Fully turbulent friction factor

Approximate fully turbulent friction factors for motionless mixers are given in Table 2. (Note These figures are approximate and for comparison purposes only they should not be used for design. The true friction factors vary slightly with Re and scale.)... 

Hydrodynamically Developing Flow. Hydrodynamically developing turbulent flow in concentric annular ducts has been investigated by Rothfus et al. [114], Olson and Sparrow [115], and Okiishi and Serouy [116]. The measured apparent friction factors at the inner wall of two concentric annuli (r = 0.3367 and r = 0.5618) with a square entrance are shown in Fig. 5.17 (r = 0.5618), where / is the fully developed friction factor at the inner wall. The values of/ equal 0.01,0.008, and 0.0066 for Re = 6000,1.5 x 104, and 3 x 104, respectively [114]. 

Effects of Eccentricity. Jonsson and Sparrow [119] have conducted a careful experimental investigation of fully developed turbulent flow in smooth, eccentric annular ducts. The researchers have provided the velocity measurements graphically in terms of the wall coordinate h+ as well as the velocity-defect representation. From their results, the circumferentially averaged fully developed friction factor is correlated by a power-law relationship of the following type ... 

The fully developed friction factor and heat transfer coefficients for turbulent flow in an asymmetrically heated rectangular duct have been reported by Rao [59]. In this investigation, the experimental region of the Reynolds number was from 104 to 5 x 104. 

FIGURE 5.31 Fully developed friction factor for turbulent flow in smooth right-angled and equilateral triangular ducts [45]. 

FIGURE 5.40 Fully developed friction factor for longitudinal turbulent flow between a triangular and a rectangular array [261]. 

This behavior can be seen in Fig. 10.22, which shows the fully established turbulent friction factor as a function of Reynolds number Re for concentrations ranging from 10 to 1000 wppm of polyacrylamide in Chicago tap water. This series of measurements, which were taken in a tube 1.30 cm in diameter, revealed that the hydrodynamic entrance length varied with concentration, reaching a maximum of 100 pipe diameters at the higher concentrations. Therefore, the friction factors shown in Fig. 22 were measured at values of xld greater than 100. The asymptotic friction factor is reached at concentrations of approximately 50 wppm of polyacrylamide in tap water for the tube diameter used in the test program [50, 93]. The... 

FIGURE 10.24 Fully established friction factors for aqueous polyacrylamide solutions in turbulent pipe flow as a function of the Weissenberg and Reynolds numbers. 

It is recommended that either Eq. 10.71 or Eq. 10.72 be used to predict the fully developed friction factor (that is, for xld greater than 100) of viscoelastic aqueous polymer solutions in turbulent pipe flow for Reynolds numbers greater than 6000 and for Weissenberg numbers above critical value. The critical Weissenberg number for aqueous polyacrylamide solutions based on the Powell-Eyring relaxation time is on the order of 5 to 10 [50]. In the absence of experimental data for other polymers, this value should be used for other viscoelastic fluids with the appropriate caution. 

The fully established friction factor for turbulent flow of purely viscous nonnewtonian fluids in rectangular channels may be determined by the modified Dodge-Metzner equation [72,110] ... 

The fully established friction factor for turbulent flow of a viscoelastic fluid in a rectangular channel is dependent on the aspect ratio, the Reynolds number, and the Weissenberg number. As in the case of the circular tube, at small values of Ws, the friction factor decreases from the newtonian value. It continues to decrease with increasing values of Ws, ultimately reaching a lower asymptotic limit. This limiting friction factor may be calculated from the following equation ... 

The fully developed turbulent friction factor seems to be in disagreement with the Blasius equation for smooth microcharmels and with the Colebrook correlation for rough microchannels. 

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. 

A constant value of the friction factor f = 0.009 is assumed, for fully developed turbulent flow and a relative pipe roughness e = 0.01. The assumed constancy of f, however, depends upon the magnitude of the discharge Reynolds number which is checked during the program. The program also uses the data values given by Szekely and Themelis (1971), but converted to SI. 

For the Bingham plastic, there is no abrupt transition from laminar to turbulent flow as is observed for Newtonian fluids. Instead, there is a gradual deviation from purely laminar flow to fully turbulent flow. For turbulent flow, the friction factor can be represented by the empirical expression of Darby and Melson (1981) [as modified by Darby et al. (1992)] ... 

A first estimate for the pipe friction factor and the AT/s can be made by assuming that the flow is fully turbulent (and the a s = 1). Thus,... 

Although Eq. (9-17) appears to be explicit for G, it is actually implicit because the friction factor depends on the Reynolds number, which depends on G. However, the Reynolds number under choked flow conditions is often high enough that fully turbulent flow prevails, in which case the friction factor depends only on the relative pipe roughness ... 

At high Reynolds numbers (high turbulence levels), the flow is dominated by inertial forces and wall roughness, as in pipe flow. The porous medium can be considered an extremely rough conduit, with s/d 1. Thus, the flow at a sufficiently high Reynolds number should be fully turbulent and the friction factor should be constant. This has been confirmed by observations, with the value of the constant equal to approximately 1.75 ... 

For fully developed turbulent flow in rough pipes,/is independent of the Reynolds number, as shown by the nearly constant friction factors at high Reynolds number in Figure 4-7. For this case Equation 4-33 is simplified to... 

Thus the velocity of the liquid discharging from the pipe is 3.66 m/s. The table also shows that the friction factor/changes little with the Reynolds number. Thus we can approximate it using Equation 4-34 for fully developed turbulent flow in rough pipes. Equation 4-34 produces a friction factor value of 0.0041. Then... 

Determine the Fanning friction factor / from Equation 4-34. This assumes fully developed turbulent flow at high Reynolds numbers. This assumption can be checked later but is normally valid. 

The excess head loss terms 2 Kt are found using the 2-K method presented earlier in section 4-4. For most accidental discharges of gases the flow is fully developed turbulent flow. This means that for pipes the friction factor is independent of the Reynolds number and that for fittings Kf = and the solution is direct. 

Assume fully developed turbulent flow to determine the friction factor for the pipe and the excess head loss terms for the fittings and pipe entrances and exits. The Reynolds number can be calculated at the completion of the calculation to check this assumption. Sum the individual excess head loss terms to get 2 Kf. 

Taylor (T4, T6), in two other articles, used the dispersed plug-flow model for turbulent flow, and Aris s treatment also included this case. Taylor and Aris both conclude that an effective axial-dispersion coefficient Dzf can again be used and that this coefficient is now a function of the well known Fanning friction factor. Tichacek et al. (T8) also considered turbulent flow, and found that Dl was quite sensitive to variations in the velocity profile. Aris further used the method for dispersion in a two-phase system with transfer between phases (All), for dispersion in flow through a tube with stagnant pockets (AlO), and for flow with a pulsating velocity (A12). Hawthorn (H7) considered the temperature effect of viscosity on dispersion coefficients he found that they can be altered by a factor of two in laminar flow, but that there is little effect for fully developed turbulent flow. Elder (E4) has considered open-channel flow and diffusion of discrete particles. Bischoff and Levenspiel (B14) extended Aris s theory to include a linear rate process, and used the results to construct comprehensive correlations of dispersion coefficients. 

Some simple methods of determining heat transfer rates to turbulent flows in a duct have been considered in this chapter. Fully developed flow in a pipe was first considered. Analogy solutions for this situation were discussed. In such solutions, the heat transfer rate is predicted from a knowledge of the wall shear stress. In fully developed pipe flow, the wall shear stress is conventionally expressed in terms of the friction factor and methods of finding the friction factor were discussed. The Reynolds analogy was first discussed. This solution really only applies to fluids with a Prandtl number of 1. A three-layer analogy solution which applies for all Prandtl numbers was then discussed. 

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. 

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. 

For fully developed turbulent flow in smooth tubes, a simple relation for the Nusselt number can he obtained by substituting the simple power law relation / - 0.184 for the friction factor into Eq. 8-66. It gives... 

C How does the friction factor f vary along the flow direction in the fully developed region in (a) laminar flow and (6) turbulent flow ... 

The Moody chart for the friction factor for fully developed flow in circular pipes for use in the head loss relation -----. Friction factors in tlie turbulent flow... 

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