Theoretical basis of continuous drying of dispersed materials

A mathematical model, a numerical method, and the calculations results of the heat and mass transfer dynamics and phase transformations in the processes of continuous drying dispersed materials are presented. The calculation of continuous drying a chopped wood was carried out. The calculation results testify to the adequacy of the mathematical model and the efficiency of the calculation method.


Introduction
Drying processes are an integral part of most production technologies in the energy, chemical, food, construction, paper, pharmaceutical and other industries. The trend of modern production processes is the introduction of continuous automated technological cycles, in which it is advisable to use continuous dryers. When developing effective modes of continuous drying, it is important to have the results of the dynamics of heat and mass transfer in a porous material, taking into account changes in the parameters of the coolant along the length of the drying chamber. If a dispersed porous material is to be dried, then the dispersion characteristics of the wet layer should be taken into account. The method of mathematical modeling makes it possible to solve these problems. In [1], for the first time, a mathematical model was built and a numerical method was developed for calculating the dynamics of heat and mass transfer and phase transformations during the dehydration of consolidated capillary-porous materials in continuous convective dryers. In [2], shrinkage was taken into account during continuous drying of colloidal capillary-porous materials. In this paper, a mathematical model of the dynamics of continuous convective drying of dispersed capillary-porous materials is presented.
Results and discussion The wet dispersed layer can be considered as a multicomponent heterogeneous system, including the skeleton, liquid, and gas-vapor mixture. When constructing a mathematical model of the dynamics of drying a dispersed layer, it was assumed that the liquid phase moves only in the particle capillaries, while vaporous moisture moves along the particle capillaries and in the space between them. The system of equations was built on the basis the differential equation of substance transfer W (energy, mass, momentum) [ where wspeed of substance movement, where wLspeed of conveyor belt, wψ filtration speed of substance's components (liquid  = fl, vapor  = v, air  = ai) relative to its skeleton; jWsubstance fluence rate W; ІWpower of internal sources of the substance; Vrelative volumetric strain. Transfer of the substance W is carried out by diffusion and filtration: In the steady state operation of the dryer: W/t = 0, V /t = 0 and wL = const.
When the filtration transfer of the substance takes place, the components  move relative to the body's skeleton with the speed wψ, and relative to the body of the apparatus with the speed w. For this case, equation (1) can be written as (2) At moderate temperature conditions of drying, heat and mass transfer in the dispersed layer is carried out by diffusion. Then wψ = 0 and (1) takes the form Usually, the width and length of the tape is much greater than the height of the layer of material on it. In the Cartesian coordinate system, where the x-axis is perpendicular to the working surface of the belt, and the z-axis is parallel to the wL vector, Equations (2), (3) can be simplified. Then the mathematical model for the case of continuous diffusion drying of a layer of a dispersed colloidal capillary-porous body where Ufl, Uvvolume concentrations of liquid and steam; λеf = λbb + λflfl + (λv + λai)g; сеf=сbρb(1-П)(1-εla)+сflUfl+сvUv+сaiaigeffective thermal conductivity and heat capacity of the dispersed layer; where, the volume fractions of the body b=(1-П)(1-εla), liquid fl = Ufl/fl and gas g=1-b-fl in the dispersed layer, Пmaterial porosity; εlalayer porosity; Lis the heat of vaporization; еhe relative volume deformation εV is based on the differential equation of the thermal-concentration deformation [1]. The diffusion coefficients of the liquid and gas phases is determined by the formulas: , where Runiversal gas constant, АDactivation energy, Pggas phase pressure, γDfl , γDv = Const.
To determine the effect of the porosity εla on the desired functions and the intensity of evaporation ІW, the control volume ΔV, which includes one particle, is considered. If the number of particles in a unit volume of np, then the average value of the control volume ΔV=1/np, and the average volume of the particle Vp = ΔV (1 -εla). In this case, relations were found for the volume concentrations of the components of the dispersed layer: The average area of the outer surface of the particle is determined by its effective size. The intensity of the phase transition in a unit volume of the dispersed layer is represented by [5] ( )  The solution of differential equations (4) -(6) under conditions (7) -(14) was performed by a numerical method based on a three-layer explicit difference scheme Nikitenko N.I. [1] and algorithm splitting procedures for physical factors. As an example, a thin layer of Xm=5,6 mm, П = 0,585, εla = 0,65 of crushed energy willow was considered, which was blown with a drying agent with initial parameters Te.m0 = 200 °С, Pe.m0 = 103 kPa, Pve.m0 = 1,6 kPa, we.m0 = 4,5 m/s, Ge.m0 = 8 kg/s. The calculation results are presented in Fig.1.

Conclusion
The values of the parameters of the coolant in each section z along the length of the channel are related to each other as in the diagram of the state of moist air. This testifies to the adequacy of the proposed approach to the calculation of the continuous drying process. As can be seen from the figure, the drying agent cools and moistens rather quickly, which significantly slows down the process of the material reaching equilibrium moisture content. In some cases, it is not possible to dry the material to a low moisture content in a continuous convective drying unit without additional heating and drying of the drying agent.