A mathematical model of a special class of solid/fluid chemical reactions has been developed for use in analyzing the pyrolysis of particles of cellulose, wood, and similar biomass materials. Although it involves some simplifying assumptions (as do all models of chemical and physical processes), this model is nevertheless sufficiently detailed to yield spatially and temporally resolved approximations of chemical compositions and temperatures of particles and of liquid and gaseous products as they evolve during the pyrolysis process.

In the model, each particle is assumed to be spherical and porous. The particle is assumed to be chemically reactive, with known volumetric reaction rates for the various chemical species present. It is assumed that the porosity (pore volume ÷ total volume) and the permeability of the particle are large enough to allow for continuous flows of gases. For these and other purposes, the detailed pore structure is not analyzed and instead, only the bulk or "effective" properties associated with porosity are represented. Unlike in some other models, conditions in the environment of the particle are not assumed to be quasi-steady. Conditions at the surface of the particle are allowed to evolve according to the equations of the model and the far-field temperature and pressure.

The model provides for the formation of solid (char), liquid (tar), and gaseous products. Both convection and diffusion of the liquid product(s) are neglected; this assumption is justified, provided that either the viscosity of the liquid substantially exceeds that of the gases or the characteristic reaction time of the liquid is much smaller than its characteristic diffusion and convection times.

The dynamics of flows of gases and of chemical reactions within the particle are represented by coupled equations for conservation of mass, momentum, and energy. For the gaseous chemical species, the equations for conservation of mass include both flow-divergence terms and source terms proportional to the products of partial densities and reaction rates. For the solid and liquid species, only the source terms are included. The reactions are assumed to be irreversible and of first order.

For modeling conservation of energy, all species are assumed to be in local thermal equilibrium in a mixture. The partial internal energy of each chemical species is assumed to be proportional to the its partial density × its specific heat at constant volume × the local temperature. The rate of change of local internal energy is represented by a sum of (1) heat-release rates proportional to the chemical-reaction source terms, (2) conduction proportional to the local gradient of temperature × an effective thermal conductivity according to a mixture-and-porosity-based submodel, and (3) convection of heat via the flow of gaseous species.

The conservation of momentum for the gas phase is expressed by a single equation for radial flow of a compressible gas mixture through pores, with an effective molecular viscosity proportional to the porosity × the sum of mass-fraction-weighted viscosities of all gas species. The set of equations is completed by an equation of state, based on the perfect gas law, wherein the total local pressure is expressed in terms of the effective local total gas density, porosity, local temperature, and local volume fractions of the gas species.

The model was applied in a parametric study of the effects of initial reactor temperature, heating rate, porosity, and initial particle size on the char yields and conversion times in the pyrolysis of spherical biomass particles in initially quiescent superheated steam. In this study, reaction rates (see figure) were calculated from previously published data on the kinetics of chemical reactions for cellulose and wood. The equations of the model were solved numerically. The solutions qualitatively reproduced aspects of pyrolysis as observed in previous mathematical-modeling and experimental studies.

In particular, the numerical results indicated that one can reduce the production of char by decreasing the initial particle size or by increasing the reactor temperature and heating rate, but that the achievable decrease in the production of char is limited by endothermic reactions, heat capacities, and thermal diffusion. Three pyrolysis regimes were identified: (1) initial heating followed by (2) primary reaction at an effective pyrolysis temperature followed by (3) final heating.

A Few Principal Chemical Reactions are included in the model of the pyrolysis of cellulose and wood. The reaction rates are calculated by the Arrhenius equation

*K _{i}* =

*A*exp

_{i }*(-E*,

_{i }/R̄T)where *K *is a reaction rate, *A* is a rate frequency constant, *E *is an activation energy, *R̄ *is the universal gas constant, *T *is the absolute temperature, and the subscript *i *denotes the *i*th reaction path or chemical species.

*This work was done by Josette Bellan and Richard S. Miller of Caltech for *NASA's Jet Propulsion Laboratory*. For further information, access the Technical Support Package (TSP) **free on-line at www.techbriefs.com** under the Physical Sciences category, *or circle no. 192*on the TSP Order card in this issue to receive a copy by mail ($5 charge). NPO-20070*

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###### Mathematical model of pyrolysis of biomass particles

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