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Data reconciliation—An industrial case study
Computers chem. EngngVol. 20, No. 12, pp. 1441-1449, 1996 Copyright© 1996ElsevierScienceLtd Printed in Great Britain. All rights reserved 0098-1354(95)00241-3 0098-1354/96$15.00+ 0.00
Pergamon
D A T A R E C O N C I L I A T I O N - - A N I N D U S T R I A L CASE STUDY G. H. WEtss,~t J. A. ROMAGNOUZt and K. A. ISLAM3 11CI Australia Engineering Pty Ltd, 16-20 Beauchamp Rd, Matraville, NSW 2036, Australia 2Joint University of Sydney-ICI Engineering Chair of Process System Engineering, NSW, Australia 3Department of Chemical Engineering, University of Sydney, NSW 2006, Australia (Received 28 December 1993; final revision received 23 August 1995) Abstract--In this paper the application of data reconciliation to an industrial pyrolysis reactor is discussed. The reconciliation problem was developed around simplified mass and energy balances, and was solved via successive linearisation, although true linear and non-linear methods were briefly explored. The approach was tested using plant data collected at regular intervals over a full operational cycle of the reactor. Presence of gross errors in the data set was determined using a global test, and these were detected and rectified using serial elimination. An independent analysis of the process measurements also identified the source and magnitude of the gross errors and these agreed with results obtained form the data reconciliation. The overall heat transfer coefficient, one of the operating parameters of the pyrolysis reactor, calculated using reconciled data showed a trend consistent with plant experience and could be used to determine better regeneration cycle time of the reactor. Copyright © 1996 Elsevier Science Ltd
INTRODUCTION
gross errors. The techniques to detect and eliminate gross errors are well known (Mah, 1990; Romagnoli, 1983; Romagnoli and Stephanopoulos, 1981).
Reliable process data are the key to efficient operation of chemical plants. With the increasing use of computers in industry numerous data are acquired and used for on-line optimisation and control. Many process control and optimisation activities are based on small improvements in process performances; errors in process data or unreliable methods of dealing with these errors can easily exceed or mask actual changes in process performance. It should be common practice to adjust raw measurements taken from a process, so that known errors and measurement noise are eliminated. This procedure is called data reconciliation. A good survey of the available methods for data reconciliation is in the monograph by Mah (1990). The most common approach formulates a least squares problem, but the successful application of a least squares solution relies on the errors being normally distributed with zero mean. In practice, the process data may also contain other types of error which are caused by non-random events, These are called gross errors. Common examples of these gross errors are calibration errors and instrument malfunctions. The presence of gross errors alters the criteria of data reconciliation via the least squares approach. So in order to safely use the least squares approach it is essential to check for the presence of gross errors in the measurement data, and to eliminate the measurements that contain the t To whom all correspondence should be addressed,
This paper looks at the application to an industrial pyrolysis reactor. It begins with a brief description of the process under consideration, followed by a description of the techniques applied during the investigation. It concludes with a discussion of the reconciliation of measurements from the reactor, and an independent validation of results of the data reconciliation. PROCESS DESCRIPTION
The pyrolysis reactors are a key step in the manufacture of ethylene and propylene. The cracking reactions occur inside tubes passing through the radiant zone of a gas fired furnace. Each reactor consists of a preheat section in the convection zone of the firebox and a cracking section in the radiant zone. There are two sides to the furnace. The feed which is a mixture of hydrocarbon and steam is split between eight parallel passes, or coils. Each of the four coils on one side combine at the outlet to the reactor and are fed to a heat exchanger (called the transfer line exchanger), where the rapid reduction in temperature halts the cracking reactions. During the operation of the reactors, coke builds up on the inside of the tubes, which leads to a gradual increase in the temperature of the metal in the tube walls. Eventually, the reactor must be taken off-line for a decoke.
1441 CArE 20-12-F
G.H. WEiss etal.
1442 Hj~:Irocarbon
Craeso~r
Cr~
Steam Howmeters Temperatures Ternperatu-res Coil Outlet Fiowmeters . . . . . . . . . . . . . . Temperatures
I .....
Convection Section
............
Radianl Section
Fig. 1. Overview of pyrolysis reactor.
There are a number of inportant measurements around each reactor. Naturally, the flow of steam and hydrocarbons to each coil is measured, along with the temperature within the cracking zone. Maintaining identical cracking temperatures in all coils maximises the time between successive decokes, and is clearly desirable. This leads to a requirement for accurate measurement of the cracking temperatures. The crossover temperature for each coil, which is the temperature of the feed as it leaves the preheat section and enters the cracking zone is also measured. Differences in the crossover temperatures which are not present in the cracking temperatures are used by plant engineers as a guide to possible errors in the cracking temperatures. After the four coils on each side of the reactor are combined, and prior to the gas entering the transfer line exchanger, its temperature is measured and used to adjust the fuel gas flowrate. This measuremerit is referred to as the coil outlet temperature, Good reactor operation calls for the two sides to be fired at the same rate, which in turn means that the coil outlet temperatures must be accurately measured. Measurements of the overall hydrocarbon flow and the firebox temperature are also available, Figure 1 shows a simple layout of a pyrolysis reactor,
Several issues confront the operators of the pyrolysis reactors. Firstly, it is highly desirable to maintain equal cracking temperature in all the reactor coils as this maximises the period between decokes. Secondly, conditions in the reactor should maximise the yield of ethylene through the control of the severity of cracking, which in turn depends upon the firing rates. Lastly, an estimate of the rate of coking of the cracking coils assists in setting the future operating strategy of the reactor. In each case, accurate measurement of reactor conditions is needed. Therefore, a pyrolysis reactor was an ideal candidate for data reconciliation.
The model o[ the reactor The first step in the development of the data reconciliation package was the preparation of a model of the reactor. It was based on an overall hydrocarbon mass balance and several energy balances. Details of the model equations may be found in the Appendix. There were 36 variables in total and 11 equations. All variables in the model equations were measured, with the exception of the heat transfer coefficients. The coil and side heat transfer coefficients were known to change with time as a result of coke build up on the tubes, so needed
Data reconciliation
1443
to be estimated as part of the reconciliation process.
Successive linearisation methods
Fortunately, it was known from plant history, that the two heat transfer coefficients were closely related. This meant that only one needed to be estimated from the measurements. Two methods of parameter estimation were employed during the work; direct use of one of the equations to calculate the heat transfer coefficients, and joint parameter estimation and data reconciliation. The latter technique is briefly outlined in the next section.
A shortcoming of a linear solution is that the solution does not necessarily satisfy the non-linear constraints. Successive linearisation sees the linear problem iterated until an optimal point is obtained satisfying the non-linear constraints. It retains the advantages of relative simplicity and fast calculation.
These methods directly solve (2) as a general non-linear programming problem. The non-linear
STEADY-STATEDATARECONCILIATION Process measurements are subject to error. These errors give rise to discrepancies in material and energy balances. Data reconciliation is the process of adjusting the process measurements to obtain values that are consistent with the material and energy balances. A simple case is a process operating at steady-state where all desired variables are measured. The measurement vector (y) can be written as
y=x+t, (1) where x is the vector of the true values of the variables, and t is a vector of random measurement errors that are normally distributed with zero mean, and possessing a covariance matrix Q. The data reconciliation problem can be stated as a constrained least squares estimation problem where the weighted sum of errors is to be minimised, subject to constraints: Minx (y -x)TQ-I(Y - x )
(2)
such thatf(x) = 0. The constraints arise because the mass balances, energy balances and any other performance equations must be satisfied, and are encapsulated in the term f(x). Several methods have been used to solve the optimisation problem.
Linear solution Usually the constraints are linear or almost linear, and by linearising the latter, it is possible to reduce (2) to an unconstrained QP problem which can be solved analytically. The solution is obtained by means of Lagragian multipliers, and is given by:
x=y-QAr(AQAr)-I~, (3) where ~ is the residual of the unsatisfied balances and A is the Jacobian of the constraint equations. ~ is described by: = A t = Ay, (4) since Ax is zero.
Non-linear methods
programming solution makes it simple to augment (2) with upper and lower bounds on the variables, which may lead to a better formulated problem. The additional constraints are
Xl.i~Xi~Xu.i Vi, xt, i and xu.i refer to the lower and upper constraints on variable x,. In this work, the upper and lower bounds were taken as 110 and 90%, respectively, of measured values. A non-linear programming code that uses the successive quadratic programming algorithm, as described by Biegler and Cuthrell (1985), was used. As the focus of this work was on the industrial application of data reconciliation, it is worthwhile examining the implementation of the three alternatives in an industrial environment. The linear method can easily be implemented within the environment of a distributed control system (DCS) which is commonly used to control industrial chemical processes. The matrix inversion required to solve (3) can be performed off-line. Successive linearisation and the non-linear methods are less suitable for implementation on a DCS, but the use of a data processing computer interfaced to the DCS is a possible approach. The sequential processing of the balance equations described by Romagnoli and Stephanopoulos (1981) may provide a method to implement successive linearisation using a DCS. The three methods were applied during this work.
Gross errors handling In the previous section it was assumed that only errors present in the data are normally distributed measurement errors, with zero mean and known covariances. In practice, the process data may also contain other types of error, which are caused by non-random events such as calibration errors and malfunctions. The presence of gross errors alters the statistical basis of data reconciliation using least squres minimisation (Mah, 1990), so the successful application of the least squares approaches requires that the presence of gross errors be detected, and
G.H. WEISSet al.
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that the variables with gross errors be identified, and the gross errors be estimated. The approach used to deal with gross errors in the data reconciliation package which was developed as part of this work is described below,
Detection of gross errors. The presence of bias in the measurement is checked by global test (Mah, 1990), using the following test function:
value of h is removed from the measurement vector, and the procedure returns to step 1. Details of algorithms used to implement this approach to the identification of gross errors may be found in Romagnoli (1983). Estimation of the magnitude of the bias. Gross
(6)
errors were treated by the method proposed by Romagnoli (1983). They were considered as con-
is the covariance matrix of the residual of the balances (6), which has the value $ =AQA r. If all
stants to be calculated during the data reconciliation. If a gross error is identified in a measurement,
errors are normally distributed with zero mean then
it is e~/pressed as
h =6Tq~-]6,
h will have a chi-squared distribution with the number of degrees of freedom equalling the number of balance equations. Comparing the value of h against a critical value for X2 provides a test for the assertion that the measurement errors are normally distributed with zero mean and known covariance, Should the test fail, then at least one gross error is present,
Identification of the measurements with gross errors. If the presence of a gross error is found by the global test, it is necessary to eliminate that measurement from the reconciliation so that the least squares approach can be safely used. The
y=x+e+BxAe,
(7)
where/~ possesses as many rows as there are measurements and a column for each gross error. The element of each column corresponding to the measurement with the gross error contains the value 1. All other elements are 0. Ae is a vector containing the gross errors. The data reconciliation problem slightly modified: Min (y--x)TQ-I(y--x) x.~
(8)
such t h a t f ( x ) = 0 .
sequential approach proposed by Romagnoli (1983) was used for the identification of gross errors. It is outlined below. If the measurement Yi has a gross e r r o r (el), then the assertion that ei is zero fails at its nominal variance, qi. However, if qi is made arbitrarily large, then the assertion that ei is statistically no different from zero becomes true. A t this large value of qi,
The solution for the linear case (f(x)=-Ax) is
the test for gross errors (6) will indicate that no gross errors are present. This suggests the following approach for the identification of the gross errors:
The reconciliation of the pyrolysis reactors required the estimation of a heat transfer coefficient from the measurements. An error-in-variable
Step 1. Set a counter i to 1. Step 2. Increase the value of the variance of measurement i to a large value, Step 3. Perform the global test.
method (EVM) was implemented which provides both parameter estimates and reconciled data estimates that are consistent with the model. For the EVM the model of the reactor can be expressed as
Step 4. If the global test indicates that no gross errors are present, then the error is in element i, and the procedure is termihated, Step5. Reset the variance of element i to its original value and increment i by 1. Return to step 2 if there are more elements to be tested. Step 6. If all the elements of the measurement vector have been tested and if the global
f(x, 0 ) = 0.
test still indicates the presence of a gross error, then it means that more than one measurement contains a gross error. The measurement that contributes most to the
Ae = [BT~- 1B]-1BTq~-16 e = [QAX~ -]] [6 - BAt]
B=AB.
(9)
Joint parameter estimation-data reconciliation
(10)
As before x is the state vector, and 0 is a vector of parameters to be estimated. In the case when several sets of process measurements are available, the joint data reconciliation and parameter estimation problem can be expressed as m
Min ~'~ (x~-yi)TQ-t(x~-yi) x,O ~.d ~ such thatf(x~, 0 ) = 0
(11)
i = 1, 2 , . . . m,
where i refers to the ith data set. There are a total of m such data sets. The most straightforward
Data reconciliation
1445
Table 1. Raw measurements used in reconciliation Coil
Hydrocarbon flows
Coil steam flows
Crossover temperature
Cracking temperature
0.87 0.90 0.90 0.90 0.90 0.90 0.90 0.89
604 600 622 626 618 619 601 608
846 846 846 846 842 842 842 841
1 1.91 2 2.18 3 2.00 4 2.14 5 2.13 6 2.11 7 2.02 8 2.00 Total hydrocarbon flow Side B outlet temperatures Side A outlet temperatures Firebox temperature
16.7 860 860 1056
All flows are in tonnes/h and all temperatures are in °C. approach for solving the non-linear E V M problem is to use non-linear programming to estimate 0 and x simultaneously. H o w e v e r , in this work a two-stage non-linear E V M was implemented which involves separating the p a r a m e t e r estimation and data reconciliation steps, using the m e t h o d described by Valko and Vajda (1987). T h e two stage algorithm implemented here can be written as: 1. A t j = 1, set ,U equal to the measurements y. ls the optimal estimate for x. 2. Find the minimum 0 (j+l) of the function "
SLO)=2.~,, x-'~ [f(xi, O)-Aj(y-,U)] r i= ~
linear and successive linearisation methods for the remainder of the study. The small difference in the results did not justify the extra complexity assoelated with the non-linear method. The global test, when applied to these data yielded a value of 33.2 for the term h which suggested with better than 99.5% certainty, that a gross error was present. S a m e key results of the reconciliation carried out with no gross error treatment are in Table 2. Each result shown is the change in the measurement value divided by the standard deviation of the measurement. Values for the other measurements were not significantly affected by the reconciliation. The results of the reconciliation suggested a problem with coil 2 and possibly an additional problem with coil 3. Therefore, a search for
× (AQAX)-l[f(xi, O)-Aj(y
-,~J)], s
(12) 2
where Aj is the linear approximation t e l ( x , 0 j) at y. 3. I f j > l and II0 ~j+' 0ill is less than a suitable tolerance then stop. 4. A t fixed 0 j solve the data reconciliation problem, increment j and return to step 2. The same successive linearisation algorithm was used to solve the inner loop for data reconciliation,
DATA
RECONCILIATION
OF PYROLYSIS
REACTOR
E
[] [] ~ ' ~ ,
[] [] [] miD[] = [] ~=. . .. .... .
[] == ila= . i
[]
mm m
[]
gB
[]
z tll ~ [] ~
c3)
o Successive Linear solution linearisation
(4)
[] Non-linear
is)
=
~
I
I
I
solution I
I
I
t
I
I
I
=
i
I
Readings
MEASUREMENTS
Fig. 2. Comparison of solution methods.
Application of the data reconciliation techniques
T a b l e 2 . Key
reconciliation results:
no
gross e r r o r
treatment
applied The methods of solving the data reconciliation problem were applied to a set of raw measurements which are listed in Table 1, and the results are displayed in Fig. 2. The figures shows the error for each measurement, scaled by its standard deviation. No gross error corrections were applied to the measurements. The three methods gave similar results, which provided encouragement to use the
C h a n g e in v a lu e
Parameter T o t a l h y d r o c a r b o n flow
Coil hydrocarbon flows Crossover
temperature
Crackingtemperature
after reconciliation - 1.49
Coil 2 3 Coil 23
1.62 1.11 -0.711'88
Coil2 3
-3.95 1.36
-
G . H . WEzss et al.
1446
Table 3. Effect of standard deviation Measured value Standard deviations Coil flowmeters Coil temperatures Firebox temperature Total hydrocarbon flowmeter Global test result Reconciliation results Hydrocarbon flow Coil 2 3 Cracking temperatures Coil 2 3
2.18 2.00 846 846
the gross error was undertaken using the serial elimination technique. Prior to discussing the results of the serial elimination, it is valuable to examine the impact of the standard deviations on the reconciliation results, Selection of the values for the standard deviations of the measurements is never easy. The values used in the study were determined from the known accuracy of the sensors, in consultation with the plant instrumentation engineers. Some indication of the effect of the standard deviation on the reconciliation results may be seen in Table 3. Altering the values of the standard deviations had the expected effect on the reconciliation results. Again, only key results are shown. As the relative value of the standard deviations changed, the magnitude of adjustment to the hydrocarbon flows and the cracking temperature altered in a m a n n e r inversely proportional to the respective standard deviations. H o w e v e r , in all cases, the presence of a gross error was detected and the problem was with the measurements for coil 2. The results of the reconciliation suggested that the standard deviations used during the study were realistic. The fact that the global test found one or perhaps two gross errors meant that the standard deviations were not too small. Furthermore, with the elimination of the measurements with gross errors, the resultant global test results were not so
0~01 4 15 0.1 23.4
0,02 3 15 0.1 33.2
0~03 2 15 0.1 42.6
2.17 2.01 832 851
2,14 2.03 834 850
2.07 2.06 839 848
small as to imply that the standard deviations were too large. Table 4 lists the results of the serial elimination. R e m o v i n g the cracking temperature yieding the largest improvement in the X2 statistic, and if the gross error was assumed in the cracking temperature of coil 2, then applying the gross error estimation technique resulted in a value of - 18,9~C for the gross error. The results for the reconciliation on this basis are in Table 5. The effect of the gross error correction being applied to the coil 2 cracking temperature was to reduce the necessary corrections to the other measurements; so much so that all the resultant corrections satisfied a null hypothesis test at their respective standard deviations. These results would be satisfactory, but for the arbitrary selection of the cracking temperature as the location of the gross error. Eliminating any of the measurements for coil 2 reduced the value of gross error test result below the critical value, so any of these measurements could have contained the gross error, and the fact that the removal of the cracking temperature caused the maximum reduction in X2 may have been due to random events. This impasse could only be resolved by way of additional information, in the form of another relationship between the flows and temperatures for each coil. Fortunately, this extra information was obtained through further analysis of the system.
Resultant global There was only one balance per coil and a residual test result in that equation may have been due to an error in
Parameter eliminated
Coil 2 steam flow Coil 2 crossover temperature Coil 2 cracking temperature Coil 2 & 3 hydrocarbon flow
Selected Temperature values SD low
Analysis of daily averages
TaMe 4. Results of serial elimination
All imrameters present Coil hydrocarbon flow
Flow SD low
Coil 1 2 3 4 5 6 7 8
33.2 28,0 18,0 27.0 32.9 33.0 32.1 32.8 30A 17.8 9.0 8.0 11.0
Table 5. Key reconciliationresults: coil 2 temperature corrected Change in value after reconciliation
Parameter Total hydrocarbon flow Coil hydrocarbon flows Crossover temperature Cracking temperature
Coil 2 3 Coil 2 3 Coil 2 3
- 1.49 0:24 0:82 0.05 -0.47 - 6.30 1.03
Data reconciliation 40
1447
40
° ~= 1 0 .
~
~ 10
o e
o
,0 -lo
=~-
>=
"" " ' .
i.20 O -30
~-20
o
-40 -300
I -200
-10 •
O -30
I i i -100 0 100 Flow Difference
i 200
300
-40 -300
O I -200
I I J -100 0 100 Flow Difference
i 200
300
Coil 1 Coil 2 Coil 3 Coil 4 C' o •
Coil 1 Coil 2 Coil 3 Coil 4 @ o •
Fig. 3. Crossover temperature correlation--side B without flow adjustment,
Fig. 5. Crossover temperature correlation--side B after adjustment of flows.
one of several variables. A further set of equations describing the coils was required and they were found analysing the past operation of the reactor, Plant engineers use the relative value of the crossover temperatures for the coils as an indication of errors in measurement of the coil flowrates or cracking temperatures. With this in mind, an analysis of a years worth of daily averages of the reactor data was undertaken. A n expression relating the relative value of the crossover temperature with the relative value of the flow through the coil was sought, The results are presented in Figs 3 and 4 Each set of points shows the deviation of a coil crossover temperature from the average crossover temperature for the reactor plotted against the deviation of the coil thermal mass flow from the average for the reactor. Two features are apparent. Firstly, a common pattern is seen; the trends for each coil do have similar slopes. Secondly, the two sides behaved in a different way. In the case of side A, coils 5 and 6 were paired, as were coils 7 and 8. Now coils 5 and 6
follow a similar path through the convection section of the reactor, which is different from the path followed by coils 7 and 8. So it was expected that results for coils 5 and 6 would be grouped together, and that the results for coils 7 and 8 would be grouped together. Side B did not show this pairing even though coils i and 2 have similar paths through the reactor, as do coils 3 and 4. The discrepancy between the two sides was eliminated if 160 kg/h was deducted from the measured hydrocarbon flow for coil 2 and 210 kg/h added to the hydrocarbon flow for coil 3. Figure 5 shows the effect that these adjustments to the coil flow have on the crossover temperature correlations. Further evidence of errors in the measurements of the coil 2 and coil 3 hydrocarbon flowrates may be seen in Table 6, which shows the variations in the coil overall heat transfer coefficients. There was no statistical deviation between the coil heat transfer coefficient for side A but as before, coils 2 and 3 gave anomalous results for side B. If the same
3o 20
~ ~
e
• 0
,,
~
-1o -
o
~ ,
-20 .30
-200
i
i
I
I
-loo
o
loo
20o
30o
F~, D~ereme Coil 5 Coil 6 Coil 7 CoIl 8
o o • Fig. 4. Crossover temperature correlation--side A.
corrections were made to the coil 2 and 3 hydrocarbon flowrates then the differences in the side B heat transfer coefficients disappeared. All of this pointed to the gross errors being in the hydrocarbon flowrate measurements for coils 2 and 3, and elimination of these measurements yielded a data set with no gross errors present. The gross error identification technique described earlier found gross errors in these two measurements of - 200 and + 90 kg/h, respectively, which compared quite well with the corrections of - 160 and + 210 kg/h obtained by the analysis of the daily averages. The trends seen in Figs 3 and 4 led to a futher seven equations being added to the problem. These were used in a new reconciliation of the sensor data. After elimination of the hydrocarbon flows for coils
G . H . WEISSet al.
1448
Table 6. Coil heat transfer coefficientsform yearly averages Thermal mass flow Coil 1 2 3 4 Side B Coil 5 6 7 8 Side A
1.82 1.98 1.80 1.92 1.88 1.94 1.93 1.84 1.84 1.89
Coil heat Deviation transfer from side coefficient averagehtc 0.111 0.122 0.103 0.11l 0.112 0.114 0.114 0.111 0.111 0.112
Deviation Student's t value
Deviation
t Value
-0.0002 0.010 - 0.0092 -0.0008
-0.03 2.7 - 2.5 -0.2
-0.0002 0.13010 0.0020 -0.0008
-0.03 0.3 0.6 -0.2
0.0017 0.0016 -0.001l -0.0022
0.6 0.7 -0.4 -0.9
2 and 3, the gross error test yielded a X 2 value of 15.6 which was acceptable for the 17 degrees of freedom now available meaning that no further gross errors were present. Key results are in Table 7. The outcome of this reconciliation was very different from the reconciliation which considered the coil 2 cracking temperature as having the gross error. Incorrectly eliminating this cracking temperature caused the reconciliation to estimate a cracking t e m p e r a t u r e approximately 20°C lower than the true temperature, and if this erroneous result had been used for control purposes, coil 2 would have overheated and the resultant accelerated coking rate would have significantly reduced the run length of the reactor. W h e n the two coil flows were correctly Table 7. Key reconciliation results: coil 2 & 3 flows corrected Change in value after reconciliation
Parameter Total hydrocarbon flow Coil hydrocarbonflows Crossover temperature Cracking temperature
Coil2 3 Coil 2 3 Coil 2 3
-2.33 -9.50 4.89 0.70 0.12 -0.52 - 0.42
118 t 110 ................................... , ................ [_... ~ ......... ". . . . . . . . 108 ~e.• •°% o• . . . . . . . . . . . . . . . ...........
~ 100 k............... . .................................... 98
..................................................
~. 90 ..................................................... as ........................................... • ..... a°o
200~
400J 800~ aoo~ 1,000~ 1,aooJ 1,400 Timeof Ol~mtion(Houm)
Fig. 6. Time evolution of the estimated overall heat transfer coefficient,
After flow correction
eliminated, the reconciliation found no significant error in the cracking temperature.
Analysis over the whole operating cycle The overall heat transfer coefficient, calculated using the joint data reconciliation-parameter estimation procedure described before, is shown in Fig. 6. The overall heat transfer coefficient remained fairly constant throughout the whole operating cycle of the pyrolyis reactor, but near the end of the cycle the heat transfer coefficient dropped to a comparably low value signifying that the reactor needs to be regenerated. CONCLUSIONS Both linear and non-linear methods were used to solve the data reconciliation problem. The linear methods, which included successive linearisation, yielded results very similar to those from the nonlinear method. The large computational time required by the non-linear method could not be justified, and the majority of the study used only the successive linearisation method. These general comments can be made. Firstly, the analysis of the daily averages for a year pointed to a consistent error in the m e a s u r e m e n t of the hydrocarbon flow through coils 2 and 3. Similar biases were found using the data reconciliation methods, although the hydrocarbon flows for coils 2 and 3 were just one of several possible locations for the gross errors. H o w e v e r , the results did provide an independent verification of the least squares approach to data reconciliation. Secondly, the assumption that all coils were equal was supported by the analysis of the yearly results. Furthermore, it meant that the residuals in the balances were due to sensor errors and not errors in the balance equations. Furthermore, the reconciled plant measurements highlighted deficiencies in reactor operation which were reducing its effectiveness. The identification of these situations is the ultimate purpose of data reconciliation. Finally, overall heat transfer
Data reconciliation
1449
coefficients, calculated using reconciled data s h o w e d a t r e n d c o m p a r a b l e to plant e x p e r i e n c e and thus could be used to d e t e r m i n e b e t t e r r e g e n e r a t i o n cycle time o f the reactor. D a t a reconciliation was s h o w n to be capable of identifying errors in process m e a s u r e m e n t s , and
Table A1. Model parameters Parameter
Value
Specificheat of hydrocarbon feed Specificheat of steam Heat transfer area per coil Heat t r a n s f e r a r e a p e r side
0.93 kcal/kg *C 0.1 kcal kl~*C 12.2 m' 62.4m 2
Coil heat transfer coefficient
0.116 kcal/s m2°C
r e m o v i n g t h e s e errors to give a b e t t e r view of the true state o f the process than is p r o v i d e d by the raw m e a s u r e m e n t s . It is t h e r e f o r e an i m p o r t a n t adjunct to a d v a n c e d control and optimisation,
Acknowledgement--The
authors wish to thank ICI Australia Operation Pty Ltd for their assistance during this project and for allowing us to publish the results. REFERENCES Biegler L. T. and J. E. Cuthrell, Improved infeasible path optimisation for sequential modular simulators---II. The optimisation algorithm. Computers chem. Engng 9,257267 (1985). Mah R. S. H., Chemical Process Structures and Information Flows. Butterworths. Romagnoli J. A., On data reconciliation: constraints processing and treatment of bias. Chem. Engng Sci. 38, 1107-1117 (1983). Romagnoli J. A. and G. Stephanopoulos, Rectification of process measurement data in the presence of gross errors. Chem. Engng Sci. 36, 1849-1863 (1981). Valko P. and S. Vajda, An extended Marquardt-type procedure for fitting error-in-variables models, Computers chem. Engng II, 37-43 (1987).
were lumped into an "effective" specific heat for the hydrocarbons. First, the side energy balance: I-
UsA~[TFa-
Tn+TXnq 2 ]-FrM.,,(T,,-TX,,),
Fru.n, thermal mass flow for side n. The values for n are A and B. Side A consists of coils 5-8, and side B contains coils 1-4; T~, the coil outlet temperature for side n; TX~, the crossover temperature for side n, which is the average of the crossover temperatures for the four coils for side n; T~, the firebox temperature; Us, the heat transfer coefficients for the reactor sides; A~, the heat transfer area of a reactor side. The coil energy balances are:
UA [| Tva- Tp+2TX.] r| - FTM.p(Tp- TXp), I-
(A3)
d
F~M.p, thermal mass flow for pass p; Tp, the cracking temperature for pass or coil p; TXp, the crossover temperature for coil p; U, the heat transfer coefficients for the reactor coils; A, the heat transfer area of a reactor coil. The two heat transfer coefficients are related by: U= 1.154Us.
APPENDIX
(A2)
(A4)
(A1)
There are 11 equations in total and 36 measured variables. The heat transfer coefficients are not known so one of the equations must be used to calculate values for the heat transfer coefficients. The additional equations added by the analysis of the daily averages are:
FH,7, total flow of hydrocarbons to the reactor; FH. p, flow of hydrocarbons to coil p. Note that p = 1-8. In deriving the energy balances, it was assumed that the two sides are identical, and that all passes are identical and the two sides are are identical. In addition, it was assumed that the heat transfer coefficient for the side is related to the heat transfer coefficient for the passes. Effects such as the heat of reaction were not considered separately but
a i is an offset for coil i and b is the regression coefficient for the relationship and is common to all coils. TX and Fxu are average values for the reactor. The values of some model parameters are Table A1. The effective specific heats were determined by means of a rigorous cracking simulator (PHENICS), and the specific heat for the hydrocarbons includes the effect of the enthalpy change due to the cracking reactions.
PYROLYSIS REACTOR MODEL
The overall hydrocarbon balance is straightforward, 8
F n ' r = X FH'p' p = 1
TXp
-
T X = a~ + b ( F ~ . p - Fa~4),
(A5)
Pergamon
D A T A R E C O N C I L I A T I O N - - A N I N D U S T R I A L CASE STUDY G. H. WEtss,~t J. A. ROMAGNOUZt and K. A. ISLAM3 11CI Australia Engineering Pty Ltd, 16-20 Beauchamp Rd, Matraville, NSW 2036, Australia 2Joint University of Sydney-ICI Engineering Chair of Process System Engineering, NSW, Australia 3Department of Chemical Engineering, University of Sydney, NSW 2006, Australia (Received 28 December 1993; final revision received 23 August 1995) Abstract--In this paper the application of data reconciliation to an industrial pyrolysis reactor is discussed. The reconciliation problem was developed around simplified mass and energy balances, and was solved via successive linearisation, although true linear and non-linear methods were briefly explored. The approach was tested using plant data collected at regular intervals over a full operational cycle of the reactor. Presence of gross errors in the data set was determined using a global test, and these were detected and rectified using serial elimination. An independent analysis of the process measurements also identified the source and magnitude of the gross errors and these agreed with results obtained form the data reconciliation. The overall heat transfer coefficient, one of the operating parameters of the pyrolysis reactor, calculated using reconciled data showed a trend consistent with plant experience and could be used to determine better regeneration cycle time of the reactor. Copyright © 1996 Elsevier Science Ltd
INTRODUCTION
gross errors. The techniques to detect and eliminate gross errors are well known (Mah, 1990; Romagnoli, 1983; Romagnoli and Stephanopoulos, 1981).
Reliable process data are the key to efficient operation of chemical plants. With the increasing use of computers in industry numerous data are acquired and used for on-line optimisation and control. Many process control and optimisation activities are based on small improvements in process performances; errors in process data or unreliable methods of dealing with these errors can easily exceed or mask actual changes in process performance. It should be common practice to adjust raw measurements taken from a process, so that known errors and measurement noise are eliminated. This procedure is called data reconciliation. A good survey of the available methods for data reconciliation is in the monograph by Mah (1990). The most common approach formulates a least squares problem, but the successful application of a least squares solution relies on the errors being normally distributed with zero mean. In practice, the process data may also contain other types of error which are caused by non-random events, These are called gross errors. Common examples of these gross errors are calibration errors and instrument malfunctions. The presence of gross errors alters the criteria of data reconciliation via the least squares approach. So in order to safely use the least squares approach it is essential to check for the presence of gross errors in the measurement data, and to eliminate the measurements that contain the t To whom all correspondence should be addressed,
This paper looks at the application to an industrial pyrolysis reactor. It begins with a brief description of the process under consideration, followed by a description of the techniques applied during the investigation. It concludes with a discussion of the reconciliation of measurements from the reactor, and an independent validation of results of the data reconciliation. PROCESS DESCRIPTION
The pyrolysis reactors are a key step in the manufacture of ethylene and propylene. The cracking reactions occur inside tubes passing through the radiant zone of a gas fired furnace. Each reactor consists of a preheat section in the convection zone of the firebox and a cracking section in the radiant zone. There are two sides to the furnace. The feed which is a mixture of hydrocarbon and steam is split between eight parallel passes, or coils. Each of the four coils on one side combine at the outlet to the reactor and are fed to a heat exchanger (called the transfer line exchanger), where the rapid reduction in temperature halts the cracking reactions. During the operation of the reactors, coke builds up on the inside of the tubes, which leads to a gradual increase in the temperature of the metal in the tube walls. Eventually, the reactor must be taken off-line for a decoke.
1441 CArE 20-12-F
G.H. WEiss etal.
1442 Hj~:Irocarbon
Craeso~r
Cr~
Steam Howmeters Temperatures Ternperatu-res Coil Outlet Fiowmeters . . . . . . . . . . . . . . Temperatures
I .....
Convection Section
............
Radianl Section
Fig. 1. Overview of pyrolysis reactor.
There are a number of inportant measurements around each reactor. Naturally, the flow of steam and hydrocarbons to each coil is measured, along with the temperature within the cracking zone. Maintaining identical cracking temperatures in all coils maximises the time between successive decokes, and is clearly desirable. This leads to a requirement for accurate measurement of the cracking temperatures. The crossover temperature for each coil, which is the temperature of the feed as it leaves the preheat section and enters the cracking zone is also measured. Differences in the crossover temperatures which are not present in the cracking temperatures are used by plant engineers as a guide to possible errors in the cracking temperatures. After the four coils on each side of the reactor are combined, and prior to the gas entering the transfer line exchanger, its temperature is measured and used to adjust the fuel gas flowrate. This measuremerit is referred to as the coil outlet temperature, Good reactor operation calls for the two sides to be fired at the same rate, which in turn means that the coil outlet temperatures must be accurately measured. Measurements of the overall hydrocarbon flow and the firebox temperature are also available, Figure 1 shows a simple layout of a pyrolysis reactor,
Several issues confront the operators of the pyrolysis reactors. Firstly, it is highly desirable to maintain equal cracking temperature in all the reactor coils as this maximises the period between decokes. Secondly, conditions in the reactor should maximise the yield of ethylene through the control of the severity of cracking, which in turn depends upon the firing rates. Lastly, an estimate of the rate of coking of the cracking coils assists in setting the future operating strategy of the reactor. In each case, accurate measurement of reactor conditions is needed. Therefore, a pyrolysis reactor was an ideal candidate for data reconciliation.
The model o[ the reactor The first step in the development of the data reconciliation package was the preparation of a model of the reactor. It was based on an overall hydrocarbon mass balance and several energy balances. Details of the model equations may be found in the Appendix. There were 36 variables in total and 11 equations. All variables in the model equations were measured, with the exception of the heat transfer coefficients. The coil and side heat transfer coefficients were known to change with time as a result of coke build up on the tubes, so needed
Data reconciliation
1443
to be estimated as part of the reconciliation process.
Successive linearisation methods
Fortunately, it was known from plant history, that the two heat transfer coefficients were closely related. This meant that only one needed to be estimated from the measurements. Two methods of parameter estimation were employed during the work; direct use of one of the equations to calculate the heat transfer coefficients, and joint parameter estimation and data reconciliation. The latter technique is briefly outlined in the next section.
A shortcoming of a linear solution is that the solution does not necessarily satisfy the non-linear constraints. Successive linearisation sees the linear problem iterated until an optimal point is obtained satisfying the non-linear constraints. It retains the advantages of relative simplicity and fast calculation.
These methods directly solve (2) as a general non-linear programming problem. The non-linear
STEADY-STATEDATARECONCILIATION Process measurements are subject to error. These errors give rise to discrepancies in material and energy balances. Data reconciliation is the process of adjusting the process measurements to obtain values that are consistent with the material and energy balances. A simple case is a process operating at steady-state where all desired variables are measured. The measurement vector (y) can be written as
y=x+t, (1) where x is the vector of the true values of the variables, and t is a vector of random measurement errors that are normally distributed with zero mean, and possessing a covariance matrix Q. The data reconciliation problem can be stated as a constrained least squares estimation problem where the weighted sum of errors is to be minimised, subject to constraints: Minx (y -x)TQ-I(Y - x )
(2)
such thatf(x) = 0. The constraints arise because the mass balances, energy balances and any other performance equations must be satisfied, and are encapsulated in the term f(x). Several methods have been used to solve the optimisation problem.
Linear solution Usually the constraints are linear or almost linear, and by linearising the latter, it is possible to reduce (2) to an unconstrained QP problem which can be solved analytically. The solution is obtained by means of Lagragian multipliers, and is given by:
x=y-QAr(AQAr)-I~, (3) where ~ is the residual of the unsatisfied balances and A is the Jacobian of the constraint equations. ~ is described by: = A t = Ay, (4) since Ax is zero.
Non-linear methods
programming solution makes it simple to augment (2) with upper and lower bounds on the variables, which may lead to a better formulated problem. The additional constraints are
Xl.i~Xi~Xu.i Vi, xt, i and xu.i refer to the lower and upper constraints on variable x,. In this work, the upper and lower bounds were taken as 110 and 90%, respectively, of measured values. A non-linear programming code that uses the successive quadratic programming algorithm, as described by Biegler and Cuthrell (1985), was used. As the focus of this work was on the industrial application of data reconciliation, it is worthwhile examining the implementation of the three alternatives in an industrial environment. The linear method can easily be implemented within the environment of a distributed control system (DCS) which is commonly used to control industrial chemical processes. The matrix inversion required to solve (3) can be performed off-line. Successive linearisation and the non-linear methods are less suitable for implementation on a DCS, but the use of a data processing computer interfaced to the DCS is a possible approach. The sequential processing of the balance equations described by Romagnoli and Stephanopoulos (1981) may provide a method to implement successive linearisation using a DCS. The three methods were applied during this work.
Gross errors handling In the previous section it was assumed that only errors present in the data are normally distributed measurement errors, with zero mean and known covariances. In practice, the process data may also contain other types of error, which are caused by non-random events such as calibration errors and malfunctions. The presence of gross errors alters the statistical basis of data reconciliation using least squres minimisation (Mah, 1990), so the successful application of the least squares approaches requires that the presence of gross errors be detected, and
G.H. WEISSet al.
1444
that the variables with gross errors be identified, and the gross errors be estimated. The approach used to deal with gross errors in the data reconciliation package which was developed as part of this work is described below,
Detection of gross errors. The presence of bias in the measurement is checked by global test (Mah, 1990), using the following test function:
value of h is removed from the measurement vector, and the procedure returns to step 1. Details of algorithms used to implement this approach to the identification of gross errors may be found in Romagnoli (1983). Estimation of the magnitude of the bias. Gross
(6)
errors were treated by the method proposed by Romagnoli (1983). They were considered as con-
is the covariance matrix of the residual of the balances (6), which has the value $ =AQA r. If all
stants to be calculated during the data reconciliation. If a gross error is identified in a measurement,
errors are normally distributed with zero mean then
it is e~/pressed as
h =6Tq~-]6,
h will have a chi-squared distribution with the number of degrees of freedom equalling the number of balance equations. Comparing the value of h against a critical value for X2 provides a test for the assertion that the measurement errors are normally distributed with zero mean and known covariance, Should the test fail, then at least one gross error is present,
Identification of the measurements with gross errors. If the presence of a gross error is found by the global test, it is necessary to eliminate that measurement from the reconciliation so that the least squares approach can be safely used. The
y=x+e+BxAe,
(7)
where/~ possesses as many rows as there are measurements and a column for each gross error. The element of each column corresponding to the measurement with the gross error contains the value 1. All other elements are 0. Ae is a vector containing the gross errors. The data reconciliation problem slightly modified: Min (y--x)TQ-I(y--x) x.~
(8)
such t h a t f ( x ) = 0 .
sequential approach proposed by Romagnoli (1983) was used for the identification of gross errors. It is outlined below. If the measurement Yi has a gross e r r o r (el), then the assertion that ei is zero fails at its nominal variance, qi. However, if qi is made arbitrarily large, then the assertion that ei is statistically no different from zero becomes true. A t this large value of qi,
The solution for the linear case (f(x)=-Ax) is
the test for gross errors (6) will indicate that no gross errors are present. This suggests the following approach for the identification of the gross errors:
The reconciliation of the pyrolysis reactors required the estimation of a heat transfer coefficient from the measurements. An error-in-variable
Step 1. Set a counter i to 1. Step 2. Increase the value of the variance of measurement i to a large value, Step 3. Perform the global test.
method (EVM) was implemented which provides both parameter estimates and reconciled data estimates that are consistent with the model. For the EVM the model of the reactor can be expressed as
Step 4. If the global test indicates that no gross errors are present, then the error is in element i, and the procedure is termihated, Step5. Reset the variance of element i to its original value and increment i by 1. Return to step 2 if there are more elements to be tested. Step 6. If all the elements of the measurement vector have been tested and if the global
f(x, 0 ) = 0.
test still indicates the presence of a gross error, then it means that more than one measurement contains a gross error. The measurement that contributes most to the
Ae = [BT~- 1B]-1BTq~-16 e = [QAX~ -]] [6 - BAt]
B=AB.
(9)
Joint parameter estimation-data reconciliation
(10)
As before x is the state vector, and 0 is a vector of parameters to be estimated. In the case when several sets of process measurements are available, the joint data reconciliation and parameter estimation problem can be expressed as m
Min ~'~ (x~-yi)TQ-t(x~-yi) x,O ~.d ~ such thatf(x~, 0 ) = 0
(11)
i = 1, 2 , . . . m,
where i refers to the ith data set. There are a total of m such data sets. The most straightforward
Data reconciliation
1445
Table 1. Raw measurements used in reconciliation Coil
Hydrocarbon flows
Coil steam flows
Crossover temperature
Cracking temperature
0.87 0.90 0.90 0.90 0.90 0.90 0.90 0.89
604 600 622 626 618 619 601 608
846 846 846 846 842 842 842 841
1 1.91 2 2.18 3 2.00 4 2.14 5 2.13 6 2.11 7 2.02 8 2.00 Total hydrocarbon flow Side B outlet temperatures Side A outlet temperatures Firebox temperature
16.7 860 860 1056
All flows are in tonnes/h and all temperatures are in °C. approach for solving the non-linear E V M problem is to use non-linear programming to estimate 0 and x simultaneously. H o w e v e r , in this work a two-stage non-linear E V M was implemented which involves separating the p a r a m e t e r estimation and data reconciliation steps, using the m e t h o d described by Valko and Vajda (1987). T h e two stage algorithm implemented here can be written as: 1. A t j = 1, set ,U equal to the measurements y. ls the optimal estimate for x. 2. Find the minimum 0 (j+l) of the function "
SLO)=2.~,, x-'~ [f(xi, O)-Aj(y-,U)] r i= ~
linear and successive linearisation methods for the remainder of the study. The small difference in the results did not justify the extra complexity assoelated with the non-linear method. The global test, when applied to these data yielded a value of 33.2 for the term h which suggested with better than 99.5% certainty, that a gross error was present. S a m e key results of the reconciliation carried out with no gross error treatment are in Table 2. Each result shown is the change in the measurement value divided by the standard deviation of the measurement. Values for the other measurements were not significantly affected by the reconciliation. The results of the reconciliation suggested a problem with coil 2 and possibly an additional problem with coil 3. Therefore, a search for
× (AQAX)-l[f(xi, O)-Aj(y
-,~J)], s
(12) 2
where Aj is the linear approximation t e l ( x , 0 j) at y. 3. I f j > l and II0 ~j+' 0ill is less than a suitable tolerance then stop. 4. A t fixed 0 j solve the data reconciliation problem, increment j and return to step 2. The same successive linearisation algorithm was used to solve the inner loop for data reconciliation,
DATA
RECONCILIATION
OF PYROLYSIS
REACTOR
E
[] [] ~ ' ~ ,
[] [] [] miD[] = [] ~=. . .. .... .
[] == ila= . i
[]
mm m
[]
gB
[]
z tll ~ [] ~
c3)
o Successive Linear solution linearisation
(4)
[] Non-linear
is)
=
~
I
I
I
solution I
I
I
t
I
I
I
=
i
I
Readings
MEASUREMENTS
Fig. 2. Comparison of solution methods.
Application of the data reconciliation techniques
T a b l e 2 . Key
reconciliation results:
no
gross e r r o r
treatment
applied The methods of solving the data reconciliation problem were applied to a set of raw measurements which are listed in Table 1, and the results are displayed in Fig. 2. The figures shows the error for each measurement, scaled by its standard deviation. No gross error corrections were applied to the measurements. The three methods gave similar results, which provided encouragement to use the
C h a n g e in v a lu e
Parameter T o t a l h y d r o c a r b o n flow
Coil hydrocarbon flows Crossover
temperature
Crackingtemperature
after reconciliation - 1.49
Coil 2 3 Coil 23
1.62 1.11 -0.711'88
Coil2 3
-3.95 1.36
-
G . H . WEzss et al.
1446
Table 3. Effect of standard deviation Measured value Standard deviations Coil flowmeters Coil temperatures Firebox temperature Total hydrocarbon flowmeter Global test result Reconciliation results Hydrocarbon flow Coil 2 3 Cracking temperatures Coil 2 3
2.18 2.00 846 846
the gross error was undertaken using the serial elimination technique. Prior to discussing the results of the serial elimination, it is valuable to examine the impact of the standard deviations on the reconciliation results, Selection of the values for the standard deviations of the measurements is never easy. The values used in the study were determined from the known accuracy of the sensors, in consultation with the plant instrumentation engineers. Some indication of the effect of the standard deviation on the reconciliation results may be seen in Table 3. Altering the values of the standard deviations had the expected effect on the reconciliation results. Again, only key results are shown. As the relative value of the standard deviations changed, the magnitude of adjustment to the hydrocarbon flows and the cracking temperature altered in a m a n n e r inversely proportional to the respective standard deviations. H o w e v e r , in all cases, the presence of a gross error was detected and the problem was with the measurements for coil 2. The results of the reconciliation suggested that the standard deviations used during the study were realistic. The fact that the global test found one or perhaps two gross errors meant that the standard deviations were not too small. Furthermore, with the elimination of the measurements with gross errors, the resultant global test results were not so
0~01 4 15 0.1 23.4
0,02 3 15 0.1 33.2
0~03 2 15 0.1 42.6
2.17 2.01 832 851
2,14 2.03 834 850
2.07 2.06 839 848
small as to imply that the standard deviations were too large. Table 4 lists the results of the serial elimination. R e m o v i n g the cracking temperature yieding the largest improvement in the X2 statistic, and if the gross error was assumed in the cracking temperature of coil 2, then applying the gross error estimation technique resulted in a value of - 18,9~C for the gross error. The results for the reconciliation on this basis are in Table 5. The effect of the gross error correction being applied to the coil 2 cracking temperature was to reduce the necessary corrections to the other measurements; so much so that all the resultant corrections satisfied a null hypothesis test at their respective standard deviations. These results would be satisfactory, but for the arbitrary selection of the cracking temperature as the location of the gross error. Eliminating any of the measurements for coil 2 reduced the value of gross error test result below the critical value, so any of these measurements could have contained the gross error, and the fact that the removal of the cracking temperature caused the maximum reduction in X2 may have been due to random events. This impasse could only be resolved by way of additional information, in the form of another relationship between the flows and temperatures for each coil. Fortunately, this extra information was obtained through further analysis of the system.
Resultant global There was only one balance per coil and a residual test result in that equation may have been due to an error in
Parameter eliminated
Coil 2 steam flow Coil 2 crossover temperature Coil 2 cracking temperature Coil 2 & 3 hydrocarbon flow
Selected Temperature values SD low
Analysis of daily averages
TaMe 4. Results of serial elimination
All imrameters present Coil hydrocarbon flow
Flow SD low
Coil 1 2 3 4 5 6 7 8
33.2 28,0 18,0 27.0 32.9 33.0 32.1 32.8 30A 17.8 9.0 8.0 11.0
Table 5. Key reconciliationresults: coil 2 temperature corrected Change in value after reconciliation
Parameter Total hydrocarbon flow Coil hydrocarbon flows Crossover temperature Cracking temperature
Coil 2 3 Coil 2 3 Coil 2 3
- 1.49 0:24 0:82 0.05 -0.47 - 6.30 1.03
Data reconciliation 40
1447
40
° ~= 1 0 .
~
~ 10
o e
o
,0 -lo
=~-
>=
"" " ' .
i.20 O -30
~-20
o
-40 -300
I -200
-10 •
O -30
I i i -100 0 100 Flow Difference
i 200
300
-40 -300
O I -200
I I J -100 0 100 Flow Difference
i 200
300
Coil 1 Coil 2 Coil 3 Coil 4 C' o •
Coil 1 Coil 2 Coil 3 Coil 4 @ o •
Fig. 3. Crossover temperature correlation--side B without flow adjustment,
Fig. 5. Crossover temperature correlation--side B after adjustment of flows.
one of several variables. A further set of equations describing the coils was required and they were found analysing the past operation of the reactor, Plant engineers use the relative value of the crossover temperatures for the coils as an indication of errors in measurement of the coil flowrates or cracking temperatures. With this in mind, an analysis of a years worth of daily averages of the reactor data was undertaken. A n expression relating the relative value of the crossover temperature with the relative value of the flow through the coil was sought, The results are presented in Figs 3 and 4 Each set of points shows the deviation of a coil crossover temperature from the average crossover temperature for the reactor plotted against the deviation of the coil thermal mass flow from the average for the reactor. Two features are apparent. Firstly, a common pattern is seen; the trends for each coil do have similar slopes. Secondly, the two sides behaved in a different way. In the case of side A, coils 5 and 6 were paired, as were coils 7 and 8. Now coils 5 and 6
follow a similar path through the convection section of the reactor, which is different from the path followed by coils 7 and 8. So it was expected that results for coils 5 and 6 would be grouped together, and that the results for coils 7 and 8 would be grouped together. Side B did not show this pairing even though coils i and 2 have similar paths through the reactor, as do coils 3 and 4. The discrepancy between the two sides was eliminated if 160 kg/h was deducted from the measured hydrocarbon flow for coil 2 and 210 kg/h added to the hydrocarbon flow for coil 3. Figure 5 shows the effect that these adjustments to the coil flow have on the crossover temperature correlations. Further evidence of errors in the measurements of the coil 2 and coil 3 hydrocarbon flowrates may be seen in Table 6, which shows the variations in the coil overall heat transfer coefficients. There was no statistical deviation between the coil heat transfer coefficient for side A but as before, coils 2 and 3 gave anomalous results for side B. If the same
3o 20
~ ~
e
• 0
,,
~
-1o -
o
~ ,
-20 .30
-200
i
i
I
I
-loo
o
loo
20o
30o
F~, D~ereme Coil 5 Coil 6 Coil 7 CoIl 8
o o • Fig. 4. Crossover temperature correlation--side A.
corrections were made to the coil 2 and 3 hydrocarbon flowrates then the differences in the side B heat transfer coefficients disappeared. All of this pointed to the gross errors being in the hydrocarbon flowrate measurements for coils 2 and 3, and elimination of these measurements yielded a data set with no gross errors present. The gross error identification technique described earlier found gross errors in these two measurements of - 200 and + 90 kg/h, respectively, which compared quite well with the corrections of - 160 and + 210 kg/h obtained by the analysis of the daily averages. The trends seen in Figs 3 and 4 led to a futher seven equations being added to the problem. These were used in a new reconciliation of the sensor data. After elimination of the hydrocarbon flows for coils
G . H . WEISSet al.
1448
Table 6. Coil heat transfer coefficientsform yearly averages Thermal mass flow Coil 1 2 3 4 Side B Coil 5 6 7 8 Side A
1.82 1.98 1.80 1.92 1.88 1.94 1.93 1.84 1.84 1.89
Coil heat Deviation transfer from side coefficient averagehtc 0.111 0.122 0.103 0.11l 0.112 0.114 0.114 0.111 0.111 0.112
Deviation Student's t value
Deviation
t Value
-0.0002 0.010 - 0.0092 -0.0008
-0.03 2.7 - 2.5 -0.2
-0.0002 0.13010 0.0020 -0.0008
-0.03 0.3 0.6 -0.2
0.0017 0.0016 -0.001l -0.0022
0.6 0.7 -0.4 -0.9
2 and 3, the gross error test yielded a X 2 value of 15.6 which was acceptable for the 17 degrees of freedom now available meaning that no further gross errors were present. Key results are in Table 7. The outcome of this reconciliation was very different from the reconciliation which considered the coil 2 cracking temperature as having the gross error. Incorrectly eliminating this cracking temperature caused the reconciliation to estimate a cracking t e m p e r a t u r e approximately 20°C lower than the true temperature, and if this erroneous result had been used for control purposes, coil 2 would have overheated and the resultant accelerated coking rate would have significantly reduced the run length of the reactor. W h e n the two coil flows were correctly Table 7. Key reconciliation results: coil 2 & 3 flows corrected Change in value after reconciliation
Parameter Total hydrocarbon flow Coil hydrocarbonflows Crossover temperature Cracking temperature
Coil2 3 Coil 2 3 Coil 2 3
-2.33 -9.50 4.89 0.70 0.12 -0.52 - 0.42
118 t 110 ................................... , ................ [_... ~ ......... ". . . . . . . . 108 ~e.• •°% o• . . . . . . . . . . . . . . . ...........
~ 100 k............... . .................................... 98
..................................................
~. 90 ..................................................... as ........................................... • ..... a°o
200~
400J 800~ aoo~ 1,000~ 1,aooJ 1,400 Timeof Ol~mtion(Houm)
Fig. 6. Time evolution of the estimated overall heat transfer coefficient,
After flow correction
eliminated, the reconciliation found no significant error in the cracking temperature.
Analysis over the whole operating cycle The overall heat transfer coefficient, calculated using the joint data reconciliation-parameter estimation procedure described before, is shown in Fig. 6. The overall heat transfer coefficient remained fairly constant throughout the whole operating cycle of the pyrolyis reactor, but near the end of the cycle the heat transfer coefficient dropped to a comparably low value signifying that the reactor needs to be regenerated. CONCLUSIONS Both linear and non-linear methods were used to solve the data reconciliation problem. The linear methods, which included successive linearisation, yielded results very similar to those from the nonlinear method. The large computational time required by the non-linear method could not be justified, and the majority of the study used only the successive linearisation method. These general comments can be made. Firstly, the analysis of the daily averages for a year pointed to a consistent error in the m e a s u r e m e n t of the hydrocarbon flow through coils 2 and 3. Similar biases were found using the data reconciliation methods, although the hydrocarbon flows for coils 2 and 3 were just one of several possible locations for the gross errors. H o w e v e r , the results did provide an independent verification of the least squares approach to data reconciliation. Secondly, the assumption that all coils were equal was supported by the analysis of the yearly results. Furthermore, it meant that the residuals in the balances were due to sensor errors and not errors in the balance equations. Furthermore, the reconciled plant measurements highlighted deficiencies in reactor operation which were reducing its effectiveness. The identification of these situations is the ultimate purpose of data reconciliation. Finally, overall heat transfer
Data reconciliation
1449
coefficients, calculated using reconciled data s h o w e d a t r e n d c o m p a r a b l e to plant e x p e r i e n c e and thus could be used to d e t e r m i n e b e t t e r r e g e n e r a t i o n cycle time o f the reactor. D a t a reconciliation was s h o w n to be capable of identifying errors in process m e a s u r e m e n t s , and
Table A1. Model parameters Parameter
Value
Specificheat of hydrocarbon feed Specificheat of steam Heat transfer area per coil Heat t r a n s f e r a r e a p e r side
0.93 kcal/kg *C 0.1 kcal kl~*C 12.2 m' 62.4m 2
Coil heat transfer coefficient
0.116 kcal/s m2°C
r e m o v i n g t h e s e errors to give a b e t t e r view of the true state o f the process than is p r o v i d e d by the raw m e a s u r e m e n t s . It is t h e r e f o r e an i m p o r t a n t adjunct to a d v a n c e d control and optimisation,
Acknowledgement--The
authors wish to thank ICI Australia Operation Pty Ltd for their assistance during this project and for allowing us to publish the results. REFERENCES Biegler L. T. and J. E. Cuthrell, Improved infeasible path optimisation for sequential modular simulators---II. The optimisation algorithm. Computers chem. Engng 9,257267 (1985). Mah R. S. H., Chemical Process Structures and Information Flows. Butterworths. Romagnoli J. A., On data reconciliation: constraints processing and treatment of bias. Chem. Engng Sci. 38, 1107-1117 (1983). Romagnoli J. A. and G. Stephanopoulos, Rectification of process measurement data in the presence of gross errors. Chem. Engng Sci. 36, 1849-1863 (1981). Valko P. and S. Vajda, An extended Marquardt-type procedure for fitting error-in-variables models, Computers chem. Engng II, 37-43 (1987).
were lumped into an "effective" specific heat for the hydrocarbons. First, the side energy balance: I-
UsA~[TFa-
Tn+TXnq 2 ]-FrM.,,(T,,-TX,,),
Fru.n, thermal mass flow for side n. The values for n are A and B. Side A consists of coils 5-8, and side B contains coils 1-4; T~, the coil outlet temperature for side n; TX~, the crossover temperature for side n, which is the average of the crossover temperatures for the four coils for side n; T~, the firebox temperature; Us, the heat transfer coefficients for the reactor sides; A~, the heat transfer area of a reactor side. The coil energy balances are:
UA [| Tva- Tp+2TX.] r| - FTM.p(Tp- TXp), I-
(A3)
d
F~M.p, thermal mass flow for pass p; Tp, the cracking temperature for pass or coil p; TXp, the crossover temperature for coil p; U, the heat transfer coefficients for the reactor coils; A, the heat transfer area of a reactor coil. The two heat transfer coefficients are related by: U= 1.154Us.
APPENDIX
(A2)
(A4)
(A1)
There are 11 equations in total and 36 measured variables. The heat transfer coefficients are not known so one of the equations must be used to calculate values for the heat transfer coefficients. The additional equations added by the analysis of the daily averages are:
FH,7, total flow of hydrocarbons to the reactor; FH. p, flow of hydrocarbons to coil p. Note that p = 1-8. In deriving the energy balances, it was assumed that the two sides are identical, and that all passes are identical and the two sides are are identical. In addition, it was assumed that the heat transfer coefficient for the side is related to the heat transfer coefficient for the passes. Effects such as the heat of reaction were not considered separately but
a i is an offset for coil i and b is the regression coefficient for the relationship and is common to all coils. TX and Fxu are average values for the reactor. The values of some model parameters are Table A1. The effective specific heats were determined by means of a rigorous cracking simulator (PHENICS), and the specific heat for the hydrocarbons includes the effect of the enthalpy change due to the cracking reactions.
PYROLYSIS REACTOR MODEL
The overall hydrocarbon balance is straightforward, 8
F n ' r = X FH'p' p = 1
TXp
-
T X = a~ + b ( F ~ . p - Fa~4),
(A5)