The asymptotic solution of model equations for heat capacities of fluidized bed phases

This paper is aimed at solving a singularly perturbed system of equations, describing the balance of mass and heat in a two-phase fluidized bed, using asymptotic methods. The time dependencies of heat capacities and temperatures of solid and gaseous phases have been obtained. The dependencies of relaxation times of these characteristics upon kinetic and thermophysical parameters of bed phases have also been found.


Introduction. The creation of model equations
It is known, that obtaining exact solutions of equations in a rather general theoretical model of heatand mass transfer processes when drying particulate materials is quite problematic. In some cases, this is due to non-linearity of equations, caused by the dependence of coefficients upon the sought values. In other cases, this is due to impossibility to record the boundary conditions in the analytical form. In view of these circumstances, it is advisable to resort either to numerical solutions of model equations, or to creating simplified models, adapted to specific conditions of process flow or to design features of devices [1], [2].
At the same time, a number of models, describing the transfer processes during drying materials in the dynamic bed, besides numerical solutions, permit the use of approximate analytical methods [3], including asymptotic ones [4].
The present paper is devoted to constructing the asymptotic solution of the system of differential equations for one model of heat-and mass transfer processes in the dynamic bed with directed movement of moisture-containing particulate material. The model, adopted in the work, belongs to the semi-empirical class [5], [6]. The basic model equations, describing the dynamics of temperatures and heat capacities of the bed subsystems are derived from the phenomenological relations of heatmoisture balance for particulate and gaseous bed phases [1], [7], [8], as well as from the semiempirical correlation by A.V. Lykov, simulating the change in moisture content of the material during the period of declining speed of mass transfer [9]. The assumptions, simplifying the model and given in the work [7], are also accepted.
As a rule, during drying the particulate material in conditions of fluidization, the ratio between the masses of material and gas, contained in the bed, is low. The indicated mass ratio (the gas content of a bed) is contained in equations for temperature and moisture content of the particulate component as a factor for derivatives from the corresponding variables. This circumstance permits to use the method of boundary functions, developed in the paper [10], for constructing the approximate solution of model equations. Having reformulated the ratios of mass balance, based on moisture content [7], as equations The values, introduced in formulas (3), have the following meaning: The condition of low gas content in the bed 1  is subsequently assumed to be satisfied. In this case, the equation (1) represents the singularly perturbed system of four differential equations for given heat capacities 0 () k C  and dimensionless temperatures () k   ( k  1, 2) of the bed subsystems. The solution of this system will be obtained in the first approximation of the perturbation parameter  .

The asymptotic solution of model equations
Using the method of boundary functions, we will get the principal terms of the asymptotic expansion of system solution (1).
The subsystem of equations in (1) for given heat capacities of the material and gas does not include the temperature terms Herewith, the first equation of the subsystem is autonomous and its solution is as follows The approximate solution of the second equation of the system (4) will be obtained in the form of expansion: where N is the specified level of approximation; It is easy to show that the series in the expression (7) converges under condition of 1   .
Therefore, the given heat capacity of the gaseous bed phase is expressed through heat capacity of the particulate phase. The approximate solution of the subsystem of temperature equations (0) 1, is carried out in a similar way. We seek the solution of the system (8)  After substituting (9) into the equations of the system (8), and subsequent equating the coefficients for similar degrees of the  parameter (separately for functions depending on «slow» ( ) and «fast» ( s ) arguments), we will obtain the differential equations for principal and boundary functions. The equations for the first terms of expansion (8) have the form