Simulation of a Moving Bed Reactor used in the Pulp and Paper Industry
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Problem Description
The pulp and paper industry plays an important role in European economies. The chemical reactions that transform wood chips in pulp occur mainly in a complex moving bed reactor, the socalled digester. Nowadays, the use of mathematical models to simulate the transient behaviour of temperature and compound concentrations represents a real need for industry since experiments are either expensive or risky.
The digester  the most critical piece of the equipment  is an heterogeneous reactor with an almost cylindrical shape, where wood chips react with an aqueous solution of sodium hydroxide and sodium sulfide, to remove the lignin from the cellulose fibers.
Modelling & Computational Challenges
From a mathematical point of view, the dynamical behaviour of the reactor can be represented by a system of hyperbolic nonlinear partial differential equations. Among the 15 equations of the system, we can identify three main types: the equations that describe the temperature and the concentration of, respectively, the solid, the entrapped liquid, and the free liquid phase. Each one of these type of equations present a certain complexity, its numerical simulation being an hard task. Several factors contribute to the difficulty in the numerical simulation: (i) the high nonlinearity of the functions that represents the chemical reactions; (ii) the discontinuities induced by the extraction, enrichment and heat of the free liquor; (iii) the discontinuities in the convection velocity of the free liquor  positive where the liquid flown downwards and negative where the free liquid flows upwards.
Research at LCM
In this project, we develop numerical methods, with good qualitative properties, based on operator splitting, nonuniform refinement and some particular techniques to smooth discontinuities. Special attention has devoted to the effects induced in the process by discontinuities of wood chips composition. The most critical piece of equipment in a kraft pulp and paper plant is the digester.
The digester is a complex heterogeneous reactor where a moving bed of wood chips, containing cellulose, hemicellulose and lignin, reacts with sodium hydroxide and sodium sulfide, in a liquid phase, to remove the lignin from the cellulose fibers. The digester studied in this project has been described in [4][5] and it is an industrial hydraulic reactor with isothermal cooking, a further development of modified continuous cooking.
The mathematical transient model is derived from the fundamental principles of mass and energy balances and it is represented by a system of 15 nonlinear Partial Differential Equations (PDE) of convectionreaction type described in detailed in [5].
The numerical method used in the discretization of this PDE's system is based on operator splitting which essentially consists in considering separately convection and reaction phenomena. Splitting methods have been studied by several authors. A mathematical analysis of convergence for the method used in this paper is developed in [1][3].
The main aspects of our approach are the following: from an engineering point of view, the system of PDE's presented furnishes a description of the transient behaviour of the digester which allows the prediction of the quality of the pulp when some changes in the wood properties occur; from a mathematical point of view, the application of splitting methods to the system allows the use of specially tailored methods adapted to the phenomena that take place in each part of the digester.

Papers & Reports
 [1] A. Araújo, J. A. Ferreira, N. Fernandes, and P. Oliveira, Using splitting methods in continuous digester modeling, Applied Mathematical Modelling, 32(9) (2008) 18691882
 [2] A. Araújo, J. A. Ferreira, N. Fernandes, and P. Oliveira, Simulation of the transient behaviour of a digester used in the pulp and paper industry, Computer Aided Chemical Engineering, 18 (2004) 325330
 [3] A. Araújo, J. A. Ferreira, and P. Oliveira, The role of abstract numerical properties in the change of character of reactive flows, Journal of Mathematical Fluid Mechanics, 7 (2005) 141163
 [4] A. Araújo, J. A. Ferreira, P. Oliveira, F. Patrício, and P. Rosa, The use of splitting methods in the numerical simulation of reacting flows, Computing and Visualization in Science, 6(23) (2004) 5966
 [5] N. C. Fernandes and J. A. A. M. Castro, Steadystate simulation of a continuous moving bed reactor in the pulp and paper industry, Chemical Engineering Science, 55 (2000) 37293738.
 [6] N. C. Fernandes, Modelização de um digestor contínuo de cozimento kraft, PhD Thesis, University of Coimbra, 2007
Software
 A Matlab code has been developed and is available under request.
Project Team
 Adérito Araújo, LCM/CMUC
 José Augusto Ferreira, LCM/CMUC
 Maria Fernanda Patrício, LCM/CMUC
 Paula de Oliveira, LCM/CMUC
 Paulo Rosa, LCM/CMUC
 Natércia Fernandes, Department of Chemical Engineering
Project Reference
FCT Research Project  POCTI/35039/MAT/2000