For laminar non-Newtonian flow

On substituting for tw in equation 2.10, the Fanning friction factor for laminar non-Newtonian flow becomes... 

For laminar non-Newtonian film flow, see Bird, Armstrong, and Hassager Dynamics of Polymciic Liquids, vol. 1 Fluid Mechanics, Wiley, New York, 1977, p. 215, 217), Astarita, Marrucci, and Palumbo (Jnd. Eng. Chem. Fundam., 3, 333-339 [1964]) and Cheng (Jnd. Eng. Chem. Fundam., 13,394—395 [1974]). 

The velocity profile for isothermal, laminar, non-Newtonian flow in a pipe can sometimes be approximated as... 

Figure 9.1.3 Laminar velocity profile for the non-Newtonian flow of a transparent colloidal slurry in a 51 mm diameter glass tube at a mean velocity of 1.37 ms (data from Park et al. 1989).

In this chapter the general equations of laminar, non-Newtonian, non-isothermal, incompressible flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity of presentation, vector notations are used and all of the equations are given in a fixed (stationary or Eulerian) coordinate system. 

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian hquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic flmd described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

Economic Pipe Diameter, Laminar Flow Pipehnes for the transport of high-viscosity liquids are seldom designed purely on the basis of economics. More often, the size is dictated oy operability considerations such as available pressure drop, shear rate, or residence time distribution. Peters and Timmerhaus (ibid.. Chap. 10) provide an economic pipe diameter chart for laminar flow. For non-Newtouiau fluids, see SkeUand Non-Newtonian Flow and Heat Transfer, Chap. 7, Wiley, New York, 1967). 

The solution to the problem of determining the wall shear rate for a non-Newtonian fluid in laminar flow in a tube relies on equation 2.6. 

A stability analysis made by Ryan and Johnson (1959) suggests that the transition from laminar to turbulent flow for inelastic non-Newtonian fluids occurs at a critical value of the generalized Reynolds number that depends on the value of The results of this analysis are shown in Figure 3.7. This relationship has been tested for shear thinning and for Bingham... 

For a non-Newtonian fluid, the viscosity is not constant even for the laminar flow. Therefore, shear rate is easier to estimate than shear stress. 

For most pure liquids and for many solutions and dispersions, t) is a well-defined quantity for a given temperature and pressure which is independent of other solutions and dispersions, especially if concentrated and if the particles are asymmetric and/or aggregated deviations from Newtonian flow are observed. The main causes of non-Newtonian flow are the formation of a structure throughout the system and orientation of asymmetric particles caused by the velocity gradient. 

A method for predicting pressure drop and volume fraction for non-Newtonian fluids in annular flow has been proposed by Eisen-berg and Weinberger (AlChE J., 25, 240-245 [1979]). Das, Biswas, and Matra (Can. J. Chem. Eng., 70,431 37 [1993]) studied holdup in both horizontal and vertical gas/liquid flow with non-Newtonian liquids. Farooqi and Richardson (Trans Inst. Chem. Engrs., 60, 292-305, 323-333 [1982]) developed correlations for holdup and pressure drop for gas/non-Newtonian liquid horizontal flow. They used a modified Lockhart-Martinelli parameter for non-Newtonian liquid holdup. They found that two-phase pressure drop may actually be less than the single-phase liquid pressure drop with shear thinning liquids in laminar flow. 

A large body of literature is available on estimating friction loss for laminar and turbulent flow of Newtonian and non-Newtonian fluids in smooth pipes. For laminar flow past solid boundaries, surface roughness has no effect (at least for certain degrees of roughness) on the friction pressure drop of either Newtonian or non-Newtonian fluids. In turbulent flow, however, die nature... 

LAMINAR FLOW OF NON-NEWTONIAN LIQUIDS. Because of the difference in the relation between shear stress and velocity gradient, the shape of the velocity profile for non-newtonian liquids differs from that of a newtonian liquid. In the more complicated situations of non-newtonian flow the shape of the profile is determined experimentally. For the simpler cases such as the power-law model [Eq. (3.7)] or the Bingham model [Eq. (3.6)] the same methods used for determining the flow parameters of a newtonian fluid can be used for non-newtonian fluids in these categories. 

---