Reynolds number Newtonian pipe flow

Equation (94) provides the means for rearranging all of the theoretical expressions for v) given above into expressions involving the friction factor. For example, when Eq. (75) for Newtonian pipe flow is so rearranged and one eliminates (v) in terms of the Reynolds number, Re = D v)p/fi, one obtains... 

The most important case of this transition for chemical engineers is the transition from laminar to turbulent flow, which occurs in straight bounded ducts. In the case of Newtonian fluid rheology, this occurs in straight pipes when Re = 2100. A similar phenomenon occurs in pipes of other cross sections, as well and also for non-Newtonian fluids. However, just as the friction factor relations for these other cases are more complex than for simple Newtonian pipe flow, so the criteria for transition to turbulence cannot be expressed as a simple critical value of a Reynolds number. 

Other expressions which deal with specific cases also are reported in the literature. For example, Adusumilli and Hill [6] give some equations for pressure rise, energy lost, and eddy length for non-Newtonian low Reynolds number (Re<80) flows through sudden expansions. They use an approach based on a macroscopic mass and energy balance across a sudden pipe expansion, as detailed originally by Bird et al. [7]. The expansion ratio (p) is defined as ... 

In order to predict Lhe transition point from stable streamline to stable turbulent flow, it is necessary to define a modified Reynolds number, though it is not clear that the same sharp transition in flow regime always occurs. Particular attention will be paid to flow in pipes of circular cross-section, but the methods are applicable to other geometries (annuli, between flat plates, and so on) as in the case of Newtonian fluids, and the methods described earlier for flow between plates, through an annulus or down a surface can be adapted to take account of non-Newtonian characteristics of the fluid. 

However, this expression assumes that the total resistance to flow is due to the shear deformation of the fluid, as in a uniform pipe. In reality the resistance is a result of both shear and stretching (extensional) deformation as the fluid moves through the nonuniform converging-diverging flow cross section within the pores. The stretching resistance is the product of the extension (stretch) rate and the extensional viscosity. The extension rate in porous media is of the same order as the shear rate, and the extensional viscosity for a Newtonian fluid is three times the shear viscosity. Thus, in practice a value of 150-180 instead of 72 is in closer agreement with observations at low Reynolds numbers, i.e.,... 

Measurements with different fluids, in pipes of various diameters, have shown that for Newtonian fluids the transition from laminar to turbulent flow takes place at a critical value of the quantity pudjp in which u is the volumetric average velocity of the fluid, dt is the internal diameter of the pipe, and p and p. are the fluid s density and viscosity respectively. This quantity is known as the Reynolds number Re after Osborne Reynolds who made his celebrated flow visualization experiments in 1883 ... 

Under normal circumstances, the laminar-turbulent transition occurs at a Reynolds number of about 2100 for Newtonian fluids flowing in pipes. 

For Newtonian flow in a pipe, the Reynolds number is defined by... 

Turbulent flow of Newtonian fluids is described in terms of the Fanning friction factor, which is correlated against the Reynolds number with the relative roughness of the pipe wall as a parameter. The same approach is adopted for non-Newtonian flow but the generalized Reynolds number is used. 

A general time-independent non-Newtonian liquid of density 961 kg/m3 flows steadily with an average velocity of 2.0 m/s through a tube 3.048 m long with an inside diameter of 0.0762 m. For these conditions, the pipe flow consistency coefficient K has a value of 1.48 Pa s0,3 and n a value of 0.3. Calculate the values of the apparent viscosity for pipe flow p.ap, the generalized Reynolds number Re and the pressure drop across the tube, neglecting end effects. 

In highly developed turbulent flow Govier and Winning (Gl) found experimental data to fall within 10% of the conventional Newtonian curve. This means that the effect of the Hedstrom number may be neglected beyond the end of the transition region. Their data were for five clay slurries in two pipe diameters and in. I.P.S.) for Reynolds numbers to 200,000 and Hedstrom numbers to 560,000. 

For Newtonian fluids flowing in smooth pipes, the friction losses can be estimated for laminar flow (Re < 2100) using the Fanning friction factor, f. The Reynolds number, Re, is given by ... 

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. 

Dodge and Metzner (16) presented an extensive theoretical and experimental study on the turbulent flow of non-Newtonian fluids in smooth pipes. They extended von Karman s (17) work on turbulent flow friction factors to include the power law non-Newtonian fluids. The following implicit expression for the friction factor was derived in terms of the Metzner-Reed modified Reynolds number and the power law index ... 

The flow regime in a pipe is determined by the Reynolds number. For a Newtonian fluid ... 

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