(550f) A Generalized Semi-Analytical Method for Solution of Ordinary Differential Equations Applied to a Model of Heat Transfer in a Biomass Pyrolysis Reactor

Authors: 
Ender, L., Regional University of Blumenau
Bertoli, S., Regional University of Blumenau
Almeida Junior, J., Regional University of Blumenau
Bastos, J. C. S. C., University of Blumenau
Fernandes de Carvalho, L., Regional University of Blumenau
Silvana, L., Furb
The optimization and improvement of industrial processes depends on the development and solution of mathematical models that describe the phenomena involved. In this way, the rational understanding and the modeling of heat transfer phenomena in industrial equipment is fundamentally important to the scale up and optimization of processes.

Pyrolysis is considered a promising form of energy generation because it is an independent and self-sustaining process that can use raw materials from renewable sources and it is generally conducted in moving bed reactors. This process has attracted the attention of the scientific community in recent decades due to its potential as one of the solutions of the energy crisis. The process consists of the thermal rupture of chemical bonds in the absence of oxygen, with the formation of solid (coal), liquid (tar and other organic) and gaseous products.

In this context, a general semi-analytical method for solution of ODE´s is proposed and applied to a pyrolytic reactor model. The model is based on the following hypotheses and restrictions: wall considered as an isothermal black body, uniform physical properties, spherical particles uniformly distributed in the cross-section of the tube, fluid transparent to thermal radiation, steady-state operation, flow developed in both phases and velocity profiles of the fluid and particles uniform in the cross section of the tube, particles and fluid may have different velocities, uniform fluid temperature in the cross section of the tube, and diluted bed.

The numerical solution is developed through the Finite Analytical Method – FAM, which is based on decomposing a total region of a problem described by differential equations into a number of small elements in which local analytical solutions are obtained. In the intervals, the terms are made linear and evaluated at the temperature conditions in the beginning of the interval in question. The generalized numerical solution is applicable to systems described by first order ordinary differential equations and tested in the model in order to verify the results with known analytical solution. For the case in question the comparison with the analytical solution shows excellent agreement.