Kinetic energy correction

If the fluid velocity is uniform over the cross section at a value equal to the average velocity V (i.e., plug flow ), then the rate at which kinetic energy is transported would be 

Therefore, a kinetic energy correction factor, a, can be defined as the ratio of the true rate of kinetic energy transport relative to that which would occur if the fluid velocity is everywhere equal to the average (plug flow) velocity, e.g., 

True KE transport rate 1 Plug flow KE transport rate A 

The Bernoulli equation should therefore include this kinetic energy correction factor, i.e., 

As will be shown later, the velocity profile for a Newtonian fluid in laminar flow in a circular tube is parabolic. When this is introduced into Eq. (5-38), the result is a = 2. For highly turbulent flow, the profile is much flatter and a 1.06, although for practical applications it is usually assumed that a = 1 for turbulent flow. 

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The Cannon-Fenske viscometer (Fig. 24b) is excellent for general use. A long capillary and small upper reservoir result in a small kinetic energy correction the large diameter of the lower reservoir minimises head errors. Because the upper and lower bulbs He on the same vertical axis, variations in the head are minimal even if the viscometer is used in positions that are not perfecdy vertical. A reverse-flow Cannon-Fen ske viscometer is used for opaque hquids. In this type of viscometer the Hquid flows upward past the timing marks, rather than downward as in the normal direct-flow instmment. Thus the position of the meniscus is not obscured by the film of Hquid on the glass wall. 

Some orifice viscometers, such as the Shell dip cup and the European ISO cup, which resembles a Eord cup with a capillary, have long capillaries. These cups need smaller kinetic energy corrections and give better precision than the corresponding short-capiHary viscometers. However, they are stiU not precision instmments, and should be used only for control purposes. 

Example 5-2 Kinetic Energy Correction Factor for Laminar Flow of a Newtonian Fluid. We will show later that the velocity profile for the laminar flow of a Newtonian fluid in fully developed flow in a circular tube is parabolic. Because the velocity is zero at the wall of the tube and maximum in the center, the equation for the profile is... 

This can be used to calculate the kinetic energy correction factor from Eq. (5-38) as follows. First we must calculate the average velocity, V, using... 

Evaluate the kinetic energy correction factor a in Bernoulli s equation for turbulent flow assuming that the 1/7 power law velocity profile [Eq. (6-36)] is valid. Repeat this for laminar flow of a Newtonian fluid in a tube, for which the velocity profile is parabolic. 

The a s are the kinetic energy correction factors at the upstream and downstream points (recall that a = 2 for laminar flow and a = 1 for turbulent flow for a Newtonian fluid). 

That is, we now aim to describe in a more appropriate way the interaction part of the kinetic energy that is introduced to the ex-change-correlation functional in the Kohn-Sham scheme. Including the kinetic energy corrections increases the computational requirements substantially, but the accuracy is also much improved compared with conventional gradient-corrected functionals. 

In all cases, intrinsic viscosities were measured at 25 C in constant temperature baths controlled to +0.1°C or better, using suspended level Ubbelohde dilution viscometers with solvent flow times of at least 100 sec.. No kinetic energy corrections were made. Solution flow times were measured at four concentrations for each sample, and intrinsic viscosities were obtained from the classical double extrapolation of hg /c vs. c and (In hj.)/c vs. c to a single intercept value. Concentration ranges were varied somewhat with the molecular weights of the samples, but were chosen such that both functions were straight lines in all cases. 

Fig. 9. Dependence of kinetic-energy correction factor upon flow-behavior index.

Viscosity measurements were made with two Cannon-Ubbelohde viscometers, and timing was by an optical device actuating an electronic timer (Wescan Instruments, Inc.). An air thermostat was used. The viscometers were calibrated with redistilled air-saturated water over the range 10°-50°C. The kinetic-energy correction was used in the form ... 

It was noted that the velocity in a channel approaching a weir might be so badly distributed as to require a value of 1.3 to 2.2 for the kinetic energy correction factor. In unobstructed uniform channels, however, the velocity distribution not only is more uniform but is readily amenable to theoretical analysis. Vanonil has demonstrated that the Prandtl universal logarithmic velocity distribution law for pipes also applies to a two-dimensional open channel, i.e., one that is infinitely wide. This equation may be written... 

Kinetic-Energy Correction. In deriving Eq. (10) it was assumed that end effects are negligible. Such end effects are most likely to be important at the inlet of the capillary, where a pressure drop occurs as a result of the necessity of imparting kinetic energy to the fluid. 

This quantity is called the kinetic-energy correction. In practice, it is subtracted from the overall pressure difference — pi, since no comparable pressure change need ordinarily be considered at the outlet end. This results from the fact that the kinetic energy of the fluid carries it in a jet stream far out into the body of fluid beyond the outlet, so that the kinetic energy is dissipated (as heat rather than as potential energy) at a considerable distance from where it could otherwise be effective in producing a pressure change at the outlet. 

With Eq. (17) find 17 pp for dry air and use this value in Eq. (16) to find the value of K. Now calculate 17 pp for the other gases from Eq. (16) (one can use the approximate value of rj in evaluating the kinetic-energy correction term). Finally, apply the slip correction to obtain the true viscosity. The gas pressure p that appears explicidy in Eq. (19) and implicitly in Eq. (16), where p = pMRT, can be replaced by the average inlet pressure (pj + p()l2. 

The integral is the kinetic energy correction where p is the density, Q = V/t the volume flow rate of the fluid, and u the velocity of the flow at point r from the axis of the capillary. For Newtonian liquids it can be shown that the integral term becomes noting that dQ = 2vrdr-u, —du/dr —... 

In the MEB equation, kinetic energy losses can be calculated easily provided that the kinetic energy correction factor a can be determined. In turbulent flow, often, the value of a = 2 is used in the MEB equation. When the flow is laminar and the fluid is Newtonian, the value of a = 1 is used. Osorio and Steffe (1984) showed that for fluids that follow the Herschel-Bulkley model, the value of a in laminar flow depends on both the flow behavior index ( ) and the dimensionless yield stress ( o) defined above. They developed an analytical expression and also presented their results in graphical form for a as a function of the flow behavior index ( ) and the dimensionless yield stress ( o)- When possible, the values presented by Osorio and Steffe (1984) should be used. For FCOJ samples that do not exhibit yield stress and are mildly shear-thinning, it seems reasonable to use a value of a = 1. 

The value of Cd depends on conditions of flow and shape of the constriction. For a well-shaped constriction (notably circular cross section), it would vary between 0.95 and 0.99 for turbulent flow. The value is much lower in laminar flow because the kinetic energy correction is larger. The return of the fluid to the original velocity by means of a diverging section forms a flow-measuring device known as a Venturi meter. 

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