Directed Fluidized Bed in Drying Machines: Main Stages of Optimization Calculation

The article deals with the study of main hydrodynamic characteristics of the directed fluidized bed in gravitation shelf dryers. The algorithm to calculate hydrodynamic characteristics of the directed fluidized bed in the dryer’s workspace is described. Every block of algorithm has theoretical model of calculation. Principles of disperse phase motion in various areas in the gravitation shelf dryer are established. The software realization of author’s mathematic model to calculate disperse phase motion trajectory in free and constrained regime, disperse phase residence time in the dryers’s workspace, polydisperse systems classification is proposed in the study. Calculations of disperse phase motion hydrodynamic characteristics using the software product ANSYS CFX, based on the author’s mathematic model, are presented in the article. The software product enables to automatize calculation simultaneously by several optimization criteria and to visualize calculation results in the form of 3D images. The data on the construction of a computational model and computational grid for modeling the gas flow motion is presented. The results of determining the gas flow velocities field in the workspace of a gravitational shelf dryer distinguishing the specific zones of the drying agent’s motion are demonstrated. High and low-intensity places of gas flow are established in the dryer.The residence time of particles in the dryer is calculated depending on the technological and design parameters of the drying process, in particular, the constraint degree of the particle flow. The influence degree of the dryer’s features and its operating mode on the expected "hydrodynamic" residence time of the material in the device is established. The obtained value of the "hydrodynamic" residence time of the material in the device is compared with the drying time to find the optimal design of the dryer.


Introduction
Implementing a fluidized bed in heat-mass transfer processes has become widespread due to the undoubted advantages of such a hydrodynamic system [1][2][3]. At the same time, there are often difficulties in devices with a fluidized bed providing the required hydrodynamic regime. In this regime, the dispersed phase is predicted to spend in the device the estimated time needed to complete the process. In this case, a significant advantage of devices with a directed fluidized bed is the ability to control the residence time of the dispersed phase in the workspace of the device [4]. The authors of this work attempted a theoretical description [5], experimental studies of the fluidized bed configuration [6,7] and the conditions for the implementation of heat-mass transfer processes [8,9] in other devices with the directed motion of the dispersed phase -vortex granulators [10]. The device and principle of a multistage gravitational shelf dryer's operation can be found in the following studies [11]. The aim of the article is to form an algorithm for calculating the hydrodynamic parameters of the flow in devices with inclined shelves for the implementation of heat treatment and dehydration processes. The research results will complement the general algorithm for the engineering calculation of shelf units, which authors begin to study in the research [12]. The authors use the hypothesis of the possibility to control the motion trajectory and the residence time of the dispersed phase in the dryer's workspace thanks to its directed transfer mechanisms (as shown in the work [13,14]). The joint solution of the basic equations of flow motion hydrodynamics, the kinetics of change of the temperature-humidity features in the interacting flows, and the growth rates of granules will allow inventing a rational design of workspace, heat transfer agent's optimum flow rate and its temperature-humidity features in a shelf dryer. The calculation is carried out by the optimization criterion of the "minimum" hydrodynamic "residence time of the dispersed phase in the workspace of the device." "Hydrodynamic" time should be equal to "kinetic" time -the period during which the temperature and humidity features of the dispersed phase should acquire a normative value.
2 Algorithm for calculating the hydrodynamic features of the directed fluidized bed in a shelf dryer

General statements
There are multiphase flows of different nature. Their survey is observed, for example, in [15][16][17]. The following types of multiphase flows should be distinguished. In the first case, the considered volume is filled with the substance of one phase. The substance of another phase occurs in the form of discrete particles (solid phase) or bubbles (gaseous phase) where the volume rate of the substance of the other phase is low (up to 10% of the total volume). In the second case, the considered volume is partially filled with liquid and partly with gas, which do not mix and the free surface separates them from each other. In the most difficult case, substances of different phases can mix (dissolve / stand out from solution), and the volume rate of the substance of another phase is large (over 10% of the total volume). Various approaches are used to model these multiphase flows. Analysis of previous works in the field of modeling two-phase flows consisting of gas as a dispersion phase and dispersed particles shows that one of the most promising ways to calculate particle motion is the so-called trajectory method. In modeling the constrained motion of particles of large (0.5-5 mm) diameter can be based on the Lagrangian model for the force analysis of particle motion using differential equations of motion. The trajectory method with highly accurate obtained results can be used in the case of software that exports a theoretical model of the single-particle motion and considers the constrained flow degree. It is necessary to determine the conditions of the constrained motion of particles in the dryer for this method implementation. In many cases, it is possible to use the dispersed particles model, the mixing model, and the multiphase Euler model, to simulate flows in which substances of different phases can mix and do not form a free surface. Additional criteria for choosing a model include [17].
The ratio β of substance mass of the dispersed phase (d) to the substance mass of the carrier phase (c): where Fd and Fc are volume rates, γ is the ratio of the dispersed and carrier phases density, γ = ρd / ρc; this ratio can be over 1000 for solids in a gas flow, about 1 for solids in a liquid flow, and less than 0.001 for gas particles in a liquid flow. At a low β ratio, dispersed particles practically do not affect the carrier phase flow, and any of the listed models can be used. At very high values of β, dispersed particles strongly affect the carrier phase flow, and only the multiphase Euler model should be used for the proper simulation of the flow. With average βvalues, one needs to calculate the Stokes number to select a suitable model, as described below.
where td is the time describing the motion of particles, td = (ρd dd 2 ) / (18 μc),dd is the particle diameter, μc is the viscosity of the carrier phase substance, tc = Lc / Uc is the time describing the carrier phase flow, Lc is the peculiar length, Uc is the peculiar velocity.
If St << 1.0, the dispersed phase particles almost do not deviate from the streamlines of the carrier phase, and any flow model can be used (as a rule, the mixing model is the least resource-intensive). If St> 1.0, the trajectory of the dispersed phase particles completely does not coincide with the carrier phase streamlines, and the mixing model is unsuitable in this case: either the dispersed particles model or the multiphase Euler model must be used. In the case under consideration, the dispersed phase motion trajectory is applicable to the dispersed particle model. The substance forming the main phase is assumed to be a continuous space, and its flow is modeled (depending on the flow turbulence degree) by the Navier -Stokes or Reynolds equations and the flow continuity. The substance in the flow in the form of discrete particles does not form a continuous medium. Individual particles interact with the flow of the main phase and with each other discretely. The Lagrange approach is used to model the particles motion of the dispersed phase. It means that the motion of the separate particles of the dispersed phase influenced by forces from the side of the main phase flow, is controlled. It should be noted that there is a constrained motion of particles (discrete particles, a solid phase, occupies a volume greater than 10% of the total space in the device) in the industrial model of a shelf dryer. In this case, the calculation model should be supplemented with a block that allows defining how the particles' residence time changes (increases) in the dryer if the volume of particles increases more than 10%.

Modeling of the gas flow motion
The previous experimental studies [18] demonstrate that the gas flow in the shelf dryer is turbulent. Direct modelling of the turbulent flows by calculation of the Navier -Stokes equations written for instantaneous velocities is complicated. Besides, not instantaneous, but time-averaged velocity values attract attention. For the analysis of turbulent flows, the Reynolds equation and the continuity of the flow are used [2,15]: In equations (3) and (4) the simplified equations are used, i, j = 1… 3, the summing up over the same indices is assumed, x1, x2, x3 -coordinate axes, t -time. The fi term expresses the action of mass forces.
In this system of 4 equations, the independent sought parameters are 3 velocity components u1, u2, u3 and pressure p. The density ρ of the liquid and the gas, at velocities up to about 0.3 of the Mach number, can be assumed to be constant. As the boundary conditions, the adhesion condition is set on all solid walls (the velocity is zero), the distribution of all velocity components in the inlet section, and the first derivatives (in the direction of flow) of the velocity components in the outlet section are equal to zero. Besides, the direct interest is the distribution of the velocity along the length of the device in space above the shelf, where the dispersed particles motion occurs. The calculation results by formulas (3) and (4) include the components of the gas flow velocity in the dryer's workspace. Further, the obtained components of the gas flow velocity are used in the differential equations of particle motion (these equations are below) and consider the degree of gas flow effect (transfer of angular momentum from the gas flow to the particles) on the prevailing direction of the particle velocity vector and the numerical value of the particle velocity.

Modeling of the particle motion in free motion mode
Let us assume that the dispersed phase particles have a spherical shape. The forces influencing this particle are caused by the difference between the particle velocity and the flow velocity in the main phase, and the displacement of the main phase by this particle. The equation regarding the motion of such a particle is as follows [18]: . 6 12 Here mp -the mass of the particle, d -the diameter of the particle, v -velocity, μ -the dynamic viscosity of the substance in the main phase, Ccor -its viscous resistance coefficient; Index p (particle) refers to the particle, index f (fluid) refers to the main phase substance.
The left side of equation (5) is the sum of all forces influencing a particle, expressed by the mass and acceleration of this particle. According to Stokes' law, the first term on the right side represents the deceleration of the particle owing to the viscous friction against the flow of the main phase. The second term is the force applied to the particle due to the pressure drop in the main phase surrounding the particle caused by the main phase flow's acceleration. The third term is the force required to accelerate the weight of the main phase in the volume displaced by the particle. These two terms must be considered when the density of the main phase exceeds the density of particles, for example, when considering air bubbles in a liquid flow. The fourth term is an external force directly influencing the particle, for example, in this case, it is gravity. Equation (5) is a first-order differential equation, in which the only unknown quantity is the particle velocity up, and the argument is the time t. The velocity of the main phase substance ufat all points is assumed to be known. In addition to the size and properties of the particle, its position at the initial moment of time is set as the initial data. It is also indicated what should happen when a particle collides with a wall or with another particle. The terms containing vp are transferred to the left side of equation (5) to perform the calculation. The velocity and position of the particle at each subsequent moment of time is determined by numerical integration over time with some step Δt of all other terms in the equation (5) [19-24].

Modeling of particle motion in the constrained motion mode
Let us consider the motion of a particle in the inter-shelf space. At air velocity uf> ucr1 in the shelf space, it will be in a weighted state until it reaches uf = ucr2, causing the ablation. If the air velocity is uf<ucr2, then this difference of velocities 2 f cr f u u u    will make the particle to move from top to bottom. If uf<ucr1, the particle will move in the gravitational falling layer mode with a sharp decrease in the residence time on the shelf. Given that the gas flow transmits up to 95% of the momentum of the dispersed material, we suppose that the difference in particle velocities

   and the length
Lsh on the i-th stage of the shelf device will be equal to: In the case when the tilt angle of the shelf is small (in practice within 10-35°), the velocity f u  that describes the motion of the particle from top to bottom, will actually have one roll sin which is simplified with 90   to the previous expression (6). Obviously, the ratio of the particle's motion time along the shelf is inversely proportional to the sines of the tilt angle of the shelf: According to these considerations, it is possible to define the possible constructive influence on the residence time of the particle in the inter-shelf space and the regulation of the drying process. The residence time of the dispersed particle (drying time) at this stage is increased by reducing the tilt angle of the shelf. Formula (7) allows defining the residence time of a particle on the shelf that moves independently of other particles, i.e. its free motion is considered. Such free motion is observed only at small volumetric contents of the dispersed phase in a two-phase system, when there is such a distance between the particles where collisions or mutual influence of the particles are absent. If δ≥0.1 (constrained particle motion), there are some changes in the system: the distances between the particles' surfaces or the size of the passages between the particles become smaller than their diameter, and the particle cannot slip freely between the other two particles [25]. It is necessary to consider the collision effect of particles with each other. Besides, the collision of particles in a two-phase system can also occur when the dispersed phase consists of polydisperse particles or particles with different densities. It is possible to consider the limited particle motion phenomenon and the interphase interaction force when introducing the constraint coefficient of the particle χ. Various formulas are obtained to identify the particles constraint coefficient, based on different schemes of the dispersed phase particles location. In particular, the following formula considers the scheme with random free filling [25]: where δ = 0.6 (the case of random free filling [16]); m = 3-5 (accepted as preliminary data and should be specified by the experimental studies and computer modelling. Thus, expression (7) will be as follows: . sin

The gas flow motion
The gas flow motion in the dryer's workspace is modeled based on the above mathematical model. The ANSYS CFX software solves the stationary problem of turbulent airflow in a shelf dryer. While forming the computational grid, the design features of the device are considered (the need to thicken the computational grid in the shelves and near the walls). The calculation results are demonstrated in the form of velocity vectors and the flow lines around the velocities defining the velocity field by pouring over the dryer's workspace.
In the computational model of the device, it is possible to change the construction of the cascade of shelves as a whole and each shelf, in particular, the location of the shelves and their number, and the drying agent's technological features. Fig. 1 shows the model of the dryer in the space for creating a three-dimensional model and in the calculation sphere, as well as the construction of a computational grid for modeling. The computational grid of the constrained model is created in the ANSYS Meshing program. As a result, the computational grid has approximately 7.5 million elements. The computational grid in the wall layers was condensed. The grid quality is controlled by the dimensionless parameter y + on the shelf dryer's walls, which varied in the range from 20 to 50 for the k-ε model of turbulence; growth rate; proportionality, and orthogonality coefficient of the model grid. The task is statically considered. The model uses two inputs and two outputs.  . 2) allow us to establish the following patterns: -the flow rate of the drying agent and the motion velocity significantly affect the gas flow motion mode, the peculiarities of the gas bending around the shelves and its passing through the shelf perforation. Further, the parameter is decisive for calculating the particle motion features; -an increase in the length of the shelf and a decrease in the outloading gap cause the additional vortex formation at the end of the shelf, negatively affecting the nature of the particle motion and complicating the calculation of their residence time due to the stochasticity of the vortex formation process; -an increase in the shelf's tilt angle leads to the formation of vortices under the shelf.
Besides, a significant excess of the tilt angle of the shelf over the angle of repose of the material leads to a decrease in the contact time of particles with the gas flow; -the perforation of the shelves enables to evenly redistribute of the gas flow over the device's work-space and ensure an increase in particles' contact time with the gas flow. Based on the calculation results, at this stage, the critical velocities of the gas flow are selected (the first critical velocity is the beginning of particle fluidization, the second critical velocity is the removal of particles from the device) for particles with a specific diameter or a polydisperse system with calculated average particle diameter. This data is included in calculating the "hydrodynamic" residence time of the particles in the dryer.

The dispersed material motion
A software package "Multistage fluidizer © " [26] is created to determine the factors that influence the residence time of particles in the dryer. It is based on the above model for calculating a two-phase flow motion in a free and constrained mode. A brief analysis of the calculation results is given below.
1. Influence of tilt angle of shelf on the residence time of a particle ( fig. 3). The shelf tilt angle increase reduces the particle residence time on the stage of the dryer. It can be critical if the tilt angle of the shelf is significantly greater than the angle of repose of the material. 2. Influence of radius of the granule on the residence time of a particle ( fig. 4).
Particle size defines the critical gas flow rates. With a constant gas flow rate, particles of larger diameter have a shorter residence time on the shelf and in the device. 3. Influence of volumetric content of a dispersed phase in a two-phase flow on the residence time of a particle ( fig. 5). An increase in the volumetric content of particles in a two-phase system significantly affects an increase in their residence time in the device. In this case, it is significant to prevent overheating of particles if their properties are changed during prolonged contact with a hot heat transfer agent. It is necessary to combine the calculations of the "hydrodynamic" residence time of particles in the device and the drying time (it will be shown in the conclusions). 4. Influence of degree of perforation (free area) on the residence time of a particle ( fig.  6). An increase in the perforation degree of the shelf leads to an increase in the residence time of the particles in the device. In the case of varying the shelf perforation degree, it is important to select the perforation degree and the size of the perforation holes. The latter parameter has a significant effect on the pressure loss in the dryer.
5. Influence of shelf length on the residence time of a particle ( fig. 7). The residence time of a particle on the shelf is directly proportional to the length of the shelf. However, an increase in the shelf length can cause vortices at the end of the shelf and in the outloading gap (the consequences of vortex formation are described above). Besides, an excessive increase in the length of the shelf can lead to overheating of the particles and their destruction due to significant temperature stresses from the hot heat transfer agent. 6. Influence of tilt angle of shelf and radius of the granule on the residence time of a particle ( fig. 8).
The simultaneous increase in these parameters significantly reduces the residence time of particles in the dryer. However, in the constant drying stage, this combination is possible. In other cases, with an increase in the particle radius, it is necessary to decrease the tilt angle of the shelf. 7. Influence of radius of the granule and volumetric content of a dispersed phase in a two-phase flow on the residence time of a particle ( fig. 9). The change in the volumetric content of particles in a two-phase flow determines the regularity of the simultaneous change in these two parameters. However, if it is necessary to select the optimal operating mode of the dryer for polydisperse materials (with a calculated average particle radius), the particle size can have a significant effect.

Conclusions
The research results show that by varying the design of the main element in the dryer -shelf contacts -it is possible to create an algorithm for controlling the residence time of particles in the device. In this case, it becomes possible to synchronize the periods of the drying velocities and the particle motion modes along the shelves and in the inter-shelf space. The algorithm for "combining" the periods of the drying velocities and the particle motion modes in the dryer is as follows: 1. Calculation of the drying time in each drying period: warming up the material, a period of constant drying velocity, a period of decreasing drying velocity. 2. Preselection of the number of drying stages and the design of the shelves. 3. Calculation of the "hydrodynamic" residence time of particles in the dryer for the selected shelf design and the number of shelves. 4. Correction of the shelf design, the number of shelves and the technological features of the gas flow to ensure the required residence time of the particle in each drying period. The calculation enables to select an individual design for each shelf and even divide the shelf into several sections with different designs (for example, a shelf with a variable tilt angle of various sections or with a variable perforation degree along its length).