Reynolds Number and Friction Factor

Friction Factor and Reynolds Number For a Newtonian fluid in a smooth pipe, dimensional analysis relates the frictional pressure drop per unit length AP/L to the pipe diameter D, density p, viscosity p, and average velocity V through two dimensionless groups, the Fanning friction factory and the Reynolds number Re. 

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Noncircular Channels Calciilation of fric tional pressure drop in noncircular channels depends on whether the flow is laminar or tumu-lent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter shoiild be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraiilic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraiilic diameter for a circiilar pipe is = D, for an annulus of inner diameter d and outer diameter D, = D — d, for a rectangiilar duct of sides 7, h, Dij = ah/[2(a + h)].T ie hydraulic radius Rii is defined as one-fourth of the hydraiilic diameter. 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

The friction factor depends on the Reynolds number and the surface conditions of the pipe. There are numerous charts and equations for determining the relationship between the friction factor and Reynolds number. The friction factor can be calculated by [63]... 

Figure 3.40. Metzner and Reed correlation of friction factor and Reynolds number...

The Blasius relation between friction factor and Reynolds number for turbulent flow is ... 

Derive the relation between the friction factor and Reynolds number in turbulent flow for smooth pipe [Eq. (6-34)], starting with the von Karman equation for the velocity distribution in the turbulent boundary layer [Eq. (6-26)]. 

Thus, in laminar flow in a straight channel, the density of the fluid does not affect the pressure drop. To use the correlation for friction factor, or Fig. 8.1(a), one must know the density. This problem is avoided if one plots the product of friction factor and Reynolds number versus Reynolds number, since from their definition ... 

The product of fully developed friction factor and Reynolds number in laminar flow / Re obtained by Ratkowsky and Epstein [296] for polygonal ducts with centered circular cores (see the inset in Fig. 5.65) are shown in Fig. 5.65. The fully developed NuHi obtained by Cheng and Jamil [297] are given in Fig. 5.66. It can be observed that as -> >, the value of/Re approaches 6.222 for a = 1 (annular duct) /Re approaches 16 for a =0 (circular duct). 

Figure 26. Effect of the polymer CMC on the relationship between Fanning friction factor and Reynolds number in laminar and turbulent pipe flow. (Reproduced with permission from reference 90. Copyright 1959 American Institute of Chemical Engineers.)...

As in the case of flow in pipes, there are several different friction factors in common usage for flowing porous media, all differing by a constant. The choice between these is completely arbitrary in this text we drop the 5 in Eq. 12.9 and the in Eq. 12.10 to find our working forms of the friction factor and Reynolds number for porous media ... 

Despite the fact that equation (3.37) is applicable to all kinds of time-independent fluids, numerous workers have presented expressions for turbulent flow friction factors for specific fluid models. For instance, Tomita [1959] applied the concept of the Prandtl mixing length and put forward modified definitions of the friction factor and Reynolds number for the turbulent flow of Bingham Plastic fluids in smooth pipes so that the Nikuradse equation, i.e. equation (3.37) with n = 1, could be used. Though he tested the applicability of his method using his own data in the range 2000 < Reg(l — 4>f 3 — )< 10, the validity of this approach has not been established using independent experimental data. 

Irvine [11] proposed an empirical explicit relation between the friction factor and Reynolds number as... 

Combining Eq. (2.10-5) with the left-hand side ofEq. (3.11-15), the result obtained shows that the friction factor is a function of the Reynolds number (as was shown before in the empirical correlation of friction factor and Reynolds number) and of length/diameter ratio. In pipes with L/D 1 or pipes with fully developed flow, the friction factor is found to be independent of LID. 

Whereof and Reef are constriction based friction factor and Reynolds number respectively, and Po is the Poiseuille number specific to the cross-sectional geometry of the channel, 16 for circular pipes, 14.23 for a square channel and 24 for flow between two parallel plates. 

The correlation between friction factor and Reynolds number in Figure 5.17 has three distinct regions. There is a linear region for Re < 1, a curved region for 1 < Re < 1,000, and a flat region for Re > 1,000. 

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