Experimental Closed-Loop Control of a Distillation Column Malcolm C. Beaverstock' and Peter Harriott" Cornell University, Ithaca, New York 1@50
A 15-plate distillation column for the continuous separation of methanol and water was controlled b y feedback manipulation of the reflux. The critical frequency, maximum controller gain, and peak errors for vapor and feed composition loads were measured and compared with predictions based on a simplified model. Because of the hydraulic lag of 6.5 sec per plate, there was a substantial decrease in critical frequency and controller gain as the control point was moved down the column.
V a r i o u s mathematical models have been proposed for predicting the dynamic response of distillation columns. I n many cases these models haiTe been developed from data for openloop transient,s, which are dominated by the large major time constants of the column and often show strong nonlinear effects (lloczek, Otto, and Williams, 1965). To account for the changes in response time with plate location and type and magnitude of disturbance, the models must be quite complex arid are not convenient for control studies. However, for simulation of closed-loop control systems, where deviations from the set point are generally small and nonlinearities relatively unimportant, simpler models may be satisfactory. ;In approximate model based on a linearized analysis was presented by TVahl a n ' i Harriott (1970), and the model was used to predict typical column transfer functions and the effect of control plate location on critical frequency, maximum controller gain, and steady-state error in product composition. Such predictions wouId permit determination of the best coritrol plate and quantitative evaluation of alternate control schemes. Predictions based on a similar model are compared with some published data in recent studies by Toijala and workers (1971, 1972). ai practical test of the proposed model requires closed-loop studies on columiis large enough in diameter to have realistic hydraulic behavior and. with enough plates to give a reasonable range of time con:itants. Many studies of large columns have been published, but the information needed to predict the control characteristics is usually incomplete. The major time constants for in'dustrial columns range from a few minutes (Harriott, 196t) to as much as 23 hr (McNeill and Sacks, 1969). Xs expected, the largest values are for columns with many plates or a high reflux ratio, but to make a quantitative comparison with theory, the holdups in the column, reboiler. and condenser, the flow rates, and the vapor-liquid equilibria have to be known. In some studies, the ultimate period of the closed-loop coiitrol system is given, and this ranges from 1 to 2 min (Luybeii. 1971) to more than 1 hr (McNeill and Sacks, 1969). However, since the composition response of the column has a limiting phase lag of 90' (Rijnsdorp, 1959), the valve lag, hydraulic lags.. and mee surement lags, which are usually not specified, are important in determining the ultimate period or critical frequency (180" phase lag). Since the hydraulic lags and valve lags are generally less than 1 min, a period of many 1
Preqent addres4, UniRoyal, Geismar, La.
'
minutes or more suggests a large time delay in the anal system, which makes the data less valuable in checkiri model for the column. Theory and results were compared in two recent studies of closed-loop control. Shoneman and Gerster (1970) controlled the temperature near the bottom of a 10-tray column regulating the steam rate. The column response ivas first order with a predicted time constant of 14 mifi. While the closed-loop data agreed with predictions, the critical frequeiicy de )ended mainly on the assumed reboiler dynamics. For this would have been interesting to see the effect of a control plate; because of negligible vapor flow lag. the critical frequency should be almost the same, but the masimum gaiii should change. Shunta and Luybeii (1971) used a 24-l)late column with reflus as the manipulated variable. The critical frequency increased as the sensor was moved up the coluni~i, but an arbitrary time delay had to he iricluded in the model to match the experimental results. This apparent delay i h a t least partly due to use of the Francis weir equation to predict the hydraulic lag, since this gives too small a value. The olijectives of this study were to measure the closed-loop response of a column for various control plate location> and different disturbances and to predict the response using the Wahl and Harriott (1970) model for the column combined with experimental values of the secondary lags iii the coiitrol system. This required measurement of the hydraulic lags, the valve lag, and the response of the vapor sense composition. The methanol-~ater simplify product analysis and to give fairly pure products at moderate reflux in a 15-plate column. The relative volatility ia high and changes appreciably with composition, as doe? the heat of vaporization. These effects complicate the calculat'ion of system gains but were not expected to have much effect 011 the time constants. Equipment
The distillation unit used in this study m s designed to provide maximum flexibility in the changing of column parameters and control configuratioris. The flow diagram and principal control arrangement are shown in Figure 1. detailed description of the column, peripheral equipment. control instrumentation, and analytical techniques is given by Beaverstock (1968). The column consists of 15 sieve plates witli a tray spacing of 12 ill. and an inside diameter of 12 ill. The sieve plates are 316 Ind. Eng. Chem. Process Des. Develop., Vol. 1 2 , No. 4, 1973
401
WATER
2.0 VI
CMENSER
z z 0 0
2 3 Liauio FLOW, GAL I MIN.
4
Figure 3. Average liquid depth on a sieve plate
DISTILLATE
STEAM
k3
Figure 1. Distillation column and control system
measurement plate with the vapor pressure of a sample of known composition. The bulb, which is permanently attached to the low-pressure cavity of the cell, contains the known sample and is inserted into the column. The high-pressure side ofthe cell is connected to the vapor space above the plate. The DVP cell affords a continuous measurement of composition in the methanol-rich end of the column, where temperature measurements alone are unreliable, because the change in boiling point w-ith pressure is significant compared to t h a t caused by the small changes in composition. The plate composition was controlled with a two-mode electronic controller which adjusted the setpoint of the reflux flow controller. Pneumatic level controllers regulated the flows of top and bottom products, and the steam f l o to ~ the reboiler was controlled manually. Pneumatic equipment was used except for the primary controller. Hydraulic Lag
PLATE 15
10
20 TIME, SECONDS
I
n,
05
ONE CAPACITY MODEL
PLATE
0
10
eo
I2
30
T I M E , SECONDS
Figure 2. Hydraulic response to step change in flow to plate 15
stainless steel and have 3/16-in. diameter holes punched on a 0.75-in. triangular pitch. The downcomers are 2-in. pipe and extend 1.5 in. above the plate. The column is equipped with a vertical thermosyphon reboiler and a vertical overhead condenser and accumulator system. Most of the process lines are 0.5-in. S P S steel pipe. The entire unit is capable of continuous operation by mixing the overhead and bottom product flows in the feed tank. Orifice plates and the accompanying differential pressure transmitters indicate the flow rates, while iron-constantan thermocouples measure the temperatures on plates 1, 2, 4,6, 8, 10, 12, 13, 14, and 1.5 as well as in the reflux and bottom product lines. The main measure of composition on any plate is the Fosboro ;\lode1 13YA differential vapor pressure cell transmitter. The cell compares the vapor pressure in the column above the 402
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 4, 1973
The liquid flow to intermediate plates does not respond a t once to a change in reflux because the holdup on the plates and in the downcomers increases with liquid flow. The hydraulic lag has usually been treated as a series of non-interacting first-order systems, though the downcomers and plates really form an interacting system, because the downcomer level depends on the level on the plate and on the pressure drop across the plate above. To see if these interactions were significant, a column section with six trays was simulated on a computer. The doxncomer holdup responded more rapidly than the plate holdup, but the response of each plate was only slightly slower than the response obtained with a model that lumped the capacities of the downcomer and plate for each stage. The plates were then considered as a series of noninteracting, equal, first-order stages. The time constant per plate was determined by operating the column with air and water and making a step change in the water rate to the top plate, A low-range differential pressure transmitter Tyas used to follow the change in level on different plates. The response of the top plate (15) and the fourth plate from the top are shown in Figure 2. The different symbols are for duplicate runs, The time constants for 28 similar runs ranged from 5.8 to 9.5 see with a n average value of 6.5 see. There was no change with number of plates tested and no significant difference betr\-een the response for increasing or decreasing flow. The floms of xater and air Ivere 1.8-3.0 g m and 1.3-1.9 scf,’sec, too small a range to have a significant effect on the time constant. l l t h o u g h the experimental results compare well v i t h the simple model, the time constant of 6.5 see is much greater than the value of 2 see predicted from linearization of the Francis weir equation. Xs shown in Figure 3, the average depth is much less than the calculated depth a t the weir, but it changes more rapidly n i t h liquid rate. Similar conclusions and more extensive holdup data for sieve plates are presented by Ber-
FRACTION
0.0
DEVIATION -0.0035
0.0035 FRdCTlON
-
e
l
3
,--t’Y
I
/
0.0
DEVIATION
e
DEVIATION
3
V
I
-0.0035
I
I
I
I
-0.0035
MOLE FRACTION METHANOL I N LlOUiD
I 3
2 MINUTES
Figure 4. Maxirnum gain determination, plate 14
Figure 6. McCable-Thiele diagram for control plate 13 change in vapor rate from 8.7 to 8.3 Ib/min 0.0035
8.7
MOLE FRdCTlON
to
I
1
-0.0035 0
1
CONTROL
PLATE
15
- - _ -- - - - - .
8.3 lbs/min
OC
DEVIATION
-
MOLE
0 003fi
MINUTES 2 3
4
DEVIATION
,
I
-0,0035
0
2
4
6
, 8 10 MINUTES
l
I
I
12
14
IS
I 18
5 WEIGHT
MOLE
8 3 to 87 Ibs
FRACTION
PLATE
/min
00
8
FRACTION
04
DEVIATION
-00035
Figure 5. Response to vapor rate change with control plate 13
nard and Sargent (196’6) and Thomas and Campbell (1966). Hydraulic time coiistaiits reported for other types of trays range from 2 to 10 sec (‘Toijala, 1971), but there are no data for very large columiis, and the time constant is predicted to increase with the length of the liquid flow path (Harriott, 1964).
Figure 7. Response to change in feed composition with control plate 15 step change from 0.198 to 0.161 mol fraction methanol 0 0035 I CONTROL
FRACTION 0 0 DEVIATION 0 0035
-
Closed-loop tests \\-ere carried out with the differential vapor pressure transmitter on each of the top five plates in the column. *lfter the column had reached steady state, the masimum coiltroller gain and critical frequency were determined. Typical response curves are showi in Figure 4. In some found by interpolation to a damping cases the critical gain coefficient of 1.0, since the coiltroller gain could only be varied in steps. The masimum gains and critical frequencies are shown in Figures 11 and 12. To determine the response to load changes, the controller settiiigs were adjusted to the Ziegler-Sichols recommendations for proportional-integral control, using the measured masirnuni gain and critical frequency. Changes in vapor rate (5-10%) were imposed 011 the column by making step changes in the air iiressure to the yalve controlling steam flow t o the reboiler. Changes in feed composition were made by switching from one feed tank to a~iother.The composition on the control plate was contiiiuously :*ecorded, and samples of the feed and distillate were taken every 5 min and analyzed by gas chromatography. The apl)rosimate compositions on other plates were followed by therniocoupies. The response to a change in yapor flow with control plate 13 is shown in Figure 5 , and the corresponding JlcCabe-Thiele diagram is Figure 6. The Ziegler-n‘ichols control settings
I
,
I
I
I
PLATE
11
I
I
I
10
12
I
-- I I
I
14
16
I8
1
I
I
MINUTES 0
0 6
Control Tests
1
MOLE
2 I
4
6
8
WEIGHT FRACTION
Figure 8. Response to change in feed composition with control plate 1 1 step change from 0.288 to 0.1 80 mol fraction methanol
gave a slightly underdamped response with negligible error after two cycles. The peak error was about 0.004 mol fraction for a 5% change in vapor rate. The peak error was about half as great for control on plate 14 and could hardly be distinguished from background noise for control on plate 15. The peak error occurred a t times increasing from 45 see for plate 14 to 100 sec for control on plate 11, reflecting the increased period of oscillation. For all yapor loads, an early peak, smaller than and opposite in direction to the main peak, appeared in the response. This sniall peak can be attributed to the change in vapor flow, which moves rapidly up through the column, and the effect of this change on plate holdup. For low liquid rates, which exist in the top of the methanol column, a n increase in vapor flow decreases the liquid holdup and causes a momentary increase in liquid flow (Harriott. 1964). In this case, the increased flow Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 4, 1973
403
Table 1. Column Transfer Function Constants
Plate
0.5 MOLE FRACTION METHANOL IN LIOUlD
Condenser 15 14 13 12 11
I. 0
Figure 9.McCabe-Thiele diagram for control plate 1 1
1.3 2.57 4.00 5.60 7.45 9.42
Normal tlows, mol/min
Feed Distillate Bottoms Vapor
TOP
PRODUCT
1
Reflux gain, Vapor goin, K,, mol Kv, mol fraction/ fraction/ moI/min moI/min
0.65 0.12 0.53 0.30
I
Feed flow gain, Kr, mol fraction/ moI/min
Feed comp. gain, K x , mol fraction/ mol fraction
0.0188 0.037 0.059 0.086 0.122 0.165
0.331 0.649 1.05 1.53 2.15 2.90
-0.786 -1,55 -2.42 -3.40 -4.54 -5.77 Time constants from Wahl-Harriott correlations, min
Ti = Tz = Ta = T4 = T, = Tzp = T,, =
9.65 5.25 4.85 2.48 2.1 -0.05 -2.1
Measured time constants, sec
Th = Tml = Tm2 T, =
6.5 4.1 1.4 2.5
LOA3 VAWR LOAD MAIN CONTROLLER
Y
Figure 10. Control system block diagram
to the control plate is richer in methanol, causing a slight increase in concentration on the plate and a decrease in the controlled reflux flow. The decrease in reflux, which is opposite to the eventual change, makes the peak error greater than for a vapor rate change by itself. The response of the control plate to changes in feed composition is shown in Figures 7 and 8 along with the response on plate 8, as determined from t,emperature measurements. For control on the top plate (no. 15) the maximum error was about 0.001 mol fraction, only slightly greater than the noise level, and the response \vas essentially over in 3 t'o 4 min. For other runs with plate 15, the noise was much greater, and the effect of the load change was not evident. With plate 11 as the control point, the peak error was somewhat greater and the response a little slower than with top plate control. Similar results were obtained with control on plates 12, 13, or 14, but random fluctuations made it hard to correlate the peak error with control plate location. The composition on plate 8 started to change after 2 min and continued to change for 15-20 min, reflecting the major time constant of the column. The drift in composition on plate 8 indicates that the control plate composition should also tend to change and that the reflux should be continuously changed by the control loop to compensate, but after the first few minutes, these changes were too small to detect. In most cases, a decrease in methanol content of the feed gave an initial increase in eoncentration on the control plate, which is opposite t,o the expected behavior. This vias probably caused by a change in vapor rate in the upper section of the column. A heat balance around the feed plate indicated a 2% change in vapor rate on switching feeds, which could account' 404
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. A, 1973
for most of the initial deviation. The feed entered as hot liquid, and when the temperature control was poor, the signal from the control plate was much noiser than usual, which was also attributed to changes in vapor flow above the feed plate. The feed cornposition changes did not have as much effect on the control plate as vapor loads, but they did lead to larger changes on other plates, as showi by the McCabe-Thiele diagrams in Figures 6 and 9. The product compositions must also change if an intermediate plate is controlled, but the changes are quite small for this system. The calculated changes in distillate composition are given in Table I1 and discussed later. .1few runs u-ere made with 10-20% changes in feed rate and control on plate 11 or 14. The maximum deviation was less than 0.002 niol fraction. The experimental results indicated that feedback control of the column reflux was satisfactory for the methanol-water system for all load changes tested. Changes in vapor rate resulted in larger peak deiriations than corresponding changes in feed flow or methanol content of the feed. The tests showed some increase in peak error and response time as the control plate was moved down the column, but the effect was not very striking, because the errors were small even vihen controlling five plates from the top. Control of the top plate composition worked quite well in this study because the product purity was only about 98%) and the DVP cell could detect small differences in concentration. When much purer product is required, the best control plate location mould depend on the measurement error, as well as on the changes in dynamic response and steady-state offset as the control plate is moved down t'he column. Analog Simulation
The closed-loop control system was simulated on a combination of TR-48 and TR-20 analog computers. The block diagram is showii in Figure 10 and the parameters given in Table I. The transfer fuiictions for the column were based on the measured hydraulic lag and the correlations for concentration lag given by Wahl and Harriott (1970). d s sholvn by .irmstroiig and Wood (1961), there is little interaction be-
-"I
I
0.4
15
14
12
13
CONTROL
I1
PLATE
Figure 1 1. Control loop maximum gain
tween these lags because of the much slower concentration response. The transfer function relating the composition a t control platen to reflus changes has n - 1hydraulic lags and a first-order concentration lag of 9.66 min, taken from the correlation for T1, T , and a calculated value of T , = 11.5 min. The same time coiistaiit is used in the transfer function for vapor loads, and the lag in transmission of vapor flow vlianges is assumed negligible. The transfer functions relating toil liroduct composition to reflus and vapor flows have a n addit,ioiial time constant T,, the holdup time in the condenser aiid accumulator, because the limiting phase angle is 180' with a total condenser. The parameters TP, T,,, and T,,, for feed composition loads lvere taken from the corresponding correlations. The gains for sill column transfer functions were obtained from digital computer calculatioiis of the steady states. The valve time constant was determined from frequency respoilre tests. The response of the DVP cell was predicted from the resiFtaiice aiid capacity of the bulb wall and the internal fluid niid a n estimated film coefficient of 750 Btu/hr f t 2 O F . The bulb and fluid form a n interacting system with predicted effective time constants of 4.1 and 1.4 see. One transient response test showed that these values were approsiniately correct. The control loop alsone was simulated first and the maximum gaiii and critical frequency were determined. Figures 11 and 12 compare the predicted and measured values. The esperimeiital critical frequencies decrease about as espected with change iii control plate, and the esperimental points show only random scatter. This is iiot unespected, however, as the critical frequency is primarily influeiiced by the minor time coilstants in the control loop, which were determined esperi. mentally. The masimum gains decrease as the control plate is moved down the columii, but the agreement with theory is not as good. The esperimental point for the critical gain of plate 15 is likely to be in error because of the high noise level on that plate. TT'heii the DVP tlulh was on plate 15, it was sporadically hit with cold reflus which showed up as noise and made it difficult to see whether the system was stable or iiot. The remaiiiing four points still indicate a smaller slope than predicted, and a change iii T1would only shift the predicted curve up or down without changiiig the slope. The difference in slope is attributed to the use of a single time constant to represent the column coiiceiitratioii dynamics. This is the simplest transfer fuiictioii whicli has the correct limiting phase angle of 90' and beenled a good approsimation for most of the cases esnniiiied by K a h l (1967). However, n-heii the relative vola.-
2.5 15
14 13 12 CONTROC PLATE
II
Figure 12. Control loop critical frequency
CONDENSER
FRACTION DEVIATION
CONTROL
80
m so
PLATE
5040
-0 005
M to IO
o
MINUTES
Figure 13. Analog simulation for control on plate 12 simultaneous feed composition and vapor loads: Axf = 0.02 mol fraction, Av = 0.004 mol/min
tility is high aiid the reflus is low, the time constants in the complete transfer function are bunched together more closely, and a better approsimation would have two time constants and one zero.
Based on the pole-zero maps for similar columns, Tb would be about 4 min, while Ta might vary from 5 t o 2.5 miii on moving the control plate down the column. Using these values, the predicted gain would be loivered for top plate control and raised for control on plate 11, which would more closely agree with the esperimental results. The analog computer was then used to simulate the closedloop response to changes in vapor rate, feed composition, and both a t once. The effect of vapor flow on plate holdup was not included in the simulation, arid the response curves for a vapor load did iiot show the initial inverse response that was noted esperimentally. The peak error increased as the coiitrol plate was moved down the columii. The peak error for a 5% vapor load with control 011 plate 13 was 0.0030 mol fraction, slightly leas than that shown in Figure 5. X typical analog result for simultaneous load changes is given in Figure 13. The increase in vapor flow produces a n inverse response a t the control plate with a peak negative error Ind. Eng. Chem. Process Des. Develop., Vol. 12, No.
4, 1973 405
__
~~
Table II. Control Characteristics from Analog Simulation for Feed Composition Change from 0.20 to 0.2375 Mol Fraction Control plate
Maximum gain
15 14 13 12 11
11 1 3 6 1 7 0 9 0 56
Critical frequency, rods/min
12 7 4 3 3
8 3 9 6 0
slightly greaber than the posit’ive error due to the composition change. The same concentration load alone produces a normal response with a peak error two thirds as great. The initial negative peak in the condenser response is an artifact caused by the negative value of T,, in the numerator of the transfer function. The main response in top product concentration is an increase caused by the richer feed followed by a slow drift t o a negative offset, as the controller decreases the reflux to maintain the concentration of the control plate. The effect of control plate location for composition changes only is summarized in Table 11. ;\loving the control plate away from the top greatly increases the predicted peak overshoot because of the decrease in controller gain and bhe increase iii process gain. The experimental overshoot was generally smaller than predicted even though a change in vapor floiv occurred a t t’he same time. Perhaps the change in holdup, which acts opposite to the expected effect of a vapor load, makes the net effect of simultaneous disturbances less than for a feed concentration load. Table I1 also shows the changes in top product (condenser) error as the control point is niored down the column. The peak positive error actually decreases with this shift because of the increasing influence of the steadystate error in the opposite direction. I n this example, plate 14 is the preferred control plate because the steady-state error for other cases is almost as large or larger t,lian the peak error when using plate 14. If the steady-state error were much smaller, as might be true for some systems, the control plat’e could be selected after calculating the integrated errors in product composition, assuming that the steady-state error was reduced to zero a t intervals by manual adjustment or another control loop. Discussion
The experimental results are generally in good agreement with the predictions based on the model of Wahl and Harriott (19TO). If the valve and measurement lags are small, there is an appreciable decrease in critical frequency and in maximum gain as the control point is shifted further from the top of t’he column. These changes are caused by having a n increasing number of hydraulic lags in the control loop, and for a large number of such lags, their effect is almost as detrimental as a pure time delay. -1much smaller effect of control plate location was fourid in the computer study of Shunta and Luyben (1971): because their system included a constant time delay of 2.97 miii as well as too small a value for the hydraulic lag per plate. The experimental results afford only a partial check of the correlations presented for columii transfer functions. The critical frequency depends mainly 011 the hydraulic lags and other small time constaiits and would be practically the same for a twofold change in the major time constant, TI. However, a change in T I would cause a proportional change in the maximum gain, and the fair agreement in Figure 11 supports the predicted value for TI. The value of TI can be roughly 406
Ind. Eng. Chem. Process Des. Develop,, Vol. 1 2 ,
No. 4, 1973
Control plate peak error
0 0 0 0
Condenser peak error
00083 00175 00375 0061
0 0 0 0
00133 00112 00094 00095
Condenser steady-state error
-0 -0 -0 -0
0004 00105 00168 0023
checked from the slow response of intermediate plates after a load change, and the 20-mi11 response time for plate 8 is consistent with a major time constant of about 10 min. The correlations for the secondary time constants and zeroes do not affect the stability of the closed loop but do influence the initial response aiid peak error. -1lthough the noisy signals aiid m a l l errors made close comparison difficult, the predicted errors for composition changes seem somewhat, too high. Toijala and Fagervik (1972) used a model that has larger time constants and no zeroes in the transfer functions for product and control plate response to feed composition and, as a result, smaller peak errors were predicted for the methanol-water system. However, the difference between predicted and measured response could be caused by the complex effect of vapor flow changes, a subject which deserves further study. Conclusions
(1) For control of top product composition by feedback manipulation of the reflux, moving the control point down the column decreases the critical frequency and the maximum controller gain because of the increasing number of hydraulic lags, and it also increases the steady-state error in product composition. These effects suggest a control point as close to the top as possible, consistent with measurement accuracy. ( 2 ) When the measurement and valve lags are small, twomode feedback control can correct for moderate (10-20%) changes iri feed rate or feed composition with barely noticeable changes in product composition. Similar changes in vapor flon. produce greater upsets, partly because rapor flow changes are trarismitted more rapidly through the columii and partly because of the effect of vapor flow on plate holdup. (3) A linearized model seems satisfactory for predicting the effect of control plate location aiid for comparing control schemes. The absolute value of the peak error was not accurately predicted in this study because the column model had too few time c o n h i i t s and because the dynamic effect of changes in vapor flow was not included. More work is needed to predict the hydraulic lags and the effect of vapor rate changes. literature Cited
A4rmstrong,W. D., Wood, R. bl., Trans. I m t . Chcm. Eng., 39, 80 (1961). Beaverstock, M. C., Ph.D. Thesis, Cornel1 University, Ithaca, S . I?.,1968.
Bernard, J. 11. T., Sargent, It. W. H., Trans. Inst. Chem. Eng., 45, T33 (1966);t Harriott, P., Process Control,’’ 31cGraw-Hill, X e w York, S . Y., 1964, pp 287-209.
Luvben, W. L., AIChE J . , 17, 713 (1971). ll&eill, S. A . , Sacks, .J. D., Chem. Etrg. Progr., 65, 33 (1969). bloczek, J. S.,Otto, li. E., Williams, T. J., Chcm. Eng. Progr. Symp. Ser., 5 5 , 136 (1963).
Rijnsdorp, J. E., LIaarleveld, A4., “Instrumentation and Computation in Process Development and Plant Design,” 135, In,titute of Chemical Engineers, London, 1959.
Shoneman, K. F., Gerster, J. A., A I C h E J., 16, 1080 (1970). Shunta, J. P., Luyben, W. L., AZChE J., 17, 92 (1971). Thomas, W. J., Campbell, M., T r a m . Inst. C h m . Eng., 44, T314, IlRIX'I.
Wahl, E. F., Ph.D. Thesis, Cornel1 University, Ithaca, N . Y., 1967. Wahl, E. F., Harriott, P., Ind. Eng. Chem., Proc. Des. Develop., 9,
_ _ _ (imn).
296
T&jala,'K., Acta Acad. Aboensis $lath. Phys., 31, Xo. 5 (1971); 32, Xo. 2 (1972). Toijala, K,, Fagervik, K., Acta Acad. Aboensis JIath. Phys., 32 KO. 1 (1972). Toijala, K., Jonasson, D., Acta Acad. Aboenszs X d h . Phys., 31 No. 11 (1971).
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RECEIVED for review October 7, 1969 RESUBMITTED January 24, 1973 &lay 17, 1973 ACCEPTED Presented at the JACC meeting, Boulder, Colo., Aug 1969.
Two-Stage Catalytic Converter. Transient Operation George 1. Bauerle and Ken Nobe" School of Engineering and Applied Science, Cniversity of California, Los Angeles, California 90024
A mathematical model for two-stage catalytic exhaust converters has been developed b y extending the work of others for single-stage converters. It has been shown that computations of the operational characteristics of the first stage are not in serious error by assuming temperatures of the gas and solids phases as equal. On the other hand, computations for the second stage are more in error but not substantially. The effectiveness of two-stage, axial-flow converters to decrease pollutant emissions has been shown to vary with reactor diameter; the effect of secondary air rate was not as appreciable. Furthermore, warm-up time varied with diameter much more for the second stage than for the first stage.
F e d e r a l standards foil 1975-1976 auto exhaust emissions have spurred considerable research and development work in catalytic control methods. The purpose of this work is to extend the mathematical modeling of single-stage catalytic exhaust converters by Vardi aiid Biller (1968), who studied the thermal response of catalyst beds without reaction and by Kuo and coworkers (1971), who examined the transient behavior of the oxidatjon converter to a two-stage catalytic converter. The analysis of steady-state, two-stage catalytic Converters has been given in an earlier paper (Bauerle and Xobe, 1973) and will be referred to as part I. Mathematical Model
I n deriving the present mathematical model for a trvo-stage converter, it is assumed that t x o cylindrical reactors in series are employed. Only axial flow of gases is considered. I t rras shown in p i r t I that the longitudinal arid radial diffusioii terms are negligible in practical design applications. Thus, the mass balance simplifies to
to be operating adiabatically. Thus, the energy balances for the gas phase and solids phase, respectively, can be expressed as
and
It has been assumed that the heats of reactions are developed initially 111 the solids phase. The model included a computational scheme suggested by Kuo, et al. (1971). Temperature was assumed to remainat an average value over each time arid length step. It was also assumed that the species in excess remained a t the initial value during the time step. Equation 1 can now be written for the j t h length step
Khere the a\ erage temperatures are taken as Vardi and Hiller (19168) showed that the heat capacity of the solids is about three orders of magnitude greater than the heat capacity of the gas aiid that the thermal conduction terms are very small comliared t o gas-solid heat exchange aiid the bulk flow t e r m . Kuo, et al. (1971); found that the thermal behavior of the solid phase dominated the dynamic response of the entire system. K i t h these consideratioils the transient energy term for the gas phase and all thermal conduction terms can be neglected, Furthermore, in part I it was showii that for all practiral piirposes the converter can be considered
T
=
(T
+ 2"')
2
(5)
For illuatratir e purpo>es, the catalyst i n the fir3t qtage nil1 be assumed to be higlilj selective for the reaction
so
+ co
=
?S?t co,
(6)
I n addition. it 1.; a5wmed that the oudation reactions in the second .tage can be represented by the reaction
co +
?02
=
co2
Ind. Eng. Chern. Process Des. Develop., Vol. 12, No. 4, 1973
(7) 407
