# The Impact of Increasing the Length of the Conical Segment on Cyclone Performance Using Large-Eddy Simulation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Setup

_{s}= 0.1.

#### 2.1. The Governing Equation

_{i}are the components of velocity in the i- coordinate direction, ρ is the density and µ is the dynamic viscosity of the gas, x

_{i}is the position in the i-direction (I = 1, 2, 3), p is the fluid pressure, and S

_{ij}is the extra-stress tensor, which is represented as [27]

_{sgs}denotes the viscosity of the turbulent sub-grid scale. The strain rate of the resolved field is represented as:

#### 2.2. Modeling the Discrete Phase

_{p}denotes particle diameter, µ represents fluid molecular viscosity, and Re indicates the Reynolds number, represented by

_{D}).

## 3. Description of the Model, Validation, and Methodology

#### 3.1. The Details of Cyclone Geometry

_{c}) also influenced the bottom opening (B

_{c}). For instance, increasing the cone length value (H

_{c}/D) from 1.5 to 3.5 reduced the bottom opening value (B

_{c}/D) from 0.3125 to 0.0625. The details have been provided in Table 1. Hence, the cyclone performance was an interaction of the two geometrical entities viz. the cone length and cone tip diameter. A complete detail of all the cyclone models is represented as a 2-D sketch in Figure 2a and the respective 3-D geometry is shown in Figure 2b.

#### 3.2. Mesh Generation

#### 3.3. CFD Methodology

^{−5}was used for the conservation equations. The developed field served as the starting point for the LES simulations, saving a considerable amount of effort and time. The LES simulations were carried out until the monitored flow variables reached a (quasi) steady state. The collection efficiency was then determined by the number of particles being retained in cyclones.

#### 3.4. Numerical Settings and Boundary Conditions

#### 3.4.1. Continuous Phase

^{3}and viscosity of 1.834 ∙ 10

^{−5}kg/m·s was considered as a working medium for the continuous phase. A uniform profile of a velocity inlet boundary condition was provided at the inlet. The inlet velocity was taken as U

_{in}= 10 m/s, 15 m/s, and 20 m/s, resulting in Reynolds numbers Re = 1.23 ∙ 10

^{5}, 1.84 ∙ 10

^{5}, and 2.45 ∙ 10

^{5}. The pressure outlet boundary condition was considered at the outlet of the cyclone. A no-slip boundary state was allotted for the respective boundaries, and the Smagorinsky constant C

_{s}= 0.1 was utilized for modeling the sub-grid scales. As discussed in the earlier section, a 0.1 ms time step size was applied for the simulation of all numerical models.

#### 3.4.2. Discrete Phase

^{3}, were injected into the cyclone model in the face-normal direction. Ten different particle diameters ranging from 0.5 μm to 10 μm were considered. A time-step size equivalent to that of a continuous phase, i.e., 0.1 ms, was used. All the simulations were performed at three inlet velocities viz. 10 m/s, 15 m/s, and 20 m/s. The particles reaching the bottom plate of the cyclone were considered trapped. Hence, a trap boundary condition was applied at the bottom plate. The outlet plane was assigned as the escape boundary condition. For the remaining boundary walls, a rebound boundary condition was applied, and the coefficient of restitution was taken as unity for both wall-normal and tangential directions. Therefore, the solid particles reaching these surfaces rebounded without any losses. Several studies have already utilized these boundary conditions [7,9,10,17,18,24].

#### 3.4.3. Near-Wall Treatment

_{sgs}denotes the subgrid scale mixing length, and $\left|S\right|=\sqrt{2{\overline{S}}_{ij}{\overline{S}}_{ij}}$ is the resolved deformation rate [30]. A similar approach was used in several prior studies [7,31,32].

#### 3.5. Validation and Methodology

^{5}. Two levels of the grid have been evaluated. The average values of the axial and tangential velocity components are shown in the first and second rows of Figure 4. Differences in results between grid levels are minimal, indicating that the solution is grid independent. Moreover, the coarse and fine mesh results agree very well with the experimental data. However, the coarse mesh results at Z/D = 1.0 showed the smallest deviation in the mean axial velocity, whereas at Z/D = 2.0 and Z/D = 2.5, the local dip near the axis overshot moderately. The third and fourth lines of Figure 4 represent the rmse values of the axial and tangential velocities, respectively. A good agreement between the rmse values and experimental data was observed, particularly in the outer vortex region. However, in the core region, a mild increase in the rmse axial velocity was observed. It becomes apparent from the last row of Figure 4 that the nature of rmse tangential velocity was different from the reference data near the central axis of the cyclone, whereas the former exhibits a local shoot-up rather than a dip. The mesh refinement could be responsible for the small errors (e.g., in the mean velocity profiles) getting amplified in the precession-induced fluctuation levels. Such discrepancies in the mean and rmse velocity profiles have also been reported in several previous studies [33,34,35]. Hence, despite the known issues with resolving the flow anisotropy using advanced closure LES, the simulation procedure may be applied to all future simulations.

## 4. Results and Discussion

#### 4.1. The Quality of LES

_{s}is the Smagorinsky constant (taken as 0.1), and Δ is the filter size, which is equal to the cubic root of the cell volume (i.e., V

^{1/3}; V is the local cell volume).

^{+}values is warranted. Figure 5 represents the TKE from the SGS model implemented in LES; both the instantaneous (to the left) and mean (to the right) values of S are represented. The latter is calculated as

^{+}values (cf. Figure 6) only for cyclone variant E. It becomes apparent that the global y

^{+}value does not exceed 40, which indicates a fair figure with the wall-modeling approach.

#### 4.2. The Mean Flow Field

#### 4.3. The Fluctuating Flow Field

#### 4.4. The Representation of the Vortex Core

_{2}criterion. The vortex core resembles a twisted rope-like structure extending throughout the cyclone length. The λ

_{2}criterion represents the coherent structures based upon the iso-surface values: at different values, eddies with different length scales appear. However, we selected a value of 0.35, which was found to be a good balance for the visualization of the vortex core. Interestingly, in the core region, the diameter of the coherent structure seemingly reduces from models A–E.

_{in}, with f, the frequency (in Hz); D, the cyclone main body diameter (in m); and U

_{in}, the inlet velocity (in m/s). Multiple peaks with a very low power were observed in the frequency range of 0 to 50 Hz in nearly all the models. However, in the frequency band ranging from 50 to 100 Hz, a prominent spectral peak was observed that corresponds to the principal frequency of the flow system. The power related to the lower harmonic is considerably less than the higher harmonic in cyclone model E. For cyclones A–E, the corresponding frequencies of the lower harmonic were calculated as St

_{1}= 0.61, 0.61, 0.69, 0.61, 0.61, and 0.60, whereas the frequency of the higher harmonic in cyclones from models A–E was calculated as St

_{2}= 1.56, 1.54, 1.50, 1.35, and 1.30, respectively. Interestingly, the principal frequency reduced with an increase in the cone length for all inlet velocities, as represented in Figure 13.

#### 4.5. Performance of the Cyclone

_{50}). The pressure drop is defined as the difference in total pressure values across the inlet and outlet, while the cut-off diameter is the particle size that corresponds to 50% collection efficiency on the grade efficiency curve (GEC) curve. The separation efficiency is the ratio of the number of particles retained by the cyclone separator to the number of particles injected at the inlet of the cyclone separator. In cyclone separators, the pressure losses are known to take place mainly inside the vortex finder tube due to a high swirl intensity of the fluid leaving the cyclone, followed by the losses at the walls as well as due to fluid friction. Furthermore, due to a nonlinear increase in the tangential velocity peak values and a significant increase in the velocity fluctuations, the pressure losses increase significantly with an increase in Re [14].

_{50}values reduce consistently with length, whereas at the lower and upper values of inlet velocities, the results are dramatic. It is observed that at 10 and 20 m/s, the lowest d

_{50}value exists for cyclone model B, and for all U

_{in}, d

_{50}is the largest for model A. Figure 16b illustrates the total collection efficiency for all models at different U

_{in}. At the lower and upper limits of velocity, cyclone model B has the highest value of η, whereas cyclone model A has the lowest value of η. However, Brar et al. [17] reported a consistent reduction in d

_{50}, and an increase in η with an increase in the cone length (cf. Figure 13, p. 675; and Figure 15 p. 676). The difference in the results is due to the fact that in the present study, given that the apex cone angle is held fixed, the cone tip (a lower bottom opening) undergoes a significant reduction in the diameter with an increase in the length of the conical segment, whereas in another study [17], the cone-tip diameter remained the same for each increment in the cone length. There is a contradictory nature in the mean tangential velocity and the velocity fluctuation fields: the former acts as a source of centrifuging the solid particles towards the outer walls, while the latter has a strong tendency to disperse these particles (which causes a reduction in the collection efficiency) [39]. Hence, the variations in the collection efficiency and cut-off particle size are due to the combined effect of the mean as well as the fluctuating flow field, particularly in the lower part of the cyclone, where a large variation in the latter exists (cf. Figure 9).

## 5. Conclusions

- The pressure drop decreases by increasing the conical height of the cyclone separator and decreasing the cone-tip diameter.
- The magnitude of the tangential velocity decreases as the length of the cyclone separator increases along with the diameter of the cone tip.
- The velocity fluctuations were suppressed in models A–E.
- The frequency with which the vortex core precesses about the geometrical axis was lowest in model A and it decreased gradually from cyclone models A–E.

^{5}, the following was observed:

- The pressure drop reduced from 1017 Pa to 877 Pa.
- The collection efficiency increased from 59.63% to 63.90%.
- The cut-off size reduced from 2 μm to 1.54 μm.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) The first row represents the 2-D profile of different models (on the Y-plane), and (

**b**) the second row represents the 3-D geometry of all models generated by revolving their respective 2-D profiles about the axis (of revolution). Thereafter, a tangential inlet has been attached to each cyclone.

**Figure 3.**Details of the mesh for model E, consisting of nearly 1.88 million hexahedra; (

**a**) mesh representation on the cyclone surface, (

**b**) mesh refinement in the lower part of the cyclone, (

**c**) a cut plane at Z = 0.1 m, representing the mesh, and (

**d**) a cut plane at Z = 0.4 m, representing the mesh.

**Figure 4.**Comparison of simulation results against experimental data [2] at Re = 2.8 ∙ 10

^{5}at two grid levels. (a) From left to right: velocity profiles at axial locations Z/D = 0.75, 2.0, and 2.5 (R = D/2 is the cyclone radius), and (b) from top to bottom: mean axial velocity, mean tangential velocity, rmse axial velocity, and rmse tangential velocity, respectively.

**Figure 5.**Representation of the instantaneous (to the left) and mean (to the right) fractional values of the unresolved (SGS) TKE for the model E.

**Figure 6.**Representation of the instantaneous wall y+ values from two different angles for the model E.

**Figure 7.**Contour plots on Y = 0 plane at U

_{in}= 15 m/s (a) from left to right: cyclone models A, B, C, D, and E, and (b) from top to bottom: mean static pressure, mean axial velocity, and mean tangential velocity, respectively.

**Figure 8.**Radial profiles at U

_{in}= 15 m/s (a) from left to right: at axial locations Z/D = 1.0, 2.0, and 3.0, and (b) from top to bottom: mean static pressure, mean axial velocity, and mean tangential velocity, respectively, in cyclone models A, B, C, D, and E.

**Figure 9.**Contour plots on Y = 0 plane at U

_{in}= 15 m/s (a) from left to right: cyclones A, B, C, D, and E, and (b) from top to bottom: rmse static pressure, rmse axial velocity, and rmse tangential velocity, respectively.

**Figure 10.**Radial profiles at U

_{in}= 15 m/s (a) from left to right: at axial locations Z/D = 1.0, 2.0, and 3.0, and (b) from top to bottom: rmse static pressure, rmse axial velocity, and rmse tangential velocity, respectively, in cyclone models A, B, C, D, and E.

**Figure 11.**The vortex core representation based on λ

_{2}criteria was set at a level of 0.35 in all the cyclone models. The vortex core was colored with the velocity magnitude. The data are extracted at an inlet velocity, U

_{in}= 15 m/s.

**Figure 12.**PSD plots derived by FFT of the recorded tangential velocity time-series data at X/D = Y/D = Z/D = 0 for all models. The data were extracted at an inlet velocity, U

_{in}= 15 m/s for (

**a**) model A, (

**b**) model B, (

**c**) model C, (

**d**) model D, and (

**e**) model E.

**Figure 13.**Variations in the frequency values: (a) at U

_{in}= 10, 15, and 20 m/s, and (b) in different cyclone models A, B, C, D, and E. A linear fit was applied, and R

^{2}values are shown in the graphs.

**Figure 14.**The pressure drop values in all the cyclone variants at the inlet velocity (a) U

_{in}= 10 m/s, (b) 15 m/s, and (c) 20 m/s.

**Figure 15.**The grade efficiency curves of all cyclone variants at the inlet velocity, (

**a**) U

_{in}=10 m/s, (

**b**) 15 m/s, and (

**c**) 20 m/s (from left to right).

**Figure 16.**(

**a**) From left to right: cut-off particle size and (

**b**) the total collection efficiency at different inlet velocities.

**Figure 17.**Particle traces (colored with particle diameters) in cyclone model E at U

_{in}= 15 m/s and at different time intervals.

Cyclone Geometry | Cyclone Model | Symbols (Normalized with D) | Dimensions [-] |
---|---|---|---|

Vortex finder diameter | D_{e}/D | 0.5 | |

Inlet duct height | a/D | 0.5 | |

Inlet duct width | b/D | 0.2 | |

Vortex finder insertion length | L_{v}/D | 0.5 | |

Height of cylindrical section | H/D | 1.5 | |

Height of conical segment | A | H_{c}/D | 1.5 |

B | 2.0 | ||

C | 2.5 | ||

D | 3.0 | ||

E | 3.5 | ||

Cone-tip diameter | A | B_{c}/D | 0.3125 |

B | 0.2500 | ||

C | 0.1875 | ||

D | 0.1250 | ||

E | 0.0625 |

Model | A | B | C | D | E |
---|---|---|---|---|---|

Mesh count | 0.685 M | 0.880 M | 1.5 M | 1.6 M | 1.8 M |

LES (Coarse) | LES (Fine) | |
---|---|---|

Mesh count | 0.5 million | 1.0 million |

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**MDPI and ACS Style**

Pandey, S.; Brar, L.S. The Impact of Increasing the Length of the Conical Segment on Cyclone Performance Using Large-Eddy Simulation. *Symmetry* **2023**, *15*, 682.
https://doi.org/10.3390/sym15030682

**AMA Style**

Pandey S, Brar LS. The Impact of Increasing the Length of the Conical Segment on Cyclone Performance Using Large-Eddy Simulation. *Symmetry*. 2023; 15(3):682.
https://doi.org/10.3390/sym15030682

**Chicago/Turabian Style**

Pandey, Satyanand, and Lakhbir Singh Brar. 2023. "The Impact of Increasing the Length of the Conical Segment on Cyclone Performance Using Large-Eddy Simulation" *Symmetry* 15, no. 3: 682.
https://doi.org/10.3390/sym15030682