Differential Bernoulli

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. 

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

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)... 

Because the fluid velocity and properties change from point to point along the pipe, in order to analyze the flow we apply the differential form of the Bernoulli equation to a differential length of pipe (dL) ... 

The pitot tube is a device for measuring v(r), the local velocity at a given position in the conduit, as illustrated in Fig. 10-1. The measured velocity is then used in Eq. (10-2) to determine the flow rate. It consists of a differential pressure measuring device (e.g., a manometer, transducer, or DP cell) that measures the pressure difference between two tubes. One tube is attached to a hollow probe that can be positioned at any radial location in the conduit, and the other is attached to the wall of the conduit in the same axial plane as the end of the probe. The local velocity of the streamline that impinges on the end of the probe is v(r). The fluid element that impacts the open end of the probe must come to rest at that point, because there is no flow through the probe or the DP cell this is known as the stagnation point. The Bernoulli equation can be applied to the fluid streamline that impacts the probe tip ... 

The flow of fluids is most commonly measured using head flowmeters. The operation of these flowmeters is based on the Bernoulli equation. A constriction in the flow path is used to increase the flow velocity. This is accompanied by a decrease in pressure head and since the resultant pressure drop is a function of the flow rate of fluid, the latter can be evaluated. The flowmeters for closed conduits can be used for both gases and liquids. The flowmeters for open conduits can only be used for liquids. Head flowmeters include orifice and venturi meters, flow nozzles, Pitot tubes and weirs. They consist of a primary element which causes the pressure or head loss and a secondary element which measures it. The primary element does not contain any moving parts. The most common secondary elements for closed conduit flowmeters are U-tube manometers and differential pressure transducers. 

For potential flow, ie incompressible, irrotational flow, the velocity field can be found by solving Laplace s equation for the velocity potential then differentiating the potential to find the velocity components. Use of Bernoulli s equation then allows the pressure distribution to be determined. It should be noted that the no-slip boundary condition cannot be imposed for potential flow. 

Integration of Eq. 2.9-11 leads to the macroscopic mechanical energy balance equation, the steady-state version of which is the famous Bernoulli equation. Next we subtract Eq. 2.9-11 from Eq. 2.9-10 to obtain the differential thermal energy-balance... 

When fluid flows around the outside of an object, an additional loss occurs separately from the frictional energy loss. This loss, called form drag, arises from Bernoulli s effect pressure changes across the finite body and would occur even in the absence of viscosity. In the simple case of very slow or creeping flow around a sphere, it is possible to compute this form drag force theoretically. In all other cases of practical interest, however, this is essentially impossible because of the difficulty of the differential equations involved. 

Head-type flowmeters include orifice plates, venturi tubes, weirs, flumes, and many others. They change the velocity or direction of the flow, creating a measurable differential pressure, or "pressure head," in the fluid. Head metering is one of the most ancient of flow detection techniques. There is evidence that the Egyptians used weirs for measurement of irrigation water flows in the days of the Pharaohs and that the Romans used orifices to meter water to households in Caesar s time. In the 18th century, Bernoulli established the basic relationship between the pressure head and velocity head, and Venturi published on the flow tube bearing his name. 

Venturi tubes, flow nozzles, and flow tubes, similar to all differential pressure producers, are based on Bernoulli s theorem. Meter coefficients for venturi tubes and flow nozzles are approximately 0.98-0.99, whereas for orifice plates it averages about 0.62. Therefore, almost 60% (98/62) more flow can be obtained through these elements for the same differential pressure (see Figure 3.82). 

A very useful equation to deal with phenomena associated with the flow of fluids is the Bernoulli equation. It can be used to analyse fluid flow along a streamline from a point 1 to a point 2 assuming that the flow is steady, the process is adiabatic and that frictional forces between the fluid and the tube are negligible. Various forms of the equation appear in textbooks on fluid mechanics and physics. A statement in differential form can be obtained ... 

Differential pressure devices consist of two elements—one causes a change in the flow rate of a flowing fluid, which creates a pressure difference (Bernoulli effect) between two sections of the tube or pipe, and the second element measures the resultant Ap. Such a device is nonlinear, since Ois proportional to As a result, the range of Rvalues... 

Thus it follows that d1 2x will be equal to 2 fdx x. John Bernoulli seems to have told you of my having mentioned to him a marvelous analogy which makes it possible to say in a way the successive differentials are in geometric progression. One can ask what would be a differential having as its exponent a fraction. You see that the result can be expressed by an infinite series. Although this seems removed from Geometry, which does not yet know of such fractional exponents, it appears that one day these paradoxes will yield useful consequences, since there is hardly a paradox without utility. 

For expanding gas flow, Vj V, with horizontal pipe, Z2 = Zj. Hence, the differential form of Bernoulli s equation can be expressed as... 

If there is no relative motion within the fluid, the differential form of the energy balance (Bernoulli equation) reduces to... 

For isothermal flow in a pipe, the mass flux (G = till A) can be determined by integrating the differential form of the Bernoulli to give... 

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). 

EQUATIONS FOR BLOWERS AND COMPRESSORS. Because of the change in density during compressible flow, the integral form of the Bernoulli equation is inadequate. Equation (4.32), however, can be written differentially and used to relate the shaft work to the differential change in pressure head. In blowers and compressors the mechanical, kinetic, and potential energies do not change appreciably, and the velocity and static-head terms can be dropped. Also, on the assumption that the compressor is frictionless, / = 1.0 and hf 0. With these simplifications, Eq. (4.32) becomes... 

Draft is a differential pressure between the outside atmosphere and the internal pressure in a furnace. Differential pressure such as units of draft are sometimes referenced as psid. However, draft pressures are ordinarily so low that smaller units are needed. Two common units are mm w.c. or in w.c. (also w.c. or i.w.c.) that is, millimeters, water column or inches, water column, respectively. For example, in order to displace a fluid from a glass to one s mouth three inches above the liquid surface via a straw, one must cause a differential pressure of 3 in. w.c. Draft pressures are often smaller. For example, a typical natural draft refinery furnace operates anywhere from 0.25 in. w.c. to 0.50 in. w.c. This seemingly small pressure difference can cause great volumes of air flow. Gas pressure and velocity are related by Bernoulli s equation ... 

Suppose that our problem concerns a complicated fluid flow system in which we suspect that Bernoulli s equation, along with other equations, would apply. Then we can write Bernoulli s equation in differential form (without pump or compressor work) and integrate to find... 

According to Bernoulli and L Hospital both the numerator and denominator are differentiated with respect to t. As soon as the time t approaches zero the numerator becomes zero. Then the denominator is given by... 

Since the layer is extremely thin, Bernoulli s differential equations (see Section 3.4.1) reduce to the normal photokinetic equations (see eq. (5.107)). The boundary conditions of constant /abs.A as well as constant a z,t) within the z-direction of the measurement beam can be used as a first approximation. 

Ventnri-indnced convective flow is driven by a pressure differential created by wind blowing across tall dead plant cnlms (Armstrong et al., 1992). This pressure differential results in mass flow of air into the nndergronnd system via broken cnlms closer to the water level (Figure 7.18). The Venturi effect of wind blowing across an open tnbe was described by Bernoulli s equation ... 

This velocity measurement device uses pitot tube to measure differential pressure. It s an indirect way to measure flow rate. In accordance with Bernoulli equation, resulting in ... 

Bernoulli, Daniel (1700-82) Swiss mathematician. In 1724 he published a work on differential equations, which earned him a professorship at St Petersburg. He returned to Basel, Switzerland, in 1733 and began researches on hydrodynamics (see Bernoulu tueorem), the work for which he is best known. He also initiated the kinetic theory of matter. 

This is the differential equation for the time rate of change of pressure in tank 2. Correspondingly, from Bernoulli s equation, the rate of flow of lox in moles/sec from 1 to 2 will be given as. 

Using an equation of equilibrium or motion, which determines the deformation of the solid skeleton, and (5.58), a system of differential equations for specifying the mean velocity v (i.e., the conventional consolidation problem) is achieved. Note that in (5.58)

total head excluding the velocity potential), k is the hydraulic conductivity tensor, p is the pore pressure of the fluid, g is the gravity constant, and is the datum potential. Thus by starting with the mass conservation laws for both fluid and solid phases, we can simultaneously obtain the diffusion equation and the seepage equation which includes a term that accounts for the volumetric deformation of the porous skeleton. 

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