is assumed where all of the overhead vapor is liquefied, this condensate being divided into two portions, one of which is returned to the column for reffyrc and the other withdrawn as overhead product. The bottoms are of!itinuously withdrawn from the still or reboiler. (See end of chapter for nomenclature.)
Consider the region bounded by the dotted line in Fig. 7-1., The only material entering this section is the vapor from the nth plate F„, while leaving the section is the distillate D and the overflow from the (n + l)th plate 0n+i. By material balance,
Considering only the more volatile component, the mols entering this section are the total mols of vapor from the nth plate multiplied by the mol fraction of the more volatile component in this vapor VnVn. Likewise, the mols of the more volatile component in the distillate are Dxd; and in the overflow from the (n + l)th plate are On+iXn+i. A material balance on the more volatile component for this section therefore gives
Fig. 7-1.. Diagram of continuous distills tion column.
Fig. 7-1.. Diagram of continuous distills tion column.
Thus, starting at the condenser, the composition of the reflux to the tower, with the type of condenser employed, is the same as the composition of the distillate, which makes the composition of the vapor to the condenser the same as that of the distillate. The mols of vapor from the top plate are equal to Ob + Z>, and the reflux to this plate is Or, Sorel's assumption of the vapor and liquid leaving the plate being in equilibrium, called a theoretical plate, makes it possible to calculate the composition of the liquid leaving the top plate of the tower from the composition of the overhead vapor and vapor-liquid equilibrium data.
In the design of such a tower, it is generally customary to set or fix certain operating variables such as the composition of the distillate antl of the bottoms, the reflux ratio Or/D, and the composition and thermaFcondition of the feed. With these values and a known quantity of feed per unit time, by over-all material balances it is possible
from the plate below the top plate by Eq. (7-2a), it is necessary to know the mols of overflow from the top plate and the mols of vapor from the plate below as well as the known quantities D, xD, and the mol fraction of the more volatile component in the liquid overflow from the top plate. Sorel obtained the mols of overflow from, and the mols of vapor to, the plate by heat (enthalpy) balances. Thus the heat brought into the plate must equal that leaving.
Equation (7-3) gives a relation between Vt-i and Ot; if the enthalpies are known, this equation can be solved simultaneously with Eq. (7-1) to give Vt~i and Ot. This, in general, involves trial-and-error solutions, since i?*-! is not known and must be assumed until the conditions of the next plate are known. The values of V^i and Ot obtained in this manner are used in Eq. (7-2a) to give yt~i. From this compo sition the value of xt~i is obtained from vapor-liquid equilibrium data as well as the temperature on this plate. The value of Ht~i can then be accurately checked, and the calculations corrected if necessary. This operation is continued plate by plate down the tower to the feed plate.
A similar derivation for the plates below the feed gives
These equations are used in the same way as Eq. (7-2).
Because of the complexity of Sorel's method, it is usually modified by certain simplifying assumptions. The heat supply to any section of the column above the feed is solely that of the vapor entering that section. This supply of heat to the next plate goes to supply vapor from this plate, to heat loss from the section of the tower that corresponds to this plate, and to heating up the liquid overflow across this plate. In a properly designed column, the heat loss from the column should be reduced as far as is practicable and is generally small enough to be a negligible quantity relative to the total quantity of heat flowing up the column. Thus the enthalpy of the vapor per unit time tends to be constant from plate to plate, and in order to simplify the calculations for such systems, Lewis (Ref. 9) assumed that the molal vapor rate from plate to plate was constant except as changed by additions or withdrawals of material from the column. This assumption also leads to a constant overflow rate for such a section. In the case illustrated in Fig. 7-1, this simplifying assumption would give constant vapor and overflow rates above and below the feed plate, but the rates in the two sections of the tower would be different due to. the introduction of the feed. This assumption, together with the theoretical plate concept, has been of great assistance in the analysis and design of fractionating column. The validity of these two assumptions will be considered in later sections.
On the basis of Lewis' assumption On+1 and Vn are constant in the section above the feed plate, and the relation between yn and xn+\ becomes a straight line with the slope equal to O/V. Similarly, below the feed, ym is linear in xm+i. On the basis of the operating variables previously fixed, On+i, Fn, D, and xD are known, and the equation between yn and xn+i is completely defined; likewise for ym and xm+\.
A plate on which Sorel's conditions of equilibrium are attained is defined as a "theoretical plate/' i.e., a plate on which the contact between vapor and liquid is sufficiently good so that ffie vapor leaving the plate has the same composition as the vapor in equilibrium with the overflow from the plate. For such~a plate the vapor and liquid leaving are related by the equilibrium y,x curve (see page 18). Rectifying columns designed on this basis serve as a standard for comparing actual columns. By such comparisons it is possible to determine the number of actual plates equivalent to a theoretical plate and then to reapply this factor when designing other columns for similar service.
Sorel-Lewis Method. As an illustration of the Sorel-Lewis method, consider the rectification of a 50 mol per cent benzene and 50 mol per cent toluene mixture into a product containing 5 mol per cent toluene and a bottoms containing 5 mol per cent benzene. The feed will enter as a liquid sufficiently preheated so that its introduction into the column does not affect the total mols of vapor passing the feed plate; i.e., such that F» ■» Fm. A reflux ratio On/D, equal to 3, will be employed, and the column will operate with a total condenser and indirect heat in the still. The y,x equilibrium curve is given in Fig. 7-2.
Taking as a basis 100 lb. mols of feed mixture, an over-all benzene material balance on the column gives gives Since
Vn - (15%oo)*«+i + (5%oo)(0.95) - 0.75zn+1 + 0.2375 (7-5)
Since a total condenser is used, yt ~ xd — xr ~ 0.95
from the equilibrium curve at IT" 0-95, x - 0.88; i.e., x% in equilibrium with yt is 0.88.
Equation (7-5) then gives yM - 0.75®* + 0.2375 - 0.75(0.88) + 0.2375 - 0.8975
by equilibrium curve, Xt~i at yt-i 0.8975 is 0.77 and yt.* - 0.75(0.77) + 0.2375 » 0.8145 xt~* » 0.64
yt-. - 0.753,-2 + 0.2375 - 0.75(0.64) + 0.2375 - 0.7165 « 0.505
Since the valueof xt-s is close to the composition of the feed, this plate will be taken as"" the feedl>I^ portion of the tower must be used. Since the feed was preheated such that
?//_! « 1.25(0.505) - 0.0125 « 0.615 Xf-\ » 0.392 from equilibrium curve
The desired strength of the bottoms was xw « 0.05; is too high, and is too low. Thus it is impossible to satisfy the conditions chosen and introduce the feed on the fourth plate from the top with an even number of theoretical plates. Hdwever, by slightly reducing the reflux ratio it would be possible to make x/-t equal to xw, or by increasing the reflux ratio to make x/-& equal to xw* In general, such refinements are not necessary, and it is sufficient to say that between eight and nine theoretical plates are required in addition to the still, three plates above the feed plate, the feed plate, and four or five plates below the feed, and the still, approximately 8M- The percentage difference between eight and nine is much less than the accuracy with which the ratio of actual to theoretical plates is known; whichever is used, a sufficient factor of safety must be utilized to cover the variation of this latter factor.
McCabe and Thiele Method (Ref. 11). By the Sorel-Lewis method, the relation between yn and x»+1 is a straight line, and the equation of this line may be plotted on the y,x diagram. Thus, for the example worked in the preceding section, yn = 0.75#n+i + 0.2375
This is a straight line of slope 0.75 « On/Vn which crosses the y « x diagonal at yn » zn+i - 0.95 « x». On the y,z diagram for benzene-toluene, a line of slope 0.75 is drawn through y « x « xD (see line
This represents a straight line of slope 0m/Vm = 1.25 and passe* through the y = x diagonal at x = xw = 0.05 (see line CD, Fig. 7-3)
This represents a straight line of slope 0m/Vm = 1.25 and passe* through the y = x diagonal at x = xw = 0.05 (see line CD, Fig. 7-3)
mined by the tower operating conditions, AB being the operating line for the enriching section and CD the operating line for the stripping, or exhausting, section. To determine the number of theoretical plates by l?ig. 7-3, start at xD; as before, yt = xD == 0.95, and the value of xt is determined by the intersection of a horizontal line through yt = 0.95 with the equilibrium curve at 1, giving xt = 0.88. Now, instead of using Eq. (7-2a) algebraically as in the Sorel-Lewis method, it is used graphically as the line AB. A vertical line at xt = 0.88 intersects this operating line at 2, giving « 0.89. By proceeding horizontally from intersection 2, an intersection is obtained with the equilibrium curve at 3. Since the ordinate of intersection 3 is yt-h the abscissa must be the composition of the liquid in equilibrium with this vapor; i.e., xt-i = 0.77. As before, the intersection of the vertical line through the point 3 with the operating line at 4 gives the y on the plate below, or yt~-i = 0.815. This stepwise procedure is carried down the tower. At intersection 8, xt-s is approximately equal to xF; and at this plate, the feed will be introduced. The stepwise method is now continued, using the equilibrium curve and the operating line CD.
Such a stepwise procedure must yield the same answer as the previous calculations, since it is the exact graphical solution of the algebraic equation previously used. It has a number of advantages over the latter method: (1) It allows the effect of changes in equilibrium and operating conditions to be visualized. (2) Limiting operating condi-tiwlS~afe easily determined, and if a column contains more than two ox thTee~plafes, it is generally more rapid than the corresponding algebraic procedure. Because of the importance of this diagram it will now be considered in further detail.
Intersection of Operating Lines. In Fig. 7-3, the operating lines intersected at x = xF. This intersection is not fortuitous, since the positions of the two operating lines are not independent but are related to McF^therlby the composition and thermal condition of the feed. This~~relation is irtff&TeasitylAowri by writing a Heat balance around the feed plate. Let p be the difference between the mols of overflow to and from the feed plate divided by the mols of feed.
Let Xi and yi be the coordinates of the intersection of the operating lines. At this intersection, yn must equal ym, and xn must equal xm. An over-all material balance on the more volatile component gives Dxd + Wxw = Fx*, where zF is the average mol fraction of this component in the feed. Writing Eqs. (7-2a) and (7-4) for the intersection and using the values y% and x%,
and subtracting,
- (0,+i - of)xi + fzf (Vf - Vf-i)yi _ (O/4.1 - Of\ F ~ V ^ /
Substituting values of p and p + 1 gives the point on the diagram at which the intersection must occur.
Equation (7-7) together with Eq. (7-26) gives
This line of intersections crosses the y = x diagonal at
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