Ind. Eng. Chem. Res. 1995,34,3078-3082
3078
Liquid-Phase Resistance to Mass Transfer on Distillation Trays Guang X Chen* and Karl T. Chuang T6G 2G6
Department of Chemical Engineering, University of Alberta, Edmonton, Canada
The liquid-phase resistance to mass transfer was determined for the distillation of six binary mixtures. It was found that the liquid-phase resistance to mass transfer is significant, ranging from 20 to 55% of the total mass-transfer resistance. The results also indicated t h a t the traditional assumption of & = OL is not valid for distillation systems. If this assumption is applied, the fraction of liquid-phase resistance to mass transfer will be significantly underestimated. Tray efficiency models developed from data obtained from absorption and stripping systems (e.g., the AIChE method) were suggested to greatly overpredict the number of liquidphase mass-transfer units for distillation systems. Thus, careful examination of the use of various efficiency models for tray design is required.
Introduction The most important parameter in the design and analysis of tray-type columns is tray efficiency. Most tray efficiency models were developed from the experimental data of systems whose mass-transfer resistance was confined entirely to either the liquid phase or the gas phase (e.g., AIChE, 1958). These models tend to predict a large value for the number of liquid-phase transfer units for distillation systems and therefore to give a small value to the fraction of liquid-phase resistance over the total mass-transfer resistance. Based on the AIChE model (AIChE, 19581, many researchers have assumed that distillation is a gas-phase-controlled operation. However, experimental data indicate that the liquid-phase resistance in distillation is significant (Zuidenveg, 1982; Moens and Bos, 1972; Lockett and Plaka, 1983; Chen, 1993; Chen and Chuang, 1994). To date, the discrepancy between the models and experimental results has not been resolved. The objective of this study was to examine the extent of the liquid-phase resistance in distillation. In addition, it is to be determined if the distillation systems and the stripping systems can share the same tray efficiency model.
Tray n
Figure 1. Two-phase dispersion on tray n used to define EOGand NOG.
Integration of eq 3 yields
Let NOGbe defined as hf
NOGE
If (K0calus)is independent of hf, eq 5 becomes
Theory Figure 1 shows a two-phase dispersion on tray number n. The Murphree gas-phase point efficiency can be defined as
where
hf -
t,
(7)
US
+
where y* = mx b and x is the local liquid concentration at the point on the tray. y n and yn-l are the local mole fractions of the more volatile component of the gas leaving and entering the tray. If it is assumed that the gas is in plug flow through the froth and the liquid is completely mixed in the vertical direction, a steady state mass balance over an elemental strip of the froth gives
Combining eqs 5 , 4, and 1 produces
From the two-film theory, the following equation can be obtained: 1 -=-
from which it follows that
Dividing eq 9 by
1 'G
KOG
+-me&L 'L@LMG
( u ~ G yields )
1 - 1 NOG
* Author to whom correspondence should be addressed. Present address: Fractionation Research, Inc., P.O. Box 2108, Stillwater, OK 74076.
NG
+-NLm
(10)
where
NG = k G U t G
0888-5885l95I2634-3Q78$09.QQlQ 0 1995 American Chemical Society
(11)
Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 3079
and
0.8
0.6
It can be considered that ( l / N o ~is ) the total masstransfer resistance, ( ~ / N Gthe ) gas-phase mass-transfer resistance, and (m/NL)the liquid-phase mass-transfer resistance. The fraction of the resistance which lies in the liquid phase (LPR) can be defined as
,
I 1 c6lC7
I
-
MeOHIn-PrOH EtOH/n-PrOH
A
AcHIH20 Benz.lC7 MeOH/H20
a
4 0.4 0.2
-
\
(13) and the fraction of gas-phase resistance (GPR) can be given as
I
1
I
I
I
0.2
0.4
0.6
0.8
I
Average tray conc., x (mole)
Figure 2. LPR as a function of concentration for six distillation systems.
Substituting eq 10 into eq 13 yields
LPR =
m
m
-~
+ NLING
6 -
Combining eqs 15, 11, and 12 gives
"
: C6lC7 A
F2 4 (16)
-
MeOHIn-PrOH EtOH/n-PrOH AcH/H20 Benz./C7 MeOH/H20 0
2-
It can be found from eq 16 that the LPR depends on system physical properties only and is independent of tray design and the ratio of the gas t o liquid flow rate (GIL). Equation 16 also shows that the LPR increases as m increases and decreases as the ratio of (k,gNd KG g&L) increases. The penetration theory (Higbie, 1935) gives (17) where 6L is defined as the time during which the liquid element has been in contact with the gas phase at the interface. On the basis of the results of Lockett et al. (1979) and Mehta and Sharma (1966), KG may be given by
DGO'~
(18) A dimensional analysis to eq 18 gives following equation: kG a
(19) where 6 G can be considered as the time during which the gas element has been in contact with the liquid phase at the interface. Substituting eqs 17 and 19 into eq 16, one obtains
Given 6 G and &, the LPR can be calculated from eq 20. However, 6 G and 6L have not been determined by experiments. Consequently, a rather arbitrary assumption of 6 G = OL has been made by many investigators (Calderbank and Pereira, 1977; Lockett and Uddin, 1980; Neuburg and Chuang, 1982; Stichlmair, 1978; Dribika and Biddulph, 1986). If 8 G = OL, eq 20 becomes
0
:
-
2
7 -I
LPR =
1
I
I
m I
(21)
@LMG(DL)05
@&L
DG
Given m = 1,MG = ML, (@d@G)= 500, (DIJDG)= for a typical distillation system, eq 21 would give LPR = 0.17 and GPR = 0.83, which might indicate that distillation is a gas-phase-controlled operation. Since the assumption of OG = OL is arbitrary, the resulting conclusion that distillation is a gas-phase-controlled operation is also arbitrary and not reliable.
Results and Discussion Recently, Chen and Chuang (1994) determined NG and N L from tray efficiency data for six binary distillation systems, acetic acid/water (AcWH20), benzene/ "heptane (Benz/C7), cyclohexaneln-heptane (C6/C7), ethanolln-propanol (EtOWn-PrOH), and methanoll water (MeOWH20). These data can be used directly to calculate LPR using eq 15. The calculated LPR values as a function of composition, shown in Figure 2, indicate that the liquid-phase resistance in distillation is significant. Except for the methanollwater system, the percentage of liquid-phase resistance for the other five systems ranges from 20 to 55%, depending on the value of m. The liquid-phase resistance (LPR)decreases with increasing concentration because m decreases as concentration increases. To exclude the effect of m on the LPR, the (NIJNG)ratio was calculated and is given in Figure 3. It can be seen that the (NIJNG)ratios for the five systems lie between 1 and 2, which results in
3080 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 7w
,
I
1
0
I
Geometrical interfacialarea, a'
0.2
0.4
0.6
0.8
1
Point efficiency, E,,
tems. This point efficiency corresponds t o an effective interfacial area of 75 l/m (see Figure 4). In another paper, Lockett and Uddin (1980)found a point efficiency (EoG)around 0.02 for their liquid-phase-controlled masstransfer system, which corresponds to an interfacial area of 560 l/m. If it is assumed that the froth structures on sieve trays for these two systems (Lockett et al., 1979; Lockett and Uddin, 1980) are similar, the effective interfacial area for the NLmeasured from the liquid-phase-controlled system is about 7.5 times that for the NG measured from the gas-phase-controlled system. If it is further assumed that the mass-transfer mechanism in distillation is similar to absorption, the following calculations can be made to obtain the ( k L e d k G e G ) ratio: For the liquid-phase-controlled system:
Figure 4. Calculated effective interfacial area as a function of EOG.
an LPR = 0.33-0.5 at m = 1. However, eq 21 calculates a much lower LPR value. These results indicate that the assumption of OG = OL made in eq 21 is wrong, at least for systems presented in this study. Lockett and Uddin (1980) found that when m = 1, the liquid phase contributes about 10% of the total resistance to mass transfer. Their calculations assumed that both NG and NL obtained from absorptions have the same effective interfacial area. This assumption is incorrect as indicated by Lockett (1986). In fact, the effective interfacial area can be expressed as a function of the point efficiency or NOGon the basis of the twobubble-size model for a given froth structure (Lockett and Plaka, 1983;Ashley and Haselden, 1972; Prado and Fair, 1990). For example, a froth has a small bubble diameter d l = 5 mm, a large bubble diameter d2 = 50 mm, a small bubble rising velocity u1 = 0.6 d s , a froth void fraction E = 0.8, a local froth void fraction for small bubbles ~1 = 0.7, and a gas velocity us = 1.0 d s . Then, the rising velocity for large bubbles is (Lockett, 1986)
For the gas-phase-controlled system: and
a , = 7.5a2
(28)
Combining eqs 26, 27, and 28 yields (29) Substituting the experimental results obtained by Lockett and Uddin into eq 29 gives
(30)
If this same ratio of the mass-transfer coefficients is maintained under distillation conditions, eq 16 gives the result that the liquid phase contributes about 44% of the total resistance to mass transfer at m = 1. These results are in good agreement with those from the distillation data shown in Figures 2 and 3. Clearly, the N L correlation of the AIChE method (AIChE, 1958) is and the fraction of gas carried by small and large based on an effective interfacial area many times larger bubbles f1 and f2 (Lockett, 1986) than that for distillation systems. As a result, the AIChE method would overpredict N Lby many times for distillation systems. On the other hand, the efficiency models developed from results for distillation systems (Zuiderweg, 1982; Chen and Chuang, 1993) would For simplicity, it is assumed that the mass-transfer underpredict NL for low-efficiency, liquid-phase-concoefficients of small bubbles and large bubbles are the trolled systems. same. Then, the effective interfacial area ( a ) can be Recently, Kister (1992) suggested that the O'Connell obtained (Lockett and Plaka, 1983) by equation (O'Connell, 1946)should be used for predicting -US 6h&OG 6h&OG tray eEciency. Lockett (1986) expressed the O'Connell In fl exp - plot in equation form for distillation: d,u, )+f2exp(-d,u,)) a = -K O G h f
( (
(24) whereas the geometrical interfacial area ( a '1 is
From eq 24, the effective interfacial area can be calculated as a function of the point efficiency EOG. The results are shown in Figure 4. For a given froth structure and geometrical interfacial area, the effective interfacial area for mass transfer decreases significantly as the gas-phase point efficiency (EoG)increases. Lockett et al. (1979) obtained a point efficiency (EoG)around 0.75 for their gas-phase-controlled mass-transfer sys-
E, = 9.06(pu,a)-0.25
(31) where Eo is the overall column efficiency (%), p~ the liquid viscosity in (N s/m2),and a the relative volatility. It is interesting that the OConnell equation contains only two terms, the liquid viscosity and the relative volatility. If the gas-phase mass-transfer resistance in distillation is assumed to be negligible, the O'Connell equation can also be obtained by the following steps. With the assumption of negligible gas-phase resistance (~/NG = 0), eqs 10 and 12 can be combined to give
Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 3081
:q
1.25
' I
x=o.29 ....... Eq 36
1.2
....
Eq 35
c
E
y- 1.15
.-
2, m
0"
a. 1.1
j,
0
1.05
1
1
1.2
1.4 1.6 Relative volatility
1.a
2
0'41
C6lC7 MeOH/n-PrOH EtOHIn-PrOH Ng2.22
0.2
0
0.2
0.4
0.6
0.8
1
LPR
Figure 6. Gas-phase point efficiency as a function of LPR at a fixed NG.
Figure 5. Relationship between m and a when x = 0.29.
Zuidenveg (1982) obtained
(33) The mole fraction of the more volatile component in the gas phase can be related t o the mole fraction of more volatile component in the liquid phase by:
(34) From eq 34, m can be given as
m = d2 =
a
(35) (1 (a - 1 ) ~ ) ~ When a is in the range from 1 to 2 and x = 0.29, eq 35 is approximately equivalent to the following equation:
dx
+
= 1.02aO.~~ (36) Figure 5 compares the results from eqs 36 and 35. Substituting eqs 36 and 33 into eq 32 gives
N o , a (C~/-J-L)-O'~~ (37) When NOGis small, eq 8 can be approximately given as = NOG
(38)
If the liquid on the tray is totally mixed, then (39) If A, mGIL, is not too far from unity, the overall column efficiency can be approximated by =
E , =E,,
(40)
Combining eqs 40, 39, 38, and 37 yields
E , = (pLa)-0.25 (41) Equation 41 is the same as eq 31. The above analyses indicate that the main implicit assumptions made by the O'Connell equation are that mass transfer in distillation is controlled by the liquid phase. It can also be realized that the O'Connell equation only gives a rough estimation of the overall column efficiency, considering the various assumptions made and shown in the above analyses. For example, the approximation of eq 36 to eq 35 is good only at x = 0.29. NOGand hence the point efficiency (EoG)can also be expressed in terms of the LPR. Rearranging eq 14 gives N O G = N G ( 1 - LPR) (42) Therefore, the point efficiency can be expressed in terms
of the LPR at a fured NG, which is shown in Figure 6. Figure 6 suggests that gas-phase-controlled systems (LPR < 0.2) normally have a higher gas-phase point efficiency than distillation systems (0.2 < LPR < 0.8) and liquid-phase-controlled systems (LPR > 0.8). The stripping of trace volatile organic compounds (VOC) where m is extremely large is an example of a liquidphase-controlled system, and the gas-phase point efficiency (EoG)for this kind of application can be very low. It has often been assumed that in distillation 8 G and OL in eq 20 are equal (Calderbank and Pereira, 1977; Lockett and Uddin, 1980; Neuburg and Chuang, 1982; Stichlmair, 1978; Dribika and Biddulph, 1986). This rather arbitrary assumption has not been verified by experimental results. From the ( N ~ N Gratio ) shown in Figure 3, the (OrjOG) ratio can be calculated by combining eqs 11, 12, 17, and 19, which gives (43) For a common distillation system, using (@d@G) = 500, (DJDG)= (MI/MG)= 1,and the results, (NI/NG) = 1-2 (for the five systems shown in Figure 2), eq 43 yields
-0_, - 25-6.3 0,
(44)
These results indicate that the exposure time of gas elements is about 25-6.3 times shorter than the exposure time of liquid elements, if the penetration theory is equally applicable to both the gas and liquid phases. The shorter exposure time of gas elements results from a low gas viscosity and a more violent mixing in the gas phase compared with the liquid phase (Chen, 1993). When OG is used instead of O L in the calculation of k L , the k L value would be overestimated by 5-2.5 times. In other words, N L or kL might be 5-2.5 times smaller than that calculated by the penetration theory using O L = 6 G . It is also clear that if B L = e G , the LPR in eq 18 will be significantly underestimated. It is worth noting in Figure 3 that the N ~ N ratio G for the methanoywater system is about 3-4 times larger than that for the other five systems used in this study. Chen et al. (1994) attributed this to an enhanced NL from the surface renewal effect resulting from a surface tension gradient. Equations for estimating the effect of surface renewal on NL,kL, and OL have been obtained (Chen, 1993).
3082 Ind. Eng. Chem. Res., Vol. 34,No. 9, 1995
Conclusions The fraction of liquid-phase resistance over the total mass-transfer resistance for five distillation systems presented in this study was shown to be about 0.330.50 at m = 1. The results also indicate that efficiency models based on absorption and stripping systems (e.g, the AIChE model) may greatly overpredict NL for distillation systems. It was also shown that the assumption of & = & is not valid, a t least for the systems involved in this study and may require further study. If 6~ = 6~ is assumed, the LPR will be significantly underestimated. It is recommended that the AIChE model (AIChE, 1958)should be used for predicting the tray efficiency for systems whose mass-transfer resistance is confined entirely to either the liquid phase or the gas phase, whereas the efficiency models based on distillation systems (Zuiderweg, 1982;Chan and Fair, 1984;Chen and Chuang, 1993)should be applicable only to distillation systems. Nomenclature Ab = tray bubbling area, m2 a = effective interfacial area, m2/m3 a’ = geometrical interfacial area, m2/m3 d l = small bubble diameter, m dz = large bubble diameter, m DG = vapor molecular diffusion coefficient, m2/s DL = liquid molecular diffusion coefficient, m2/s E m = Murphree gas-phase tray efficiency Eo = overall column (tray) efficiency EOG= Murphree gas-phase point efficiency f l = the fraction gas carried by small bubbles f 2 = the fraction gas carried by large bubbles G = molar gas flow rate, GPR = fraction of gas-phase mass-transfer resistance hf = froth height, m kG = gas-phase mass-transfer coefficient, d s kL = liquid-phase mass-transfer coefficient, d s
KOG= overall gas-phase mass-transfer coefficient, d s L = molar liquid flow rate LPR = fraction of liquid-phase mass-transfer resistance m = slope of the vapor-liquid equilibrium line M G = gas molecular weight ML = liquid molecular weight NG = number of gas-phase transfer units N L = number of liquid-phase transfer units NOG= number of overall gas-phase mass-transfer units NOC,= number of overall gas-phase transfer units for liquid-phase-controlled systems NOGZ= number of overall gas-phase transfer units for gasphase-controlled systems t G = contacting time defined by eq 7, s u1 = small bubble rising velocity, d s u2 = large bubble rising velocity, d s us = gas velocity based on bubbling area, m/s x = mole fraction of more volatile component in liquid phase y = mole fraction of more volatile component in gas phase y n = mole fraction of more volatile component in gas phase yn-l = mole fraction of more volatile component in gas phase y* = mole fraction of more volatile component in gas phase in equilibrium with liquid x
Greek Letters a = relative volatility E = gas hold-up fraction 61 = local gas hold-up fraction & = gas-phase interfacial contact time, s O L = liquid-phase interfacial contact time, s 1 = mG/L p~ = liquid viscosity, N s/m2 QG = gas density, kg/m3 QL = liquid density, kg/m3
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Abstract published in Advance ACS Abstracts, August 1, 1995. @
