ENGINEERING AND PROCESS DEVELOPMENT
w
Taylor (11) and Koffolt, Duncan, and Withrow (16). His method of calculating theoretical plates may account for this variation. Some of the data of Furnas (11)for ring packings were not used because they did not agree with other results or cannot be reconciled with the work of other investigators-for example, height of equivalent theoretical plate is lower with a/&nch rings as the packed height is increased, and is lower with 1-inch than with 3/8-in~hrings. For systems such as ethyl alcohol-water or methanol-water, where a varies widely with composition, the average a was obtained by a graphical integration method similar to t h a t used by Koffolt, Duncan, and Withrow (16) to evaluate m, the slope of the equilibrium curve. The author’s data on methanol-water, used to evaluate the effect of column diameter, were obtained a t 40 to 1 reflux ratio rather than total reflux. Koffolt and Withrow ( 1 6 ) and Furnas ( 1 1 ) present a large amount of data obtained a t low reflux ratios, but th’e results show very high values at low vapor-liquid ratios. The reason for this is unknown. Colburn and Pigford (7) suggest t h a t liquid and vapor compositions are so close together that poor distribution at the ends of the column results in low efficiency. Another explanation might be t h a t with low reflux ratios the operating line is very close to the equilibrium curve. The assumption of a straight operating line may not be valid, and if the operating line actually curves toward the equilibrium curve the true number of theoretical plates would be much greater than if a straight operating line is assumed, and hence the values for height of equivalent theoretical plate would be lower. literature Cited
(1) Aston, J. G . , Lobo. W. E.. and Williams. B.. IND. ENG.CHEM., 39, 718 (1947).
Berg, L., and Popovac, D. O., Chem. Eng. Progr., 45,683 (1949). Bowman, J. R., IND. ENG.CHEM.,39, 745 (1947). Bragg, L. B., Ibid., 33, 279 (1941). Cannon, M. R., Manning, R. E., and Heinlein, A. C., C h m . Eng. PTOgT., 47, 344 (1951).
Colburn, A. P., and Chilton, T. H., IND. ENG.CHEM.,27, 255 (1935).
Colburn, A. P., and Pigford, R. L., “Chemical Engineers’ Handbook.” 3rd ed.. New York. McGraw-Hill Book Co.. 1950. Deed, D. W., Schutz, P. W., and Drew, T. B., IND. ENG. CHEM., 39, 766 (1947).
Fisher, A. W., and Bowen, R. J., Chem. Eng. Progr., 45, 359 (1949).
Forsythe, W. L., Stack, T. G., Wolf, J. E., and Conn, A . L., IND. ENG.CHEM.,39, 714 (1947). Furnas, C. C., and Taylor, M. L., Trans. A m . Inst. Chem. Engrs., 36, 135 (1940).
Gilliland, E .R., “Elements of Fractional Distillation,” 4th ed., New York, McGraw-Hill Book Co., 1950. Hawkins, J. E., and Brent, J. A,, IND.ENG.CHEM.,43, 2611 (1951).
Heinlein, A. C., M. S. thesis, Pennsylvania State College, 1949. Kirschbaum, E., “Distillation and Rectification,” pp. 313, 315, 316, New York, Chemical Publishing Co., 1948. Koffolt, J. H., Withrow, J. R., and Duncan, D. N., Trans. A m . Inst. Chan. Engrs., 38, 259 (1942).
McCabe, W. L., and Thiele, E. W., IND.ENG.CHEM.,17, 605 (1925).
Manning, R. E., M.S. thesis, Pennsylvania State College, 1949. Myles, M., Wender, I., Orchin, M., and Feldman, J., IND. ENQ. CHEM.,43,1452 (1951). Peters, M. S., Ph.D. thesis, Pennsylvania State College, 1951. Ryan, J. F., M.S. thesis, Pennsylvania State College, 1950. Sherwood, T. K., and Holloway, F. A. L., Trans. Am. Inst. Chem. Engrs., 36, 39 (1940).
Sprague, E., and Fletcher, J., Dow Corning Corp., private communication. Struck, R. T., and Kinney, C. R., IND. ENG.CHEM.,42,77 (1950). Weimann, M., Chem. Fabrik, 6 , No. 40, 411 (1933). RECEIVED for review June 14, 1952. ACCEFTED bugust 26, 1953. Prasented at the Meeting-in-Miniature, New Jersey Section, AMERICAN CHEMICAL SOCIETY, Newark, N. J., January 1961.
Application of Computing Machines to Exchange Column Calculations ASCHER OPLER f i e Dow Chemicol Co., Pittrburg, Calif.
T
HIS report represents the results of an intensive exploration of the possibilities of applying digital computing machines to the mathematical treatment of ion exchange column performance. Using a simple, slow speed, punched card operated computer, several different types of calculation were programmed and typical ion exchange problems were solved. This report should be interpreted as indicative of the potentialities of applying computing machines to this problem rather than as an exhaustive study. Rosen and Winsche (18)and Rose, Lombardo, and Williams (19, 17) previously have applied computing machines t o similar problems. Rosen and Winsche used an analog computer in investigating the performance of a coIumn whose feed concentration is a sinusoidal function of time. Rose, Lombardo, and Williams described a finite difference method for calculating the performance of adsorption columns. Recent books have reviewed mathematical treatment of adsorption and ion exchange ( 4 , IO, 16). Although the equations, examples, and discussions will be restricted to ion exchange, in all cases the techniques may be modi-
December 1953
fied for use in adsorption column calculations. Chromatography, selective adsorption, and similar phenomena may be treated by each of the methods described. Computing machines can play a large role in extending the number of practical problems that can be treated mathematically. They may be applied to various theoretical and empirical equations for the purpose of obtaining large quantities of numerical solutions; applications to the equilibrium plate and the kinetic treatment are illustrated. These machines are useful in connection with a finite difference technique t h a t requires the machine to operate as an analog of the column. The probabilistic Monte Carlo method may also be handled by automatic computing machines and may ultimately lead to the most complete treatment of parameters. With one minor exception, all calculations reported here were performed on the 602-A calculating punch of the International Business Machines Co. Other supplementary machines were used in originating and manipulating the more than 50,000 punched cards involved in this study. The 602-A was programmed using both the standard wire-programming method, the
INDUSTRIAL AND ENGINEERING CHEMISTRY.
2621
ENGINEERING AND PROCESS DEVELOPMENT use of control panels, and the card-programming method developed in this laboratory by the author. This latter method, which uses an instruction code on each card, proved useful in carrying out complex programs. Wire-programming proved most suitable for operations that were relatively simple and for probability calculations. The work described here was limited by the speed and capacity of the machine to problems of modest length and complexity. The extension to more difficult, longer problems on a high speed computing machine should be straightforward.
Table I.
( p = 39,
Results of Equilibrium Stage Treatment Prove Adaptable to Machine Computation
Using a set of simplifying assumptions, blayer and Tompkins ( 1 3 ) have derived equations for ion exchange column perform-
ance. Their assumptions are:
A column may be considered to be a vertical array of discrete, uniform layers or plates containing equal masses of adsorbent. Equal unit volumes of developer are added in discontinuous stepwise fashion displacing previously fed volumes to successively lower plates. In the special case treated by these authors Flow and Darticle size are such that eauilibrium occurs on each plate. A linear isotherm relates the equilibrium on any plate, that is q * / c * = C. The top plate contains 1.000 unit of ion A a t the start of the operation. Under these conditions the concentration of A in the nth unit volume leaving plate p is given b y Cn,p
=
(n
+p
c-1
- l)!
(n - l)! p ! (1
+ C)"+P
(1)
Elution Calculated Using Mayer and Tompkins Equilibrium Stage Treatment
c
= 1)
n
cn,p
n
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
0.00013 0,00024 0.00041 0.00068 0.00107 0,00164 0.00242 0,00346 0,00480 0.00647 0.00849 0,01087 0 01358 0.01661 0.0198'i 0.02329 0 02679 0.O3025 0.03356 0 03661 0 03930 0,04154 0,04328 0.04445 0 04505 0 ,04604 0.04447 0.04338 0,04184 0,03989 0.03762
46 47 48 49 50 51 52
44 45
Cn,p
0 . 0 3 5 11 0.03245 0.02968 0.02689 0.02415 0,02149 0.01896 0.01659 0.01440 0.01240 0.01060 0.00982 0.00757 0,00633 0.00525 0.00433 0.00355 0.00289 0.00234
53
54 55 56 57 68 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
Q.00188 0.00150 0.00119 0.00114 0,00074 0.00058
O.OO046 0.00035 0,00027 0.00020 0.00015 0.00012
76
sponding c and p values located and punched automatically. The results are listed in the desired order by the tabulating machine. Table I gives the reeults of euch a calculation when C = I, and the column consists of 39 equilibrium plates. In accordance with Mayer and Tompkins, emax occurs when p = n / C and is close to the expected value
and the concentration of ion A on the exchange resin is given by q71.p
=
+
0 p - I)! ( n - l ) ! p ! (1 C P + p
(72
+
These two formulas may be evaluated rapidly and accurately using a punched card calculator. In general, one would wish t o p constant), an calculate a column profile a t any instant ( n elution curve for a column p plates long ( p constant), or the concentration history of the nth volume as it passes from plate to plate (n constant) As is typical of computing machine applications, an entire set of functions is calculated as a single operation To evaluate Equations 1 and 2, they are rearranged to logarithmic (base 10) form
+
log
Cn,p
+ p - l)! - log (n - l)! + ( n - 1) log c - (n + p ) log ( I + C) = log (a + p - 1)' - log ( n - l)! Iogp! + n l o g C - ( n + p j l o g i l + C j = log ( n
log p !
log qn p
(3) (4)
This may be considered equivalent to operations of the type
X
-Y
-
Z +EG - F H
(5)
which may be performed by the 602-A machine in one operation For any calculation log C and log (1 C ) are placed in internal storage. The punched cards carry (n - I), ( p ) , (n - 1 p), and the logarithm of the factorial of each of these 3 parameters. As no punched card tables of log (n!)were reported available, a deck was prepared here. Values to log IOO! were keypunched from a table. Stirling's approximation to n! ( = n n e n c n )was transposed to log ( n ! )= n log n 1/z log 2 n n log e and this was evaluated for 100
