Drag coefficient changes

Close inspection of Fig. 4.3 shows that the drag coefficient changes very little for Reynolds numbers greater than 2 x 10" and smaller than 5 x 10 and is almost equal to 4/9. There have been various correlations offered for the drag coefficient for flows past solid spheres, each derived for a particular range of the Reynolds number. Some of these relations are... 

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. 

In the droplet drag sub-modelT572 598 the effects of droplet distortion and oscillation due to droplet-gas relative motion on droplet drag coefficient are taken into account. Dynamical changes of the drag coefficient with flow conditions can be calculated with this submodel. Applications of the sub-models to diesel sprays showed that... 

Equations (8-10) to (8-12) have been confirmed many times [e.g. (D4, W7)]. For M > 10, bubbles and drops change directly from spherical to spherical-cap, as noted in Chapter 2. The drag coefficient is then closely approximated by... 

Data predictions for droplets moving freely in turbulent gas streams are confounded by the problem of ballistics of droplets. Until the droplet is essentially accelerated or decelerated to the gas stream velocity, Reynolds number, thus Nusselt number, and thus X are changing constantly, and precise calculations require very small steps. The drag coefficient is of considerable importance. El Wakil, Uyehara, and Myers (117) em-... 

The relationship between drag coefficient and Reynolds number is shown in Figure 6. Changes in drag coefficient correspond to changes in... 

For relatively flat leaves that can align with the wind direction, cd is usually 0.02 to 0.2, but it can be 0.5 to 0.9 for bluff bodies such as branches and tree trunks that block and hence substantially change the airflow. For flexible objects the drag coefficient defined by Equation 7.9 can decrease as the wind speed increases. In particular, the areas of leaves and small branches perpendicular to the wind direction can decrease threefold as the wind speed increases (Fig. 7-5), leading to a more streamlined shape, as occurs for many trees. Indeed, changes in shape to minimize drag have evolved many times, as is especially apparent for the fusiform shape of various marine animals. 

Measurements have been made on a wide variety of molecules adsorbed on Au, Ag, or Pb surfaces [3,4,131,132]. The phase of the adsorbed layer changes from fluid to crystal as the density is increased. As expected, motion of fluid layers produces viscous dissipation that is, the friction vanishes linearly with the sliding velocity. The only surprise is that the ratio between friction and velocity, called the drag coefficient, is orders of magnitude smaller than would be implied by the conventional no-slip boundary condition. When the layer enters an incommensurate phase, the friction retains the viscous form. Not only does the incommensurate crystal shde without measurable static friction, the drag coefficient is as much as an order of magnitude smaller than for the liquid phase ... 

If the change in drag coefficient is small, which is usually the case for large rotameters with low- or moderate-viscosity fluids, the maximum velocity stays the same with increasing flow rate, and the total flow rate is proportional to the annular area between the float and the wall ... 

However, the curve of the sphere drag coefficient has some marked differences from the friction factor plot. It does not continue smoothly to higher and higher Reynolds numbers, as does the / curve instead, it takes a sharp drop at an of about 300,000. Also it does not show the upward jump that characterizes the laminar-turbulent transition in pipe flow. Both differences are due to the different shapes of the two systems. In a pipe all the fluid is in a confined area, and the change from laminar to turbulent flow affects all the fluid (except for a very thin film at the wall). Around a sphere the fluid extends in all directions to infinity (actually the fluid is not infinite, but if the distance to the nearest obstruction is 100 sphere diameters, we may consider it so), and no matter how fast the sphere is moving relative to the fluid, the entire fluid cannot be set in turbulent flow by the sphere. Thus, there cannot be the sudden laminar-turbulent transition for the entire flow, which causes the jump in Fig. 6.10. The flow very near the sphere, however, can make the sudden switch, and the switch is the cause of the sudden drop in Q at =300,(300. This sudden drop in drag coefficient is discussed in Sec. 11.6. Leaving until Chaps. 10 and 11 the reasons why the curves in Fig. 6.22 have the shapes they do, for now we simply accept the curves as correct representations of experimental facts and show how to use them to solve various problems. 

When the liquid phase exhibits non-Newtonian behavior, the mass transfer coefficient will change due to alterations in the fluid velocity profile around the submerged particles. The trends for both mass transfer and drag coefficient are analogous. As before for Newtonian fluids, two types of interfacial behavior need to be considered. 

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