Continuity Equation

Conservation of mass in the flow is represented by continuity equations for the gas phase and the solid phases,

∂ ( α g ρ g ) ∂ t + ∇ ⋅ ( ρ g α g v → g ) = 0 (1)

∂ ( α s ρ s ) ∂ t + ∇ ⋅ ( ρ s α s v → s ) = 0 (2)

where α is the volume fraction, ρ is density, and ν is velocity. The g and s refer to the gas and solids phases, respectively [21].

Momentum Equation

Momentum for the Gas Phase:

The influence of viscous, pressure, and gravity forces on the dynamics of the gas and solid particles are captured by the momentum equations as follows [21]:

∂ ∂ t ( α g ρ g v → g ) + ∇ . ( ρ g α g v → g v → g ) = − α g ∇ P + ∇ τ g + ρ g α g g + B g s ( v → g − v → s ) (3)

τ g = v g [ ∇ v g + ( ∇ v g ) T − 2 3 μ g ( ∇ ⋅ v g ) I ] (4)

where ( v → g − v → s ) is the slip velocity between the phases, B_gs denotes the drag force coefficient relevant to the phases g and s, P stands for the pressure, and τ g ¯ ¯ is the deviator effective stress tensor of fluid phase g gravity [21]:

Momentum for the Solid Phase:

τ s = ( − P s + ζ s ∇ ⋅ v s ) I + μ s { [ ∇ v s + ( ∇ v s ) T ] − 2 3 ( ∇ ⋅ v s ) I } (5)

Here, P_s, µ_s, ζ_s, and I are the solids pressure [Pa], the solids viscosity [Pa·s], the solids bulk viscosity [Pa·s] and the unit tensor, respectively. P_s represents the solid pressure (normal forces) [Pa] created due to particle-particle collisions in a flow due to presence of several solid phases [22].

P s = ε s ρ s θ s + ∑ n = 1 N 2 d n s 3 d s 3 ( 1 + e n s ) g 0 , n s ε n ε s ρ s θ s (6)

e_ss is the particle-particle restitution coefficient between phase s and n [-], d_s is the particle diameter [m], d_ss is the mean diameter of the particles in phase n and s [m], g_0,ss and are the radial distribution function [-] and the granular temperature θ_s [m²/s²], respectively [22].

The bulk viscosity of the solids, in Equation (7), given by ζ_s

ζ s = 4 3 ε s 2 ρ s d s g 0 , s s ( 1 + e s s ) θ s π (7)

The solids shear viscosity in Equation (8) is given as,

μ s = 4 3 ε s 2 ρ s d s g 0 , s s ( 1 + e s s ) θ s π + 10 ρ s d s π θ s 96 ( 1 + e s s ) ε s g 0 , s s [ 1 + 4 5 ε s g 0 , s s ( 1 + e s s ) ] 2 (8)

The radial distribution g_0,ss for solid phases can be expressed by Equations. (9) and (10) [16]:

g 0 , s s = 1 + 2.5 ε s + 4.59 ε s 2 + 4.52 ε s 3 [ 1 − [ ε s ε s , max ] 3 ] 0.678 + 1 2 d s ∑ n = 1 N ε n d n (9)

ε s = ∑ n = 1 N ε n (10)

This study assessed three drag models for the gas-solid interphase exchange coefficient: the Syamlal-O’Brien, Gidaspow, and Schiller-Naumann drag models.

Syamlal-O.Brien Drag Model

The Syamlal O.Brien drag model, given in Equation (11), was derived for a single spherical particle in a fluid, and modified with a relative velocity correlation (v_r). The relative velocity correlation is the terminal settling velocity of a particle in a system divided by the terminal settling velocity of a single sphere [23]. The main idea behind this model is the assumption that the Archimedes number in a single particle and a multi-particle system is the same. The Archimedes number relates the gravitational forces to the viscous forces [23].

K s g = 3 α g α s ρ g 4 d s v r 2 C D | u s → − u g → | (11)

In this model, α_g is the volume fraction of fluid or in this case gas, α_s is the particle volume fraction, ρ_g is the fluid or gas density, d_s is the particle diameter and |u_s-u_g| is the absolute relative interracial velocity of the particles compared to the fluid. Since this model is derived for a single spherical particle, the drag factor C_D is also a single particle model from Dalla Valle [24]. This C_D is modified with the relative velocity correlation as shown in Equation (12).

C D = [ 0.63 + 4.8 R e v r ] 2 (12)

Richardson and Zaki [25] analytical model based on experimental data was used for the relative velocity correlation v_r. This model is given by [26], as shown in Equation (13).

v r = 1 2 [ A − 0.06 R e ] + 1 2 [ ( 0.06 R e ) 2 + 0.12 R e ( 2 B − A ) + A 2 ] (13)

A = α g 4.14 (14)

B = { 0.8 α g 1.28         α g ≤ 0.85 α g 2.65                 α g > 0.85 (15)

The Reynolds number used in this model is the particle Reynolds number as given in Equation (16) [26].

R e = ρ g d s | u s → − u g → | μ g (16)

Gidaspow Drag Model

The Gidaspow drag model is a combination of the Wen and Yu drag model Equation (17) and the Ergun model Equation (19) [23]. The Wen and Yu drag model uses a correlation extracted from Richardson and Zaki’s experimental data [25]. This correlation is valid when the internal forces are negligible, which implies that the viscous forces dominate flow behavior. The Ergun equation is derived for a dense bed and relates the drag to the pressure drop through porous media.

K s g = 3 α g ρ g ( 1 − α g ) 4 d p C D | u s → − u g → | α g − 2.65 (17)

The drag factor C_D is the drag factor for a spherical particle given by Equation (18) [23].

C D = 24 α g R e s [ 1 + 0.15 ( α g R e s ) 0.687 ] (18)

The Ergun equation is shown in Equation (19).

K s g = 150 μ g ( 1 − α g ) 2 α g ( d s φ ) 2 + 1.75 ρ g ( | u s → − u g → | ) ( 1 − α g ) d s φ (19)

The Ergun equation is a combination of the Kozeny Carman and the Burke Plummer equations [27]. The Kozeny Carman is the first part of the Ergun equation and describes the viscous flow with low Reynolds number. The Burke Plummer equation describes the kinetic flow with high Reynolds number and is the second part of the Ergun equation. The constant is a shape factor for the particles. The combination of the two drag models (Equations (17) and (19)) in the Gidaspow drag model is shown in Equation (20) [23]

K s g = { K s g ( WenYu )               α g > 0.8 K s g ( Ergun )                     α g ≤ 0.8 (20)

Schiller-Naumann Drag Model

The Schiller-Naumann model is the default method in the ANSYS Fluent, and it is acceptable for general use in all fluid-fluid multiphase simulations. When using the homogeneous population balance models, the interfacial area will be calculated directly from the population balance variables [21]. This model is used to simulate the drag between fluid phases in the multiphase flow. The drag function f is given as Equation (21) [21]. The relative Reynolds number Re for the primary phase g and secondary phase s is given as in Equation (21).

f = C D R e 24 (21)

The drag coefficient C_D is given as:

C D = { 24 ( 1 + 0.15 R e 0.687 ) / R e               R e ≤ 1000 0.44                                                                 R e ≥ 1000 (22)

In the Space Claim, a circle with 0.6 m diameter was created to represent the transverse plan of the rotating pan coater. A computational mesh created with 20,500 elements as shown in Figure 3.

The main boundary condition of the transverse plane in the rotary pan coater is the relative motion between the bed material and the rotating wall. A no-slip condition was assumed, with the relative velocities of the gas and particles at the wall set to zero and particles subject to wall friction and gravity. Initially, all dispersed particles were located at the bottom of the pan coater. The bed height was 0.2 m at t = 0 s, with two granular solids of particles (urea and sand) and a filling limit of up to 40%. The physical properties of the gas and solid phases are given in Table 1.

The governing equations of the model were solved by the finite volume method. The computational domain was partitioned into a finite number of control volumes. The second order upwind discretization method was used for the discretization. Discretization is the method that converts the partial differential equations of the model to algebraic equations for numerical solution. This set of algebraic equations was solved using CFD-based code (Fluent 19.1). The “SIMPLE” algorithm was used to do the pressure-velocity coupling. Two solid phases were used to describe the binary mixture of particles. Volume fraction was discretized according to the QUICK scheme. Applying unsteady state conditions and Eularian model for laminar flow, the solver ran for 60 seconds with 10⁻³ s time step and 10⁻³ residual value for the convergence. Table 2 summarizes the simulation parameters and their corresponding values.

To achieve the statistical steady-state (pseudo steady-state) for solution, the time evolution of particles volume fraction in different region along the bed depth is monitored at varied restitution coefficients of 0.7, 0.8, 0.9, and 0.95. The restitution coefficient (e_ss) characterizes the energy dissipated during particles collision and is defined as in Equation (23). The lower the value of the restitution coefficient, the more kinetic energy is lost up on collision [19].

e s s = Particles speed of separation after collision Particles speed of approach before colision (23)

As seen in Figures 4-7, as time passes, the urea particles travel downward to the bed mid-region while the sand particles move upward to the bed top-region. The volume fractions vary rapidly at the start of the simulation (5 to 20 s) for the cases with e_ss = 0.7 and 0.8, but slowly after 30 s. On the other hand, with e_ss = 0.9 and 0.95 the simulation did not converge to a stationary state until the time exceeded 65 s. As a result, for the scenario when e_ss = 0.95, the time-averaged variables are computed between 65 and 75 s. The results show that as restitution coefficient values increase, so does the simulation time required for conforming to the stationary state. However, when the restitution coefficient increases, there is less dissipation of particle kinetic energy due to a more substantial elastic particle-particle collision. This may explain why, as the restitution coefficient increases, a longer simulation time is required before the mixing pattern reaches a stationary state. Furthermore, higher restitution coefficient value indicates the existence of a strong segregation phenomenon in the rotating process of the binary particle system.

Figures 4-7 also show that when the restitution coefficient increases, more urea particles accumulate at the mid-section of the bed indicating a stronger tendency for segregation. However, when the restitution coefficient increased to 0.95, the volume fraction value increased dramatically. A similar finding has been reported by Taghizadeh et al. [20] and Zhao et al. [31]. The restitution coefficients of 0.7 and 0.8 may be assigned to one class (A) and the 0.9 and 0.95 can be assigned to another class (B). The simulation results were also produced in the form of velocity vectors profiles of the granular phases, as shown in Figure 8. In Figure 8, irregular velocity vectors can be seen in the active region of the granular bed for the lower restitution coefficient values (class A), but the higher restitution coefficient values (class B) resulted in excellent rolling regime behaviour obtained.

The classical drag models available in Fluent 19.1 of Gidaspow, Syamlal-O’Brien, and Schiller-Naumann drag models were studied to find an optimal drag model for improved rotating bed hydrodynamics. To allow for complete segregation, the simulations were run for 60 s of real flow time. Figures 9-11 show the results obtained from the simulated three drag models. Considering the particle segregation, the simulations indicate that the Gidaspow and Syamlal-O’Brien drag

models can simulate the segregation contrary to Schiller-Naumann model. The results obtained with the Syamlal-O’Brien model shown in Figure 10 deviate from the results of the Gidaspow drag models. Although the model was able to simulate an acceptable rolling flow mode for the bed with larger active-region and less particle collision in the passive-region, it did not show the complete separation of the two particulate phases which was found by applying Gidaspow model (Figure 9). Even Though, Syamlal-O’Brien model shown is capable to predict the particle segregation in a rolling flow as in agree with [7] The minor deviation shown could be due to assumption that the Archimedes number is the same in a single particle and a multi-particle system for this model as predicted by [7] [23].

The Schiller-Naumann model (Figure 11), however, are not the proper models for this case, where there is no rolling flow mode was obtained, particles in the passive-region shows much collision when they move, and no completed segregation is expected to occur. This finding was in contrast with Karunarathne et al. [7] finding, were Schiller-Naumann model and the Morsi-Alexander are able to model the particle segregation phenomena showing a good agreement with the experimental results while the Syamlal-O’Brien-Symmetric model had some deviations. Were Santos et al. [4] demonstrated that, the influence of different drag models on particle velocity profile can be neglected in the case of a rotating drum operated in the rolling regime where there is no fluid entering or leaving the system. Disputes in the drag force results, reveal that further investigation is required.

The effect of particle size on mixing of granular particles was studied using two particles of different size (100 μm and 1000 μm) and same densities. The density of the particles was the same as the density of urea solid granular. Figure12 shows the volume fraction of each phase of the mixture at 60 second. Variances in the particles size show a significant effect on each phase volume fraction. As predicated from Jha [32] study the particle size is a dominated parameter for segregation of small particles at different size ratio and mixing ratio. Where the smaller particles concentrated in the mid-section of the moving bed, while the larger particles sink towards the bottom of the rotating disk, similar results have been reported elsewhere [7] [32] [33].

Mixing due to density variations was investigated using a constant particle size of 1000 μm. The study considered densities of 1300 kg/m³ and 1500 kg/m³. Figure 13 shows the volume fractions of the two particle types. From the figure, no

major separation appears due to density differences whilst good mixing could be achieved under different particles density, this could be due to the insignificant different in the tested values. Moreover, a uniform mixing would be achieved at relative higher densities, similar to what was shown by Aissa et al. [12].