Reynolds number impact

Optimal hydraulics is the proper balance of hydraulic parameters (flowrate and equivalent nozzle size) that satisfy chosen criteria of optimization. Hydraulic quantities used to characterize jet bit performance include hydraulic horsepower, jet impact force, jet velocity, Reynolds number at the nozzle, generalized drilling rate or cost per foot drilled. While designing the hydraulic program the limitations due to cuttings transport in the annulus and pump performance characteristics must be included. 

The results also show that the normalized slip length and the average friction factor-Reynolds number product exhibit Reynolds dependence. Furthermore, the predictions reveal that the impact of the vapor cavity depth on the overall frictional resistance is minimal provided the depth of the vapor cavity is greater than 25% of its width. 

The collection efficiency of particles at a stage of an impactor is based on curvilinear motion and assumes Reynolds numbers for flow greater than 500 but less than 3000. Figure 8A illustrates the principle of inertial sampling in which particles with high momentum travel in the initial direction of flow of an airstream impacting on an obstructing surface and those with low momentum adjust to the new direction of flow and pass around the obstruction. The efficiency of this phenomenon can be described as follows ... 

Fig. 14 shows the comparison of the photographs from Chandra and Avedisian (1991) with simulated images of this study for a subcooled 1.5 mm n-heptane droplet impact onto a stainless-steel surface of 200 °C. The impact velocity is 93 cm/s, which gives a Weber number of 43 and a Reynolds number of 2300. The initial temperature of the droplet is room temperature (20 °C). In Fig. 14, it can be seen that the evolution of droplet shapes are well simulated by the computation. In the first 2.5 ms of the impact (frames 1-2), the droplet spreads out right after the impact, and a disk-like shape liquid film is formed on the surface. After the droplet reaches the maximum diameter at about 2.1ms, the liquid film starts to retreat back to its center (frame 2 and 3) due to the surface-tension force induced from the periphery of the droplet. Beyond 6.0 ms, the droplet continues to recoil and forms an upward flow in the center of the... 

Impaction When an air stream containing particles flows around a cylindrical collector, the particle will follow the streamlines until they diverge around the collector. The particles because of their mass will have sufficient momentum to continue to move toward the cylinder and break through the streamlines, as shown in Figure 8.3. The collection efficiency by this inertial impaction mechanism is the function of the Stokes and the Reynolds number as ... 

Typical cascade impactors consist of a series of nozzle plates, each followed by an impaction plate each set of nozzle plate plus impaction plate is termed a stage. The sizing characteristics of an inertial impactor stage are determined by the efficiency with which the stage collects particles of various sizes. Collection efficiency is a function of three dimensionless parameters the inertial parameter (Stokes number, Stk), the ratio of the jet-to-plate spacing to the jet width, and the jet Reynolds number. The most important of these is the inertial parameter, which is defined by Equation 2) as the ratio of the stopping distance to some characteristic dimension of the impaction stage (10), typically the radius of the nozzle or jet (Dj). 

The characteristic behavior of impactors depends on factors such as nozzle-to-plate distance, nozzle shape, flow direction, and Reynolds numbers for both the jet and the particle. Other factors of importance include the probability that the particles will stick to the impaction surface and particle loss to the walls of the impactor. It is not surprising that with such a variety of possible variables it is quite difficult, if not impossible, to accurately predict impactor characteristics on purely theoretical grounds. 

Although they used droplets with diameters of 2 mm and more, the work of Park et alP is interesting on account of the fact that they used four different substrates and four different hquids. They observed the impact of droplets of distilled water, n-Octane, n-Tetradecane or n-Hexadecane onto glass slides, sihcon wafers, HMDS (Hexamethyl dishazane) coated sihcon wafers or Teflon, for Reynolds numbers from 180 to 5513 and Weber numbers from 0.2 to 176. A model was constructed to predict the maximum spreading ratio, which is the ratio of the maximum spreading diameter to the initial droplet s diameter, for low impact velocities. 

This is still a low power level of 6.2/1.74 = 3.55 HP/1,000 gal. With this agitator, a reasonable upper hmit for agitator speed would be 100 rpm, for which the impeller power would be 22 HP with a specific power input of 13HP/l,000gal and Npe = 9. This change would move up into the Reynolds number near the lower limit recommended by Hemrajani and Tatterson (in Paul (2004), 345). This example illustrates the great impact of fluid viscosity on (1) the power requirement of a 6BD and (2) the choice of an impeller style between a turbine and a helical ribbon impeller. 

The extension of the theory to non-uniform flows and to high Reynolds number flows is still controversal. Nevertheless, Clift et al [22] reviewed numerous investigations and claimed that the history term has a negligible effect on the mean motion of a particle in a turbulent fluid. The impacts on the fluctuations in particle motion might be more severe at high frequencies. 

For Stokesian particles, rvi o impaction regimes are similar when the Stokes, interception, and Reynolds numbers are the same. The impaction efficiency. as in the case of diffusion, is defined as the ratio of the volume of gas cleared of particles by the collecting element to the total volume swept out by the collector. (Refer to Fig. 4.5 for the case of the cylinder.) If all panicles coming within one radius of the collector adhere, then we obtain... 

Flow around single cylinders is the elementary model for (he fibrous filter and is the geometry of interest for deposition on pipes, wires, and other such objects in an air flow (Chapter 3). The flow patterns at low and high Reynolds numbers differ significantly, and thi.s affects impaction efficiencies. For Re > 100. the velocity distribution outside the velocity boundary layer can be approximated by inviscid flow theory. This approximates the velocity distribution best over the front end of the cylinder which controls the impaction efficiency. The components of the velocity in the direction of the mainstream flow, x, and normal to the main flow, y, are... 

Experimental studies have been made of the impaction of nearly monodi-sperse sulfuric add particles in the. size range 0.3 < dp <. 4 pm on a wire 77 m in diameter over a Reynolds number range 62 to 500 (Ranz and Wong, 1952 Wong and Johnstone, 1953). In Fig. 4.7, these data are compared with numerical computations based on inviscid flow theory for point particles (Brun el al., 1955). Agreement between measured and calculated values is fair. The experimental results fall somewhat below the calculated values for 0.8 < Stk < 3, At values of Stk above about 3, the data fall above theory. Dam were not taken at sufficiently small values of Stk to test the theoretical value (Stkiznt = 1/8) al which the efficiency is expected to vanish. The results of these measurements are not directly applicable to high-efficiency fibrous filters, which usually operate at much lower Reynolds numbers based on the fiber diameter. 

Figure 4.8 Comparison of experiment and theory for impaction of monodisperse droplets on single solid spheres (after Hahner el al., 1994). Points correspond to different collector diameters. The numerical simulations were for a Reynolds number of 500 based on sphere diameter. The two cases considered, potential flow and boundary layer (low, gave similar results. Both agree well with experiment for Slk > 0.2. However, the experimental results indicate that measurable deposition occurs at values of Stk < Stkcrii = 1/12.

As in Section 11.1, we consider the reflection of a particle from a flat boundary at normal sedimentation. This assumption can be used for a bubble when we are interested in impacts close to the pole, at < ,. Thus, we can simplify the expression for the normal flow of liquid setting cosine to unity and assuming that over whole section of the surface at < j the length of recoil is characterized by a constant value. In dimensionless form, the equation for calculating the inertia path change due to the opposite motion of the liquid which has a velocity distribution expressed by a linear relationship (differs from a linear second-order differential equation with constant coefficients only due to a variation of Reynolds number for a retarded particle). It reads... 

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