Mathematical modeling of the dynamics of heat and mass transfer, phase transformations and thermal decomposition at high temperature drying of biomass

A mathematical model and method for calculating heat and mass transfer, phase transformations and shrinkage during drying of colloidal porous bodies in the form of a final cylinder are presented. A mathematical simulation of high-temperature drying of willow accompanied by the destruction of hemicellulose was carried out using the data obtained on the values of the activation energy of various types of biomass.

The use of biomass for the production of solid fuels has a high potential for the transition to renewable energy sources in Ukraine and in Europe. Emissions from combustion of biomass are recognized as CO 2 -neutral. Quality fuel pellets are obtained from raw materials with humidity of 8 -12%. In Ukraine, drying of crushed biomass is carried out mainly in drum driers at coolant temperatures Tc = 120 -170 ºС. The organization of high-temperature (Tc = 300 -500 ºС) drying of biomass intensifies the process and is accompanied by thermal decomposition of biomass. At the initial stage of thermal decomposition, hemicellulose decomposes with the release of oxygen-containing gases and pyrogenic moisture. This contributes to increasing the caloric content of biofuels. At temperatures above 270 ° C, cellulose and lignin begin to decompose, and these processes in the presence of air are exothermic for all types of biomass. Their passage contributes to the loss of the combustible component of biofuels. Consequently, the development of technologies for high-temperature drying of biomass is important for compliance with the condition of passing only the first stage of thermal decomposition. The possibilities of experimental methods for investigating high-temperature dehydration and thermal destruction in the particles of small size are substantially limited. More promising is the use of the method of mathematical modeling.
Biomass particles are colloidal capillary-porous bodies of cylindrical shape. Drying in high-temperature coolant involves the passage of heat and mass transfer through diffusion, filtration and phase transformations. Pyrogenetic water is removed along with residues of free and bound biomass moisture. The mathematical model [1] of dynamics of dehydration of biomass was based on the equation of transport of substance (energy, mass) [2] V y r LI y  Here U fl , U v , U aivolumetric concentrations of liquid, vapor and air phases; Тtemperature; ttime; c efeffective heat capacity, ; w efeffective filtration rate, ; w fl , w gare the velocity vectors of the filtration motion of the liquid and gas phases, which, according to the Darcy law are proportional to gradient the phases fl P  and g P  ; L is the heat of phase transformation.
The effective diffusion coefficient of the liquid D fl was determined by the formula obtained by Nikitenko N.I. [3], and a pair of D v -from the kinetic theory of gas: Here А D is the activation energy of the liquid molecules for the diffusion process. The pressures of fluid P fl and gas P g are expressed in terms of the functions U fl , U v , U ai and T. Volume fractions of the solid component  s , liquid  fl and gas  г in porous body are defined:  s = 1-П,  fl = U fl / fl ,  g = 1 - s - fl , partial densities of vapor and air:  v =U v / g and  ai =U ai / g , partial pressures of vapor P п and air P в : P п = п R у T/ п and P в = в R у T/ в . The pressures of gase and liquid phases: , r min <r*< r max . (6) Here, (r) is the volume fraction of the capillary occupied by the liquid; f(r) is the differential function of pore size distribution; σ(Т) is the coefficient of surface tension; r* is the characteristic parameter of the dispersion of pore sizes; r min and r max are the minimum and maximum pore radii of a unit volume.
Intensity of phase transition on the outer surfaces of the biomass particles [4] is as the difference between the evaporating liquid streams and the condensed vapor where ε is the radiation coefficient;  fl is the density of the liquid; δ* is the thickness of the condensate layer in which the evaporation process takes place; φ b is the body moisture, which is determined from the sorption isotherm equation, depending on U fl ; А is the activation energy; R is the universal gas constant; φ en is the relative humidity of the medium, eq v e.m. / P P   , P v is the water vapor partial pressure, P eq is the saturation pressure. The expression for the intensity of evaporation I V in the unit volume of the body follows from formula (6) under the condition of a local thermodynamic phase equilibrium where φ is the relative humidity of the gas in the pores of the body, )] ; S is the area of the contact surface of the liquid and gas phases. To determine the function S in pores of the unit volume of a body, which is not completely filled with liquid, the formula [5] was where the derivative ∂U fl /∂φ b is found from the sorption isotherm equation. If the isotherm equation is given in the form , g = const, then . The data on the equilibrium moisture content W eq for wood [6] accurately describes the equation , W max corresponds to φ=1 and at 100°C equals 16%, U fl = 0,01Wρ b .
Volumetric deformation ε V was determined on the basis of the thermoconcentration deformation equation of Nikitenko N.I. [5], that was solved in [7] analytically for the case of an osseo-symmetric stressed cylinder state due to the heterogeneity of the temperature fields and the concentration of the components of the bound substance.
On the surfaces of the particles in contact with the drying agent, the boundary conditions of the heat-mass transfer of the third kind are given.
The solution of differential equations (1) -(4) under the conditions (10) -(11) was carried out by a numerical method developed on the basis of an explicit three-layer recalculated difference scheme of Nikitenko N.I. [5] and procedures for splitting the algorithm into physical factors. Difference approximation of the transfer equations (1) -(4) are presented in [1].
To confirm the adequacy of the mathematical model and the efficiency of the numerical method, a physical modeling of the kinetics of drying of energy willow particles of cylindrical form in the air flow and mathematical modeling of the process at the same initial data was carried out: Т 0 = 303 К; W 0 = 1,3 kg/kg; А = А D = 0,420510 8 J/kmol; П = 0,58. The results of the calculation and the experimental data presented in Figure 1 are well-coordinated. As shown in [8], the onset of the thermal decomposition of hemicellulose in the drying process of different types of biomass is characterized by a sharp change in the effective energy of activation of the microparticles of the bound substance. Some results obtained in [8] are given in Table 1. Since the process of thermal decomposition, like diffusion and evaporation [5], is an activating, in mathematical modeling of joint processes of drying and thermal destruction in the calculation program developed on the basis of (1) -(4), in expressions (6), (7) and for D fl after material temperature has reached the beginning of thermal decomposition, the value of the activation energy of water is changed to a value corresponding to the temperature interval of decomposition of hemicellulose [8] (Tab.1). For hardwood species of trees A ef = A Def = 0,7525 ·10 8 J / kmol. Figure 2 shows the results of numerical experiments. а) T e.m =300 °С b) ) T e.m =350 °С Fig. 2. Change in time of the average moisture content W, temperature T, and maximum temperature T max on the surface of energy willow particles in the sizes d/h = 5,6/10 mm (curve 1) and d/h = 5,6/10 mm (curve 2) at drying without and with taking into account thermal decomposition (W ', T', T max ') in flue gases with parameters: w e.m = 2 m/s, d e.m = 18 g/ kg

Conclusion
The process of thermal decomposition, as well as the processes of diffusion and evaporation, is activating. Taking into account the mathematical model of this phenomenon will allow to organize the process of biomass drying effectively and improve the quality of biofuels.