Applications of the Bernoulli equation

The other method is the velocity head method. The term V2/2g has dimensions of length and is commonly called a velocity head. Application of the Bernoulli equation to the problem of frictionless discharge at velocity V through a nozzle at the bottom of a column of liquid of height H shows that H = V2/2g. Thus II is the liquid head corresponding to the velocity V. Use of the velocity head to scale pressure drops has wide application in fluid mechanics. Examination of the Navier-Stokes equations suggests that when the inertial terms dominate the viscous terms, pressure gradients are expected to be proportional to pV2 where V is a characteristic velocity of the flow. 

The solution to the problem once again requires the application of the Bernoulli equation ... 

Application of the Bernoulli Equation to the simple apparatus which is an air amplifier implements what is known as the Coanda Effect (named after Romanian aerodynamics pioneer Henri Coanda, who, in 1934, was the first to recognize the practical application of the phenomenon in development of aircraft which would have adequate lift at low air speed). Specifically, in an air amplifier, compressed air expands across a specially configured surface. A high velocity jet is produced. The jet clings to the walls of the surface (the Coanda Effect). A zone of low pressure is created. Available air at a higher pressure is pulled into the zone of lower pressure. The level of low pressure is dependent upon the dimensions of the surface configuration (as predicted by the Bernoulli Equation). 

Equation (4.23) is the point form of the Bernoulli equation without friction. Although derived for the special situation of an expanding cross section and an upward flow, the equation is applicable to the general case of constant or contracting cross section and horizontal or downward flow (the sign of the differential dZ corrects for change in direction). 

Application of the Bernoulli and continuity equations will enable the ratio of transverse area to total longitudinal area and number of sections required for any degree of mixing to be determined. 

A typical piping application starts with a specified flow rate for a given fluid. The piping system is then designed with the necessary valves, fittings, etc. and should be sized for the most economical pipe size, as discussed in Chapter 7. Application of the energy balance (Bernoulli) equation to the entire system, from the upstream end (point 1) to the downstream end (point 2) determines the overall net driving force (DF) in the system required to overcome the frictional resistance ... 

For one component fluids the Bernoulli equation for inviscid flow along a streamline can either be formulated by direct application of Newton s second law to a fluid particle moving along a streamline [114] [10] or derived projecting the generalized momentum equation (1.78) onto a streamline. Applying the latter approach, the Navier-Stokes equation for non-viscous fluids becomes ... 

Now consider the pressure drops that occur during rapid expansion. Applying the Bernoulli equation for an inviscid, incompressible fluid, we can readily show that the pressure drop from pre-expansion conditions to the initiation of rapid expansion (i.e., from Pq to Fini) is smaller by a factor of 10 than the pressure drop from pre-expansion conditions to the nozzle outlet (i.e., from Po to P2). Now for a typical RESS application, the pre-expansion pressure Po is about 200 bar. Because of the choked-flow conditions that exist at the nozzle exit, the pressure drop is about 100 bar. Accordingly, the pressure at Zini (i.e., Fini) is only about 0.01 bar less than Fq for incompressible flow, and the pressure drop... 

The first step in an application of the first law is to identify the system. This is conveniently done by drawing a line around the system, which is called the control volume. The quantities Q, W, and AE are then introduced into Eq. (11.5). The change in thermal energy AE) in thermal problems usually involves changes in temperature T) and pressure P). Tables of relative values of specific energy per unit mass (m) in kJ/Kg for different materials such as air, water, steam, refrigerant, etc. are available. These tables also give values of other properties at different combinations ofP and T, such as volume per unit mass. The Bernoulli equation [Eq. (5.15)] is a special application of the first law for applications where there are no losses, no heat transfer, and no work done. 

Bernoulli s equation results from the application of the general energy equation and the first law of thermodynamics to a steady-flow systan. 

As an example of a simple application of Bernoulli s equation, consider the case of steady, fully developed flow of a liquid (incompressible) through an inclined pipe of constant diameter with no pump in the section considered. Bernoulli s equation for the section between planes 1 and 2 shown in Figure 1.5 can be written as... 

An important application of Bernoulli s equation is in flow measurement, discussed in Chapter 8. When an incompressible fluid flows through a constriction such as the throat of the Venturi meter shown in Figure 8.5, by continuity the fluid velocity must increase and by Bernoulli s equation the pressure must fall. By measuring this change in pressure, the change in velocity can be determined and the volumetric flow rate calculated. 

Applications of Bernoulli s equation are usually straightforward. Often there is a choice of the locations 1 and 2 between which the calculation is made it is important to choose these locations carefully. All conditions must be known at each location. The appropriate choice can sometimes make the calculation very simple. A rather extreme case is discussed in Example 1.1. 

These expressions for the losses due to sudden expansion and sudden contraction can be derived by application of Bernoulli s equation and the momentum equation. Figure 2.3 shows a sudden expansion. 

In the earlier days of the petroleum age, many pipe experiments were conducted. In the quest for the magic formula, one was found to be the closest to utopia even to this day, called the Darcy formula. The Darcy formula is derived manually from the Bernoulli principle, which simply describes the energy balance between two points of a fluid flowing in a pipe. This energy equation is also applicable to a static condition of no flow between the two points. The classic Bernoulli energy equation [1] is ... 

The method of moments is directly applicable with Equations 47 and 56, and the sum rules are automatically satisfied. In the first part of the present section, results of these investigations will be discussed the various Jacobi expansions will be compared with the Bernoulli series, and with the G(w)-method. 

DEVELOPED HEAD. A typical pump application is shown diagrammatically in Fig. 8.5. The pump is installed in a pipeline to provide the energy needed to draw liquid from a reservoir and discharge a constant volumetric flow rate at the exit of the pipeline, Zj, feet above the level of the liquid. At the pump itself, the liquid enters the suction connection at station a and leaves the discharge connection at station b. A Bernoulli equation can be written between stations a and b. Equation... 

The most interesting applications of Bernoulli s equation include the effects of friction. Before we can solve these, we must learn how to evaluate the term, which we do in Chap. 6. However, in many flow problems the friction heating terms are small compared with the other terms and can be neglected. We can solve these by means of Bernoulli s equation without the friction heating term. A good example of this type of problem is the tank-draining problem, which leads to Torricelli s equation. 

For inherently two- or three-dimensional flows, such as the flow around an airplane, the simple application of Bernoulli s equation from one point to another in the flow, which we have used here, is applicable only if the two points chosen are on a single streamline, as discussed in Chap. 10. 

This formula gives good results for low-velocity gas flow (less than about 200ft/s), hut for high-velocity gas flows Bernoulli s equation is no longer applicable. For all velocities between zero and sonic velocities, we can assume that the part of the mainstream which is stopped by the impact tube is stopped practically isentropically. If that is correct, then the pressure measured at F, is the reservoir pressure for the flow. Thus, we can use Eq. 8.17, solved for... 

In deriving Bernoulli s equation, we assumed that the flow into and out of the system was of uniform velocity, etc. In general this will not be true of any stream tube, because the velocity may be different from one streamline to the next. However, if we make our stream tube smaller and smaller, then the nonuniform flow across its entrance (and exit) becomes negligible. In the limit the stream tube can be thought of as being so small that it shrinks down to just a streamline. Forj such a stream tube, Bernoulli s equation, as derived in Chap. 5, is obviously applicable. This result is true for any kind of incompressible flow if the flow is frictionless (i.e., an ideal fluid), then the friction term may... 

The Pitot tube measures pressure, from which the speed of the fluid can be deduced by application of Bernoulli s equation. It consists of a tube whose tip face is perpendicular to the direction of the fluid flow, as shown in Figure 6.9, and an additional pressure tap to measure the static pressure. The kinetic energy... 

The simpler Euler-Bernoulli theory which considers zero transverse shear deformation Yxz has also been tested. Using Hamilton s principle the equations of motion of the beam are derived. This model has been used in various investigations of our group (see, among others, Stavroulakis et al. 2005, 2007). Further applications of piezoelectric layers in control can be found in the review article (Irschik 2002). 

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