Hoftyzer, P. J., Trans. ZnJt. Chem. Engrs. 42, T109 (1964). Jesser, B. W., Elgin, J. C., Trans. A.Z.Ch.E. 39, 277 (1943). Kolbezen, M. J., Eckert, J. W.,Wilson, C. W., Anal. Chem. 36, 593 (1964). Larkins, R. P., “Two-F’hase Cocurrent Flow in Packed Beds,” Ph.D. thesis, University of Michigan, 1959. Larkins, R. P., White, R. R., Jeffrey, D. W., A.Z.CI1.E. J . 7, 231 (1961). Leva, M., “Tower Packings and Packed Tower Design,” 2nd ed., pp. 28-37, U. S. Stoneware Co., 1953. Lockhart, R. W., Martinelli, R. C., Chem. Eng. Progr. 45, 39 (January 1949). Lovelock, J. E., Anal. Chcm. 33, 162 (1961). Mcllvroid, H. G., “Mass Transfer in Cocurrent Gas-Liquid Flow through a Packed Column,” Ph.D. thesis, Carnegie Institute of Technology, 1956. Otvos, J. W., Stevenson, D. P., J . A m . Chem. SOC.78, 546 (1956). Porter, K. E., Jones, M. C., Trans. Znst. Chem. Engrs. 41, 240 (1963). Shulman, H. L., Ullrica, C. F., Wells, N., Proulx, A. Z . , A.Z.Ch.E. J . 1, 259 (1955). Standart, G., Chem. Eng. Sci. 19, 227 (1964).
Standish, N., Nature 202, 587 (May 9, 1964). Sternling, C. V., private communication, 1963. Wen, C. Y., O’Brien, W. S., Fan, L. T., J . Chem. Eng. Data 8, 42 (1963a). Wen, C. Y., O’Brien, W. S., Fan, L. T., J . Chem. Eng. Data 8, 47 (1963b). Weekman, V. W., Myers, J. B., A.Z.Ch.E. J . 10, 951 (1964). Weekman, V. W., Myers, J. B., A.Z.Ch.E. J . 11, 13 (1965). RECEIVED for review December 5, 1966 ACCEPTED May 1, 1967
Material supplementary to this article has been deposited as Document 9495 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington, D.C. A copy may be secured by citing the document number and by remitting $2.50 for photoprints or $1.75 for 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress.
C A L C U L A T I O N OF EFFECT OF V A P O R MIXING O N T R A Y EFFICIENCY D A V I D A. DIENER
ESSOResearch and Engineering Co., Florham Park, N.J .
The relationship between the point efficiency and the Murphree tray efficiency has been developed for the case of partial mixing of the liquid and no mixing of the vapor. This is an extension of previous studies which have assumed the vapor entering each tray to be completely uniform in composition. The lateral concentration gradient in the vapor rising from the liquid is assumed to be similar to the longitudinal concentration gradient in the liquid flowing across the tray. This assumption probably approximates the actual physical situation best at high per cents of flood. The principal application of the results of this work is to systems where point efficiencies are greater than about 0.80. HE relationship between the point efficiency and the TMurphree tray efficiency has in general been described in terms of the degree of backmixing of the liquid as it flows across a tray. This has been done using a number of different models to characterize the liquid mixing. These models include mixed pools :in series (Gautreaux and O’Connell, 1955), eddy diffusion (Gerster et al., 1958), recycle stream (Oliver and Watson, 1956), splashing of the liquid (Johnson and Marangozis, 1958)., and measurement of residence times of liquid elements (Foss et al., 1958). In the use of each of these models to develop the relationship between point and tray efficiencies, however, the assumption has always been made that the vapor entering successive trays is well mixed. Only in ‘the extreme case of no backmixing of the liquid has the relationship between the point efficiency and the tray efficiency been developed without assuming the vapor to be completely mixed. This case, for plug flow of the liquid and unmixed vapor, has been presented (Lewis, 1936) for two different flow situations. In this work, partial mixing of the liquid with no mixing of the vapor is considered for the same two flow situations considered by Lewis in his development. T h e model used to describe the partial liquid mixing is that of eddy diffusion. In carrying out the development, it is assumed that a lateral concentration gradient exists in the vapor similar to the longitudinal concentration gradient which exists in the liquid flowing across the tray. T h e vapor is assumed to rise directly (without lateral mixing) from the point where it leaves the aerated liquid on one tray to the corresponding point on the tray above.
General Relationships
Application of the eddy diffusion model to describe the partial mixing of aerated liquid on a distillation tray yields, in terms of a material balance on a small vertical slice of liquid,
with w varying from 1 to 0 as the liquid flows across the tray. b’, and the Use of the equilibrium relationship, yn* = m x , definition of the point efficiency, Eo, = ( y , - Y n - l ) / ( Y n * ynv1), to substitute in Equation 1 then gives the relationship between the point efficiency and the vapor compositions as
+
-
T o obtain the relationship between the point efficiency and the Murphree tray efficiency, however, it is also necessary to relate the tray efficiency to the vapor compositions. This can be done by considering the definition of the Murphree tray efficiency, EIMV= (g, - gn- l)/(yo* - gn- l ) . Using this definition, the desired relationship may be expressed as
1’ (Yn
EMV =
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- Yn-
ddw
(3)
1
NO. 4 O C T O B E R 1 9 6 7
499
Equations 2 and 3 are the basic equations which are used in this work to develop the relationship between the point efficiency and the tray efficiency for the two specific flow situations. T h e first development is for the case of aerated liquid flow in the same direction on successive trays; the second is for aerated liquid flow in opposite directions on successive trays. I n both cases, the assumption is made that the vapor, in addition to passing through the liquid without backmixing, also rises directly from a given point in the liquid to the corresponding point on the tray above (no lateral mixing).
where
+ = e”[(b2 + c2) sinh G - 2cb cosh c ]
For /3 > 1 the corresponding expression, obtained using Equation 7, is
where
liquid Flow in Same Direction on Successive Trays
#
As in Lewis’ Case 11, for liquid flow in the same direction on successive trays (Lewis, 1936), the assumption of straight operating and equilibrium lines plus constant Eo, over the composition range covered by trays n and n - 1 leads to similar Y us. ZL’ curves on these trays. This condition may be expressed mathematically by the relationship 2,
= a
2,-1
- ’)
- bn)o = yn-1 - bn- 1)1; = Yn
+
(4)
[(-a + Eo,
-
1)
]
= 0
(5)
This equation has been solved using the boundary conditions that both zn-l = 0 and dz,-l/dw = 0 a t w = 0, the latter expression following from the condition d x / d w = 0 a t w = 0 (Wehner and Wilhelm, 1956). T h e solution to Equation 5 , however, also depends on the magnitude of the term /3, which is defined as 4(1 - a ) XEo,/(a EO, - l)(Pe). For (3 < 1
+
kn
zn-i = ___ {ebzc 1- a
[; b
sinh
(CW)
-
cosh (cw)
with
b = -Pel2 and c = For p
- (Pe/2)
eb[(b2 - a2) sin a
-
2ab cos a ]
+ 2ab
Since a has been introduced as a new variable, it is now also necessary to have an additional relationship. This relationship has been derived previously (Lewis, 1936) and is expressed as
Substituting these relationships in Equation 2 and defining k n as bn)o - bn- d o yields
[(a
=
Combination of Equation 11 with Equation 9 or 10, depending on the value of p , yields EAuv for given values of Eo,, A, and Pe. For the special case of X = 1, the above equations are not directly applicable. The relationship between E,, and Eo0 in this case is given as
2,-1
with 2,
+ 2cb
dl
E.w = [2 Eo, ( e - p e
+
2 EOGPe2 Pe - 1) Pe2 (2
+
- EO,)]
(12)
-
This expression is obtained by solving Equation 1 1 for X 1 and substituting for EiuVusing Equation 9. As indicated by Lewis, X - 1 is then expressed as a Taylor series in powers of a - 1. Since a = 1 when X = 1, division of the Taylor series by a - 1 and evaluation of the resulting expression a t X = 1, a = 1 then gives Equation 12. The results of the computer calculations for liquid flow in the same direction on successive trays are presented in Table I. liquid Flow in Opposite Directions on Successive Trays
For the case of liquid flow in opposite directions on successive trays, the assumption is again made that Eo, is constant a n d the operating and equilibrium lines are straight over the composition range covered by trays n and n - 1. This leads to y us. w curves which are again similar on trays n and n 1, except that now the profiles are reversed on successive trays. T h e relationship presented in Lewis’ Case I11 to express this condition is (Lewis, 1936)
-
- /3
>1
(1 3)
( z n ) w = a (Zn-111-w
with
with b = -Pe/2
and a =
- (Pe/2)
zn = Y n
d F
Rearranging the expressions given in Equations 4 and substituting in Equation 3 yields
zn- 1 = ~ n
- (Yn)o 1- - b n - 1) 1
Additional relationships also needed for the present development are
JO 1- zc
and which upon integration, using Equation 6, yields for (3
(14)
2)
0.706 0.787 0.849 0.936 1.048 1.132 1.249
EOG= 0.800 1 .o 2.0 3.0 5.0 10.0 20.0 50.0 m
1. o 2.0 3.0 5.0 10.0 20.0 50.0 m
0.847 0.881 0.907 0.942 0.980 1.004 1.020 1.031
0,895 0.945 0,967 1.059 1.023 1.147 1.099 1.273 1.190 1.436 1.254 1.560 1.299 1.656 1.333 1.732 EOG= 1.000 1.152 1.233 1.276 1.427 1.374 1.585 1.515 i ,818 1.695 2.141 1.826 2.402 1.925 2.616 2.000 2.794
1.075 1.134 1.180 1.246 1.324 1.374 1.407 1.431
0.996 1.154 1.281 1.467 1.725 1.936 2.110 2.260 1.317 1.587 1.815 2.164 2.682 3,142 3.551 3.922
The solution to Equation 16 could thus be expressed in general form as
where A i and Bj are functions of a, Pe, EOG,A, a n d j . T h e two boundary conditions used to evaluate ao' and a1 were (Zn)u=-1/2
=
0
which follows fromy, = ( y n ) o when zu = 0
which follows from the condition dx/dw = 0 a t w = 0 (Wehner and Wilhelm, 1956). After the determination of ao' and al, Equation 3 was evaluated as
EM, Shifting the origin from the new abscissa, [a b n ) u - (Zn1-u
+a
ZL'
(1-1
=
0 to zu =
l/2
then gives, with u as
=
I'+l"z rqn +
-
1
=
(19 )
This equation can now be solved by series solution with N
(Zn)u =
a5 u5 j=O
and
As in the parallel flow situation, this equation was combined with Equation 11 to yield values of E M , for given values of Eoa,A, and Pe. For the case of X = 1, Equation 11 was solved for X 1 and Equation 19 substituted for E.Mv. Then A 1 was expressed as a series in powers of a - 1, and the same procedure used to obtain Equation 12 yielded an expression valid at A = 1 for EM, in terms of the zn series. Since the Peclet number appears in the numerator of the expressions for the a, coefficients in Equations 17, the solution just described was found to converge for low ( 5 2 0 ) but not for
-
VOL. 6
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OCTOBER
1967
-
501
high Pe numbers (250). This solution was therefore used for the calculation of the results shown in Table I1 for which Pe was less than 20. Solution for High Peclet Numbers. I n order to obtain a solution which would converge for Pe > 20, alternative expressions for the coefficients in Equations 17 were developed which had Pe in the denominator rather than the numerator. These expressions are : a0 = ao’
a1
= ao’
[
+, 1-CY
(a - 1) (a
+ 1 - Em)
f o r j even in general ( j > 1)
Qn
-
T h e point efficiency value of 0.80 was chosen for the curves in Figures 1 and 2, because above this Em level the effects of vapor mixing in general become more significant. This can presented be seen from the typical values of (EMV),/(E.wV)M
Table II. Tray Efficiencies for Partially Mixed Liquid and Unmixed Vapor
(Liquid flow in opposite directions on successive trays) ao‘
[
arbitrary
+ Eod(2) + 1 - Eoa)(Pe)
(a - 1 (a
EMV
1
Pe
X = 0.5
1. o 2.0 5.0 20.0
0.203 0.204 0.207 0.209 0.210
0.205 0.209 0.214 0.219 0,221
0.411 0.418 0.428 0.438 0,442
0,421 0.435 0.457 0.479 0.490
0.624 0.639 0.649 0 662
0.648 0.679 0.700 0 727 0.759 0.780
m
X = 7.0 X = 1.5 Eo0 = 0.200 0.208 0.213 0 I221 0.230 0.233
X = 2.0 0.211 0.218 0.229 0.240 0.246
EOG= 0.400 1. o 2.0 5.0 20.0 m
f o r j odd in general ( j > 1) 2.0
3.0
s o
10.0 20.0 m
aN arbitrary
+
1 is the number of terms in the truncated series. where N Thus, the solution to Equation 16 in this case can be expressed as
0.443 0.472 0.519 0.574 0 604
Eo@= 0.600 1. o
and
0,432 0.453 0,487 0.524 0.543
0.iii 0.685 0.696
0,697 0,763 0.810 0.874 0.954 1.014 1.099
0.672 0.720 0.754 0.797 0.851 0.889 0.940
0.800
EOG= 0.800 1 .o 2.0 3.0 5.0
10.0 20.0 50.0 m
0.841 0.867 0.884 0.906 0.930 0,946 0.957 0.966
0.883 0.936 0.972 1.018 1.073 1.111 1.140 1.161
0.926 1.009 1.067 1.144 1.239 1.310 1.365 1.409
1.061 1.095 1.117 1.143 1,171 1.190 1.202 1.212
1.125 1.200 1.250 1.313 1.385 1.435 1.472 1,500
1.192 1.314 1.399 1.511 1.650 1.754 1.835 1,901
I
I
I
0.971 1.086 1.170 1.285 1.436 1.554 1.651 1,732
Eo0 = 1.000
where the C j and D jare functions of CY, Pe, EOG, A, and j . ‘The boundary conditions used to evaluate ao’ and aN in this equation were the same as those used for the low Pe number solution. After evaluation of these constants, Equation 20 was substituted in Equation 19 to evaluate EYV a t high Pe numbers for all values of X except X = 1. Again, for this particular value of A, the method previously described was used to obtain the necessary expression. T h e solution for high Pe numbers was found to converge for Pe numbers as low as 20. I t was thus used to calculate the results in Table I1 for Pe 2 20.
1. o 2.0 3.0 5.0 10.0 20.0 50.0 m
1.15
1.263 1.438 1.567 1.744 1.979 2.170 2.330 2.467
1
Discussion
T h e results presented in Tables I and I1 for both flow situations indicate that, as X or Pe approaches zero, the equations properly predict that Enlv approaches EOG. Also, as P e becomes very large the values a t Pe = [calculated from Lewis’ solution (Lewis, 1936) for each flow situation] are approached. I n Figures 1 and 2 the ratio of EdMVfor unmixed vapor to E.hlv for completely mixed vapor is plotted for both flow situations as a function of X a t EOG= 0.80. These figures indicate the error involved when the model for completely mixed vapor is applied to a flow situation where the vapor is actually unmixed. T h e (E.MV)uvalues used in these figures were calculated from the relationships developed in this work. The values were obtained using the relationship previously developed for this case in terms of the eddy diffusion model (Gerster et al., 1958). 502
I&EC
PROCESS DESIGN A N D DEVELOPMENT
A
Figure 1. (EMv)u/(EMv)M for liquid flow in same direction on successive trays
tremes of the and (E.Mv)M values-Le., in the direction of the completely mixed vapor case. T h e lower the per cent of flood, the more time the vapor has to mix laterally. Acknowledgment
T h e author expresses appreciation to Esso Research and Engineering Co. for permission to publish this work and to C. J. Colwell and R. K. Neeld for helpful criticism of the manuscript.
0.95---
0.90
0.85
L
Nomenclature aj
= =
b b‘
= =
C
=
a
EMV = EOG = G = 0
0.5
1.0
2.0
1.5
kn
Figure 2. (EMv)u/(E,wv)M for liquid flow in opposite directions on successive trays
m Pe Qn
U W
Table 111.
Pe
Selected Vlnluer of (E,VV)U/(€MV),I.I at h 0.60
=
L
A
EOG 0.80
=
1.0
1 .oo
X
Y 2
defined in Equation 7 defined in Equation 17 defined in Equation 6 constant in equilibrium relationship defined in Equation 6 Murphree tray efficiency point efficiency molar vapor flow rate
b n ) o - bn-l)o
= molar liquid flow rate = constant in equilibrium relationship = Peclet number = = w - ‘/2 = fractional distance across tray from outlet
bn)o- bn-l)l
Liquid Flow in Same Direction on Successive Trays
GREEKLETTERS
5
a
m
1.025 1.043
1.048 1.088
1.082 1.164
Liquid Flow in Opposite Directions on Successive Trays 5 m
0.990 0.982
0.972 0.948
0.937 0.873
in Table I11 for h = 1, €’e = 5 and m , and EOGvalues between 0.60 and 1.00. From Figures 1 and 2 it can also be seen that the effect of vapor mixing is more pronounced a t higher values ~ ) M greater than 1.0 for of A . Further, ( E . ~ v ) ~ ~ ( E .isMalways liquid flow in the same. direction on successive trays, but is always less than 1.0 for liquid flow in opposite directions on successive trays. O n the basis of the results presented here, the principal application of the equations developed in this paper, which take into account unmixed vapor, is to those systems which have Eo, values above aboui: 0.80. Such systems might, for example, be found in the area of air distillation, where EoG values above 0.90 have been reported. I n most systems, however, Eo, values are lesr: than 0.80; thus, the effect of vapor mixing on tray efficiency is generally not too important. T h e assumption in this work of partial longitudinal mixing of the liquid with no lateral mixing of the vapor probably represents the actual physical situation fairly well for towers operating a t high per cents of flood-perhaps above 70%. For lower per cents of flood, the vapor is probably mixed laterally to some extent, giving EMv values between the ex-
P
h
$I
#
weir towards
tray inlet = liquid mole fraction = vapor mole fraction = defined in Equations 4 and 13 constant defined in Equations 4 and 13 4 (1 - a) E o G / ( ~ EOG- 1)(Pe) = m G/L = defined in Equation 9 = defined in Equation 10 = =
+
SUPERSCRIPTS * = equilibrium value - = average value SUBSCRIPTS = completely mixed vapor n = tray number, measured from bottom of column U = unmixed vapor 0 = evaluated a t w = 0 1 = evaluated a t w = 1 u = evaluated a t u w = evaluated a t w
M
literature Cited
Foss, A. S., Gerster, J. A., Pigford, R. L., A.Z.Ch.E. J . 4, 231 (1958).
Gautreaux, M. F., O’Connell, H. E., Chem. Eng. Progr. 51, 232 (1955).
Gerster, J. A., Hill, A. B., Hochgraf, N. N., Robinson, D. G., “Tray Efficiencies in Distillation Columns,” Final Report from University of Delaware, A.I.Ch.E., New York, 1958. Johnson, A. I., Marangozis, J., Can. J . Chem. Eng. 36, 161 (1958). Lewis, W. K., Jr., Znd. Eng. Chem. 28, 399 (1936). Oliver, E. D., Watson, C. C., A.Z.Ch.E. J . 2, 18 (1956). JVehner, J. F., Wilhelm, R. H., Chem. Eng. Sci. 6 , 89 (1956). RECEIVED for review January 23, 1967 ACCEPTEDJune 19, 1967
VOL. 6
NO.
4 OCTOBER 1 9 6 7
503
