Friction factor — laminar 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. 

The convective heat-transfer coefficient and friction factor for laminar flow in noncircular ducts can be calculated from empirically or analytically determined Nusselt numbers, as given in Table 5. For turbulent flow, the circular duct data with the use of the hydrauhc diameter, defined in equation 10, may be used. 

Fig. 5. Moody diagram for Darcy friction factor (13) (-----), smooth flow (----), whoUy turbulent flow ( ), laminar flow.

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 ). 

In laminar flow,/is independent of /D. In turbulent flow, the friction factor for rough pipe follows the smooth tube curve for a range of Reynolds numbers (hydrauhcaUy smooth flow). For greater Reynolds numbers,/deviates from the smooth pipe cui ve, eventually becoming independent of Re. This region, often called complete turbulence, is frequently encountered in commercial pipe flows. The Reynolds number above which / becomes essentially independent of Re is (Davies, Turbulence Phenomena, Academic, New York, 1972, p. 37) 20[3.2-2.46ln( /D) ... 

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 ... 

In laminar flow, the friction factor for curved pipe/ may be expressed in terms of the straight pipe friction factor/= 16/Re as (Hart, Chem. Eng. ScL, 43, 775-783 [1988])... 

Figure 6-40 shows power number vs. impeller Reynolds number for a typical configuration. The similarity to the friction factor vs. Reynolds number behavior for pipe flow is significant. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In turbulent flow, where inertial forces dominate, the power number is nearly constant. 

Laminar flow after transition usually turns into turbulent flow when Re > 2000. It has been shown that the pressure loss of a turbulent flow is caused by a friction factor with the magnitude of... 

This is the basis for establishing the condition or type of fluid flow in a pipe. Reynolds numbers below 2000 to 2100 are usually considered to define laminar or thscous flow numbers from 2000 to 3000-4000 to define a transition region of peculiar flow, and numbers above 4000 to define a state of turbulent flow. Reference to Figure 2-3 and Figure 2-11 will identify these regions, and the friction factors associated with them [2]. 

VTien the Reynolds number is below a value of 2000, the flow region is considered laminar. The pipe friction factor is defined as ... 

Flow is <2000, therefore, flow of viscous or laminar system consists of friction factor, fp, for 4-in. pipe = 0.017 (Table 2-2). 

If the calculated value of the Reynolds number is below 2,000, the flow will generally be laminar, that is, the fluid particles will follow parallel flow paths. For laminar flow the friction factor is... 

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. 

Transition from laminar to turbulent flow occurs when the friction factor exceeds the low ARe range. In Fig. 2.20a the results obtained for a mbe of diameter 705 pm by Maynes and Webb (2002) are compared against the value accepted for laminar flow A = 64/Re. Based on the above data, one can conclude that the transition occurs at Tie >2,100. 

The influence of compressibility was assessed by varying the Mach number in the range 0 < Ma < 0.38, while Kn and ks/H were kept low. Friction factor data were reported only with Ma < 1 at the exit, to ensure the flow rate was controlled by viscous forces alone. A mild increase in the friction factor (8%) was observed as Ma approached 0.38. This effect was verified independently by numerical analysis for the same conditions as in the experiment. The range of relative surface roughness tested was 0.001 < ka/H < 0.06, yet there was no significant influence on the friction factor for laminar gas flow. 

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. 

Glass and silicon tubes with diameters of 79.9-166.3 iim, and 100.25-205.3 am, respectively, were employed by Li et al. (2003) to study the characteristics of friction factors for de-ionized water flow in micro-tubes in the Re range of 350 to 2,300. Figure 3.1 shows that for fully developed water flow in smooth glass and silicon micro-tubes, the Poiseuille number remained approximately 64, which is consistent with the results in macro-tubes. The Reynolds number corresponding to the transition from laminar to turbulent flow was Re = 1,700—2,000. 

Wu and Cheng (2003) measured the friction factor of laminar flow of de-ionized water in smooth silicon micro-channels of trapezoidal cross-section with hydraulic diameters in the range of 25.9 to 291.0 pm. The experimental data were found to be in agreement within 11% with an existing theoretical solution for an incompressible, fully developed, laminar flow in trapezoidal channels under the no-slip boundary condition. It is confirmed that Navier-Stokes equations are still valid for the laminar flow of de-ionized water in smooth micro-channels having hydraulic diameter as small as 25.9 pm. For smooth channels with larger hydraulic diameters of 103.4-103.4-291.0pm, transition from laminar to turbulent flow occurred at Re = 1,500-2,000. 

The existence of roughness leads also to decreasing the value of the critical Reynolds number, at which transition from laminar to turbulent flow occurs. The character of the dependence of the friction factor on the Reynolds number in laminar flow remains the same for both smooth and rough micro-channels, i.e., X = const/Re. 

A study of forced convection characteristics in rectangular channels with hydraulic diameter of 133-367 pm was performed by Peng and Peterson (1996). In their experiments the liquid velocity varied from 0.2 to 12m/s and the Reynolds number was in the range 50, 000. The main results of this study (and subsequent works, e.g., Peng and Wang 1998) may be summarized as follows (1) friction factors for laminar and turbulent flows are inversely proportional to Re and Re ", respectively (2) the Poiseuille number is not constant, i.e., for laminar flow it depends on Re as PoRe ° (3) the transition from laminar to turbulent flow occurs at Re about 300-700. These results do not agree with those reported by other investigators and are probably incorrect. 

For single-phase gas flow in micro-channels of hydraulic diameter from 101 to 4,010 pm, in the range of Reynolds numbers Re < Recr, the Knudsen number 0.001 < Kn < 0.38, and the Mach number 0.07 < Ma < 0.84, the experimental friction factor agrees quite well with the theoretical one predicted for fully developed laminar flow. 

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