correlation of valve tray efficiency data

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-propanol- toluene 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 Underwoods (1932) or Smokers (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 Murphrees 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 mVn/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 Murphrees equations using temperatures instead of compositions. In an attempt to remove cer-

Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972 589

LN +I u X N t I

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:

( 5 ) V n Y n , j - Vn+lYn+l.j

vn*Yn, j* - V n + ~ ~ n + l , j EV,j =

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 Standarts 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;

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 :

(9) 1 / N o G = 1 / N G + mV/LNL where

KVaPZ V

KLaPLZ L

N G = ~

N L = ~

The overall point efficiency was found to be related to the overall gas transfer unit by

(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

E o G = 1 - exp (-AoG)

In 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.

590 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972

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, addi- tional 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 satura- tion 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 V = - KLA ~ 1 , ~ + where

In 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 homogene- ous phase is practically nil. From his work, Ruckenstein presented a complicated theoretical mass transfer model involving the error function, the Jakob number, and Danck- werts 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 re- searchers explained these phenomena in terms of the RIara- goni 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 volatil- ity 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 (com- mercial 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 drop- lets 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 to 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/z times 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


where VAPOR PHASE

N: 4

N,V - L L

L I Q U I D PHASE

A - N,L

* N2L

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.

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. In accordance with the above simplifying assumptions, one may write Fick's law of diffusion for the vapor phase as

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,, = -q (16)

Substituting N2,,' from Equation 18 into Equation 15 and solving for Nlviv, one obtains

A similar expression can be written for the liquid phase:

A material balance a t the interface leads to the following equations:

N1,tL = N1,tv (21)

Therefore

4 XZ

- - Y1,r + coKo(!/l,i - ! / l , b ) - -

1 - (1 - Xl/hZ)Yl,i P A 2

- - x1,i f CLKL(x1,b - X I , $ ) (22) 1 - (1 - X l / X 2 ) X l , i

If we assume that the vapor and liquid a t the interface are related by the equilibrium relationship, then

Y L i = 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:

XI/^ # 1) (24) -B f (B2 - 4 AC)"*

2 A X l , , =

where

rl = m(l - h / X 2 ) ( 1 + KL'/K,')

(1 - h l / X z ) ( ~ ~ l , & ~ ' / K , ' -k Y1.d c = Y 1 , b f xi,&~'/Ko'

K,' = 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

(25) 9 N1,iv -- A2 y1,i + c & ~ ( Y I , , - ~ l , d and

(26) 4 Ni,iL = -x, ~ 1 , i + cJ(~(zl,a - $ I , < ) 592 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972

I C

1 7 1 5 -

0 9 - 1 3 : I I

0 7 - 0 5 - 0 3 .

I

0, I c .- - V

N

Figure 3.

I I 1 I I . BENZENE- N-PROPANOL -

I

-1 - - - -

I 1 I I I I

I . 7 I I I I I I 1 .5 N - PROPANOL -TO LU EN E

Height weir = 2 in., L/V = 1.0, valve leg lengths = 0.3125, 0.3750, 0.4375 in.

As before, solving for x1 , i leads to the following equation:

(X,/Xz = 1) X l , b K L / K v + y l , b

(27) Once the interfacial concentrations and consequently the interfacial temperatures are determined, the molar flux N1 can be calculated by either Equation 19 or 20. The total number of moles of component 1 that is exchanged between the phases can now be calculated by multiplying the molar flux by the total interfacial area. The total number of moles of component 2 can be computed from Equation 18.

Data Analysis. To develop tray efficiency prediction methods based upon the actual mass transfer between phases, the tray froth heights and clear liquid depths had to be correlated first so tha t the average residence times of the phases and the total volume of bed on the tray could be determined. The correlation of these two variables, as well as all other variables correlated, was accomplished by using a nonlinear least squares regression program which utilized a simplex optimat opti- mization routine in n-dimensions for fitting the nonlinear coefficients. As pointed out in the previous article, the operation of the column a t the two extremes of the operating range (15 and 97y0 of column flood) were not stable and the results a t these two extremes were difficult to reproduce. Because of this and because the original scope of work did not include these two points of operation, the 15 and 97% of column flood were not included in the analysis of the data.

Analysis of Tray Clear liquid bepth Data Examination of all the possible linear and second-order

effects of the independent variables produced the following equation for the prediction of the clear liquid depth on the valve tray.

2, = 0.71 + 0.11 H, + 1.80 How + 1.76 F,(Fs - 1.63) + 2.67 p ( F s - 1.32) (28)

. I

N 0.61 I I I I I I J

1.61 I I I I I

0.8 0 0.6 I I I I I I

( f t / s e c ) ( I b / f t 3 ) 2 0.3 0.5 0.7 0.9 I I 1.3 1.5

F FACTOR

Figure 4. Clear liquid depth results for benzene-n-propanol system

L/V = 1 .O, valve leg length = 0.31 25,0.3750,0.4375 in.

I I I I I

L/ = 1.0

- 1.0

N 0.6 0.0

1.2

0.8

03 0 5 0 7 0 9 I 1 1 3 1 5 F FACTOR ( f t / s e c ) ( Ib / f t 3

Figure 5. Clear liquid depth results for benzene-n-propanol system

Weir height = 2 in., valve leg lengths = 0.3125, 0.3750, 0.4375 in.

where

2, H, = weir height, in. H,, = weir height crest, in. F , p = viscosity, CP

= clear liquid depth, in. of water

= superficial flow factor, (ft/sec) (lb/fta)12

To predict clear liquid depths for nonwater systems, the value from Equation 28 should be multiplied by the ratio of the


Table 1. Calculated Heat and Mass Transfer

Run no.

101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 201 202 203 204 205 206 207 208 209 210 21 1 212 213 214 215 216 217 218 219 220 22 1 222 223 224 225 226 227 l A l b 1A2 1x3 1A4 1A5 1A6 1A7b l B l b 1B2 1B3 1B4 1B5 1B6

N I , T

0.85 0.99 1.87 6 .18 1.89 2.09 3.83 0.94 2.83 0.43 3.63 3.31 2.22 2.00 5.55 1.82 2.27 1.91 4.79 1.42 3.84 1.92 6.42 2.02 2.43 4.48 3.45 7.20 6.07 3.98

10.01 4.60 1 .63 6.99 2.45 3.12 0.59 6.68 6.71 4.62 2.02

10.26 3.91 4.11 1.78 9.48 3.54 6.92 3.97

10.06 5.00 1.58 7.34 6.26 0.86 1.87 2.84 3.07 4.29 6.18 6.57 1.10 2.73 3.04 3.63 3.89 4.83

N 2 , T

0.96 1 .13 1.63 6.56 1.88 2.32 4.10 0.79 3.82 0.57 3.67 3.29 2.22 2.44 5.83 1.57 2.29 2.46 4.97 1.21 3.92 1.69 6.92 2.03 3.33 4.74 3.44 5.48 4.60 2.78 7.52 3.52 1.69 5.55 1.76 3.09 0.73 4.89 4.97 3.38 2.06 7.75 2.80 3.01 1.82 7.03 2.47 5.00 2.72 7.68 3.81 1.64 5.53 4.65 0.54 1.63 2.76 3.03 4.40 6.56 7.09 0.75 2.86 3.05 3.67 3.95 5.01

N I , T N Z , T 0.88 0.87 1.15 0.94 1.01 0.90 0.93 1.18 0.74 0.76 0.99 1.01 1 .oo 0.82 0.95 1.16 0.99 0.78 0.96 1.17 0.98 1.14 0.93 1.00 0.73 0.95 1 .oo 1.31 1.32 1 .43 1.33 1.31 0.96 1 .26 1.39 1.01 0.80 1.37 1 .35 1.37 0.98 1.32 1.40 1.37 0.98 1.35 1.44 1.38 1.46 1.31 1.31 0.96 1 .33 1.35 1.59 1.15 1 .03 1.01 0.97 0.94 0.93 1.47 0.95 0.99 0.99 0.99 0.96

Q

-1222.2 -1267.1 -9762.3 - 14863.7 - 6483.5 -3693.9 -8680.4 - 5274.5

5169,l 587.4

- 11605.5 -11352.8 -7373.8 -249.2

- 14317.1 - 9800.9 -7334.6

1657.0 - 13328.5 -7861.2 - 11544.. 9 -9796.9 - 13961.4 -6687.1

- 10983.7 - 11596.1

7236.4 5753.4

-537.0 8110.6 4931.7 9852.7

11026.2 633.1

15761.1 5739.9 3201.8 4267.8 2102.1

11342.9 8975.0 780.6

1942.0 10067,4 5951.1

2105.4

10396.6 5098.4 9450.0 6219.8 4135.0

5148.3

-634.9

-1466.7

- 7585.4 - 9762.3 - 10684.2 - 10831.8 - 12641.8 - 14863.7 - 13980.8 - 8742.0 -7064.4 -9797.8 - 11605.5 - 12144.2 - 13438 5

a

244.7 195.9 109.8 498.4 237.1 277.4 434.4 107.8 521.7 99.6

261.7 259.9 239.8 276.1 510.7 114.4 197.8 230.6 417.9 84.7

267.0 113.1 513.8 267.4 322.9 305.6 234.5 306.2 238.4 122.5 532.5 260.5 281.6 470.8 122.6 503.7 118.4 284,3 283.7 275.3 302.3 563,7 136.0 215.2 236.7 456.8 97.4

290.8 125.2 526,3 263.1 273,8 302.0 237.5 70.6

109.8 171.3 255.0 369.6 498.4 567.2 75.0

121.3 170.2 261.7 359.9 485.4

9

-5 .0 -6 .5

-88.9 -29.8 -27.3 -13.3 -20.0 -48.9

9.9 5 .9

-44.3 -43.7 -30.7 -0 .9

-28.0 -85.7 -37.1

-31.9 -92.8 -43.2 -86.7 -27.2 -25.0

15.9 -35.9 -49.5

23 6 24.1

- 4 . 4 15.2 18.9 35.0 23.4 5 . 2

31.3 48.5 11.3 15.0 7 . 6

37.5 15.9 5.7 9 . 0

42.5 13.0

-6.5 7 . 2

-11.7 19.8 19.4 34.5 20.6 17 4

-107.5 -88.9 -62.4 -42.5 -34.2 -29.8 -24.6

-116.6 -58.2 -57.6 -44.3 -33.7 -27.7

7 . 2

tL'Q

0.0760 0.0319 0.1009 0.1681 0 I 0372 0.0658 0.0805 0.0616 0.3460 0.3177 0.0439 0.0443 0,0474 0.0979 0.1962 0.1413 0.0215 0.0294 0.1026 0.0661 0.0440 0.0869 0.1677 0,0358 0.0635 0.0648 0.0270 0.1019 0.0440 0.1305 0.2420 0.0566 0.1254 0.1118 0.0890 0.3735 0.4289 0.0655 0.0658 0.0719 0.1820 0,3094 0.1580 0.0350 0.0650 0.1601 0.0822 0.0659 0.1124 0,2088 0.0522 0.1239 0,0919 0.0396 0.3446 0.1009 0.0514 0.0473 0.0738 0.1681 0.2695 0.2555 0.0697 0.0468 0.0439 0.0660 0.1459

594 Ind. Eng. Chem. Process Der. Develop., Vol. 1 1 , No. 4, 1972

Table 1. (Continued)

Run no.

1B7b 1C1b 1 c 2 1C3 1C4 1C5 1 C6 1 C7* 2Dlb 2D2 2D3 2D4 2D 5 2D 6 2D7b 2Dlb 2D2 2D3 2D4 2D5 2D6 2D7b 2Elb 2E2 2E3 2E4 2E5 2E6 2E7b 2Elb 2E2 2E3 2E5 2E6 2E7b 2Flh 2F2 2F3 2F4 2F5 2F6 2Fib 2Flh 2F2 2F3 2F4 2F5 2F6 2F7b

NI,T

5.89 0.82 1.92 2.59 3.74 4.72 6.42 6.74 1.87 3.95 5.50 7.33 7.99

10.12 9.40 1.53 3.97 5.59 7.11 8.39

10.06 8 .50 1.62 3.83 5.15 6.68 8 .26

10.02 10.92 1.71 3.93 5.22 8.63 9.61

11.09 1 .80 3.98 5.52 6.75 8.10

10.01 10.66 1.78 4.18 5.51 6.99 7.94 9.49

10.18

N ~ , T NZ .T

6.21 0.45 1.69 2.46 3.77 4.89 6.92 7.34 1.18 2.82 3.90 545 6.39 8.14 7.81 0.88 2.72 4.04 5.18 6.24 7.68 7.07 0.91 2.76 3.76 4.89 6.08 7.74 8.58 1 .30 2.81 3.73 6.32 7.31 8.45 1.07 2.78 3.96 4.90 6.02 7.52 8.35 1.17 3.02 4.03 5.26 6.25 7.64 8.14

NZ.T

0.95 1.83 1.14 1.05 0.99 0.96 0.93 0.92 1.59 1.40 1.41 1.34 1.25 1.24 1.20 1 .75 1.46 1.38 1.37 1.34 1.31 1.20 1.78 1.38 1.37 1.37 1.36 1.29 1.27 1.31 1.40 1.40 1.37 1.31 1.31 1.68 1 .43 1.39 1.38 1.34 1.33 1.28 1.52 1.38 1.37 1 .33 1.27 1.24 1 .25

Q

- 14774.2 - 8223.0 -9796.9 - 10493.9 - 11932.3 - 13094.9 - 13961.4 - 13353.2 -2637.8

564.4 289.0

5018.9 13429.8 17595.1 20917.0

-3805.8 -1466.7

1644.6 2943.6 5688.3

10396.6 18883.1

1155.4 2319.9 3201.8 4441.9

11944.8 15523.3 1743.0 690.5 816.9

4053.1 9329.6

11103.1

- 4257.6

- 3660.4 -537.0 1207.8 2503.1 5474.4 8110.6

14601.0

1233.1 2607.3 5974.2

11586.1 16673.2 16917.4

- 1598.1

a

543.8 68.2

113.1 178.9 282.1 394.3 513.8 546.6 76.4

124.9 197.0 290.9 394.7 500.3 574.4 71.3

125.2 197.9 300.6 413.5 526.3 560.1 71.8

119.9 194.2 284.3 392.0 538.0 577.3 75.9

122.1 191.4 406.7 523.7 597.8 76.6

122.5 196.5 295.0 412.4 532.5 578.0 76.1

124.8 194.8 292.1 398.3 519.3 582.5

4

-27.2 -120.5 -86.7 -58.7 -42.3 -33.2 -27.2 -24.4 -34.5

4 . 5 1 . 5

17.3 34.0 35.2 36.4

-53.4 -11.7

8 .3 9 . 8

13.8 19 .8 33.7

-59.3 9.6

11.9 11 .3 11 .3 22.2 26.9 23.0

5.7 4 . 3

10.0 17.8 18.6

-47.8 -4 .4

6 . 1 8 .5

13 .3 15.2 25.3

-21.0 9 . 9

13.4 20.5 29.1 32.1 29.0

t L Q

0.2161 0.3477 0.0869 0.0464 0.0450 0.0768 0,1677 0.2097 0.3591 0.1170 0.0657 0.0642 0.0964 0.1748 0.2890 0.4304 0.1124 0.0663 0.0649 0.1045 0.2088 0.2533 0.4326 0.1309 0,0706 0.0655 0.0971 0.2387 0.3072 0.4179 0.1261 0,0701 0,1046 0.2157 0.3559 0.3750 0.1305 0.0729 0.0702 0.1149 0.2420 0.3240 0.4043 0,1297 0.0743 0.0712 0.1073 0.2184 0.3398

Liquid contact times are reported in units of seconds. b These data points were not used in the data analysis, but are included for completeness.

densities (62.4/pL). The value of How was estimated from the Francis weir formula with the correction for segmental downcomers suggested by Bolles (1946).

H O W = 0.48 F,(Q,/Lm)2a (29) where

F , = Bolles correction factor Q L = liquid flow, gal/min L, = length of weir, in.

The least-squares regression analysis of the data shows a n average deviation of 0.08 in. of water (8.79%), a maximum error of 0.22 in. (23.65%), and a multiple correlation coeffi-

cient of 0.923. The average absolute deviation of 0.08 in. compares well with the value of the reproducibility deviation between duplicates of 0.07 in.

The experimental data are compared graphically with those calculated by the predictive equation in Figures 3-5. Figure 3 illustrates the data for the two different binary systems, and Figures 4 and 5 illustrate the benzene-n-propanol binary data for different values of weir heights and L/V ratios.

Analysis of Tray Froth Height Data

The expression U : , 2 p , / ( p L - p u ) has been used successfully in other studies (Hughmark, 1965; Lowry, 1967) to correlate bed heights and thus was included as a correlating variable in


2.0 1 1 0.0 0.02 0.04 0.06 0.08 0.10 0.12

F F ( f t / s e c

Figure 6. Tray froth height results for both binary systems

this study. The form of this expression (referred to hereafter as the frothing factor, F F ) was derived first by Souders and Brown (1934). An arithmetic average of the vapor flows ( U a ) , vapor densities ( p , ) , and liquid densities ( p L ) about the test tray were used in computing the frothing factors. A least- squares regression analysis of the data resulted in the following equation:

(30)

The average absolute deviation of the fitted equation from the fitted data was 0.43 in. (4.4773 with a maximum error of 1.22 in. (16.11%). The multiple correlation coefficient for the regression analysis was 0.986. This average absolute deviation of 0.43 in. is equivalent to the calculated deviation between duplicates of 0.43 in. Values of the bed heights predicted by this equation are graphically compared with the experimental data points in Figure 6.

Analysis of Mass Transfer Data

As proposed in the theory discussion, the calculation of the total moles of each component exchanged between the liquid and vapor phases was to be used as the basis for the prediction of tray efficiencies. The number of moles of component 1 ( N 1 , T ) that is exchanged between phases can be calculated from a knowledge of the molal flux of component 1 a t the interface (N1,J and of the total interfacial area (a) by the following equation:

Z, = 3.00 + 119.39 F F + 0.82 Z,(hot liquid)

N L T = aN1,, (31) As derived previously, the molal flux of component 1 a t the

interface is given by either

9 A 2

- - Yl,C + cvK,(!h,t - y 1 , b ) (19) N19iu = 1 - (1 - Xl/AB)?JI,I

or

since N1,tu = NlZiL a t the interface. Equation 23 shows yl,{ and XI,^ are related by the equi-

librium constant m, and xl,( is defined by Equation 24 in terms of the mass transfer coefficients K , and KL. There- fore, if a, q, K,, and KL are known, Ni,T can be calculated by Equation 31.

The values of the mass transfer coefficients are defined by the penetration theory (Higbie, 1935) as:

K , = 2 (3): KL = 2 ( % ) I 2 (33)

where

D, = vapor diffusion coefficient, ft2/hr DL = liquid diffusion coefficient, ft2/hr t, = vapor contact time, hr tL = liquid contact time, hr Hughmark (1935) presented the following empirical equa-

tions for the determination of the interfacial area per unit volume of tray bed and of the liquid contact times.

a = (91 + 266 F , - 92 F S 2 - 75/H, + 40 Horn) X

and

tL = tL(Hza/2)1iS(SA/18)1/8 exp (-1.065 +

(900 p o / p L ) 16 (0.67/~,) ll2(70/~) (34)

1.8 How + 1.517 F , - 0.67 HozaF,) (35) where

p , = vapor density, lb/ft3 p L = liquid density, lb/fta pL = liquid viscosity, CP u = surface tension, dyn/cm tL = liquid residence time, hr S A = percent free area per bubbling area

In the study a t hand, the liquid residence time was defined

t L = (volume of clear liquid on the tray)/(average flow rate as

of liquid onto the tray)

where

-4, = active or bubbling tray area, ft* QL = liquid flow rate, ft3/hr Z c = hot clear liquid depth, in. By definition, the units of a are square feet per cubic feet.

To calculate the actual interfacial area, a must be multiplied by the total volume of the bed, which results in the following equation.

a = aZ,Aa/12.0 (37)

Examination of the above equations shows that two unknown quantities remain, namely the heat flux a t the interface, q , and the vapor contact time, t,. The determination of these two unknowns is discussed below.

Calculation of Heat Flux at Interface

following equation An earlier enthalpy balance a t the interface yielded the

(16) p = AiNi.1 - XzNa,t


Table I I . Experimental Run Numbers and level of Variables

Valve leg Weir Run no.a length height LIV ?& Flood

01 0 4375 3 1 60 02 0.4375 1 1 60 03 0.4375 2 1 30 04 0.4375 2 1 90 05 0.4375 2 1 . 5 60 06 0.4375 2 0 . 5 60 07 0.3750 2 1 . 5 90 08 0.3750 2 1 . 5 30 09 0.3750 2 0 . 5 90 10 0 3750 2 0 . 5 30 11 0.3750 2 1 60 12 0.3750 2 1 60 13 0.3750 3 1 . 5 60 14 0.3750 3 0 . 5 60 15 0.3750 3 1 90 16 0.3750 3 1 30 17 0.3750 1 1 . 5 60 18 0.3750 1 0 . 5 60 19 0.3750 1 1 90 20 0.3750 1 1 30 21 0.3750 2 1 60 22 0.3125 2 1 30 23 0.3125 2 1 90 24 0.3128 2 1 . 5 60 25 0.3125 2 0 . 5 60 26 0.3125 3 1 60 27 0.3125 1 1 60

Last two characters of a run number.

Multiplication of both sides of this equation by the interfacial area results in

Q = A I N I ~ T - A ~ K ~ , T (38) which permits the direct calculation of the total heat flux (Btu/hr) from the experimental data. Values of and L V ~ , ~ were computed for the second tray from the experimental data (Todd and Van Winkle, 1972), and the values of the molal latent heats of vaporization were computed a t the average tray temperature. Table I contains the total number of moles of the two components exchanged between the phases, the ratio of divided by I Y ~ , ~ , &, a, and q. The values of the liquid contact times (fL', sec) were calculated by Equation 35 and are tabulated also.

Table I1 contains the values of the experimentally independent variables associated with the different run numbers. The data sets labeled A , B, C, D, E, and F are supplementary data collected a t 15, 30, 45, 60, 75, 90, and 97.5% of the column flooding conditions for both binaries a t each length of tray valve legs (A, B, and C sets with n-propanol-toluene binary and D, E, and E' sets with benzene-n-propanol binary; -1 and F sets-valve leg lengths of 0.4375 in., B and E sets- valve leg lengths of 0.3750 in., and C and D sets-valve leg lengths of 0.3125 in.). d weir height of 2 in. and a L/ V ratio of 1 .O were used for these runs.

Inspection of the tabulated values of the ratio ~ Y I , T / S ~ , T , shows that the value is generally greater than one for the thermally negative benzene-n-propanol system and less than one for the thermally positive n-propanol-toluene system. This result is in agreement with the postulation made earlier. It is interesting to compare these values with the ratio of the

15.01

-20.0 I I I I I I I 1 0.3 0.5 0.7 0.9 1 . 1 1.3 1.5

F FACTOR ( f t /sec ) ( Ib /f t 3 ) ''2

Figure 7. Calculated heat transfer at interface A n-Propanol-toluene system 0 Benzene-n-propanol system

1.4

1.2-

1.0-

-

I

u 2 0.8 - d

. > L 0.6 -

0.4 -

0 .2 -

01 I I I I I 0.3 0.5 0.7 0.9 1 . 1 1.3 1.5

F FACTOR ( f t / s e c ) ( I b / f t 3

Figure 8. Calculated vapor contact times A n-Propanol-toluene (B data set) 0 Benzene-n-propanol (E data set)

latent heats. If Q were zero, Equation 38 would reduce to

l171,T A 2

N?,T A1 _ _ _ - -

where

(39)

A2/X1 = 0.81 for the n-propanol-toluene system A2/hl = 1.41 for the benzene-n-propanol system

In general, the experimental ratio of moles and the ratio predicted by the above equation are in closer agreement for the benzene-n-propanol system than for the n-propanol-toluene system.

The calculated values of the total heat transferred for the A, B, C, D, E, and F data sets are illustrated in Figure 7. As the plotted data indicate, the total heat transferred across the interface for the thermally negative benzene-n-propanol system is greater than zero.

Ind. Eng. Chem. Process Des. Develop., Vol. 11, No. 4, 1972 597

Run no. 101. 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 1Alb 1A2 1A3 1A4 1A5 1A6 1A7b lBlb 1B2 1B3 1B4 1B5 1B6 1B7b

K L 2.08 3.19 1.67 1.33 2.77 2.20 2.04 2.13 1.28 1.09 2.56 2.52 2.51 2.05 1.22 1.42 3,87 4.03 1.66 2.07 2.58 1.80 1.36 2.84 3.04 2.19 3.24 1.79 2.71 1.47 1.17 2.51 1.89 1.83 1.95 1.11 1.01 2.16 2.17 2.16 1.57 1.03 1.35 3.10 2.60 1.41 1.85 2.13 1.57 1.27 2.60 1.89 1.86 2.80 0.91 1.67 2.36 2.43 1.96 1.33 1.07 1.05 2.23 2.55 2.56 2.07 1.40 1.16

A x 0.0025 0.0024 0.0140 0.0142 0.0043 0.0054 0.0067 0.0058 0.0071 0.0066 0.0081 0.0074 0.0056 0.0058 0.0135 0.0154 0.0045 0.0034 0.0103 0,0109 0.0084 0.0131 0.0141 0.0040 0,0042 0.0102 0.0067 0.0152 0.0108 0,0247 0.0186 0.0082 0.0037 0,0095 0.0117 0,0068 0.0062 0,0124 0.0124 0.0089 0,0052 0.0203 0.0240 0.0071 0,0035 0.0168 0.0219 0.0126 0.0224 0.0174 0.0085 0.0037 0.0150 0.0107 0,0160 0.0140 0.0103 0.0072 0.0088 0.0142 0.0166 0.0174 0,0154 0.0105 0.0081 0.0077 0,0106 0.0141

Table 111. Calculated Vapor Contact Times AY

0.0366 0.0406 0.1139 0.1427 0.0699 0.0998 0.0964 0.0514 0.1357 0.0855 0.1424 0.1320 0.0756 0.1082 0.1351 0.1123 0.0973 0.1223 0.1210 0.0947 0.1498 0.1168 0.1588 0.0663 0.1331 0.1649 0.1334 0.2809 0.2829 0.2501 0.2703 0.1823 0.1416 0.1755 0.1584 0.1585 0.1173 0.2683 0.2735 0.1626 0.1321 0,2694 0.2540 0.1679 0.1504 0.2644 0.2480 0.2628 0.2415 0.2740 0.1841 0.1314 0.2761 0.2744 0.0862 0.1139 0.1365 0.1170 0.1308 0.1427 0.1597 0.1062 0.1816 0.1614 0.1424 0.1268 0.1198 0.1363

ATL 0.0248 0.0233 0,0981 0.1094 0.0382 0.0558 0.0983 0,0484 0.3164 0.1223 0,0599 0.0531 0.0585 0.1200 0.0999 0.1082 0.0600 0.1025 0.0735 0,0809 0.0651 0.0915 0.1250 0.0368 0,1908 0.0929 0.0482 0.5698 0.3937 0.6457 0.6768 0.3995 0.3310 0.5126 0.5158 0.5840 0.5350 0.3843 0.4091 0.3679 0.4583 0.7524 0.6592 0.2963 0.2948 0.5599 0.5553 0.3702 0.5371 0.6720 0.4002 0.3161 0.5270 0.3523 0.1206 0,0981 0.0743 0,0508 0.0631 0.1094 0.1517 0.1234 0.1796 0.0906 0.0599 0.0553 0,0765 0,1059

ATv 1.04 1.09 4.40 7.31 2.89 4.95 5.50 1.86 9.65 4.80 6.83 5.86 3.50 6.58 6.59 4.32 5.17 7.98 5.22 2.91 7.53 4.62 8.94 2.79 9.30 9.12 6.00 15.41 15.54 14.14 14.98 10.17 7.12 9.94 9.03 8.12 5.96 14.99 15.19 9.30 6.77 14.95 14.31 9.55 7.63 14.79 14,OO 14.76 13.70 15.13 10.33 6.74 15.29 15.24 2.44 4.40 6.27 4.57 5.88 7.31 9.10 3.78 10.70 8.62 6.83 5.53 5.13 6.79

K 47.34 62.28 64.99 40.47 52.53 36.44 42.63 72.81 20.25 27.15 45.16 44.29 56.59 34.00 37.16 61.13 54.75 35.22 43.60 76.12 44.83 63.56 36.50 52.28 29.23 41.72 50.07 31.86 34.13 47.32 24.95 37.45 18.57 32.15 47.74 16.40 21.48 32.30 32.16 38.27 22.86 24.19 41.75 42.43 22,56 27.67 52.98 33.01 46.81 24.99 39.48 19.86 33.19 35.67 52.74 64.99 50,79 47.07 41.18 40.47 33.70 53.11 59 I 44 51.62 45.16 39.02 37.89 36.16

tu&

0.3680 0.2126 0.2003 0.5023 0.3019 0.6234 0.4508 0.1583 1.9840 1.1482 0,4108 0.4257 0,2605 0.7229 0.5926 0.2255 0.2795 0.6776 0.4303 0,1450 0.4180 0,2092 0.6129 0.3020 0,9629 0.4789 0.3294 0,8732 0.7614 0.4012 1.3571 0.6306 2.5800 0.8253 0.3931 3.1346 1.9778 0.8465 0.8541 0.5974 1.7059 1.4368 0.5135 0.4879 1.7451 1,0955 0.3186 0.8090 0.4068 1.3378 0.5635 2.2457 0.8038 0.6927 0.3028 0.2003 0.2715 0.3764 0.4897 0.5023 0.7199 0.2989 0.2430 0.3186 0.4108 0.5413 0.5650 0.6172


Table 111. (ConfinuedJ Run no.

1Clb 1 c 2 1C3 1 C4 1C5 1C6 1C7b 2Dlb 2D2 2D3 2D4 2D5 2D6 2D7b 2Dlb 2D2 2D3 2D4 2D5 2D6 2D7b 2Elb 2E2 2E3 2E4 2E 5 2E6 2E7b 2Elb 2E2 2E3 2E 5 2E6 2E7b 2Flb 2F2 2F3 2F4 2F5 2F6 2F7b 2Flb 2F2 2F3 2F4 2F5 2F6 2F7b

K L

0.90 1 .80 2.46 2.52 1.94 1.36 1 .23 0 .86 1.57 2.09 2.21 1.92 1.46 1.16 0.76 1 ,57 2.11 2.16 1 .74 1.27 1 .25 0 .75 1.50 2.06 2.16 1.79 1.20 1 .08 0.85 1.52 2 .03 1.72 1.24 0 .98 0.82 1.47 2.00 2.07 1.66 1.17 1 .05 0.82 1 .51 2.02 2.12 1.81 1.30 1.04

Ax

0.0154 0.0131 0.0084 0.0078 0,0092 0.0141 0.0154 0.0307 0.0226 0.0150 0.0130 0.0124 0.0164 0.0167 0.0290 0.0224 0.0151 0.0124 0.0133 0.0174 0.0145 0.0304 0.0242 0.0146 0.0124 0.0134 0.0180 0.0205 0.0306 0.0239 0.0151 0,0141 0.0171 0.0220 0.0299 0.0247 0,0158 0.0125 0.0135 0.0186 0.0205 0.0313 0.0251 0.0159 0.0130 0,0129 0.0165 0.0198

AY

0.0979 0.1168 0.1229 0.1352 0.1403 0.1588 0.1673 0.2315 0.2568 0.2533 0.2737 0.2916 0,2850 0.2851 0.2108 0.2415 0.2618 0.2641 0.2704 0.2740 0.2853 0,2064 0.2590 0.2668 0.2683 0.2682 0.2784 0,2782 0.2728 0,2563 0.2571 0.2645 0.2739 0.2699 0.2146 0,2501 0,2603 0.2647 0,2724 0.2703 0.2772 0.2409 0,2620 0.2684 0.2792 0.2892 0.2854 0.2816

ATL

0.1120 0.0915 0.0594 0,0564 0.0691 0.1250 0,1525 0.6432 0.6277 0.3963 0.4371 0.5729 0,7992 0.9293 0,5038 0.5371 0.4366 0.3722 0.4469 0.6720 0,7915 0.5126 0,6934 0.4398 0.3843 0,4320 0.7445 0.9240 0.9422 0,6665 0.4130 0.4397 0.6449 0.8620 0.5464 0,6457 0.4493 0.3749 0.4604 0.6768 0.9131 0.7137 0.7479 0.4951 0,4630 0.5685 0.8136 0.9528

A Tv 3.15 4.62 5.06 6.22 6.85 8.94 9.89

13.17 14.46 14.30 15.22 15.64 15.26 15.03 11.99 13.70 14.68 14.81 15.07 15.13 15.03 11.76 14.54 14.92 14.99 15.00 15.24 15.11 15.14 14.42 14.48 14.83 15.14 14.94 12.23 14.14 14.62 14.81 15.12 14.98 15.04 13.64 14.67 14.96 15.38 15.56 15.18 15.22

K V

43.06 63.56 53.13 45.17 39.28 36.50 34.18 36.03 45.46 40.06 34.12 26.74 26.39 21.70 32.28 46.81 39.62 32.60 27.10 24.99 19.61 34.28 45.87 37.09 32.30 28.51 24.54 24.76 31.99 46.46 38.83 28.85 24.23 24,37 35,64 47.32 39.72 31.58 26.39 24.95 24.05 33.97 47.79 39.28 32.32 26.33 23.95 23.06

fua

0.4540 0,2092 0.2973 0.4082 0.5334 0.6129 0,6969 0.6908 0.4349 0.5543 0.7541 1.2157 1.2026 1.7718 0.8514 0.4068 0.5653 0,8240 1,1666 1.3378 2.1192 0.7576 0.4270 0,6510 0.8465 1 ,0663 1.4076 1,3574 0.8869 0.4166 0.5913 1.0357 1 ,4395 1.3931 0.7013 0.4012 0.5658 0.8822 1,2457 1.3571 1,4321 0,7766 0.3949 0 5784 0,8476 1 ,2552 1.4667 1.5780

Vapor contact times are reported in units of seconds. * These data points were not used in the data analysis, but are included for completeness.

Calculation of Vapor Contact Times

By solving Equation 32 for to , the vapor contact times can be calculated directly from values of the vapor phase mass transfer coefficient, K ,. The form of the resulting equation is

To calculate the values of K,, one must determine the interfacial film composition so that the vapor phase composition driving force can be evaluated. It is then necessary to determine the value of K L from Equation 33 using the values of tL from Table I. Once KL was computed, the liquid film composition, xl,i, was solved by using Equations 20 and 31, and then a bubble point calculation was performed to determine yl,( .

Equation 19 was then used to solve for values of K,. The computed values for the liquid mass transfer coefficient ( K L ) , liquid phase composition gradient (xl , b - x1 ,%), vapor phase composition gradient (yl ,{ - yl , a ) , liquid phase temperature gradient ( T , - TL), vapor phase temperature gradient ( T , - Ti), vapor phase mass transfer coefficient (Ku) , and the vapor contact time ( t u , sec) are contained in Table 111. I n the above computations, arithmetic averages of the compositions and temperatures of entering and exiting streams of tray two were used for the respective bulk phase compositions and temperatures.

The effect of the two different thermal binary systems on the heat flux a t the interface can be analyzed from the data contained in Table 111. The average value of the K L / K , ratio for

Ind. Eng. Chern. Process Des. Develop., Vol. 1 1 , No. 4, 1972 599

Table IV. Predicted Valve Tray Efficiencies Run no.

101 102 124 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 125 126 127 201 202 203 204 205 206 207 208 209 210 21 1 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 l A l a 1-42 1A3 1A4 1A5 1A6 1A7a lB la 1B2 1B3 lB4 1B5 lB6 1B7a

% Flood Exp. NI,T 58.10 0.85 59.32 0.99 62.52 2.02 29.64 1.87 90.00 6.18 56.76 1.89 61.91 2.09 91.02 3.83 28.27 0.94 89.83 2.83 28.30 0.43 60.70 3.63 60.79 3.31 60.02 2.22 60.01 2.00 90.55 5.55 27.74 1.82 62.30 2.27 61.60 1.91 91.55 4.79 30.27 1.42 61.52 3.84 29.71 1.92 91.13 6.42 61.50 2.43 62.62 4.48 62.37 3.45 60.55 7.20 60.11 6.07 29.64 3.98 89.46 10,Ol 61.25 4.60 60.80 1 .63 90.50 6.99 31,62 2.45 89.56 3.12 30.00 0.59 59.22 6.68 59.44 6.71 59.86 4.62 60.90 2.02 90.04 10,26 30.43 3.91 59.86 4.11 60.90 1.78 89.48 9.48 30.43 3.54 59.78 6.92 30.26 3.97 89.80 10.06 61.88 5.00 60.00 1 ,58 59.78 7.34 59,56 6.26 15.01 0.86 29.64 1.87 44.92 2.84 60,35 3.07 75.68 4.29 90.00 6.18 97.25 6.57 16,78 1.10 32.22 2.73 44.25 3.04 60.70 3.63 75.17 3.89 90.81 4.83 97.25 5.89

Pred. N I , T 1 .03 1.04 2.03 1.79 5.54 1 .98 1 .81 3.84 0.96 3.48 0.47 3.73 3.44 2.22 1.68 5.31 1.79 2.51 1.83 4.30 1.40 3.97 1.87 6.32 2.52 4.76 3.43 7.17 6.46 3.79

10.13 4.69 1.87 6.62 2.51 3.31 0.73 6.75 6.81 4.65 1 .84

10.49 4.07 4.04 1.92 9.39 3.56 6.78 3.80

10.17 4.81 1.60 7.08 6.38 0.95 1.79 2.82 3.01 4.17 5.54 6.76 1.22 3.02 3.32 3.73 3.99 4.55 5.63

Exp. N Z . T 0.96 1.13 2.03 1.63 6.56 1 .88 2.32 4.10 0.79 3.82 0.57 3.67 3.29 2.22 2.44 5.83 1.57 2.29 2.46 4.97 1.21 3.92 1.69 6.92 3.33 4.74 3.44 5.48 4.60 2.78 7,52 3.52 1 .69 5.55 1.76 3.09 0.73 4.89 4.97 3.38 2.06 7.75 2.80 3.01 1.82 7 .03 2.47 5.00 2.72 7.68 3.81 1.64 5.53 4.65 0.54 1.63 2.76 3.03 4.40 6.56 7.09 0.75 2.86 3.05 3.67 3.95 5.01 6.21

Pred. N Z . T Exp. E,MV 0.86 0.88 1.92 1.64 5.89 1.87 1.99 3.94 0.71 4.79 0.50 3.81 3.51 2.11 2.04 5.68 1.65 2.39 2.36 4.56 1.26 4.06 1.72 6.73 3.36 4.92 3.49 5.50 4.94 2.62 7.64 3.65 1.93 5.18 1.84 3.35 0.73 4.95 5.06 3.42 1.86 7.92 2.86 2.98 1.99 6.95 2.45 4.90 2.56 7.75 3.70 1.65 5.34 4.76 0.74 1.64 2.75 3.08 4.38 5.89 7.25 1 01 2.87 3.25 3.81 4.18 4.88 6.03

67.35 68.99 72.72 78,21 74.42 70.57 62.61 73.77 82.20 59.47 43.00 66.94 65,48 77.62 68.23 71.77 81.12 64.83 60.34 68,69 70,76 67.17 76.90 70,62 74.15 68.99 64.43 59.11 50,24 64.38 56.30 59.86 53.26 64.93 64.93 73.27 48,64 55.33 55.47 63.18 90.80 57.62 62.46 56.19 51.69 53.31 56.97 56.55 63.59 57.23 63.22 63.76 60.60 51.94 80.81 78.21 71.34 67.67 68.40 74.42 68.04 78.88 75.87 68.25 66.94 65.27 69.80 70.32

Pred. EMV 68.22 61.00 70.39 74.58 66.10 71.58 53.52 71.92 77.95 74.04 42.41 68.29 68.11 75.12 56,66 68.55 79.72 69.09 57.63 61.76 69.96 68.62 74.56 68.51 75.95 71,65 63.94 58.71 53.18 60.65 56.85 61.24 61.10 61.20 66.63 78.21 55.83 55.65 56.04 63.60 82.51 58.64 63.58 55.33 55.90 52.60 56.40 55.26 60.01 57.61 60.91 64.42 58.27 52.73 93.07 74.58 69.68 66.83 66.68 66.10 69.19 90.70 78.80 72.50 68.29 67.36 66.21 67.12

70 Dev. -1.30 11.58 3.21 4.65

11.17

14.52 2.52 5.17

-24.49 1.39

-2.01 -4.02

3.23 16.96 4.49 1.72

-6.57 4.49

10.09 1.13

-2.16 3.05 2.99

-2.42 -3.86

-1.43

0.78 0 .68

-5 .85 5.79

-0.97 -2.29

-14.72 5.74

-2.62 -6.75

-14.76 -0 ,56 -1.02 -0.66

9.13 -1.76 -1.79

1.53 -8.15

1.33 1 .oo 2.29 5.63

-0.66 3.65

-1.04 3.85

-1.53 -15.17

4.65 2.33 1.25 2.52

11,17 -1.69

-14.98 -3.85 -6.23 -2.01 -3.21

5.14 4.55


Table IV. (Continued)

Run no.

1Cl" 1C2 1 C3 1C4 1C5 1C6 lC7" 2Dla 2D2 2D3 2D4 2D5 2D6 2D7a 2Dla 2D2 2D3 2D4 2D5 2D6 2D7" 2El" 2E2 2E3 2E4 2E5 2E6 2E7a 2El" 2E2 2E3 2E5 2E6 2E7" 2Fla 2F2 2F3 2F4 2F5 2F6 2F7" 2FIu 2F2 2F3 2F4 2F5 2F6 2F7"

% Flood 13.66 29.71 45.55 62.37 76.97 91.13 94 t 45 15.00 30.46 45.33 60.68 75.52 88.64 97.78 13.04 30.26 45.84 61.29 75.85 89.80 96.89 13.11 29.22 45,33 59.22 72.93 91.05 96.44 15.18 29.64 44.44 74.59 88.89 97.56 14.89 29.64 45.44 60.44 75.44 89.46 96,61 14,96 30.17 45,56 60.89 75.57 90.56 97.22

Exp. NI.T

0.82 1.92 2.59 3.74 4.72 6.42 6.74 1.87 3.95 5.50 7 .33 7.99

10.12 9.40 1.53 3.97 5.59 7.11 8.39

10.06 8.50 1.62 3.83 5.15 6.68 8.26

10.02 10.92 1.71 3.93 5.22 8.63 9.61

11.09 1.80 3.98 5.52 6.75 8.10 10,Ol 10.66 1.78 4.18 5.51 6.99 7.94 9.49

10.18

Pred. N ~ , T

1 . 0 3 1.87 2.60 3.70 4.69 6.32 6.94 2.40 3.89 5.25 6.94 8.49 9.87

10.58 2.13 3 .80 5.36 6 .95 8.61

10.17 10.51 2.12 3.78 5.32 6 .75 8.26 0.30 0.80 2.50 3.82 5.19 8.41 0.08

10.99 2.32 3.79 5.30 6.86 8.57

10.13 10,80 2.43 3.92 5.35 6.97 8.53

10.05 10.80

Exp. N Z . T Pred. N ~ , T Exp. Euv

0.45 1.69 2.46 3.77 4.89 6.92 7.34 1.18 2.82 3.90 5.45 6.39 8.14 7.81 0.88 2.72 4.04 5.18 6.24 7.68 7.07 0 .91 2.76 3.76 4.89 6.08 7.74 8.58 1.30 2.81 3.73 6.32 7.31 8.45 1.07 2.78 3.96 4.90 6.02 7,52 8 .35 1.17 3.02 4 .03 5.26 6.25 7.64 8.14

0.81 1.72 2.56 3.81 4.93 6 .73 7.43 1.49 2.74 3.69 5.17 6.81 7.93 8.73 1 .23 2.56 3 .85 5.05 6.40 7.75 8 .65 1.21 2.69 3.87 4.95 6.09 7.96 8.48 1.79 2.69 3.68 6.16 7.66 8.37 1.37 2.62 3.80 4.99 6.40 7.64 8.46 1.55 2.82 3.93 5.29 6.76 8.10 8.61

73.60 76,90 68.92 68.22 68.03 70.62 68.87 61.30 62.60 59.10 59,47 55.75 62,21 54.70 59.59 63.59 59.25 56.89 55,09 57.23 49.67 61 . 7 l 63,41 55.51 55.33 55.96 56.59 59.87 56.34 63.76 57.23 57.24 54.85 58.20 60.30 64,38 58.88 54,75 53,31 56.30 58.16 58.74 65,62 58.81 56.73 54.44 56.76 56.19

a These data points were not used in the data analysis, but are included for completeness.

the 27 design points of the thermally negative benzene-n- propanol system is 0.0546. The same average value for the thermally positive n-propanol-toluene system is 0.0436. The greater value of the thermally negative system indicates, as speculated earlier, that the importance of the liquid phase resistance is accentuated for thermally negative systems. As a result of this increased liquid phase resistance, the composition and temperature gradients in the liquid phase are greater for the benzene-n-propanol system. Recalling the definition of the heat flux a t the interface,

4 = 4L - 4 , = h ( T , - TL) - h" (T , - T,) (17) it would be expected that the values of the heat flus would be greater for the benzene-n-propanol system because of the

Pred. E.UV

99.39 74.56 69.24 67,31 67.41 68.51 69.76 74.69 60.82 55.93 56.19 58.92 60.54 61.04 78.04 60.01 56.46 55.38 56.25 57.61 60,64 77.13 61,60 56.79 55.65 55.82 57. 92 59.06 76.61 61.08 56.35 55.64 57.18 57,53 73.92 60.65 56.29 55.38 56.15 56.85 58.75 75.09 61.05 56.90 56.48 58,19 59.80 59.21

% Dev. -35.04

3.05 -0.46

1 .35 0.92 2.99

-1.30 -21.85

2.84 5.36 5.51

-5.69 2.68

-11.60 -30.96

5 .63 4.72 2.67

- 2 . 1 1 -0.66

-22.09 -24.99

2.85 -2.31 -0.56

0.24 -2.36

1 . 3 5 -35.97

4.22 1 , 5 5 2.79

-4 .26 1.14

-22,59 5.79 4.41

-1.14 -5.33 -0.97 -1.00

-27.85 6.96 3.24 0.44

-6.89 -5 .36 -5.38

greater liquid phase temperature gradients. This is verified by the heat data plotted in Figure 7 which shorn that the total heat transferred across the interface for the benzene-n-propanol system is always greater than that of the n-propanol- toluene system.

Inspection of tabulated values of t,' leads to the following observations: The vapor contact times increase with increased vapor flow rates and the vapor contact times for the n- propanol-toluene system are smaller than those for the benzene-toluene system. These effects are illustrated in Fig- ure 8 for the 3 and E data sets.

It is interesting to compare the vapor contact times with the average vapor residence times. In this study, the ratio of the two times (contact time divided by residence time) had


8 5 . 0

8 0 . 0 A SET I I I I I I

- L c E 85.01 I I I I 1 I 1

U W

C SET

7 0 . 0 6 5 . 0 - 60.0 I I I I I I I

25.0 35.0 45.0 55.0 65.0 75.0 85.0 95.0 PERCENT FLOOD

Figure 9. Predicted efficiencies n-propanol-toluene system

D SETS e .

55.0 50.0 45 .0 I I I 1 I I I - L

0 45.0 LL LL u

7n n, I I I I

F SETS

4 5 50 0 O* 2 5 0 3 5 0 450 5 5 0 6 5 0 7 5 0 8 5 0 9 5 0

PERCENT FLOOD

Figure 10. Predicted efficiencies benzene-n-propanol system

values ranging from 1.5 to 6.0. Hughmark (1965) originally assumed the two times were equal, but in a later study (1970) hypothesized that the ratio of times should have a range of values from 4.0 to 15.0.

Analysis of Calculated Heat Flux at Interface

The heat flux a t the interface is fundamentally defined as

q = k,(T, - TL) - h, (T , - Ti) (17) where

hL = liquid heat transfer coefficient, Btu/ftz hr, O F h, = vapor heat transfer coefficient, Btu/ft2 hr, O F

The heat transfer coefficients were estimated by the Chilton- Colburn analogy (1934) given below.

hL = ( p ~ , ) ~ ( s c / P r ) , ~ ~ K ~ (41)

h, = (p~,),(sc/Pr)*/~K,, (42) where

c, = heat capacity, Btu/lb, O F Sc = dimensionless Schmidt number Pr = dimensionless Prandtl number

The Schmidt and Prandtl numbers are defined by the following equations:

P sc = - PD

C P P Pr = - k

(43)

(44)

where

k = thermal conductivity, Btu/ft hr, O F

In correlating the heat data, the total heat transferred (Q) was easier to predict than the heat flux (9). A regression analysis of the tabulated data resulted in the following equation:

Q = -5744 + a0.[2.25 hL*(T, - T,5,b)] - 21.88 [1.0 - 0.284 ( Z . J / L ) ~ ] ~ , * ( T , , , - Ti) (45)

where

Q = total heat transferred, Btu,hr h ~ * = ( P C , ) ~ ( S C / P ~ ) , ~ ~ K ~ Btu/ft2 hr, O F h,* = ( ~ C , ) ~ , ( S C / P ~ ) ~ ~ ~ K , Btu/ft2 hr, OF The multiple correlation coefficient for the regression was

0.986. The average absolute deviation of the fitted data from the predicted values was 1031 Btu/hr (15.5%). This compares well with the standard deviation of the heat data of 8238 Btu/hr.

Analysis of Calculated Vapor Contact Times

Examination of all the possible first- and second-order effects of the independent variables produced the following equation for the prediction of the vapor contact times:

1.0 - 1.946 )] (46) V where

ab = interfacial area per square foot of bubbling surface = a 2 f/ 1 2 * 0

The least-squares regression analysis of the data shows an average absolute deviation of 0.033 sec (4.40%), a maximum deviation of 0.095 sec (25.25%), and a multiple correlation coefficient of 0.980.

Prediction of Tray Efficiency

computed for the test tray by the following procedure: The Murphree tray efficiencies for the vapor phase were


1. Values of K L and K , were computed by using Equations 32 and 33.

2. The heat flux, q, was initially assumed to be equal to zero and the interfacial liquid composition computed from Equation 24.

3. A bubble point calculation a t the interface was made to determine the interface vapor composition and temperature.

4. The value of the heat flux, q, was determined by using Equation 45.

5. If we use the predicted value of q, steps 2, 3, and 4 were repeated until the difference in the computed interface temperatures between successive iterations was less than 0.002"F.

6, The value of the molal flux for component one was then calculated by using Equation 20.

7. Equation 16 was used to compute the molal flux of the second component.

8. The molal fluxes were multiplied by the computed total interfacial area, Equation 37, to yield the total moles exchanged between the phases.

9. The compositions of the vapor and liquid streams leaving the second tray were then computed by making a material balance around the second tray.

10. When we used these compositions, the Murphree tray efficiency for the vapor phase was computed from Equation 1.

The experimental and predicted total number of transferred moles of each component and the Murphree tray efficiencies are reported in Table IV. The percent deviation between the experimental and predicted efficiencies is reported also. The average absolute value of the deviations reported in Table IV is 4.1770, (The 15 and 97Y0 flood points were not included.) The deviation of data point 109 (24.49%) represents the maximum value reported. If this point is neglected, the average absolute deviation drops to 3.96%. The predicted efficiencies for the 97y, of column flood points, with the exclusion of run 207, show good agreement with the experimental values. The average absolute deviation of these points is 3.25%.

The experimental efficiency data are compared graphically with the predicted values in Figures 9 through 12. The supplementary data are illustrated in Figures 9 and 10, and the experimental design data are illustrated in Figures 11 and 12 for the benzene-n-propanol system a t the different values of weir heights and L/V ratios.

Conclusions

The experimental results of this study provide a clearer understanding of the heat and mass transfer mechanisms occurring between the distilling phases on the valve tray. I n particular, the effects of the thermal properties of the distilling components (molar latent heats of vaporization) and the heat transfer across the interface of mass between the phases were determined.

The experimental results show that the tray efficiencies for thermally positive systems (as classified by the unique definition presented in this work with respect to the ratio of the molar latent heats of the components) are larger than those for thermally negative systems. Also, as postulated, the liquid phase resistance to mass transfer was accentuated for thermally negative systems which, in turn, had a marked effect on the direction of the heat flux a t the interface.

The proposed two-film theoretical mass transfer model, which incorporated the effects of the molar latent heats of vaporization of the components and the heat transfer across the interface, successfully predicted the experimentally determined number of moles exchanged between the distilling

80.0- 70.0-

H, = 1.0 - -

c C : 90.0 I I I I 1 I

80.0 H w 2.0

50.0 0 40.0 LL

- b 70.01

LL w

H, = 3.0 ;;;;I 50.0 40.0

25.0 35.0 45.0 55.0 65.0 75.0 85.0 95.0 PERCENT F L O O D

Figure 1 1 . Predicted efficiency for different weir heights Benzene-n-propanol system L/V = 1 .O

c 4-

I I 1 I I I 1

3 4001 I I I I I I LL L w

90.0 I I I I I I I I L i V = 1.5

50.0 40.0

25.0 35.0 45.0 55.0 65.0 75.0 85.0 95.0 PERCENT F L O O D

Figure 12. Predicted efficiency for different L/V ratios Benzene-n-propanol system, H w = 2 in.

phases. KO significant effect on the transfer of mass could be detected for the two different surface tension gradient systems (as defined by Zuiderweg) ; consequently this physical property was not incorporated into the model. The computed Murphree tray efficiencies based upon the predicted transfer showed good agreement with the experimental tray efficiencies.

The vapor phase contact time, liquid phase residence time, heat flux a t the interface, and total interfacial area used in the mass transfer model were computed from empirical equations developed in this study.

The next logical step in this continuing study is to extend the correlations presented in this article to larger size columns


operating at nonatmospheric pressures. This step is currently under investigation.

Nomenclature

a = interfacial area, itZ a interfacial area per unit volume of gas and liquid

ab = interfacial area per square foot of bubbling surface,

c = molar concentration, lb mol/lt3 cD = heat capacity, Btu/lb, OF DL = liquid diffusion coefficient, ft2/hr D , = vapor diffusion coefficient, ft2/hr E = efficiency E.ML = RIurphree tray efficiency in liquid terms

Murphree tray efficiency in vapor terms EAMVc = Murphree tray efficiency in vapor terms, cor-

rected for heat transfer through downcomer walls Eoc = overall point efficiency in vapor terms F , = C s ( P v 10 .5 = F,-factor, ( f t / s e~ ) ( lb / f t~ )~ ~ F , = Bolles correction factor for segmental downcomers F F = raZp,,/(pL - p v ) = F F , average frothing factor h = heat transfer coefficient, Btu/ft2 hr, O F H,, = crest over the weir, in. H , = weir height, in. k = thermal conductivity, Btu/ft hr, O F KL = liquid phase mass transfer coefficient, ft/hr K , = vapor phase inass transfer coefficient, ft/hr L = liquid flow, lb mol/hr L, = length of weir, in. m = vapor-liquid equilibrium constant m = slope of equilibrium line, dy/dx N c = gas film transfer unit S L = liquid film transfer unit NoG = overall gas transfer unit P = pressure, atm Pr = dimensionless Prandtl no. q = heat flux a t the interface, Btu/ft2 hr Q = total heat transferred across interface, Btu/hr Q L = liquid flowv, gal./min QL = liquid flow, ft3/hr Sc = dimensionless Schmidt 110. L L = average liquid residence time, hr tL = liquid contact time, hr t, = vapor contact time, hr T = temperature, OF AT = temperature gradient between bulk of phase and

U , = vapor velocity based on active or bubbling tray

V = vapor flow, lb mol/hr x = liquid composition, mole fraction Ax = composition gradient between bulk liquid phase

y = vapor composition, mole fraction Ay = composition gradient betrveen bulk vapor phase

2 = height in column, ft 2, = clear liquid height, in. 2, = froth height, in.

= holdup, l/ft

ftZ/f tZ

=

its interface

area, ft/sec

and its interface, mole fraction

and its interface, mole fraction

GREEK LETTERS

X = molar latent heats of vaporization, Btu/lb mol 1.1 = viscosity, cP T = constant, 3.14 p = density, Ib/ft3 u = surface tension, dyn/cm

SUPERSCRIPTS .4SD SUBSCRIPTS

b = average bulk value i = interface j = componentj L = liquid phase n = tray n T = total V,v = vapor phase 1 = component 1 2 = component 2

literature Cited

A.1.Ch.E. Bubble-Tray Design Manual, American Institute

Bainbridge, G. S., Sawistowski, H., Chem. Eng. Sci., 19, 992-3

Bolles, W. L., Petrol. Refiner, 25, 613 (1946). Chilton, E. H., Colburn, A. P.,Ind. Eng: Chem., 26, 118: (1934). Danckwerts, G. C., Sawistowski, H., Smith, W., in Intern. Symp.

on Distillation, Inst. Chem. Eng., London, pp 7-12, May 1960.

Everitt, C. T., Hutchinson, H. P., Chem. Eng. Sci., 21, 883 (1966).

Fane, B., Sawistowski, H., zbid., 23, 943-5 (1968). Haselden, G. G., Sutherland, J. P., Intern. Symp. on Distillation,

Inst. Chem. Eng., London, pp 27-32, Nay 1960. Higbie, R., Trans. Anter. Inst. Chem. Eng., 31, 365 (1935). Holland, C. D., lIcMahon, K. S., Chem. Eng. Scz., 25, 431-6 Hughmark, G. A., Fifty-fifth Xat. Meeting, Symp. on Distilla-

Hughniark. G. A . AIChE J . . 1611). 147-8 11970).

of Chemical Engineers, New York, N.Y., 1958.

(1964).

(1970 j.

tion, Houston, Tex., February 1965. I ~ - \ I , ~ \ - . -,-

Le&, W. k., In&. Eng. Chem., 14, 492 (1922). Liang, S. Y., Smith, W., Chem. Eng. Sci., 17, 11-21 (1962). Lowry, R., Foaming and Frothing Related to S p t f m Physical

Properties in a Small Perforated Plate Distillation Column, PhD dissertation, University of Taas , June 1967.

Murphree, E. V., Ind. Eng. Chem., 17, 747 (1925). Nord, M., zbzd., 38, 637 (1946). Norman, W. S., Cakaloz, T., Fresco, A Z., Suthchffe, D. H.,

31i, Bull. Inst. Politechnzc Bacuresti, 23, 129 (1961 ). Ruckenstein, Eli, Smigelischi, O., Chem. Eng. Scz., 20, 66-69

Ruckenstein, Eli, Smigelschi, O., Can. J . Chem. Eng., 45, 334-40

Ruckenstein, Eli, AIChE J . , 16, 144-6 (1970). Sawistowski, H., Smith, W., Ind. Eng. Chem., 51, 915-18 (1959). Smoker, E. H., Trans. Amer. Inst. Chem. Eng., 34, 165 (1938). Souders, Mott, Jr., Brown, G. G., Ind. Eng. Chem., 26,98 (January

Trons Tnst. Chem. Eng., 41, 61 (1963).

(1965).

(1967).

19x41 Stand&, G., Chem. Eng. Scz., 20, 611-22 (1965). Todd, W. G., Van Winkle, Matthew, Ind. Enq. Chem. Process

Des. D e d o p . , 17(4), 578 (1972). Underwood, A. J., Trans. Inst. Chem. Eng., 10, 112 (1932). Zuiderweg, F. J., and Harmens, 9. Chem. Eng. Sci., 9,89 (1958)

RECEIVED for review March 13, 1972 ACCEPTED June 5, 1972

604 Ind. Eng. Chem. Process Des. Develop., Vol. 11, No. 4, 1972

correlation of valve tray efficiency data

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