February 1953
INDUSTRIAL AND ENGINEERING CHEMISTRY
(5) Clarke, B. J., “Water-Gas-Shift Reaction in the Presence of an Iron-Type Catalyst,” unpublished Ph.D. thesis, University of Michigan, 1952. (6) Damkohler, G., “Der Chemie Ingenieur,” Bd. 111, S. 413, Leipzig Akademische, Verlagsgesellschaft,1937. (7) Forster, Th., and Geib, K. H., A n n . P k y s i k , 20, 250 (1934). (8) Frenkenburg, W. G., Komarewsky, V. I., and Rideal, E. D., eds., “Advances in Catalysis and Related Subjects,” Vol. 2, New York, Academic Press, 1950. (9) Gratz, L., Ann. Physik, 18, 79 (1883). (10) Hougen, 0. A., and Watson, K. M., “Chemical Process Principles,’’ New York, John Wiley & Sons, 1947.
Lyman, T.,ed., “Metals Handbook,” Cleveland, Ohio, American Society for Metals, 1948. Paneth and Herzfeld, 2.Elektrochem., 37, 577 (1931). Reed, R. M., Trans. Am. Inst. Chem. Engrs., 41,453-62 (1945). Sabatier, P., and Senderens, J. B., Compt. rend., 124, 1358 (1897). (15) Storch, H. H., Golumbic, N., a n d Anderson, R. B., “The Fischer-Tropsch and Related Syntheses,” New York, John Wiiey & Sons, 1951. (16) Tsohernitz, J. L., Barnstein, S., Beckmann, R. B., and Hougen, 0. A., Trans. Am. Inst. Chem. Engrs., 42,883-906 (1946). REICEIVED for review April 23, 1952.
Solution of Transient Operations on an Computer ANDREW ACRIVOS
AND
467
ACCEPTEDOctober 23, 1952.
0 0
NEAL R. AMUNDSON
University of Minnesota, Minneopolis 74, Minn.
T IS a well-known and a quite distressing fact that most
1
*
chemical engineering problems, even some of the more idealized ones, are extremely difficult if not impossible to solve exactly by means of present day mathematical tools. In most instances it is necessary, in order to arrive at some answer, to oversimplify the problem to such an extent that the mathematical model so constructed bears little or no relation to the actual physical situation. Unfortunately, however, this is not the whole story. Consider, for example, the ideal absorption or extraction unit, operating without reflux. There is indeed little in common between an actual column and the ideal, but, despite the fact that present day chemical engineers have simplified enormously the physical and engineering problems connected with the design and the operation of such a unit by the introduction of the ideal stage or by superficial expressions for efficiency, even so few of those oversimplified problems have been satisfactorily solved to date analytically. Most of the cases which have been treated deal with the steady-state operation of sueh units; in addition, i t has been found very convenient to use a simple equilibrium relation between the compositions of the two phases leaving a stage or plate. This is especially true in multicomponent extraction or absorption. However, the transient behavior of such equipment has received very little attention. In a recent paper, Lapidus and Amundson (1)have applied themselves to the problem of finding a solution to the equations which represent the behavior of an extraction or absorption ideal plate column in which one component distributes itself between two immiscible phases. It was found that the system under consideration could be described by a set of differential-difference equations; but, in order to be able to produce analytic solutions, i t was necessary to assume a linear equilibrium relation. This, of course, restricts the usefulness of the results even further, as very few systems are obliging enough to possess linear equilibrium curves up to concentrations which are not uselessly ?mall. Even so, the final formulas obtained-viz., Equations 20 and 21 in (I)-are quite complex and not easy to use, in spite of the fact that they are solutions to a problem, which, compared to the general casea i.e., in which the Y us. X relation is not linear-represents rather trivial special chapter. However, i t is found that if any other equilibrium relation is substituted, the differential equations
become nonlinear and therefore difficult if not impossible to solve. What is proposed now is to show that those differential-difference equations which describe the state of a given extraction or absorption equipment at any time can be solved numerically even for the case of a nonlinear equilibrium relation which may or may not change in form from stage to stage, to take into account changes in temperature, composition, etc., by means of t h e electronic analog computers. This is an extention of the problem treated by Lapidus and Amundson in that the equilibrium curve is given a more complex form. Otherwise, the equipment is still assumed to be ideal, and the problem considered is still the removal of one component from a phase by another phase immiscible in the first. The introduction of an efficiency causes no difficulty but will be omitted for brevity.
Analog Computers An electrical analog computer is a computing device in which an electrical analog representing a particular mathematical equation is set up, and various unknown voltages are measured a t various points in the electrical circuit. It is clear then that those machines turn out numerical solutions to a particular problem. The work to be presented in this paper was performed on the Reeves electronic analog computer (REAC) shown in Figure 1 which was obtained by the University of Minnesota for its Computing Center. The REAC contains 14 summing amplifiers each with seven inputs of amplification 10, 10, 4,4, 1, 1, 1. Intermediate values t o those fixed gains can be obtained by means of one of the 48 multiplying potentiometers which multiply a given voltage in the circuit by a fraction. There are also 14 integrating amplifiers with the same input circuit arrangement as the summing amplifiers. Twelve of those integrators have separate initial condition potentiometers associated with them. In addition, there are 12 phase inverters which are summing amplifiers with only three inputs, each of unity gain. Finally, the REAC contains four servo systems, each of which drives three multiplying potentiometers. A brief discussion of the types of problems which the REAC is capable of solving numerically is in order. It can solve up to seven linear independent simultaneous algebraic equations, b u t
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
468
I
Table 1.
=
I T l
0.125 a = 0.004
a = 0.008
t
t
S I
t
s
t
5.34 5.77 6.05 6.25 6.41 6.61 6.73 6.81 G 90 7.03
0.0
I
l ' ! $ ! -
L
-
I
L
a = O.OOO_
a = 0.002 X4
h
(Corresponding to Figure 1) p
x 4
,
Vol. 45, No. 2
I
0.2 0.4 0.6 0.8 1.2 1.6 2 0 2.8 m
Figure 2.
Wiring Diagram for a Four-Plate Ideal Absorption Column Urins an equilibrium relation of the form Y = aX3 pX2 yX
+
Table II.
+
(Corresponding to Figure 2) a = 0.000
p
=
X< 5.82 6.25 6,s; 6.78 6.95 7.17 7.32 7.42 7.54 7.61 7.7F 6 = 0.025 X1 t
Figure 1.
6
0.100 t
0 0 0 0 0 1 1
= 0.060
t
X4
7.25 8.25 8.76 9.29 9.S2
0 2 4 6 8 2 6
0.0 0.5 1.0 2.0 3.0 4.0 5.0 6.0 8.0
9.68 9.78 9 85 9.94 10.06
2 0
2 8 3 6
m
m
6
i3 = 0.012;_
XI
t
X4
= 0.000
~
t
Front V i e w of a Reeves Analog Computer (center), the Servo Unit (left), and the Recorder
i t a main usefulness lies in the fact that it can give solutions t o many high order, linear or nonlinear, homogenrous or non-homogeneous ordinary differential equations with constant or variable coefficients; that is, the REAC can give answers to many problema which cannot be solved analytically and otherwise can be treated numerically only through extremely long, tedious, complicated, and often inaccurate methods. It must be borne in mind, however, that the machine should by no meam be expected t o solve any ordinary differential equation whatsoever, and only certain types of problems can be h a t e d readily on said computer. If an ordinary equation is to be solved with comparative ease on the REAC, i t must satisfy the following: 1. All the specified conditions must be initial values. This excludes, therefore, boundary value problenis which could conceivably, and often are, solved h y trial and error. 2. Usually all the coefficients and other specified functions of the independent variable in the equation must in general remain finite in the range in which the dependent variable is to be calcu1ated. 3. All the coefficients must be specified beforehand. Sturm-
Liouville problems are then not among those which can be handled with relative ease by the computer. The above general conditions place, to be sure, limitations on the types of differential equations which the REAC can successfully handle. There remain, however, many instances where the computer will prove to be of great value. The remaining part of this paper will consider in detail a set of differential-difference equations which describe the transient behavior of a countercurrent absorption or extraction column and demonstrate a method by means of which the REAC can be used to solve this problem. Countercurrent Absorption in a n Ideal Plate Column
Let L, be the moles of inert material in the denser phase leaving the nth plate per unit time, and X , be the molesof solute which
INDUSTRIAL AND ENGINEERING CHEMISTRY
February 1953
is absorbed or extracted per 100 moles of inert material in this heavier phase; also let G, be the moles of inert material in the lighter phase leaving the nth plate per unit time, and Y , the moles of solute which is absorbed or extracted per 100 moles of inert material. A material balance gives
where h, and H , are the holdups of the two phases a t the nth plate. It is aAsumed next that the relation between Ynand X, is given by
Y, =
anXn3
+ PnX,2 + Y n X *
469
A typical wired panel board is shown in Figure 3. The numbers refer to the particular units used on one of the two connected computers and the primes denote the corresponding unit on the other computer-e.g., 2 (encircled) refers to potentiometer 2, while 3’ denotes potentiometer 3 on the other computer. It is to be noted that in the following example the feed composition Ys is known; this quantity will then be varied either stepwise or continuously; and changes in the composition of the liquid leaving the absorption column-Le., X4-will be determined as a function of the time t.
(2)
where the empirical coefficients am,p,, -yn can be made to vary from plate to plate, in order to compensate for changes in temperature, composition, etc. Then the equation becomes
Ln-IXn-1
-
+
+
+
(Ln 7nGn)Xa yn+~Gn+~Xn+l ~ ~ + I G ~ + I X ? + I -anGnX? Pn+lGn+iX:+~- PnGnX?
+
The above represents n nonlinear first order differential equations. I n order to solve this system numerically on the REAC, i t is essential to make the following simplifications:
1. Since a servo is needed to multiply two functions together, and since the REAC contains only four servos, the column can have at most four ideal plates. A machine with more servos and more computers would be needed when n > 4. 2. For the same reason-Le., because of a lack of servos-it is necessary to set H , = 0. Thus the problem becomes one in absorption only, since in such a case H,, the vapor holdup, is very small, indeed. I n the following treatment, i t will be assumed for simplicity that the parameters L, G, a,p, y , and h are constant throughout the calumn, but this is not necessary. The set of equations then becomes
bX, = KIX,l bt
-
K2X,
+ KIX,+I + KdXL1 - K4X: + KjXl+i
- KbX?
(4)
for n = 1, 2, and 3, while for n = 4
Figure 3.
Typical V i e w of a Wired Panel Board
The wiring diagram will be followed by some numerical resuIts which will be presented mainly in the form of graphs. It will be seen that data were taken for various forms of the equilibrium curve from the complicated cubic down to the linear. It must be noted that although in this particular instance only the feed compositiod was being varied and only Xd was measured as a function of t , one could have also altered the problem in the following ways:
1. By having X o # 0, and in general letting X , = X,(t). 2. Letting L, G, a, p, y, and h vary from late t o p!ate. 3. By varying L and G with time; this, fiowever, is a more difficult variation as it will necessitate changing the values on most of the potentiometers. This, of course, cannot be done continuouslp, so that it follows that L and G can be varied only in steps. 4. Instead of measuring X4, one could of course determine Y I ,or the composition on any plate as a function of time. Also a similar setup would result if the equilibrium curve were expressed as X , = a’Y; p’Y; yfYn.
+
+
A Numerical Problem Next the complete wiring diagram (Figure 2 ) used to solve the above problem will be presented. The following symbols are used :
Let
x,
= 0 = 40.0 pounds per minute
h
= 75.0 pounds
a
= 0.008
y
=
L G
H
13
= 70.0 pounds per minute
= o
= 0.125
0.25
Then, by means of Equations G , one finds that
lnte rator Summin a m p h e r or inverter Potentiometer Servo unit and potentiometer
K, Kz
K.3
2.5Kd lOKs KO
= 0.1066 = = = =
0.1532 0.0468 0.0584 0.01494 0.1805
Vol, 45, No. 2
INDUSTRIAL AND ENGINEERING CHEMISTRY
470
1.0 15 ' TIME IN MINUTES
.5
Figure 4. Y6
= 5.0 for
Plot of t
< 01 Y5
X4
20
25
3.0
vs. Time
= 8.0 lor t
> 0.
Y s is next varied discontinuously from 5.0 (yo by weight) to 8.0, and the variation of X 4 with respect to time is obtained by the recorder. X,,is here in 7 0 by weight. As seen in Figures 4 and 5, X , ( t ) can easily be determined for various values of the parameters CY and 6. Of course, one could have performed similar measurements and have varied the other y, G, L , h, X, and in addition determined parameters-i.b., Y l ( t ) ,or X and Yon any other plate.
TIME IN MINUTES Figure 6.
Ys
r
Plot of X4 vs. Time = 5.0 f sin t
I
X,
0
I
2
3
4
5 6 7 8 9 TIME IN MINUTES
Figure 5.
Ys =
5.0 for t
1011
12131415
Plot of X4 vs. Time
< Oi
Y6 = 8.0 for t
> 0.
Figure 6 represents X , ( t ) when Y , ( t ) is a trigonometric function. Now, &(t) is given by the recorder as a continuous curve, but it might be wise t o present here a simple table which could be found useful. In it t i s given in minutes. Countercurrent Extraction in a n Ideal Plate Column
So far i t has been assumed that Hn-i.e., the holdup of the lighter phase above a plate-is negligible. This assumption can be made safely in countercurrent absorption only, and obviously not in extraction. If one desires to solve a problem in extraction on the REAC, then one is forced to use a t most a parabolic equilibrium curve, unless one prefers t o cut the column down to' two plates; when n > 4 or when LY = 0, a larger computer will be needed and more servos. Let Y , = 0 X i
+ y X r z ; then the set of equations becomes
I ~
Figure 7.
I
~
Wiring Diagram for a Four-Stage Ideal Extraction Column Using an equilibrium relation of the form = PXZ r~
r
+
INDUSTRIAL A N D ENGINEERING CHEMISTRY
February 1953 a x 4
and
bt
-
L h+Hr
x3-
y G x 4 + -~ G h Hy ' bXc ___ PG -- 2pH h Hy x ' h H y x 4dt
L
f
+
+
+
47 I.
Figure 9 represents Bessel's equation of zeroth order-i.e.,. J o ( z )plotted us. 2. It is seen that this numerical solution compares very favorably indeed with Jo(x)obtained from standard! tables.
if i t is assumed for simplicity that y, p, h, and H are constant throughout the column; but this is hardly necessary, The wiring diagram follows (Figure 7 ) ; no data of any importance were taken for this case, but the remarks made earlier about the possibility of varying parameters other than the feed composition apply here, too.
- COMPUTER RESULTS
015
Evaluation of Results
The solutions given by the REAC computer will be compared with those obtained analytically for two problems. The first has been presented and solved analytically in ( 1 ) ; this is the simplified version of the problem treated in this paper. Consider for the moment, illustration B in (1). The numerical results presented there have been recalculated and are presented in Table 111,which gives these corrected values.
40.q
I I I 'l l l l l l i
I I I 1 I
L
Figure 9.
As a conclusion it can be stated'that the analog computer i s capable of solving equations which could otherwise be treated only in approximate ways, rather rapidly and with a satisfactory degree of accuracy as seen by the above two examples, I t should be noted, however, that invariably great care must be exercised in setting up the problems on the analog, for many pitfalls exist which the experienced operator must learn to avoid..
A L Y TIC ALLY
15
16 17
18
Nomenclature
moles (or pounds) of inert material in the lighter phase leaving the nth plate holdup of the lighter phase on the nth plate, in moles (or pounds) of inert holdup of the heavier phase on the nth plate, in moles (or pounds) of inert L/h L YG
+
19 2 0 21 22 23 24 25 2 6 2 7 2 8 2 9 30 TIME IN MINUTES Figure
Y7 = 90.0 for t
8.
Plot of
< 0 and t > 15;
X6
h
YG h
vs. Time
Y7 = 30.0 lor 0
Plot of JO ( x ) vs. x
PG
< t < 15.
h
OtG -
h
Glh
Since now the equilibrium relation is linear, no servos are needed so that N can be made greater than 4. It will be seen from Figure 8 that the agreement between the two methods is remarkably good-i.e., within 1yo. It might be wise to note here that the results given by the computer to all the problems discussed in this article are reproducible, a t best, to within 1%.
Table 111.
Stagewise Absorption and Extraction Equipment Correction to Table I in (1) t , minutes
0
1 3 5 10 15 20
1s
20 25 30 35 OD
a,P, Y
Acknowledgment
m
Correction t o Table I1 in (I). t , minutes 15 10
Yn+1
moles (or pounds) of inert material in the heavier phase leaving the nth plate. composition of the heavier phase entering a t the top of the column in moles (or pounds) of solute per 100 moles (or 100 pounds) of inert material composition of the heavier phase leaving the nth plate in moles (or pounds) of solute per 100 moles (or 100, pounds) of inert material composition of the lighter phase leaving the nth plate in moles (or pounds) of solute per 100 moles (or 100 pounds) of inert material = composition of the lighter hase entering a t the bottom of the column in moles &r pounds) of solute per 100 moles (or 100 pounds) of inert material parameters in the equilibrium relation (see Equation 2) =
For 0
