Fully Established Friction Factor

The asymptotic nature of the friction factor is clearly brought out in Fig. 10.24, which shows the measured fully established friction factors taken in three tubes of differing diameters as a function of the Weissenberg number based on the Powell-Eyring relaxation time for fixed values of the Reynolds number for aqueous solutions of polyacrylamide. The critical Weissenberg number for friction (Ws), is seen to be on the order of 5 to 10. When the Weissenberg number exceeds 10, it is clear that the fully developed friction factor is a function only of the Reynolds number. 

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

FIGURE 10.25 Fully established friction factor versus Re with concentrated polyethylene oxide and polyacrylamide solutions [100]. 

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 ... 

This Reynolds number has had wide use because all fully established laminar pipe flow laminar friction factor data for power law fluids lie on the line/= 16/Re. ... 

FIGURE 10.7 Experimental friction factor measurements for nonnewtonian fluids in fully established laminar flow through rectangular channels. Results of Wheeler and Wissler [50] (O), Hartnett et al. [54] (A), and Harnett and Kostic [55] ( ). 

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... 

Initially, commissioning tests were performed on the test facility using the pumped flow arrangement to demonstrate that everything was operating satisfactorily. Friction factors determined from pressure drop measurements made under conditions of isothermal flow through the test section were compared with values calculated using the well-established correlation equation of Blasius for fully developed turbulent flow in a smooth tube. The maximum discrepancy was less than 6%. 

---