Bernoulli flow

As the potential energy term has an essential meaning in hydromechanics, the static head is selected as a comparison quantity. When the energy equation (4.32) is divided by g and integrated, it gives the Bernoulli flow tube equation... 

To prevent overheating, the covering could be ventilated naturally by using the physical effects of the Bernoulli flow and also by means of an air-conditioning system within the envelope. The inner membrane was... 

Pitot Tubes. The fundamental design of a pitot tube is shown in Eigure 9a. The opening into the flow stream measures the total or stagnation pressure of the stream whereas a wall tap senses static pressure. The velocity at the tip opening, lA can be obtained by the Bernoulli equation ... 

Head-Area Meters. The Bernoulli principle, the basis of closed-pipe differential-pressure flow measurement, can also be appHed to open-channel Hquid flows. When an obstmction is placed in an open channel, the flowing Hquid backs up and, by means of the Bernoulli equation, the flow rate can be shown to be proportional to the head, the exact relationship being a function of the obstmction shape. 

Two types of floater aozzles are curreafly ia use and they are based on two different principles. The Bernoulli principle is used ia the airfoil flotatioa aozzles, ia which the air flows from the aozzle parallel to the web and the high velocities create a reduced pressure, which attracts the web while keeping the web from touching the nozzles. The Coanda effect is used to create a flotation nozzle when the air is focused and thus a pressure pad is created to support the web as shown ia Figure 19. 

Here 4 = F,Jfn is the energy dissipation per unit mass. This equation has been called the engineering Bernoulli equation. For an incompressible flow, Eq. (6-15) becomes... 

The Bernoulli equation can be written for incompressible, inviscid flow along a streamhne, where no shaft work is done. 

Example 6 Losses with Fittings and Valves It is desired to calculate the liquid level in the vessel shown in Fig. 6-15 required to produce a discharge velocity of 2 m/s. The fluid is water at 20°C with p = 1,000 kg/m and i = 0.001 Pa - s, and the butterfly valve is at 6 = 10°. The pipe is 2-in Schedule 40, with an inner diameter of 0.0525 m. The pipe roughness is 0.046 mm. Assuming the flow is tiirhiilent and taking the velocity profile factor (X = 1, the engineering Bernoulli equation Eq. (6-16), written between surfaces 1 and 2, where the... 

For homogeneous flow in a pipe of diameter D, the differential form of the Bernoulli equation (6-15) rearranges to... 

The other ease is when there is too niiieh flow through the pump. The pump is operating to the right of the BEP on its eurve (Figure 9-8). The same problem oceurs, but now in the other direction. With the severe increase in velocity through the pump, the pressures tall dramatically in the H-F-G-H arc of the volute circle (Bernoulli s Law-says that as velocity goes up, pressure comes down). Now the shaft deflects, or even breaks in the opposite direction. .. at approximately 240° around the volute from the cutwater. 

Single gas bubbles in an inviscid liquid have hemispherical leading surfaces and somewhat flattened wakes. Their rise velocity is governed by Bernoulli s theory for potential flow of fluid around the nose of the bubble. This was first solved by G. I. Taylor to give a rise velocity Ug of ... 

This example demonstrates the dimensioning of a duct with a frictional incompressible fluid flow. Now the Bernoulli equation can be written as... 

The Pitof-static tube is a basic instrument that predicts flow velocity based on Bernoulli s equation ... 

For fluid flow in the (r, 6) plane, it is reasonable to assume that the fluid is inviscid, as the Reynolds number of the fluid flow usually exceeds O(IO ). Thus Eq. (13.1), with /i, = 0, may be integrated along the streamlines to give the Bernoulli equation as follows ... 

Bernoulli effect At any point in a conduit through which a liquid is flowing, the sum of pressure energy, potential energy, and kinetic energy is constant. 

Bernoulli and Euler dominated the mechanics of flexible and elastic bodies for many years. They also investigated the flow of fluids. In particular, they wanted to know about the relationship between the speed at which blood flows and its pressure. Bernoulli experimented by puncturing the wall of a pipe with a small, open-ended straw, and noted that as the fluid passed through the tube the height to which the fluid rose up the straw was related to fluid s pressure. Soon physicians all over Europe were measuring patients blood pressure by sticking pointed-ended glass tubes directly into their arteries. (It was not until 1896 that an Italian doctor discovered a less painful method that is still in widespread... 

Flow through chokes and nozzles is a special case of fluid dynamics. For incompressible fluids the problem can be handled by mass conservation and Bernoulli s equation. Bernoulli s equation is solved for the pressure drop across the choke, assuming that the velocity of approach and the vertical displacement are negligible. The velocity term is replaced by the volumetric flow rate times the area at the choke throat to yield... 

The theory of pressure losses can be established by developing Bernoulli s theorem for the case of a pipe in which the work done in overcoming frictional losses is derived from the pressure available. For a fluid flowing in a pipe, the pressure loss will depend on various parameters. If... 

The Bernoulli theorem can be used as the basis for a means of determining the flow through an orifice. The equation will be of the form ... 

At all points in a system, the static pressure is always equal to the original static pressure less any velocity head at a specific point in the system and less the friction head required to reach that point. Since both the velocity head and friction head represent energy and energy cannot be destroyed, the sum of the static head, the velocity head, and the friction head at any point in the system must add up to the original static head. This is known as Bernoulli s principal, which states For the horizontal flow of fluids through a tube, the sum of the pressure and the kinetic energy per unit volume of the fluid is constant. This principle governs the relationship of the static and dynamic factors in hydraulic systems. 

That the stream velocity does not change in the direction of flow. On this basis, from Bernoulli s theorem, the pressure then does not change (that is, dP/dx — 0). In practice, 3P/ dx may be positive or negative. If positive, a greater retardation of the fluid will result, and the boundary layer will thicken more rapidly. If dP/ dx is negative, the converse will be true. 

The Bernoulli effect—In a rapidly expanding flow, the two phases accelerate differently. For low initial velocities, the ratio of final velocities at the end of expansion, Va/VL, can be approximated by (pL/pG)u2-... 

The conditions at two different positions along a pipeline (at points 1 and 2) are related by the Bernoulli equation (see Problem 11). For flow in a pipe,... 

For plug flow, the Bernoulli equation for this system is... 

A 4 in. diameter open can has a 1/4 in. diameter hole in the bottom. The can is immersed bottom down in a pool of water, to a point where the bottom is 6 in. below the water surface and is held there while the water flows through the hole into the can. How long will it take for the water in the can to rise to the same level as that outside the can Neglect friction, and assume a pseudo steady state, i.e., time changes are so slow that at any instant the steady state Bernoulli equation applies. 

Consider a section of uniform cylindrical pipe of length L and radius R, inclined upward at an angle 0 to the horizontal, as shown in Fig. 6-2. The steady-state energy balance (or Bernoulli equation) applied to an incompressible fluid flowing in a uniform pipe can be written... 

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 ... 

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