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Identification of Kinetic Parameters

  • Fabrizio Caccavale
  • Mario Iamarino
  • Francesco Pierri
  • Vincenzo Tufano
Part of the Advances in Industrial Control book series (AIC)

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.

Keywords

Objective Function Kinetic Model Hessian Matrix Steep Descent Method Implicit Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of Principal Symbols

A

reactant phenol

C

concentration [mol m−3]

Co

confirmation of a theory

d

experimental data

D

matrix of experimental data

e

experimental error

Ea

activation energy [J mol−1]

Ex

experimental results

f

function in an implicit mathematical model

fp

probability density function

G

positive definite matrix

H

Hessian matrix

ΔHR

molar enthalpy change of reaction [J mol−1]

I

reaction intermediate

I

identity matrix

k0

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

kc

rate constant [(mol m−3)1−n  s−1]

n

reaction order

NC

number of dependent or state variables or of components

ND

number of data in the sample

NL

number of lumped chemical reactions

NM

number of measured variables

NP

number of adjustable parameters

NU

number of input variables or constants

NZ

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

Greek Symbols

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

σ2

universe variance

σC

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)

Subscripts and Superscripts

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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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • Fabrizio Caccavale
    • 1
  • Mario Iamarino
    • 2
  • Francesco Pierri
    • 3
  • Vincenzo Tufano
    • 4
  1. 1.Dipartimento di Ingegneria e Fisica dell’AmbienteUniversità degli Studi della BasilicataPotenzaItaly
  2. 2.Dipartimento di Ingegneria e Fisica dell’AmbienteUniversità degli Studi della BasilicataPotenzaItaly
  3. 3.Dipartimento di Ingegneria e Fisica dell’AmbienteUniversità degli Studi della BasilicataPotenzaItaly
  4. 4.Dipartimento di Ingegneria e Fisica dell’AmbienteUniversità degli Studi della BasilicataPotenzaItaly

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