November 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY
-
( c ) The b y s stem may now be considered as a linear system for t i e 4 b’s on the left side, the b’s
Table I. Numerical Example Constanta for s Specified Temperature, Pressure, and Atomic Composition KI = 1.444 X 10-6 T = 3000° K. P 20.42 atm. K I = 1.130 X 10-1 K8 = 1 548
$ H
-
P
0.7143
0.2381
K‘
-
0 1579
K I = 1.373 X 10-2
K6 = 8.81 K I = 9.61
= 0.3333
Iterative Cycle No. Assumed initial values $1 = &IF fr faa 21 $4
--
far fN2
E values calcd. from right-hand
...
19a. side of Equation 18a In the more general notation thew are Eq. 1, 2, 3, 4 BI = II(#I, A, fa, 24) Bz = A Br fa Bi = fc New z valuea calcd. from solutions of Eq. 168. 19a of the form of Equations l’, 2’, 3‘. 4’)
-
..
ZI 51 XI ZI
1
2
35
0.6250 0.2080 0.0208 0*1460
0.8124 0.1690 0.0356
0.0302
-0.0038 +0.0202 -0.0021 +0.9707
-0.0033 +0.0238 -0.0019 f0.9877
-0.0035 +0.0340 -0.0016 fO.9598 0,8124
$8!:
0.1429
2449
0.6145
0.1806
0.6150
:;:;8:
4
-
0.6140 0.1797 0~3308
-0.0033 +-0.0235 -0.0019 +0.9679
0.6140
0.8140
0.1434
0.1434
8:igg; 8:A:X:
0.1436
also occurring in the r’s. The y’s are considered now as constants as obtained in step (b). Assume 4 6’s and perform iterations on the b’s until approximate convergence. This can be done again on the IBM machines as before. This completes one subcycle of the entire iteration process. ( d ) From the conservation of mass equations, linear between the 4 a’s and 4 b’s, the a’s can be exressed as linear combinations of the b’s once and For ali. Substituting the values of the 4 b’a obtained from ate (c), calculate new values of the 4 a’s. (e) &th the new values of the a’s and the final values of the x’s obtained in step (a), repeat steps (a), ( b ) (c), and { d ) until convergence in the 4 u‘a is reac6ed. This finishes the com lete iteration cycle and the problem is solved. $his procedure seems to a large extent, but with certain modifications, analogous to the procedure outlined in ( 1 ) .
ACKNOWLEDGMENT The authors wish t o acknowledge S. W. Dunwell of the International Business Machines c o . who programed the iteration method for use on digital aomputers.
As a result of prior ex erience the assumed values for thi’ cycle were taken to be those of cycle 2 plus 0.8 orthe i n c r k n t to the x’s obtained in cycle 2. The number of oyclea required for convergence was thereby decreased. b When the z values in the third step agree with the assumed values witbin the allowable error, the computation is complete. 0
cumberaome system than the single phase roblem but the iterations procedure in the former case can &e generalized to the present case as follows: (a) Assume 4 a’s and consider them fixed. Calculate the x sysother 12 a’s. Assume 4 5’s. Perform iterations on a tems as in the single phaae method until approximate convergence. This can be done on the IBM machines aa usual. (b) From the chemical potential equations solve for the logarithms of the y’s in terms of the x’s obtained in (a). This can be done once and for all in so far as solving the system of simultaneous equations goes. Only substitution is necessary. Calculate the y’s from the logarithms.
-
LITERATURE CITED
(1) Brinkley, Stuart R., Jr., J. C h m . phys., 15, 107-10 (February 1947). (2) Fraaer, R. A., Duncan, W. J., and Collar, A. R., “Elementarv Matrices.” New York. Macmillan Co., 1946. (3) International Business Maohines Co., “A Demonstration of the Applioation of the IBM Type 604 Electronic Calculator to Sequence Calculations,” Aug. 16, 1948. (4) M. W. Kellogg Co., “Bimonthly Progress Report No. 2-Propellant Investigation, Appendix ‘A’,” SPD 155 (April 1, 1948). (Confidentiai.) (6) Whittaker, E., and Robinson, G., “The Caloulus of Observationa,” London, Blackie and Son, Ltd., 1944,
RWIUIVED Maroh 5, 1951.
Absorption Calculations by Punch Card Calculators J. W. DONNELL
AND
R. W. DRAPER
Michigan State College, East Lansing, Mich. T h i s work was undertaken to eliminate the long trial and error calculations involved in (previous) accurate methods of calculating the number of absorber trays and/ or the amount of absorber medium required to produce a given absorption and to reduce errors by making the calculations mechanical. An iteration process has been developed such that when preliminary estimates-however approximate-are subjected to one iteration cycle, an answer is obtained closer to the cdrrect answer than the preliminary one. These iteration cycles are arranged so that no engineering judgment is required and the cycle is performed mechanically. A working table included in the article outlines in detail the steps to be taken. The IBM 602 calculating machine was adapted to this table, thereby greatly speeding up the work.
This process will enable engineer. to carry out many accurate absorption calculations (including cooler locations) to arrive at the most economical number of plates versus the absorber medium rate.
T
HE absorption work of Kremser (1) in 1930 paved the way for a systematic approach to multicomponent absorption tower design. Kremser made the assumptions that the liquidto-gas ratio does not change throughout the absorption tower and that the equilibrium curve, y = kx, is a straight line. His work gave fairly accurate tower design for lean hydrocarbon mixturesi.e., small amount of absorbed material as compared to total gas. For rich mixtures the absorbed material itself begins t o act as an absorber oil and Kremser’s calculutions do not accurately apply. Some modifications of the method have been developed (2) to take into coneideration this additional absorber oil effect : how-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
2450
ever, these modifications give only approximate results. Certainly this is the rase when one has to take into consideration the temperature effect of the absorbed gapes and provide coolers to keep the tower temperature down. Of course, one can have recourse to the long and tedious but rigorous method of estimating the individual gases absorbedi.e., estimate analysis of the outgoing gas-and then make tray-totray calculations to the bottom plate. If the gas entering the bottom plate does not agree with the calculated analysis then a new estimate of the gases absorbed must be made and the plateto-plate calculations must be done again. This procedure is repeated until the calculated analysis coincides with that of the actual incoming gases. This procedure is long and tedious, although accurate. No doubt the IBM punch card calculator could be used to make the above calculations with considerable saving in time due to the great rapidity with which additions, subtractions, multiplications, and divisions can be made. However, under these conditions considerable unnecessary IBM machine time would be lost and an engineer would have to be in attendance to help make the new estimate of the amount of gases absorbed. By the proper rearrangement of the equations used in the plate-to-plate calculations more time can be saved and in addition no engineer would have to be in attendance. The rearrangement of the material and heat balance equations used in plate-to-plate calculations will first be carried out followed by the adaptation of the IBM machines to these equations.
Vol. 43, No. 11
The system of equations derived from material balances around each plate (set A ) is as follows:
Yo YI Yz
AZYZ- y1(1 + A I ) = 0 +++ Asp3 - Yz(1 + A ? ) = 0 AdY, - Ya(1 + A3) = 0
............................
+
YN-1 -k AN+IY.x+I - Y N ( ~ A.v) = 0
+
By dividing the first equa.tion by ( 1 A , ) and adding the seeond to it, Y1 is eliminated and the equations are:
Yg
+
.44E’4
-
Y3(1
+ As)
=
0
Similarly Y2 may be eliminated by dividing the resulting equa-
+ AS - &)
If a sequence of numbers is defined as follows: 61 = (1 AI), bz = (1 Az , bh. = (1 AN then elimination
tion by (1
+
and adding the third equation.
e). . .
+
+
&)
+
yo AN+IYN+I bibs . . . bN-1 AN+I Y N +I For the m e where bN
o f Y I ,Y2. . . YN--Ileaves the equation
- YNbNOr Y.v = bibtb~YO... b*v
.
+
YN+, = 0
EQUATION REARRANGEMENT The first prerequisite for this arrangement will be that when the engineering data*-g., operating pressure, number of plates, maximum allowable tower temperatures, eta-are available for use with these equations no further engineering data or judgment will be required regardless of the number of trial and error calculations or coolers required. In other words, the equations must be arranged 80 that the use of these is purely mechanical. Using the standard nomenclature for absorption calculations the rearrangement of equations is carried out as follows: By definition K , = yn/xn and A , = L,/G,K, and so L,X, =
L n G , K, yn G,
=
A n y , . A J , , is the amount of a component in the
liquid from the nth plate. From a material balance around the nth plate for one component:
Yn-I + A n + ]
17w+1
=
Yc
+ AnY,
(Set A)
From a material balance around the top of the column for one component:
Yo
+ Anl’n +
AIYI
A N + I Y Y + I Yn-1
+
YN
Yn-I
+ AnYn
(Set B) (Set C)
N is the total number of trays and n is any plate below the top plate.
or for any plate n:
Yo l7,$ = -____
bil12b~. . .bn
From a material balance around the entire column
+
or
Yo f A N + I Y N += I YN AIYI Yo - Y1.41 = YN - -4N+iYNfi
When A N + I Y N +=I 0, Y N = Y O- YIA1. After YNhas been determined the Y,’s can be found by a sequence of calculations using the system of equations derived froni niaterial balances around each plate. However, the calculations are slightly simpler if the equation from a material balance around the top or bottom of the column is used. The equations from a material balance around the bottom of the column (set B ) are:
++ + ..............................
(Yo - Y I A I ) A2Yz - Y I = 0 (Yo - YIAI) AsYa - Yz = 0 (Yo - YiAi) A,Y, - Yz = 0
+
(Yo - Y I A I )
- YN
3
0
JVhen A.v+IYN+I= 0, Y N = (Yo - Y I A I and ) the equations may be written
YO - YN Yl = ___ AI
SOLUTION OF MATERIAL BALANCE EQUATIONS It is necessary to use all three of the above equations in the absorber tray calculations. (The position of the plate in the tower determines which one of the three equations will be used.) This is necessary in the interest of accuracy-e.g., when very small concentrations of a given component are used to calculate a much larger concentration on another plate large accumulative errors are involved unless excessive decimal places are used in the small concentrations. This condition is then eliminated by switching to one of the other equations. A simple rule will be given later stating just when to use each equation. Each of the above sets of material balances form a system of N equations, containing N unknowns, if the compositions of the input streams to the column are known where N is the number of plates in the abfiorber tower.
AN+lYN+I
................. Or for any platen:
For components for which the A’s are small and Ye is only slightly larger than Y Nthis system of equations is not practical
INDUSTRIAL AND ENGINEERING CHEMISTRY
November 1951
to use and the equation derived from a balance around the top of the column is preferable. The equations derived from material balances around the top of the column (set C) are as follows:
- A N + I Y N + I+) A N Y N - A w + I Y N ++~ )A N - I Y N - I 22 = t2 (YN - A N + ~ Y N + + ~AN-ZYN-2 ) ................................
+
YI= ( Y N - A N + I Y N + I ) AZYI
for cases where A N + I Y N +=~ 0 these equations become: Y N - I= Y N Y N - Z= Y N YN-3 = Y N
- to)
=
- to) ( L N C +~ ~GNCp,,) - ( ~ N - I - to) GNC+ - H,AG.v __. LN+~CD~
-
(3)
For components for which YNis very small these equations are not practical to use and the former system derived from material balances about the bottom of the column are preferable. It is not necessary to develop heat balance equations in order that the heat of absorption be taken into consideration. This heat absorption is considerable when gases that contain large concentrations of absorbable material are being processed and manifests itaelf in an increase in temperature of the absorber oil. In many cases coolers have to be located in or on the tower to keep this temperature down. These heat balance equations help locate these coolers in the tower. The gas rate from each plate is found by the equation G , = XY,. The decrease in gas for each plate is Gn-l G, = AG,. The liquid rate from each plate is L. = L,+ 1 AGn.
+
(tN+I
( 1 ~
The above equation niust equal zero nhen the cooler cools the product down to temperature of fo. These equations can all be easily solved directly with the e\ception of those plates between the top cooler and the tower top. Theee particular equations can best be solved by trial and error. Specifically the trial and error procedure for these few plat- is carried out as follows:
YN-8
NYN +++ AAAN-SYN-S N-IYN-I .................. Yi = Y N + AzYz Yo = I'N + AiYi Or for any plate n: Yn-j = Y N + A n y n
0 =
245 1
-
HEAT BALANCES AROUND PLATES The datum temperature is t o o F.
Trial (t, - t o ) and (k lo) can be assumed and the respective sequences of equations solved until trial tl and to are found for which to and t ~ are + equal ~ to to. When the temperatures have been determined, equilibrium constants corresponding to the respective temperatures are used to calculate new absorption coefficients to be used in another series of calculations.
DETERMINATION OF ABSORPTION COEFFICIENTS For the first calculation the absorption coefficients for the various plates are unknown. A satisfactory means is to msume constant LIGratio and constant temperature with values in the range in which these quantities are likely to vary. The resulting temperature and liquid and gas rates from the first results are ueed t o obtain the absorption factors for the second calculation. The use of these results will correct the trial absorption factors in the direction of the true values-e.g., if the liquid and gas rates found in the first calculations are too high the L/G ratio will be high for the liquid and gas rates, normally used; if the liquid rate is high the amount of gas condensed is high which indicates that the calculated heat liberated and thus the temperatures are high. Increased temperature causes an increase in equilibrium constant K. As the change in K is more pronounced than the change in L / G the absorption coefficientA = L/GK decreases. The smaller valuep result in small liquid and gas results in the succeeding calculation. This method corrects the absorption coefficient in the right direction. STEPS IN CALCULATION PROCEDURE The follov&g procedure for the analysis of the gas leaving a given tower-Le., number of plates-with a given oil and gas rate summarizes the calculations. 1. Assume some average operating temperature, say 100' F., and at the operating tower pressure calculate the A values-i.e., L/KG for each component. The value of L/G for these A values will be that of the incoming oil rate (moles) divided by the incoming gas rate. 2. Calculate the amount of gas leaving the top plate by Equation I-Le., Y N for each component. Also calculate the amount of e ent in the gas leaving each plate by means of Equation 2-i.e., Yn-l = Yn--2 - y N for a11 An- 1 values of A greater than 1.0. For those coniponents whose A values are less than 1.0 use Equation 3-Le., Y,-1 = Y,v -IAY,. 3. By substitutin in the heat balance equations locate the liquid coolers such t f a t the maximum temperature rise is not exceeded. A trial and error calculation is involved in this step in calculatin tray temperatures between the top cooler and the top tray. T i e temperature of the tray above the last cooler must be chosen so that when calculating tray-to-tray upward the calculated temperature of the liquid to the top tray must coincide with the temperature of the incoming absorber oil. The tray temperatures are also obtained in the cooler location calculation. 4. Repeat items one, two, and three using values obtained in step three until the individual plate, L/G, becomes constant from one calculation cycle to the next.
-
LCCP, Similarly the equations for the plates above the cooler are: (lc+2
- to) = (h+l t o ) (LetlCp,f G*ICP,) (Is Lo+ZCP,
- to) GaCp,, - HvAGo+i
(tots
- to) = - to) (Lo+~Cpl+ G c + d J p e ) - ( t a + l
- to) GoCp, - HsAGc+,
-
(te+2
Lo+8 C P I ..................................................
USE OF WORK SHEET Table I is a work sheet that allows the calculations to be carried out mechanically. This is done by numbering each column and in each column detailed instructions are given as to how the cal-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
2452
culation for this column is to be carried out. The vertical line in the center indicates that the calculation for some intermediate plates has been omitted from this sheet but these omitted calculations are carried out in exactly t h e same fashion as the calculations immediately preceding the line. The data for the example used on the work sheet are contained in columns A and B and it will be noted that only four out of the 20 plate calc u l a t i o n s a r e shown. The remaining calculations were carried out s e p a r a t e l y from t h e sheet. Only the first calculation cycle and the start of the second cycle is shown on the work sheet. For thia problem only three ca 1cu la t i on cycles were required to arrive a t t h e c o r r e c t answers. A discussion of the heading of each column follows:
u
I
I
Vol. 43, No. 11
ml I
k
1. For the first calculation constant values are assumed for L,fGand t-columns C (1, , .N)' D, and E (1.. .N). 2 . K values corresponding to the temperature are selected from a F table-columns (1.. . N ) . 3. The a b s o r p t i o n factor A = L / K G is calculated-col~ns G (I, 2 . . .N). 4. The sequence of calculations for the b'a is calculated-column H (1, 2 . . .N). 5. Y Nis calculatedcolumns I and J. 6. YlY,. YN--1are calculated for components for which A is less than 1.0-column K (1. ~, 2 . . .N). 7. Y I Y ~ . ..YN-1 are calculated for components for which A is g r e a t e r t h a n 1.0column L (1, 2 . . . N). 8. The gas rate is G, = ZY,-column M (1, 2 . . .N). 9. The decrease in G = AGn Gn-l-column
..
d
E:
N (1, 2 . . .N).
10. The sequence of liquid rates is calculated from the to down Ln = L,+l A8,---column 0 (1, 2 . . .N).
+
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
x b e a r i k 1951
2453
~~
Functioning of Punch Card Calculator for Absorption Calculations
Table I!. ~Contrul Bmandl
Algebraic E uations to Be Ca?cd.
NO.
&/sa
r(L/@J9 dl Ai
9
X
Source of External Data L and Q from cards being punched
Where Calcd. Data Are Punched
On respective cards being
processed AiArAa eta., from cards being proc- brbsbs, eto., on cards being eased. 1 from digit emitter prooessed
= A
++ 11 =- bf,?;; - b.
Y N on trailer card
bihba, eto., from cards being prooessed. Yo from trailer cards
+ 0+ Y N A N + YN-IAN-I l" + YN-rAN-a, etc.
I N
YN
YN-2
= YN
PN-
'I"-, =
I"
, on YN from oard preoedinr set of cards Y N - I Y N- r Y ~ - r etc., being prooessed. A N A N - ,AN -s cards being processed from oar& being prooessed Ya and YN from cards recedin set of Y I Y I Y I ,etc., on cards being processed cards being processex AiAi%r, eto., from oards being prooessed YN4 YNbYNo from series of sets of cards a, AQ L on respective traildr cards being rooked YN-lo?N-rbYN - l o . . . YN -r.YN -8bYN t o . LN + I from oard following cards giving Y N- I O Y N - i b 4- YN o . . . C p o C p l + H v from a oard preceding the Ha, Ha, X L L on cards set of cards being prooessed. 0, L, being processed AG from cards being prooessed Ln+ 1Cpi; PUn = HvQn+ I from cards (tn + I to) on cards being being processed. (tl - to) from first processed card being processed
- . -.
- tO)(LfiCPl + &CP" Ln + LCPZ (tn - - to)& - I C P ~ H v AQn (In
.L
-
I
tn+i
+
-
- to
11. H, LW,GCp, and Lep, are the products of the respective uantities-wlwnns P (1,2. N), Q (1, 2. . .N), and 8 (1, 2.
..
I
..
3,. 12. The equations for the Issloeessive temperatures are solved
directly. However, the piak temperatures between the to plate and top cooler are mlved by trial and error until the cap culated top temperature equals the actual temperature-columns D and E. 13. Usin these L, 0,and t values, K values may be selected and these ca%ulatione repeated. Each of the above series of calculations can be worked on the IBM 602 calculating punch machine.
APPLICATION OF IBM PUNCH CARD CALCULATORS The principle involved in the IBM punch calculators is that of rapidly taking data recorded on punch holes in 31/4 X 7*/4 inch cards and multiplying, dividing, subtracting, or adding this data in a predetermined fashion a t a speed of several hundred operations per minute. In the application of the IBM calculator to this problem it is assumed that K values for various components at various temperatures and pressures are available or can be made available in the form of punched cards. Since the calculators must be wired to perform certain calculations and, furthermore, since the wiring mechanism is in the form of easily removable completely wired boards the functioning of the boards for the absorption calculation is outlined. The first column in Table I1 shows that the rearranged material and heat balance equations have been calculated entirely by the calculating machine. The wiring of the control boards to take
care of such items as the trial and error calculations involved between the top cooler and top plate is left to the machine operator aa is the sorting of the cards to enable the machine to get the proper information for a given calculation.
NOMENCLATURE
L
G z g
= liquid rate, mole/hour
= gas rate, mole/hour = mole fraction of a component in liquid = mole fraction of a component in gas
Lz = amount of a component in liquid mole/hour Y = amount of a component in gae (&), mole/hour K = equilibrium constant, y/z a t equilibrium
A = absorption factor L/KG Cp, = heat capacity of fiquid, B.t.u./" F./mole CpE= heat capacity of gas, B.t.u./O F./mole
H, = heat of vaporization, B.t.u./mole t
= temperature,
O/F.
Subscripts 1, 2 . . . N = plate number in absorber columns counting from the bottom. (All plates are theoretical-i.e., equilibrium is assumed to be attained on each plate.) c = plate above cooler; also liquid from cooler N = top plate
LITERATURE CITED (1) Kremser, A . , Natl. Petroleum News, 22, No. 21, 42 (May 21, 1930). (2) Sherwood, T. K., "Absorption and Extraction," p. 115, New York, MoGraw-Hill Book Co., 1937.
RECEIVED February 28, 1851.
One of the papers from this symposium, "Computing Machines for Spectrochemical Analysis," by D. E. Williamson and D. Z. Robinson of Baird Associates, Inc., Cambridge, Mass., will be published in the November 1951 issue of Analytical Chemistry.
