State profile estimation of an autothermal periodic fixed-bed reactor

Pergamon PII: S0009 - 2 5 0 9 ( 9 7 ) 0 0 0 9 0 - 0

Chemical Engineerin9 Science, Vol. 53, No. 1, pp. 47 58, 1998 ~() 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0009-.2509/98 $19.00 + 0.00

State profile estimation of an autothermal periodic fixed-bed reactor X. Hua,* M. Mangold,t A. Kienle and E. D. Gilles Institut fiir Systemdynamik und Regelungstechnik, Universit~it Stuttgart, Pfaffenwaldring 9, D-70550 Stuttgart, Germany (Received 20 November 1996; accepted in revised form 5 June 1997) Abstract--In this paper, the state profile estimation of a novel autothermal fixed-bed reactor (called circulation loop reactor, CLR), which can be operated as an autonomous periodic system, is studied. Since in autothermal operation the reactor shows sustained large oscillations, its dynamics have to be modelled as a strongly nonlinear distributed parameter system. The generally used lumping techniques (e.g. the orthogonal collocation method) for designing the state estimator of fixed-bed reactors are not applicable in this case. An approach for estimating the state profiles of the CLR based on a nonlinear distributed parameter observer is presented. To decrease the computational effort of the observer in real-time applications, the reduction of the detailed model of the CLR is investigated. In order to achieve asymptotic stability and fast convergence of the observer, the observability of the CLR is analysed, and use of features of the sustained oscillatory system and physical insight are made. A method based on exponential weighting functions is proposed to approximate the observation error profile. The temperature and concentration profiles are estimated from a small number of temperature measurements. The obtained observer was successfully tested with the dynamic simulator DIVA in the cases of errors in the initial condition and the model parameters, random noisy measurements, and unknown disturbances, also including the different choice of sensor locations, and different observer parameters. It shows excellent dynamic tracking of the reactor profiles, strong robustness, and easy tuning. It forms a basis for developing appropriate control strategies of the CLR. © 1997 Elsevier Science Ltd Keywords: Fixed-bed reactors; parameter estimation; distributed parameter system; nonlinear oscillations.

1. INTRODUCTION Autothermally operated fixed-bed reactors have been widely used in processing industries (Nieken et al., 1994). In particular, the development of new dynamic autothermal operation strategies has received increasing attention in chemical reaction engineering during the last years. Recently, a novel dynamic reactor concept--the so-called circulation loop reactor (CLR) (Lauschke and Gilles, 1994; Kienle et al., 1995)~has been proposed for autothermal reactor operation. In contrast to the well-known reverse flow reactor (Matros, 1989), which is operated as a periodically forced system, the CLR is intended to work as an autonomous periodic system. The CLR makes use of the phenomenon of self-exciting travelling reaction zones, and is a highly integrated process. Bifurcation

analysis, simulations and experimental studies in a laboratory-scale plant have shown that the autonomous periodic operation of the CLR allows total conversion for weakly exothermic and equilibrium limited reactions. A first industrial application example is concerned with autothermal combustion of volatile organic compounds in exhaust air streams of ambient temperature emitted by polymer plants. So far, very few investigations are concerned with the control of that new type of autothermal fixed-bed reactors (e.g. Hansen and Jergensen, 1995). In order to apply advanced concepts for the control of these processes, the reconstruction of the unknown temperature and concentration not only pointwise but as whole spatial profiles is often required. In industrial fixed-bed reactors, concentration measurements are often not available, and temperature measurements are taken only at a few locations. Therefore, this work will concentrate on the estimation of temperature and concentration profiles of the CLR from limited temperature measurements. The special utilization of nonlinear phenomena in the CLR introduces new methodological problems to

* On leave from the Institute of Automatic Control, East China University of Science and Technology, 200237 Shanghai, China. t Corresponding author. Tel: +49 711 685 6501; fax: +49 711 685 8371; e-mail: [email protected]. 47

48

X. Hua et al.

be solved. Because of the integration of the heat exchanger into the fixed-bed reactor, the CLR has some rather intricate characteristics: (i) it is strongly nonlinear; (ii) there are high-interactions between the distributed parameter systems of different bed sections; (iii) very steep temperature profiles result from large sustained nonharmonic oscillations over a relatively wide operation range for the considered cases. To our knowledge, no results have been reported in the open literature on the state estimation of such a complicated nonlinear distributed parameter system with sustained oscillations. Standard state estimation approaches for nonlinear distributed parameter fixedbed reactors based on linearization technique and lumping, e.g. via the orthogonal collocation method, were found to be not suitable. Further, the application of the extended Kalman filter to the full nonlinear distributed parameter model of fixed-bed reactors, will result in a large number of differential equations for filter algorithm (Windes et al., 1989), and thus, lead to a considerably large computation burden. Therefore, the design concept of Zeitz (1977) is used to design the nonlinear distributed parameter state observer for the CLR. The important advantages of this method are its feasibility for general nonlinear problems, the possibility to use physical considerations in the design process, as well as its availability for on-line implementation. The paper is organized in the following way: In a first step, a reduced distributed parameter model for describing the dynamic behaviour of the CLR is developed through some reasonable physical assumptions. Further reduction of the model and the main dynamic characteristics of the CLR are analysed. Using the reduced model, an approach for estimating temperature and concentration profiles of the CLR with a nonlinear distributed parameter observer from only limited temperature measurements is presented. Further, the observability of the CLR is discussed. The asymptotic stability of the observer in the neighbourhood of a periodic orbit is shown by approximating the observation error. Fast convergence of the observer can be achieved by tuning a parameter. In order to improve the approximation of the observation error profile in the design procedure, a method based on exponential weighting functions (Mangold et al., 1994) is proposed. Finally, efficiency and properties of the proposed observer are tested extensively with the dynamic simulator DIVA (Kroener et al., 1990). It is shown that the observer is robust against noisy measurements, model errors and initial observation errors.

O < .-~_ _ _

2

I

3

Fig. 1. Circulation loop reactor: (1) heating, (2) co-current heat exchanger,(3) reactor loop; parts 2 and 3 are filledwith catalyst. CLR inlet is only used for the start-up procedure. Figure 1 refers to a laboratory-scale reactor with a double tube as a heat exchanger. An industrial design would be of a more compact shape. In Kienle et al. (1995), a detailed one-dimensional model of the CLR in which ethene oxidation takes place has been given by making some assumptions, of which the most essential are (i) no temperature difference between catalyst particles and gas phase; (ii) negligible radial temperature gradient; (iii) constant heat capacities. This model comprises 12 partial differential equations and a set of appropriate boundary conditions, and can describe the steady state and dynamic behaviour of the CLR. It shows a good qualitative agreement with the real process. The operation conditions and physical parameter values in the model of the reactor are given in Kienle et al. (1995). However, for the purpose of developing appropriate state estimation algorithms and control strategies of the reactor it is necessary to reduce this detailed model. Here some further reasonable assumptions are made: (i) the model of the inlet heating is not included; (ii) heat capacities of inner and outer walls can be neglected; (iii) material transport is in quasi-steady state, and thus the resulting model equations read as: Balance equations for the heat-exchanger inner tube (0 < z < I1):

~Tr

~TI

(pc~)b - T f - = - ~v~(pc~)~ ~z 2. M O D E L

OF THE

CLR

The CLR shown in Fig. 1 consists of two main parts: a catalytic fixed-bed co-current heat exchanger serving as a heat feed-back unit and a tubular catalytic fixed-bed loop reactor connecting the inner and the outer tube of the heat exhanger for the evolution of the travelling reaction zones. The heating at the

, 632T1 + Aeff.l"~'-Z2 + 20~h'w(Ts -- T1) ri, 1

+ ( - A h R ) r l ( x t , T~) c3xt c~2xl 0 = - v~ ~ + D~,I ~3z2

Mg rx(xl, TI). ~Pg

(1) (2)

49

State profile estimation Balance equations for the reactor loop (It < z < lL):

~TL ~vg(p%)° &

(pcp)b O_~L _

0 2 TL + 2.mL-'-~Z z + ~

M° rL(XL, TL). ePo

(4)

Balance equations for the heat-exchanger jacket

fiT,

~ Ts 8 2 Tj ~vo(pep) o ~ + 2e,,j &2

20~h,vro, 2 2~h wro 1 -~ .5---T2- (Tv - Ts) + r2~.2"" ~ ] ( T , -- Tj) ri,2 -- ro, 1

+ ( -- A h R ) r j ( x j ,

", --

,

T j)

(5)

c~xs c32xs 0 = - vo ~ z + D,x,z OZ2

Mg rj(xj, Tj). epo

(6)

The coupled boundary conditions are as follows: At the reactor inlet:

•eff,I ~-Z OTI o,,

"Jr-ev,(pcp)o(Ti. -- Tt(0, t)) = 0

Dox,, ~~xx o.t + ~V~(X~n- -

X , ( 0 , t)) = 0

(7)

(8)

r2 1 ,~eff,, ~63TI - Z I,,t = r21}~eff, L ~O- -TzL O,t

(9)

o .~,L--~Z < 1 [o,,

(10)

_

&,l

,

TL(0, t) = T1(ll, t)

(11)

xL(0, t) = x1(ll, t)

(12)

At the reactor loop outlet and the heat-exchanger jacket inlet: , 2

6qTL z~,t

2

(pe,)b = e(p%) o + (1 -- e)(pCp)s.

(21)

c~Tj

r ~ °"'~-~z

= (r~: - ro.~),~eff.~--~Z

2 ~ OXL IL, t ri'll")ax'L-ffZ-2

= ( r 2 2 - r ° ' 2x ) O " x " ~~XJ[ z o,, (14)

0,t

The definitions of the symbols in the above equations are listed in the Notation section. The values of the model parameters are the same as those in Kienle et al. (1995), and are given in the Appendix. Compared with the simulation results in Kienle et al. (1995) we find that the dynamic behaviour of the above reduced model is very close to that of the detailed model. A typical simulation result for one operation period (approximately 1 h for this reactor) of the reduced model, when the CLR is in autothermal operation, is depicted in Fig. 2, which shows spatial temperature profiles and the corresponding concentration profiles. This figure shows that the CLR has very steep spatial profiles. In particular, the temperature profiles in the loop reactor show a very large temperature rise (larger than 500 K). 3. DESIGN OF A NONLINEAR DISTRIBUTED PARAMETER OBSERVER

At the heat-exchanger inner tube outlet and the reactor loop inlet:

r ~ , , O ~ j ~ T l t , , , = r2,

(19)

(3)

(1L < Z < l j):

(pc~)b

Vg = l?o/eA

(Tv - TL)

+ ( - AhR)rL(XL, TL) ~XL ~2XL 0 = -- vo -~z + D~., L (~Z2

where

(13)

From the theoretical point of view, a real-time simulation based on the above model can be used for the on-line estimation of the unmeasured state profiles of the CLR, but this simulation is unable to compensate for errors of the initial condition, unknown disturbances, model parameter uncertainties, and measurement errors. Therefore, it is necessary to design a state estimator based on the above model, the available reactor inlet conditions and the temperature measurements Y(t) = [yT(t), yL(t), T y~(t)] T

(22)

where yi(t) = [Ti(zl, t) . . . . . Ti(z,n, t)] T, (i = I, L, J), the superscript T denotes the transpose of a vector, m is the number of measurement points in every section of the CLR. Because of the features of the CLR, a nonlinear distributed parameter observer is considered here. In the design procedure, the concept of Zeitz (1977) will be adopted.

3.1. Structure of the observer

Tj(O, t) = TL(IL, t)

(15)

xj(O, t) = XL(IL, t)

(16)

At the reactor outlet:

c3Ts l,., Oz = 0

(17)

&~

(18)

[~Z ]j,t

= 0

According to the general theory of nonlinear observers, a nonlinear distributed parameter observer can be a straightforward extension of the Luenberger observer. The observer is derived from the model equation by adding a suitable injection term, which consists of the weighting errors between the measured states and the observed ones at the sensor locations. In our reactor, since no concentration measurement is available, and the material balance is in quasi-steady state, only the energy balance equations are corrected and form the main body of the observer. The material balance equations are corrected indirectly, by the

50

X. Hua et al. 1000

800

g N

600

I400

200

~ 0.5

0

~ 1

1.5

z [m] 10 "3

N"

-1

i 0.5

0

t

t

1

1.5

z [ml Fig. 2. Temperature and concentration profiles of the CLR in autothermal operation, T~, = 295 K, xi. = 0.004, 120= 10 m3/h.

coupling to the energy balances. Thus, the observer equations read as follows:

20:h, wro, 1

^

+ (-- AhR)rs(£s, T j) + K~(yj - Ys) (pCp)b ~

= -- gl)o(pCp) 0 ~

-~- /~¢ff, l aZ2

(27)

+ 2ah, w ( ~ j _ T l ) + ( -- AhR)r1(2~, TI) ri, 1

T

+ K, (y, - ~,) ~2t

022I

0 = -- vg ~c3 + D,:,,~ c~z2

OTL (PCP)b ~ -

0TL

(23) M° ri(2i, ~Pl) (24) epg

#zT~L

cxj 02xj 0 = - v o 6-~ +Dax,J Oz 2

The form of the boundary conditions of the observer is the same as in eqs (7)-(18). Because of the linear boundary conditions, no error injection terms are added to these equations (Zeitz, 1977). It is assumed that the initial state profiles ~(z, 0) = 7~o,,(z)

= -- F~uo(PCP)g~-Z "}- /~eff,L ~22

:~i(z, O) = 2o, i(z), +

20~h,U

( T v -- TL) + (--AhR)rL(2L, TL)

Fi, 1

+ KT(yL -- YL) ~FcL 32XL 0 : -- t)g ~ ~- Oax ' L (~Z2

(25) M ° rc(~L, TL) (26) ePo

OTJ OTj d2~s (pCp)b ~ = -- ~'UO(PCp)O~Z'z "~ Aeff.J OZ2 2~h,vro, 2 ( T v -- T j )

+ r 2 2 - to,2 1

M 9 rs(fcs, 7Fj). (28) ePo

i = I, L, J,

are adequately determined. In the above observer equations, the feed temperature T~. and the feed rate I?0 are assumed to be known. The weighting factors or gains K T (i = I, L, J) have to be determined in the design process of the observer. In general, they are nonlinear functions of time and space as well as of input and state variables. If the observability of the system is given, and the gains are properly designed, the observed states converge to the true states of the CLR. Hence, an important task is to investigate the observability of the CLR for selecting sensor locations, and to determine suitable gains.

State profile estimation 3.2. Observability and choice o f sensor locations Observability is a structural property of a system requiring all states to be reflected in the measurements differently. The choice of the sensor locations can have a great influence upon the performance of the nonlinear distributed parameter observer of fixed-bed reactors. In ordinary fixed-bed reactors in steadystate operation, the reaction zone extends usually only over small ranges of the reactor tube length. Obviously, only temperature measurements within the zone can guarantee observability of the temperature profile. In order to obtain observability of the reactor, at least one measurement point has to lie within the zone. Because the concentration profile is coupled strongly to the temperature profile, it seems plausible that, if the temperature profile is observable, the concentration profile is also observable. The principle of the CLR is that a travelling reaction zone ignites a new one by using a fixed-bed heat-exchanger as a thermal feed back. This results in a 'circulating' reaction zone shown in Fig. 2. In contrast to ordinary fixed-bed reactors, the travelling reaction zones of the CLR pass every temperature sensor at any location during one period. Therefore, the CLR is observable over the whole length of the reactor. The choice of sensor locations can be somewhat arbitrary, and temperature measurements at the boundaries of the reactor sections alone can guarantee its observability. In addition, it should be pointed out that the temperature measurements of the CLR nicely reflect the internal nonlinearities since there are sustained strong oscillations in the reactor. This is also very helpful for state observation of the CLR. 3.3. D e t e r m i n a t i o n o f the gain The determination of the gain in the observer eqs (23)-(28) should meet some important requirements on an observer such as asymptotic stability and fast convergence to the real process. Due to the existence of sustained oscillations in the CLR, the method presented by the stability theory for nonlinear equilibrium point systems is not suitable for the observer design of the reactor. Hence, the design method proposed here is based on physical insight to the observer model (Zeitz, 1977). Because of errors in the initial conditions, unmeasured disturbances and model errors, usually there is a difference between the observed states and the true states of the process. A possible approach to correct the observer profiles is to interpret the correction term as a heat transfer between process and observer: The observer profiles will converge to the profiles of the real process, if the vector of weighting functions K~X(z,t) is chosen in such a way

51

that KT(z, t)(yi(t) - ~i(t)) ~- 8i(Ti(z, t) - Ti(z, t)) (29) where i = I, L, J, and 81 are adjustable heat transfer coefficients, which determine the heat transfer velocity of the observer, i.e. its convergence. When there is a temperature difference between the process and the observer, the fictitious heat transfer term causes an increase or decrease of the temperature 7~i(z,t) (i = 1, L, J) of the observer until Ti(z, t ) - , Ti(z, t). Thus, if the error profile is known, eq. (29) can be used for determining the gain. The above physical analysis can be mathematically justified by studying approximate observation errors in the neighbourhood of a periodic orbit. By defining the observation errors ei(z, t) = Ti(z, t) - Ti(z, t) el(Z, t) = Ri(z, t) -- xi(z, t),

i = l, L, J

by subtracting eqs (23)-(28) from eqs (1)-(6), respectively, and by a linearization near an oscillatory state, we obtain a set of partial differential equations and corresponding boundary conditions for the observation errors. Since the reaction rate (20) and its derivatives with respect to the states are positive, the consideration of the error dynamics of ¢g(z, t) can be neglected (Zeitz, 1977). Thus, through some algebraic manipulations, the following error equations are obtained: 0ei

0et

-c~t - ~

02et

- al, 1 "~Z + al.2 -Oz - 2 - al,3el + al,4ej T

+ 9i(~t, T~)er + Ke.t(yt - $'i) C~eL O~f ~

(30)

OeL 02eL - a2' 1 ~ Z ~- a2` 2 ~Z 2 -- a2, 3eL _}_ gL(~L ' ~.L)eL _j_ Ke, T L(YL -- YL)

Oej

Oej

(31)

~2ej

~ - ~ -- a3,1 ~-z + a3'2 --'-T Oz -- a3,3ej + a3.4ex T

+ gs(2J, T j ) e s + Ke.j(yj - ~'j).

(32)

The corresponding b o u n d a r y conditions are linear homogeneous equations for the observation errors. Here, aj, k (j = 1 . . . . . 3; k = 1 . . . . . 4) and K~,~ (i = I, L, J) are coefficients which are easy to get from eqs (1)-(6), and ( - AhR) ori ;,, t,"

(33)

g ' - (pc.)~ ~T,

The above equations can be compactly written as (34)

~ A e + Ge + K~e r

where 0 -- aLx ~z +

A =

02 a L 2 ~0z

0

a3,4

--

al'40

0

al'3 -

a2'l ~2 +

62 a2'2 ~0z 0

-

a2'3

0

02

-- a3,1-~~z q- a3,2 ~z2 -- a3,3

/

X. Hua et al.

52 and G = diag[gt, gL, gs](3 ×3), •

T

T

K~ = dlag[K~,1, K e . L ,

T

Ke,2](3

× 3n),

e T -~ [el, eL, e j ] ( 1 × 3), ef = [(yl

-- ~I)T,

(YL -

~rL)T,

(YJ -

~rs)T](1 x 3n).

The injection term can be defined as

(35)

Kee r = - (~eI + G)e

precisely. In (Zeitz, 1977), a polygonal approximation of e~(z, t) based on the measurements and the houndary conditions information at each sampling time is proposed. However, this method requires a high number of sensor locations in the case of the CLR. Here, according to the investigation in (Mangold et al., 1994), a method based on exponential weighting functions is proposed to improve the approximation of the error profile. The error profile is approximated by the observation errors at the sensor locations zk and weighting functions wk(z), k = 1 . . . . . m, i.e.

where ~e = a/(p%)b, or K/T(yi -- )'i) = -- (~ + (pCp)bgi)ei(z, t),

Ti(z, t) - Ti(z, t) ~ ~ (Ti(Zk, t) -- Ti(Zk, t))Wk(Z),

i=I,L,J

k=l

(36) where e is a positive tuning parameter. Then the observation errors will be described by a set of linear homogeneous partial differential equations and boundary conditions. When the parameter e is properly selected, their solution will asymptotically tend to zero for arbitrary initial conditions in the considered cases, i.e. the asymptotic convergence of the observer near a periodic orbit is achieved. The rate of convergence is also determined by this parameter. Thus, the objective of the observer design is to determine the tuning parameter e such that the observation error asymptotically and rapidly tends to zero for all relevant initial errors. It should be noted that for simplicity the same tuning parameter ~ is used in eq. (35) or eq. (36) for three different sections of the reactor. It is also possible to use different tuning parameters for different sections, but this is not necessary as shown by the subsequent results•

i = I, L, J.

(37)

where similar to the case of the tuning parameter ~, the same weighting functions WR(Z) are used for all three sections of the reactor. Then the key problem becomes the choice of the weighting functions Wk(Z), which is in general difficult. In (Mangold et al., 1994), some heuristic considerations and simulation studies for an adsorption column are made using different weighting functions between sensor locations Zk-1 and zk, e.g. linear interpolation. This concept is applicable to the CLR. The state profiles can be well corrected at the sensor locations, because the observation errors at these locations are exactly known. The observation errors are small immediately behind a sensor location; they increase with inreasing distance from the next sensor location in flow direction. Therefore, an exponential weighting function seems to be suitable, e.g. exp We(Z) =

-

Zk-I
0

otherwise

4. IMPLEMENTATION OF THE OBSERVER

Once a nonlinear distributed parameter observer has been designed, its algorithmic implementation is a standard problem. There exist some available techniques to solve nonlinear partial differential equations, as well as commercial software for general or special application purposes. In this case, the method of lines with finite-difference approximations is used to transform the partial differential equations of the observer into a DAE system, and then a sparse version of the extrapolation method LIMEX (Deuflhard et al., 1987) is adopted to integrate the resulting equations. On a modern computer, the observer of the CLR based on the reduced model is calculated 10-100 times faster than real time, since the CLR is a periodic process with a period of approximately 1 h. Therefore, the observer is able to track the transients of the real process. Moreover, the computation time may be decreased by a coarser spatial grid, or further reduction of the reactor model. However, from practical point of view, some problems need further investigations. 4.1• Approximation of the observation error profile In eq. (36), the errors el(z, t); i = I, L, J must be known. Obviously, it is impossible to obtain them

(38) where/3 is a positive adjustable parameter. With this approximation, although the observer has one parameter more than the above design, the tuning of the observer becomes more flexible, and an improvement of the observer performance (e.g. observation accuracy) is achieved. 4.2. Approximation of the 9ain For a linear distributed parameter system, the gain of its observer is only a function of the spatial variable z. In our case, the gain of the obtained nonlinear distributed parameter observer is nonlinear due to gi(~, T). Hence, for more convenient on-line implementation of the observer, it is interesting to investigate the possibility of approximating the gain. Some simulation studies show that gi(~, T) are zero apart from some narrow peaks. Thus, the gi only have a weak influence on the asymptotic stability of the observer, and are neglected in this case, i.e. K~(yi - Yi) = - ctei(z, t),

i = I, L, J.

(39)

Therefore, the design of the observer concerns only and Wk(Z), and the gain of the nonlinear distributed

State profile estimation parameter observer of the CLR becomes similar to that of a linear distributed observer. 4.3. Tuning of the design parameters Using eqs (36)-(38), the observer contains two design parameters e and ft. In the tuning procedure, it can be helpful to remember the physical meaning of the parameter. According to the definition in eq. (36), the observer model describes a heat conduction problem. The parameter c~ can be interpreted as a heat transfer coefficient, and determines the convergence of the observer. In this case, it is found that a suitable range of the ~ is from 1000 to 5000 W/m2K. The proper choice of the fl is helpful for improving the approximation accuracy of the observation error profile. Here, the B ranges from 0.1 to 0.8. Due to the features of the nonlinear oscillatory process, the influence of the parameter tuning in the above ranges on the observer performance is not very sensitive. 5. SIMULATIONRESULTS In order to test the proposed observer, the CLR and the observer are implemented in the dynamic flowsheet simulator DIVA (Kroener et al., 1990). The simulation results of the reactor based on the model (1)-(6) are taken as measurement information, i.e. they serve as the 'real system'. The simulated measurements are fed to the observer. The estimated state profiles can be compared to the simulated 'true' ones of the process. Thus, the estimation quality can be easily judged. Except the other capabilities of such

53

a dynamic flowsheet simulator, DIVA has some important features: e.g. in DIVA the topological structure of the flowsheet is altered by changing the coupling information between unit blocks. Therefore, the influence of different conditions on the performance of the observer, e.g. changing sensor locations, adding random noise to the 'measurement', can be systematically investigated without additional programming. A large number of simulation experiments have been performed to study the characteristics of the obtained observer such as stability, convergence, accuracy and robustness. The following aspects have been investigated: (1) Influence of initial state profiles; (2) Influence of random measurement noise n(t); (3) Influence of errors in the model parameters, e.g. ~h,w, ko; (4) Effect of unknown disturbances, e.g. xi,; (5) Tuning of the observer design parameters c~ and/~; (6) Approximated gain of the observer, eq. (39); (7) Number and location of sensors. Some simulation results are given in the following. In order to design an observer of the fixed-bed reactor, it is important to choose a reasonable starting profile for the temperature and concentration. In the first test case, the influence of different initial deviations between the state profile of the observer and that of the 'real' process on the performance of the observer is tested. Figure 3 shows typical simulation

1000 t=1800 s

t=O s

8O0

600 I40(

20~ 0

0.5

1 z

1.5

[m]

x 10 -3

6 5 \

4

"if2 1

0

0

0.5

1.5 z

[m]

Fig. 3. Dynamic temperature and concentration profiles of the CLR ( - -

) real, ( - - - ) estimate.

X. Hua et al.

54

results. In this example, the CLR is in autothermal operation, and its inlet conditions are known, i.e. T~, = 295 K, xi, = 0.004. It is assumed that the initial state profile of the real process is the periodic solution in this autothermal operation, whereas the initial state profile of the observer is the periodic solution for ~i~in = 295 K, 2i, = 0.0045. The input variables of the observer are the same as the inlet conditions of the reactor, and temperature measurements are available from two sensors in each section of the CLR, located at nearly equidistance. The parameters of the observer are chosen as c~ = 1500 and/~ = 0.6. A step change of the inlet temperature Tin from 295 to 350 K is exerted. Figure 3 illustrates that with such a large error in the initial state profile, the estimates of the observer can rapidly converge to the real states of the reactor, and the amplitude and wave form of the oscillation can be excellently matched. Figure 4 gives simulation results of another example, in which the C L R is in an autothermal operation with Ti, = 295 K, xin = 0.005. It is assumed that the initial state profile of the real process is the periodic solution for this operation conditions, whereas the initial state profile of the observer is simply guessed as 7~o,i(z)= 295K and 2o.i(z) = 0.004 (i = I, L, J). Temperature measurements and tuning parameters are the same as in the above example. F r o m Fig. 4, it is found that even in such a poor initial state profile guess, the observer is able to track the sustained oscillations of the real process, while the convergence speed of the estimates is a little slower. However, larger values of the tuning

1000

.............................

parameters can accelerate the convergence speed of the observer. In order to test the applicability to industrial processes, it is necessary to investigate the observer robustness against noisy measurements. In test case 2, a white noise n(t) is added to the temperature sensor signals from a Gaussian random number generator with zero mean and a standard deviation, i.e.

yi(t) = y,(t) + n(t),

i = I, L, J.

(40)

Figure 5 shows the temporal evolution of the real and estimated temperatures at the reactor outlet when the reactor is in the same operation as in Fig. 3, and when the same design conditions of the obserever are used. In contrast to Fig. 3, the temperature measurements are disturbed by a band limited (time constant 100 s) Gaussian random noise with a relatively large standard deviation of 60 K. This figure illustrates that the observer can give satisfactory estimates despite large random noisy measurements. In practice, some characteristics of the fixed-bed reactor may change with time, because of e.g. the decay of the catalyst activity and the decrease of the heat-transfer coefficient in the heat exchanger due to fouling. Therefore, in test case 3, the influence of plant-model/observer-model mismatches on the observer performance is investigated. Figure 6 shows the simulation experiment result when the reactor is in the same autothermal operation, and using the same design conditions of the observer as in Fig. 3 but

t=bs ....... t=3600s ..............

t : : i 80iJS . . . . . . . . . . . . . . . . . . . . . . .

800

N I--

600

400

_

i 0.5

200

0

1

1.5

z [m] x 10 -3 6,

- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

×

3S • -I ' t = 3 6 0 0 S ] ........ l. . . . . . . . . . . . . . . . . . . . . .

2

..........

0

t ...............

0.5

't=1800S i ~ ............................

i

i

!

i

..........................

1

1.5

z [m] Fig. 4. Dynamic temperature and concentration profiles of the CLR, - -

) real, ( - -

) estimate•

State profile estimation

55

800 ~- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

600

..................

i ....................

-~ 500

...........................

...........

f ....... 200[ 0

.

.

.

.

.

.

tJ

.

.

.

.

.

.

.

.

.

.

.

.

.

" i 5000

.

.

.

.

.

.

.

u

.

.

.

.

.

.

.

.

.

.

.

:~ 10000

15000

time [s] 800 700

......

• -

~" 600

i ................. I

I

~ ...............

i .......................... I

? I

i........................

:

............................ :

~! . . . . . . . . . . . . . . . . . . . . . . . .

: :

i

~_1500 4OO 30£

I

0

5000

10000

15000

time [s] Fig. 5. Influence of measurements with random noise n(t) on observer performance, left: noisy measurements; right ( ) simulated without noise, ( - - - ) estimate.

Ti. keeps constant i.e. 295 K, and the reaction constant ko in the reactor model is diminished by 10%. This figure shows that the observer has strong tracking ability to the real process when its model has a biased error. In the above test cases, it has been assumed that the reactor inlet concentration xin is known. But in industrial processes it may not be true, since e.g. diluted volatile organic compounds in exhaust air are usually not measured. In test case 4, the influence of erroneous inlet concentration 2i, on the observer performance is studied. Some simulation experiment results when the reactor is in the same autothermal operation (i.e. x~, = 0.004), and using the same design conditions of the observer as in Fig. 3 but 2i. = 0.005 is given in Fig. 7. Note that the temperature estimates converge quickly to the real states, while the concentration estimates are less accurate in the oscillation amplitude despite the correct temperature estimates. Moreover, the large the bias value of 2i, from the true condition is, the large is the estimate error for the concentration. The concentration measurement would help to improve the quality of the concentration estimates, but in our case it is assumed to be not available. A method for the estimation of this unknown disturbance will be suggested elsewhere (Hua et al., 1996). In the simulation results given above, the same tuning parameters c~ = 1500 and fl = 0.6 of the observer have been adopted. In fact, as mentioned above, different tuning of the parameters in a range of

values is available, and robust to the convergence of the observer. But it may have some influences on the convergence speed of the estimate. In test case 5, some investigations in the case of different tuning parameter values have been made. For example, using the tuning parameters, ~ = 1000 and fl = 0.2, the estimates of the observer have slower convergence speed, but are still considerable satisfactory. In order to test the influence of the observer gain proposed above, in test case 6 the simulation experiments were performed with an approximated gain according to eq. (39). It showed that with the approximated gain the observer can also excellently follow the dynamic behaviour of the reactor, apart from larger deviations at some positions, e.g. the reactor loop outlet, only in a short time span at the beginning. Finally, the influence of the number and location of temperature sensors on the performance of the observer was studied in test case 7. Through some investigations, it is found that the location of the temperature sensor within the section of the reactor has no strong influence. Moreover, the tracking of the estimated states to the real ones only based on three boundary temperature measurements, i.e. the heatexchanger inner tube outlet, the reactor loop outlet and the C L R outlet, is shown to be still rather good, while the convergence speed becomes slower. The simulation results with such three temperature sensors but when other conditions are the same as in Fig. 3 is given in Fig. 8.

X. H u a et al.

56 800

.......................................................................

700 600 O

H j 500

..... I .................. I/,' 1',

i ............................ i

i ....................... ~

"

! . . . . . . . . . . . . . . . . . . . . .

! ..........................

L. . . . . . . . . . . . . . . . .

i i i

400 300 0

5000

10000

15000

time [s] x 10 -3 2 ................................................................................................... h

1.5 . . . . . . . . . . . . . . . . . . . . . . • , 0

×

I

I./

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

!~ '

1 .............................................................................. i ! il I

I

'tl

.........

oi0 ];"

5000

10000

15000

t i m e [s] Fig. 6. O b s e r v e r p e r f o r m a n c e fter 1 0 % d e c r e a s e of k0 in r e a c t o r m o d e l , (

) real, ( - - -

estimate.

800

700

.....

71 .................

;~ ......

;

.........

I

*~" 6 0 0

i

0

,t ................

I

~.

~

I

:

. . . . . . .

.I

. . . . . . . . . . .

I

................

i ..... !.........................

i! .............

i ..............

!

I

I

I

I

:

i

I

~.t 5 0 0 400 300 0

5000

10000

15000

time [s] X 10 -3 3 ............................................................................................ 2.5 . . . . . . . . . . . . . . . . . . . . 2

: .........................................................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

° t 1.5 X

0

5000

10000

15000

SIM_TIME Fig. 7. I n f l u e n c e of e r r o n e o u s inlet c o n c e n t r a t i o n 21, o n o b s e r v e r p e r f o r m a n c e , ( estimate.

) real, ( - - - )

State profile estimation 900 ............

800

:............ :

............

t - 3 6 o o s !

700............

i.'.~. /

=600

i

............

: ........... : ............ : q~rl~OOs .........

t!.

,

57

:. . . . . . . . . . . . f - -t=O

i!..

: .....................................

......

...........

...... :- ........ '.. li .... "~\!::

i

.

........ .

:............

~.i. ?.,\ ...... ::............ . . .

. . . . . . . . . . .

i

:. . . . . . . . . . . .

:. ...........

i

!

500

i

400

i : .....

"

301~

1

0

-

0.2

F-

-

j

0.4

~,

.~! .........

I

0.6

0.8

~ 7

~

r

i

1

1.2

~-

-

" .....

-

1.4

:

t

1.6

z [m] 5

X 10 -3 ...........

. ........................

~............

4

:

3

....................

. ..........

i. . . . . . .

!i!It

! t=3600!s

: i l l

it=1800s

0

0.2

0.4

.

.

.

.

.

.

•. . . . . . . . . . . .

li 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I::t=0s f

I

i

0

~ .

t I

:

. . . . . . . .

i..................................................

~

0.6

0.8

I

1.2

1.4

1.6

z [m] Fig. 8. Dynamic temperature and concentration profiles with three temperature sensors at z = 0.43, 1.08, 1.51, ( - - ) real, ( ) estimate. 6. CONCLUSIONS The state profile estimation of a novel autothermal fixed-bed reactor (CLR) has been studied in this paper. In contrast to reverse flow fixed-bed reactors, this reactor can be operated as an autonomous periodic system. In autothermal operation this reactor shows sustained large oscillations which result in very steep temperature profiles, and thus its dynamics have to be modelled as a highly nonlinear distributed parameter system. The linearization is not adequate in this case because of highly nonlinear oscillations of the reactor. The c o m m o n design techniques based on lumping (i.e. spatial discretization), e.g. via the orthogonal collocation method, for nonlinear distributed parameter fixed-bed reactors are very difficult to use for the design of a state estimator in this case. This work presents an approach for the estimation of state profiles of the CLR based on a nonlinear distributed parameter observer. The reduction of the detailed model of the CLR forms a basis for designing the observer. The analysis of the observability of the CLR is very helpful, and it shows that the CLR has 'strong' observability, and exit temperature measurements of each reactor section would give sufficient information. The design method associated with physical consideration of the process is successfully used for the design of the CLR observer. The approximation of the observation error profile based on exponential weighting functions can be helpful for the design of the observer. The tuning of the observer with two

parameters is relatively easy and robust. Excellent tracking of temperature and concentration profiles can be achieved from only a limited n u m b e r of temperature measurements in different cases, e.g. errors in initial state profiles, in model parameters, and random noisy measurements. When the information about the inlet concentration of the reactor cannot properly be obtained, the observer can still give rather satisfactory estimates of the temperature profile, but the estimation of the concentration profile may be not sufficiently good. Hence, it is necessary to estimate the u n k n o w n disturbance for improving the concentration estimation. Finally, the proposed observer is a necessary prerequisite for further work on developing the suitable control strategy for the desired periodic operation of the CLR (Hua et al., 1996).

Acknowledgements The first author is indebted to the National Science Foundation of China and the Chinese Education Commission for partial support of this research. This work is financially supported by the Bundesministerium fiir Bildung und Forschung. Special thanks are given to Professor M. Zeitz for his helpful suggestions and comments. NOTATION

a A A

elements of A cross-section area, m 2 matrix of eq. (34)

58

cp Dax e e EA g G AhR ko K K~

K~ l m M n

r ri ro

R t

T

¢ w x 2 Y Z

X. Hua et al. specific heat capacity, J/kg K dispersion coefficient, m2/s observation error of temperature, K error vector of eq. (34) activation energy, J/mol nonlinear functions eq. (33) matrix of eq. (34) reaction enthalpy, J/mol reaction constant, mol/m3 s injection coefficient in observer matrix of eq. (34) elements of Ke distance of each section outlet, m n u m b e r of temperature sensors molar mass, kg/mol r a n d o m noise reaction rate, mol/s radius of inner tube, m radius of outer tube, m gas constant, J/mol K time, s temperature, K observed temperature, K velocity, m/s volume flow, m3/h weighting functions molar concentration observed molar concentration temperature measurement, K axial coordinate, m

Greek letter o:h

E 0 2 P

tuning coefficient of the observer heat transfer coefficient, W / m 2 K coefficient of weighting function observation error of concentration void fraction fabricated heat transfer coefficient heat conductivity, W / m K density, kg/m 3

Subscripts 1

2 b eft g I in J L out s U

inner wall outer wall of heat exchanger bed effective gas inner tube of heat exchanger reactor inlet jacket of heat exchanger reactor reactor loop outlet solid ambient

w y

reactor wall measurement

REFERENCES

Deuflhard, P., Hairer, E. and Zugck, J. (1987) One-step and extrapolation methods for differential-algebraic systems Numer. Math. 51,501-516. Hansen, J. E. and Jorgensen, S. B. (1995) Control of forced cyclic process. Proceeding of IFAC Symposium on DYCORD + '95, Denmark, pp. 21-26. Hua, X., Mangold, M., Kienle, A. and Gilles, E. D. (1996) Inferential control of an autonomous periodic fixed-bed reactor. Submitted to J. Proc. Control. Kienle, A., Lauschke, G., Gehrke, V. and Gilles, E. D. (1995) On the dynamics of the circulation loop reactor--numerical methods and analysis. Chem. Engng Sci. 50, 2361 2375. Kroener, A., Holl, P., Marquardt, W. and Gilles, E. D. (1990) DIVA - - and open architecture for dynamic simulation. Comput. Chem. Enyng 14, 1289-1295. Lauschke, G. and Gilles, E. D. (1994) Circulating reaction zones in a packed-bed loop reactor. Chem. Engng Sci. 49, 5359-5375. Mangold, M., Lauschke, G., Schaffner, J., Zeitz, M. and Gilles, E. D. (1994) State and parameter estimation for adsorption columns by nonlinear distributed parameter state observers. J. Proc. Control 4, 163-172. Matros, Y. S. (1989) Catalytic Process under Unsteady-State Conditions. Elsevier, Amsterdam. Nieken, U., Kolios, G. and Eigenberger, G. (1994) Control of the ignited steady state in autothermal fixed-bed reactors for catalytic combustion. Chem. Enyng Sci. 49, 5507 5518. Windes, L. C., Cinar, A. and Ray, W. H. (1989) Dynamic estimation of temperature and concentration profiles in a packed bed reactor. Chem. Engng Sci. 44, 2087-2106. Zeitz, M. (1977) Nichtlineare Beobachter fur chemische Reaktoren. VDI-Fortschr.-Ber., Reihe 8, No. 27, VDI-Verlag, Dtisseldorf. APPENDIX: MODEL

PARAMETER

=0.5 (pcp)b = 7.98541 × 105J/m 3 K Da~ = 10 2 v~mZ/s Mg = 2.86 x 10 -2 kg/mol pg = 0.823 kg/m 3 c,.~ = 1017.0 J/kg K "~eff, l, )*eff,L : 0.85600 W/m K 2eff.S = 0.73912 W/m K ~h,w = 100.0 W/mZK k0 = 2.75 x 101° mol/m 3 s Ea = 9.915 × 104 J/mol R = 8.314 J/mol K (--Aha) = 1.3 x 106 J/tool ri, 1 = 42.5 x 10

3m

ro. 1 = 44.5 × 10 3m ri, z = 62.5 × 10 -3 m r0,2 = 66.5x 1 0 - 3 m It = 0.43 m IL = 1.08 m Ij = 1.51 m Tv = 295.0 K l?q = 10.0 m3/h

VALUES