Reynolds number drag force

Flow Past Bodies. A fluid moving past a surface of a soHd exerts a drag force on the soHd. This force is usually manifested as a drop in pressure in the fluid. Locally, at the surface, the pressure loss stems from the stresses exerted by the fluid on the surface and the equal and opposite stresses exerted by the surface on the fluid. Both shear stresses and normal stresses can contribute their relative importance depends on the shape of the body and the relationship of fluid inertia to the viscous stresses, commonly expressed as a dimensionless number called the Reynolds number (R ), EHp/]1. The character of the flow affects the drag as well as the heat and mass transfer to the surface. Flows around bodies and their associated pressure changes are important. 

Viscous Drag. The velocity, v, with which a particle can move through a Hquid in response to an external force is limited by the viscosity, Tj, of the Hquid. At low velocity or creeping flow (77 < 1), the viscous drag force is /drag — SirTf- Dv. The Reynolds number (R ) is deterrnined from... 

Based on such analyses, the Reynolds and Weber numbers are considered the most important dimensionless groups describing the spray characteristics. The Reynolds number. Re, represents the ratio of inertial forces to viscous drag forces. 

Forveiy thin hquids, Eqs. (14-206) and (14-207) are expected to be vahd up to a gas-flow Reynolds number of 200 (Valentin, op. cit., p. 8). For liquid viscosities up to 100 cP, Datta, Napier, and Newitt [Trans. In.st. Chem. Eng., 28, 14 (1950)] and Siems and Kauffman [Chem. Eng. Sci, 5, 127 (1956)] have shown that liquid viscosity has veiy little effec t on the bubble volume, but Davidson and Schuler [Trans. Instn. Chem. Eng., 38, 144 (I960)] and Krishnamiirthi et al. [Ind. Eng. Chem. Fundam., 7, 549 (1968)] have shown that bubble size increases considerably over that predic ted by Eq. (14-206) for hquid viscosities above 1000 cP. In fac t, Davidson et al. (op. cit.) found that their data agreed veiy well with a theoretical equation obtained by equating the buoyant force to drag based on Stokes law and the velocity of the bubble equator at break-off ... 

Lyachshenko number, dimensionless left hand side, dimensionless particle mass, kg pressure, N/m or force, N mass feed rate, kg/s or volumetric flowrate in mVhr drag or resistance force, N physical properties correction factor for slurries Reynolds number, dimensionless right hand side hydraulic radius, m... 

The use of a Reynolds number based on relative velocity rather than superficial velocity in setting these limits was suggested by Horio (1990). In setting viscous or inertial limits, it is the interphase drag which is characterized as being dominated by viscous or inertial forces. The particle inertia is important even if the interphase drag is viscous dominated. This is because of the typically large solid-to-gas density ratio. 

Fig. 10. Normalized drag force at arbitrary Reynolds numbers and gas fractions. The symbols represent the simulation data, the solid line the Ergun correlation Eq. (18), the dashed line the Wen-Yu correlation Eq. (46) for e = 0.8, and the grey line the correlation by Hill et al. (2001a,b) Eq. (47) and the long-dashed line Eq. (19), both for e = 0.5.

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). 

The Reynolds number for mixing ReM represents the ratio of the applied to the opposing viscous drag forces. 

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. 

As a result of the changing flow patterns described above, the drag coefficient Cd is a function of the Reynolds number. For the streamline flow range of Reynolds numbers, Rep<0.2, the drag force F2 is given by. 

Equation 3.1, which is known as Stokes law is applicable only at very low values of the particle Reynolds number and deviations become progressively greater as Re increases. Skin friction constitutes two-thirds of the total drag on the particle as given by equation 3.1. Thus, the total force F is made up of two components ... 

Rowe and Henwood(26) made similar studies by supporting a spherical particle 12.7 mm diameter, in water, at the end of a 100 mm length of fine nichrome wire. The force exerted by the water when flowing in a 150 mm square duct was calculated from the measured deflection of the wire. The experiments were carried out at low Reynolds numbers with respect to the duct (< 1200), corresponding to between 32 and 96 relative to the particle. The experimental values of the drag force were about 10 per cent higher than those calculated from the Schiller and Naumann equation. The work was then extended to cover the measurement of the force on a particle surrounded by an assemblage of particles, as described in Chapter 5. 

The friction factor, which is plotted against the modified Reynolds number, is Pi/pu, where R is the component of the drag force per unit area of particle surface in the direction of motion. R can be related to the properties of the bed and pressure gradient as follows. Considering the forces acting on the fluid in a bed of unit cross-sectional area and thickness /, the volume of particles in the bed is /(I — e) and therefore the total surface is 5/(1 — e). Thus the resistance force is R SH — e). This force on the fluid must be equal to that produced by a pressure difference of AP across the bed. Then, since the free cross-section of fluid is equal to e ... 

Thus, for any voidage the drag force on a sedimenting particle can be calculated, and the corresponding velocity required to produce this force on a particle at the same voidage in the model is obtained from the experimental results. All the experiments were carried out at particle Reynolds number greater than 500, and under these conditions the observed sedimentation velocity is given by equations 5.76 and 5.84 as ... 

For a spherical particle with radius a moving at low Reynolds number, the drag force is Stokesian,... 

When there is gas flow upward through the balance chamber, an aerodynamic drag force is exerted on the suspended mass. If the particle Reynolds number (Re = laV lv) is sufficiently small so that creeping flow may be assumed, Eq. (14) may be written to include the aerodynamic drag, and it becomes... 

As the Reynolds number increases further, vortex shedding takes place (Figure 1.13) in what is known as the Newton region, in the range 500 < Re <2 X where the drag coefficient has a value of approximately 0.44. Consequently equation 1.28 gives the drag force as... 

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