In many cases there are considerable differences in the compositions of the liquids at various points on a plate, and the conditions assumed in the derivation are not satisfied for the whole plate. However, for convenience, the value of the Murphree plate efficiency is defined as

where yi, y0 = average composition of vapor entering • and leaving plate, respectively ' '

y* » composition of vapor in equilibrium witji liquid flowing to plate below

Equations (17-3) and (17-4) appear similar, but the former uses the values for a small region of the plate, while the latter uses average values for the various streams entering and leaving a plate. E%y is not equal to 1 —

The derivation of Eq. (17-3) should apply to a limited region of the plate, and this has been termed the Murphree point efficiency, E%v. In this case "

Vi - Ve where y\ y0 — vapor compositions entering and leaving the local region y9 = vapor composition in equilibrium with the liquid in the local region ihe Murphree plate efficiency is the integrated effect of all the Murphree point efficiencies on the plate.

The derivation of the Murphree equation is based on a very qualitative picture of the mass transfer. At low vapor rates individual bubbles are obtained, but a study of mass transfer for such systems indicates that it cannot be expressed as a simple rate equation involving a constant multiplied by a driving force in mol fraction units. The experimental data indicate that mass transfer is very rapid while the bubble is being formed and then is relatively slow while the bubble is rising through the liquid. At higher vapor rates channels are blown through the liquid, and a large quantity of .spray is thrown up into the vapor space giving additional mass transfer. The simple rate Eq. (17-1) cannot be any more than a crude expression of the phenomena involved, and KQa must be a complicated function of a large number of variables including Ay and 0. /

The value of E°} E°uvr and E%y are the most commonly used design factors for plate efficiencies. The over-all tower plate efficiency is simpler to apply than the Murphree efficiencies because only terminal conditions are required; whereas in the calculation of the Murphree plate efficiency, plate-to-plate compositions are required, and for the Murphree point efficiency, complete liquid- and vapor-composition traverses are required on each plate. However, the Murphree efficiencies are probably on a more fundamental basis than the over-all efficiencies.

By definition, a theoretical plate is one on which the average composition of the vapor leaving the plate is the equilibrium value for the liquid leaving the plate. If the vapor and liquid upon a plate were completely mixed, it would be impossible to obtain better separation, than that given by a theoretical plate. However, when there is a concentration gradient in the liquid across the plate, the average concentration of the more volatile component in the liquid on the plate may be appreciably greater than the concentration of the liquid leaving the plate; as a result of this greater concentration, the vapor actually leaving the plate may exceed the concentration of the vapor in equilibrium with the liquid leaving. It is thus possible for the concentration-gradient effect to give over-all and Murphree plate efficiencies greater than 100 per cent; but since such gradients do not apply to the Murphree point efficiency, this latter efficiency should never exceed 100 per cent. The theoretical effect of the concentration gradient has been studied by a number of investigators (Refs. 15, 19, 21). Three cases were considered by W. K. Lewis, Jr.: Case I, vapor completely mixed, liquid unmixed; Case II, vapors do not mix, and the overflows are arranged such that the liquid flows in the same direction on all plates; Case III, the vapor rises from plate to plate without mixing, and the liquid flows in the opposite direction on successive plates.

Lewis assumed that (1) E%v was constant over all of the plate, (2) the equilibrium curve is a straight line over the concentration range involved, ye — Kx + b} and (3) the liquid flows across the plate without mixing.

The results of this analysis for the three cases are given in Fig. 17-2 in which the ratio (E°MV/E%V) is plotted as a function of E%v and the ratio of the slope of equilibrium curve, K, to the slope of the operating line, (O/V). The slope of the equilibrium curve, K, should be the average slope, dyjdx, over the concentration region involved. These calculated values indicate that it should be possible to obtain high plate efficiencies by preventing the liquid from mixing. The usual bubble-cap plates probably fall between Cases I and III as far as the vapor is concerned, but they give considerable liquid mixing which would lower the value of (E0MY/E%V) compared to the values given by the plot. Case II has the possibility of giving higher plate efficiencies than the other two cases, but in practice it is difficult to arrange the downflow pipes such that the liquid flows in the same direction on all plates. The circumferential flow plate, Fig. 16-41?, gives essentially this type of flow, but it is not a desirable construction in most cases.

The relationship between the over-all column efficiency and E°MY can be derived in a similar manner and assuming

1. Constant O/V

2. Constant slope of the equilibrium curve, i.e., dy€/dx » K

3. E°MY same for all plates considered, Lewis (Ref. 21) obtained no _ In {1 4- E°mv[K/{0/V) - 1]} * ~ In [K/(0/V)] I17*'

This equation is plotted in Fig. 17-3. It will be noted that in general E°/E0MV is close to 1.0. In cases where the rectifying system

operates from low to high concentrations, the value of K/(0/V) will average out to around 1.0, making E° approximately equal to E°MV.

For cases where K/(0/V) is widely different from 1.0 and E°uv is low, the ratio of E° can be either much larger or smaller than E°ur.

Thus Fig. 17-4 illustrates a case where K/(0/V) is very small, and one theoretical plate would go from yn to a composition of almost 1.0. If the over-all column efficiency for this region were 0.5, two actual plates would give the same increase, but the diagram indicates that about five plates with E°MV « 0.5 would b6 required. Actually the efficiencies employed were incompatible, and if E°uv = 0.5 then E°

would be small for the low value of K/(0/V). The first step with E°mv = 0.5 does make a change in the vapor composition equal to one-half of that for a theoretical plate but, because of the convergence of the equilibrium curve and the operating line, the available potential decreases and the succeeding plates do not make so large a change in the vapor composition.

Murphree also gave a derivation for a plate efficiency based on liquid-phase compositions. The basic differential equation was

where x = mol fraction in liquid x* « liquid in equilibrium with Vapor leaving Kh = mass-transfer coefficient V « liquid in slug Under consideration The equation was integrated with Kl, a, Z/, and x* constant to give eo sss ^-% « 1 - q-IXiaW)

where x0i Xi = liquid composition to and leaving plate, respectively E°ml — liquid-phase plate efficiency L'/O = liquid rate It is difficult to picture any mass-transfer process on a bubble-type plate that corresponds to the derivation of Eq. (17-8). For example, consider the application of these equations to a local section of the plate. The liquid flowing across this section contacts the vapor rising through it, but the vapor composition varies with the liquid depth while the integration assumed that x* was constant. The mol fraction ratio of Eq. (17-8) will be used as the definition of EMl, but the masstransfer portion of the equation does not apply to bubble-cap plate conditions.

Equations (17-3) and (17-8) can be related by the operating line and the equilibrium curve. Assuming that the equilibrium curve is such that Ke = y0/x* = y*/x0 (Ke is equal to K if the equilibrium curve is a straight line through the origin) gives

The assumptions made in the derivation of Eq. (17-3) appear to be on a somewhat sounder mass-transfer basis than Eq. (17-8). If this is true, E%v depends on the operating conditions only to the extent that they affect the mass-transfer conditions while Eml in addition is a function of the ratio Ke/(0/V).

Table 17-1

Emv/^M7

Location in column

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