NUMERICAL SIMULATION OF THREE DIMENSIONAL GAS-PARTICLE FLOW IN A SPIRAL CYCLONE

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1 Applied Mathematics and Mechanics (English Edition), 2006, 27(2): c Editorial Committee of Appl. Math. Mech., ISSN NUMERICAL SIMULATION OF THREE DIMENSIONAL GAS-PARTICLE FLOW IN A SPIRAL CYCLONE WANG Can-xing (fi (), YI Lin ( ) (College of Mechanical and Energy Engineering, Zhejiang University, Hangzhou , P. R. China) ψψψ (Communicated by LIN Jian-zhong) Abstract: The three-dimension gas-particle flow in a spiral cyclone is simulated numerically in this paper. The gas flow field was obtained by solving the three-dimension Navier-Stokes equations with Reynolds Stress Model (RSM). It is shown that there are two regions in the cyclone, the steadily tangential flow in the spiral channel and the combined vortex flow in the centre. Numerical results for particles trajectories show that the initial position of the particle at the inlet plane substantially affects its trajectory in the cyclone. The particle collection efficiency curves at different inlet velocities were obtained and the effects of inlet flow rate on the performance of the spiral cyclone were presented. Numerical results also show that the increase of flow rate leads to the increase of particles collection efficiency, but the pressure drop increases sharply. Key words: spiral cyclone; numerical simulation; particle trajectory; collection efficiency Chinese Library Classification: O359; TQ Mathematics Subject Classification: 76T20 Digital Object Identifier(DOI): /s Introduction The spiral cyclone is a new type of cyclone and its schematic structure is given in Fig.1. Its body is made up of a continuous spiral channel with some turns; particles are collected on the side wall due to the centrifugal force when the gas mixed with particles flows through the spiral channel. Experimental studies have demonstrated that spiral cyclones have the significant advantages, such as lower pressure loss, higher collection efficiency and more sampling flow rate than conventional cyclones, etc [1]. Due to the very low volumetric concentration of the dispersed particles in cyclones, the effects of particles on the fluid flow are usually ignored. In this circumstance, the fluid and particle flows may be considered separately in the numerical simulation. A common approach is to solve the fluid flow firstly without considering the presence of particles, then the particles flow can be calculated based on the solution of the fluid flow. This method is used in this paper. Fig.1 A schematic diagram of a spiral cyclone Received Jun.11, 2004; Revised Aug.29, 2005 Corresponding author WANG Can-xing, Doctor, mecwangcx@cmee.zju.edu.cn

2 248 WANG Can-xing and YI Lin 1 Simulation of Gas Flow 1.1 Governing equations for fluid flow in spiral cyclone Most fluid flows in cyclones are turbulent and incompressible. The Reynolds-averaged continuity and momentum equations are shown as u i =0, (1) x i u i t + u u i j = 1 P + υ 2 u i x j ρ x i x ( ρu i u j ), (2) j ρ x j where ρu i u j is known as the Reynolds stress tensor which represents the effects of turbulent fluctuations. Equations (1) and (2) are not closed, so turbulent models, which relate the Reynolds stress tensor to the mean flow, must be provided. 1.2 Turbulent models The selection of turbulent model in the numerical simulation is vital for a correct prediction. The standard k ɛ model has been shown to be inadequate for the simulation of swirling flows in cyclones. This is due to the isotropic assumption on the turbulence which can not properly describe the anisotropic behavior of swirling flow. In the RNG (Renormalization grop) k ɛ model, the effect of rotation is included in the turbulent viscosity. This model gives an improved prediction but the improvement is still limited. The RSM (Reynolds stress model) is the most complicated and accurate model. It takes into account the six independent components of the Reynolds stress tensor and solves their transport equations, and has been shown to be more suitable for the simulation than k ɛ model and RNG k ɛ model [2]. In this paper, RSM was used in the numerical simulation. The kinetic energy equation, dissipation rate equation and Reynolds stress equation in the RSM are [3] D Dt (u i u j )= Dk Dt = Dɛ Dt = where P ij = (u i u k u j u k k 2 [(C k x l ɛ k 2 [(C k x l ɛ + υ) u i u j x l ]+P ij + ϕ ij 2 3 ɛδ ij, (3) + υ) k x l ]+P k ɛ, (4) k 2 ɛ ɛ [(C ε + υ) ]+C ɛ1 P k x l ɛ x l k C ɛ2 ɛ 2 k, (5) + u u j u i k u k ), P k = u i u u i ɛ l x l, ϕ ij = C 1 k (u i u j 2δ ij k/3) C 2 (P ij 2δ ij P k /3) the constants take the values: C k 0.10, C ɛ =0.09, C ɛ1 1.44, C ɛ2 1.92, C 1 1.8, C Simulation of Particle Flow in Spiral Cyclone The trajectory of a dispersed particle can be determined by solving their equations of motion in a Lagrangian reference frame. The equation of particle motion is given by m dui p = F i, (6) dt where m is the mass of particle, u i p is the particle velocity and F i is the forces which act on the particle. Although there are a number of different forces acted on the particle, they have different influence on the motion of the particle. In most cyclones, when the density of the particle is much greater than the density of the gas, the drag force is dominant force and the others may be ignored. For a sphere particle, the drag force is given by F D i = m 18μ ρ p d p 2 C D Re p (u i u i 24 p), (7)

3 Numerical Simulation of Gas-Particle Flow in Spiral Cyclone 249 where μ is the dynamic viscosity of the gas, u i p and u i are the particle and gas velocity respectively, d p is the particle diameter, ρ p is the particle density, Re p is the particle Reynolds number, given as Re p = ρd p u u p /μ, C D is the coefficient of drag force and it can be expressed as a function of the particle Reynolds number: C D = a 1 + a 2 /Re p + a 3 /Re 2 p,where a 1, a 2 and a 3 depend on particle Reynolds number [4]. Due to the low volumetric concentration of the dispersed particles, the effect of the particles on the gas may be neglected, and we assumed that the particles do not interfere with each other. In this paper, particles are generated by cooking, their dimension are micro level, so we can assumed that they are spherical and they can be collected once if they reach the cyclone wall. 3 Results and Discussion 3.1 Computational domain and boundary conditions The dimensions of the spiral cyclone are given in Fig.2, where cyclone height h=0.3 m, inlet width b=0.1 m, outlet diameter D e =0.28 m, maximum diameter of body R max =0.55 m. The Archimedes spiral is selected in this paper and it can be described as r = K θ in polar coordinates, where the constant K= The channel has 3 turns in cyclone body and the out wall is from 5π to 11π, the inside wall is from 3π to 9π. The finite volume method [5] (FVM) has been used in the numerical simulation. After domain discretization, the grids used in the calculation are shown in Fig.3. The control volume integration method is used in the discretization of governing equations, and for the term of convection the QUICK interpolation scheme is used. Further, the SIMPLE is used for the pressure-velocity coupled equations. The boundary conditions of the gas flow are as follows: At the inlet, the velocity of air flow into the cyclone is 10, 15, 20 and 25 m/s, respectively. The turbulent kinetic energy is k = 3 2 C2 t v2 in where turbulence intensity C t is set to be 6%, subscript in denotes inlet. The dissipation rate is ɛ = C 3/4 μ k 3/2 /(0.07L), where the constant C μ =0.09 and the characteristic length L take the value of the width of inlet [6]. The Reynolds stress are u i u j =2k/3 (ifi = j), u i u j =0(ifi j). At the outlet, the value of static pressure is set to be zero. Fig.2 Dimensions of the spiral cyclone Fig.3 The grids used in the simulation 3.2 Features of flow field The numerical results show that gas flow in the spiral channel is a steadily tangential flow which is advantageous for particles separation. When reaching the centre of the cyclone, the gas flow becomes to have a complicated structure. The results also show that there is a backflow zone near the axis in outlet, and this is due to a negative pressure gradient caused by high tangential velocity.

4 250 WANG Can-xing and YI Lin The radial profiles of tangential velocity at different axial positions are shown in Fig.4, when the inlet velocity is 20 m/s. It can be seen that there exists a combined vortex structure in the centre region of the cyclone, that is, the confined vortex in the core and the free vortex near the core. The tangential velocity increases with the increase of radial distance R in the confined vortex region and reaches a maximum, then decreases with increasing radius in the free vortex region. Further more, the results show that the effect of axial positions on the tangential profiles is little. Figure 5 shows the axial velocity profiles. It can be seen that the axial velocities fluctuate in the spiral tunnel, and they have opposite trends in the upside and underside of channel, that is, the axial velocities have the same magnitude but their directions are opposite at the same radial distance R. In the center region of cyclone body, the axial velocity varies strongly and this is due to the entrance of the reverse flow while the gas leaves the outlet away. Figure 6 shows the static pressure profiles. It can be seen that the pressure loss in the spiral channel is smaller than that in the center Fig.4 Tangential velocity profiles region. This is due to the strongly swirling flow in center region. Fig.5 Axial velocity profiles Fig.6 Static pressure profiles 3.3 Particle trajectories and collection efficiency When the spiral cyclone operates at inlet velocity of 20 m/s, Fig.7 shows the typical particle trajectories which are injected from the same point (the center of the inlet plane). The particle with a diameter of 5 μm escapes from the outlet, and the particles with a diameter of 10 μm are collected on the wall. It can be seen that from the same initial position, the particles of larger diameter take less turns in the cyclone before they reach the wall to be collected. Figure 8 shows the trajectories injected from different initial position, the diameters of the particles are all 11 μm. It is found that the initial position of the particle at the inlet plane significantly affects the collection probability of the particle. Particles injected from the outer side of the inlet plane have a greater chance to be collected than those particles injected from the inner side, see Figs.8(a) and 8(b). This is because the particles injected from outer side of the inlet plane more easily reach the side wall of the cyclone. Further more, it can be observed that the particle trajectory injected from the upside of the inlet is an ascending spiral line and that particle trajectory injected from the underside is a descending spiral line, see Figs.8 (c) and 8(d). This is due to the distribution of axial gas velocity. In the same radial distance, the

5 Numerical Simulation of Gas-Particle Flow in Spiral Cyclone 251 (a) Particle diameter: 5 µm (b) Particle diameter: 10 µm Fig.7 Two typical particle trajectories (a) Outside (b) Inside (c) Upside (d) Downside Fig.8 Particle trajectories injected from different initial positions (the particle diameter is 11 µm) axial velocities at the upside and underside region have the similar magnitude but their directions are opposite. For a given particle diameter, the particles trajectory injected from different locations along the inlet are calculated, then the collection efficiency for this particle size can be obtained. If the calculations are repeated for different size, it is possible to construct the entire efficiency curve and determine the 50% cut-off diameter d 50 and the critical diameter d [7] 100. In this paper, 300 particles are placed in the inlet plane uniformly, and their diameters gradually increase from 4 μm to the critical diameter. Figure 9 shows the efficiency curves of collection at different inlet velocities. It can be seen that a higher inlet velocity results in higher collection efficiency. Furthermore, a sharp efficiency Fig.9 Particle collection efficiency curves curve can be observed with a high inlet velocity, and this shows that the effect of particle

6 252 WANG Can-xing and YI Lin size on the collection efficiency is more remarkable when there is a higher inlet velocity. The key performance parameters of cyclone are: gas flow rate, collection efficiency and pressure loss. Figure 10 shows the relationship between efficiency of collection and gas flow rate, and Figure 11 shows the relationship between pressure loss and gas flow rate. In Fig.10, the cut-off diameter d 50 is employed to indicate the collection efficiency. It can be seen that an increase in the flow rate leads to an increase in collection efficiency, while the pressure loss sharply increases with the increasing of gas flow rate, see Fig.11. So it is important to choose an appropriate inlet velocity. The coefficient ξ is usually used to represent the pressure loss of cyclones, where ξ = Δp/(ρu 2 in /2). For this spiral cyclone, results of the numerical simulation show that ξ is about 5, which is smaller than that of the conventional cyclones. Fig.10 Relationship between collection efficiency and gas flow rate Fig.11 Relationship between pressure drop and gas flow rate 4 Conclusions In this paper, three-dimension of two-phase flow in spiral cyclone has been numerically simulated. Because of the highly rotational turbulent flow in the spiral cyclone, the RSM turbulent model has been selected in the simulation. The results show that there is a steady fluid flow in the spiral channel, which is advantageous for particle separation. In the centre of cyclone body, the tangential velocity profiles show a combined vortex structure, and the highly rotational flow results in a large pressure loss. The simulating results of particle trajectories show that the initial position of the particle at the inlet plane substantially affects its trajectory in the cyclone. In this paper, some performance parameters at different inlet velocities have been studied; the corresponding efficiency curves of particle collection, depended on the inlet velocity and particle diameter, have been obtained; the relationships, between collection efficiency and flow rate, and between pressure loss and flow rate, have been constructed. The results show that an increase in the flow rate leads to an increase in collection efficiency, while the pressure loss increases. So it is important to choose an appropriate inlet velocity. Results of the numerical simulation also show that for the spiral cyclone considered in this paper, the coefficient of pressure loss is about 5, which is smaller than that of the conventional cyclones. References [1] Zhao Jialin, Gao Qinyou. Analysis on the structure characteristic of continuous spiral cyclones[j]. Journal of Shandong Institute of Building Materials, 1997, 11(1):71 74 (in Chinese).

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