I
B. A. DILIDDO' and T. J. WALSH2 Case Institute of Technology, Cleveland, Ohio
Computer Simulation of Pulse Columns This method may be used to study the effects of design, operating, and extraction system variables on pulse column dynamics and controllability
XE
dynamic characteristics of chemical process equipment have been studied increasingly in the past few years. Results of such studies are useful in designing automatic control systems of chemical process plants. A simulation technique is described for studying the behavior of pulse columns for liquid-liquid extraction. This work was done on an IBM-650 digital computer. While the results presented in this article are dependent on the extraction system, the simulation technique is general in nature. Pulse columns for extraction are substantially more efficient than conventional columns and their method of operation is unique in the field of chemical process equipment (7, 77). I t appears that only in the field of heavy metals separation have pulse columns received real application. Consequently, the extraction system used in this work corresponds to one used in the Purex process (8).
Pulse Column Simulation The system selected for study (below) involves the extraction of uranium from organic phase entering the bottom of the column by aqueous phase entering the top. T h e contacting zone consists of a series of evenly spaced perforated plates. A composition controller serves to maintain the exit organic solute concentration a t the desired value by manipulating aqueous feed rate. A flow controller maintains the average outlet aqueous flow rate equal to the inlet aqueous flow rate. Major assumptions are as follows:
during a pulse cycle and leave the column only during an UPsurge.
used for simulating the pulse column shown in the figure were:
The simulation technique essentially of these steps:
0 Hydrodynamic equations for describing the nature of flow within the column, and predicting column flooding Transient material balance and mass transfer equations for calculating the rate of accumulation of solute 0 Controller equations for describing automatic control characteristics
1. The pulse column is considered as a series of connected stages or cells, defined by the top disengagement chamber, the spaces between plates, and the bottom disengagement chamber; 2. After each downsurge pulse, a set of material balance and mass transfer equations is solved to determine the new solute concentrations in each phase in each cell; 3. After each succeeding upsurge pulse, a setbf material balance equations and mass transfer equations is solved to calculate the new solute concentrations in each phase in each cell; 4. A set of controller equations is utilized to adjust aqueous feed rate after each pulse cycle. Since the solute concentrations are calculated after each pulse, the transient response of the column is specified. In this article, the material balance equations are difference-differential equations which were solved numerically with the aid of a digital computer. T h e mass transfer relationships consist of a combination of equilibrium and efficiency equations.
0 T h e liquids are immiscible, no volume changes occur through the column, and the aqueous phase is continuous Pulse wave-form is sinusoidal Column is operating in the mixersettler region Only aqueous is pulsed through the plates on a downsurge, but both aqueous and organic may be pulsed through the plates on a n upsurge 0 Liquids are fed at constant rates
1 Present address, B . F. Goodrich Research Center, Brecksville, Ohio. Present address, Thompson Ram0 Wooldridge Inc., Cleveland, Ohio.
consists
n:-, CELL n
6
SINE
WAVE PULSE GENERATOR
A perforated plate pulse column was simulated
Hydrodynamic Equations. Examples of some of the downsurge hydrodynamic equations are given below. Time duration of downsurge td
= (1/2F) - (Q/aF)
where 9 = sin-' ( E / n v F )
(1)
Volume of fluid pulsed downward with each downsurge
+
q d = vcosq5 (E/2nF)(24- n) ( 2 ) Fraction of each downsurge which is recycjed
x
= 1
-
(S/q#)
(3)
Flooding due to inadequate pulsation ( S 4- E ) F ~A VF Emulsion-type flooding
(s +
+ (E/2)
(4)
E)Fc = ~ F , ( S / E ) 0 . 0 ' 4 ( a F ) - 0 . 6(5) 21
Note. Each of the listed factors, such as time duration of a downsurge, is written as a function of operating variables only. Consequently, only the effects of operating variables were studied in this work. Equations 1 through 4 were originally derivrd by Edwards and Beyer ( 5 ) and were verified by experimentation. Equation 5 is a simplification of a generalized flooding equation obtained by Smoot, Mar, and Babb (73) by dimensional analysis. Only one set of extraction system physical property variables and column geometrical design variables was used in this work and it is embodied in KFcrwhich was evaluated by fitting Equation 5 to Geier's data ( 7 ) for a similar system. Transient Material Balance Equations. Transient material balance equations were written to calculate the rate of accumulation of solute in the column of each liquid downsurge and each liquid upsurge. Consider the development of such an equation for cell n on a liquid downsurge. VOL. 53, NO. 10
OCTOBER 1961
801
If l d is the instantaneous volume rate of liquid flow through the plates down the column, and U,,= initial amount of solute in cell n, then input = I&+, Output accumulation
= !!dXy 3XC
= ---2 L,
dl
nature of these equations, an approximate iterative method of solution was used in this work (4). Mass Transfer Relationships. The efficiency of each mixer-settler contact occurring in each cell after each downsurge and upsurge was calculated from:
+ a)at
2' G,
or (HTU)on
Equation 6 is a difference-differential equation having two independent variables, cell number n and Lime t. A mass transfer relationship is required to solve Equation 6. For the extraction system used in this work, y: does not vary linearly with x,". Consequently, Equation 6 could not be solved analytically. However, it was solved by the numerical technique of setting up a time-space grid. The time intervals were equal to the downsurge durations, and the space intervals were equal to the heights of the individual cells. Thus, the accumulation of solute in cell n for a single downsurge was estimated by observins that t i
If aqueous streams leaving a cell are assumed to be unmixed with entering streams from the next highest cell, Equation 6 may be written as
E,vV = 1
EMV = (Y.
D E . S
= 'u (t€TU)oc
- e [ Z / ( H T L ' ) ~ ~ ~ ~ I (12)
- Y ' ~ ) / ( Y-~ Y:) for downsurge
EMV = (yn-
(11)
I
(13)
Equation 10 is a simplified generalized performance equation derived by dimensional analysis by Smoot, Mar, and Babb (73). For application in this work: it was fitted to a set of experimental data given by Sege and Woodfield ( 1 7). The embodies all the exconstant R(HTU,OC traction system physical property variables and all the column geometrical design variables. The derivations of Equations 11 and 12 are given in references (2) and (72). EAMV is the Murphree plate efficiency. Controller Equations. The controllability of the pulse columns was simulated by utilizing the standard equations for proportional control Asp = Kpe
(14)
The total amount of solute in cell n after the downsurge is given by =
Un f
AC'n,td
(8)
The split of solute between the liquid phases after the liquid downsurge was determined by solving Equation 8 simultaneously with a combination of equilibrium and efficiency equations. Solution of the transient material balance equation
is required to calculate the solute accumulation on a liquid upsurge. The solute concentration as a function of time was solved numerically by Equations 6 and 9 with the mass transfer equations successively for each liquid downsurge and upsurge. Equilibrium Relationships. The extraction system used in this work is wateruranium-nitric acid-tributylphosphate (in kerosine). Both uranium and nitric acid are extractable components. The equilibrium relationships used in this work have been discussed by others (7, 9 ) . Because of the complex
802
Data
- ~ n ) / ( ~ nI -- Y?) for upsurge
and for integral control
b'n,td
The second kind of computer study was that of observing the uncontrollcd response of the pulse column to step changes in operating variables. That is, after the column had achieved some steady operating condition, a step change in an operating variable was made and the path to a new operating level was observed. The third kind of study was to observe the controllability of the pulse column to step changes in feed concentration. Column performance with proportional and integral control modes individually and in combination was observed.
Derivative control was not used. Operation of valves and other mechanical components was assumed to be instantaneous. However, a first-order mixing lag was assumed in the contro!ler feed-back signal.
Pulse column specifications are: Physical Dimensions of Pulse Column to Be Simulated D
4
1 d,
2
Column diameter, inches Plate spacing, inches Diameter of holes in plates, inch 7 Free area in plates, 70 VCVoeli Column cell volume, liter VT Top chamber volume, liter VB Bottom chamber volume, liter
Once all equations for the mathematical model were assembled, they were programmed on an IBM-650. Studies were conducted on a 3-plate ideal model and a 9-plate real model. Extraction efficiencies of 1 0 0 ~ owere used in the 3-plate runs and actual stage efficiencies in the 9-plate runs. Three types of studies were conducted with each of these models. The first studies were to observe the effects of operating variables on equilibration path and time. This is the so-called "start-up problem." I n this series of runs, the column was assumed to be filled initially with aqueous phase; then a t time zero, both aqueous and organic feeds are started instantaneously at some specific rate.
INDUSTRIAL A N D ENGINEERING CHEMISTRY
23.0
0.412 0.824 0.824
Runs 14 and 45 were the reference runs for studies on the 3-plate and 9-plate models, respectively. The same operating data were uscd in both runs (Table I), with the exceptions of plate number and column efficiency. Results
Results of some typical computer studies are illustrated in Figures 1 through 6 . The performance and con-
Table I. Reference Operating Conditions for Runs 14 and 45 Variable P
6 o
Computer Studies
0.125
E S
p
Pulse frequency, cycles/ minute Pulse amplitude, inch Pulse volume, liter Organic feed rate, liters/ minute Aqueous feed rate, liters/ minute Ratio of organic to aqueous feed rates
Organic Feed Cornposition Uranium concentration, mole/liter yg Hydrogen ion concentration, mole/liter & Nitrate ion concentration, mole/liter ga Tributylphosphate concentration, mole/liter
&
Aqueous Feed Composition a$! Uranium concentration, mole/liter :T Hydrogen ion concentration, mole/liter z2 Nitrate ion concentration, mole/liter
Value 20
1.0 0.206 1.67 1.67 1.00
0.323 0.060 0.706 0.758
0.00
0.01 0.01
PULSE COLUMNS trollability of the pulse columns were observed by plotting the solute concentration in the organic phase as a function of time. Studies conducted on the 3plate ideal model are discussed first. Figure 1 shows the equilibration path in each cell of the 3-plate model for column start-up. The response curve of cell 1 is that of a simple capacitance unit with very small time constant. T h e S-shaped response curves of cells 2, 3, and 4 are typical of processes having two or more capacitance units in series. A definite time lag is apparent for cells 3 and 4 indicating the time necessary for organic phase to work its way u p the column. T h e equilibration curves for various throughputs with the 3-plate ideal model are shown in Figure 2. Fastest response and best extraction are obtained a t highest throughput. Although 100% extraction efficiency was used in each of these runs, a variation in column performance is evident. This effect was caused by increased aqueous backmixing occurring a t decreased throughputs. Degrees of back-mixing, as defined by Equation 3, for runs 13, 14, and 15 were 0.0, 0.5, and 0.9, respectively. The small variations in final yy values (about 0.02 mole/liter) indicate that back-mixing does not affect column performance greatly. The uncontrolled transient responses of the 3-plate ideal model to 25% step increases in organic feed solute concentration (run 18), organic feed rate (run r
I
7
I
1 I
CELL I
i a
t
*PLATE 01
MODEL
I
I
I
I
I
I
RUN 3 3 , AE ~ 2 5 %
3-PLATE MODEL
3 028 RUN 61. AyE = 10%
RUN 47, PE = 2 5 %
R U N 34. A F i 2 5 %
.=
I .O
20
30
TIME, MINUTES
Figure 1. The equilibrium paths for each cell illustrate the effects of series capaciiance units and time lag
7
-
-
w
TIME, MINUTES
Figure 3. The paths of the uncontrolled response curves to step changes depend greatly upon the variable which i s changed
33)) and pulse frequency (run 34) are illustrated in Figure 3. The final values (time = 3.6 minutes) from run 14 were used as initial conditions in this series. The usual S-shaped response curves were obtained in runs 18 and 33 with a slight time lag being apparent in that of run 18. Although the solute input rate was the same for both runs, the effect of a 25% step increase in organic feed rate was much greater than a similar increase in feed concentration. I n run 33, the ratio of aqueous to organic feed rates was decreased causing the operating and equilibrium lines to approach one another, producing poorer column efficiency. The response curve to the step increase in pulse frequency (run 34) is interesting, The decrease in column performance was caused, in part, by increased aqueous back-mixing occurring a t increased pulse frequency and, in part, by decreased “double-contacting” obtained in a pulse column. The unusual shape of the response curve is due probably to the nonlinear equations used in the model. Controllability studies with the 3plate ideal model indicated that fast response without offset could be obtained by using proportional and integral control modes together. T h e effects of column efficiency relationships on dynamic behavior are of particular interest in the series of studies with the 9-plate real model. Runs were made to observe the effects of column throughput, pulse frequency, pulse amplitude, and ratio of aqueous to organic feed rates on column equilibration path and time. T h e following results were obtained from these studies.
7 A
RUN 14. S t E =3.34
~
3-PLATE I
2.0
40 TIME, MINUTES
MODEL I
6.0
Figure 2. Fastest column response and best extraction performance are obtained a t highest liquid throughput
RUN 48, A F : 2 5 %
9 - U T E MODEL
m
E CELL 4
022
0 The initial transients of equilibration responses vary in a complex fashion with column throughput. At low column efficiencies, the effects of column throughput on column performance and equilibration time are relatively minor. 0 For the conditions studied, increased pulse frequency and pulse amplitude cause increased column efficiency but d o not affect column equilibration time markedly. 0 Column performance improves
0 20
20 30 T I M E , MINUTES
40
Figure 4. The uncontrolled response curves are affected greatly by the efficiency equations
substantially with increased ratio of aqueous to organic feed rates, but equilibration time is affected very little. 0 The relatively minor effects of operating variables on column equilibration time were due to low stage effivaried from 2 to 5y0. ciencies. EAMV’s Figure 4 shows the uncontrolled transient response curves of the 9-plate real column to 25y0 step increases in organic feed concentration (run 46), organic feed rate (run 47), and pulse frequency (run 48). The response to a 10% step increase in organic feed concentration also is shown (run 61). The final values from run 45 (t = 3.6 minutes) were the intial conditions in this series. The response curves are quite different from those obtained for similar runs with the 3-plate ideal column. I n particular, the effect of an increase in organic feed concentration (run 46) was much greater than that of an equivalent change in organic feed rate (run 47). This result io contrary to that observed with the 3-plate model. It is interesting since not only does decreased S / E ratio reduce the driving force for mass transfer, but also stage efficiencies are reduced. The observed result shows that a column of low stage efficiencies reacts greatly to changes in feed concentration but virtually ignores changes in feed rate. The response curve of run 48 shows that pulse column efficiency increases with increased pulse frequency. This result is contrary to the result observed with the 3-plate ideal model, and shows the overriding effect of the efficiency equations compared to the back-mixing equations. The peculiar shape of this response curve can only be attributed to the nonlinear relationships used. Figure 5 shows the controlled response curves to a 25Oj, step increase in organic solute feed concentration with proportional (run 51), integral (run 52), and proportional plus integral (run 53) control modes. With proportional control, y y leveled off at a value substantially higher than the desired value. This offset may be eliminated by incorporating integral control. However, in runs 52 and 53. VOL. 53, NO. 10
OCTOBER 1961
803
U
026
I!
e
I
1
IO
20
30
4 0
024
J T I M E , MINUTES
TIME, MINUTES
Figure 5. The controlled response curves for AYE = 25% show a large offset with proportional control mode and column flooding with integral control
the column flooded because of demand for excessive aqueous rate. These runs indicate that simple manipulation of aqueous rate does not provide satisfactory control for large upsets. Runs 62, 63, and 64 illustrate in Figure 6 a situation in which simple manipulation of aqueous feed rate does provide suitable control. Because of the need for excessive computer time, it was not possible to obtain a K, - K I surface diagram to illustrate the region of suitable control operation. However, it appears that for greatest stability, small values of I;? and K , should be used. Attempts to overcontrol surely would cause flooding. Discussion
T h e most significant assumptions made in this simulation were those concerning the liquid flow pattern within the pulse column. It was assumed that mixersettler column operation existed, but in some computer studies, the column went into the emulsion-type operation region. When column operation changes from mixer-settler to emulsion-type, organic phase back-mixing occurs and organic phase hold-up increases. Neither of these effects was acknowledged in the described models, but both should be included in more advanced models. The error resulting from assuming mixer-settler operation is difficult to ascertain since the performance equations inherently include the effects of liquid back-ndxing and column hold-up. The bearing of the performance relationships on pulse column behavior is vividly demonstrated by comparing the results obtained on the 3-plate ideal and 9-plate real models. Perhaps the only way to establish the validity of the results ohtained is by actual experimentation. \7ery little information appears in the literature concerning the equilibration time of pulse columns. Ellison ( 6 ) studying thr extraction of uranium from HN03-aqueous solution into TBP solution with pulse column indicated that 2.5 to 3 column throughputs are necessary to establish equilibrium conditions. Rubin and Lehman (70), working with a different system, indicated that about five column throughputs usually were necessary to obtain constant product compositions.
804
INDUSTRIAL
Figure 6. Suitable control is obtained for a small upset, AYE = 10%
The results obtained with the 9-plate real model indicate that about two column throughputs are required to obtain essentially constant y$ values. This result probably is due to the low column efficiencies employed. Only the effects of operating variables on pulse column dynamics and controllability were studied. By using the generalized forms of Equations 5 and 10, the effects of extraction system variables and column design variables could be studied. Although the described computer technique was designed to study pulse column dynamics. it may be used to design plate-type pulse columns. T h e usual steady-state stagewise calculation techniques are not strictly valid for pulse column design due to the oscillating flow existing within pulse columns. By employing a computer program in which the total number of plates is a variable. the number of plates required to produce a given solute concentration may be determined. Pulse column design has been reported in very few studies ( 3 ) . Nomenclature -
pulse amplitude, inches distribution coefficient of uranium E = volumetric feed rate of organic phase, liters/min. EM\, = Murphree plate efficiency F = pulse frequency, cycles/min. G = volume of organic phase, liters = instantaneous volume rate of g7L flow of organic phase on an upsurge, liters/min. (HTU) = height of a theoretical transfer unit, feet K = flooding constant, liters/inch K ( , ~ ~= ,performance ~ ~ constant, feeti inches/min. = integral reset rate = proportional gain factor = volume of aqueous phase, liters = instantaneous volume rate of flow of aqueous phase on a downsurge, liters/min. = instantaneous volume rate of flow of aqueous phase on upsurge, liters/min. = volume of fluid pulsed downward during a single downsurge = volumetric feed rate of aqueous phase, liters = time, min. = time duration of downsurge, min.
U
DC
AND ENGINEERING CHEMISTRY
= =
= total
amount of uranium, moles u = pulse volume, liters x = solute concentration in aqueous phase, moles/liter 3’ = solute concentration in organic phase, moles/liter = concentration of solute in Y* organic phase after contact if equilibrium had been obtained 2” = concentration of solute in organic phase after downsurge contact = distance betwren plates, feet Superscripts H = hydrogen ion N = nitrate ion T = tributylphosphate u = uranium Subscripts d = downsurge E = organic feed E = flooding, emulsion-type F, = flooding due to inadequate pulsation I = integral control mode n = plate number or cell number OC = over-all continuous phase OD = over-all discontinuous phase P = proportional control mode S = aqueous feed ec = upsurg? Greek Letters = ratio of organic to aqueous P feed rates E = error signal. moles/liter x = fraction of aqueous recycled A = total chanqe in value of variable
z
Acknowledgment
The authors thank J. P. Speroni of the Case Institute of Technology Computing Center for his assistance with the computations, and the Union Carbide Corp. for granting financial assistance. literature Cited (1) Benedict, M., Pigford. T. H.. “Nuclear
Chemical Engineering,” McGraw-Hill, Kew York. 1957. (2) Brown, G. G.. others. “Unit Operations,” Wiley. Yew York, 1951. (,3,) Burkhardt, L. E.. Fabien, R. W.. U. S. Atomic Energy Commission, Rept. ISC-1095,1958.
(4).DiLiddo, B. A , : Ph.D. Thesis, University of Michigan, University Microfilms Inc., Ann Arbor, Mich., 1960. (5) Edwards, R.. E., Beyer, G. H., A . I. Ch. E. Journal 2, 148-52 (1956). (6) Ellison, C. V., U. S. Atomic Energy Commission, Rept. ORNL, 912,1956. (7) Geier, R. G., Ihid.,Rept. HW-49542, 1957. (8) Irish, E. R.: Reas, W. H., Ihid., Rept. 49483-A, 1957. (9) Moore, R. I>.,Zbid., Rept. AECD-3196, 1
ncr
lYJ1.
(IO) Rubin, B., Lehman, H. R.,Ihid.. Rept. UCRL-718, 1950. (11) Sege, G., Woodfield. F. W.. Chem. Engr. Progr. 50, 396-402 (1954). (12) Sherwood, T. K., Pigford, ‘R. L..
“Absorption and Extraction,” McGrawHill, New York, 1952. (13) Smoot, L. D., Mar, B. W., Babb, A. L., IND. ENG. CHEM.51, 1005-10 (1959).
RECEIVED for review November 3, 1960 ACCEPTED April 3>1961
