Using splitting methods in continuous digester modeling

Abstract

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 digester. Nowadays the use of mathematical models to simulate the transient behaviour of the digester in terms of temperature and compound concentrations represents a real need for industry because it allows simulation of experiments that can not be afforded or that might be very risky. The digester - the most critical piece of the equipment of a pulp mill - is a 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. From a mathematical point of view the dynamical behaviour of the reactor can be represented by a system of hyperbolic nonlinear partial differential equations. In this system, with 15 equations, we can identify three main types: the equations that describe the temperature and the concentration respectively of the solid, entrapped liquid and free liquid phase. Each of these type of equations present a certain complexity, its numerical simulation being a hard task. In this sense we point out the high nonlinearity of the functions that represent the chemical reactions; the discontinuities induced by the extraction and injection of the free liquor; the discontinuities in the convection velocity of the free liquor - positive where the liquid flown downwards and negative where the free liquid flows upwards. Numerical methods based on operator splitting, nonuniform refinement and some particular techniques to smooth discontinuities, are studied from a qualitative and quantitative viewpoint. Several simulations on temperature and concentrations of organic and inorganic compounds are presented. Special attention will be devoted to the effects induced in the process by discontinuities of wood chips composition.