AN INCREMENTAL APPROACH FOR THEIDENTIFICATION OF REACTION KINETICS

Marc Brendel ∗, Adel Mhamdi ∗,Dominique Bonvin ∗∗, Wolfgang Marquardt ∗,1

∗ Lehrstuhl fur Prozesstechnik, RWTH Aachen University,D-52064 Aachen, Germany

∗∗ Laboratoire d’Automatique, EPFL, CH-1015 Lausanne,Switzerland

Abstract: This paper proposes an incremental approach for the identificationof complex reaction kinetics in chemical reactors. The reaction fluxes for thevarious species are first estimated on the basis of concentration measurementsand balance equations. This task represents an ill-posed inverse problem requiringappropriate regularization. In a further step, the reaction rates are estimatedwithout postulating a kinetic structure. Finally, the dependency of the reactionrates on concentrations, i.e. the kinetic laws, are constructed by means of feed-forward neural networks. This incremental approach is shown to be both efficientand flexible for utilizing the available process knowledge. The methodology isillustrated on the industrially-relevant acetoacetylation of pyrrole with diketene.

Keywords: Identification, reactor modeling, input estimation, regularization,neural networks

1. INTRODUCTION

The description of reaction kinetics often repre-sents the most challenging part in the model-ing of chemical reactors. A reliable description israrely available a priori. For example, it is wellknown that reaction kinetics cannot necessarily bederived from stoichiometries (Connors, 1990), inparticular in the case of catalyzed reactions. Thus,a reliable kinetic model needs to be identified fromexperimental data.

The model-based techniques used in process con-trol and optimization require a model that ade-quately describes the process dynamics, i.e. alsothe kinetics in reaction systems. For the casewhere a kinetic structure is not available, Psi-chogios and Ungar (1992) proposed a hybrid ap-

1 Corresponding author. E-mail: marquardt@lfpt.rwth-aachen.de

proach to process modeling as an alternative torecurrent neural networks for describing the dy-namic system. The hybrid model combines priorknowledge on mass and energy balances with afeed-forward neural net model that serves as asubstitute for the constitutive equations that can-not be determined from first principles. Theseauthors found that the hybrid model has bet-ter properties than standard black-box neuralnet models, i.e. interpolation and extrapolationare more accurate and the model is easier toanalyze and interpret. Parameters in the neuralnet part of the hybrid model can be estimatedfrom experimental data. Recently, Tholudur andRamirez (1999) used a two-step approach for theidentification of kinetics: Reaction rates are firstidentified, assuming known curve characteristics,and subsequently correlated with the independentstate variables using a feed-forward neural netapproximation. Van Lith et al. (2002) combined

an extended Kalman filter for the estimation ofstates and rates with subsequent fuzzy submodelidentification.

In this work, an incremental approach for theidentification of reaction kinetics is proposed whenno prior kinetic knowledge is available. The ap-proach is applicable to all reactor types, i.e. also tothose exhibiting transient behavior and possiblyvariable feed and effluent streams. The reactionfluxes for the various species are estimated fromnoisy concentration data using the approach ofMhamdi and Marquardt (1999). Then, the in-dividual reaction rates can be calculated usingknowledge of reaction stoichiometry. These reac-tion rates and the concentration data serve asinput to a Bayesian algorithm to train a feed-forward neural network yielding the kinetic model.The approach is especially suited for nowaday’shigh resolution measurement techniques such asIR (Alsmeyer et al., 2002) or Raman spectroscopy(Bardow et al., 2003), where concentration datacan be obtained continuously in-situ.

2. PRELIMINARIES

2.1 Model of the reaction system

Consider a homogeneous, not necessarily isother-mal, chemical reaction system with R reactionsinvolving S species. The time evolution of thenumber of moles of species i, ni [mol], is given by:

dni

dt= f in

i − fouti + f r

i , i = 1, .., S (1)

where f ini and fout

i [mol/min] are the molar flowrates of species i into and out of the reactor andf r

i [mol/min] is the reaction flux of species i, i.e.the net molar flow rate of species i produced orconsumed by the various chemical reactions.

The reaction flux of species i can be expressed interms of the individual reaction rates:

f ri = V

R∑j

νijrj , i = 1, .., S (2)

where νij is the stoichiometric coefficient forspecies i in the jth reaction, rj [mol/l min] therate of the jth reaction, and V [l] the volume.

In vector form, equation (2) reads:

f r = V Nr (3)

where f r is the S-dimensional reaction flux vector,r the R-dimensional reaction rate vector and Nthe S × R stoichiometric matrix.

Equation (3) indicates that, if S ≥ R, the reactionrate vector can be calculated from the reactionfluxes as follows:

r =1V

N+f r (4)

where N+ is the Moore-Penrose inverse of N.

For a constant-density semi-batch reactor with avolumetric feed of rate F [l/min] and concentra-tion cin

i [mol/l] and no outflow, the mole balanceequation (1) expressed in terms of the molar con-centration ci = ni/V [mol/l], and the total massbalance give

dci

dt=

F

V(cin

i − ci) +f r

i

V(5)

dV

dt= F, (6)

implying no volume change by the reactions.

2.2 Estimation of reaction fluxes

The reaction fluxes f ri (t) can be estimated inde-

pendently for each species. A generic model of theproblem is developed as follows. Let

yi(t) = ni(t) − ni(t0) −∫ t

t0

ui(τ) dτ, (7)

where ui(τ) = f ini (τ) − fout

i (τ). This transforma-tion, applied to (1), leads to

dyi(t)dt

= f ri (t), yi(t0) = 0, (8)

where f ri (t) is considered as an unknown input

that must be determined on the basis of a noisymeasurement

yi(t) = yi(t) + εyi(t). (9)

Here, the superscript (·) is used to denote a noisyquantity and εy represents the measurement noisecontained in y.

This estimation problem represents an ill-posedinverse problem according to the definition ofHadamard (Engl et al., 1996). Since the measure-ment is noisy, the estimate f r

i (t) of f ri (t) can be

arbitrarily large if no regularization of the solu-tion is considered. Mhamdi and Marquardt (1999)used Tikhonov-Arsenin filtering for the estimationof f r

i (t). The quality of the estimation is greatlyinfluenced by the choice of the regularization pa-rameter that weighs the tradeoff between noisereduction and bias in the estimate. Adequate reg-ularization parameters can be determined by theL-curve criterion (Hansen, 1998), for example.

Another approach to filtering is the use of smooth-ing splines (Craven and Wahba, 1979). Splinesare piecewise polynomial functions that possesscertain smoothness and differentiability propertiesat the nodes. General cross validation (GCV) isoften used to select a suitable regularization pa-rameter (Craven and Wahba, 1979).

3. INCREMENTAL IDENTIFICATIONAPPROACH

The incremental identification approach mirrorsthe steps taken when developing a model for agiven process. During model development, the bal-ance equations are set up first and the unknownfluxes are then described by constitutive equa-tions. If needed, variable parameters in the con-stitutive equations can be modeled as functions ofthe system states. Transferring this procedure tothe identification process, the incremental identi-fication approach features the stepwise identifica-tion of quantities as they are used in the modelingprocess. In an adaptive model identification con-text (Marquardt, 2002), the incremental approachallows the utilization of as much information ascan be safely provided by first-principle modelingor sound empirical approaches. The process ofidentification then reduces to modeling uncertain-ties, i.e. unknown parameters in a given structureor the model structure itself. This way, the iden-tification procedure is split up into a sequence ofdecoupled identification problems. This offers twomain advantages: i) the solution at a given stepbecomes more simple as e.g. process dynamics areconsidered in the first step and can be omittedsubsequently, and ii) physical insight is providedfor tackling the following steps.

The incremental identification approach for theidentification of reaction kinetics is depicted inFigure 1. It includes the following steps:

(1) The fluxes f ri (t), i=1, .., S are estimated using

mole balances (Model 1). Use equations (7)-(9) and ni = ciV .

(2) With additional information on stoichiom-etry (Model 2), the reaction rates rj(t),j=1, .., R are then calculated using (4).

(3) Furthermore, if the rate laws (e.g. r =kcAcB) are known (Model 3), (time-variant)rate constants kj(t) are calculated from thereaction rate rj(t) and concentrations ci(t).

(4) Model 4 in addition assumes a temperaturedependency of k such as the Arrhenius law(k = k0e

−ERT ). The rate constant parameters

(k0j , Ej) can then be estimated from kj(t)and T (t).

If parts of the kinetics are unknown, such as theArrhenius law, the outputs of Model 3 can betaken as inputs to a data-driven approach fordescribing k = k(T ). For an unknown rate law(Model 2 known), ci, T and rj serve as inputsto the data-driven models rj = rj(ci) and rj =rj(ci, T ) for the isothermal and non-isothermalcases, respectively. If the reaction stoichiometryis unknown, target factor analysis (Bonvin andRippin, 1990) can help identify the stoichiometrybased on the estimated fluxes.

Measurements

BalancesModel 2

Model 1 Balances

Stoichiometry

Model 3Rate law

(conc. dep.)StoichiometryBalances

Model 4Rate law

(conc. dep.)StoichiometryBalances

Arrhenius law(temp. dep.)

Model structures and parameters for kinetic phenomena

VuTc ii , , ,~

Tci ,ˆ

Tci ,ˆ

Tci ,ˆ

Tci ,ˆ

rif

jrrif

rif

rif

jr

jr jk

jk jj Ek ˆ ,0

Fig. 1. Incremental approach for reaction kineticsidentification

For describing functional relations in a data set,methods can be grouped in two categories basedon the quality and amount of prior knowledge.If a model structure is available, the unknownmodel parameters can be identified from the data.Lacking such a model structure, black-box ap-proaches are usually used, with the choice of basisfunctions based on some prior knowledge. Theirability to approximate any function arbitrarilywell, given a sufficient number of parameters,may imply overfitting where the error on thetraining set is small, but a large error results ifnew data are presented to the model. To avoidoverfitting and improve the predictive capabilityof the model, regularization, model discrimina-tion or data validation techniques are commonlyused. Feed-forward neural networks with Bayesianregularization (MacKay, 1992) may serve as anautomated regularization procedure for training.

4. ILLUSTRATIVE EXAMPLE

4.1 Simulated reaction system

The incremental approach for identifying reactionkinetics is illustrated on the acetoacetylation ofpyrrole with diketene (Ruppen et al., 1997):

P+D K→ PAA (10a)

D+D K→ DHA (10b)

D → oligomers (10c)

PAA+D K→ F (10d)

In addition to the desired reaction of diketene (D)and pyrrole (P) to 2-acetoacetyl pyrrole (PAA)(10a), there are several undesired side reactions(10b)-(10d). These include the dimerization andoligomerization of diketene to dehydroacetic acid(DHA) and oligomers as well as a consecutivereaction to the by-product F. The reactions takeplace isothermally in a laboratory-scale semi-batch reactor with an initial volume of 1 liter,to which a diluted solution of diketene is addedcontinuously.

Reactions (10a), (10b) and (10d) are catalyzed bypyridine (K), the concentration of which continu-ously decreases during the run due to addition ofthe diluted diketene feed. The dilution of catalystis modeled by normalizing the corresponding rateconstants with respect to the volume. Reaction(10c), which is assumed to be promoted by otherintermediate products, is not normalized. Hence,the effective reaction rates are described by thefollowing constitutive equations

rj(t) =V0

V (t)rj (t), j = {a, b, d}, (11)

rc(t) = rc (t), (12)

with the formal reaction rates

ra(t) = kacP(t)cD(t), (13a)

rb (t) = kbc

2D(t), (13b)

rc (t) = kccD(t), (13c)

rd(t) = kdcPAA(t)cD(t), (13d)

where ka, kb, kc and kd represent the rate con-stants and V0 the initial volume.

The mole balances for the species D, P, PAA andDHA read

dcD(t)dt

=F (t)V (t)

[cinD − cD(t)] +

f rD(t)V (t)

, (14a)

dcP(t)dt

= −F (t)V (t)

cP(t) +f rP(t)

V (t), (14b)

dcPAA(t)dt

= −F (t)V (t)

cPAA(t) +f rPAA(t)V (t)

, (14c)

dcDHA(t)dt

= −F (t)V (t)

cDHA(t) +f rDHA(t)V (t)

, (14d)

with the initial conditions cD(0) = cD0, cP(0) =cP0, cPAA(0) = cPAA0 and cDHA(0) = cDHA0. Thereaction fluxes f r

D, f rP, f r

PAA and f rDHA can be re-

lated to the reaction rates using the stoichiometry:

f rD = (−ra − 2rb − rc − rd)V, (15a)

f rP = −raV, (15b)

f rPAA = (ra − rd)V, (15c)

f rDHA = rbV. (15d)

4.2 Experimental design

To assess the capability of the incremental iden-tification approach and allow a comparison of themodeled and true kinetics, concentration trajecto-ries are generated using the model described aboveand the rate constants given in Table 1.

The measured concentrations are assumed to stemfrom a high-resolution in-situ measurement tech-nique such as Raman spectroscopy, taken at a

Table 1. Values of rate constants

ka kb kc kd

[ lmolmin ] [ l

molmin ] [ 1min ] [ l

molmin ]

value 0.053 0.128 0.028 0.001

Table 2. Range of independent variables

cD0 cP0 cPAA0 cDHA0 F cinD

[ moll ] [ mol

l ] [ moll ] [ mol

l ] [ lmin ] [ mol

l ]

min 0.07 0.40 0.10 0.02 0.5e-3 4.0max 0.14 0.80 0.20 0.04 1.5e-3 6.0

sampling frequency fs = 60 min−1 and corruptedwith normally distributed white noise of standarddeviation σc = 0.01 mol/l. The batch time istf = 60 min. Concentration data are available forthe species D, P, PAA and DHA, but not for theoligomers and the side product F since the latterare difficult to obtain.

The reaction system (10a)-(10d) suggests that rb

and rc are univariate functions of cD, whereas ra

and rd are expected to be bivariate functions ofcP, cD and cPAA, cD, respectively.

To obtain reliable approximations of the reactionrates, in particular for the bivariate functions,experiments are designed so as to obtain concen-tration data over a large domain. Six indepen-dent variables can be considered: the four initialconditions cD0, cP0, cPAA0 and cDHA0, feed rateF chosen to be constant during a run, and feedconcentration cin

D . The possible ranges of theseindependent variables are given in Table 2. SincecD0, cP0, F and cin

D have the largest impact onthe resulting transient behavior, a 26−2 factorialdesign consisting of 16 experiments is selected.Fewer experimental runs would reduce the valid-ity range and/or the predictive capability of themodel, while additional runs would improve them.

4.3 Various modeling scenarios

In the following, three different modeling scenariosare presented, each differing in the amount of priorknowledge regarding the reactions. The fluxes,reaction rates and reaction kinetics are identifiedfrom noisy concentration measurements.

Scenario 1 In the first scenario, we assumeknowledge regarding the existence of reactions(10a)-(10d), including their stoichiometric coeffi-cients. Moreover, it is known that the rates of thereactions (10a), (10b) and (10d) are proportionalto the catalyst concentration, see (11).

The reaction fluxes for the various species are ob-tained from (8) using appropriate regularization.Here, smoothing splines with GCV are used fordetermining the regularization parameters.

From the time-dependent reaction fluxes f ri , i =

{D,P,PAA,DHA}, the time-dependent reaction

rates rj , j = {a, b, c, d}, can be calculated using(15a)-(15d). Since the influence of the catalyst onthe reaction rates is known, the formal reactionrates r

j are determined from (11) and (12).

Finally, the concentrations and the reaction ratesfrom one or several runs are correlated as r

a =ra(cP, cD), r

b = rb (cD), r

c = rc (cD) and r

d =rd(cPAA, cD), as proposed by stoichiometry. A

feed-forward neural net with Bayesian regular-ization as training algorithm and 3 nodes in thehidden layer is utilized.

Scenario 2 In the second scenario, no informa-tion regarding the effect of catalyst on the kineticsis postulated. This corresponds to r

j = rj . Other-wise, the procedure is identical to that of Scenario1.

Scenario 3 We consider the case where little isknown a priori about the reaction system. Besidesthe known desired reaction (10a), there is evidencethat diketene (D) and pyrrole (P) are involvedin other reactions, including the formation of thedimerization product DHA. Hence, the stoichio-metric model

P+D → PAA (16a)

D + ν1PAA → ν2DHA + G (16b)

is postulated, where the possible side reactionsare lumped into reaction (16b) with the unknownstoichiometric coefficients ν1 and ν2 and someunknown side products G.

From the estimated reaction fluxes, the reactionrates r

a(t) for Reaction (16a) and rlump(t) for

Reaction (16b) as well as the stoichiometric co-efficients ν1 and ν2 can be determined as solutionof the reconciliation problem:

f rD(t) = [−r

a(t) − rlump(t)]V (t) (17a)

f rP(t) = −r

a(t)V (t) (17b)

f rPAA(t) = [r

a(t) − ν1rlump(t)]V (t) (17c)

f rDHA(t) = ν2r

lump(t)V (t). (17d)

The rates ra(t) and r

lump(t) can subsequently becorrelated with the concentrations, as discussed inScenario 1.

4.4 Identification results

Reaction fluxes and concentrations Exemplarily,the true and estimated reaction fluxes for speciesD are shown in Figure 2 (right). Integration of(14a) yields an estimate of the concentration cD,as shown in Figure 2 (left).

Reaction rates For Scenario 1, the estimated raterb in the univariate case is shown in Figure 3,

along with training data and true rate.

0 20 40 600

0.02

0.04

0.06

0.08

0.1

Time t [min]

Con

cent

ratio

n c D

[mol

/l]

0 20 40 60−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5x 10

−3

Time t [min]

Rea

ctio

n flu

x f r D

[mol

/min

]

estimated reaction fluxtrue reaction flux

concentration dataconcentration estimate

Fig. 2. True and estimated reaction flux f rD (right),

measured and estimated concentration cD (left)

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16−0.5

0

0.5

1

1.5

2

2.5

3x 10

−3

Concentration cD [mol/l]

Rea

ctio

n ra

te r

b* [mol

/l/m

in]

Reaction 2: D+D → DHA

data setsapproximated ratetrue rate

Fig. 3. Estimated reaction rate rb

Kinetic model The validity range of a model isdefined as the smallest n-dimensional (e.g. n = 2for the bivariate case) box containing all concen-tration combinations taken for training. The meanand maximum values of the neural net predictionsin the validity range are compared to the true(simulated) reaction rates in Table 3.

The predictions obtained in Scenarios 2 and 3 arecomparable which can be accredited to the factthat the volume change remains small during theruns (the increase in volume does not exceed 3%for the low and 9% for the high feed rate). Pre-sumably, the importance of the catalyst dilutionbecomes more obvious for large volume changes.Rate r

d, whose value is small compared to theother rates, is mainly influenced by noise andcannot be identified satisfactorily.

For the lumped model in Scenario 3, the mainreaction rate r

a is identified with reasonable accu-racy despite the error introduced by lumping allside reactions in (16b). Here, the stoichiometriccoefficients ν1 and ν2 were calculated as 0.0028and 0.2227, respectively. Since the rates r

b , rc and

rd were not modeled, they are not identified in this

case.

Table 3. Reaction rate prediction errors

r�a r�

b r�c r�

dScenario 1 Mean error 2.95 6.15 5.16 185

Max. error 11.24 26.42 20.92 3245

Scenario 2 Mean error 4.48 7.33 4.88 117Max. error 15.69 25.84 18.58 466

Scenario 3 Mean error 4.64 - - -Max. error 62.99 - - -

4.5 Validation of the hybrid model

To check the predictive capability of the hybridmodel consisting of the mole balance equationsand the neural-net-based kinetic laws, concentra-tion trajectories are simulated using (13a)-(13d)and the neural net approximations used to predictthem. Ten runs were simulated with experimen-tal conditions chosen randomly within the rangesgiven in Table 2. The mean and maximum valuesof the prediction errors are listed in Table 4.

Table 4. Hybrid model prediction errors

[%] cD cP cPAA cDHA

Scenario 1 Mean error 0.81 0.11 0.40 1.10Max. error 2.80 0.63 1.48 5.58

Scenario 2 Mean error 0.79 0.24 0.61 1.10Max. error 2.30 0.43 1.19 5.04

Scenario 3 Mean error 0.60 0.20 0.53 4.13Max. error 1.91 0.42 1.28 15.58

Hybrid model predictions show excellent agree-ment with the true (simulated) values. The factthat the model predictions are good even for apoorly estimated reaction rate r

d illustrates thedifficulties experienced in estimating this rate.These results suggest omitting reaction (10d) inthe postulated reaction scheme. Comparison ofthe three scenarios also indicates that the hybridmodel derived from the lumped model equationsperforms nearly as well as the detailed models.

5. CONCLUSIONS

This work has proposed an incremental approachfor the identification of unknown kinetics in achemical reactor. The approach consists of: (i)model-free estimation of the reaction flux asso-ciated with each species, (ii) calculation of thereaction rates using the (partially) known systemstoichiometry, and (iii) identification of kineticmodels using neural nets to represent the corre-lation between reaction rates and concentrations.Here, information on stoichiometry helps choosingthe independent variables.

The predictive capability of the hybrid modelwas very satisfying as were the kinetics identifiedin the case of known stoichiometry. The errorsobserved were largely caused by missing initialinformation regarding the reaction fluxes, a phe-nomenon that requires further investigation.

It should be emphasized that the proposed incre-mental modeling approach is by no means limitedto the use of neural net submodels. Mechanisticmodels using target factor analysis, multidimen-sional sparse grids or multigrid methods may alsotake advantage of the incremental approach.

ACKNOWLEDGEMENTS

This work was partially funded by the DeutscheForschungsgemeinschaft (DFG) within the Col-laborative Research Center (SFB 540) ”Model-based experimental analysis of kinetic phenomenain fluid multiphase reactive systems”

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