Method for Designing Packed Columns M. J. SIMON' AND M. A. GOVINDA RAU2 Indian Institute of Science, Bangalore, India
At present columns are designed using over-all transfer coefficients and over-all H.T.U. values. This method is defective, as it assumes a constant value for the over-all transfer coefficients, whereas it is a well known fact that the over-all coefficients are a function of the slope of the equilibrium curve and are hence variable functions. By assuming that the slope of the equilibrium curve is practically constant over a short length of the curve in an equilibrium unit, the authors have been able to derive a general equation for ddigning all types of packed columns in which the method of material transfer from one phase to the other is by diffusion across interfacial films. The equation can be used either for evaluating the height of a packed column or conversely for determining the individual transfer coefficients.
The equation for the rate of enrichment in the differential seotion of a packed distillation column is: Gdy = koa(yi
- y)dz
= kLa(x
-~
) d = z Ldx
(2)
As shown in Figure 1, x and y are the compositions of the liquid and vapor which contact each other in a differential height dz of the column: xi and 2/t are the interfaciak compositions which are in equilibrium, as suggested by the Lewis-Whitman film theory (6). Equation 2 is of little practical valve as the interfacial compositions xi and yi are not directly known. However it may be reformulated in terms of the over-all compositions x, y, x* and y*. These compositions are represented by the four points A , B, C, and D on the McCabe-Thiele diagram ( 8 ) ehown in Figure 2.. Let the equation of the straight line passing through B and D be of the general form y = m,x kl,where m1 is the slope and kl its constant.
+
P
ACKED columns have been designed usingalmostexclusiveli over-all transfer coefficients and over-all H.T.U. values rathor than the individual film coefficients and the individual H.T.U.'s. Since the over-all transfer coefficients and over-all H.T.U.'s are defined for distillation (1)as follows:
- 1= - + 1Koa
1 - + -1 -=
KLa
1 mkoa
kLa
(H.T.U.)oo = (H.T.U.)c
yi = mixs
Hence (y*
m kLa
koa
+ m . GZ . (H.T.U.)L
(H.T.U.)OL= (H.T.U.)L
L + -.(H.T.U.)o mG
+ kl
Then y * = ,mix
- yi)
=
+ kl
ml(x - zi)
Equation 2 may be now modified in such a way that the interfacial compositions are eliminated from it. Multiplying both sides by ml,
(1B)
mlkGa kLa
(IC)
- m1t.C - 2 % )- Y* - yi (vi - Y) vi - y
Adding the denominators to the numerators on both sides:
+
mlkca kLa kLa
(ID)
their use for design purposes can be justified only when they are constant. For example, this happens to be so both when the resistance of one of the films predominates over that of the other, so that the resistance of the second can be neglected, and when t,he value of m remains constant-that is, the vapor-liquid equilibrium relation is of a linear nature. When both films are controlling and the vapor-liquid equilibrium curve is not a straight line, then KZia, &a, (H.T.U.)OG,and (H.T.U.)OL are functions of the slope of the equilibrium curve and are therefore not suitable for design purposes. They are nevertheless used (incorrectly) for designing packed columns as no alternative method of approach has been presented to the chemical engineer. The purpose of this paper is to suggest a new method for designing packed columns, which overcomes the aforesaid difficulty by the use of the individual film coefficients rather than the over-, all coefficients. The equations in this paper have been formulated in terms of the rate equations involved in distillation; but they can be easily extended to other diffusional operations in which the two films are controlling.
Substituting for(yi Gdu
- Y* - y Yi
-Y
- y) in Equation 2, = koa(yi - l/)dz
and
(3) In the H.T.U. method (9)the function G
(:a
- + kta)
is
assumed to be constant throughout the distillation range and is therefore substituted by a coefficient which is equal to the height of a single transfer unit. This assumption would be valid only in the limiting case when the equilibrium curve is a straight line; under all other circumstances the height of R transfer unit is not constant, as it is a function of the slope of the equilibrium curve. If, however, the reasonable assumption be made that the equilibrium curve coincides with the straight line BD,then the rela-
1 Present address, c / o Works Manager, I. C. I. Ltd., Mossend Co., Lanarkshire, Scotland. * Present address, University of Madras A C. College of Technology, Baidapet P. O., Madras.
93
94
INDUSTRIAL AND ENGINEERING CHEMISTRY
n
y * on the absicssa against
Vol. 40, No. 1
1
(Y* - Y) ~
on the ordinate and measuring
the areas under the curve bounded by the ordinates. The two values will be different. If the values of the integrals are N o , and No2, then the height of the column will be:
&
The height of the column may therefore be expressed as the sum of two heights ZG, and Z G ~ , each of which may be pictured as assisting separately in the transfer across each of the two films. 1 When the gas film is controlling, - becomes negligible and kLa
Equation 5 reduces to the standard form z = -1 =-
1
Koa
&a
G Kca
- N Q ,because
m kLa
, as -is negligible.
When the vapor-liquid equilibrium relation is linear with constant slope m, then integrals A and ,B of Equation 4 may be evaluated directly. No, and NG,are found to be related to each other thus:
LNc,
mGNG,
=
Under these circumstances Equation 5 again reduces to the standard form:
L kLa
C '-Figure 1. Distillation Column Rates and Compositions
tion between x and g* would be identical with the equation of this otraight line. Therefore y * = m1x
rm
dx
=
2(7 from Equation 2.
+ kl
Substituting for
ml
in Equation 3
we have
e
-.G
dy
koa (Y*
- y)
+-.L
dy* h a (Y* - Y)
Integrating between the respective limits A
B
Equation 5 is therefore the general equation for all diffusional operations, and its use may be illustrated by an example. EXAMPLE
A continuous still used for enriching ethyl alcohol-water mixtures was fitted with a 10-foot high column of 1 square foot cross section. It was packed with 1 X 1 inch Raschig rings and operated on a vapor rate of 15.0 pound moles per hour and a reflux ratio of 2. Under steady conditions it was capable of enriching vapors of 27.0 mole % composition to a product of 50.0 mole yo alcohol. Assuming that the value of kca is 2.98 pound moles/ hr. X sy. ft. X Ay at this particular vapor velocity, calculate what should be the height of the column if it is desired to have, a t the same vapor and liquor rate, a distillate of 70.0 mole Yo composition instead of the usual 50.0 mole yo. There are two methods of solving this problem; one using the H.T.U. method and the other by using Equation 5. METHOD1. If the H.T.U. method of design is adopted, then the H.T.U. for the existing column is given by H.T.U. =
where y1
Integrals A and B may be evaluated graphically by plotting y or
h
=
10 = 8.5 ft. 1.176 dy
27.0
= composition of vapors at bottom of enriching sec-
tion of column yl* = composition of vapors in equilibrium with liquid of composition 21 at bottom of column YD = composition of distillate = XD YO* = composition of vapors in equilibrium with distillate of composition XD
G =
Assuming that this value of the H.T.U. does not alter, t,he height of the column required to enrich the vapor from 27.0 to 70 mole %would be h = H.T.U. X
.-&.-
=
8.5
x 4.44
=
37.8 Et.
INDUSTRIAL AND ENGINEERING CHEMISTRY
January 1948
METHOD 2. To use Equation 5'it is essential to know the value of k ~ in a addition to koa. This may be done by Substituting for
95
I
and solving for k ~ a . Therefore
k ~ = a 1.92 lb. moles/hr. X sq. ft. X Ax. For the new column:
Evaluating graphically: 70.0
Therefore h =
(l5
dy
75.0
= 4.44, and
zg:984*44) + ;g24'04) (lo
=
dy*
22.4
4.04
+ 21.1 = 43.5 ft.
As is seen from the result, the H.T.U. method gives an answer in this particular case which is 13% below the height obtained by using Equation 5. The validity of Equation 5 may be tested as follows: Assume koa and kLa have been correctly determined and are 2.98 and 1.92 lb. moles/hr. X sq. ft. X Ay or Ax, Knowing the value of koa and k ~ the a correct height of the column required to enrich the vapor from 27.0 to 70.0 mole % composition may be determined by using the integrated form of Equation 2.
&MPCIStTrON
Figure 2.
- kLa -
The point of intersection of this line with the
equilibrium curve gives the point (z;,y;). 1
-
y is plotted against
and the area under the curves bounded by the ordinates
(Vi VI' y = 27.0 and y = 70.0 gives the value of:
70.0
dy
= 9.097.
Therefore z =
15.00 X 9.097 2.98 45.7 f t .
Equation 5 gives a value which is only 4.8% below this figure; this shows that it is sufficiently accurate for engineering purposes. Unfortunately, Equation 5 cannot be used at present for design purposes as very few data exist on the magnitudes of the transfer coefficients koa and k ~ a . It is therefore essential to gather more data, and methods of doing this are described in the following section. For the case of distillation or absorption where the equilibrium curve is a straight line, the method developed by Haalam, Ryan, and Weber ( 4 ) is quite suitable. They assumed that the gas transfer coefficient koa varies as the 0.8th power of the vapor velocity just as the heat transfer coefficient ha does. Hence
- 1= - + -1 = Koa
koa
1 1 H ~ L U +(Go.') + HkLa 1
where H = Henry's law constant for cases of absorption
If
m.v. =
with a positive intercept of
1 us. Koa
1
1
It H~LQ'
should be a straight line is therefore possible to
-
L (H.T.U.)oG KLa G (H.T.U.)oL = Koa
For experiments in which the vapor and liquid velocities are constant, the values of koa and kLa may be assumed to be constant. If 1/KLa computed from such experiments is plotted against l/mav.,then from Equation 1B the slope of the straight line would give us the value of l/koa, while the intercept on the y-axis would give the value of l / h a . Another and perhaps a more direct method of evaluating the film coefficients would be by setting up a second equation involving koa and &a similar to Equation 5 and solving the two aa simultaneous equations. I n the case of experiments performed with a view to determining the value of koa aqd h a , side streams of the vapor and liquid may be tapped and analyzed at some other depth z' from the top of the column. If 2; and y i are the compositions of the liquid and vapor, then
(7) where y:*
=
composition of vapor in equilibrium with liquid of composition 2;
Equations 6 and 7 may be solved as simultaneous equations, end the values of h a and kLa are:
1 is assumed to be independent of the gas velocity, then a HkLa
plot of the over-all resistance
Equilibrium Diagram
evaluate the film coefficient k ~ a ;and by substituting its value in Equation 6, koa can also be determined. A method of evaluating the individual film coefficients when the vapor-liquid equilibrium curve is not linear has been devised by Furnas and Taylor (8),particularly for problems on distillation. They substituted m of Equation 1B by the average slope ma". obtained from the relation.
To solve this equation, any point (z, y) is selected on the operating line, and a straight line is drawn from this point having a slope
O F LIQUID X-P-
and
96
INDUSTRIAL AND ENGINEERING CHEMISTRY
This method of evaluating kGa and b ~ ahas two distinct advantages: The conditions of flow of the vapor and the liquor, on which the magnitudep of k,a aud k ~ depend, a are maintained constant during the experiment; and in a sing!c run, involving the tapping and analysis of five samples and the evaluation of four graphical integrals, sufficient data t o ctrlc:*1atethe values of k ~ a a n d kLa are obtained. The values of the film coefficients are useiul not only for design purposes but also for correlations. For instance, kca may be correlated against the characteristic dimensions and properties of the gas film as defined by Reynolds group d f i p I p and the Bchmidt group p , / p D . This correlation could be also used to verify the Chilton and Colburn theoretical equation (2)for mass transfer. Moreover, new correlations between kLa and other dimensionless groups which define the thickness and properties of the fluid film can also be attempted.
K&
=
Vol. 40, No. 1
number of theoretical units defined by
h’c, = number of theoretical units defined by IV& = number of theoretical units defined by ?,!*
(?/* -
Y)
P R
= rate of withdrawal of product, Ib. mole:/;. ft. X hr. = reflux ratio L I P u = linear velocity of vapors, ft./sec. x = mole fraction of more volatile component, in liquid phase 5* = mole fraction of more volatile component in equilibrium with vapors of composition y xi = mole fraction of more volatile component at interface in equilibrium with vapor of composition y, 1 1 = mole fraction of more volatile component in vapor phase y* = mole fraction of more volatile component in equilibrium with liquid of composition z Vi = mole fraction of more volatile component a t interphase Ay = small finite change in y 2 = height of column, ft. 2’ = height of column required to enrich vapors from y i to
ACKNOWLEDGMENT
The authors are highly grateful to Jnan Zhandra Gosh, director of Indian Institute of Science, for his constant encouragement of investigations into the principles of chemical engineering. They are also thankful to R. L. Pigford and E. M. Schoenborn for their criticism and valuable suggestions.
YD.
P C
NOMENCLATURE
c
PI
density, Ib./cu. ft. coefficient of viscosity of main body of fluid, lb./sec. X ft,. = average coefficient of viscosity of film, lb./sec. x ft,.
= =
a = area of interphase contact, sq. ft./cu. ft.
d = equivalent diameter, ft. D = coefficient of diffusion, sq. ft./hr. G = rate of flow of vapors, lb. moles/hr. X sq. ft. H = Henry’s law constant, lb. moles/cu. ft. X atm. km = gas film coefficient, lb. moles/cu. ft. X hr. X Ay k~ = liquid film coefficient, lb. moles/cu. ft. X hr. X Ax KQU= over-all gas transfer coefficient, lb. moles/cu. ft. X hr. X AII KLU = over-all liquid transfer coefficient, lb. moles/cu. ft. X hr. X A x L = rate of flow of reflux, lb. moles/sq. ft. X hr. m = sloue of eauilibrium curve
No, = number of theoretical units defined by
LITERATURE CITED (2)
Colburn, A . P., Trans. Am. Inst. Chem. Engrs., 35,211 (1939). Colburn, A. P., and Chilton, T. B., IND.ENQ.CHEM.,26, 1183
(3)
Furnas, C. C., and Taylor, M. L., Trans. Am. Znst. Chem. Engre..
(4)
Haslam. R. T.. Rvan. fV. P.. and Weber. H. C.. Ibid.. 15. Pt. 1.
(1)
(1934).
36, 135 (1940). I
“
177 (1923). (5) Lewis, W. K., and Whitman, W. G., IND.ENG.CHEM., 16,121520 (1924).
(6) McCabe and Thiele, Ibid., 17, 605 (1926)
LYD?,.” Y)
RECEIVED March 3 , 1945.
Kinetics of an Esterification with Cation-Exchange Resin Catalyst LJ
*/
CHARLES L. LEVESQUE AND ANDREW M. CRAIG Resinous Products and Chemical Co., Philadelphia, Pa.
T h e esterification of butanol and oleic acid has been studled in the presence of an acid-form cation-exchange resin as catalyst. The reaction is essentially second-order after on initial slow period. The velocity constant for the reactbn is directly proportional to the surface area of the catalyst per unit weight of reactants.
D
URING the war years, German chemists discovered the usefulness of acid-form cation-exchange resins as catalysts for esterification reactions, and apparently developed the process to the point of large-scale continuous operation (1, 5). Recently Bussman (6)reported data on the use of such resins in a number of reactions susceptible to acid catalysis. In the course of work in this laboratory, considerable data have been obtained on the butanol-oleic acid system used in most of
Sussman’s experiments. This paper reports the results of the application of kinetic theory to these data. It is unfortunate that the study of the kinetics was not the primary purpose of the R-ork. Had it been, refinements in experimental conditions could have yielded data better suited to kinetics analysis. However, the results obtained seem of considerable interest from b?th a theoretical and practical point of view. CATALYST PREPARATION
The cation exchanger used was a phenol-formaldehyde-sulfonic acid resin (Q), with an exchange capacity of 2.7 milliequivalents per gram of dry weight. (A similar resin is commercially available, sold under the name Amberlite IR-100.) Two samples were prepared:
