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Korean J. Chem. Eng., 18(5), 652-661 (2001)

652

†To whom correspondence should be addressed.

E-mail: cnkim@khu.ac.kr

On the Feasibility and Reliability of Nonlinear Kinetic Parameter Estimation

for a Multi-Component Photocatalytic Process

Linxiang Wang* and Chang Nyung Kim†

*Industrial Liaison Research Institute, College of Mechanical & Industrial System Engineering,

Kyung Hee University, Yongin 449-701, Korea

(Received 23 April 2001 • accepted 18 June 2001)

Abstract− − − −Nonlinear kinetic parameter estimation plays an essential role in kinetic study in reaction engineering. In

the present study, the feasibility and reliability of the simultaneous parameter estimation problem is investigated for a

multi-component photocatalytic process. The kinetic model is given by the L-H equation, and the estimation prob-

lem is solved by a hybrid genetic-simplex optimization method. Here, the genetic algorithm is applied to find out,

roughly, the location of the global optimal point, and the simplex algorithm is subsequently adopted for accurate con-

vergence. In applying this technique to a real system and analyzing its reliability, it is shown that this approach results in

a reliable estimation for a rather wide range of parameter value, and that all parameters can be estimated simultaneously.

Using this approach, one can estimate kinetic parameters for all components from data measured in only one time

experiment.

Key words: Photocatalytic Oxidation, Kinetic Model, Parameter Estimation, Nonlinear Optimization, Hybrid Method

INTRODUCTION

Photocatalysis is a promising approach to efficient destruction

of environmental pollutants [Alberici and Jardim, 1997; Herrmann,

1999; Ollis, 2000; Chai et al., 2000], and many experimental inves-

tigations have been reported in this field [Anpo et al., 1997; Sirisuk

et al., 1999]. Because an experimental approach is usually costly

and time consuming, some other investigations with computer sim-

ulations have also been carried out, which are based on a mathe-

matical kinetic model for the photocatalytic oxidizing process. Re-

searches have shown that most photocatalytic reactions follow the

L-H (Langmuir-Hinshelwood) equation [Fox and Dulay, 1993]. In

fact, numerical simulation of the L-H equations is not a difficult

task in itself, and here various numerical integral methods can be

applied to the equation to solve concentration profiles of each reac-

tant and product with time. The obstacle is that kinetic parameters

in the L-H equations, that is, reaction rate constants and adsorption

equilibrium constants, are not measurable, and there is no way to

deduce an analytical formula to theoretically estimate its value. The

only feasible approach to obtaining their values is to estimate them

based on experimental data. The reason is that the L-H equations

are a set of implicit nonlinear differential equations that are cou-

pled with each other since some reactants are the resultants of others

in a multi-component system.

During a long period, kinetic parameters have been estimated

by the so-called initial rate method [Levenspiel, 1972], which uses

linear regression method, based on the reciprocal form of a single

L-H equation. But, this method cannot yield satisfying results owing

to the fact that a nonlinear equation is merely replaced by a linear-

ized equation in this method [Mehrab et al., 2000]. When more than

one component is being oxidized simultaneously and the L-H equa-

tions are coupled with each other, which is very often encountered

in photocatalytic reactions, this method is no longer applicable be-

cause reciprocals of the L-H equations are still nonlinear.

Taking into account the nonlinearity of the L-H equations, sev-

eral approaches have been suggested for nonlinear parameter esti-

mation in reaction engineering [Biegler et al., 1986; Kim et al., 1990;

Farza et al., 1997; Park and Froment, 1998; Oh et al., 1999; Balland

et al., 2000]. For kinetic model governed by the L-H equations, Fro-

ment [1987] has shown that nonlinear regression can be applied to

perform nonlinear parameter estimation. Mehrab et al. [2000] have

adopted the Box-Draper nonlinear regression method to find the

best point estimates, in which a variable metric algorithm is em-

ployed with an improved gradient calculation. Although the local-

convergence methods mentioned above do have a potential to yield

a better estimation of kinetic parameters and are expected to be us-

able in multi-component systems, there is still a rigorous limitation

that a rather good initial parameter value should be given. Because

an objective function for nonlinear model often contains more than

one optimum, a local-convergence method is highly prone to fall

into non-global optima [Press, 1986] owing to their downhill (hill-

climbing) algorithm. To protect the parameter estimation from re-

garding a local optimum as a global one, Belohlav et al. [1997] have

applied a random search method in nonlinear regression. This ap-

proach does work, but is computationally less efficient because of

its random search algorithm. Especially when more than one com-

ponent is oxidized, it hardly results in a satisfying estimation.

To locate the global optimum confidently, various approaches

under the term “evolutionary algorithm” have been also investi-

gated recently. Wolf and Moros [1997] have estimated rate con-

stants in oxidizing methane to C2 hydrocarbons by the Genetic Al-

gorithm (GA); Park and Froment [1998] have used the GA esti-

mated kinetic parameters and tested a heterogeneous catalytic reac-

tion; Balland et al. [2000] have estimated kinetic and energetic pa-

rameters in the saponification process of ethyl acetate using the GA.

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On the Feasibility and Reliability of Nonlinear Kinetic Parameter Estimation653

Korean J. Chem. Eng.(Vol. 18, No. 5)

Although, the kinetic models studied in these works are not given

by the L-H equation, their nonlinearities are not as strong as that of

the L-H equation, and their final results are not so accurate as those

of local convergence method, these works do show us that the GA

has a potential to find out, roughly, the location of the real global

optimum for the nonlinear estimation problem.

In the present study, we will show it is feasible and reliable to

estimate simultaneously all kinetic parameters in the L-H equations

by a hybrid genetic-simplex optimization method, even when sev-

eral oxidation processes are coupled with each other in the reac-

tion. The proposed hybrid optimization method is set up in mating

the modified GA with the simplex algorithm. The role of the mod-

ified GA is to find a rough estimation for kinetic parameters, and

this assures us that subsequent local search will converge to the glo-

bal optimum when the result of the rough estimation is used as a

starting point of the simplex algorithm. The simplex algorithm is a

local convergence method used to refine the rough estimation and

to make the estimation more accurate. By analyzing the sensitivity

of the simulated concentrations with respect to the estimated param-

eter values, we will show that the hybrid optimization method is

able to estimate parameters accurately and reliably. By applying it

to a process about which previously published results are available,

we will show the hybrid method gives a more accurate estimation.

PARAMETER ESTIMATION OF

MULTICOMPONENT SYSTEM

In the present study, a process of photocatalytic purification of

three VOC components, which has been extensively investigated

by Turchi and Rabago [1995], Turchi et al. [1996] and Wolfrum et

al. [1997], will be adopted for discussion. In the system, there are

three kinetic-significant reactants (acetone, isopropanol and metha-

nol) and one product (carbon dioxide), where acetone is also the

product of isopropanol oxidation. The three reactants existing in

the same contaminated air stream, say, representing exhaust stream

from semiconductor plants, are oxidized simultaneously. The reac-

tion stoichiometry can be depicted as [Turchi et al., 1996]:

Acetone→3 CO2+xH2O

Isopropanol→acetone→3 CO2+xH2O

Methanol→CO2+xH2O

The L-H equations for this process can be expressed as:

(1)

where subscript 1 denotes acetone, 2 isopropanol, 3 methanol and

4 carbon dioxide. c is the concentration, k the reaction rate con-

stants and K the adsorption equilibrium constants.

In the previous literature [Turchi et al., 1996; Wolfrum et al., 1997],

ki and Ki, i=1, 2, 3, have been estimated separately from experi-

ments with each individual reactant having its various initial con-

centrations. In the present study, with only one set of data, the kinetic

parameters in the above equations will be estimated by a hybrid

optimization method.

To fulfill the nonlinear estimation, let’s construct object func-

tions for four components in the system as follows:

(2)

where vector k={k1, k2, …, km, …} denotes all reaction rate con-

stants and vector K={K1, K2, …, Km, …} denotes all adsorption

equilibrium constants. ci(m) is experimental concentration of the i-

th component at the m-th instance with time, while

ulated concentration of the i-th component at the m-th instance. Ap-

parently, the object function is a square summation of difference be-

tween the simulated and experimental concentrations (Least Square

Estimation). To make the problem solvable, it is convenient to com-

bine all object functions into a single total object function by the

following weighted average (or summation) method.

is the sim-

(3)

where wi is the weight for the i-th component. Thus, the kinetic pa-

rameter estimation problem is expressed as a nonlinear optimiza-

tion problem as follows:

(4)

Apparently, since the L-H equations are a set of implicit nonlinear

ordinary differential equations, the object function must be a multi-

peak function. Therefore, the only practical way to solve this prob-

lem is numerical search method.

HYBRID OPTIMIZATION METHOD

It has been understood that the GA has the potential to locate the

global optimum but its final result may not be accurate enough, while

the local-convergence method has the potential to locate the local

optimum accurately but is highly prone to fall into non-global op-

timum. Therefore, it is natural to expect that a hybrid algorithm set

up by mating the GA with the local convergence method should

be a promising approach for nonlinear kinetic parameter estima-

tion. In the following paragraph a brief description will be given

about the GA and the simplex algorithm, which are hybridized for

the present study. Detailed information about these algorithms can

be found in many previous investigations [Cheney and Kincaid,

1985; Winston, 1991; Gen and Cheng, 1997].

The GA maintains a population of individuals, say P(n), for gen-

eration n and each individual consists of a set of genes, where each

gene stands for a parameter to be estimated. One individual repre-

sents one potential solution to the problem at hand. Each individual

is evaluated to give some measure of its fitness. Some individuals

r1 = dc1

dt

------ - = − k1K1c1 − k2K2c2

1+

Kici

i= 1

3

∑

----------------------------------- -

r2 = dc2

dt

------ - = − k2K2c2

1+

Kici

i= 1

3

∑

--------------------- -

r3 = dc3

dt

------ - = − k3K3c3

1+

Kici

i= 1

3

∑

--------------------- -

r4 = dc4

dt

------ - = 3k1K1c1 + k3K3c3

1+

Kici

i= 1

3

∑

-------------------------------------- -

Jik K

(,) =

cim

( ) − cˆim

( )[]2

m= 1

p

∑

c ˆim

( )

J =

wiJi

i= 1

n

∑

wi

i= 1

n

∑

⁄

min J

dci

dt

------ = fik1K1c1k2K2c2Λ, , ,(,, ,)

(k, K)

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654L. Wang and C. N. Kim

September, 2001

undergo stochastic transformation by means of genetic operations

to form new individuals. There are two transformations: crossover,

which creates new individuals by combining parts from two indi-

viduals, and mutation, which creates a new individual by making

changes in a single individual. New individuals, called offspring

S(n), are then evaluated. A new population is formed by selecting

fitter individuals from the parent population and the offspring popu-

lation. After some generations, the algorithm converges to the best

individual, which hopefully represents an optimal or sub-optimal

solution to the problem. A general structure of the GA is as follows:

begin

n=0;

initialize P(n);

evaluate P(n);

while (not termination condition) do

begin

recombine P(n) to yield S(n);

evaluate S(n);

select P(n+1) from P(n) and S(n);

n=n+1;

end

end

To be applied to this concrete problem, some operations of the

GA are modified here.

1. Modified Genetic Algorithm

In the present study, variables to be optimized are reaction rate

constants k={k1, k2, …, km, …}, and adsorption equilibrium con-

stants K={K1, K2, …, Km, …}, which are all float values, and this

fact makes the coding procedure different from that of the tradi-

tional GA, whose individuals are all coded by binary figures.

1-1. Representation

Using real possible values of the reaction rate constants and the

adsorption equilibrium constants as its gene, the jth individuals Vj

can be encoded as:

Vj: {k1

j, k2

j, …, kn

j, K1

j, K2

j, …, Kn

j}

lk≤ki

j≤uk; lK≤Ki

j≤uK

where ki

ith component, while Ki

equilibrium constant of i-th component in the current generation. lk

and uk are the lower and upper limit, respectively, for reaction rate

constants, and lK and uK are the lower and upper limit, respec-

tively, for adsorption equilibrium constants. It is also feasible to set

different limits for each variable.

1-2. Crossover and Mutation

In order to explore the search space, some randomly chosen in-

dividual pairs are recombined by crossover operation, which is clear-

ly sketched by the following two in- dividuals

j is the j-th possible value of reaction rate constant of the

j is the j-th possible value of adsorption

Vj: {k1

j, k2

j, …, kn

j, K1

j, K2

j, …, Kn

j}

Vj+1: {k1

j+1, k2

j+1, …, kn

j+1, K1

j+1,K2

j+1, …, Kn

j+1}

New individuals resulting from the above may be:

Vj

*: {k1

j, k2

j, …, kn

j, K1

j, K2

j+1, …, Kn

j+1}

V*

j+1: {k1

j+1, k2

j+1, …, kn

j+1, K1

j+1, K2

j, …, Kn

j}

The position where an individual is cut off for recombination is ran-

domly chosen, and the number of individuals chosen to be recom-

bined is set by crossover probability. After crossover, all entries

(genes) in all individuals are given a chance of undergoing muta-

tion operation with a certain mutation probability. If a gene is se-

lected for mutation operation, it will be assigned a random value in

the given range.

1-3. Evaluation and Selection

The evaluation function plays the role of the environment in natu-

ral evolution, and it rates individuals in terms of their fitness. For

the minimization problem here, the fitness of each individual is de-

fined over object function values as:

(5)

where Fj is the fitness of the j-th individual vj, m is the number of

individuals in the present generation, and J(Vj) is the object func-

tion value of the j-th individual. It is apparent that the individual

having the smallest object function value will have the highest fit-

ness. In constructing the next generation, the probability of the se-

lection of the jth individual is calculated by the following equation:

(6)

The selection is implemented by adopting a roulette wheel approach

[Gen and Cheng, 1997].

2. Simplex Algorithm

From the mathematical point of view, the simplex algorithm is a

relatively simple algorithm, but is effective for many optimization

problems, especially in case it is difficult to deduce an analytical

gradient formula [Cheney, Kincaid, 1985]. Its principle is as follows:

When the simplex algorithm starts with a given starting point in

an m dimensional space, it will choose arbitrarily one point differ-

ent from the starting point along each dimensional axis in a small

neighborhood of the starting point, and define a simplex in the search

space with the given point (starting point) together with the m points

chosen along each dimensional axis (m is the number of parame-

ters to be optimized, i.e., the dimension of search space, and the sim-

plex has m+1 points). Then a downhill (or hill-climbing) method

is applied to update the simplex iteratively and to make the algo-

rithm finally achieve the optimum. Assume that vj is the worst one

among the m+1 points v1, v2, …, vm+1, which means vj has the big-

gest object function value (in case of minimizing); a new point can

be created as follows:

(11)

where λ is the step size which can be optimized in each iteration.

By replacing the worst point vj with the newly created point v*, the

simplex can be updated. In the next iteration, the new worst point

in the new simplex is identified and replaced by a new point. If this

updating operation is repeated, all the m+1 points will come closer

to the local optimum, and the step size will become smaller [Win-

ston, 1991]. When the step size becomes smaller than a given small

value (convergence criteria), each one of the m+1 points can be

Fj =

J vj

( )

j= 1

--------------- -

m

∑

J vj

( )

Pj = Fj

Fj

j= 1

m

∑

---------

v* = 1− λ()vj + λ

vi − vj

i= 1

------------------- -

m+ 1

∑

2n

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On the Feasibility and Reliability of Nonlinear Kinetic Parameter Estimation655

Korean J. Chem. Eng.(Vol. 18, No. 5)

regarded as the optimum.

3.Solution Strategy

In the simplex algorithm, the simplex is updated by replacing the

single worst point (the current point) in the search space iteratively.

During a single iteration, a new point is selected from the neigh-

borhood of the current point. If the new point provides a smaller

value of the object function, the current point is deleted and the new

point will be used in the process of the simplex algorithm; otherwise,

another neighbor is selected and tested. It is clear that this search

strategy provides local optimum values only, and these values depend

on the selection of the starting point. Considering the fact that the

optimization problem in the present study is implicit and nonlinear,

which means that the object function generally has more than one

extrema, we have to produce a reasonable starting point to ensure

the simplex algorithm converges to a global optimum rather than a

local optimum. On the other side, the GA is a compromise between

an accurate local convergence method and a robust random meth-

od; it combines elements of directed and stochastic search. It has

been shown by many investigations that the GA will finally con-

verge to the best individual roughly with a random initial popula-

tion, but it is hard to improve its accuracy. In our hybrid method

for kinetic parameter estimation, the genetic algorithm is used not

to find the final best solution to the problem, but to yield a rough

guess of parameters, which will be used as the starting point of the

simplex algorithm; therefore. it is unnecessary to let the GA oper-

ate a long time till it finally converges to the best solution.

In our hybrid optimization method, the GA is adopted firstly with

its initial individuals created randomly in a given bound of param-

eter values. About 2000 generations are evolved, and the fittest in-

dividual in the population (from the first generation to the last gen-

eration) is chosen as the favorite point. Here, the number of genera-

tions used in the GA is an empirical value based on the experience

of the author because there is no general value for this purpose. Suc-

cessively, the simplex algorithm is started to refine the solution by

using the point given by the GA as its starting point. When the step

size of the simplex algorithm is smaller that 1.0e-6, the search pro-

cess is stopped and the optimization is regarded as converged.

PERFORMANCE OF THE HYBRID METHOD

By the hybrid optimization method mentioned in the above sec-

tion, all kinetic parameters in the system defined by Eq. (1) have

been estimated simultaneously by only one set of data, which is

read carefully from the published experimental figure [Turchi et

al., 1996]. For the sake of convenience, the units of concentration,

reaction rate constants and adsorption equilibrium constants are all

changed into SI system. In this test with the modified GA, the pop-

ulation has 200 individuals, the crossover probability is set 0.2 and

mutation probability is set 0.01. The convergence criterion for the

simplex algorithm is that its step size is smaller than 1.0×10−6, which

means all points of the final stage of the simplex algorithm come

close enough and wander in a very small region (approximated by

a polyhedron whose diameter is smaller than 1.0×10−6). With the

assumption that 0<ki<10000 and 0<Ki<10, i=1, 2, 3 (it is a rea-

sonable guess judging from our knowledge of reaction engineer-

ing, and if these bounds do not work, we can easily change these

bounds), the hybrid optimization method finally yields the esti-

mated values as follows:

k1=140.8013766138399, K1=0.02038825898700673

k2=7275.7 08194557086, K2=0.01252516703250473

k3=68.65429354582977, K3=0.2569758737264554

Comparison is made between the simulated concentrations ob-

tained from our estimated kinetic parameters and the experimental

concentrations in Fig.1. Because the exact values of the kinetic pa-

rameters are unknown, the only way we can show the perfor-

mance of the hybrid optimization method is to compare our esti-

mated results with those in other published investigations. Here, for

the comparison the estimated results made by Turchi et al. [1996]

and Wolfrum et al. [1997] are reproduced in Fig.2, where their pa-

rameter estimation was based on single component data (separate

estimation for each component). For clarification, data are arranged

in the same style in the two figures, where it is very clear that our

hybrid optimization method gives a more accurate estimation of

the kinetic parameters in the system, because the simulated con-

centrations based on the hybrid optimization method fit more ac-

curately with the experimental data than those based on separate

estimation though the mathematical kinetic models are the same.

Furthermore, only 27 experimental concentrations of each reactant

are used for our estimation, while it has been reported in the previ-

ous literature that a series of experiments for each component are

Fig. 1.Comparison of experimental observations with the simu-

lated concentrations based on the estimated parameters by

the hybrid optimization method (All experimental data are

the same as those in Fig. 2).

Fig. 2.Comparison made by Turchi et al. [1996] between experi-

mental observations of photocatalytic process given in Eq.

(1) and simulated concentrations based on parameter esti-

mation by single component data (T denotes temperature,

RH denotes relative humidity).

Page 5

656L. Wang and C. N. Kim

September, 2001

necessary for their parameter estimation.

This application indicates that the hybrid optimization is feasible

for simultaneous estimation of kinetic parameters in the L-H equa-

tions for a multi-component system, and this method does give a

more accurate estimation than what the separate estimation method

does.

RELIABILITY OF THE ESTIMATION

In order to show the reliability of the kinetic parameter estima-

tion, let’s analyze the sensitivity of the simulated concentrations with

respect to the estimated kinetic parameters, which determines error

transfer during the estimating process. To clarify the problem, a single

component system is used for discussion, whose L-H equation is:

(6)

where c is concentration, k is the reaction rate constants and K is

the adsorption equilibrium constants.

1. Sensitivity

It is known that there must be some measurement error intro-

duced into the measured concentration during experiments, and the

measurement error will affect the estimation results. The ratio of

error in the estimated parameters to that introduced in measurements

is determined by the sensitivity of the estimated parameter with re-

spect to the value of the object function. Because the experimental

concentration profile with time in the given system is fixed in the

process of parameter estimation (it was used as input), we can con-

sider the sensitivity of the estimated parameters with respect to the

simulated concentrations instead. It is not a difficult task to obtain

the sensitivity of simulated concentrations from the L-H equations.

(7)

(8)

where sk,m is the sensitivity of k with respect to the m-th simulated

concentrations, while sK,m is sensitivity of K with respect to the m-

th simulated concentrations. is the simulated concentrations at

the j-th instance with time and ∆t is time interval between concen-

tration sampling.

Generally, k>>1 while K<<1; it makes sk,m>sK, m in most cases.

This fact means that error in the estimated reaction rate constant is

larger than that in the estimated adsorption equilibrium constant,

where both the errors are caused by the experimental concentra-

tions error. However, it does not mean that the estimated K is more

reliable than estimated k, because their absolute magnitudes are dif-

ferent. If “relative sensitivity” is to be used, we can easily find the

error transfer:

(9)

(10)

where rsk, m and rsK,m are the relative sensitivity of the estimated k

and K with respect to the concentrations, respectively, and cref is the

reference concentration (we use the initial concentration as the re-

ference here). Because K is small,

and rsk,m<rsK,m, we can deduce that the estimated k has a little smaller

relative error than the estimated K when the same experimental con-

centrations are used, which means the estimated reaction rate con-

stant is a little more reliable than the estimated adsorption equilib-

rium constants.

Because the relative sensitivity is a function of the kinetic pa-

rameters (to be estimated) and the concentrations, this fact makes it

difficult to abstract a general formula of sensitivity analysis for all

reaction processes. Instead, we use a concrete numerical test to show

how the error introduced into the experimental concentrations will

affect the accuracy of the estimated parameters. Let k=100, K=0.01,

and the initial concentration is c(0)=100 in the system given by Eq.

(6). It is easy to simulate this system and the obtained concentra-

tions are sketched in Fig.3. To test the reliability of kinetic param-

eter estimation by the hybrid optimization method, we sample the

simulated concentration at 30 different instances along time span

evenly from 0 to 9 seconds, which are marked as discrete small cir-

cles in the figure. Here we pretend that we know nothing about the

values of k and K, and they have to be estimated with the sampled

concentrations. In an ideal case (no error exists in the experimental

concentrations), the estimated k and K by the hybrid optimization

method are

is a little larger than 1,

k=100.0000000360223

K=0.009999999993527

The obtained parameters are almost the same as their real values,

and the test indicates the hybrid optimization method has the po-

tential to recover kinetic parameters accurately. Now, let’s introduce

a random error into the input concentrations:

cinput=cexact+ε

(11)

where cexact is concentration without any error (calculated with given

parameter values), ε is the mean zero random error introduced into

the concentrations with |ε|<2 in this test, and cinput is concentration

r = dc

dt

----- = kKc

1+ Kc

------------- -

sk m

,

= 1dcˆm

-------------- 1

⁄

( )

dk

≈∆t

Kcˆj ( )

1+ Kcˆj ( )

------------------- -

j= 1

j= m

∑

⁄

sK m

,

= 1dcˆm

-------------- 1

⁄

( )

dK

≈∆t

kcˆj ( )

1+ Kcˆj ( )

[]2

------------------------- -

j= 1

j= m

∑

⁄

cˆj ( )

rsk m

,

= 1dcˆm

----------------------- 1

⁄

( ) cref

dk k ⁄

⁄

≈∆t

kKcˆj ( )

1+ Kcˆj ( )[]cref

-------------------------------

j= 1

j= m

∑

⁄

rsK m

,

= 1dcˆm

----------------------- 1

⁄

( ) cref

dK K

⁄

⁄

≈∆t

kKcˆj ( )

1+ Kcˆj ( )[]2cref

-------------------------------- -

j= 1

j= m

∑

⁄

1+ Kcˆj ( )

Fig. 3.Simulated concentration of the system given in Eq. (6) and

its sampling.

Page 6

On the Feasibility and Reliability of Nonlinear Kinetic Parameter Estimation657

Korean J. Chem. Eng.(Vol. 18, No. 5)

used for parameter estimation.

Considering the randomness of the introduced error, we run the

estimation 500 times to abstract an approximate statistical character-

istic. The distributions of estimated reaction rate constant are marked

in Fig.4, while distributions of estimated adsorption equilibrium

constant are in Fig.5. It is very clear from the figures that the es-

timated k and K are distributed in small bounded regions, respec-

tively, with their average values:

k=100.5615342848486

K=0.010061512537634

Here, the estimated parameters are very close to their real values. This

result shows that the errors in the estimated parameters are bounded,

and if enough data are available, the parameters can be recovered

accurately.

If only one set of data is available, the reliability of the parame-

ter estimation can be shown by the distribution of the relative error

defined in the following equation:

(12)

where e is the relative error and x denotes either k or K. denotes

the average of x. Here, to show the reliability of the estimated re-

sults, we use a probability density function with respect to the relative

error of the estimated results, where the probability density func-

tion is obtained by counting the number of points and by smooth-

ing the resulting probabilities to a curve. The probability density

functions of k and K are sketched together in Fig.6 for convenience

of comparison.

From the probability density functions, we can easily draw two

bottom lines. The first one is that each of estimated k and K has a

high and narrow peak around the zero relative error point with bound-

ed relative errors. It assures us that, when only one set of data is

used, the estimations are reliable even though some experimental

error exists in the concentration. The second one is that the reac-

tion rate constant can be estimated more reliably than the adsorp-

tion equilibrium constants, because the probability density function

of the former has a higher peak around the zero relative error point,

and this agrees with the result of the qualitative sensitive analysis

carried out before.

2. About the Object Function

In the kinetic parameter estimation, the role of the object func-

tion is to make the simulated concentrations as close to the exact

concentrations (no experimental concentration error) as possible. If

the exact concentrations are available (which is not the case that

we can expect), the ideal object function should be

(13)

where cexact(m) is the ideal exact concentrations while csimulated(m) is

the simulated concentrations, and m denotes m-th instance with time.

Apparently, the function’s minimum is zero, which means there is

no difference between the simulated and the exact concentrations

when the function is minimized, and in this case the estimated pa-

rameters should be accurate, as shown in the above numerical test.

But, in practice, the experimental errors always exist and are un-

known, which makes the exact concentrations unavailable; there-

fore, the object function given by Eq. (2) is commonly used instead.

e = x − x

() x ⁄

x

J =

cexactm

( ) − csimulatedm

( )[]2

m= 1

p

∑

Fig.4. Distribution of the estimated reaction rate constant with

500 times estimation including random error.

Fig.5. Distribution of the estimated adsorption equilibrium con-

stants with 500 times estimation including random error.

Fig. 6.Distributions of the probabilities of the estimated parame-

ters with respect to relative error when random error is in-

troduced into the system (based on 500 estimations).

Page 7

658L. Wang and C. N. Kim

September, 2001

But, this substitution (replace) causes the simulated concentration

to deviate lightly from the exact one. To show it clearly, the object

function is rewritten here as:

(14)

in which the experimental concentration is decomposed into two

components: the one is the exact concentration and the other is the

experimental error ε(m). If the simulated concentrations and the

exact concentrations are the same, the object function defined in

Eq. (14) will have the value of

this situation in a real application, it is assumed in the estimation

algorithm that the simulated concentration is the same as the exact

one when the object function defined in Eq. (14) is minimized. In

fact, it is not what we can get in a real application.

Let’s consider the following inequality:

. Because we cannot find

(15)

It indicates that the minimum of object function defined in Eq. (14)

should be smaller than , where csimulated(m) are not equal to

cexact(m). Of course, the estimated parameters are also influenced

by the errors in the experimental concentrations.

Here, we can consider the “overshoot” of the object function,

which can be defined as the value by which the minimum of the

object function defined in Eq. (14) is smaller than

the overshoot of the object function is generally small, the error in

the estimated parameters is also small and tolerable. On the other

hand, the “overshoot” is not caused by the optimization algorithm

itself, but by the fact that no detailed information about the experi-

mental error is available, which forces us to use the object function

defined in Eq. (14) instead of the one defined in Eq. (13). From the

information point of view, it is reasonable that we cannot recover

the kinetic parameters exactly when some information about the

reaction is lost in the experiment.

3.Further Verification

To show further the reliability of the hybrid optimization method

for kinetic parameter estimation, here the fitness of the estimated

concentrations (calculated based on the estimated kinetic parameters)

to the given exact concentrations is tested, where the initial values

used to calculate the estimated concentrations are different from

those used to estimate kinetic parameters. Let’s consider the fol-

lowing two component system:

. Because

(16)

where subscripts A and B denote components A and B, respectively.

Given the kinetic parameters as kA=100, kB=40, KA=0.04 and KB=

0.01, the exact concentration profile with different initial values of

components A and B can be easily calculated as lines in Fig.7, where

three sets of initial concentrations are chosen for the test. In Fig.

7(a), the initial concentrations are cA(0)=180 and cB(0)=180, in

Fig.7(b) cA(0)=120, cB(0)=120, while in Fig.7(c) cA(0)=60, cB(0)=

60. To estimate the kinetic parameters, the exact concentration pro-

files are sampled evenly from the time range of 0 to 20 seconds with

J =

cexactm

( ) + ε m

( ) − csimulatedm

( )[]2

m= 1

p

∑

ε m

( )

2

m= 1

p

∑

cexactm

( ) + ε m

( ) − csimulatedm

( )[]2≤

m= 1

p

∑

cexactm

( ) − csimulatedm

( )[]2

+

ε m

( )2

m= 1

p

∑

m= 1

p

∑

ε m

( )

2

m= 1

p

∑

ε m

( )

2

m= 1

p

∑

dcA

dt

------- = − kBKBcB − kAKAcA

1+ KAcA + KBcB

------- = −

kAKAcA

1+ KAcA + KBcB

---------------------------------- -

dcB

dt

------------------------------------- -

Fig. 7.An artificial experiment of a two-component system. (Exact

concentrations are denoted by lines, while sampled concen-

trations with random error are denoted by discrete mark-

ers).

(a) Initial concentrations of cA(0)=180 and cB(0)=180 (mol/L).

(b) Initial concentrations of cA(0)=120 and cB(0)=120 (mol/L).

(c) Initial concentrations of cA(0)=60 and cB(0)=60 (mol/L).

Page 8

On the Feasibility and Reliability of Nonlinear Kinetic Parameter Estimation659

Korean J. Chem. Eng.(Vol. 18, No. 5)

an interval of 0.5 second. A bounded random relative error, whose

maximum will not exceed 5% of the concentrations, is superposed

on the exact concentration to simulate the experiment error. For com-

parison, concentration samples are also sketched in the figure as di-

screte markers.

Choosing the concentration samples in Fig.7(a) as input, the hy-

brid optimization method estimates the four kinetic parameters as

kA=94.64923483415211, kB=42.39506915461477

KA=0.03972790629826880, KB=0.008864935373366311

If the concentration samples in Fig.7(b) are chosen as input, the

estimated parameters will be:

kA=111.2879861550922, kB=38.65915807659286

KA=0.03671313198303160, KB=0.001036427370199021

While the concentration samples in Fig.7(c) are used, the estimated

result will be:

kA=98.99980578291010, kB=37.01748896537582

KA=0.04292663036394419, KB=0.001120936475488305

It is clear that all three sets of the estimated parameters are close to

the exact parameter values with a very small error, which indicates

that the hybrid optimization method is independent on the initial

concentrations.

To check the reliability of the hybrid optimization method in an-

other way, one of the three sets of estimated parameters mentioned

above is chosen to calculate the concentration profile with different

initial values. Here, the estimated parameters related to the initial

concentrations cA(0)=120, cB(0)=120 are chosen, but the concentra-

tion profiles are calculated with the initial values cA(0)=180, cB(0)=

180 and cA(0)=60, cB(0)=60. The goodness of the approximation

to the exact concentrations is sketched in Fig.8, where the good-

ness is denoted by a concentration difference between the exact one

(given in Fig.7(a) and (c)) and the currently calculated one. The

concentration differences shown in Fig.8(a) are calculated with cA

(0)=180, cB(0)=180, while in Fig.8(b) initial values are cA(0)=60,

cB(0)=60.

When the initial concentrations are cA(0)=180, cB(0)=180 in this

figure, the maximal difference is only about 5; when cA(0)=60, cB

(0)=60, the maximal difference is only about 1.2. The maximal re-

lative errors in both cases are less than 3%, which is less than the

relative errors introduced into the concentration samples. It is clear

in Fig.8 that the concentration differences between the exact and

the estimated value are very small with both sets of initial concen-

trations. The small concentration difference indicates that the esti-

mated kinetic parameter is a good approximation to the real exact

value. Considering the fact that the initial concentrations used for

comparing concentration difference are different from that used to

estimate kinetic parameters, it can be concluded that the hybrid op-

timization method has the capability to estimate kinetic parameters

reliably with only one set of concentration samples, provided that

the mathematical model is a reasonable description of a focused

reaction. Of course, when many sets of data are available, the esti-

mation can be improved by some related statistical method.

CONCLUSION

For a photocatalytic reactions, either single component or multi-

component systems, it has been customary to estimate kinetic pa-

rameters separately based on single component data, even when the

L-H equations are coupled where more than one reactants are in-

volved. In this study, for a multi-component photocatalytic system,

it has been illustrated that simultaneous parameter estimation is fea-

sible in solving the relevant multi-objective optimization problem

by a hybrid genetic-simplex method. In this method the genetic al-

gorithm is used to find roughly an optimum in a rather wide range,

while the simplex algorithm is used sequentially to refine the rough

optimum and make it accurate. Applying it to a real reaction, we

found the estimated results by the hybrid optimization method are

more accurate than those of existing investigations. By sensitivity

analysis and numerical verification, it has been also shown that the

estimation is reliable even when only one set of experimental data

(with unknown error) is available. This investigation proposes an

effective approach to abstract kinetic parameters from very few ex-

perimental data.

ACKNOWLEDGEMENT

This work was supported by Korea Research Foundation Grants

Fig.8. (a) The concentration difference between the exact one

[given in Fig.7(a) and (c)] and the currently calculated one.

(a) Initial concentrations of cA(0)=180 and cB(0)=180 (mol/L).

(b) Initial concentrations of cA(0)=60 and cB(0)=60 (mol/L).

Page 9

660L. Wang and C. N. Kim

September, 2001

(KRF-1999-005-E00025).

NOMENCLATURE

ci

cexact(m)

: concentration of component i [mol/L]

: the ideal (without error) concentration at the mth in-

stance [mol/L]

: the mth observation of concentration of the compo-

nent i [mol/L]

csimulated(m) : the simulated concentration at the mth instance [mol/

L]

: the mth simulated concentration of component i [mol/

L]

e : the relative error

Fj

: the fitness of jth chromosome

J : object function

Ji

: object function of component i

k

: collection of all reaction rate constant

ki

: reaction rate constant of component i [mol/(L)(s)]

ki

: the jth possible value of ki

K

: collection of all adsorption equilibrium constants

Ki

: adsorption equilibrium constant of component i [L/

mol]

Ki

: the jth possible value of Ki

lk : lower limit of the reaction rate constant

lK: lower limit of the adsorption equilibrium constant

Pj

: possibility of the jth chromosome

ri

: reaction rate of component i [mol/(L)(s)]

uk : upper limit of the reaction rate constant

UK: upper limit of the adsorption equilibrium constant

Vi

: the ith chromosome

wi

: average weight of component i

x : one of estimated reaction rate constants or adsorption

equilibrium constants

: average of x

x

ci(m)

j

j

Greek Letter

ε(m) : random error at the mth instance

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