Drag coefficient definition

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.,... 

This result can also be applied directly to coarse particle swarms. For fine particle systems, the suspending fluid properties are assumed to be modified by the fines in suspension, which necessitates modifying the fluid properties in the definitions of the Reynolds and Archimedes numbers accordingly. Furthermore, because the particle drag is a direct function of the local relative velocity between the fluid and the solid (the interstitial relative velocity, Fr), it is this velocity that must be used in the drag equations (e.g., the modified Dallavalle equation). Since Vr = Vs/(1 — Reynolds number and drag coefficient for the suspension (e.g., the particle swarm ) are (after Barnea and Mizrahi, 1973) ... 

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... 

Due to various relationships of the drag coefficient, Parameter k, in Eq. (2.32) has different definitions for various flow regimes, as follows. 

The drag force across the float is defined as equal to the product of the drag coefficient, Co, the pressure drop across the float, pi - p2, and a characteristic area for the float, Ap. After substituting this definition and expressions for the gravitational and buoyant forces into Equation 8.19, the force balance becomes... 

The literature in the field of fluid mechanics [8] provides us with an expression as the definition of the fiictional drag coefficient For spheres, disks, cylinders, and similar bodies, the diameter is... 

The absolute value enforces that the drag force acts opposite to the direction of fluid motion. Here Cd is the drag coefficient for the array, and it may differ in magnitude from the drag coefficients of isolated elements of the same form (see, e.g., discussions in [88,462, 505]). Note that this definition uses a factor 1/2, which is conventional for hydraulic studies. Many air studies exclude this factor as in equation (4.4) in Chapter 4. Cd is a function of stem Reynolds number, as well as canopy density, a. The dependencies of Cd are discussed below. 

The drag coefficient is not only related to the shape and orientation of an object, as shown in Figure 6.28, it is also affected by the Reynolds number (Re) of the flow gas. This can be explained by inserting this statement into the definition of Reynolds number in Equation (2.58). The thickness of the boundary layer is decreased as the Reynold number increases, therefore reducing the contribution of the velocity boundary layer to the drag force. 

By applying the F definition of Benyahia et al [12], the closure can be expressed as a function of the friction coefficient j3 instead of the dimensionless drag coefficient F [92] ... 

Here Cf is the drag coefficient of the body, d is the center section diameter, and b(X) is the local half-width of the wake. The coordinate X is measured from the rear point of the body and Y is the transverse coordinate. The definition of the local half-width b(X) of the wake is a matter of convention it can be estimated as... 

The electromechanical analogy provides for simple equivalents of a resistor, an inductance, and a capacitance, which are the dashpot (quantified by the drag coefficient, p), the point mass (quantified by the mass, mp), and the spring (quantified by the spring constant, /cp). The ratio of force and speed is the mechanical impedance, Z - For a dashpot, the impedance by definition is... 

With this definition all the steady-state drag data on single, smooth spheres moving in infinite , quiescent, newtonian fluids at moderate velocities can be represented by aj single curve on Fig. 6.22. This figure shows also drag coefficients for disks and cylinders, to be discussed later. It is limited to steady velocities of less than about one half the local speed of sound velocities higher than this are discussed elsewhere [18]. 

From the definition of the drag coefficient q of a single bubble, the following expression for the drag force can be derived ... 

Without yet introducing any assumptions, this can be written as follows, which serves as a definition of the drag coefficient Ca)... 

It is necessary to emphasize certain special features and certain assumptions that were made in deriving Eq. (X.33). The first assumption is in the definition of the relationship between the drag coefficient and Reynolds number. This assumption was made when we chose Eq. (X.29), which is still an averaged sort of relationship, for Reynolds numbers over a comparatively narrow range. If the range of Reynolds numbers is less than 1-100, more precise expressions can be selected for the determination of the coefficient Cx- The use of such expressions, however, restricts the calculations to special cases so that the approach to drag calculations loses all generality. 

A Definition of Drag Coefficient for Flow Past Immersed... 

We will here adopt the buoyancy convention incorporated in Eqs. (2) and (6) rather than (3) and (7). However, since the latter convention has made significant inroads into the literature, the reader is referred to Table 1 of Khan and Richardson (1990) for useful drag coefficient relationships based on the above alternate conventions, as well as on alternate definitions of the characteristic liquid velocity, and to conversions by Jean and Fan (1992) of several equations incorporating buoyancy as defined by Eq. (3) [derived by Foscolo et al. (1983, 1989), Foscolo and Gibilaro (1984, 1987) and Gibilaro et al. (1985a, 1986)] to the corresponding equations based on buoyancy as defined by Eq. (2). 

As shown in Eqn. (6), the drag coefficient of a cylindrical fiber imder cross flow condition is a function of the Reynolds Number, which is generally expressed as Re = pUp,hd/p (i.e. the ratio of inertial force to viscous force). This definition holds true for Newtonian fluids, where shear stress < shear rate. However, the fluids that are often utilized in fiber sweep applications are non-Newtonian. Hence, the Reynolds Number must be redefined using the apparent viscosity function as Re = pUp>d/papp. The viscosity for Newtonian fluids is independent of the shear rate. However, for non-Newtonian fluids, the apparent viscosity varies with shear rate. Applying the Yield Power Law (YPL) rheology model, the apparent viscosity is expressed as ... 

The design of the flocculator of Figure 6.11 may be made by determining the power coefficients for laminar, transitional, and turbulent regime of flow field. We will, however, discuss its design in terms of the fundamental definition of power. Consider Fd as the drag by the water on the blade Fd is also the push of the blade upon the water. This push causes the water to move at a velocity Vp equal to the velocity of the blade. 

Note that, during the drag evaluation, the size moments mo, are constants,and thus A and B are constant matrices. Moreover, these coefficient matrices are both symmetric and positive definite, implying that the eigenvalues for the first-order ODE system in... 

The foregoing pertains entirely to spheres. We can use Eq. 6.53 for other shapes, if we agree on what area A represents. Generally, in drag measurements it refers to the frontal area perpendicular to the flow that is the definition on which the coefficients in Fig. 6,22 are based. Moreover, we must decide on which dimensions to base the Reynolds number in our correlation of Cj versus in Fig. 6.22 the Reynolds number for cylinders takes the cylinder diameter as /), and that for disks takes the disk diameter. 

To illustrate this point, one must look at the derivative of drag with respect to chord length. We wish to treat velocity, area, and lift coefficient as constants, so the span must be a function of aspect ratio. Using the definition for aspect ratio and defining the average chord length c =A/b, we obtain... 

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