Application of the Bernoulli

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). 

Here, we analyze what can be learnt from the application of the Bernoulli s and momentum theorems when they are appUed to the emptying of a tank in which the water level is H. 

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. 

Propellers for the marine environment appeared first in the eighteenth centui y. The French mathematician and founder of hydrodynamics, Daniel Bernoulli, proposed steam propulsion with screw propellers as early as 1752. However, the first application of the marine propeller was the hand-cranked screw on American inventor David Bushnell s submarine, Turtle in 1776. Also, many experimenters, such as steamboat inventor Robert Fulton, incorporated marine propellers into their designs. 

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 ... 

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). 

Newton and Leibnitz. The foundations of calculus of variations were laid by Bernoulli, Euler, Lagrange and Weierstrass. The optimization of constrained problems, which involves the addition of unknown multipliers, became known by the name of its inventor Lagrange. Cauchy made the first application of the steepest descent method to solve unconstrained minimization problems. In spite of these early contributions, very little progress was made until the middle of the 20th century, when high-speed digital computers made the implementation of the optimization procedures possible and stimulated further research in new methods. 

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. 

Missen, Ronald W., Applications of the THdpital-Bernoulli Rule in Chemical Systems , J. Chem. Educ. 54,448 (1977). 

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

To illustrate the major idea of the finite difference method, the solution of a end loaded bar as shown in O Fig. 26.19 will be considered. A good introduction to the application of the finite difference method to Bernoulli beams can be found for example in (Bathe 1996). 

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. 

The pioneers in mathematical statistics, such as Bernoulli, Poisson, and Laplace, had developed statistical and probability theory by the middle of the nineteenth century. Probably the first instance of applied statistics came in the application of probability theory to games of chance. Even today, probability theorists frequently choose... 

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. 

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 ... 

---