Frictional pressure drop friction factor — laminar flow

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 ... 

As indicated earlier, non-Newtonian characteristics have a much stronger influence on flow in the streamline flow region where viscous effects dominate than in turbulent flow where inertial forces are of prime importance. Furthermore, there is substantial evidence to the effect that for shear-thinning fluids, the standard friction chart tends to over-predict pressure drop if the Metzner and Reed Reynolds number Re R is used. Furthermore, laminar flow can persist for slightly higher Reynolds numbers than for Newtonian fluids. Overall, therefore, there is a factor of safety involved in treating the fluid as Newtonian when flow is expected to be turbulent. 

The investigation shows agreement between the standard laminar incompressible flow predictions and the measured results for water. Based on these observations the predictions based on the analytical results of Shah and London (1978) can be used to predict the pressure drop for water in channels with as small as 24.9 pm. This investigation shows also that it is insufficient to assume that the friction factor for laminar compressible flow can be determined by means of the well-known analytical predictions for its incompressible counterpart. In fact, the experimental and numerical results both show that the friction factor increases for compressible flows as Re is increased for a given channel with air. 

Pressure drop measurements. For the majority of experiments the instrumentation was relatively similar. Due to limitations associated with the small size of the channels, pressures were not measured directly inside the micro-channels. To obtain the channel entrance and exit pressures, measurements were taken in a plenum or supply line prior to entering the channel. It is insufficient to assume that the friction factor for laminar compressible flow can be determined by means of analytical predictions for incompressible flow. 

In Table 6.7, C is the Martinelli-Chisholm constant, / is the friction factor, /f is the friction factor based on local liquid flow rate, / is the friction factor based on total flow rate as a liquid, G is the mass velocity in the micro-channel, L is the length of micro-channel, P is the pressure, AP is the pressure drop, Ptp,a is the acceleration component of two-phase pressure drop, APtp f is the frictional component of two-phase pressure drop, v is the specific volume, JCe is the thermodynamic equilibrium quality, Xvt is the Martinelli parameter based on laminar liquid-turbulent vapor flow, Xvv is the Martinelli parameter based on laminar liquid-laminar vapor flow, a is the void fraction, ji is the viscosity, p is the density, is the two-phase frictional... 

In laminar flow the friction factor is f = 1/Re = p/Dhup Pressure drop,... 

A number of equations have been proposed for use in the calculation of pressure drop in coils of constant curvature [Srinivasan et al (1968)]. The latter are known as helices. For laminar flow, Kubair and Kuloor (1965) gave an equation for the Reynolds number range 170 to the critical value. In terms of the Fanning friction factor, their equation can be written as... 

This second method does not lend itself to the development of quantitative correlations which are based solely on true physical properties of the fluids and which, therefore, can be measured in the laboratory. The prediction of heat transfer coefficients for a new suspension, for example, might require pilot-plant-scale turbulent-flow viscosity measurements, which could just as easily be extended to include experimental measurement of the desired heat transfer coefficient directly. These remarks may best be summarized by saying that both types of measurements would have been desirable in some of the research work, in order to compare the results. For a significant number of suspensions (four) this has been done by Miller (M13), who found no difference between laboratory viscosities measured with a rotational viscometer and those obtained from turbulent-flow pressure-drop measurements, assuming, for suspensions, the validity of the conventional friction-factor—Reynolds-number plot.11 It is accordingly concluded here that use of either type of measurement is satisfactory use of a viscometer such as that described by Orr (05) is recommended on the basis that fundamental fluid properties are more readily determined under laminar-flow conditions, and a means is provided whereby heat transfer characteristics of a new suspension may be predicted without pilot-plant-scale studies. 

A simpler method due to Kem (1950, pp. 147-152) nominally considers only the drop across the tube banks, but actually takes account of the added pressure drop through baffle windows by employing a higher than normal friction factor to evaluate pressure drop across the tube banks. Example 8.8 employs this procedure. According to Taborek (HEDH, 1983, 3.3.2), the Kern predictions usually are high, and therefore considered safe, by a factor as high as 2, except in laminar flow where the results are uncertain. In the case worked out by Ganapathy (1982, pp. 292-302), however, the Bell and Kem results are essentially the same. 

Brauer (B18), 1960 Treatment of film flow in vertical tubes with gas streams (countercurrent, cocurrent up and down). Pressure drops and gas stream friction factors are calculated for the various cases. Fully developed gas flow and laminar film flow (smooth) are assumed throughout. 

Table I. Asymptotic dimensionless laminar flow heat or mass transfer coefficients Nu = a dh/V, (constant wall conditions) and Fanning friction factor / for pressure drop Ap = 2f r 7.ijd ) g for ducts of different cross section [10).

Related Calculations. Helical Coils. The same procedure can be used to calculate the pressure drop in helical coils. For turbulent flow, a friction factor for curved flow is substituted for the friction factor for straight tubes. For laminar flow, the friction loss for a curved tube is expressed as an equivalent length of straight tube and the friction factor for straight tubes is used. The Reynolds number required for turbulent flow is 2100[1 + 12(Dj/Dc)1/2], where Dt is the inside diameter of the tube and Dc is the coil diameter. 

Figure 8-3 illustrates the friction frictor versus GRe relationship for power law fluids under laminar flow conditions. It can also be used for Newtonian fluids in laminar flow with the Reynolds number being used in place of GRe. In fact, the Newtonian/ versus Re relationship was established much earlier than extension to non-Newtonian fluids. Once the magnitude of the friction factor is known, the pressure drop in a pipe can be estimated from Equation 12. 

B Obtain analytic relations for the velocity profile, pressure drop, friction factor, and Nusselt number in fully developed laminar How, and B Determine the friction factor and Nusselt number in fully developed turbulent flow using empirical relations, and calculate the pressure drop and heat transfer rdte. 

Equation (7) is based on the universal flow profile for turbulent flow. Values of A and G were obtained from the theoretical analysis of the laminar and turbulent pressure drop characteristics of assemblies of circular channels with different diameters. These values also proved to give accurate friction factor predictions for noncircular channels, such as symmetric and asymmetric annuli and rod bundles. 

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