Fanning friction factor, turbulent flow

The transition from laminar to turbulent flow occurs at Reynolds numbers varying from ca 2000 for n > 1 to ca 5000 for n = 0.2. In the laminar region the Fanning friction factor (Fig. 2) is identical to that for Newtonian fluids. In the turbulent region the friction factor drops significantly with decreasing values of producing a family of curves. 

For laminar flow (Re < 2000), generally found only in circuits handling heavy oils or other viscous fluids, / = 16/Re. For turbulent flow, the friction factor is dependent on the relative roughness of the pipe and on the Reynolds number. An approximation of the Fanning friction factor for turbulent flow in smooth pipes, reasonably good up to Re = 150,000, is given by / = (0.079)/(4i e ). 

Equation (6-41) adequately represents the Fanning friction factor over the entire range of Reynolds numbers within the accuracy of the data used to construct the Moody diagram, including a reasonable estimate for the intermediate or transition region between laminar and turbulent flow. Note that it is explicit in /. 

The model for turbulent drag reduction developed by Darby and Chang (1984) and later modified by Darby and Pivsa-Art (1991) shows that for smooth tubes the friction factor versus Reynolds number relationship for Newtonian fluids (e.g., the Colebrook or Churchill equation) may also be used for drag-reducing flows, provided (1) the Reynolds number is defined with respect to the properties (e.g., viscosity) of the Newtonian solvent and (3) the Fanning friction factor is modified as follows ... 

Determine the Fanning friction factor / from Equation 4-34. This assumes fully developed turbulent flow at high Reynolds numbers. This assumption can be checked later but is normally valid. 

The Fanning friction factor may be determined either from a chart for both rough and smooth tubes or from a variety of correlations (Knudsen and Katz, 1958, pp. 173,176). The following correlation applies for turbulent flow in smooth tubes and for Reynolds numbers between 3,000 and 3,000,000 ... 

For turbulent flow, White (1932) gave an equation for the Reynolds number range 15000 to 100000. In terms of the Fanning friction factor, White s equation can be written as... 

Turbulent flow of Newtonian fluids is described in terms of the Fanning friction factor, which is correlated against the Reynolds number with the relative roughness of the pipe wall as a parameter. The same approach is adopted for non-Newtonian flow but the generalized Reynolds number is used. 

Experimental results for the Fanning friction factor for turbulent flow of shear thinning fluids in smooth pipes have been correlated by Dodge and Metzner (1959) as a generalized form of the von Karman equation ... 

Taylor (T4, T6), in two other articles, used the dispersed plug-flow model for turbulent flow, and Aris s treatment also included this case. Taylor and Aris both conclude that an effective axial-dispersion coefficient Dzf can again be used and that this coefficient is now a function of the well known Fanning friction factor. Tichacek et al. (T8) also considered turbulent flow, and found that Dl was quite sensitive to variations in the velocity profile. Aris further used the method for dispersion in a two-phase system with transfer between phases (All), for dispersion in flow through a tube with stagnant pockets (AlO), and for flow with a pulsating velocity (A12). Hawthorn (H7) considered the temperature effect of viscosity on dispersion coefficients he found that they can be altered by a factor of two in laminar flow, but that there is little effect for fully developed turbulent flow. Elder (E4) has considered open-channel flow and diffusion of discrete particles. Bischoff and Levenspiel (B14) extended Aris s theory to include a linear rate process, and used the results to construct comprehensive correlations of dispersion coefficients. 

The Darcy friction factor is four times the Fanning friction factor, fp, i.e., fp = 4fp. For fully developed turbulent flow regime in smooth and rough pipes, the Colebrook [5] equation or the Chen [6] equation can be used. 

The Sieder-Tate term has been added, and fp is the Fanning friction factor for smooth tubes in turbulent flow.)... 

Transition flow and fully developed turbulent flow Fanning friction factors for a circular duct are given by Bhatti and Shah [46] as... 

In turbulent flow the frictional pressure drop down the drill pipe must be calculated from equations that have been determined empirically. The commonest method in drilling fluid hydraulics (91-93) is to use a friction factor/, the so-called Fanning friction factor, defined by the ratio of the wall shear stress rw to the kinetic energy per unit volume of the flowing fluid 0.5 pvm2 (94)... 

Figure 24. Dependence of Fanning friction factor on Reynolds number for various values of exponent n for laminar and turbulent pipe flow. (Reproduced with permission from reference 90. Copyright 1959 American Institute of Chemical Engineers.)...

Several studies have been reported to determine friction losses in turbulent flow of slurries. Hannah et al. (29) presented an approach in which they compared expressions for the friction pressure of the slurry and clean fluid. In their analysis, they assumed Blasius (30) turbulent Fanning friction factor versus Reynolds number equation for Newtonian fluids. The following expression for estimating slurry friction pressure knowing the clean fluid friction pressure is proposed. 

The Fanning friction factor,/, equals 16/Re for laminar flow in empty tubes. An expression that is satisfactory for Reynolds numbers between 5000 and 200,000 (i.e., for turbulent flow) is... 

For turbulent flow at > 10,000 with a smooth wall,/o is given by Eq. (13.15) or a Fanning friction factor chart can be used to obtain/. 

For turbulent fully developed flow (Re 10,000) m a sfnooth microtube, the Fanning friction factor can be calculated from the Karman-Nikuradse relation ... 

Use of friction factor for friction loss in laminar flow. A common parameter used in laminar and especially in turbulent flow is the Fanning friction factor, f which is defined as the drag force per wetted surface unit area (shear stressr, at the surface) divided by the product of density times velocity head or pu. The force isApy times the cross-sectional area t R and the wetted surface area is 2nR AL. Hence, the relation between the pressure drop due to friction and/is as follows for laminar and turbulent flow. 

In turbulent flow, as in laminar flow, the friction factor also depends on the Reynolds number. However, it is not possible to predict theoretically the Fanning friction factor/ for turbulent flow as it was done for laminar flow. The friction factor must be determined empirically (experimentally) and it not only depends upon the Reynolds number but also on surface roughness of the pipe. In laminar flow the roughness has essentially no effect. 

Skin friction losses in flow through straight pipe are calculated by using the Fanning friction factor. However, if the velocity of the fluid is changed in direction or magnitude, additional friction losses occur. This results from additional turbulence which develops because of vortices and other factors. Methods to estimate these losses are discussed below. 

In turbulent flow of time-independent fluids the Reynolds number at which turbulent flow occurs varies with the flow properties of the non-Newtonian fluid. Dodge and Metzner (D2) in a comprehensive study derived a theoretical equation for turbulent flow of non-Newtonian fluids through smooth round tubes. The final equation is plotted in Fig. 3.5-3, where the Fanning friction factor is plotted versus the generalized Reynolds... 

Fanning friction factor,/, for turbulent flow of an incompressible fluid in a smooth pipe is... 

In turbulent flow, the Fanning friction factor ff for the slurry described by a power-law fluid model depends on both the generalized Reynolds number Re and the flow behaviour index n. 

For turbulent flows in smooth pipes, the Fanning friction factor depends on both the Reynolds number, defined in terms of the plastic viscosity... 

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