Keywords

1 Introduction

Graphite has good thermal shock resistance, high self-lubricating strength, good thermal conductivity and electrical conductivity. It finds extensive application across various sectors including metallurgy, machinery, national defense, military industry and so on. It is an important strategic resource. Flake graphite is an important branch of graphite, which has few resources, wide uses and high value, the research on its high purification technology is very important.

Ma F Y et al. [1] proposed a new grinding flotation process for the beneficiation of flake graphite, which can improve the coarse graphite ore to a higher grade graphite concentrate. Thair A et al. [2] used ball mill and mill to grind raw flake graphite ore, and obtained high grade flake graphite through alkaline roasting and acid leaching process. Mustafa A [3] applied FLUENT software to analyze the jet characteristics of three kinds of nozzles with different shapes, including rectangular, circular and two-dimensional profiles, and obtained the central velocity distribution laws of the three kinds of nozzle jets within the range of 0.2 ~ 13 mm from the target distance, the correctness of the numerical simulation results is proved by experiments. Juan W et al. [4] used CFD method to explore the atomizing performance of fan shaped nozzle, and obtained the relationship between nozzle spray drift and wind speed. Chunzhao Z et al. [5] used FLUENT software to conduct numerical simulation on the fan-shaped nozzle with round head, truncated cone and innovative outlet, it was found that the comprehensive performance of the innovative nozzle was better than that of the other two nozzles. Mireia A et al. [6] applied CFD method to analyze the internal and outward flow fields of three industrial fan-shaped atomizing nozzles, and obtained the characteristic values such as nozzle flow coefficient.

To sum up, some scholars have studied the grinding and floating of flake graphite and the jet characteristics of fan shaped nozzles and have achieved some research results [7, 8]. However, there are few reports on the numerical simulation research of the whole flow field of the fan-shaped liquid–solid two-phase jet formed by the ejection of flake graphite slurry through the fan-shaped nozzle. In this article, the self-designed fan-shaped nozzle is used as the energy conversion element, and the whole flow field of flake graphite slurry flowing through the fan nozzle is numerical simulated by FLUENT software. The flow properties of water flow field and flake graphite flow field are obtained, which lays a further technical foundation for the separation of flake graphite by jet.

2 Basic Model and Calculation Method

2.1 Geometric Model

The geometric model for numerical simulation of the fan shaped nozzle with premixed water jet is established by using SolidWorks software, which is consisted of four parts: the inlet cone convergence section, the middle cylinder section, the outlet spherical V-groove and the outward flow field cylinder. In this case, the inlet diameter of the conical convergence section of the nozzle is 7 mm, the convergence angle is 30º, and the length of the convergence section is 10.3 mm; The diameter of the cylindrical section is 2 mm and the length is 5 mm; The diameter of the outlet spherical end is 2 mm, and the V-groove chamfer is 45º. The outward flow field is set as a cylinder with a diameter and length of 20 mm.

2.2 Finite Element Model

The geometry model is meshed by using the preprocessing module Gambit in FLUENT software. In order to enhance the stability and accuracy of the calculation process, the tetrahedral unstructured mesh is selected, and the local mesh refinement is carried out at the connection position of the internal flow field and the outward flow field of the nozzle. Through grid independence test, the number of grid cells after division is 1331257, and the number of nodes is 237238. Figure 1 shows the finite element model of the fan shaped nozzle.

Fig. 1
A 3 D finite element model of the nozzle. It is composed of a cylinder section with an inlet cone and an outlet with a spherical-shaped V groove. The pressure inlet to the V groove and outlet to the cylindrical field.

Finite element model

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2.3 Mathematical Model

2.3.1 Governing Equation

The premixed water jet of flake graphite is a liquid–solid two-phase high speed jet, and the liquid–solid two-phase medium in the jet flow field is similar to a continuous medium, and the volume fraction of solid particle flake graphite in the conveying process of flake graphite is more than 10%, which is suitable for Euler multiphase flow model. Therefore, Euler model is selected to analyze the liquid–solid two-phase flow of water and flake graphite, and its control equation is [9].

The mass conservation equations of water and flake graphite are

$$\frac{\partial }{\partial t}\left( {\alpha_{w} \rho_{w} } \right) + \nabla \cdot \left( {\alpha_{w} \rho_{w} v_{w} } \right) = \sum\limits_{s = 1}^{n} {\left( {\vec{m}_{sw} - \vec{m}_{ws} } \right)} + S_{w}$$
(1)
$$\frac{\partial }{\partial t}\left( {\alpha_{s} \rho_{s} } \right) + \nabla \cdot \left( {\alpha_{s} \rho_{s} v_{s} } \right) = \sum\limits_{w = 1}^{n} {\left( {\vec{m}_{ws} - \vec{m}_{sw} } \right)} + S_{s}$$
(2)

The momentum equations of water and flake graphite are

$$\begin{aligned} & \frac{\partial }{{\partial t}}\left( {\alpha _{w} \rho _{w} v_{w} } \right) + \nabla \left( {\alpha _{w} \rho _{w} v_{w} v_{w} } \right) = \alpha _{w} \nabla p - \nabla p_{w} + \nabla \cdot \tau w + \alpha _{w} \rho _{w} g \\ & + \sum\limits_{{s = 1}}^{n} {\left[ {K_{{sw}} \left( {v_{s} - v_{w} } \right) + \mathop m\limits^{ \to } _{{sw}} v_{{sw}} - \mathop m\limits^{ \to } _{{ws}} v_{{ws}} } \right] + \left( {F_{w} + F_{{lift,w}} + F_{{wl,w}} + F_{{vm,w}} + F_{{td,w}} } \right)} \end{aligned}$$
(3)
$$\begin{gathered} \frac{\partial }{\partial t}\left( {\alpha_{s} \rho_{s} v_{s} } \right) + \nabla \cdot \left( {\alpha_{s} \rho_{s} v_{s} v_{s} } \right) = - \alpha_{s} \nabla p - \nabla p_{s} + \nabla \cdot \tau_{s} + \alpha_{s} \rho_{s} {\text{g}} \hfill \\ + \sum\limits_{w = 1}^{n} {\left[ {K_{ws} \left( {v_{w} - v_{s} } \right) + \vec{m}_{ws} v_{ws} - \vec{m}_{sw} v_{sw} } \right]} + \left( {F_{s} + F_{lift,s} + F_{wl,s} + F_{vm,s} + F_{td,s} } \right) \hfill \\ \end{gathered}$$
(4)

The interphase velocity of water to flake graphite and the interphase velocity of flake graphite to water vws and vsw are respectively

$$v_{ws} = \left\{ {\begin{array}{*{20}c} {v_{ws} = v_{w},\, \vec{m}_{ws} > 0} \\ {v_{ws} = v_{s},\,\vec{m}_{ws} < 0} \\ \end{array} } \right.$$
(5)
$$v_{sw} = \left\{ {\begin{array}{*{20}c} {v_{sw} = v_{s},\, \vec{m}_{sw} > 0} \\ {v_{sw} = v_{w},\, \vec{m}_{sw} < 0} \\ \end{array} } \right.$$
(6)

The energy equations of water and flake graphite are

$$\begin{aligned} &\frac{\partial }{\partial t}\left( {\alpha_{w} \rho_{w} h_{w} } \right) + \nabla \cdot \left( {\alpha_{w} \rho_{w} v_{w} h_{w} } \right) \\ & = \alpha_{w} \frac{{\text{d}p_{w} }}{{\text{d}t}} + \tau_{w} :\nabla v_{w} - \nabla \cdot \vec{q}{}_{w} + S_{w} \\ & + \sum\limits_{n}^{s = 1} {\left( {Q_{sw} + \vec{m}_{sw} h_{sw} - \vec{m}_{ws} h_{ws} } \right)} - \nabla \cdot \sum\limits_{j} {h_{j,w} \vec{J}_{j,w} } \end{aligned}$$
(7)
$$\begin{aligned} &\frac{\partial }{\partial t}\left( {\alpha_{s} \rho_{s} h_{s} } \right) + \nabla \cdot \left( {\alpha_{s} \rho_{s} v_{s} h_{s} } \right) \\ & = \alpha_{s} \frac{{{\text{d}}p_{s} }}{{{\text{d}}t}} + \tau_{s} :\nabla v_{s} - \nabla \cdot \vec{q}_{s} + S_{s} \\ & + \sum\limits_{n}^{w = 1} {\left( {Q_{ws} + \vec{m}_{ws} h_{ws} - \vec{m}_{sw} h_{sw} } \right)} - \nabla \cdot \sum\limits_{j} {h_{j,s} \vec{J}_{j,s} } \end{aligned}$$
(8)

2.3.2 Turbulence Model

The liquid–solid two-phase jet of flake graphite slurry with fan-shaped nozzle is a high turbulent flow, and the internal and outward flow fields have high Reynolds numbers, so the turbulence model selection standard k- ε Model. The turbulence model contains two basic equations, which are turbulent kinetic energy k and dissipation rate ε of mixed fluid transport equation [10].

Turbulent kinetic energy k of mixed fluid is

$$\frac{\partial }{\partial t}\left( {\rho k} \right) + \frac{\partial }{{\partial x_{i} }}\left( {\rho ku_{i} } \right) = \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\mu + \frac{{\mu_{t} }}{{\sigma_{k} }}} \right)\frac{\partial k}{{\partial x_{j} }}} \right] + G_{k} + G_{b} - \rho \varepsilon - Y_{M} + S_{k}$$
(9)

Dissipation rate ε of mixed fluid is

$$\frac{\partial }{\partial t}\left( {\rho \varepsilon } \right) + \frac{\partial }{{\partial x_{i} }}\left( {\rho \varepsilon u_{i} } \right) = \frac{\partial }{{\partial x_{i} }}\left[ {\left( {\mu + \frac{{\mu_{t} }}{{\sigma_{\varepsilon } }}} \right)\frac{\partial \varepsilon }{{\partial x_{j} }}} \right] + C_{1\varepsilon } \frac{\varepsilon }{k}\left( {G_{k} + C_{3\varepsilon } G_{b} } \right) - C_{2\varepsilon } \rho \frac{{\varepsilon^{2} }}{k} + S_{\varepsilon }$$
(10)

2.4 Calculation Method

The numerical simulation of fan-shaped nozzle jet flow field is solved by a separation solver. The first-order accuracy upwind difference scheme is selected, and the turbulence control equation is based on the volume fraction of liquid–solid multiphase flow. The convergence criterion of numerical simulation is residual R ≤ 10–4, the relaxation factor is the default value.

2.5 Boundary Condition

The pressure inlet and outlet for the fan-shaped nozzle are positioned at the nozzle’s inlet and the right end face of the outward flow field’s outlet, respectively. The pressure of the fan-shaped nozzle outward flow field is designated as the standard atmospheric pressure in its respective domain. Additionally, the standard wall function without velocity slip is applied to the inner wall surface of the nozzle. The density of the main phase, water, is established at 1000 kg/m3, whereas its viscosity is 1 × 10–3 Pa·s. For the secondary phase, flake graphite, its density is set at 2100 kg/m3. The viscosity of this phase is measured at 1 × 10–5 Pa·s, and it has a diameter of 0.1 mm. In order to analyze how variations in the inlet pressure affect the properties of the jet flow field.

3 Results and Discussion

3.1 Effect of Inlet Pressure on the Flow Field of Flake Graphite

Figure 2 shows the axial velocity distribution cloud diagram of the flake graphite flow field in the fan-shaped nozzle when the inlet pressure p is 2 MPa, 4 MPa, 6 MPa, 8 MPa, and 10 MPa.

Fig. 2
4 cloud diagrams of the fan-shaped nozzle with the distribution of velocity along the flake graphite at p = 2, 4, 6, and 8 megapascals. The inlet pressure region has the highest axial velocity distribution, and the outflow field has the lowest velocity distribution.figure 2

Cloud diagram of axial velocity distribution in the flow field of flake graphite

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From Fig. 2, it can be seen that the axial velocity distribution of the flake graphite flow field has significant surface symmetry at different inlet pressures, and the jet plane diffuses outward at a certain angle. Due to the acceleration of the conical contraction structure of the fan nozzle, the axial velocity of the flake graphite flow field increases continuously. In the cylindrical section of the middle of the nozzle, the axial velocity of the flake graphite flow field remains unchanged due to the constant cross-sectional area. At the nozzle outlet, due to the joint action of the spherical structure at the fan-shaped nozzle outlet and the V-groove, the cross-sectional area of the jet decreases rapidly, and the flow field of flake graphite is accelerated for the second time, and its axial velocity reaches the maximum at the V-groove at the nozzle outlet. Similarly, the axial dynamic pressure distribution rule of flake graphite flow field obtained by analysis is similar to the axial velocity.

3.2 Effect of Target Distance on the Cross-Sectional Shape of the Flow Field of Scaled Graphite

Figure 3 shows the shape distribution of axial velocity section of flake graphite flow field of fan-shaped nozzle when the inlet pressure p is 10 MPa and the target distance l is 0, 5, 10, 15 and 20 mm respectively.

Fig. 3
3 cloud diagrams of the flake graphite with the distribution of velocity at l = 0, 5, and 10 millimeters. The circular region has the lowest axial velocity distribution, and the center region has the moderate and highest velocity distributions.figure 3

Axial velocity section shape of flake graphite outward flow field

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From Fig. 3, it can be seen that when the target distance is 0~10 mm, the axial velocity of flake graphite is relatively large, the regular flat structure of the jet can be basically maintained and gradually developed. When the target distance is 0 mm, 5 mm and 10 mm respectively, the width b and height h of the axial velocity flat section are 3.84 mm and 0.61 mm, 9.70 mm and 1.62 mm, 14.75 mm and 2.02 mm respectively. Because the flat structure of the jet is in contact with the air, with the increase of the target distance, the disturbance of the outflow field air to the jet becomes more intense, and the uniformity of the flow field decreases. When the target distance is 15 mm, the jet begins to fluctuate obviously, the shape of the flat section of the jet changes, and the height direction gradually becomes a symmetrical “groove” structure with high ends and low middle. At this time, the width b and height h on both sides of the axial velocity section are 18.99 mm and 3.05 mm respectively. When the target distance is 20 mm, the air moves with jet entrainment to the outlet of the outward flow field. At this time, the axial velocity section width of flake graphite reaches the maximum, and its width b and height h are 19.60 mm and 4.02 mm, respectively.

3.3 Flake Graphite Flow Field Axial Velocity Change Rule of Law

Figure 4 shows that the axial velocity curve of the flake graphite flow field on the nozzle axis changes with the abscissa x when the inlet pressure is 10 MPa.

Fig. 4
A line graph plots axial velocity versus abscissa. The y axis ranges from 0 to 120, and the x axis ranges from 0 to 35. A curve begins at (0, 5), follows an increasing trend with some fluctuations, reaches a peak at (15, 106), again decreases gradually, and ends at (35, 20). Values are estimated.

Axial velocity curve of flake graphite flow field

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From Fig. 4, it can be seen that in the jet flow field, due to the small volume fraction of flake graphite, the water flow field has a good acceleration effect on the flake graphite flow field, and the kinetic energy of the water flow field is fully utilized by the flake graphite flow field. In the inner flow field, the abscissa is 0 mm (at the entrance of the fan-shaped nozzle), and the axial velocity of the flake graphite flow field is 4.45 m/s. Due to the increase of abscissa, the axial velocity of flake graphite flow field increases rapidly and gets the first acceleration. At the abscissa position of 9.3 mm (the outlet of nozzle cone convergence section), the axial velocity of flake graphite flow field is 40.53 m/s. When the abscissa is in the range of 9.3~14.3 mm (the cylindrical section of the nozzle), the axial velocity of the flake graphite flow field remains basically unchanged due to the constant cross-sectional area of the cylindrical section. When the abscissa is in the range of 14.3~15.3 mm (the spherical V-groove at the outlet of the nozzle), due to the joint action of the spherical structure at the outlet of the fan-shaped nozzle and the V-groove, the jet changes from cylindrical to flat fan-shaped jet, the cross-sectional area of the jet decreases rapidly, the axial velocity of the flake graphite flow field increases sharply, and the second acceleration is obtained. At the abscissa of 15.3 mm (the V-groove outlet), the axial velocity of the flake graphite flow field is 101.24 m/s. In the outward flow field, in the range of 15.3~15.8 mm on the abscissa, because the flow field of flake graphite at the nozzle outlet lags behind the water flow field, the flake graphite gets the third acceleration under the action of water. At the position of 15.82 mm on the abscissa, the axial velocity of the flow field of flake graphite reaches the maximum, which is 104.38 m/s. Then, with the increase of target distance, the axial velocity of flake graphite flow field decreases rapidly.

4 Conclusion

  1. (1)

    The axial velocity distribution cloud diagram of flake graphite flow field of fan-shaped nozzle jet has obvious plane symmetry, and the jet plane diffuses outward at a certain angle. The axial velocity of the internal flow field has two accelerations, and the axial velocity of flake graphite reaches the maximum at the V-groove at the nozzle outlet, and the maximum axial velocity is positively correlated with the inlet pressure.

  2. (2)

    The section shape of the flow field of flake graphite can basically maintain the regular flat structure of the jet when it is ejected that the target distance is less than 10 mm.When the target distance is 15 mm, the cross-section shape of the flow field presents a symmetrical “groove” structure with high ends and low middle. When the target distance is 20 mm, the cross-section shape of the flow field section shape reaches the maximum.

  3. (3)

    The axial velocity of flake graphite flow field of fan-shaped nozzle jet has two acceleration in the internal flow field and one acceleration in the outward flow field. When the jet inlet pressure is 10 MPa, the axial velocity reaches the maximum at the abscissa of the outward flow field of 15.82 mm, and its value is 104.38 m/s. After that, the axial velocity decreases rapidly with the increase of target distance.