Model Reduction for Nonlinear Systems with Distributed Parameters

Copnight © IF,'\C COlltrol of Distillatioll Columns and Chemical Reactors. Bournemouth. L'l\. l~IHtl

MODEL REDUCTION FOR NONLINEAR SYSTEMS WITH DISTRIBUTED PARAMETERS U. Epple* Imtitut ji"ir SystmlllYlI{lIl1ik lllld RegelulIgstecllllik, L'lIi,'ersitiit Stuttgart, Stuttgart, FRG

R'O'duced J"odel'" ('f nC'n\inear distributed £),stenls which Are ec(~urate wice range of '-'I',8Liltirlg conditions, are diHi':~IH, te obtain, This pape~ . 'r:'1I~ '.:ith 11 method which u~,(:s t.he detailed -jiEerent-.i,91 '::lI\1'lnce eC'juJtions 'Cl 1\ st-,<'["_'n'} point B.nd result3 finally in a low ord,,:, lumped parametet' .-.C":lel, T'1e ;r,ain idea is to pr-oUt by a selforgsflization process, which "rysi cl111), corfines the varieLy c-f the possible svstembehaviour consid:?rably, !'le r-eo'.'lt are structured s::.l'.lt.ions with a characteristic form and dynamic. 8:1e of ~he widespread and most simple types is the movir.g wa""! [ront solution, It is surprising, that such a moving wave front is a ch~::-ac~eristic structure not- only for one, but for many totally different systems. starting with a short phenomenological view of some examples of moving wave front solutions the mathematical procedure to carry out the reduction is described in detail. In a very special case it is possible to show the physical meaning of the used assumptions by an analytical investigation. This property is very helpful for the clarity of the method and the resulting reduced model. It is well known, that the reactionzone in a fixed-bed reactor can be interpreted as a moving wave front solution. As an application example for the reduction technique a complete reduced model of a catalytic fixed-bed reactor is implemented. ~.QstracL:

::or

11

Keywords: Modelling; model ['eduction; fixed-bed reactor; parameter systems; nonlinear systems; waveform analysis.

I NTRODUCTI ON The mathematical modelling of a physical system often results in nonlineor partial differential equations, Due to the infinite dimension of a PDE, a wide variety of solutiontypes would be expected. But, from an empiric point of view, it is well known, that in r.1any cases the behaviour of such a complex system can be characterized by only a few parameters, The sta~e of a fixed-bed reactor for example is mainly determined by the height and the location of the temperature maximum. The reason of this restriction of the system behaviour is a selforganization ir the system. The £elforganization leads te the fonnation of a dynamic asymptotic solution, This me[\w;, that the systembehaviour splits into two parts, a fast epP['oach from iniLia1, or disturbed cc.nditions to the Ilsymptotic :~oltJtion and a slow motion accor-ding to ~·.he slow ~~Flamic dwnges of the asymptotic solution. The prior condition for the existence oE su~h Cl qp:ittirlY is a noticeable 'lorlinear ter!n and an "f't-,~r-ral feedood: mechanism In the system. The decree :,f the splitting is directly coupled to the ::-afe of rc.'11ino,)["ity, !n the case of a very strong nonl i neat' i ty , the approach c,o th:? asymptoti c solution i3 very fGst. Assumirg the deviations of t.he asymptotic solution vanish immediate;y, the

(*) Present address:

IN PLT, Bayer AG, 4047 Dormagen, Germany. The research was performed when the author was a member of the Institut fur Systemdynamik und Regelungstechnik, Universitat Stuttgart ,Ger--my.

distributed

solutions can be seen as to be quasistationary coupled to the asymptotic solution. This situation is of course an ideal base for a reduced description of the system behaviour. A great number of models now find a later physical justification in this property. But moreover, the structuring of the system can be taken as a starting point of a reduction method. In this way it is possible to develop a reduced model systematically. The first step is a description of the characteristic form of the asymptotic solution ~ by a nonllncar expression of the spacecoordinates ~ and the limedependent formparameters Q(t). (1)

Each of these forms is generally applicable to a \-,'101e class:lf solutions. The wavc[ront is in the fonT: of an S-profile movir>g along the z-coordinate. To get the dynamic equations of the formparamete,s Q(t) a method ar:alogous to the method of weighted resid'~als car be used. The procedure results ir. low order lumped parameter models. it becomes evident, that the new state variableC'. Q(t) show only a slightly nonlinear depen'jency :In the input ·:~riables. The strong n0nlinearity in ~ is mainly introduced by the formexpressior (1). The example used here is the catalytiC fixeG-bed reactor. The fixed-bed reactor is of great technical importance and shows all the characteristic propel'ties - but also the difficulties - arising with the concrete application of the reduction method.

DCCR-S

279

C. Epple

280

PHENOMENOLOGY Wave propagation processes occure in different forms. " vert frequent one is the moving front. Pigure 1 sho~s the principle shape of this structure.

zu ~::;;;;;:;:;;::;;;::===============1

T

T(z,t) ,c(Z,t)

czu~~~================~ Z

Fig. 2: The catalytic fixed-bed reactor

zFig. 1 Hoving

~al!efLont

In a small LC'::!":: of the space-coordinate z Lhe variable K changes monotonically from a low t.o a high level . The transition takes place in the form of an 5-profile. Due to this 5-profi.le, the first derivative of K. dx/az shows a high maximum at the middle of t.he front and the second derivative d' x/az' a maximum at the left and a minimum at the right side of the front. The main dynamic behaviour of a wave front is the movement along the space coordinate with a constant velocity w. During this shift, the shape of the front remains nearly unchanged. The velocity is completely determined by the parameters of the system and independent of the initial conditions. Moving wavefronts can be seen as the source of the characteristic behaviour of important systems in different disciplines of science.

A gas flow streams through the tube which is tilled with catalytic pellets. The reactant, carried by the gas-flow, diffuses into the pores of the pellets and reacts on the catalytic layer. In the case of an exothermic reaction, the turnover causes a heatproduction in the corn. This heat is lead off by heat exchange lIith the gasstream and by direct r.eat-conduction to neighbouring pellets. Due to the direct heat-conduction an internal feedback mechanism ' arises. The interaction of the strongly nonlineur reactionrateterm with the internal heat feedback 1n the fixed-bed can result in marked reaction zones. Reaction zones aLe small regions of space in ~hich th9 reactant is nearly completely consumed. Fig. 3 5 show the typical lIuvefront behaviour of the reactionzone in different situations. The decrease of the temperature behind the reaction front is caused by the cooling of the lIall of the tube. 400r-----------------------~

Well known are the wave fronts in population genetics (Fisher 1937), nerve propagation (Hodgkin and Huxley 1952), or the shockwaves in fluiddynamics (Burgers 1948). Some of the most impressive and technically most important examples arise in the region of chemical engineering. The limitation of energy and mass-transport causes gradients in temperature and concentrations. Consequently the systems have to be regarded to have distributed parameters. "dditionally the nonlinearities arising for example with the reactionrateterms or the expressions of the thermodynamic equilibriwn effect the behaviour of the system essentially. Well known are the fronts in adsorption columns especially in chromatographic packages, ignition fronts in flame pt'opagation, thermic fronts on catalytic wires, or concentration fronts in reaction-diffusion systems. Lately it has become obvious, that the regions of mass exchange in distillation and absorptioncolumns even can be interpreted as wavefronts (Marquarrlt 1986). " typical example fO[' a lIavefront is the moving reaction zone in a fixed-bed reactor . The catalytic fixed-bed reactor is the most important conti nuousl y operated chemi ca 1 reactor. It is technically seen as the obvious means to carry out heterogeneous catalysis. In industry the fixed-bed reactor is applied to acid-catalyed reactions, hydrogenation, oxidation and other reactions. For example the synthesis of ammonia, methanol or vinylacetate and the oxidation of methanol, So" ethylene or naphthalene are well knolln. Fig. 2 sholls one tube of a multitube reactor,

20 0 t:::~====::::::~::::::::~_

z-

o

Fig.3: A mov1ng reactionzone after a decrease in the feed temperature In Fig. 3, after a step decrease of the inlet temperatut'e, the reactor goes from an ignited state to an extinct state. The dynamic transition occurs in form of a reaction-~ront moving dOllnstream through the fixed-bed. I

cntoJyst section

T

z-Fig.

4: Extinction lIave caused by a deactivation of the catalyst (Weng, Eigenberger, Butt 1975)

Fig. shows another form of an extinction-lIave. In this case, the extinction is caused by a deactivation of the catalyst.

Model Reduction for ;\Ion linear Systems with Distributed Parameters

281

lIith the experimental data. Problems arise if the calculation methods of the control theory are applied to such a model. It becomes obvious, that the methods of control theory don't conform to a set of nonlinear partial differential equations. Thus. if lie lIant to design a controller, to construct an observer or to carry out an optimization, it is desirable to have 8 model lIith the property E additional to 11 - D.

500

E: structure: - The mathematical model should consist of 8 1011 order lumped parameter system. - The dependency of the state variables on the input variables Ehould be just barely nonlinear. Fig. 5: Transient behaviour after a stepdecrease the feed temperature

in

Fig. 5 demonstrates, that even in the cas~ of the transition be(lIeen tllO stationary ignited stat.es, the lIavefront dynamic remains the dominant part of the system ~)ehaviour . Although the temperature and the velocit, ) F ~he front a:-e changing due to the strong illfluP. lC:c' o[ the upstream boundary condition, tr.c !!la'.!l fo':r. r CICairs r.early unchanged.

MATHEMATICAL MODELLING The taste of a mathematical model is to describe the beha\iour of a system in a formal lIay. such a model has to fulfil several demands: A: Qualitiy: - The main qualitative behaviour of the system should be reproduced by the model. Qu~ntitatively a sufficient accuracy is required. These tllO properties have to be valid in a lIide range of operating conditions, B: Systematic: - It should be possible to deduce the model by a general method. C: Continuity: - The model should alloll to reconstruct the IIhole space-profile in t.he case of a distributed system. D: Ilelatedness:- To understand the systems behaviour it is important, that the state variables of the model correspond to quantities lIith a physical meaning. The basic method to get a mathematical model is to use the physical lalls of energy-, impuls-, and masspreservation . Formulating the balances of a concrete system results in a set of differential equations . In the case of the lIavefront system considered here, the set consists of nonlinear partial equations of the relaxation type.

To fill the gap bet~'een the or.'XIel (2) - presumed to be knolln - and the desired 1011 order lumped parameter system, a model reduction has to be carried out. The basic idea of the reduction method applied here is to take advantage of the selforganization process occuring in systems lIith lIave propagation . Due to this selforganization process, soll'tions of different initinl conditions converge quickly to a lIavefront solution. This asymptotic aspired lIavefront solution is characterised by a constant shape end only slollly changing formparametcrs. As lie lIill see in the n'!xt chapter , this splitting of the dynamic into a fast part - the asymptotic approach to the lIavefront - and a sloll part - the changing of the formparameters of the lIavefront - can be proved analytically, It is evident that the physical selforganization process for a modelreduction is to be used. The reduction is carried out in three steps: I Quasistatlonary asumption

In most applications, the fast dynamic approach to the lIavefront solution is of minor interest. Neglecting these quickly vanishing parts reduces ext['emely the variety of the possible systembehaviour. Under the quasistationary asumption the behaviour of the system is approximately characterized by the behaviour of the lIavefront solution. This means the problem of finding a model describing the variety of solutions reduces to the problem of finding a model of the 118vefront solution. 11. Description of the form 11 model of the lIavefront solution consists of tllO parts, an expression describing the constant shape of the solution and a set of ordinary differential equations describing the timedependency of the formparameters.

The main problem is the mathematical description of the shape of the front. The shall~ depends on the special nonlinearity and is in almost all cases not knolln. As a good approximation. the expression (3) can be used.



N.

1£

(~(~,t)

= 0

(2)

(2,Z)= - - - - -

-4q, (z-z, )

CP(z]

e

+1

at

(3) T

Mathematical models of this type are knolln for the most important systems of technical interest. By high order simulation it can be sholln, that the results given by these models are in good accordance

(z, , q, ) Zz

Z-

This expression contains only tllO parameters z the location of the front and q " the gradient in the front.

U. Epple

282

Ill. Dynamic equations

for the

the

A possibility to keep the calculations easy is to choose the weightfunction p(z) as a sum of H /5-functions.

(4 )

p(z)

formparameters

In the scalar case, the approximation wavefront gives an expression of the form

of

H

x '"



(.!;l(t) ,z)

The timedependent formparameters .!;l(t) have to be interpreted as the state va('i
n L

=

x

a

I

(t.~.g

I

(5)

(z)

i=l I (z) bebr:;:; to a system of orthogonal functions. The soluti~·:~ ;; in (5) is a superposition of items whi ch are I i r.o,,:rl y dependent on the statevariables

9

a

I

L /5ez-z;)

(12)

j=l

In this case the components of the vector RS and the matrix A can be calculated by sums of algebraic expressions. H must be at least equal t.o . But to achieve results which are independent of the choice of the collocationpoints z,·, H should not be under 10-20. This number of points does not complicate the procedure at all - only some more components of the summation need to be calculated but this number of points has an essential stabilising effect. Solving the linear problem (9) we get the desired system of ordinary differential equations for the formpara~eters .!;l. (13)

(t).

The

dynl'mic r:C:'.l"tions for the state variables of I (t) ::~~I be achieved systematically by the method of '~eighted residuals (Finlayson 1972) . This methoc. has been primarily developed for an expansion of the form (5), nevertheless it can be also applied to nonlinear expressions of the form

STRUCTURE ANALYSIS

(5) 11

One of the basic equations producing a wave front solution is given by (14)

ox

(4) .

02 -

The procedure is as follows: Substituting equation (4) into ferential equation we get. -

.!;l - N(
N(~)

at the partial

dif-

(6)

Res(.!;l,.!;l, z)

at To determine the 11 free variables Q we minimize the quadratic error function Res(g,.!;l,z) with respect to Q. 1

J(Res

o

2 (z,Q,Q))

. p(z)dz

,

Hin ... .!;l

p is a »eightfunction to be chosen. an exact solution , the Integral identically. (Res sO!).

X

f(x)

C

(14 )

az2

where fex) is a nonlinear function . A series of papers deal with the existence, stability and dynamic behaviour of the solutions of (14). Famous contributions are for example Kolrnogorov, ?etrovski and Piskunov (1937), Aronson and Weinberger (1975) or Fife (1979). From their work, we will use the important fact, that there exists a unique stable wave front solution if f(x) is a nonlinearity of type (15).

(7)

If

<)I

»ould

»ould be vanish (1S)

By a differentiation of (7), it is possible to obtain 11 conditions to determine the cornponents of the vektor 6.. 1 0<1> J • Iles (z,Il,p).p(z)dz

o ap,

o

i=l, ..

,ll

x--

(8)

Fig. 6

Due to the relaxationform of the partial differential equation, the residuum (6) depends linearly on the components of Q. Thus equation (8) can be written as (9)

Form of f(x) for an equation of the Huxley type.

This equation is known as Huxley type. It is used for example to model the nerve propagation process (Hodgkin a, Huxley 1952), or ignitionfronts on catalytic wires (Barelko et al (1978)). It is a very helpful attribute that in the special case

»ith: 1 04>

04> T

Ae.!;l,p(z))= J -- . (--).p(z)dz

(10)

o an all

fex) = - k x(l-x)(a-x)

(16)

the wavefront solution is explicitly known.

i a4>

RS(.!;l,p(z))= J --.N(4)).p(z)dz o a.!;l and .!;l can be determined by problem (9).

of this equation,

(11)

solving

4>(z)= -

the

J;'

(17)

-- (z-z..-w.t) 2c

e

+ 1

283

l\lodel Reduction for :'-ionlinear Systems with Distributed Parameters 1IPPL l

The velocity of the \.li\vefront is given by

C., TI ON

(18)

11 good example to test the efficiency of the reduction technique is the catalytic fixed-bed reactor.

with help of this solution it is possible to investigate, analytically, the spectrum of the op-el-atC'r liniarised around the solution. The ~'esu 1ts are summar i zed:

In the easiest case of a reaction 11 --) products the behaviour of the reactor can be described by an energy - and a massbalance.

1. Using

-+

~, ",!2£k-' (a-O.S)

El

quadratic norm \lith \leight func-

at

tion

az' (20)

(2a-1) p = 1l>

-B(T-T.) + Da.r (T,c)

-c,

az

ac

(1-2a) •

(1-1l»

(1 (3)

(This \leight. function corresponds to the \leight [unction p' = 1 in the self-adjoint system), the spectrum consists of a continuous an0 a

- Da.r(T,c) az The normalized state variables T and c are defined in the region O
discrete par-to

2. The discrete part of the spectrum exists of the eigenvalue ).t=o and - in the region 0,25 < a < 0,75 - of a second eigenvalue A,. Except for At and .\, the discrete part of the spectrum is empty .

A2 X

Fig. 8 Spectrum for a

Re 1/2

3. The eigenfunction corresponding to At causes a shift along the z-coordinate. The value of At =0 is characteristic for all \lave front solutions. It reflects the fact, that a \lavefront solution can be displaced at any locati.on along the infinit spacecoordinate '.I .i thout changing for'm or velocity. The eigenEu;1ction corresponding to A2 leads to a ',ariation of the gradient in the front. compar.i~,on oE this analytic result \lith mathematical model of the last chapter sho\ls:

11

1. The chosen expression

T

I=

:~az

T,.

z=O

I

o

I= c'" z=O

z=l

(21)

Comparing simulation results and measured data a good agreement is achieved (Ruppel 1980, Canavas 1984). This quantitative agreement exists also in the region of moving reaction front solutions. It is necessary to emphasize that the nonlinear effects of selforganization ace dominant only in the front region. Outside the fcont, the behaviour of the reactor is mainly influenced by additional efEects like boundacY conditions, cooling and the residual reaction of the extinct state. In these regions it is to be expected that the nonlinearities are of little significance. With this in mind it seems suitable to split up the model system into t\lO subsystems, a basesystem Sl and a \lavefronr.system S2.

T

the wove system

the shupc of the ",avefront model becomes an exact solution in the special case (16). (3)

c

fo~'

splittinCj of the dynamic behaviour of the model into the East approach and the 510'" changes, corresponds to the decomposition of t.he spect['um of the linearized system into continuous a;~ discrete ~~rts. Moreover there is a one to one correspondence IJet\leen the dynamic of z, and the first eigenfunction and the dynamic of q, and the second eigenfunction.

S2

Tl base system Sl

I., The

Without entering into details it should also be mentioned that in the case of the Bur'gers equa':ion an exact solution is kno\ln having the same form as (17) , The discrete spectrum oE the \la'le Eront solution here too contains the dominant eigenvaluc A\ =0 corresponding to a shift oE the pr.ofile. The BUl'gers equation is important as a basic equation of flu .~d dynamics and very similar to the basic equations of countercurrent exchange processes. The results oE the structure analysis can be seen as a hint that there is a physical meaning in the chosen form and Eormparameters of the reduced model,

z----

r~

\at

,~Tl

,T 1

+az-cTzT

51\

(22)

IT1z=oI = Tzu

aT 1 ; az

r

H z + ~_ dT' dZ

\

= 0 ;

z=1

,2T"2

£

azr-

=-B,T 2+Oa , (r(T I+ T2'c 1+c 2) -r(T ,c )} 1 1

1

(23)

52\ I

o

o

o

Fig. 8 Splitting oE the system in a basesystem Sl and a \lavefrontsystem S2.

284

C. Epple

It is easy to see, that lIith X=X ,;X " y=y I +y , , by an addition of the tllO pa['ts 51 and 52 the ~,plit system passes exactly t o the origlnal s ystem, The consequence of the nonlinearity is a unidirectional coupling of the basesystem to the lIavefrontsystem, This coupling occurs via the Sl dependent parameters of the react i :)f) rate term and transfer!} lhe i nput- i nf luences to the lIavefrontsi's lem 52, To model the sli')hty nonlinear basesystem is no p[-oblr.m, In mosl~ cases simple linear t.erms ,:Ire trllnderfunctions, s ufficient 'co desc:::ibe the Here lie take a model of oder tllO to describe t.he :::espc,nse of T I (z~z , ) to
T,

Tz 0

•

(2 1)

.. - - - - . - - - - _ . _-

-y( z-z , )

e

In the case of the lIavefront systems the characlerislic feature of the asymptotic solution is an S-shaped profile IIhich moves along the space coordinate, While the physics and the balance equatiO:1S of the different lIavefront systems are totally different, the existence of the asymptotic solution lIith the S-for.m and the moving along the space coordinate is a common property of all these systems, This gives the possibility to develop a reduction method generally applicable to the IIhole class of lIavefront systems. In the first step the part of the system determined by the lIavefront i~ separated from the rest-system, The rest-system or base system contains effects influencing the behaviour mainly outside the front IIhere the nonlinearities are of small magnitude, The lIavefront system can be described by its form and the dynamic ch~r~e of the formparameters, To produce the S-shaped front the expression for the form contains an exponential term in the denomimtor , The dynamiC dependency of the formparameters can be achieved ' by a generalization of the method of lIeighted resi.duals,

+1

The mlr.liti ona l f ormparnmeters T,o find fJ describe the muqnitude of the tempe['uture rise In the front and th e decrease of the t.emperature behind t.he fr-ont IIhich is caused by t.he cooling of the lIall. VIi th z , ,r, T, 0 ,n a1JCl t.he tllO state variables of the basesystem lie get a model of order 6, Figure 9 sholls a comparison betlleen the reduced model ancl the detailed model, Form ancl velocity of the front is reproduced ver-y lIell by the 1011 or'der model

As sholln .i n t.he example of a fixed - bed reClctor, the l:esul l.ing ['edue-ed model is in good accorda nce lI i th cea ll t y , 11 very helpful property of the model i s that the r e remains only a slightly nonlinear dependency of the s tate variables frorn the operaling conditions, The strong nonlinearity, visible in the solution x, is introduced by the transformation x = ~(~,z) in the formexpression, Thus such a model seems to be an ideal base for controller design , modelbased measurement techniques or optirnization calculations,

4

t T

2.5L--,---- - - - - - '

O.

z--

1

Fig. 9 Temperature profiles of a moving front reducetl model dctai led model CONCLUSION We sta['ted '.Jith a phenomenolo']ical Vlell to systems lIiU, 1.J
REFERENCES Aronson D G" Weinberger H,t. ( 1975), Nonlinear cliftuslon in population genetics, combustion nerve propagation, Lecturenotes 1 n and mathematic 44b Springer, Berlin, ~-49, Barelko V" KurochKa 1,1,) Merzhanov A,G., Shkadlnskii K,G, (1978 Investigation or travelling lIaves on catalytic lIires, Chem, Eng , Sci . 33 80~-811 , Burgers ' J,M, (1948), A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech,) 1, 171-199, Canavas C, (1984, Mode 11 ing and simulation of a fixed-bed (,·eador. International 81 AMSE Conference "Modelling and simulation", Athen, ' Fife P,C, (1978). Asymptotlc states for equatlons of reaction and dlffuslon, Bulletln of the American Mathematical Society, Vol, 84, N[' , ~ 693-726 Finlayson. B,A, (1972), The method of lIeighted reslduals and variational prinCiples , Academic Press, Nell York, r'j,t;iler R,A. (1937), The advance of advantageous genes. Ann, Eugen, 7, 35~-369, Gilles I': , D, (1974). Quasi-stationares Verhalten von wanderncten Brennzonen, Chem , Eng , Sci" 29 1211-1216 . Hodgkin A.L. , Huxley A.F, (19~2), A quantitative description of membrane current ancl its application to conduction and excitation in nerves, Journal of physiology, 117 , ~00-541 . Kolmog{)rov A,N" Petrovskl I.G" Plskunov (193 7 ). A st.udy of the equation of diffusion lIith increase in the quantity of matter arrl its application to a blOlogical problem , B)ul . Moskovskovo Gos , Univ, 1 no, 7, 1-72, Ma>:quardt \I, (1986), Nonlinear model reduction for bi nary distillation . DYCORD 86, Bournemouth, RCippel W, (980), Eine mathematische Beschrelbung lIanclernder Reaktionszonen in Schuttschichten , Dissertation, Universitat Stuttgart, Weng H.S., Eigenberger G" Butt J.B, (1975). Catalyst poisoning and lixed-bed reactor clynamlcs, Chem, Eng, Sci., 30, 1341-1351,