Plate Efficiencies

The numbers of theoretical or equilibrium stages for a vapour-liquid separation process can be evaluated quite precisely when the equilibrium data are known. But in practice equilibrium is not attained completely on trays, and the height of packing equivalent to a theoretical stage is a highly variable quantity. In columns of large diameter (≥ 3 m), significant concentration gradient exists along the path of liquid flow so that the amount of mass transfer may correspond to more than that calculated from the average terminal compositions. Mass transfer performance of packed towers is most conveniently expressed in terms of HETS (height equivalent to a theoretical stage), particularly when dealing with multicomponent mixtures to which the concept of HTU is difficult to apply. In addition to the geometrical configuration of the tray or packing, the main factors that affect their efficiencies are flow rates, viscosities, relative volatilities, surface tension, dispersion, submergence, and others that are combined in dimensionless groups.

The efficiency of mass transfer is expressed as the ratio of the actual change in mole fraction to the change that could occur if equilibrium were attained

Efficiency, E = Figure 6.8 Concentration changes along plate n.(6.8)

Three kinds of efficiencies are usually found to be in use (i) Point efficiency or Murphree point efficiency, EP (ii) Murphree efficiency or Murphree plate efficiency, EM and (iii) Overall efficiency or Overall column efficiency, EO.

A theoretical plate is defined as a plate in which the liquid and gas or vapour leaving the plate are in equilibrium in respect of the component being transferred. No real plate can achieve this because of the limited time of contact. Performance of a real plate is expressed in terms of Murphree plate efficiency defined as the ratio of actual enrichment on a plate to the theoretical enrichment that would occur if the system attains equilibrium.

Figure 6.8 Concentration changes along plate n.

Referring to Figure 6.8, this may be expressed as

EM = (6.9)

where,

yn = Average bulk concentration of solute in gas/vapour leaving the nth plate yn+1 = Average bulk concentration of solute in gas/vapour entering the nth plate

yn* = Equilibrium solute concentration in gas/vapour leaving the nth plate corresponding to its average bulk concentration xn in the liquid on the plate

Equation (6.9) represents the actual change in gas concentration on the nth plate as a fraction of the change that would have occurred if equilibrium were established.

Rewriting Eq. (5.59) for dilute systems when (1-y) may be assumed to be equal to unity, the number of overall transfer units between nth and (n +1)th plate is given by

NtoG = (6.10)

On integrating and algebraic manipulation of Eq. (6.10), we have

or 1 - exp (-NtoG) = (6.11)

Equating Eqs. (6.9) and (6.11), one can write 1 - EM = exp(-NtoG) (6.12)

or - ln(1 - EM) = NtoG (6.13)

whence, = (6.14)

Comparing Eqs. (5.66), (5.67) and (5.71), we have

Z = HtG$NtG = HtL$NtL = HtoG$NtoG = HtoL$NtoL

Hence,

Substituting the expression for HtoG from Eq. (5.77) expressed as

HtoG = HtG + HtL we have,

Hence,

(6.15)

For large plates, Murphree plate efficiency which represents average performance of a plate is rather over simplification since it does not give any idea about performance of different parts of the plate, which may differ significantly. In such cases, point efficiency, defined by Eq. (6.9) may be used,

EP =

where, the concentrations are at a particular point on the plate.

Murphree plate efficiency may be correlated with Murphree point efficiency as

EM = (6.16)

If the liquid on the plate is of uniform concentration every where due to complete mixing of the liquid, x = xn for the liquid, and y* = y will also be constant over the plate. In such situation, the Murphree plate efficiency and Murphree point efficiency become equal

EM = Ep (6.17)

An estimate of the overall plate efficiency although too much over simplification, is needed when the design method used gives an estimate of the number of ideal stages required for the separation. A relationship between the overall tray efficiency, EO and Murphree plate efficiency, EM is represented as follows:

EO = (6.18)

Because of concentration gradients along the tray, primarily in the liquid phase, the overall efficiency is different from point efficiency. Since the hydraulics of the tray usually cannot be known accurately, point and overall efficiencies are difficult to relate. Walas (1988) has shown selected values of efficiencies of some types of trays for various systems operating under different process conditions.

Since the efficiency may vary with the position of individual tray and with the position of the tray in the tower, three kinds of efficiencies are not the same. In a situation if values of more than one type of efficiency are available, the lowest value of the efficiencies should be taken as the overall efficiency when required.

For estimation of tray efficiencies a number of simpler correlations/equations are available in literature (O’Connell 1946, Chu et al. 1951, AIChE 1958, Bakowski 1963, 1969, McFarland et al.

1972). A critical survey on the available correlations and or equations has been dealt by Vital et al.

(1984). The relations proposed by Bakowski and McFarland et al. give a good fit to the experimental values while others give conservative values. However, the method of O’Connell is popular because of its simplicity. It expresses the efficiency in terms of the product of viscosity and relative volatility for fractionators, and the equivalent term HP/n for absorbers and strippers, H being the Henry’s constant while P and n represent total pressure and viscosity, respectively. Nevertheless, the collected experimental data and the several correlations mentioned just give a background on the basis of which judicious decision can be made for solution of specific problems.