Bernoullis Equation

A useful energy approach to the solution of fluid mechanics problems employs the Bernoulli equation. This equates the total energy per unit mass m) between two points on the same stream line when energy loss due to friction is negligible. A stream line gives the path of a fluid particle. 

F /2 = kinetic energy per unit mass, (KE)/m gh = elevation energy per unit mass As an example, consider the water clock used by Galileo to measure time (Fig. 5.14). This shows the path of a fluid particle from a point on the upper surface (1) that is maintained at a constant elevation (Aj) to a point (2) just beyond the orifice in the side of the vessel. Substituting into Eq. (5.15), and noting thatp- =Pj, Pi = Pi, and Fj = 0  

He next discusses the equilibrium of fish and how the entrainment of more or less air may compensate for changes in the density of water. This anticipates the submarine. The basis for the hydrometer is also contained in the discussion of neutral equilibrium of bodies. The extreme sensitivity of a ball of wax impregnated with sand used by physicians to establish neutral equilibrium in the measurement of density is also described. 

In searching for possible forces on falling bodies, Galileo considers the force that enables a droplet of water to stand high on a cabbage leaf. He reasons that this force is not of buoyant origin, since the drops should then 

Galileo indicates that all bodies (lead and feathers, for example) should fall with the same velocity in vacuum. In orderto study the speed of fall in a perfect vacuum, one should approach this situation as closely as possible. Hence, motions in air should be studied. The resistance the medium offers to being pushed aside (drag) will then be small. 

Consider the following isothermal parallel reactions in a constant-volume batch reactor with the reaction order of 1 for the first reaction and order n for the second reaction  

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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. 

The energy state of soil water can be defined with respect to the Bernoulli equation, neglecting thermal and osmotic energy as... 

Head. The tme meaning of the total developed pump head, H, is the amount of energy received by the unit of mass per unit of time (14). This concept is traceable to compressors and fans, where engineers operate with enthalpy, a close relation to the concept of total energy. However, because of the almost incompressible nature of Hquids, a simplification is possible to reduce enthalpy to a simpler form, a Bernoulli equation, as shown in equations 1—3, where g is the gravitational constant, SG is specific gravity, y is the density equivalent, is suction head, is discharge head, and H is the pump head, ie, the difference between H, and H. 

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. 

Unlike the momentum equation (Eq. [6-11]), the Bernoulli equation is not easily generahzed to multiple inlets or outlets. 

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... 

Still more mileage can be gotten out of Ah = uV2g when using it with Equation 2, which is the famous Bernoulli equation. The terms are... 

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

The diameter rfj is solved analogously. The Bernoulli equation at the interval 1-2 is... 

Another procedure for design of an air curtain is proposed by Tamm based on the Bernoulli equation. Recently Partyka proposed another procedure based on the model of Schlichting previously described. 

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 ... 

As an example of the contrast between analysis and design, consider the column buckling problem. To analyze the buckling resistance of a simply supported, axially loaded column, we use the Euler-Bernoulli equation,... 

Equation (3.14.2.17) shows the form of the Bernoulli equation that is a first-order differential equation. By substituting (3.14.2.18)... 

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,... 

Note that if each term of Eq. (5-35) is divided by g, then all terms will have the dimension of length. The result is called the head form of the Bernoulli equation, and each term then represents the equivalent amount of... 

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

The pressure P2 is determined by Bernoulli s equation. If the diffuser is horizontal, there is no work done between the inlet and outlet, and the friction loss is small (which is a good assumption for a well designed diffuser), the Bernoulli equation gives... 

Comparing this with the Bernoulli equation [Eq. (5-33)] shows that they are identical, provided... 

We see that there are several ways of interpreting the term ef. From the Bernoulli equation, it represents the lost (i.e., dissipated) energy... 

Looking at the Bernoulli equation, we see that the friction loss (ef) can be made dimensionless by dividing it by the kinetic energy per unit mass of fluid. The result is the dimensionless loss coefficient, K ... 

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

This can be solved for (P2 — Pi), which, when inserted into the Bernoulli equation, allows us to solve for ef. 

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 note first that the Bernoulli equation can be written... 

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