On the other hand, in the case of a converging channel symmetrical and peak-valley-peak type pattern emerges downstream of bifurcation whose nature depends on the inlet velocity and concentration. The velocity profile in the inlet branches was fully developed, but blunted near the converging section and shifted towards the lower wall. 11 Figure 1.4 The macroscopic flow behavior of a suspension will depend on the dynamics at the microstructural level (Uit Deshpande et al. 2010).
Introduction
Suspensions
In industry, suspensions are frequently handled during the production of processed foods (juices, sauces, etc.), pharmaceuticals, cosmetics, paints, etc. In the petroleum industry, suspensions are encountered in various operations such as drilling, well simulation, fracturing and enhanced oil recovery (Schramm, 1996).
Forces on particles
- Non-hydrodynamic Forces
- Hydrodynamic forces
Neutral buoyancy means that the densities of the particles and the liquid are equal and thus settling of the particles due to gravity does not occur. In this study, to understand the dynamics of suspensions we have made some simplifications in the flow equations.
Viscosity
In the above equation, η is the effective viscosity of the suspension, ηo is the viscosity of the suspending fluid, and φ is the particle volume fraction. For more concentrated suspensions it is necessary to consider viscosity corrections that are of higher order in the volume fraction.
Shear induced migration
- Shear induced migration through general geometries
- Shear induced migration through bifurcation geometries
NMR images taken by Hampton et al. 1997) of the particle volume fraction profile (φ ) for pressure-driven suspension flow in a circular tube. The shear-induced migration phenomenon through bifurcation channels can be important in many of the above applications.
Objectives
Organization of the thesis
Shear Induced Migration Models
Introduction
In the cone and plate geometries, the diffusive flux model predicts no migration of the particles, while inward migration of the particles in the parallel plate geometries. They studied migration in one-dimensional and two-dimensional flows with arbitrary geometry and boundary conditions. 1995) used the diffusive flux model to find the particle migration in eccentric flows. Both the diffusive flux and the suspension balance model predict the wall concentration. 2012) in the mixing model suggested the adjustment of the maximum packing close to the wall, which corrects the depletion of particles near the wall.
We have chosen diffusive flux model since several previous studies have confirmed that for rectilinear channel flow, predictions of diffusive flux model and suspension balance model are similar. The mass, momentum and particle conservation equations for the diffusive flux model were solved using the finite volume method (FVM).
Diffusion models
- Original Diffusive Flux Model [Phillips et al. 1992]
- The stress tensor
- Continuity and Momentum equations
- Particle conservation equation
- Flow-aligned Tensor Model
The second term on the right-hand side of equation (2.8) states that the gradient of the particle volume fraction will cause a spatial variation in the interaction frequency. Due to the viscosity gradient, the resistance to movement on one side of the particle is greater than on the other side. As a result, the particles move in the direction of decreasing viscosity, as shown in Figure 2.1b.
At a small concentration gradient, the change in viscosity is linear in the concentration gradient. 1998) suggested that the parameter Kc is a function of the particle volume fraction in the modified Phillips model.
Suspension balance model (SBM)
- Governing Equations
- Suspension stress
The constitutive law for the particle stress is that proposed by Morris and Boulay (1999) for shear flows, is given by equation (2.28) below and contains both shear and normal parts. The anisotropy in the particle stress tensor arises from non-zero normal stress differences in Stokesian suspensions. Several experiments have measured the normal voltage differences and particle pressure for concentrated suspensions (Gadala-Maria, 1979, Zarraga et al. 2000, Singh and Nott, 2003, Dbouk et al, 2013).
The two normal stress differences and the suspension pressure ∏ are given in terms of this modeling by. The combinations of the above definitions for both the particle phase and fluid phase stresses ultimately yield bulk suspension stress in the suspension balance model of the shape.
Numerical implementation of DFM in OpenFOAM
- Model Governing Equations
To overcome this problem, a small non-local shear rate ( .. γnl) is added to the local shear rate, proposed by Miller and Morris (2006), which is a function of particle size and channel width. The applicable partial differential equations for this transport problem are non-linear (in velocity and concentration). The rate equation of momentum is non-linear, since the effective viscosity of the mixture is a function of concentration and shear velocity (velocity gradient).
This is explicitly shown in the terms containing φ, φ2 and the gradient of the concentration∇∇∇∇φ, as well as the terms containing the effective viscosity (η), which is also a function of the local concentration of solids. Below is a brief description of the code implementation procedure in the simpleFoam solver.
Boundary conditions
- Validation
The concentration profiles from our simulations agree well with the analytical solution of Phillips et al. We also compared the simulation results with the available experimental data of Leble et al. 2011) in a Y-shaped bifurcation channel. We compare the velocity profiles obtained by our simulations with the experimental values of Leble et al.
The profile from our simulation is in good agreement with the experimental data of Leble et al. A comparison plot of the velocity profiles obtained from our simulation with. the experimental data of Leble et al. 2011) and concentration profiles for different locations in the inlet branch (a) at location 1 and (b) at location 2.
Shear Induced Migration of Concentrated Suspension through Y-Shaped 2D
- Introduction
- Problem Description
- Results and Discussion
- Velocity field
- Concentration field
- Effect of particle size and flow rate
- Conclusion
The velocity profiles at the same locations for a suspension of 40% concentration for a bifurcation angle θ =60 are shown in Figure 3.4b. Therefore, in the subsequent analysis, we showed the velocity and concentration profiles of only the left branch. The velocity profiles in the inlet section at location 5 are damped for suspension and the degree of damping increases with bulk particle concentration.
This suggests that the splitting does not affect the concentration profile in the inlet section. The effect of bulk particle concentration on the velocity at location 5 in the inlet branch (c) and at location 7 in the left branch (d).
Shear-Induced Particle Migration in 3D Y-Shape Bifurcation Channel . 79
Straight Vs Bifurcation channels
The relative position in the lateral (x) and spanwise (z) directions was normalized by the half-width (B) of the channel. It should be noted that Lyon and Leal (1998) conducted experiments in 3D channels with a rectangular cross-section, where the depth of the channel was much greater compared to its width. Another point to note is that in the experiments of Lyon and Leal (1998) the depth of the channel was much greater compared to its width, while our simulations were for square cross-sections where the migration was in both the lateral and wingspan is expected. Travel directions.
However, unlike the velocity profile, the flow bifurcation has less impact on the concentration profile upstream of the bifurcation. The value of the maximum concentration in the middle of the channel increases with the increase in inlet particle concentration.
Results and Discussion
- Velocity field
- Concentration field
- Wall shear stresses
It was observed that the increase in the bifurcation angle does not show any significant effect on the bluntness of the velocity profile in the inlet section. Effect of the bifurcation angle on the velocity profile at location 6 of the side branch in lateral direction (c) and spanwise direction (d). In the inlet section, the shear-induced migration causes the particles to move towards the center of the channel.
30% and 40% particle concentration) at various locations of the inlet section in the lateral direction (a) and spanwise direction (b). Effect of bifurcation angle on concentration profile at location 6 of the tributary in the lateral direction (c) and spanwise direction (d).
Conclusion
The velocity profiles of suspension flow were observed to be different from those of Newtonian fluid with the same effective viscosity. At bifurcation, the velocity profiles of suspension in lateral and spanwise directions are blunted, and the degree of blunting is more apparent with increasing bifurcation angle and particle concentration. It was observed that the migration of particles in the downstream branches leads to asymmetric velocity and concentration profiles.
The effect of non-uniform concentration distribution including the detailed concentration and velocity profiles in each branch is well captured by our numerical simulations. In the tributary, the difference in wall shear stress level of inner and outer wall was found to be the highest for a bifurcation angle of 90°.
Shear Induced Migration in Symmetric 3D T-Shaped Channels
Introduction
Case 1: Diverging flow
- Results and Discussion
- Velocity field
- Shear rate field
- Concentration field
- Wall shear stresses
The cross-sectional views (x-z plane) of velocity contour at various locations in the inlet section for Newtonian fluid and suspension (40% and 50%) are shown in Figure 5.6a. Comparison of the velocity profiles for Newtonian fluid and suspension (30% . particle concentration) at various locations in the tributary of the symmetrical T-shaped bifurcation channel for (a) lateral and (b) spanwise direction. Comparison of the velocity profiles for Newtonian fluid and suspension (40% . particle concentration) at various locations in the tributary of the symmetrical T-shaped bifurcation channel for (a) lateral and (b) spanwise direction.
Comparison of velocity profiles for a Newtonian fluid and suspension (50% particle concentration) at different locations in a side branch of a symmetric T-shaped bifurcation channel for (a) lateral and (b) spanwise directions. Comparison of concentration profiles for slurry flow (40% particle concentration) at different locations in the side branch for (a) lateral direction and (b) spanwise direction.
Case 2: Converging flow with equal inlet concentrations
- Results and Discussion
- Velocity field
- Shear rate field
- Concentration field
- Shear stress field
Comparison of velocity profiles for fluid and Newtonian suspension (40% particle concentration) at different locations in the inlet section for the T-channel in (a) lateral direction and (b) opening direction. Comparison of velocity profiles for fluid and Newtonian suspension (50% particle concentration) at different locations in the inlet section for the T-shaped channel in (a) lateral direction and (b) spatial direction. Comparison of velocity profiles for Newtonian fluid and suspension at different locations in the outlet section for the T-shaped bifurcation channel in (a) the lateral direction and (b) the spanwise direction.
Comparison of velocity profiles for a Newtonian fluid and a suspension at different locations in the outlet section of a T-shaped channel in (a) lateral and (b) spanwise directions. Shear velocity contour planes for Newtonian flow of liquid and suspension (30% and .40% concentration) at different locations in the outlet portion of the T-shaped channel.
Case 3: Converging flow with unequal inlet concentration
- Results and Discussion
- Velocity field
- Shear rate field
- Concentration Field
- Wall shear stress
Velocity profiles of the suspension at different locations in the inlet-1 section (30% . particle concentration) in (a) lateral and (b) spanwise directions. Velocity profiles of the suspension at different locations in the inlet-2 section (20% . particle concentration) in (a) lateral and (b) spanwise directions. The cross-sectional views (x-z plane) of the velocity contour at various locations in the suspension outlet section are shown in Figure 5.50.
The velocity profiles of suspension at different locations in the exhaust section in the lateral direction are shown in Figure 5.51. The lateral velocity profiles of suspension at different locations in the exhaust section of the T-shaped channel.
Conclusion
Conclusions and Future direction
Conclusions _____________________________________________________________________ 163
