Pipe Flow of a Newtonian Fluid

Writing a momentum balance over the incremental volume 27rr ArT gives the following  

In terms of velocity gradients for a Newtonian fluid and constant p and p  

Convective flow pvzV (2irrAr) z-o —pVzVz 2nrAr) z=l 

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Wells CS, Spangler JG (1967) Injection of a drag reducing fluid into turbulent pipe flow of a Newtonian fluid Phys Fluids 10 1890... 

The Mean Velocity of Laminar Pipe Flow Use the macroscopic mass-balance equation (Eq. 2.4.1) to calculate the mean velocity in laminar pipe flow of a Newtonian fluid. The velocity profile is the celebrated Poisseuille equation ... 

An example of the use of momentum balance in pipe flow of a Newtonian fluid is... 

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 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. 

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). 

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. 

An approximate equation for the profile of the time-averaged velocity for steady turbulent flow of a Newtonian fluid through a pipe of circular... 

Some of the simplifications that may be possible are illustrated by the case of steady, fully-developed, laminar, incompressible flow of a Newtonian fluid in a horizontal pipe. The flow is assumed to be axisymmetric with no swirl component of velocity so that derivatives wrt 6 vanish and vg = 0. For fully-developed flow, derivatives wrt z are zero. With these simplifications and noting that the flow is incompressible, the continuity equation (equation A. 11) reduces to... 

Example 7.8 Residence Time Distribution Functions in Fully Developed Laminar Flow of a Newtonian Fluid in a Pipe The velocity distribution... 

Consider the flow of a Newtonian fluid in a pipe, as illustrated in Figure 9.1. The governing differential equation is (Bird et al., 2002) ... 

Solve for the entry flow of a Newtonian fluid into a pipe using the parameters in the example. Reproduce Figure 10.2 through Figure 10.5 plus Figure 10.7. 

Problem 5-11. Flow Through a Sinusoidal Pipe. Consider the pressure-driven flow of a Newtonian fluid with viscosity // and density/) through a sinusoidally varying pipe. The radius R of the pipe is given by... 

Velocity distribution in steady, laminar flow of a newtonian fluid in a circular pipe. 

A somewhat harder problem is steady flow of an incompressible newto-nian fluid in sonJe duct or pipe which is of constant cross section but not circular, such asja rectangular duct or an open channel. The problem of laminar flow of a newtonian fluid can be solved analytically for several shapes. Generally the velocity depends on two dimensions. In several cases of interest, the problems can jbe solved by the same method we used to find Eq. 6.8, i.e., setting up a force balance around some properly chosen section of the flow, solving for the sh r stress, introducing the newtonian law of viscosity for the shear stress, and integrating to find the velocity distribution. From the velocity distribution the flow rate-pressure-drop relation is found. 

As a simple example of the calculation of the weighted average total strain (WATS) consider the laminar flow of a Newtonian fluid in a pipe of length L and radius R. The velocity profile gives the local velocity v at radial position... 

FIGURE 8 9 Cumulative residence time distribution function versus reduced time for flow in an extruder, plug flow, flow of a Newtonian fluid in a pipe, and a continuously stirred tank (CST) vessel. 

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