2.1 Gas phase control equation
Based on the fact that the air in the shot blasting machine recovery unit is the carrier for transporting solid particles, the air is considered as incompressible and continuous phase, so the Navier-Stokes equation can be used to solve the gas-phase flow. Considering the small effect of temperature on the gas-solid two-phase flow and transport piping in the blast machine recovery unit, the effect of temperature in the solid particle recovery process is not considered. The Standard-k-ω model is the most widely used turbulence model. The flow of the gas phase is consistent with the conservation of air mass and momentum, then the mass and momentum equations of the gas phase in the corresponding blast machine recovery unit are:
$$\frac{{\partial {\rho _{\text{a}}}}}{{\partial t}}+\nabla \cdot \left( {{\rho _{\text{a}}}{v_{\text{a}}}} \right)=0$$
1
$$\frac{{\partial \left( {{\rho _{\text{a}}}{v_{\text{a}}}} \right)}}{{\partial t}}+\nabla \cdot \left( {{\rho _{\text{a}}}v_{{\text{a}}}^{2}} \right)= - \nabla {p_{\text{a}}}+\nabla {\tau _{\text{a}}}+{\rho _{\text{a}}}g - {f_{\text{a}}}$$
2
Where: ρa is the air density, kg/m3; t is the time, s; va is the air speed, m/s; pa is the air pressure, Pa; g is the acceleration of gravity, m/s 2; τa is the air stress tensor, Pa; fa is the average resistance of air, N.
A standard k-ε turbulence model is used to describe the rotating airflow in the recovery bin of a horizontal mobile blast machine, which is represented in the model as follows:
$$\frac{\partial }{{\partial t}}({\rho _g}{k_g})+\frac{\partial }{{\partial {x_i}}}\left( {{\rho _g}{k_g}{u_i}} \right)=\frac{\partial }{{\partial {x_j}}}\left[ {\left( {\mu +\frac{{{\mu _t}}}{{{\sigma _k}}}} \right)\frac{{\partial k}}{{\partial {x_j}}}} \right]+{G_k}+{G_b} - {\rho _g}\varepsilon - {Y_M}$$
4
$$\frac{\partial }{{\partial t}}\left( {{\rho _g}\varepsilon } \right)+\frac{\partial }{{\partial {x_i}}}\left( {{\rho _g}\varepsilon {u_i}} \right)=\frac{\partial }{{\partial {x_j}}}\left[ {\left( {\mu +\frac{{{\mu _t}}}{{{\sigma _\varepsilon }}}} \right)\frac{{\partial \varepsilon }}{{\partial {x_j}}}} \right]+{C_{1\varepsilon }}\frac{\varepsilon }{{{k_g}}}\left( {{G_k}+{C_{3\varepsilon }}{G_b}} \right) - {C_{2\varepsilon }}{\rho _g}\frac{{{\varepsilon ^2}}}{{{k_g}}}$$
5
where ,kg is the turbulent kinetic energy, ε is the turbulent dissipation rate, \(\mu\)is the dynamic viscosity of the gas, and the model constants C1ε、 C2ε、C3ε,、\({\sigma _k}\)and\({\sigma _\varepsilon }\) have the following default values: C1ε=1.44, C2ε=1.92, C3ε=1.3, \({\sigma _k}\)= 1.0, \({\sigma _\varepsilon }\)= 1.3. \({\mu _t}\) is the turbulent viscosity, which is expressed as follows
$${\mu _t}={\rho _g}{C_\mu }\frac{{{k^2}}}{\varepsilon },{C_\mu }=0.09$$
6
2.2 Solid phase control equation
Given that the air in the blast machine recovery unit carries the solid particle motion, the solid particle population is considered as a discrete phase, so the flow of solid particles can be solved by DPM. Based on the Lagrange coordinate system[37], the trajectory of the discrete phase is solved by the differential equation of particle forces to simulate the particle motion in turbulent flow. Then the expression of the differential equation of particle force in the discrete phase is:
$$\left\{ {\begin{array}{*{20}{l}} {\frac{{{\text{d}}{v_{\text{p}}}}}{{{\text{d}}t}}={F_{\text{d}}}\left( {{v_{\text{a}}} - {v_{\text{p}}}} \right)+\frac{{g\left( {{\rho _{\text{p}}} - {\rho _{\text{a}}}} \right)}}{{{\rho _{\text{p}}}}}+{F_{{\text{other~}}}}} \\ {{F_{\text{d}}}=\frac{{18\mu }}{{{\rho _{\text{p}}}d_{{\text{p}}}^{2}}}\frac{{{C_{\text{d}}}R{e_{\text{p}}}}}{{24}}} \\ {R{e_{\text{p}}}=\frac{{{\rho _{\text{a}}}{d_{\text{p}}}\left| {{v_{\text{p}}} - {v_{\text{a}}}} \right|}}{\mu }} \\ {{C_{\text{d}}}={a_1}+\frac{{{a_2}}}{{R{e_{\text{p}}}}}+\frac{{{a_3}}}{{Re_{{\text{p}}}^{2}}}} \end{array}} \right.$$
7
Where: vp is the particle velocity, m/s; Fd(va-vp)is the traction force per unit mass of particles, m/s2; ρp is the particle density, kg/m3; µis the aerodynamic viscosity, Pa·s; Cdis the traction coefficient; Rep is the relative Reynolds number of particles; dp is the particle size, m; Fotheris the other force per unit mass of particles, m/s2; a1、a2、a3 are constants.
Considering the actual working conditions, the two-way coupling method of CFD-DEM is used to simulate the transport process of gas and solid phases in the shot blasting machine recovery unit[38]. Based on the above controlling equations for the gas and discrete phases, the momentum value transmitted from the gas phase to the discrete phase is solved by calculating the momentum change of the particles when they pass through a defined spatial region in the flow field, thus realizing the two-way coupling calculation[39, 40].
The contact forces between particles a、b and the pipe wall on particle a, are shown in Fig. 1. The above rotational motion of particle a is described as:
where, \({I_a}\)and \({\omega _a}\)denote the moment of inertia tensor and rotational velocity of particle a, respectively. \({F_{c,ab}}\) is the tangential force acting on particle a by particle b. \({F_{c,ab}}\)is defined as follows:
$${F_{c,ab}}={R_{a,b}}\left( {{F_{n,ab}}+{F_{t,ab}}} \right)$$
9
$${F_{n,ab}}=\frac{4}{3}{E^ * }\sqrt {{R^ * }} \delta _{{n,ab}}^{{1.5}}$$
10
$${F_{t,ab}}=-2\sqrt {\frac{5}{6}} \frac{{\ln e}}{{\sqrt {{{\ln }^2}e+{\pi ^2}} }}\sqrt {{S_n}{m^*}} {v_{n,ab}}$$
11
$${S_n}=2{E^ * }\sqrt {{R^ * }{\delta _{n,ab}}}$$
12
Where,\({F_{n,ab}}\) is the normal contact force, \({F_{t,ab}}\) is the tangential contact force, \({R_{a,b}}\)is the vector from the center of mass to the point of contact, \({S_n}\) is the normal stiffness, \(\delta _{{n,ab}}^{{}}\) is the normal overl, p, \({E^ * }\),\({R^ * }\),\({m^*}\)\({v_{n,ab}}\)and is the equivalent Young's modulus, equivalent radius of the particle, equivalent mass, relative normal velocity and recovery factor.
\({F_{d,ab}}\) is the rolling friction moment of particle b acting on particle a, \({F_{d,ab}}\)defined as follows:
$${F_{d,ab}}= - {\mu _{r,ab}}{d_a}\frac{{{\omega _{ab}}}}{{\left| {{\omega _{ab}}} \right|}}$$
13
where, \({\mu _{r,ab}}\)and\({d_a}\) are the rolling coefficient and particle diameter, respectively, \({\omega _{ab}}\)and is the relative angular velocity of particle a to particle b. During the particle-particle collision, the forces and moments on the particle are similar to those in the particle-particle collision, That is, equations (9)-(13).。