Frictional factor forms

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 ... 

Thus, the pipe friction chart for a Newtonian fluid (Figure 3.3) may be used for shearthinning power-law fluids if Remit is used in place of Re. In the turbulent region, the ordinate is equal to (R/pu2)n 0 fn5. For the streamline region the ordinate remains simply R/pu2, because Reme has been defined so that it shall be so (see equation 3.140). More recently, Irvine(25j has proposed an improved form of the modified Blasius equation which predicts the friction factor for inelastic shear-thinning polymer-solutions to within 7 per cent. 

Equation (3.4) reflects the dependence of the friction factor on the Reynolds number, whereas Eq. (3.5) shows conformity between actual and calculated shapes of a micro-channel. Condition (3.5) is the most general since it testifies to an identical form of the dependencies of the experimental and theoretical friction factor on the... 

Numerous researchers have studied damage to micro-organisms during flow in pipes, (Fig. 11) [87,88] Most researchers use a Fanning friction factor, f, to calculate the energy dissipation rate for fully developed flow in tubular bioreactors and capillary flow devices. There are minor differences in the equations that are used but they are generally of the following form [89,901 ... 

This is a form of the Fanning or Darcy formula with friction factor = 0.0055. This friction factor corresponds to approximately the following ... 

If the wall stress (rw) in Eq. (6-11) is expressed in terms of the Fanning friction factor (i.e., tw = /pF2/2) and the result solved for/, the dimensionless form of the Hagen-Poiseuille equation results ... 

A corresponding expression for the friction factor can be obtained by writing this expression in dimensionless form and substituting the result into Eq. (6-33). Evaluating the integral and solving for / gives... 

This is the power law equivalent of the Hagen-Poiseuille equation. It can be written in dimensionless form by expressing the wall stress in terms of the friction factor using Eq. (6-5), solving for /, and equating the result to 16/ N t (i.e., the form of the Newtonian result). The result is an expression that is identical to the dimensionless Hagen-Poiseuille equation ... 

An alternative form of Equation 4-32, useful for determining the Reynolds number from the friction factor/, is... 

The option of using alternative forms of a function depending on the value of logical variables that identify the state of the process. Typical examples are the shift in the relations uSfed to calculate the friction factor from laminar to turbulent flow, or the calculation of P — V — T relations as the phase changes from gas to liquid. 

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 ... 

In developing their correlation, Lockhart and Martinelli assumed that the friction factors could be determined from equations of the same form as the Blasius equation ... 

Obviously, correlations of one of these friction factors with an analogous Reynolds number, or with two-phase pressure-drop, throws little light on the other variables concerned, and these quantities will appear as parameters in any proposed relationship. However, Govier and Omer point out that plots of such a form do give a systematic spread of data above the single phase lines and allow easy comparison of trends. 

The protective film formed by corrosion inhibitors can act as a lubricating film to help in reducing fluid friction within a pipeline. This property can be measured and is termed the C Factor. This friction factor is expressed as follows ... 

The Rouse expression for the viscosity [Eq.(4.25)] is of this form. The frictional coefficient per main chain atom Co is the friction factor in this case the remainder of the expression is the structure factor. 

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