Sphere drag force

The resistance to liquid flow aroimd particles may be presented by an equation similar to the viscosity equation but with considering the void fraction. Recall that the shear stress is expressed by the ratio of the drag force, R, to the active surface, K27td. The total sphere surface is Ttd and Kj is the coefficient accoimting for that part of the surface responsible for resistance. Considering the influence of void fraction as a function 2( ). we obtain ... 

For dense gas-solid two-phase flows, a four-way coupling is required however, the coupling between particles is managed in a natural way in DPMs. The task is, therefore, only to find a two-way coupling between the gas and the solid phases, which satisfies Newton s third law. Basically, the gas phase exerts two forces on particle a a drag force Vda due the fluid-solid friction at the surface of the spheres, and a force Vpa = -Va Vp due to the pressure gradient Vp in the gas phase. We will next describe these forces in more detail, along with the procedure to calculate void fraction, which is an essential quantity in the equations for the gas-solid interaction. 

Beetstra, R., van der Hoef, M. A., and Kuipers, J. A. M. Drag force from lattice Boltzmann simulations of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres, Manuscript submitted to AIChE J. (2006). 

As a matter of fact, one may think of a multiscale approach coupling a macroscale simulation (preferably, a LES) of the whole vessel to meso or microscale simulations (DNS) of local processes. A rather simple, off-line way of doing this is to incorporate the effect of microscale phenomena into the full-scale simulation of the vessel by means of phenomenological coefficients derived from microscale simulations. Kandhai et al. (2003) demonstrated the power of this approach by deriving the functional dependence of the singleparticle drag force in a swarm of particles on volume fraction by means of DNS of the fluid flow through disordered arrays of spheres in a periodic box this functional dependence now can be used in full-scale simulations of any flow device. 

On increasing the Reynolds number further, a point is reached when the boundary layer becomes turbulent and the point of separation moves further back on the surface of the sphere. This is the case illustrated in the lower half of Figure 9.1 with separation occurring at point C. Although there is still a low pressure wake, it covers a smaller fraction of the sphere s surface and the drag force is lower than it would be if the boundary layer were laminar at the same value of Rep. 

Consider a spherical particle of diameter dp and density pp falling from rest in a stationary fluid of density p and dynamic viscosity p.. The particle will accelerate until it reaches its terminal velocity a,. At any time t, let a be the particle s velocity. Recalling that the drag force acting on a sphere in the Stokes regime is of magnitude iirdppu, application of Newton s second law of motion can be written as... 

Fd Drag force due to moving continuous phase total drag on the sphere, dyn [Eq. (97)] g Acceleration due to gravity, cm/sec2... 

For the case of creeping flow, that is flow at very low velocities relative to the sphere, the drag force F on the particle was obtained in 1851 by Stokes(1) who solved the hydrodynamic equations of motion, the Navier-Stokes equations, to give ... 

When Re exceeds about 2 x 105, the flow in the boundary layer changes from streamline to turbulent and the separation takes place nearer to the rear of the sphere. The drag force is decreased considerably and ... 

Fig. 9.11 Stationary relative velocity, Vp, of pentachloroethane drops in motionless water is dependent on drop diameter, dp. Very small drops behave bice rigid spheres, as shown. Larger drops have an internal circulation and are bnaUy deformed ebipticaUy. When they have reached a certain diameter, the drops in the end oschlate along and across their major axis. Their axial velocity is nearly independent of the diameter. Once the drop size is higher than a maximum value dp raax., the drop will break, owing to the drag forces. (From Ref. 2.)...

Fig. 8. The ratio of the drag force to the weight of an a-pinene droplet with initial diameter 29.8 /tm evaporating in nitrogen at 293 K. The solid line is the prediction based on Stokes law for the drag force on a sphere, assuming a quasi-steady process.

In the classical contact mode (Fig. 6a) AFM measures the hard-sphere repulsion forces between the tip and the sample. As a raster-scan drags the tip over the sample surface, the detector measures the vertical deflection of the cantilever, which indicates the local sample height. A feedback loop adjusts the position of the cantilever above the surface as it is scanned and monitors the changes in the surface height, generating a 3D image—a decisive advantage of AFM over TEM [3]. 

Stokes (1851) first showed that the drag force F on a sphere was given by... 

For a rigid sphere on the axis of a tube through which a fluid moves in laminar flow (Fig. 9.1 with h = 0), Haberman and Sayre (HI) showed that the magnitude of the drag force is... 

The dynamic response of a particle in gas-solid flows may be characterized by the settling or terminal velocity at which the drag force balances the gravitational force. The dynamic diameter is thus defined as the diameter of a sphere having the same density and the same terminal velocity as the particle in a fluid of the same density and viscosity. This definition leads to a mathematical expression of the dynamic diameter of a particle in a Newtonian fluid as... 

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