Correlation of Valve Tray Efficiency Data William G. Todd and Matthew Van Winkle’ University of Texas, Department of Chemical Engineering, Austin, Tex. 78711
The purpose of this article i s to present correlations developed from experimental valve tray data. The tray froth heights and the depth of clear liquid on the tray were empirically correlated with the determining physical properties and operating variables. A theoretical model based upon the two-film theory was developed to predict the total number of moles of each component transferred across the interface separating the distilling phases. The effects of heat transfer across the interface and unequal values of molar latent heats of vaporization of the components were incorporated into the proposed model. Using the proposed model, the values of the heat flux at the interface and vapor contact times were calculated from the experimental data. Empirical equations were then developed for the prediction of the heat flux and vapor contact times. Predicted valve tray efficiencies based upon the theoretical model were in close agreement with experimentally determined efficiencies.
I n a recent article (1972) the authors presented experimental tray efficiency, pressure drop, and froth height data for the n-propanol-toluene and benzene-n-propanol binary systems. The data were obtained from the operation of an 18-in.diameter distillation column containing three valve trays of the rectangular type. The tray efficiencies of the n-propanoltoluene binary system were higher than the corresponding tray efficiencies of the benzene-n-propanol binary system. Since visual observations of the operating tray could not explain these variations in efficiencies, it mas concluded that the differences between the efficiencies of the binary systems must be dependent upon some basic difference in the heat and mass transfer mechanisms occurring between the distilling phases. The purpose of this article is t o examine some of the possible heat and mass transfer mechanisms occurring between distilling phases and t o present correlations developed for predicting valve tray efficiencies, froth heights, and clear liquid depths. Deflnition of Efficiency
The concept of a separation efficiency for vapor-liquid contactors was developed as a substitute for the direct calculation of the amount and extent of mass and heat transfer between the contacting phases. By using the separation efficiency, it is possible to calculate the separation capability of an actual contactor from the knowledge of the behavior of an ideal comparison stage. For convenience, this ideal stage has always been chosen such that the vapor and the liquid streams leaving the stage are in thermodynamic phase equilibrium. Such a concept of the equilibrium stage provides the basis for all tray-to-tray computations. Many different approaches have been used to define the separation efficiency of contacting plates or trays. Lewis (1922) defined the overall plate efficiency for binary systems as the number of theoretical plates required t o perform a given separation divided by the number of actual plates used for this separation. This definition is a very logical one but leads to difficulties for multicomponent mixtures because To whom correspondence should be addressed.
each component will, in general, have a different separability efficiency. The computation of Lelvis’ efficiency from experimental data also presents the problem of determining the number of theoretical plates required for the experimental separation obtained. Sormally, the number of theoretical plates or equilibrium stages must be computed by such methods as Underwood’s (1932) or Smoker’s (1938). Murphree (1925) defined two tray efficiencies, one for the vapor phase and one for the liquid phase, for each plate by assuming constant molal flow rates throughout the distillation column, Figure 1 illustrates a section of a counterflow plate column and the compositions referred to in Murphree’s equations. In this case the vapor phase efficiency is given by
where y, is the vapor mole fraction of a constituent and yn* is the composition of the vapor leaving the ideal plate which would be in equilibrium with the liquid leaving the tray of composition 2,. He defined the liquid phase efficiency in a similar manner with x, being the liquid mole fraction of a constituent :
where x,* is the composition of the liquid leaving the ideal stage which would be in equilibrium with the exiting vapor, g,. If the equilibrium relation is assumed linear, the two efficiencies are related by (3)
where L , and V , are the molal flow rates of liquid and vapor and m’ is the slope of the equilibrium line. Therefore, if the value of m’Vn/L, is equal to one, the vapor phase and liquid phase Murphree plate efficiencies are the same. Kord (1946) introduced the temperature or thermal efficiencies which parallel Murphree’s equations using temperatures instead of compositions. I n an attempt to remove cerInd. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972
589
and the holdup of the transferring component in the boundary layer or region near the phase boundary is negligible with respect to the amount transferred in the process. Based upon these assumptions and if we assume additivity of resistances, the following equation for the overall gas transfer units was developed : 1/NoG = 1/NG
+ m’V/LNL
(9)
where
LN +I u
N G
=
KVaPZ V
NL
=
KLaPLZ L
XNtI
~
~
The overall point efficiency was found to be related to the overall gas transfer unit by EoG
Figure 1. Section of counterflow distillation column
tain limitations of the Murphree-type efficiencies, Holland and McMahan (1970) defined a vaporization efficiency as follows:
(4) where j denotes the j t h component, n the nth plate, and m the equilibrium constant. The authors present cases showing that their defined efficiency is always a nonzero, finite, and positive number. Standart (1965) defined his component efficiencies for the phases as: EV,j =
V n Y n , j - Vn+lYn+l.j vn*Yn,j* - V n + ~ ~ n + l , j
(5)
and
where Vn* and Ln* are defined as the equilibrium streams leaving the plate. A component material balance around the nth plate yields the following equation:
Examination of Equation 6 shows that
The main advantage of Standart’s efficiency is the ease of comparing the actual and the ideal equilibrium comparison plate because the entering streams, Vn+l and Ln-,, are the same for both plates. The main disadvantage, however, is the addition of two more unknown quantities, Vn* and Ln*. Classical Mass Transfer Model
Possibly the best known method for computing the plate efficiency is based upon the classical two-film resistance theory as outlined in the “A.1.Ch.E. Bubble-Tray Design Manual” (1958). The following assumptions are inherent in this type of development: equimolar counterdiffusion exists throughout the films; the rate of mass transfer of a component within a phase is proportional to the concentration gradient that exists between the bulk of the phase and its interface; equilibrium exists between phases at the interface; 590 Ind.
Eng. Chem. Process Des. Develop., Vol. 1 1 , No.
4, 1972
=
1 - exp
(-A’oG)
(12)
.4 complex equation involving the Peclet number was then developed to correlate the Murphree tray efficiency with the overall point efficiency. This type of development suffers because the interaction of heat and mass transfer are neglected. It will be shown later that assumption (1) is invalid when heat effects are included. Effects of Heat Transfer
I n a counterflow fractionating column (Figure l), the rising vapor entering the contacting plate has a higher temperature than the entering liquid stream. Because of this temperature gradient, Danckwerts e t al. (1960) reasoned that there would be a net transfer of heat from the vapor phase to the liquid phase. -4s discussed by Liang and Smith (1962), several conditions could arise from such an exchange of heat: The heat transfer between phases is slow relative to the mass transfer and thus the vapor will leave the plate superheated and the liquid will leave subcooled; the heat transfer between phases corresponds to the rate of mass transfer in such a way as to keep the phases exactly a t their respective condensation and boiling points; the heat transfer between phases occurs more rapidly than is required for maintaining the phases at their respective condensation and boiling points and thus partial condensation of the vapor and partial vaporization of the liquid will occur. The data of Haselden and Sutherland (1960) give support to the existence of the third condition. They reported that in their study of distilling ammonia-water solutions, a very thick fog was formed in the vapor space above the tray when large temperature differences existed between the liquid and vapor streams entering the plate. It was speculated that the fog was due to condensation in the vapor phase. Other experimental observations have been reported by Haselden (1960), Ruckenstein (1967), and Sawistowski (1959) which demonstrate that for mixtures having sufficiently different boiling points, the efficiency of a distillation column depends on concentration and has a maximum value. Several different theories have been presented to describe this phenomena. Everitt and Hutchinson (1966) and Korman et al. (1963) have successfully demonstrated that the classical diffusional model explains the dependency of efficiency on concentration (but not the maximum) by taking into account the dependence of the slope of the equilibrium curve on concentration. SawistoIvski and Smith (1959) stressed that the observed maximum in their efficiency data could not be explained by the classical theory and suggested an explanation in which the heat transfer between phases was considered.
Ruckenstein (1961) and Liang and Smith (1962) considered the interaction between heat and mass transfer in the thermal theory of rectification by using the two-film theory. The thermal theory of rectification simply superimposes the effects of the heat transfer between phases, as previously discussed, upon the classical diffusional mass transfer model. Therefore by excluding the possibility of extensive superheating in the bulk of the liquid or subcooling in the bulk of the vapor, additional mass flux terms must be added to the diffusional mass flux equation to account for the condensation of vapor and the vaporization of liquid in the bulk of the respective phases. Ruckenstein and Smigelschi (1965, 1967) assumed that the quantity of heat transferred from the bulk of the vapor phase toward the interface simply maintains the vapor a t its saturation temperature and therefore no condensation in the bulk of the vapor phase occurs. The basis for this assumption is that the thermal diffusivity in the vapor phase is very small. By considering the heat transfer a t the interface and the vapor generated in the bulk of the liquid phase, Ruckenstein and Smigelschi derived the following equation:
‘ c a
dz
= - KLA
V
~
1
+
,
~
where
I n the derivation of the above equation, the molal latent heats of vaporization of the least volatile component (lvc) and that of the more volatile component (mvc) were assumed to be equal. The equation was graphically integrated to obtain the plate efficiency. When we selected the correct values of the coefficients, the predicted efficiency showed good agreement with experimental data which exhibited a maximum efficiency with respect to concentration. Ruckenstein (1970) considered the interaction between heat and mass transfer using the penetration renewal model. In this study, only the transfer processes in the liquid phase were considered. The heat transfer a t the interface was taken into account but vaporization in the bulk of the liquid phase was neglected. Ruckenstein reasoned that for moderate superheating in the liquid, the rate of nucleation in the homogeneous phase is practically nil. From his work, Ruckenstein presented a complicated theoretical mass transfer model involving the error function, the Jakob number, and Danckwert’s renewal frequency. The molal latent heats of vaporization of the diffusing components were considered to be different in the derivation of the model. The model was not compared with experimental data. Effects of Surface Tension
Zuiderweg and Harmens (1958) found experimentally that increasing or decreasing the surface tension of the column internal reflux could influence the interfacial area and consequently affect the plate efficiency quite markedly. The researchers explained these phenomena in terms of the “RIaragoni effect” which states that liquid surfaces of high interfacial tension contract when contacted with a surface of lower surface tension. Three types of systems were defined with respect to the changes in surface tension in the reflux flow: negative systems, the internal reflux decreases in surface tension as it passes down the column because the surface tension of the lvc is less than that of the mvc; positive system, the internal re-
flux increases in surface tension because the surface tension of Ivc is greater than that of the mvc; neutral systems, the surface tension of the internal reflux does not vary because the components have equal surface tensions or the relative volatility is very small and the gradients in surface tension are consequently always small. The authors found for trays with small free area and low vapor rates (laboratory apparatus) that: The neutral and negative systems are characterized by spray-type interfacial areas and constant efficiencies; the positive systems are characterized by foam-type interfacial areas and increased efficiencies. Zuiderweg and Harmens rationalized that the interfacial films of positive systems were stabilized by the Maragoni effect and thus result in foam-type interfacial areas. For higher tray free areas and vapor velocities (commercial equipment), Zuiderweg speculated that foaming tendencies would diminish and the differences in efficiency between positive and negative mixtures would be less. This would seem to be verified by the work of Bainbridge and Sawistowski (1964) and Fane and Sawistowski (1968) which indicated that positive systems have higher efficiencies than negative systems at low hole velocities but lower efficiencies a t higher velocities. Sawistowski theorized that “once the cellular foam starts breaking down, the unstable tendencies of the negative system outweigh the conservative tendencies of the positive system in the production of fresh interfacial area.’’ To justify this statement, Sawistowski postulated that the Marangoni effect accelerates the detachment of liquid drops for the bulk of the liquid for negative systems and thus increases the amount of interfacial area and the plate efficiency. In comparison, it would appear that the Zuiderweg theory is valid for the vapor-dispersed type of tray beds (characterized by relatively stable, cellular bubbles) and that the Sawistowski theory is valid for the liquid-dispersed type of tray beds (characterized by the suspension of liquid droplets in the stream of vapor). Both theories explain the dependence of plate efficiency on concentration in terms of varying interfacial areas. Effects of Molar Latent Heats of Vaporization
The assumption of equimolar counterdiffusion through the films in the classical theory of mass transfer implies that the molar latent heats of vaporization of the diffusing components are equal. Thus the heat liberated by the condensation of the lvc a t the interface is equal t o the heat required to vaporize the mvc. For most cases the molar latent heats of the components are sufficiently close so that such an assumption is valid, but for certain systems to assume such a condition may lead to serious errors, Consider the case where the latent heat of the lvc has a value that is ll/ztimes that of the mvc. If the heat that is transferred across the interface by conduction is neglected, there will be 11/* moles of the mvc component vaporized for each mole of lvc condensed. Therefore the flux of the lvc is highly dependent upon the supply to the interface of the mvc, which must diffuse through the liquid phase toward the interface. Because the molecular diffusivity of liquids is several orders of magnitude smaller than that of vapors, it is likely that the liquid phase resistance would assume a much more important role. The same type of analysis for the case in which the latent heat of the mvc is larger than that of the lvc leads to the conclusion that the importance of the vapor phase resistance would be accentuated. At this point it is convenient to define three types of systems with respect t o the possible effects of latent heats on mass transfer: thermally positive systems, the molar latent Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972
591
N:
N,V
where
L I Q U I D PHASE
VAPOR PHASE
-
A
4
N,L
-
* N2L
L
Substituting N2,,' from Equation 18 into Equation 15 and one obtains solving for Nlviv,
L
A similar expression can be written for the liquid phase:
A material balance a t the interface leads to the following equations: N1,tL = Figure 2. Interface between vapor and liquid phase
heat of vaporization of the mvc is greater than that of the lvc; thermally neutral systems, the molar latent heats of vaporization of the mvc and the lvc are equal; thermally negative systems, the molar latent heat of the mvc is smaller than that of the lvc.
Therefore
-
4 - Y1,r XZ
1
-
+ coKo(!/l,i(1
!/l,b)
- Xl/hZ)Yl,i P
A2
1 - (1
592 Ind.
(16)
= -q
Eng. Chem. Process Des. Develop., Vol. 1 1,
No. 4, 1972
- Xl/X2)Xl,i
(22)
If we assume that the vapor and liquid a t the interface are related by the equilibrium relationship, then YLi =
mx1,i
(23)
the value of zl,can be determined by combining Equations 22 and 23. The resulting equation for x ~is ,the~ quadratic form: Xl,,
=
-B f (B2 - 4 AC)"* 2A
XI/^
- h/X2)(1
+ KL'/K,')
# 1)
(24)
where rl = m ( l
(1
c K,'
=
=
Y1,b
- h l / X z ) ( ~ ~ l , & ~ ' / K-k, ' Y1.d
f xi,&~'/Ko'
KvcG
KL'
=
KLcL
Only the positive root lying between 0 and 1.0 is of interest. This equation for x1,, is valid only for values of the ratio hl/Xz which are not equal to one because when the value is equal to one, the denominator of Equation 24 becomes equal to zero. If X1/h2 is equal to one, Equations 19 and 20 reduce to
N1,iv
+
N1,t' = !/l,t(N1,iu- NZ,tv) coKo('ki - Y l J ) (15) where the direction of the fluxes are those shown in Figure 2. A thermal balance a t the interface leads to the equation X I N I ,~ XzNz,,
-
- - x1,i f CLKL(x1,b - X I , $ )
Proposed Method for Efficiency Prediction
The Murphree tray efficiency for the vapor phase as defined by Equation 1 will be used to correlate the efficiency of the valve trays because of its ease of computation and its wide acceptance by industry. To determine the compositions of the vapor and liquid streams leaving the tray, the total number of moles of each component that is exchanged between the phases must be predicted. The following analysis of the mass transfer between the phases of a binary system is presented with this goal in mind. The two-film model will be used to calculate the diffusional mass flux of the components a t the interface. The heat and mass transfer mechanisms a t the interface are illustrated in Figure 2. The following assumptions are necessary: the rate of mass transfer of a component within a phase is proportional to the concentration gradient that exists between the bulk of the phase and its interface; the rate of heat transfer within a phase is proportional to the temperature gradient fhat exists between the bulk of the phase and its interface; thermodynamic phase equilibrium exists between the phases at the interface; the volume of the transferring component in the boundary layer or region near the phase boundary is negligible with respect to the volume transferred in the process; the heat transfer between phases corresponds to the rate of mass transfer in such a way as to keep the bulk of the phases exactly a t their respective condensation and boiling point temperatures. I n accordance with the above simplifying assumptions, one may write Fick's law of diffusion for the vapor phase as
(21)
N1,tv
9 -A2 y1,i + c & ~ ( Y I , , - ~ l , d
(25)
-x,4 + cJ(~(zl,a -
(26)
and
Ni,iL =
~ 1 , i
$I,
