) because the TC procedure gives a perfect match at the feed stage in each trial. This completes the first column iteration. 

Figure 13-47 shows the concentration profiles from the final solution. Note the discontinuities at the feed stage and the fact that feed-stage composition differs considerably from feed-stream composition. It can be seen in Fig. 13-47 from the n-C4 and i-C profiles that the separation between the keys improves rapidly with stage number additional stages would be worthwhile. 

One of the components, A (not necessarily the most volatile species of the original mixture), is withdrawn as an essentially pure distillate stream. Because the solvent is nonvolatile, at most a few stages above the solvent-feed stage are sufficient to rectify the solvent from the distillate. The bottoms product, consisting of B and the solvent, is sent to the recoveiy column. The distillate from the recoveiy column is pure B, and the solvent-bottoms product is recycled back to the extractive column. 

In case B the solvents are partially miscible, and the miscibihty is nearly constant through the extractor. This frequently occurs when all solute concentrations are relatively low. The feed stream is assumed to dissolve extraction solvent only in the feed stage and to retain the same amount throughout the extractor. Likewise, the extraction solvent is assumed to dissolve feed solvent only in the raffinate stage. With these assumptions the primary extraction-solvent rate moving through the extractor is assumed to be S, and the primary feed-... 

L = Effective total molar liquid rate in top section L - Effective total molar liquid rate in bottom section M = Total equilibrium stages below tbe feed stage including reboiler... 

M = Total equilibrium stages below feed stage ineluding reboiler... 

L = Effective liquid molar rate in column top section L = Effective liquid molar rate in column bottom section M = Total equilibrium stages below the feed stage N = Total equilibrium stages including reboiler and partial condenser... 

Maas,J. H., Optimum-Feed-Stage Location in Multicomponent Distillations, Chem. Eng, Apr. 16, (1973) p. 96. 

Now it is necessary to determine how the vapor and liquid flowrates change at the feed stage. 

What happens at the feed stage depends on the condition of the feed, whether it is subcooled, saturated liquid, partially vaporized, saturated vapor or superheated vapor. To define the condition of the feed, the variable q is introduced, defined as ... 

For a saturated liquid feed q = 1, and for a saturated vapor feed <7=0. The flowrate of feed entering the column as liquid is q-F. The flowrate of feed entering the column as vapor is (1 - q)-F. Figure 9.8 shows the feed stage. An overall mass balance on the feed stage for the vapor gives ... 

An overall mass balance for the liquid on the feed stage gives ... 

Figure 9.8 Mass balance for the feed stage. (From Smith R and Jobson M, 2000, Distillation, Encyclopedia of Separation Science, Academic Press reproduced by permission).

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