(163g) Drag Force Formulation in Macroscopic Particle Model and Its Validation

Abstract: Drag formulation is improved in macroscopic particle model (MPM) that enables fully coupled fluid-particle simulations based on finite volume method. The model is applicable to large particles immersed in the computational domain. The grid, which can resolve flow field, becomes coarse compared to the particle size. In the original MPM, the drag was calculated as integral of momentum change rate for all fluid cells within the particle volume. The previous study implied that the evaluated drag largely corresponded to virtual mass force. In the present study, the viscous and pressure drags are explicitly incorporated into the drag formulation of MPM. The new MPM is examined in terms of a falling velocity of sphere in a quiescent Newtonian liquid in a circular tube. The MPM is found to accurately predict the falling velocity decreasing with the diameter ratio of particle to tube.

Keywords: Drag force, Virtual mass, Macroscopic particle model, Terminal velocity, Unsteady, CFD, Direct numerical simulation

1. INTRODUCTION

Particulate flows in which particle size is comparable to channel dimensions have gained considerable importance with growing interest in biomedical and micro-chemical technologies (Hessel et al., 2005). A usual CFD particle tracking method is not suited to such applications because the particle is usually regarded as a point mass. It is necessary to take into account particle physical volume to model both hydrodynamic and wall effects in such narrow spaces. Although direct numerical simulation associated with dynamic meshing around a moving particle is expected to be the most accurate method for this purpose, the computing cost is not practical. In our previous study (Ookawara et al., 2005), a novel MPM proposed by Agrawal et al. (2004) was validated in terms of a falling velocity of sphere in a quiescent Newtonian liquid in a cylindrical pipe. The model was subsequently applied to model particle behavior in microseparator/classifier developed by Ookawara et al. (2004), in which wall effects on particle drag and lift would be important factors (Ookawara et al., 2007). The model prediction was found to well correspond to the experimental visualization (Oozeki et al., 2008).

The falling velocity predicted by the previous MPM was larger compared with the correlation. It seemed that the drag force in the previous formulation largely corresponded to virtual mass force that was required to accelerate fluid surrounding the particle. In the present study, the pressure and viscous drags are additionally incorporated into the formulation based on pressure and shear stress distributions around the particle. The falling velocity predicted by the improved MPM will be compared with the previous predictions as well as with the correlation.

2. NUMERICAL METHODS

In the MPM approach, particle is treated in a Lagrangian frame of reference. The particle is assumed to span several computational cells. At every time step of the unsteady simulation, a solid body velocity that describes the particle motion is fixed for the fluid cells within the particle volume. By fixing the rigid body motion of the particle, momentum is effectively added to the fluid as expressed in Eq. (1). The integral of the momentum change, linear as well as angular, gives the drag force and torque experienced by each particle. These forces are used to compute the new velocities and positions of the particles at the next time step.

                                                                
                                                            (1)

where m_f, V_f,i,, V_p,i  and Dt are fluid mass in a cell, velocity of fluid and particle in i direction and time step, respectively. In the present study, additionally, the i component in Cartesian coordinates of pressure and viscous forces acting on particle surface is respectively formulated as;

                                       
                                  (2)

In the left equation, P and d² is pressure and approximated area of particle surface in a fluid cell that is partially occupied by the particle. The vectors of r and x_i are a radius vector from the fluid cell center to particle center and a unit vector for Cartesian coordinates. In the right equation, t_ji = m (∂u_i/∂x_j) x_i is shear stress in the positive i direction on a plane perpendicular to j direction and this is the force exerted by the fluid in the region of greater j coordinates on the fluid of lesser j coordinates.

The diameter and height of circular tube are defined as 0.02 m and 0.06 m, respectively. The densities of sphere and liquid are kept constant as 1,695 kg/m³ and 1,020 kg/m³, respectively. The particle diameter is varied from 0.002 m to 0.018 m. The viscosity is set so that the terminal velocity can be kept 0.003 m/s regardless of particle diameter.

3. RESULTS AND DISCUSSION

In Stokes regime, a terminal velocity V_∞ of spherical particle falling in an infinite Newtonian fluid is described as;

                                                                          
                                                                      (3)

where d_p, r_p, r_f and m are diameter and density of particle, density and viscosity of fluid, respectively. The terminal velocity V in a cylindrical tube is much reduced due to wall effect compared with in infinite media. The extent is expressed by a wall factor f as V = f V_∞. The f is a function of sphere to tube diameter ratio l in viscous regime, which was given by Haberman and Sayre (1958) expressed as Eq. (4). It is noted that the conditions examined in this study fall into the range below the upper limit of viscous regime that is dependent on l (Chhabra et al., 2003).

                                  
                              (4)

Fig. 1. Comparison of terminal velocity predicted by present and previous MPM with the correlation.

As seen in Fig. 1, the present MPM can predict the terminal velocity in the tube that is in better agreement with the correlation compared to the previous MPM. The comparison suggests that the drag expressed by Eq. (1) largely corresponds to the virtual mass force and hence the drag magnitude is not sufficient especially in the small λ range leading to the larger terminal velocity by the previous MPM. By adding the pressure and viscous drags expressed by Eq. (2), the total drag magnitude becomes appropriate and hence the terminal velocity is properly predicted in the whole λ range by the present MPM.

It should be noted that the velocity reduced due to wall effects can be predicted without any model or correction in drag evaluation. The particle existence and motion causes the flow field around the particle while the particle motion is determined by the fluid forces caused by the flow field. The fully coupled particle-fluid simulation can be realized with a sufficient accuracy over rather coarse mesh compared to particle size. For instance, the ratio of sphere to cell volume falls into the range of 1.3 to 3300 according to λ and mesh density adopted in this study. Therefore, the MPM can be regarded as implementing a quasi-direct numerical simulation of particulate flow with a reasonable computational cost. Although this model is implemented in FLUENT 6 (ANSYS Inc., 2006) using user-defined functions and a customized graphical user interface, the approach itself is not limited to FLUENT and is usable in other CFD software also.

It will be a further study to examine the model applicability to various situations, to which the ordinary particle tracking method cannot be suited. The particle motion in non-Newtonian fluids, in which the apparent viscosity is much dependent on shear rate and hence it could vary even over the particle surface, is a practical and interesting application.

REFERENCES

Agrawal, M., A. Bakker and M.T. Prinkey (2004). Macroscopic particle model - Tracking big particle in CFD -.  In: The Proceeding of AIChE 2004 Annual Meeting, Austin, USA. 268b.

Chhabra, R. P., S. Agarwal and K. Chaudhary (2003). A note on wall effect on the terminal falling velocity of a sphere in quiescent Newtonian media in cylindrical tubes. Powder Tech., 129, 53-58.

Haberman, W. L., R.M. Sayre (1958). David Taylor Model Basin Report No. 1143. Dept. of Navy, Washington, DC.

Hessel V., H. Löwe, A. Müller and G. Kolb (2005). Chemical Micro Process Engineering-Processing and Plants. Wiley-VCH, Weinheim.

Ookawara, S., R. Higashi, K. Ogawa and D. Street (2004). Feasibility study on concentration of slurry and classification of contained particles by microchannel. Chem. Eng. J., 101, 171-178.

Ookawara, S., M. Agrawal, D. Street and K. Ogawa (2005). Modeling the motion of a sphere falling in a quiescent Newtonian liquid in a cylindrical tube by using the macroscopic particle model. In: The Seventh World Congress of Chemical Engineering. Glasgow, Scotland. C39-004.

Ookawara S., M. Agrawal, D. Street and K. Ogawa (2007). Quasi-direct numerical simulation of lift force-induced particle separation in a curved microchannel by use of a macroscopic particle model. Chem. Eng. Sci., 62, 2454-2465.

Oozeki, N., S. Ookawara, K. Ogawa, P. Löb and V. Hessel (2008). Characterization of microseparator/classifier with a simple arc microchannel. AIChE J., published online: DOI: 10.1002/aic.11650.