Bernoullis theorem

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 ... 

The equation of continuity and the Bernoulli theorem together show, for a stream of incompressible fluid, that (a) where the cross-sectional area is large and the streamlines are widely spaced, the velocity is low and the pressure is high, and (6) where the cross-sectional area is small and the streamlines are crowded together, the velocity is high and the pressure is low Hence a flow net gives a picture not only of the velocity field but also of the pressure field. 

According to the Bernoulli theorem, Eq. (10.9), if at any point the velocity head increases, there must be a corresponding decrease in the pressure head. For any liquid there is a minimum absolute pressure, and this is the vapor pressure of the liquid. As has been shown, this value depends both upon the identity of liquid and upon its temperature. If the conditions are such that a calculation results in a lower absolute pressure than the vapor pressure, this simply means that the assumptions upon which the calculations are based no longer apply. 

The coefficient of velocity may be determined by a velocity traverse of the jet with a fine pitot tube in order to obtain the mean velocity. This is subject to some slight error, as it is impossible to measure the velocity at the outer edge of the jet. The velocity may also be computed approximately from the coordinates of the trajectory. The ideal velocity is computed by the Bernoulli theorem. 

The velocity of a fluid approaching an orifice or nozzle or similar device is called the velocity of approach. For example, consider a large tank, filled with liquid, with a small orifice on its wall, near the bottom. It is assumed that the area of the tank is so large relative to that of the orifice that the velocity at the surface of the liquid (point 1) is negligible. Let the Bernoulli theorem be written between point 1 at the surface and point 2 at the orifice jet discharge, assuming the pressure is the same at both points. Let h equal the height of the liquid, measured from the surface level to the center of the orifice. Then 0 + /i + 0 = 0 + 0 + Vj2/2g, from which... 

The calculations in this case are clearly analogous to those required to prove the Bernoulli theorem. In order to show the first part of the statement, all we have to do is to determine the maximum of Eq. (36), i.e., the minimum of Eq. (43), given the auxiliary condition of Eq. (45). Boltzmann makes use of the second half of the statement in all those cases when he calls the Maxwell velocity distribution overwhelmingly the most probable one." A more quantitative formulation and derivation of this part of the statement is sketched by Jeans in [2, 22-26] and in Dynamical Theory, 44-46 and 56. 

The Bernoulli theorem can also be applied to the measurement of flow rate. The passage of an incompressible fluid through a constriction results in an increase in velocity from Ui to uj, which is associated with a decrease in pressure from Pi to P2, which can be measured directly. The volumetric flow rate Q = Mifli = M2fl2 by algebraic rearrangement (which is not shown). The final linear velocity M2 can be described by Eq. (5). 

The Bernoulli theorem may be used to determine the change in pressure caused by retardation of fluid at the upstream side of a body immersed in a fluid stream. This principle is applied in the Pitot tube, shown in Fig. 1C. The fluid velocity is reduced from Ma. the velocity of the fluid filament in alignment with the tube, to zero at B, an position known as the stagnation point. The pressure, P, is measured at this point by the method shown in Fig.l C. The undisturbed pressure, P, is measured in this example with a tapping point in the wall connected to a manometer. 

A. Basic hydraulics (e.g., Manning equation, Bernoulli theorem,... 

Bernoulli theorem At any point in a pipe through which a fluid is flowing the sum of the pressure energy, the kinetic energy, and the potential energy of a given mass of the fluid Is constant. This is equivalent to a statement of the law of the conservation of energy. The law was published in 1738 by Daniel Bernoulli. 

Weak laws consider conditions under which the probability that x - t is greater than some given epsilon, 8, tends to zero. The weak law of large numbers is represented by the Bernoulli theorem. For the Bernoulli theorem, we have the following relationship ... 

According to the fundamental limit theorem (an extension of the Bernoulli theorem by Laplace) if an event A occurs m times in a series of n independent trials with constant probability p and if n —> oo, then the distribution function tends to be... 

Bernoulli theorem A theorem in which the sum of the pressure-volume, potential, and kinetic energies of an incompressible and non-viscous fluid flowing in a pipe with steady flow with no work or heat transfer is the same anywhere within a system. When expressed in head form, the total head is the sum of the pressure, velocity, and static head. It is applicable only for incompressible and non-viscous fluids. That is ... 

It is therefore seen as the equivalent height of a column of the fluid if it were brought to rest. The velocity head can be used directly in the Bernoulli theorem. The SI unit is m. 

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