Friction Factor, Drag Coefficient

Models of the polymer coil are based on the end-to-end distance, which is generally not directly available as a quantitative feature. Coils in dilute solution can be characterized in terms of the radius of gyration, Rg, which is a statistical measure of the distribution of mass about the center of gravity or in terms of the hydrodynamic radius, Rh, that is usually determined through the use of Stokes law and a measurement of a drag coefficient or friction factor, /drag/ for the coil,... 

Fanning friction factor /i for inner wall and / 2 for outer wall of annulus /l for ideal tube bank sldn friction drag coefficient Dimensionless Dimensionless... 

Here, p is the density of the fluid, V is the relative velocity between the fluid and the solid body, and A is the cross sectional area of the body normal to the velocity vector V, e.g., nd1/4 for a sphere. Note that the definition of the drag coefficient from Eq. (11-1) is analogous to that of the friction factor for flow in a conduit, i.e.,... 

The solids contribution to the pressure drop, APls, is a consequence of both the particle-wall and particle-particle interactions. The latter is reflected in the dependence of the friction factor fs on the particle diameter, drag coefficient, density, and relative (slip) velocity by (Hinkel, 1953) ... 

The dimensionless drag coefficient Cd, analogous to the friction factor, is defined by... 

A particle drag coefficient Cd can now be defined as the drag force divided by the product of the dynamic pressure acting on the particle (i.e. the velocity head expressed as an absolute pressure) and the cross-sectional area of the particle. This definition is analogous to that of a friction factor in conventional fluid flow. Hence... 

Wall thickness Channel width Acoustic velocity Friction coefficient Conductance Capillary number Discharge coefficient Drag coefficient Diameter Diameter Dean number Deformation rate tensor components Elastic modulus Energy dissipation rate Eotvos number Fanning friction factor Vortex shedding frequency Force... 

Archimedes number Bingham number Bingham Reynolds number Blake number Bond number Capillary number Cauchy number Cavitation number Dean number Deborah number Drag coefficient Elasticity number Euler number Fanning friction factor Froude number Densometric Froude number Hedstrom number Hodgson number Mach number Newton number Ohnesorge number Peclet number Pipeline parameter... 

Whitaker [234] (chap 8) explains the convention normally used to distinguish between these two types of parameters. The friction factors for dispersed bodies immersed in a flowing fluid is traditionally referred to as dimensionless drag coefficients, whereas the drag force for flow inside closed conducts is generally expressed in terms of a dimensionless friction factor. 

So fax, we have studied the friction factor and the drag coefficient associated with a number of common cases. We may now utilize the analogy between heat and momentum transfer, obtaining the heat transfer indirectly from the friction associated with these cases. Combining Eq. (6.15) with Eq. (5.63) yields a relation for the heat transfer in fully developed turbulent pipe flow,... 

The power number Np is analogous to a friction factor or a drag coefficient. It is proportional to the ratio of the drag force acting on a unit area of the impeller and the inertial stress, that is, the flow of momentum associated with the bulk motion of the fluid. 

The friction factor fc (or drag coefficient (114, 115)) is related to the cutting Reynolds number by fc = 24/Rec for Rec <0.1. Equations 72 and 75 combine to give a more general expression for the slip velocity... 

Figure 30. Dependence of cuttings drag coefficient (friction factor fc) on cuttings Reynolds number. (Data from reference 112, plot symbol X reference 116, plot symbol + and reference 114.)...

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. 

The forces of fluids flowing over bodies are ordinarily correlated by the drag equation, in which the drag coefficient plays the same role as does the friction factor in pipe flow. 

Obtain expressions for the local and mean values of the wall shear stress and friction factor (or drag coefficient) for the laminar boundary layer flow of an incompressible power-law fluid over a flat plate Compare these results with the predictions presented in Table 7.1 for different values of the power-law index. 

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