Front. Chem. Sci. Eng. 2012, 6(3): 329–338 DOI 10.1007/s11705-012-1205-5
RESEARCH ARTICLE
Simulating multiphase flow in a two-stage pusher centrifuge using computational fluid dynamics Chong PANG1, Wei TAN1, Endian SHA2, Yuanqing TAO2, Liyan LIU (✉)1 1 School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China 2 Zhejiang Qingji Ind. Co. Ltd. Inc., Hangzhou 311401, China
© Higher Education Press and Springer-Verlag Berlin Heidelberg 2012
Abstract The design of two-stage pusher centrifuges have developed rapidly, but a good understanding of the theory behind their practice is a long-standing problem. To better understand centrifugal filter processes, the computational fluid dynamics (CFD) software program FLUENT has been used to model the three-dimensional geometry and to simulate multiphase flows based on Euler-Euler, moving mesh, dynamic mesh and porous media models. The simulation tangential velocities were a little smaller than those for rigid-body motion. In the stable flow region, the radial velocities were in good agreement with the theoretical data. Additionally, solid concentration distribution were obtained and also showed good agreement with the experimental data. These results show that this simulation method could be an effective tool to optimize the design of the two-stage pusher centrifuge. Keywords two-stage pusher centrifuge, multiphase flow, CFD, dynamic mesh, porous media
1
Introduction
Two-stage pusher centrifuges are a kind of filter centrifuge with a pulse discharge that can work continuously with automatic operation. All of their operating procedures, for example, separating, washing, drying and discharging can be completed while running at full-speed. In addition, these centrifuges have many prominent advantages, which include compact structures, continuous pushing slags, low power consumptions, large capacities and low moisture contents [1]. Therefore, they have been widely used in many chemical industries to separate suspended matter from a non-uniform phase and they are especially useful for the dehydration of suspensions containing crystals or
fibrous solids [2]. Currently, there is a great deal of theoretical research on two-stage pusher centrifuges. However, predicting the performance and doing design calculations on the machine depends on experience and tests. The main reason for this phenomenon is that centrifugal separation processes are diverse and complex. Particularly, mathematical problems arise because the sizes, shapes and motions of solid particles, which are in chaotic situation, cannot be accurately determined at present. This creates great difficulties in studying the theory of this process. Optimizing parameters using only experiments is not very reliable and it is costly. To determine the laws that govern the fluid flow in the centrifuge, appropriate mathematical models must be established so simulations can be developed. Theoretical studies have been done for settling centrifuges like solid bowl and tubular centrifuges and a series of flow laws have been determined [3–8]. But research on filtering centrifuges, especially the two-stage pusher centrifuge, has rarely been reported in the literature. To optimize the geometry and process conditions, the characteristics of flow field inside the two-stage pusher centrifuge must be known. Herein, the commercially available computational fluid dynamics (CFD) software FLUENT was used as a tool to study the flow characteristics within a two-stage pusher centrifuge using multiphase flow, moving mesh, dynamic mesh and porous media models. The simulated results were compared with the theoretical model or with the experimental data. The simulated results were also used to optimize the structural design of the two-stage pusher centrifuge, which is very important for practical applications.
2
Mathematical model
Received March 23, 2012; accepted May 11, 2012
2.1
Governing equations
E-mail:
[email protected]
Currently, there are two approaches for the numerical
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calculation of multiphase flows: the Euler-Lagrange approach, and the Euler-Euler approach [9,10]. In the Euler-Lagrange approach, the primary-phase fluid is treated as a continuum by solving the Navier-Stokes equation, while the dispersed-phase fluid is solved by tracking a large number of individual droplets. However, a fundamental assumption of this approach is that the dispersed phase occupies a low volume fraction usually less than 10%. This makes the model inappropriate for two-stage pusher centrifuges. The Euler-Euler approach is universal and usually more complex. In this approach, the different phases are treated as interpenetrating continua. The continuous and momentum balance equations for each phase are written as follows: Continuous equation: ∂ðm αm Þ ∂ þ ðm αm um,i Þ ¼ 0: ∂t ∂xj
(1)
Momentum equation: ∂ ∂ ð α u u Þ ð α u Þ þ ∂t m m m,i ∂xj m m mi mj ∂p þ m xm gi þ Fmi ∂xi ∂ðαm umj Þ ∂ ∂ðαm umi Þ , þ m þ ∂xi ∂xj ∂xj
term given by R¼
η¼
k
ε
C η3 ð1 – η=η0 Þε2 , ð1 þ β0 η3 Þk
(5)
∂Uj ∂Ui 1=2 ∂Ui þ : ∂xj ∂xi ∂xj
(6)
The constants in the RNG κ-ε turbulence scheme are specified in Table 1 [11]. Table 1
The value of the parameters in the RNG κ-ε turbulence scheme
Parameter
Value
C
0.0845
k
0.7179
ε
0.7179
Cε1
1.42
Cε2
1.68
β0
0.012
η0
4.377
¼ – αm
2.2
(2)
where m and m are the density and viscosity for phase m, umi and αm are the instantaneous velocity components and phase volume fraction for phase m, respectively, and Fmi is the inter-phase momentum exchange term. Two-stage pusher centrifuges are a kind of high-speed rotating equipment that produces several concentrated vortices in its internal fluid field. In this study, the renormalization group (RNG) κ-ε turbulence scheme presented by Yakhot and Smith was used [11]. This scheme differs from the standard κ-ε turbulence scheme in that it includes an additional sink term in the turbulence dissipation equation to account for nonequilibrium strain rates and employs different values for the model coefficients [12]. The prognostic equations of turbulent kinetic energy and its dissipation rate can be written as ∂k ∂k ∂Ui ∂ Km ∂k – ε, (3) ¼ – ui uj þ þ Uj ∂xj ∂t ∂xj ∂xj k ∂xj ∂ε ∂ε þ Uj ∂t ∂xj
ε ∂U ∂ ¼ – Cε1 ui uj i þ k ∂xj ∂xj
Km ∂ ε ε2 – Cε2 – R, (4) ε ∂xj k
where k , ε , Cε1 and Cε2 are empirical constants. The last term on the right-hand side of Eq. (4) is an extra strain rate
Dynamic mesh model
The first-stage drum of a two-stage pusher centrifuge not only has a rotating motion, but also has a reciprocating motion, resulting in changes in the shape of the flow field with time. This requires using the dynamic mesh model of FLUENT. The boundary movement involved in the dynamic mesh model can be pre-defined and not in advance, meaning that the boundary movement is determined by the results of the previous step. Before the dynamic mesh model can be used, the movement of the initial grid and boundary must be defined. At the same time, the region that moves also needs to be specified, which can be defined by a boundary-type function or by user-defined functions (UDF). In dynamic mesh computations, grid dynamic changes can be calculated by three models: the smoothing spring model, the dynamic layering model and the local remeshing model [13,14]. Since the first stage of the two-stage pusher centrifuge is divided into structured hexahedral grids, the dynamic layering model was adopted. The main principle of the dynamic layering model is that the active layer is added or reduced based on changes in the grid height near the moving boundary layer. As the boundary moves, a grid is divided into two grids if the grid height increases to a certain value close to the boundary. Conversely, if the height of grid layer decreases to a certain value, the two grids are combined into one, as shown in Fig. 1. If the grid layer j expands, there is a cutoff of cell height when
Chong PANG et al. Simulating multiphase flow in a two-stage pusher centrifuge
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two-stage pusher centrifuge (Fig. 3). The technical parameters are given in Table 2. Since the structure of the accelerating disk is irregular, a T/Grid type of element was selected for this part. Hex types were chosen for the remaining parts. Finally, the whole geometry was meshed into 331023 cells, 1140789 faces and 333930 nodes.
3 Fig. 1 Dynamic layering model
hmin > ð1 þ αs Þh0 ,
(7)
where hmin is the cell’s minimum height, h0 the ideal cell height and αs is the partition factor for the layer. Using the above conditions, the grid cells can be divided using the constant height method or the constant ratio method. 2.3
Geometry and mesh
The filter separation component of the two-stage pusher centrifuge is mainly composed of a feeding pipe, an accelerating disk, a first-stage drum and a second-stage drum (Fig. 2). In this paper, Gambit, FLUENT pre-drawing software, was used to generate a 3D geometric model of
Experiment
Using current experimental conditions, we can measure the cake and filtrate solid content, while it is difficult to measure the velocity and pressure distribution inside the centrifuge, Fig. 4 shows operation flow chart of industrial experiments. A fluid enters the centrifuge with a constant flow rate of 288 L/min and a solid content of 50%. It goes through the inlet of the centrifuge into the accelerating disk which consists of eight blades (Fig. 4(b)). The accelerator rotates with the same angular velocity as the machine and the fluid is gradually accelerated to 1300 r$min–1. Then, the fluid reaches the first-stage drum and the solids form a filter cake. Under the action of the rotary and reciprocating motion of the first-stage drum, the filter cake will move some distance to reach the second-stage drum (Fig. 4(c)). The filter cake is then further dehydrated. When the filter
Fig. 2 The filter separation of the two-stage pusher centrifuge. 1-feed pipe; 2-accelerating disk; 3-first-stage drum; 4-second-stage drum
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Table 2
Technical parameters of the two-stage pusher centrifuge
Technical parameter
Value
Diameter of drums /mm
920/1000
Length of drums /mm
215/360
Speed /(r$min–1)
1300
Pusher frequency /min–1
60
Separation factor
945
Flow rate /(L$min–1)
288
which porosity, viscous resistance and inertial resistance were equal to 0.054, 5.7e6 and 1e5, respectively. When using an unsteady solver, the moving mesh model is suitable for defining the motion of the accelerator. The implementation of modified inner-outer iterative procedures is involved with the numerical solution of the governing equations in the inner zone including the blade swept region in the rotating non-inertial reference frame and that in the outer zone excluding the blade region in a fixed inertial reference frame. The interface type is necessarily used to match the flow parameters at the overlapping parts of the inner and outer zones. Layering mesh method is selected in the dynamic mesh model. According to motion form of boundary layers, zones are defined as stationary, rigid body (using compiled UDF) and deforming type separately. The discretization of the partial differential equations is achieved by control volume formulation with a staggered arrangement of primary variables, and then the pressurevelocity coupling is dealt with the Pased Coupled SIMPLE algorithm. To ensure the accuracy of the results, a thirdorder QUICK scheme is used for convective terms and a second-order central scheme for diffusive terms. 4.2
Fig. 3
3D geometric model created by Gambit
cake reaches the exit of the second-stage drum, it is discharged outside of the machine by the solid collector. At the outlet the solid content is measured by a gravimetric method in an electrically heated drying cabinet. (Fig. 4(d)).
4
Numerical model
4.1
Model setup
Since the centrifuge rotates at a high speed and the firststage drum reciprocates, an unsteady solver was selected for calculate. A Euler-Euler multiphase flow model was used to solve Eqs. (1–8), and the liquid and solid twophase flow was calculated first. Once the calculations converged, air was introduced numerically into the inlet of the flow field. Air was considered to be an incompressible gas and water to be an incompressible Newtonian fluid. The feeding inlet is defined as a velocity-inlet boundary condition and the free surface condition is valid for the top surface of the filter cake. Standard wall functions, which are widely employed for industrial flows, were applied to solve the turbulences at the walls. The sieve region was defined as a porous media, in
Model validation
It is impossible yet to measure velocity and pressure components inside the two-stage pusher centrifuge, so the results are validated by comparing with theoretical analysis. Particles sedimentate in the centrifuge form a filter cake due to centrifugal force. As this force increases with the increase of the radius, the settling velocity of particles shows the similar law. Through experiments, Reynold and Sokolov [15] obtained the settling velocity ur at the radius r in the turbulent region as follows: 2 0:5 dω rðs – l Þ ur ¼ 1:75η1 , l and η1 ¼ ð1 – xs Þ5:5 ,
(8)
where η1 is the correction coefficient of the settling velocity, xs the solid content, d the diameter of the particle, l the density of the liquid, s the density of the solid, and ω the angular velocity of the fluid-ring at the radius r. The angular velocity (ω) lags behind that of the rigid body. Hollyn revealed by experiments that ω at a radial position r can be gained as follows: k –2 – 1 ω ¼ ω0 1 – ð1 – aÞ – 2 : (9) k0 – 1 The value of a is obtained by experiment, which can be calculated by the following formula:
Chong PANG et al. Simulating multiphase flow in a two-stage pusher centrifuge
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Fig. 4 Process charts of the industrial experiments. (a) the whole two-stage pusher centrifuge machine; (b) feeding operation; (c) running operation; (d) parking and measuring operation
1 2 a ¼ 1 – 2:6 10 Re , 0 –4
and k ¼ r=r2 , k0 ¼ r1 =r2 , Re ¼ Q=r1 ,
(10)
where ω0 is the angular velocity of the drum, r1 the radius of the free surface, r2 the radius of the wall of the drum, Q the liquid flow rate inside the drum, the kinematic viscosity of liquid, and 0 the kinematic viscosity of water (25°C, 0 ¼ 1 10 – 6 m2 ⋅s – 1 ).
5
Results and discussion
5.1
Velocity distribution
5.1.1
Tangential Velocity
If the tangential velocity difference between particles and sieve is too big, particles will be destructed. To avoid this phenomenon, the accelerator is used to increase the
tangential velocity of feeding particles. A rigid body motion profile is expected for the tangential velocity. The tangential velocities are very low in the feeding inlet region, most of which are near 0m/s. When the fluid reach the accelerator (region between the two blue dashed lines in Fig. 5(b)), the tangential velocities generally increase along the rigid body motion profile. However, the values lying underneath the linear profile while the fluid leaves the accelerator. The deceleration is mainly caused by losing the friction of the blades wall, which is not big and do not make particles damage. The velocity profile of the fluid in the first-stage and second-stage drum at different axial positions is represented in Fig. 6. The values are a little lower than what expected from Eq. (10) and the deviation becomes small with the increase of the axial coordinate in the first-stage drum (Fig. 6(b)). These laws are opposite for the secondstage drum. Although the accelerator increases the tangential velocity of the fluid to a certain extent, there still exists a velocity difference between the fluid and the first-drum. When the fluid flows along the axial direction, its tangential velocity is further increased by the effect of the wall friction but still lower than that of the rigid body motion at the maximum axial coordinate. As shown in
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Fig. 5 Tangential velocity versus radial position for the center section of the accelerator at 288 L$min–1 and 1300 r$min–1. (a) contours; (b) profile
Figs. 6(a) and 6(b), there is a sudden corner location of the tangential velocity profile near the wall, because the wall function there is associated with the turbulent region flows near the wall. 5.1.2
Radial (settling) velocity
Particles, which are rotating with the drum and suffering large centrifugal force, will move along the radial direction toward the sieve wall to settle there. As shown in Fig. 7, the solid radial velocities gained from theoretical formula are compared with those from numerical simulation at surface of different z positions at T. The simulation values are very different from the analytic data at position z = 0.1 m, because this position is near outlet of the accelerator and the fluid does not flow smoothly. The same phenomenon occurred at positions z = 0.2 m and z = 0.3 m, which are located in the transition section between the first-stage and the second-
Fig. 6 Tangential velocity versus radial position in two drums at different axial positions at t = T for 288 L$min–1 and 1300 r$min–1. (a) Contours; (b) profile of the first-stage drum; (c) profile of the second-stage drum
Chong PANG et al. Simulating multiphase flow in a two-stage pusher centrifuge
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Fig. 7 Radial velocity from simulation compared with the theoretical data versus radial position at different z positions at T for 288 L$min–1 and 1300 r$min–1. (a) z = 0.10 m; (b) z = 0.20 m; (c) z = 0.30 m; (d) z = 0.40 m; (e) z = 0.50 m
stage drum. As the fluid flow became stable at positions z = 0.40 m and z = 0.50 m, the two profiles agreed well. 5.2
Solid concentration distribution
The two-stage pusher centrifuge is composed of filter
centrifuges working continuously. The material enters the machine from feeding inlet with a certain velocity and is speeded up by the accelerator to reach the first-stage drum and rotate with the similar tangential velocity. Under centrifugal force, solid components move toward the sieve wall and settle there, while most of the liquor is filtered
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Fig. 10 Solid concentration distribution near sieve wall of twostage drums at 22 s for 288 L$min–1 and 1300 r$min–1
Fig. 8 Solid concentration distribution at the axis cross section at t = T for 288 L$min–1 and 1300 r$min–1
Fig. 9 Comparison of the maximum of vertex solid concentration gained from simulation with that from experiments versus flow time at 288 L$min–1 and 1300 r$min–1
through the sieve (porous media). Because the first-stage drum reciprocates, the material moves along the axial direction to the second-stage drum during the pushing process, and suffers even greater centrifugal force. Therefore, the solid concentration increases and achieves the
maximum when the filter cake arrives at the outlet of the machine. Solid concentration distribution at the axis cross section at t = T is presented in Fig. 8, and it can be seen that there are obvious red color parts both on the top and at the bottom of left corner, which are called the dead zone. This is because that when the first-stage drum is reciprocating, the fluid cannot be driven by the movement in this region. We select one point in the surface of the outlet to study and monitor the maximum of vertex concentration over the flow time (Fig. 9). The solid concentration is nearly zero at the beginning because the outlet is full of air, but increases fast from 20.25 s to 20.4 s and reaches the maximum (98.4%) after 21.4 s, which agrees well with the experiment value (99.5%). Figure 10 shows the solid concentration distribution in the region near sieve wall of two-stage drums at 22 s. It can be seen that the solid concentration increases with the accretion of the axial position except the positions in the transition of the first-stage drum to the second-stage drum, where the fluid turbulence causes vertex motion and backflow of liquid, and leads to the decrease of the solid concentration. Additionally, the solid concentrations in the first-stage drum are generally lower than those in the second-stage drum and the maximum of the solid concentration is at the position of z = 0.45 m. Furthermore, it can be seen that there is only 0.1% solid content in the effluent from the simulation result, indicating that porous media model can effectively hold up solid particles. So it is relatively reasonable that we make such a simplification for sieve in this study.
6
Conclusions
In this study, a complex three-dimensional multiphase flow and solid settling process in the two-stage pusher centrifuge are numerically simulated using the software
Chong PANG et al. Simulating multiphase flow in a two-stage pusher centrifuge
of FLUENT, on the basis of Euler-Euler method, and moving mesh, dynamic mesh and porous media models. Compared with theoretical and experimental data, our simulation results are relatively satisfactory. The flow pattern and solid concentration distribution are acquired in the simulation. The accelerator increases the velocity of feeding material, but tangential velocities are still lower than the velocities corresponding to rigid-body motion in two-stage drums. Generally, the deviation for the first-stage drum is a little bigger than analytic data and the result is opposite for the second-stage drum. At the position where the fluid flows stably, the radial velocity profile agrees well with that of the theoretical formula. At the same time, the solid concentration distribution in twostage drums is obtained and agrees well with that from experiments at the position of the outlet. In summary, our results indicate the current simulation could be an effective tool to optimize the design of the two-stage pusher centrifuge. Acknowledgements This work has been supported by the Program for Changjiang Scholars and Innovative Research Terms in Universities of China (No. IRT0936)
Nomenclature Gk ¼ generation of turbulence kinetic energy due to the mean velocity gra dients, kg=ðm$s3 Þ Gb ¼ generation of turbulence kinetic due to buoyancy, kg=ðm$s3 Þ Cm ,C1z ,C2z ¼ turbulent parameters ↕ ↓
u ¼ velocity vector, m$s – 1 ↕ ↓
ug ¼ speed of moveming grid mesh, m$s – 1 Γ ¼ dif f usion coef f icient SΦ ¼ source term, kg=ðm$s3 Þ nf ¼ surf ace grid number of control volume ↕ ↓
Aj ¼ surf ace area vector of surf ace j hmin ¼ cells minimum height, m h0 ¼ ideal cell height, m C2 ¼ inertial resistan ce f actor, 1=m d ¼ diameter of the particle, m ur ¼ settling velocity, m$s – 1 uz ¼ axial velocity, m$s – 1 ps ¼ static pressurePa t = time just to reach the stable work condition z = axial position, m
Greek symbols κ ¼ turbulence kinetic energy, m2 $s – 2
ε ¼ turbulent dissipation rate, m2 $s – 3 eff ¼ turbulentðor eddyÞ viscosity, Pa$s
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k ,z ¼ turbulent parameters αs ¼ partition f actor f or the layer α ¼ permeability, m – 2 η1 ¼ correction coef f icient of the settling velocity ¼ f luid density, kg$m – 3 ω ¼ angular velocity, r$min
–1
g ¼ specif ic weight of the liquid, kg=ðm2 $s2 Þ
Subscripts i, j, k = Cartesian coordinate components l ¼ liquid phase s ¼ solid phase
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