Friction factor derivation

Table 7. Comparison of the friction factor derived from viscous flow and dynamic...

The friction factor derivation culminating in the relationship of equations (3-18) and/or (3-22) specified that the tube be smooth. This raises a natural question as to what the effect of tube roughness would be on frictional heating. 

Alternatively, the explicit equation for the friction factor derived by Swamee and Jain (Equation 2.13) can be solved for the absolute roughness. 

The segmental friction factor introduced in the derivation of the Debye viscosity equation is an important quantity. It will continue to play a role in the discussion of entanglement effects in the theory of viscoelasticity in the next chapter, and again in Chap. 9 in connection with solution viscosity. Now that we have an idea of the magnitude of this parameter, let us examine the range of values it takes on. 

The spherical geometry assumed in the Stokes and Einstein derivations gives the highly symmetrical boundary conditions favored by theoreticians. For ellipsoids of revolution having an axial ratio a/b, friction factors have been derived by F. Perrin, and the coefficient of the first-order term in Eq. (9.9) has been derived by Simha. In both cases the calculated quantities increase as the axial ratio increases above unity. For spheres, a/b = 1. 

We shall see in Sec. 9.9 that D is a measurable quantity hence Eq. (9.79) provides a method for the determination of an experimental friction factor as well. Note that no assumptions are made regarding the shape of the solute particles in deriving Eq. (9.79), and the assumption of ideality can be satisfied by extrapolating experimental results to c = 0, where 7=1. 

To allow for the effect of roughness one can use the results of empirical tests in ducts that have been artificially roughened with particles glued on the surface. This approach allows roughness levels to be determined as a function of the particle diameter k. The following friction factor equation has been derived for large Reynolds numbers ... 

In deriving expressions for the packed bed friction factor, three separate flow regimes are normally considered (see Figure 2.11) as follows. 

For a particular material of pipeline construction, the friction factor can be assumed. This allows a simple formula to be derived ... 

Equation 3.11 is due to Blasius(6) and the others are derived from considerations of velocity profile. In addition to the Moody friction factor / = 8R/pu2, the Fanning or Darcy friction factor / = 2R/pu2 is often used. It is extremely important therefore to be clear about the exact definition of the friction factor when using this term in calculating head losses due to friction. 

Basically, there may be three reasons for the inconsistency between the theoretical and experimental friction factors (1) discrepancy between the actual conditions of a given experiment and the assumptions used in deriving the theoretical value, (2) error in measurements, and (3) effects due to decreasing the characteristic scale of the problem, which leads to changing correlation between the mass and surface forces (Ho and Tai 1998). 

Here ff,fG, and ft are friction factors between the liquid and the wall, the gas and the wall, and at the gas-liquid interface, respectively. Of these factors, the last one, f., is hard to determine. Some crude correlations and assumptions have to be used, as suggested by Taitel and Barnea (1990). From Eqs. (3-80) and (3-81), one can derive an equation for hf(oT 8) as a function of z ... 

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)]. 

Using the Ergun (1952) equation for the interfacial friction factor, Wen and Yu (1966) derived the following general equation to estimate the minimum fluidization superficial velocity Umf for spherical particles ... 

This is consistent with the Blasius type of expression used for the friction factors in deriving the Martinelli parameter. Using the value n = 0.20 and expressing the ratio of flow rates in terms of the quality... 

Equation (3) shows that the sedimentation velocity increases with the density difference between the particle and the medium. Any situation that brings the density of the settling unit closer to that of the solvent will decrease the sedimentation velocity. To an observer who is unaware of its derivation, however, the smaller velocity would be interpreted by Equation (4) as indicating a smaller value of (m/f). Since the actual mass of colloidal material is unaffected by the solvation, it is more correct to attribute the reduced sedimentation velocity to an increase in the value of the friction factor. 

In the next section, we see how to deal quantitatively with solvation. Until now, the friction factor has been merely a proportionality factor of rather ill-defined origin. We shall not undertake a derivation of Equation (2) in any general sense. In the next section, however, we outline the derivation of an important result due to G. G. Stokes —the friction factor for an unsolvated sphere. 

Based on the form of Eq. 4.63, we may anticipate a general result in which the product of the friction factor and the Reynolds number is a constant, /Re = constant. We seek a nondimensional analysis that leads to a general friction-factor result. We choose a length scale based on the long dimension of the channel, a. The velocity scale is based on a mean velocity derived from the mass flow rate, U = m/pAc ... 

Consider the fully developed steady flow of an incompressible fluid through an annular channel, which has an inner radius of r, and an outer radius of r0 (Fig. 4.27). The objective is to derive a general relationship for the friction factor as a function of flow parameters (i.e., Reynolds number) and channel geometry (i.e., hydraulic diameter Dh and the ratio f A friction factor /, which is a nondimensional measure of the wall... 

Eq. (2.14) is identical in form to that derived by Debye (21) in his "pearl-necklace in shear model, where a Stokes law molecular friction factor was also assumed. 

The discrepancy between the theories of Cerf and of Budtov and Gotub is not yet understood. The latter authors point to the fact that Cerf omitted the contribution of hinging on the junctions. However, when is put equal to zero in eq. (5.33), the fundamental difference in the molecular weight dependencies, as given by the second terms of eqs. (5.27), (5.31) and (5.33), does not disappear. It is perhaps superfluous to remind that both types of theories are derived only for the case that the internal friction factors are small compared with the external friction factor f. 

Laminar Flow. Theoretically derived equations for volumetric flow rate and friction factor arc included for several models in Table 6.7. Each model employs a specially defined Reynolds number, and the Bingham models also involve the Hcdstrom number,... 

In 1959 Dodge s thesis under the supervision of Metzner was published at the same time as Shaver s findings under the supervision of Merrill. Both noticed unusually low friction factors for certain non-Newtonian solutions like those of sodium carboxy-methylcellulose in water. At around the same time, industrial researchers made similar observations with certain additives, the most prominent being guar gum, which is a polysaccharide derived from a plant. The gums were used to suspend sand in the sand-water mixtures utilized in oil-well fracturing operations. 

The hL variable is named the friction factor f. The friction factor is dimensionless and is nomenclated throughout this chapter and in most every other publication the world over as the symbol f. Many mathematical formulas have been derived by experimental data to calculate f, with only two or three formulas having acceptable results. 

Having now established the friction factor f for the general pressure drop equation, the dimensional analysis equation for pipe pressure drop can be derived. A simple equation is now derived from the earlier Bernoulli equation ... 

In laminar flow the velocity distribution, and hence the frictional energy loss, is governed entirely by the rheological constitutive relation of the fluid. In some cases it is possible to derive theoretical expressions for the friction factor. Where this is possible, a three-step procedure must be followed. 

Use the Reynolds analogy to derive an expression for the Nusselt number for fully developed turbulent flow in an annulus in which the inner wall is heated to a uniform temperature and the outer wall is adiabatic. Assume that the friction factor can be derived by introducing the hydraulic diameter concept. 

Using the velocity distribution for developed laminar flow in a tube, derive an expression for the friction factor as defined by Eq. 5-112. 

In using this friction factor for a tubular reactor, the Reynolds number is evaluated at Borne estimated average condition, and then the corresponding friction factor is used for the whole bed. In calculating the axial pressure profile, the average composition and temperature in a cross section are used to estimate the density of the fluid, and this density is used with the average superficial mass velocity to estimate the axial derivative of pressure. 

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