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Identification of Kinetic Parameters

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Control and Monitoring of Chemical Batch Reactors

Part of the book series: Advances in Industrial Control ((AIC))

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Abstract

This chapter provides a general introduction to the identification of mathematical models and to the methods for estimating the relevant adjustable parameters. First, the Bayesian approach is briefly discussed as compared to Popper’s falsificationism. Then, the maximum likelihood and weighted least squares criteria are derived from the concept of conditioned probability. The different optimization techniques for parameter estimation are reviewed, with a particular emphasis on the resolution of implicit nonlinear models, which are encountered in chemical kinetics when analyzing experimental data measured in batch chemical reactors. Finally, a quantitative example, based on the phenol–formaldehyde reaction introduced in Chap. 2, is presented.

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Abbreviations

A:

reactant phenol

C :

concentration [mol m−3]

Co:

confirmation of a theory

d :

experimental data

D :

matrix of experimental data

e :

experimental error

E a :

activation energy [J mol−1]

Ex:

experimental results

f :

function in an implicit mathematical model

f p :

probability density function

G :

positive definite matrix

H :

Hessian matrix

ΔH R :

molar enthalpy change of reaction [J mol−1]

I:

reaction intermediate

I :

identity matrix

k 0 :

preexponential factor [(mol m−3)1−n s−1]

k c :

rate constant [(mol m−3)1−n s−1]

n :

reaction order

N C :

number of dependent or state variables or of components

N D :

number of data in the sample

N L :

number of lumped chemical reactions

N M :

number of measured variables

N P :

number of adjustable parameters

N U :

number of input variables or constants

N Z :

number of isothermal runs

p :

probability

P:

desired product, trimethylolphenol

\(\dot{q}\) :

specific thermal power [J m−3 s−1]

R :

reaction rate [mol m−3 s−1]

\(\mathcal{R}\) :

universal gas constant [J mol−1 K−1]

\({\hat{s}}_{\mathrm{D}}^{2}\) :

corrected sample variance

T :

temperature [K]

Th:

theory

U :

objective function

u :

vector of input variables

V :

covariance

x :

vector of state variables

y :

vector of measured state variables (outputs)

Y :

matrix of computed values to be compared with the experimental data

\(y_{\dot{q}}\) :

computed value of the specific thermal power [J m−3 s−1]

w :

weights

W:

undesired product

α :

constant in (3.66)

γ :

coefficient in (3.31)

Γ :

matrix of coefficients in (3.32)

ε :

error generated by the model

ζ:

generic random variable

θ :

adjustable parameter

θ :

vector of adjustable parameters

κ :

step length

λ :

damping factor

ν:

corrective factor for the Levenberg–Marquardt method

σ 2 :

universe variance

σ C :

root mean squared errors for the concentrations

\(\sigma_{\dot{q}}\) :

root mean squared errors for the specific thermal power

ϕ :

partial sensitivity

φ :

function in an explicit mathematical model

Φ :

function in a linear model

ψ :

known term in (3.31)

Ψ :

matrix of known terms in (3.33)

av:

mean value

m:

measured value

max :

maximum

min :

minimum

r:

reactor

s :

step index in the nonlinear optimization procedure

\(\widehat{~}\) :

best estimate or optimal value

o :

reference value

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Authors and Affiliations

  1. Dipartimento di Ingegneria e Fisica dell’Ambiente, Università degli Studi della Basilicata, Viale dell’Ateneo Lucano 10, 85100, Potenza, Italy

    Prof. Fabrizio Caccavale

  2. Dipartimento di Ingegneria e Fisica dell’Ambiente, Università degli Studi della Basilicata, Viale dell’Ateneo Lucano 10, 85100, Potenza, Italy

    Mario Iamarino

  3. Dipartimento di Ingegneria e Fisica dell’Ambiente, Università degli Studi della Basilicata, Viale dell’Ateneo Lucano 10, 85100, Potenza, Italy

    Francesco Pierri

  4. Dipartimento di Ingegneria e Fisica dell’Ambiente, Università degli Studi della Basilicata, Viale dell’Ateneo Lucano 10, 85100, Potenza, Italy

    Vincenzo Tufano

Authors
  1. Prof. Fabrizio Caccavale
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  2. Mario Iamarino
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  3. Francesco Pierri
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  4. Vincenzo Tufano
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Correspondence to Fabrizio Caccavale .

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Caccavale, F., Iamarino, M., Pierri, F., Tufano, V. (2011). Identification of Kinetic Parameters. In: Control and Monitoring of Chemical Batch Reactors. Advances in Industrial Control. Springer, London. https://doi.org/10.1007/978-0-85729-195-0_3

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  • DOI: https://doi.org/10.1007/978-0-85729-195-0_3

  • Publisher Name: Springer, London

  • Print ISBN: 978-0-85729-194-3

  • Online ISBN: 978-0-85729-195-0

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