Ind. Eng. Chem. Res. 1993,32, 701-708
70 1
Prediction of Point Efficiency for Sieve Trays in Distillation Guang X. Chen and Karl T. Chuang' Department of Chemical Engineering, The University of Alberta, Edmonton, Canada T6G 2G6 A new semiempirical model for the prediction of the number of mass-transfer units, hence tray efficiency, for distillation has been developed. The interfacial area of sieve tray dispersion is estimated by the Levich theory, and the mass-transfer coefficients are determined by the penetration theory. Two constants that are required to complete the model were obtained by fitting the model t o the tray efficiency data of cyclohexaneln-heptane mixtures published by Fractionation Research Inc. (FRI). The predicted efficiency and percent liquid-phase resistance are in good agreement with the experimental values for ethylbenzene/styrene published by Billet and additional data from FRI.
Introduction In the design of sieve tray columns for distillation, tray efficiency is used to convert the number of theoretical plates to actual plates. The number of theoretical plates can be calculated accurately with modern computer methods, but the method for predicting tray efficiency does not always give satisfactory results. The problem can be divided into two parts: the prediction of point efficiency and the relation of point efficiency to tray efficiency. The latter has been a subject of many studies (Porter et al., 1972; Lockett et al., 1973; Lockett and Safekourdi, 1976;Lim et al., 1974), and it was found that reliable calculations can be made (Biddulph et al., 1991). On the other hand, the prediction of point efficiency remains uncertain and little real progress has been achieved since the AIChE Bubble Tray Design Manual was published more than thirty years ago (AIChE, 1958). Many methods for predicting mass-transfer efficiencies have been proposed. These methods are based either on empirical correlations or on semiempirical correlations. Examples of empirical correlations are those of Drickamer and Bradford (1943), O'Connell (19461, and Bakowski (1969). These correlations have not always been validated, and the results are usually very conservative. The most often used semiempirical correlation is the AIChE method, although it has been criticized by many authors for being unreliable (Lashmet and Szezepanski, 1974; Hughmark, 1971; Strand, 1963; Neuburg and Chuang, 1982). Other correlations include those of Harris (1965), Asano and Fujita (1966),Jeromin et al. (19691,and Hughmark (1965). All of these models are developed on the basis of data from absorption, stripping, and humidification, but not distillation. Chan and Fair (1984) proposed a new correlation for NGin the AIChE model by using distillation data. Zuiderweg (1982) reviewed sieve tray performance and developed a correlation to predict point efficiencies based on the FRI results (Sakata and Yanagi, 1979;Yanagi and Sakata, 1982)by using the slope and intercept method. However, the systems involved have a narrow range in the slope of the vapor-liquid equilibrium line (m). Therefore the analysis may not be accurate. Apart from the above methods, there is another group of methods aimed at individually predicting the mass-transfer coefficients and interfacial area. This approach has been reported regularly over the years (Geddes, 1946; West et al., 1952; Garner and Porter, 1960; Calderbank and Moo-Young 1960, Andrew, 1961; Goederen, 1965, Fane and Sawistowski, 1969; Stichlmair, 1978; Neuburg and Chuang, 1982), but none of them have gained practical acceptance. Most recently, Prado and Fair (1990)developed another semiempirical model for the prediction of tray efficiencyalsobased
on information from absorption and humidification in an air-water system (Fair and Prado, 1987). The purpose of this research is to develop correlations for predicting the number of individual mass-transfer units and the liquid-phase resistance consistent with masstransfer theories, and to compare the prediction with distillation data from commercial-size sieve tray columns.
Theory Based on the two-film or two-resistance theory, the following equation is obtained:
1IKOG =
+ (mPGML)/(kflLMG)
(1)
Dividing eq 1 by ( a t d gives l/NoG = 1/NG + X/NL
(2)
where
NOG= ~ K O G ~ G
(3)
NG
= UkGtG
(4)
NL
= akLtL
(5)
and tG = hf/u, tL = (GtGPLMG)/(LP&L)
(6) (7)
X = mG/L (8) It should be noted that NOG, NG,and NLare based on the same effective interfacial area a. If NG and NL are determined separately by different absorption, stripping, and humidification experiments using gas- and liquidphase-controlled systems, eq 2 is no longer valid. Since this is the case for the models reported previously, those models for NG and NL are fundamentally incorrect. Equation 2 should only be valid if NG and NL are determined simultaneously under distillation conditions. It can also be seen from eq 7 that tL and tG are mutually related. The overall number of mass-transfer units can be related to the point efficiency by
E,, = 1- exp(-NoG)
(9)
The value of EOGcan be obtained from the measured Murphree efficiencyE m as suggested by the AIChE model (1958). In this study, the eddy diffusion coefficient, DE, is predicted by the experimental correlation of Barker and Self (1962) as suggested by Chan and Fair (1984):
0888-588519312632-0701$04.00/0 0 1993 American Chemical Society
702 Ind. Eng. Chem. Res., Vol. 32, No. 4, 1993
DE = 0.00675~~“” + 0.000922hL’ - 0.00562 (10) where hL’ is estimated by equations given by Bennett et al. (1983). The experimental data for the vapor-phase Murphree efficiency E m at a different slope m can be used to calculate the number of mass-transfer units NG and N L by the AIChE model (1958) together with eq 9 and 2. Correlations for NG and N L can then be developed. The percent of liquid-phase resistance over the total mass-transfer resistance can be defined as
LPR% = ( A / N L ) / U / N G + WNL)
(11)
Model Development (A) Clear Liquid Height h ~ .The clear liquid height on a sieve tray plays an important role in mass transfer because of its influence on t L and t G . Nearly all tray performance correlations include clear liquid height as a variable. Many correlations for h~ are available (Hofhuis and Zuiderweg, 1979;Brambillaetal., 1969;Dhulesia, 1984; Colwell, 1979;Bennett et al., 1983;Hofhuis, 1980). In this study the correlation of Hofhuis (1980) as recommended by Zuiderweg (1982) is chosen. Under total reflux conditions, the correlation is given by h, = 0.6 hW0’5p0’25((pC/pL)0.5/b)0‘25 for 25 mm < h,
< 100 mm
(12)
(B) Mass-Transfer Coefficients kc and kL. On the basis of the Higbie penetration theory (Higbie, 1935), kc can be given by =~(DG/T~G)”~ (13) where 6G is the contact time of an element at the gasliquid interface. In Higbie’s original work, 6c is taken as the time for a bubble to rise through its own height. For sieve tray dispersion we assume that kG
hJu, = t,’ If it is further assumed that 6,
a
(14)
(15) then we can obtain the following equation by combining eqs 13-15: t G a tG’
(DG~G’)’’~ can be expressed by kGtG a
Similarly, k L
(16)
kL = ~ ( D J T ~ ~ ) ~ . ~ (17) If it is assumed for sieve tray dispersion that 6L 0: (hIpJugG) (hgp,Ab)/(QGP,)= ( h L A b ) / Q L = t ~ ’ (18) then
of point efficiency because of the complex hydrodynamic situation on the tray. It has been found that the size of bubbles on a sieve tray is not uniform. Ashley and Haselden (1972) proposed that there are two groups of bubbles in froth having diameters of 5 and 50 mm based on the photograph method. This result was supported by West et al. (19521,Porter et al. (19671,Lockett et al. (19791, Burgess and Calderbank (1975), and Calderbank and Pereira (1977). According to the report by Kaltenbacher (19831, about 70-90% of the vapor passing through the froth was in the form of large-size bubbles. The values reported by Porter et al. (1967) were 66-87%, and those reported by Ashley and Haselden (1972) were 68-95 5%. It is generally agreed that the existence of large bubbles results in low tray efficiency. The small bubbles are generated from the breakup of large bubbles (Prado and Fair, 1990) and have a constant diameter in the range of 4.5-5 mm. These small bubbles have no significant effect on tray efficiency (Prado and Fair, 1990). Neuburg and Chuang (1982)concluded that no satisfactory correlations existed for the prediction of bubble diameter and interfacial area. Therefore, in this study, new correlations for Sauter mean bubble diameter d32 and interfacial area are developed. If the bubble size distribution is assumed to be independent of the Sauter mean bubble diameter, then the ratio of d32/dmu is a constant. This is similar to what was found for stirred vessels (Mersmann and Grossman. 1982;Hesken et al., 1987;Parthasarathy, 1991). Therefore dm, is needed in order to obtain the interfacial area. The fundamental work in dispersion theory in turbulent flow was conducted by Kolmogoroff (1949) and Hinze (1955). Bubble size was found to be controlled by inertial forces (due to dynamic pressure fluctuations) and surface tension forces. The inertial forces tend to deform the bubbles whereas the surface tension forces resist the deformation. The maximum stable spherical bubble size dm, could be determined by the balance between these two forces. A bubble would break up if the ratio of the inertial and surface tension forces,in the form of the Weber number, exceeded a critical value. The critical Weber number is given as
We, = 7dm,/a
(21)
Levich (1962) suggested a similar force balance. He considered the balance of the internal pressure of the bubble with the capillary pressure of the deformed bubble. The dispersed-phase density is included through the internal pressure force term, and the capillary pressure is determined from the shape of the deformed bubble rather than the spherical bubble. His theory leads to a critical Weber number:
The dynamic pressure force of the continuous phase can be obtained by eq 23 (Hinze, 1955) = tL’(MGG)/(MLL)
(19)
and eq 20 is obtained: kLtL a (hfGG/hfLL)(DLtL’)0‘5 (20) where tL’ = tG’PL/PG. Equations 16 and 20 can be used to obtain NG and NL. (C) Interfacial Area a. The prediction of interfacial area, a, is probably the most difficult part in the prediction
7
= PLii2
(23)
where a2 is the mean square velocity difference over a distance equal to the maximum bubble diameter (dmm). An expression for a2 was obtained by Batchelor (1959):
a2= 2(wdm,)2/3
(24)
Substituting 7 and a2 into eqs 21 and 22, the following equations can be obtained:
Ind. Eng. Chem. Res., Vol. 32, No. 4, 1993 703 (25)
6~
~ = - a
2
(PL PG)
0.2
cr0.6P0.1
0.28
0.4
PG
0.14
0.07
d0.14cr0.07
PL
d32
or These two equations have been successfullyused to predict the maximum bubble diameter and Sauter mean bubble diameter in stirred vessels (Kawaseand Moo-Young,1990). However they have not been used to predict the bubble diameter in sieve tray dispersions. Calderbank (1958) proposed an empirical correlation for the gas-liquid interfacial area in an aerated mixing vessel: 0.4 Q a (0)
PL
0.6
0.5
0.33 0.34 0.68 PG ua
PL
ua
d0.14cr0.67P0.1
E-
1
Fs2PL
P d
(D) Number of Mass-Transfer Units NG and NL. Combining eqs 16, 20, and 33 gives the final correlations for the number of mass-transfer units as follows:
0.6
u, /a
The form of this equation is very similar to that in eq 25 and 26, and it is likely that eq 27 was based on isotropic turbulence theory. Later, Calderbank and Moo-Young (1960) found that eq 27 also applied to sieve tray dispersions. Their measured interfacial area for the airwater system using several sieve trays of different designs showed a remarkable agreement with eq 27. This indicated that the isotropic turbulence theory is also valid for sieve tray dispersion, and eqs 25 and 26 could be used to predict the maximum bubble diameter for sieve trays. Experimental observations indicated that the bubbles on sieve trays are highly deformed and not in spherical shape. Therefore, eq 26 is used in this study. The energy dissipation per unit mass of liquid on a sieve tray is given by (Calderbank and Moo-Young, 1960; Geary and Rice, 1991) w = usg
where
Fa = u s ( p G ) o ' 5 and C1 and C2 are constants to be found by fitting with experimental data. (E) Percent Liquid-PhaseResistance LPR% The fraction of liquid-phase resistance could be obtained by substituting eqs 34 and 35 into eq 11:
.
(28)
Substituting eq 28 into eq 26 gives (29) Equation 29 assumes that only surface tension forces resist dispersion, but as shown by Calderbank (19561, viscous forces also begin to resist dispersion as the ratio of viscosity to surface tension rises. Bhavaraju et al. (1978) proposed a correlation for bubble size including the viscosity effect. They found experimentally that dm, is proportional to (p)O.I in stirred vessels. If their results are assumed to apply for sieve tray dispersion, eq 29 becomes (30) Then the interfacial area can be obtained by the following equation:
where t is the mean void fraction and can be estimated by many empirical correlations (Gardner and Mclean, 1969; Kastanek, 1970;Colwell, 1979;Stichlmair, 1978). Themost recent correlation of Stichlmair (1978) is chosen as shown in eq 32:
Substituting c into eq 31:
Determination of Constants CI and CZ In order to find the values for CIand CZ,experimentally measured efficiency data have to be used. C1 and Ca are constants and independent of system properties and tray geometry. In theory, they could be determined from one data point. Since experimental data are seldom accurate enough, a series of test results would be required for the evaluation of C1 and CZ. Point efficiency from small columns is often lower than that from large columns (Dribika and Biddulph, 1986),so these data should be avoided. Many efficiency data from large columns are now available in the open literature and have been summarized by Chan and Fair (1984). However, only those datameasured as a function of the concentration of mixtures are suitable. Efficiency data published by FRI may be considered to be the most reliable. In 1979, they (Sakata and Yanagi) reported results on two test mixtures, cyclohexaneln-heptane and isobutaneln-butane, at five different pressures. Later they (Yanagiand Sakata, 1982) published results for three of five systems but with a different tray geometry. Therefore, their efficiency data of cyclohexane/n-heptane obtained in a 1.22-m-diameter column at 165-kPa pressure was selected to determine the values of CI and CZ. In order to evaluate NG and NL, the reported tray efficiency should be converted to point efficiency using the AIChE model as suggested by Chan and Fair (1984). The AIChE model assumed partial mixing of liquid on the tray and was found to agree with experimental results for column diameters below 5 m (Porter et al., 1972;Yanagi and Scott, 1973; Fair, 1987; Biddulph et al. 1991). The
704 Ind. Eng. Chem. Res., Vol. 32, No. 4, 1993 100
W C 7 , P= 165 kPa
Weeping
&
801
c
.-0
WC7, P=34 kPa 80-
0 0 I
b
c l
I
0
$80-
.-0
E
Entrainment
-'
f---
0
Froth regime
40-
Spray regime
4
- _ 200
0
I
Equation (37) Measured data /
0
C
05
,
,
1
,
1
15
,
I
2
,
I
25
- Predicted by equation (37) 0 Measured data
":
Transition point (by Loon et al ,1973) ,
0
3
0.5
1.5
1
2.5
2
F-factor,(kglrnps Figure 2. Comparison of predicted and measured point efficiency for cyclohexaneln-heptaneunder reduced pressure at 34 H a . 100
I
iC4/nCQ - Predicted by equation (37) 0 Measured data (P=1138kPa) A Measured data (P=2068kPa) Measured data (P=2758kPa)
.-0 n
(37)
It should be noted that, under total reflux conditions, MGI ML= 1. By fitting the above equation to the experimental NOGdata as a function of m, the values of C1 and CZhave been found to be 11and 14, respectively. Only data free of weeping and entrainment were used. Figure 1compares the predicted point efficiency with the experimental data as a function of F-factor.
Prediction of Efficiency To confirm the validity of the correlations, eqs 34 and 35 were used to predict the number of mass-transfer units, point efficiency, and LPR% for other systems under different operating conditions. Figure 2 shows the experimentally determined point efficiency and that calculated by the new correlations for the cyclohexanelnheptane system at 34 kPain the FRI 1.22m column (Sakata and Yanagi, 1979). It can be seen that the prediction agrees very well with the experimental data. Figure 3 shows the comparison for the isobutaneln-butane system at three different pressures. The measured efficiencies at 2068 and 2758 kPa have been corrected for the entrainment of vapor with the downflow liquid according to the method provided by Hoek and Zuiderweg (1983). The results indicated that all predicted point efficiencies agree with the experimental values within 5 % error. An additional comparison was carried out with data obtained by Billet et al. (19691, who measured sieve tray efficiency for the ethylbenzenelstyrene system under vacuum conditions (13.3 kPa) in a 0.79-m-diameter column. Figure 4 compares the point efficiency from their measured Murphree tray efficiencywith that predicted by the new correlations. Again, the absolute error is within 5 76.
ethylbenzene/styrene,P=13.3 kPa 80-
20-
0
A /
0 0
0.5
,
1
1
h,,,=0.038m h,,=0.019m ,
l
1.5
,
~
--I
2
,
1
2.5
,
)
3
This equation shows that the percent liquid-phase masstransfer resistance is a function of system properties only.
Ind. Eng. Chem. Res., Vol. 32, No. 4,1993 705 70
i
A
Bo
5
I
1
C6IC7 system (m=0.78) iC41nC4 system (m=0.99) ethylbenzene/styrene (m=0.90)
6
t
t
Thisstudy
-/,/zuiderweg
I
30
l.m
500
1,500
2,500
2,000
3,000
Pressure, (kPa) Figure 5. Calculated percent liquid-phaseresistancefor three sytems at different pressures.
0.01
,
I
,
,
0.05
,
,
1
I
1
0.1
0.2
0.3
,
I
, / , , I
0.5
1
Flow Parameter FP Figure 7. Comparisonof number of vapor-phaaemass-transferunits obtained by various models (conditions a~ Figure 6). 12
100
1
0.02 0.m
I
\
**
A***
Chan & Fair
Stichlmair Zuiderweg
01
0.01
I
Measured data ethylbenzene/styrene c6/c7 A iC4/nC4 0
20
1
I
0.02 0.03
,
1
0.05
,
, , , I
0.1
I
I
0.2
0.3
,
I
0.5
,
,
This study ,
0
,
1
Flow Parameter FP Figure 6. Comparisonof point efficiency obtained by various modela. Conditions: column diameter = 2 m; weir length = 1.5 m; tray spacing = 0.60 m; weir height = 0.05 m; hole diameter = 0.0127 m; hole pitch = 0.038 m; fractional free area = 0.1; m = 1; 80% flooding and total reflux. (Physical properties from Porter and Jenkins (1979);part of the results were obtained by Lockett (1986).
Neither the GIL ratio nor the tray geometry can affect the value of LPR% . I t is system properties that determine which phase is mass transfer controlling. Once the number of individual phase mass-transfer units is obtained by eqs 34 and 35,the LPR% can also be determined by eq 11. Figure 5 shows the calculated LPR% for the abovementioned three systems at different pressures. I t can be seen that for these three systems the liquid-phase resistance is significant, in the range of 40-60 % , and for a given distillation mixture the LPR % decreases with increasing pressure. This may be attributed to the changes in diffusivities and densities in the vapor and liquid phases. The liquid-phase diffusivity increases with increasing pressure, whereas the opposite is true for the vapor-phase diffusivity. Comparison with Other Models Porter and Jenkins (1979) pointed out that, at total reflux, all system properties can be expressed as a function ~ . ~ . it is of the flow parameter, F P = ( p ~ / p ~ ) Therefore, convenient to use the flow parameter as a variable for the comparison of point efficiency and mass-transfer units as predicted by existing models. Figure 6 shows point efficiencies calculated by eq 37 as well as those reported by Lockett (1986) using different models as a function of the flow parameter, at m = 1 in a 2-m-diameter column. I t is found that the results of the new model are close to those of the Chan and Fair model (1984) which was
001
I
I
002 O M )
,
i
005
,
, , , I
01
i
I
02
03
,
I
,
05
Flow Parameter FP Figure 8. Comparisonof number of liquid-phaaemaaa-transfer units obtained by various models (conditions as Figure 6).
obtained by using their distillation efficiency data bank to modify the AIChE model. As shown in Figure 6, the new model also gives a correct efficiency trend. When the flow parameter approaches unity, the point efficiency should equal 100% (Hoek and Zuiderweg, 1982). For comparison, the point efficienciesfrom experimental data are also shown in this figure. Figures 7 and 8 show the number of individual phase mass-transfer units calculated by the various models. The results indicate that the new model, unlike existing models, will predict an infinite value for NG and N L at FP = 1 where the surface tension disappears and mass-transfer resistance is zero. It can also be seen from Figure 8that NLpredicted by the AIChE model and Chan and Fair’s model is much higher than that by eq 35. This suggests that the AIChE model and Chan and Fair’s model may overpredict the point efficiency for liquid-phase-controlled systems. Although Chan and Fair’s model may predict a correct point efficiency for systems within the range of their data bank, these correct results may be obtained through a wrong route, and their model may not predict the individual NG and NL accurately. Their model is open to the same criticism as the AIChE model. The calculated percent liquid-phase resistance by the various models is shown in Figure 9. It can be seen that the new model predicts a trend of liquidphase resistance different from other models. This is because the new model treats contact time differently from all other models. I t seems unreasonablethatliquid-phase resistance increases with increasing pressure because the liquid-phase diffusivity is higher at a higher pressure, whereas the vapor-phase diffusivity is lower at a higher pressure. In small FP range, the new model predicts a much higher LPR% than all other models as shown in
706 Ind. Eng. Chem. Res., Vol. 32, No. 4, 1993
0
0.01
I
0.02 0.03
0.05
0.1
1
0.2
1
0.3
,
1
0.5
,
,
,
I
1
Flow Parameter FP
Figure 9. Comparison of percent liquid-phase resistance obtained by various models (conditions as Figure 6).
Figure 9. It has been known that the AIChE modelusually overpredicts the point efficiency for liquid-phase-controlled systems because it overpredicts NL. Therefore, it is expected that the new model may overcome the deficiency of the AIChE model for liquid-phase-controlled systems.
Discussion Although the model has been tested against only three systems, at six widely different pressure levels, the three systems cover a wide range of physical properties and can be considered six different systems. It is found that diffusivities, vapor densities, and surface tensions of these three systems vary more than 10 times between low and high pressures. Therefore, the model should be applicable to other systems, at least for hydrocarbons. The effect of tray geometry on point efficiency is tested for three tray designs in this study. However, the model includes the most important variable, h ~which , is mainly a function of tray geometry as indicated by eq 12 and which may include the effect of tray geometry on point efficiency. Furthermore, point efficiency is not sensitive to tray geometry, as indicated by the O’Connellcorrelation (1946). Thus, the present model should be adequate for use with most tray designs. The model was obtained based on the theory for the froth regime only; therefore, a different model will be required for the spray regime. Fortunately, efficiency changes between the two regimes are gradual according to the FRI test results and Prado and Fair’s results (1990). The calculated transition point based on the correlation of Loon et al. (1973) is shown in Figure 1. It can be seen that there is almost no efficiency difference between the two regimes. The coefficients obtained for these correlations are actually based on the experimental data for both regimes. Therefore, the new model may also be used to estimate the efficiencies in the spray regime. The new model also includes the open hole area effect as shown in eq 37. This is consistent with FRI results by Yanagi and Sakata (1982). They found that 14% open hole area sieve trays have a lower efficiency than that of 8 % open hole area sieve trays. The new model does not include the effect of hole size like most other models. It may, therefore, underpredict the point efficiency for sieve trays having very small hole diameters (less than 4 mm), which show a higher efficiency than normal hole size trays (Biddulph and Dribika, 1986). Studies, however, have shown that variation in hole size has little effect on point efficiency (Kalbassi et al., 1987).
The new model includes the surface tension effect, NG andNL 0: (aF2I3as shown in eqs 34 and 35, which is similar to that of Stichlmair’s and Zuiderweg’s models. All other models including the AIChE model do not include the surface tension effect. The inclusion of surface tension is open to debate. Although the new model shows good results for cyclohexane/n-heptane and isobutaneln-butane at different pressures where surface tension changes widely, physical properties of these two systems tend to change in unison. Thus, a definite conclusion about the surface tension effect is still uncertain. The best way to verify the surface tension effect is to determine the masstransfer units experimentally as a function of surface tension while keeping other properties constant for both surface tension positive and negative systems in distillation. This work is now being carried out in the authors’ laboratory.
Conclusion New correlations to predict the number of mass-transfer units, hence point efficiency, of sieve trays have been developed in this study. In the new model, the interfacial area of sieve tray dispersion is estimated by the Levich theory (19621, and the mass-transfer coefficients are obtained on the basis of the Higbie penetration theory (Higbie, 1935). The final model is obtained by fitting the correlations to the FRI-measured tray efficiencies in a commercial-size distillation column. It has been shown that the new model is able to predict accurately the point efficiency for hydrocarbon systems under different pressures. The correlation should generally be applicable for the prediction of sieve tray efficiency in distillation.
Acknowledgment The study was supported by the Institute for Chemical Science and Technology and the National Sciences and Engineering Research Council of Canada through PGS awards to G.X.C.
Nomenclature Ab
= bubbling area, m2
a = interfacial area per unit volume of two-phase dispersion,
l/m
b = weir length per unit bubbling area, l/m C1 = constant in eq 38 CZ = constant in eq 39 DE = eddy diffusivity for liquid mixing, m2/s DC = vapor-phase molecular diffusivity, m2/s DL = liquid-phase molecular diffusivity, m2/s d, = maximum bubble diameter, m d32 = Sauter mean bubble diameter, m EMV= Murphree vapor-phase tray efficiency E ~ =G Murphree vapor-phase point efficiency FP = (pG/pL)0’5,flow parameter under total reflux conditions F, = superficial F-factor, U & G ) O . ~ , kgO b/(m0.5 s) G = vapor flow rate, kg-mol/s g = gravitational constant, m/s2 hf = froth height, m h~ and hL’ = liquid holdup, m hw = outlet weir height, m kG = vapor-phase mass-transfer coefficient, m/s
k~ = liquid-phase mass-transfer coefficient, m/s KOG= overall mass-transfer coefficient based on vapor, m/s L = liquid flow rate, kg-mol/s LPR % = percent liquid-phase mass-transfer resistance m = slope of vapor-liquid equilibrium line MG = vapor-phase molecular weight, kg/kmol M L = liquid-phase molecular weight, kg/kmol
Ind. Eng. Chem. Res., Vol. 32, No. 4,1993 707
NC = number of vapor-phase mass-transfer unit
NL= number of liquid-phase mass-transfer unit NOG= number of overall vapor-phase mass-transfer unit p = pitch of holes in sieve tray, m QG = vapor flow rate, m3/s QL = liquid flow rate, m3/s tG = defined as eq 6, s tL = defined as eq 7, s tG' = vapor contact time defined as eq 14, s tL' = liquid contact time defined as eq 18, s a2 = (u(x+d,,,t)-u(x,t)', m2/s2 us = superficial vapor velocity based on bubbling area Ab, m/ s We, = critical Weber number Greek Letters X = mG/L e = void fraction in dispersion 0 = contact time in eqs 18 and 22, s p = liquid viscosity, N s/m2 pc = vapor density, kg/m3 p~ = liquid density, kg/m3 u = interfacial surface tension, N/m 7 = dynamic pressure fluctuations, N/m2 4 = fractional perforated tray area w = local energy dissipation per unit mass, w/kg
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