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Modeling of oxygen delignification process using a Kriging-based algorithm

  • Gladson Euler
  • Girrad Nayef
  • Danyelle Fialho
  • Romildo Brito
  • Karoline BritoEmail author
Original Research


A phenomenological model of cellulose production processes presents limitations due to the presence of species and chemical reactions of complex computational representation. Modeling based on machine learning techniques is an alternative to overcome this drawback. This paper addresses the Gaussian process regressor (Kriging) method to model the oxygen delignification process in one of the largest pulp production plants of the world. Different correlation models were used to evaluate this method; furthermore, an optimization routine, based on the constrained optimization by linear approximation method, was coupled to model to minimize the objective function, which is based on the input cost. Results have shown the good performance of using a combined Kriging method with optimization routines in the non-linear industrial processes to obtain a representative model capable of providing optimized operating scenarios. A reduction of 36.5% in consumption of NaOH was obtained, while required restrictions are obeyed.

Graphic abstract


Kraft process Cellulose Pre-bleaching Gaussian process regressor Kriging 

List of symbols


Scale mixture factor


Regression parameters for the polynomial function


Characteristic length-scale


Vector of hyperparameters (parameters of the covariance function)






Euclidean distance between x and \({\text{x}}^{*} \left( {\sqrt {\left( {x - x^{*} } \right)^{T} \left( {x - x^{*} } \right)} } \right) ;\;{\text{x}} \ne {\text{x}}^{*}\)


Matrix of fixed-base functions

\(\varvec{f}_{\varvec{p}} (\varvec{x})\)

Fixed-base functions


Median Q of fraction k of dataset (0.25 or 0.75)

\(\varvec{R}(\varvec{\theta},\varvec{ x}_{\varvec{i}} ,\varvec{x}_{\varvec{j}} )\)

Covariance function (or kernel) evaluated at points x and \({\text{x}}^{ *} :\;{\text{x}} \ne {\text{x}}^{ *}\)

\(\varvec{R}\left[ {\varvec{R}(\varvec{\theta},\varvec{ x}_{\varvec{i}} ,\varvec{x}_{\varvec{j}} )} \right]\)

Correlation matrix or spatial correlation function


Set of response values for each sampling point


Response value for one specific input


The ith estimated output


The ith correct output


The arithmetic mean of the samples




Set of deviation functions


Deviation function



The authors thank the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for financial support for this work.


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Copyright information

© Springer Nature B.V. 2020

Authors and Affiliations

  1. 1.Chemical Engineering DepartmentFederal University of Campina GrandeCampina GrandeBrazil

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