Pipe flow Newtonian fluids

Bogue, D. C. and Metzner, A. B. 1963. Velocity profiles in turbulent pipe flow, Newtonian and non-Newtonian fluids. Industrial and Engineering Chemistry Fundamentals 2 143-149. 

Clogged Pipe, Non-Newtonian Fluid, and Coupled Solids Deposition Flow Modeling, Final Technical Report, May 2000, Brown Root Energy Services, Houston, TX... 

Laminar Flow Effective Shear Rate. In laminar flow, the fluids are often shear thinning (i.e., the viscosity decreases with increasing shear rate). The apparent or effective shear rate in an empty pipe with Newtonian fluids is... 

Occasionally, piping systems are designed to carry multiphase fluids (combinations of gases, Hquids, and soflds), or non-Newtonian fluids. Sizing piping for such systems is beyond the scope of this article. PubHcations covering multiphase flow (20) and non-Newtonian flow (21) are available. 

The following analysis can be used to determine economic pipe diameters for the turbulent flow of Newtonian fluids. The working expression that can be used is ... 

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. 

HARTNETT and KOSTIC 26 have recently examined the published correlations for turbulent flow of shear-thinning power-law fluids in pipes and in non-circular ducts, and have concluded that, for smooth pipes, Dodge and Metzner S(27) modification of equation 3.11 (to which it reduces for Newtonian fluids) is the most satisfactory. 

Heywood, N. 1. and Cheng, D. C.-H. Trans Inst. Measurement and Control 6 (1984) 33. Comparison of methods for predicting head loss in turbulent pipe flow of non-Newtonian fluids. 

Water, of viscosity 1 mN s/m2 flowing through the pipe at the same mean velocity gives rise to a pressure drop of I04 N/m2 compared with 105 N/m2 for the non-Newtonian fluid. What is the consistency ("k vaiuei of the non-Newtonian fluid ... 

Cho YI, Hartnett JP (1982) Non-Newtonian fluids in circular pipe flows. Adv Heat Transfer 15 60-141... 

Fig. 1.13 Left schematic plot of the distribu- water flowing under laminar conditions in a tion of flow velocities, vz, for laminar flow of a circular pipe the probability density of dis-Newtonian fluid in a circular pipe the max- placements is constant between 0 and imum value of the velocity, occurring in the Zmax = i>ZimaxA, where A is the encoding time center of the pipe, is shown for comparison. of the experiment.

Many materials are conveyed within a process facility by means of pumping and flow in a circular pipe. From a conceptual standpoint, such a flow offers an excellent opportunity for rheological measurement. In pipe flow, the velocity profile for a fluid that shows shear thinning behavior deviates dramatically from that found for a Newtonian fluid, which is characterized by a single shear viscosity. This is easily illustrated for a power-law fluid, which is a simple model for shear thinning [1]. The relationship between the shear stress, a, and the shear rate, y, of such a fluid is characterized by two parameters, a power-law exponent, n, and a constant, m, through... 

In the steady flow of a Newtonian fluid through a long uniform circular tube, if ARe < 2000 the flow is laminar and the fluid elements move in smooth straight parallel lines. Under these conditions, it is known that the relationship between the flow rate and the pressure drop in the pipe does not depend upon the fluid density or the pipe wall material. 

This classification of material behavior is summarized in Table 3-1 (in which the subscripts have been omitted for simplicity). Since we are concerned with fluids, we will concentrate primarily on the flow behavior of Newtonian and non-Newtonian fluids. However, we will also illustrate some of the unique characteristics of viscoelastic fluids, such as the ability of solutions of certain high polymers to flow through pipes in turbulent flow with much less energy expenditure than the solvent alone. 

This can be compared with the results of the dimensional analysis for the laminar flow of a Newtonian fluid in a pipe (Chapter 2, Section V), for which we deduced that JNRe = constant. In this case, we have determined the value of the constant analytically, using first principles rather than by experiment. 

For steady, uniform, fully developed flow in a pipe (or any conduit), the conservation of mass, energy, and momentum equations can be arranged in specific forms that are most useful for the analysis of such problems. These general expressions are valid for both Newtonian and non-Newtonian fluids in either laminar or turbulent flow. 

Corresponding expressions for the friction loss in laminar and turbulent flow for non-Newtonian fluids in pipes, for the two simplest (two-parameter) models—the power law and Bingham plastic—can be evaluated in a similar manner. The power law model is very popular for representing the viscosity of a wide variety of non-Newtonian fluids because of its simplicity and versatility. However, extreme care should be exercised in its application, because any application involving extrapolation beyond the range of shear stress (or shear rate) represented by the data used to determine the model parameters can lead to misleading or erroneous results. 

Like the von Karman equation, this equation is implicit in/. Equation (6-46) can be applied to any non-Newtonian fluid if the parameter n is interpreted to be the point slope of the shear stress versus shear rate plot from (laminar) viscosity measurements, at the wall shear stress (or shear rate) corresponding to the conditions of interest in turbulent flow. However, it is not a simple matter to acquire the needed data over the appropriate range or to solve the equation for / for a given flow rate and pipe diameter, in turbulent flow. 

We will use the Bernoulli equation in the form of Eq. (6-67) for analyzing pipe flows, and we will use the total volumetric flow rate (Q) as the flow variable instead of the velocity, because this is the usual measure of capacity in a pipeline. For Newtonian fluids, the problem thus reduces to a relation between the three dimensionless variables ... 

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.,... 

The slurry behaves as a non-Newtonian fluid, which can be described as a Bingham plastic with a yield stress of 40 dyn/cm2 and a limiting viscosity of 100 cP. Calculate the pressure gradient (in psi/ft) for this slurry flowing at a velocity of 8 ft/s in a 10 in. ID pipe. 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

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. 

It is shown in Example 1.9 that the velocity profile for laminar flow of a Newtonian fluid in a pipe of circular section is parabolic and can be expressed in terms of the volumetric average velocity u as ... 

If the velocity had the uniform value u, the momentum flow rate would be mfpu2. Thus for laminar flow of a Newtonian fluid in a pipe the momentum flow rate is greater by a factor of 4/3 than it would be if the same fluid with the same mass flow rate had a uniform velocity. This difference is analogous to the different values of a in Bernoulli s equation (equation 1.14). 

Determine the shear stress distribution and velocity profile for steady, fully developed, laminar flow of an incompressible Newtonian fluid in a horizontal pipe. Use a cylindrical shell element and consider both sign conventions. How should the analysis be modified for flow in an annulus ... 

Flow of incompressible Newtonian fluids in pipes and channels... 

Although it is unnecessary to use the friction factor for laminar flow, exact solutions being available, it follows from equation 1.65 that for laminar flow of a Newtonian fluid in a pipe, the Fanning friction factor is given by... 

For turbulent flow of a Newtonian fluid, / decreases gradually with Re, which must be the case in view of the fact that the pressure drop varies with flow rate to a power slightly lower than 2.0. It is also found with turbulent flow that the value of / depends on the relative roughness of the pipe wall. The relative roughness is equal to eld, where e is the absolute roughness and d, the internal diameter of the pipe. Values of absolute roughness for various kinds of pipes and ducts are given in Table 2.1. 

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