### 1. Introduction

*k*–ɛ turbulence model is based on the viscosity concept of isotropic vortexes for Reynolds stress expressions. In the case of complex flows with gravitational forces or stress fields, this assumption seems to be oversimplified.

### 2. Numerical Simulation Methodology for Particulate Matter and Gas Flow Processes

*CAD*vector graphic editor, employing parametric object design, creating planar and spatial elements. Parameters for the numerical multi-channel cyclone were selected in line to those for the experimental bench of the advanced multi-channel cyclone. The research and analysis of the numerical model for the two-phase (gas (air), particulate matter) flow in the cyclone were performed applying the Fluent subprogram of the ANSYS software package.

*k*–ω

*(omega)*for turbulent viscosity and their modifications –

*k*–ɛ (

*epsilon*) and

*k*–ɛ

*RNG*(Re–Normalisation Group) (advanced model for turbulent kinetic energy (TKE) and kinetic energy dissipation rate) and

*k*–ω

*SST*(TKE and a comparative rate of kinetic energy dissipation) [17]. The application of the

*k*–ɛ model assists in solving a system of two nonlinear diffusion equations – TKE density (

*k*

*) and kinetic energy dissipation rate (ɛ) – the rate at which TKE converts to heat due to viscous friction. Standard*

_{ρ}*k*–ɛ (Eq. (1) and (2)) and

*k*–ɛ

*RNG*(Eq. (3) and (4)) models are expressed in the form of partial derivatives [18]:

##### (1)

$${\scriptstyle \frac{\partial}{\partial t}}(\rho k)+{\scriptstyle \frac{\partial}{\partial {x}_{i}}}(\rho k{u}_{i})={\scriptstyle \frac{\partial}{d{x}_{j}}}\left[\left(\mu +{\scriptstyle \frac{{\mu}_{t}}{{\sigma}_{k}}}\right){\scriptstyle \frac{\partial k}{\partial {x}_{j}}}\right]+{G}_{k}+{G}_{B}-\rho \varepsilon -{Y}_{M}+{S}_{k};$$##### (2)

$${\scriptstyle \frac{\partial}{\partial t}}(\rho \varepsilon )+{\scriptstyle \frac{\partial}{\partial {x}_{i}}}(\rho \varepsilon {u}_{i})={\scriptstyle \frac{\partial}{d{x}_{j}}}\left[\left(\mu +{\scriptstyle \frac{{\mu}_{t}}{{\sigma}_{\varepsilon}}}\right){\scriptstyle \frac{\partial \varepsilon}{\partial {x}_{j}}}\right]+{C}_{1\varepsilon}{\scriptstyle \frac{\varepsilon}{k}}({G}_{k}+{C}_{3\varepsilon}{G}_{b})-{C}_{2\varepsilon}\rho {\scriptstyle \frac{{\varepsilon}^{2}}{k}}+{S}_{\varepsilon};$$##### (3)

$${\scriptstyle \frac{\partial}{\partial t}}(\rho k)+{\scriptstyle \frac{\partial}{\partial {x}_{i}}}(\rho k{u}_{i})={\scriptstyle \frac{\partial}{d{x}_{j}}}\left({\alpha}_{k}{\mu}_{eff}{\scriptstyle \frac{\partial k}{\partial {x}_{j}}}\right)+{G}_{k}+{G}_{b}-\rho \varepsilon -{Y}_{M}+{S}_{k};$$##### (4)

$${\scriptstyle \frac{\partial}{\partial t}}(\rho \varepsilon )+{\scriptstyle \frac{\partial}{\partial {x}_{i}}}(\rho \varepsilon {u}_{i})={\scriptstyle \frac{\partial}{d{x}_{j}}}\left({\alpha}_{k}{\mu}_{eff}{\scriptstyle \frac{\partial \varepsilon}{\partial {x}_{j}}}\right)+{C}_{1\varepsilon}{\scriptstyle \frac{\varepsilon}{k}}({G}_{k}+{C}_{3\varepsilon}{G}_{b})-{C}_{2\varepsilon}\rho {\scriptstyle \frac{{\varepsilon}^{2}}{k}}-{R}_{\varepsilon}+{S}_{\varepsilon};$$*k*– turbulent kinetic energy, m

^{2}/s

^{2};

*G*

_{k}*–*TKE variations under the effect of the average rate gradient equal to

*G*

*=*

_{k}*μ*

_{t}*S*

^{2}applying the Businescu hypothesis; μ

*– turbulent dynamic viscosity;*

_{t}*S*– strain tensor; G

_{b}– TKE variations under the effect of buoyancy equal to ${G}_{b}=\beta {g}_{i}{\scriptstyle \frac{{\mu}_{t}}{{Pr}_{t}}}{\scriptstyle \frac{\partial T}{\partial {x}_{i}}}$; β – the coefficient of temperature expansion;

*Pr*

_{t}– the turbulent Prandtl number for energy; g

_{i}– gravity vector taking direction

*i*;

*Y*

*– fluid rate distribution under the effect of a displacement in space at turbulent compression equal to ${Y}_{M}=2\rho \varepsilon {M}_{t}^{2}$;*

_{M}*M*

*– the coefficient of Macho turbulence;*

_{t}*C*

_{1}_{ɛ},

*C*

_{2}_{ɛ},

*C*

_{3}_{ɛ}– fixed values; σ

_{k}and σ

_{ɛ}– the turbulent Prandtl number for variables

*k*and ɛ, respectively.

*S*

_{k}and

*S*

_{ɛ}– variables established by the consumer.

*k*–ɛ RNG) of the k–ɛ viscosity model, the boundary layer functions are not used, and the model applies to the entire flow area. The employment of the advanced viscosity model, as a rule, requires a more detailed computational grid in the boundary layer and all areas where fluid flow is characterized by decreasing turbulence, i.e. the Reynolds number is small. Using the standard

*k*–

*ɛ*model is sometimes enough, and the alternative may involve the selection of an automatic model in the boundary layer and the consistent details of the computational grid in these zones. The advanced model may achieve a greater accuracy of the results of gas flow distraction and simulating the interaction of multiple gas flows.

*k*–

*ω*model assists with solving the system of two nonlinear diffusion equations derived by Wilcox–TKE and a comparative rate of kinetic energy dissipation consisting of equations for turbulent viscosity and shear stress transfer. This viscosity model is widely applied in fluid flows when the Reynolds number, compressibility and discontinuity from the boundary layer are low.

*k-ω*(Eq. (5) and (6)) and

*k*–

*ω SST*(shear stress transport) (Eq. (5) and (7)) models are expressed by differential equations with partial derivatives [19]:

##### (5)

$${\scriptstyle \frac{\partial}{\partial t}}(\rho k)+{\scriptstyle \frac{\partial}{\partial {x}_{i}}}(\rho k{u}_{i})={\scriptstyle \frac{\partial}{d{x}_{j}}}\left({\mathit{\Gamma}}_{k}{\scriptstyle \frac{\partial k}{\partial {x}_{j}}}\right)+{G}_{k}-{Y}_{k}+{S}_{k};$$##### (6)

$${\scriptstyle \frac{\partial}{\partial t}}(\rho \omega )+{\scriptstyle \frac{\partial}{\partial {x}_{i}}}(\rho \omega {u}_{i})={\scriptstyle \frac{\partial}{d{x}_{j}}}\left({\mathit{\Gamma}}_{\omega}{\scriptstyle \frac{\partial \omega}{\partial {x}_{j}}}\right)+{G}_{\omega}-{Y}_{\omega}+{S}_{\omega};$$##### (7)

$${\scriptstyle \frac{\partial}{\partial t}}(\rho \omega )+{\scriptstyle \frac{\partial}{\partial {x}_{j}}}(\rho \omega {u}_{j})={\scriptstyle \frac{\partial}{d{x}_{j}}}\left({\mathit{\Gamma}}_{\omega}{\scriptstyle \frac{\partial \omega}{\partial {x}_{j}}}\right)+{G}_{\omega}-{Y}_{\omega}+{D}_{\omega}+{S}_{\omega};$$*G*

_{k}*–*TKE variations under the effect of the average rate gradient; G

_{ω}– a variable under the effect of varying ω; Γ

_{k}and Γ

_{ω}– effective diffusion variables

*k*and ω, respectively where ${\mathit{\Gamma}}_{k}=\mu +{\scriptstyle \frac{{\mu}_{t}}{{\sigma}_{k}}}$ and ${\mathit{\Gamma}}_{\omega}=\mu +{\scriptstyle \frac{{\mu}_{t}}{{\sigma}_{\omega}}}$; Y

_{k}and Y

_{ω}– turbulence-induced dissipation variables

*k*and ω, respectively; S

_{k}and S

_{ω}– variables established by consumers; D

_{ω}– a diffusion variable equal to ${D}_{\omega}=2(1-{F}_{1})\rho {\scriptstyle \frac{1}{\omega {\sigma}_{\omega ,2}}}{\scriptstyle \frac{\partial k}{\partial {x}_{j}}}{\scriptstyle \frac{\partial \omega}{\partial {x}_{j}}}$; F

_{1}– a fluid mixing function.

*SST*–model is a combination of turbulence models

*k*–ɛ and

*k*–ω: the k–ɛ model is applied for assessing the free flow of fluid, and equations for the

*k*–ω model are used for assessing the areas of the boundary layer. The detailed computational grid is required similarly to the cases of

*k*–ω and

*k*–ɛ

*RNG*models.

*k*–

*ω SST*turbulent viscosity model was more accurate and reliable for a wider class of the examined fluid flows (particularly when the model encountered inverse pressure gradient flows, aerodynamic surfaces, sudden shock waves) than the standard

*SST*and other

*SST*models.

*ANSYS CFD Fluent*software package. The grid consists of tetrahedra the length of the edge of which makes no more than 20 mm and the proximity of the formed elements of the computational grid is at least 50 μm. The height of the boundary layer is 0,1 mm. The grid of the boundary area consists of 8 layers. All three computational grids were made by doubling the size (4,440,299; 8,880,597 and 17,761,194). In additional, the mesh structure was defined according to the geometry of the cyclone. The parameters such as proximity settings, inflation parameters and others were adjusted in order to obtain the satisfactory results. In order to create the appropriate mesh near wall the grid elements were reduced there up to 10 times. The medium-accuracy computational grid was selected for the advanced multi-channel cyclone model. The latter case was identified as the most appropriate with reference to the study on the geometry of the multi-channel cyclone [21]. The length of the advanced model for the multi-channel cyclone in different axial directions is as follows: x – 1,600 mm; y – 1,600 mm; z – 3,400 mm (Fig. 1).

_{S}

^{+}) which, in the case of the smooth surface, does not exceed 2.25 (case 1). Under the semi-smooth surface, its value can be equal to 2.25–90 (case 2). The value of the high-degree roughness height coefficient is greater than 90 (case 3).

##### (8)

$$\begin{array}{c}\mathit{\Delta}B=0,kai\hspace{0.17em}{K}_{S}^{+}\le 2,25;\\ \mathit{\Delta}B={\scriptstyle \frac{1}{k}}\hspace{0.17em}ln\hspace{0.17em}\left[{\scriptstyle \frac{{K}_{S}^{+}-2,25}{87,75}}+{C}_{s}{K}_{S}^{+}\right]\xb7\mathit{sin}\{0,4258(ln\hspace{0.17em}{K}_{S}^{+}-0,811)\},\\ when\hspace{0.17em}2,25<{K}_{S}^{+}\le 90;\mathit{\Delta}B={\scriptstyle \frac{1}{k}}\hspace{0.17em}ln\hspace{0.17em}[1+{C}_{S}{K}_{S}^{+}],\\ When\hspace{0.17em}{K}_{S}^{+}>90.\end{array}$$*B*– the roughness height variable affecting the trajectory of fluid and particulate matter;

*C*

*– the roughness coefficient determining the shape of roughness equal to fine particulate matter making 0.3–0.5;*

_{s}*k*– turbulent kinetic energy, J;

*K*

_{s}*– the equivalent height of surface roughness, m.*

^{+}*C*

*coefficient equal to 0.5 with reference to the turbulence model created in line to the methodology developed by Nikuradse. The methodology assumes that wall roughness is uniform. The coefficients of this model may be adjusted analysing the deposition of coarse grains on the walls or when grains have irregularly shaped large edges.*

_{s}### 3. The Analysed Results of the Research on the Numerical Simulation of Particulate Matter and Gas Flow Processes

^{st}and 4

^{th}curvilinear elements the values of which fluctuated in the range of 12.4–15.7 m/s. In channel 2, the area of this zone was the smallest (less than 20% of the total area between the curvilinear elements), and rates were in the range of 13.1–16.6 m/s.

*k*–

*ɛ*Enhanced Wall Treatment viscosity model. Meanwhile, using medium and high-accuracy computational grids,

*k*–

*ω SST*was optimal. Employing the latter viscosity model together with the medium-accuracy computational grid showed the smallest residuals compared to experimental results, and therefore this case was preferred as optimal and analysed further in this paper. For simulating SFD processes, the optimal

*k*–

*ω SST*viscosity model has been established in the advanced multi-channel cyclone. The application of the 1

^{st}and 2

^{nd}order discretization equations and optimal relaxation coefficients lead to rate and TKE residuals equal to 10

^{−4}, and the residual of the energy variable equalled 10

^{−7}.

*k*) the residuals of which were examined in conjunction with other variables. Research on static pressure defines the potential energy of gas flow in the investigated ‘channel’, and parameter

*k*explains energy transfer of the turbulent (pulsating) gas flow. To analyse the obtained difference, TKE parameter distribution was made in the advanced multi-channel cyclone (Fig. 6).

^{−7}m

^{2}/s

^{2}to 17 m

^{2}/s

^{2}. Despite a wide range of values, most of the advanced cyclone falls into the bluish spectrum that corresponds values from 1.7 m

^{2}/s

^{2}to 4.25 m

^{2}/s

^{2}. The lowest values are set at the beginning of each gas flow inlet, e.g. Fig. 6(a), No 1 where values do not exceed 0.85 m

^{2}/s

^{2}. Thus, the flow is little turbulent in these areas, close to the direct current, and variations in kinetic energy change insignificantly. The area defined at the beginning of each channel runs the full height of the channel. In this section, the value of TKE is equal to 1.7–2.6 m

^{2}/s

^{2}. Following the first inflow, this area was found to be the largest, following the second, it became smaller, etc. It may be concluded this is influenced by inlet rate, which, in this case, is the highest in the first inlet and decreases in each subsequent inlet. Gas flow rate affects flow turbulence and hence TKE.

^{2}/s

^{2}. This can be attributed to vortex motion and intense mixing in the central channel. Also, the flow pumped at the axis of the device to the gas duct of a smaller cross-section, i.e. the gas flow outlet duct, has a profound effect compared to the inner central gas duct. For the overall assessment of the central channel, the value of TKE reaches 3.8 m

^{2}/s

^{2}. However, at the inner surface of each quarter-ring element, the established values make 1.5·10

^{−7}m

^{2}/s

^{2}. These zones are supposed to fully confine to bordering areas where no gas flow is observed or motion is very weak.

^{st}and 4

^{th}curvilinear quarter-ring elements at the inner edge of the 1

^{st}element when gas flew in the transit direction to the central channel (Fig. 6(c)). The other site was the gap between the 3

^{rd}and 4

^{th}curvilinear elements at the inner edge of the 3

^{rd}element when gas flew in the reverse peripheral direction from the central channel to outer channel 4 (Fig. 6(f)). In both cases, the values of TKE ranged from 10.2–11.1 m

^{2}/s

^{2}in the middle of the gaps. A similar trend was observed in the gaps between other curvilinear elements (Fig. 6(d) and (e)) when gas flow deviated in the peripheral direction and a vortex formed at the edge of the previous curvilinear element (Fig. 6. No 3). In these cases, the average value was 13.2 m

^{2}/s

^{2}, but in the first case (Fig. 6(d)), the area was larger due to turbulent motion. The values of TKE decreased to 8.5 m

^{2}/s

^{2}when the flow entered the area limited to the surface of a single curvilinear element, e.g. Fig. 6(d), No 4.

^{2}/s

^{2}, and the average was equal to 1.3 m

^{2}/s

^{2}. The values decrease to the lowest one set in the model below this area, including the lower section of the hopper, and therefore analysis does not apply to these areas.