The results obtained by using Equation 4 are compared satisfactorily with the experimental and literature values in Figure 4. The average deviation is found to be 1 6 % . This generalized equation, which requires only the mass flow rate of air and the column diameter as primary information, provides a simple method for estimating the spout diameter in a spouted bed under various conditions. Acknowledgment
The authors are indebted to the Sational Research Council of Canada for financial support. literature Cited
0 -
I/ 0.5
Figure 4.
ID
15 d,
213
25
3.0
35
4.0
calculated
Calculated and experimental spout diameter
(1) Becker, H. -4., Sallans, H. R., Chem. Eng. Scz. 13, 97 (1961). (2) Buchanan, R. H., Manurang, F., Brit.Chem. Eng. 6 , 4 0 2 ( 1 9 6 1 ) . (3) Cowan, C . B., Peterson, W. S., Osberg, G. L., Pulp Paper M a g . Can. 58, 138 (1957). (4) Madonna, L. A., Lama, R. F., Brisson, W. L.. Brit. Chem. Eng. 524 (1961). (5) Mathur, K. B., Gishler, P. E., A.Z.Ch.E. J o u r n a l l , 157 ( 1 9 5 5 ) . ( 6 ) Mathur, K. B., Gishler, P. E.: J . Appl. Chem. 5 , 624 ( 1 9 5 5 ) . ( 7 ) Thonley, B., Saunby, J. B., Mathur, K. B., Osberg, G. L., Can. J . Chem. Eng. 37, 184 (1959).
RECEIVED for review December 11, 1961 ACCEPTEDMay 15, 1962
EFFECT OF EQUILIBRIUM LINE CURVATURE ON MASS TRANSFER CALCULATIONS C . A . P L A N K A N D E. R . G E R H A R D Department of Chemical Engineering, University of Louisville, Louisville, Kv.
Previous methods of combining individual phase resistances have been based upon using the slope of the equilibrium curve. This article shows that when curvature of the equilibrium line i s large the procedure should b e based upon the slope of a chord connecting two specific points on the equilibrium curve. Application o f this procedure has been extended to mass transfer through bubble trays and equations have been derived whereby the point efficiencies, EoG and EoL, may b e properly estimated from individual phase resistance. An example calculation shows that for large equilibrium line curvature the previous method can be in error b y as much as 3oyO.
N RECENr
years the two-film or two-resistance theory has
I been used frequently to characterize mass transfer operations. Use of this theory in absorption and desorption has been rather extensive and recently the concept has b e m the subject of a detailed investigation in the field of distillation (I). The equations used to combine the effects of the individual films are
34
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
or in terms of transfer units
The term n which appears in these equations is a function of the equilibrium relations of the system under consideration. Many authors have described this term as a n ’.average” or “proper” slope of the equilibrium line. S o problem arises when the equilibrium data are represented as a straight line. but when the equilibrium relation is not linear some problems
haire been encountered. One procedure has been to use an integrated average slope of the equilibrium curve
The purpose of this article is to show that for all curved equilibrium lines m should properly be considered as the slope of a chord connecting specific points on the equilibrium curve and not the slope of the equilibrium curve itself. Only for the case of linear equilibrium lines can the equilibrium line slope be used, since the slope of the chord and equilibrium line are the same for such cases. No averaging is necessary here, since the value is constant.
z
Q
L
i:LT LL
w
P
Mass Transfer at a Point
For curved equilibrium lines, Bird, Stewart, and Lightfoot (2) and Larian (3) have considered the case of absorption a t a point ( x , J ) in a tower and have shown that when KO, is used, rn is the slope of the chord connecting the interfacial composition ( x l , > Jwith the point ( x . y*). I t is further shown that rn in Equation 1.4 is not the same as rn in Equation 1B. In Equation 1B rn is the slope of the chord connecting ( x i . jt) and ( x * , y) A similar argument can be shown to hold for Equations 2.4 and 2B. However, for a linear equilibrium curve all values of m are identical. A similar treatment for desorption is given below in some detail, since this ramification is also useful in distillation and the literature is not correctly explicit for these cases. I f one considers the case of desorption a t a point (x,j) in a tower. as shoivn in Figure 1. the various coefficients are defined by '\'/A
=
kc(Yl - y) = Koc(Y*
- y)
- X,)
=
~ L ( X
= &L(X
- X*)
By algebraic manipulation
.VIA
=-
-
y*
+
kL
kG
( x
-
):
and thus
CompariniEquation 5 with Equation 1A it is seen that
This chord is shown on Figure 1 as a solid line. Similarly for KO, one finds
KOL
Yi
-Y
MOCE FRACTION LOUID
Figure 1 .
Desorpiion at a point
method which should be reasonably accurate if the curvature is slight or if the resistance of one phase is very large. However. for the case where the curvature is large and the resistance in both phases is important, this approximation would be in error. Their procedure Mill be correct for these latter cases, however. if m in the various equations of (5) is considered as the slope of the chord. as defined by Equation 6 of the present article, instead of the slope of the equilibrium line. Mass Transfer through a Bubble Tray
The use of either Equation 1 or 2, with the proper value of the slope of the chord for rn. is sufficient for point to point design calculations of absorption. desorption. and distillation when carried out in packed or wetted-wall columns. Further consideration must be given to this problem \\.hen plates or trays are used. The procedure recommended at present for this type of calculation is given in much detail ( 7 ) . These calculations may be made using either gas or liquid compositions. Gas Phase Efficiency and Over-all Transfer Units. For a bubble tray through Lvhich the vapor changes from composition 1 t o j (J > j the number of over-all transfer units is
For the case of uniform liquid composition (x and y* constant) Equation 9 can be integrated and rearranged to give the relation between the point efficiency, EoG,and No, (7).
Comparing Equations 7 and lB, rn must be defined as m =
(~q or (et) Y)- Y ' X'
-x
Specifically this value of m is not the same as in Equation 6. The chord representing Equation 8 is shown on Figure 1 as a dashed line. I n both cases m should be referred to as the slope of a chord connecting two specific points and not as the slope of the equilibrium curve. The extreme right-hand terms of Equations 6 and 8 are the same as given by Bird (2) and Larian (3)for absorption. M a s s Transfer in Continuous Contact Towers
Sherwood and Pigford (5) have considered the case of a curved equilibrium line and have presented a n approximate
The use of the identity shown in Equation 10 requires that
NO,be evaluated from another relationship such as Equation 2A. Equation 2A was derived on the basis that KO, was constant between the values of y~ and y?. Actually the value of KO, can change if k,, k,, or m changes over this composition range. Over a region of equilibrium line curvature the value of m changes and this change must be considered in the application of relations such as Equation 2A. A procedure is outlined for the case where rn cannot be considered as constant ( 7 , page 45). Although this apparently allows for a variation of m as the gas composition changes, the method is based on m's being defined as the slope of the equilibrium line rather than the VOL. 2
NO.
1
JANUARY
1963
35
slope of the chord as required by Equation 6. Distillation is analogous to desorption and Equation 6 applies to this case. The development used here for the variation of m will consider the same model as used previously ( 7 ) . This model assumes that at a point or differential element on the tray the liquid has no concentration gradient in the vertical direction and that the vapor passing through the liquid changes in composition from to y2. The model is similar to a very small wetted-wall tower with constant liquid composition. For this case the rate equation becomes Gdy
=
KO&*
- y)adZ
(11)
or considering the limits of integration
Integrating the right side of the equation and separatinq KO, into its parts by Equation l A , wherein m is defined by Equation 6, the folloxving results:
Retaining the assumption that the values of k , and k , remain constant: Equation 13 becomes
Equation 14 can be further arranged by noting that the rightk,/kL is equal to G.V,/LN,, and from hand side is equal to At'G, Equation 10 the first integral is equal to - In (1 - EOG).
Equation 15 can be used to give the correct value of E,, across a bubble tray. It is desirable, however, to consider the proper average value of rn consistent with Equation 15. Thus
Combining Equations 14, 15, and 16 gives
This is the correct average value of m and is considerably more complex than the frequently used Equation 3. When the equilibrium line is straight, m is constant and Equation 17 reduces to Equation 2A. M'hen curvature of the equilibrium line is large, rn in Equation 2A should be replaced by fii from Equation 17, where m in Equation 17 is defined by Equation 6. Use of Equations 2A and 10 allows the determination of the point efficiency across the bubble tray. The significance of Equations 15 or 1 is further shown by consideration of Figure 2. For the model chosen the gas composition changes from 4'1 to 2'2 along a path of constant liquid composition (dotted line). In order to evaluate the integral shobvn in Equations 15 or 17 a series of points along this path must be chosen, each of which is associated with separate but specific values of ( x Z , j i )and m as defined by Equation 6. Liquid Phase Efficiency and Over-all Transfer Units. An equation relating the liquid point efficiency: Eo,to the number of transfer units: SO,,can be derived similar to Equation 10. The model used for this derivation assumes that x * remains constant. ,4horizontal element from inlet to outlet of the tray is assumed and the vapor composition at this level is assumed constant across the tray, thus leading to a constant valuc of x * . The resulting equation is
The models used to derive Equations 10 and 18 cannot exist simultaneously. The liquid point efficiency model is more nearly approached the higher the liquid phase resistance. Vsing the same procedure demonstrated in the previous sections, the equations based on liquid compositions are
and
The proper average value of m consistent with Equation 20 can be evaluated from
Combining Equations 19, 20, and 21 gives
MOLE FRACTION LIQUID
Figure 2. Mass transfer through a bubble tray a t constant liquid composition 36
I&EC PROCESS DESIGN A N D DEVELOPMENT
For the case of linear equilibrium curves Equation 22 collapses to Equation 2B. For all other cases rn in Equation 2B should be replaced by E. The values of m in Equation 22 are as defined by Equation 8. For the model chosen the liquid composition changes from x2 to x3 along a path of constant vapor composition such as indicated a s j I on Figure 2. To evaluate
the integral shown in Equation 22 a series of points along a horizontal line is used and the chords are constructed in accordance with Equation 8. Example
As an example, data from a tower distilling acetone-water in a region where there was large curvature of the equilibrium line have been chosen. The conditions of operation are such that proper evaluation of m is most critical. Operational data correspond to run 50 ( 4 )and are as follows: F = 0.482; liquid rate, 0.538 gallon per minute; total reflux; liquid composition 0.185 mole fraction acetone; weir height, 3 inches; weir length, 7.5 inches; length of liquid travel, 11.5 inches; NsC, = 0.75; and D , = 2.1 X lO-5sq. foot per hour For these conditions N , and can be estimated from the correlating Equations 2, 3, 9, and 10 of ( 7 ) . These equations predict N , = 1.16 and .VL = 10.4. A trial and error solution using these values in Equations 2A and 10 of this paper yields the estimated value of Eo,. A trial and error solution is necessary, since the exit value of the vapor composition is unknown. To carry out these calculations the variation in m must be considered. Using the method recommended in (71, an average m was found to be 12.4, giving an E,, of 39%. Using the present proposed method (Equation 17), the average was calculated to be 1.8, giving E,, = 62%. value of m, The experimental point efficiency reported (4) for this run is 70%, which shows reasonably close agreement with the proposed method. I n graphical evaluation of the integral ~Y2v,mdj/(,y* - j ) it has been noted that even in regions of large curvature the value of m/(y* - y ) remains essentially constant and can, a t least to a first approximation, be evaluated a t 3.1, thus avoiding the graphical integration. For this example it was also noted that the value of T% could be closely approximated by the arithmetic average of m at y l and y2. The foregoing discussion has been concerned with the Murphree point efficiencies. Methods of converting these values to the Murphree tray efficiencies which may be stepped off on the usual McCabeThiele diagram are discussed in some detail (7).
equilibrium line. Use of chord slopes is consistent with the two-film theory of mass transfrr and from this standpoint its merit is not dependent upon agreement with experimental data. Such agreement as shown in the example, however, enhances the mass transfer approach to bubble trays. Nomenclature = interfacial area per unit volume of holdup = interfacial area = = = = =
m
m,
Discussion
In the foregoing development the use of a n average value of m has been emphasized, since this coincides with previously proposed methods (7). The use of Equation 15 for EOGand Equation 20 for EoLis sufficient and actually easier for computational purposes. Use of these equations, however, does not eliminate the necessity of a trial and error solution in each case. This proposed method for calculation of mass transfer across a bubble tray is concerned with the averaging of the slope of specific chords in opposition to the averaging of the slope of the
liquid diffusion coefficient, sq. ft./hr. gas phase point efficiency, (J? - yl)/()* - yl) liquid phase point efficiency, ( x a - x ~ ) / ( x-~ x * ) F factor, up‘j2 gas rate: moles/(hr.)(sq. ft.) gas phase mass transfer coefficient, moles/(hr.) (sq. ft.) (mole fraction) = liquid phase mass transfer coefficient, moles/(hr.) (sq. ft.) (mole fraction) = over-all mass transfer coefficient based on gas phase = over-all mass transfer coefficient based on liquid phase = liquid rate, moles,’(hr.) (sq. ft.) = constants in Equations 1,4, lB, 2A, and 2B defined by Equations 6 and 8 = slope of equilibrium line, & * / d x = correct average value defined by Equation 16 for distillation or desorption in a plate tower for .TOG or EOG = correct average value defined by Equation 21 for distillation or desorption in a plate tower for AYOL or E O L = mass transfer rate, moles/hr. = number of gas phase transfer units = number of liquid phase transfer units = number of over-all transfer units based on gas phase = number of over-all transfer units based on liquid phase = Schmidt group, gas phase = superficial vapor velocity, ft./hr. = mole fraction liquid = mole fraction gas = height of liquid = vapor density, lb./(cu. ft.) =
NSC,
u X
Y
Z
P
SUBSCRIPTS ASD SUPERSCRIPTS i = interface * = in equilibrium Lvith bulk condition of x or y Literature Cited
(1) American Institute of Chemical Engineers, New York, “Bubb e Tray Design Manual,” 1958. (2) Bird, R. B., Stewart, LV. E., Lightfoot, E. N., “Transport Phenomena,” Wiley, New York, 1960. (3) Larian, M. G., “Fundamentals of Chemical Engineering Operations,” Prentice-Hall, Englewood Cliffs, N. J., 1958. (4) Plank, C. A . Ph.D. thesis in chemical engineering, North Carolina State College, Raleigh, N. C., 1957. (5) Sherwood, T. K.: Pigford, R. L., “Absorption and Extraction,” pp. 124-6, McGraw-Hill, New York, 1952. RECEIVED for review September 28, 1961 ACCEPTED April 27, 1962
VOL. 2
NO.
1
JANUARY 1963
37
