Cellulose

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# Modeling of oxygen delignification process using a Kriging-based algorithm

Original Research

### Abstract

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.

## Keywords

Kraft process Cellulose Pre-bleaching Gaussian process regressor Kriging

## List of symbols

$$\varvec{\alpha}$$

Scale mixture factor

$$\varvec{\beta}_{\varvec{p}}$$

Regression parameters for the polynomial function

$$\varvec{\sigma}_{\varvec{l}}$$

Characteristic length-scale

$$\varvec{\theta}$$

Vector of hyperparameters (parameters of the covariance function)

$${\mathbb{E}}$$

Expectation

$$\varvec{Cov}$$

Covariance

D

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

$$\varvec{F}$$

Matrix of fixed-base functions

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

Fixed-base functions

Qk

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

$$\varvec{y}$$

Set of response values for each sampling point

$$\varvec{y}(\varvec{x})$$

Response value for one specific input

$$\hat{\varvec{y}}_{\varvec{i}}$$

The ith estimated output

$$\varvec{y}_{\varvec{i}}$$

The ith correct output

$$\bar{\varvec{y}}$$

The arithmetic mean of the samples

$$\varvec{Var}$$

Variance

$${\mathbf{Z}}$$

Set of deviation functions

$$\varvec{z}(\varvec{x})$$

Deviation function

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