Fluid flow Bernoulli equation

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 ... 

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... 

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

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 ... 

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 ... 

A special condition called slack flow can occur when the gravitational driving force exceeds the full pipe friction loss, such as when a liquid is being pumped up and down over hilly terrain. Consider the situation shown in Fig. 7-5, in which the pump upstream provides the driving force to move the liquid up the hill at a flow rate of Q. Since gravity works against the flow on the uphill side and aids the flow on the downhill side, the job of the pump is to get the fluid to the top of the hill. The minimum pressure is at point 2 at the top of the hill, and the flow rate (Q) is determined by the balance between the pump head (Hp = — w/g) and the frictional and gravitational resistance to flow on the uphill side (i.e., the Bernoulli equation applied from point 1 to point 2) ... 

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 ... 

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 ... 

Consider the flow of an incompressible fluid through a two-dimensional porous medium, as illustrated in Fig. 13-2. Assuming that the kinetic energy change is negligible and that the flow is laminar as characterized by Darcy s law, the Bernoulli equation becomes... 

The Bernoulli equation relates the pressure drop across the bed to the fluid flow rate and the bed properties ... 

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. 

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. 

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. 

Notice that one end of the U-tube is connected to the smaller diameter section of the Venturi tube, whereas the other end is connected to one of the larger diameter sections. If the fluid is flowing from left to right, according to the Bernoulli equation, v2 > vl and P2 < Pr The fact that the pressure is lower in the narrow part of the tube is the primary scientific basis for the operation of an aspirator. The fluid levels in the manometer will reflect the pressure difference between Pt and Pr A measure of the height difference Ah and a knowledge of the density of the fluid in the manometer will yield the pressure difference between Pj and P2 using ... 

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 f factor is seen as the success or failure of fluid flow systems design. It requires experimental input, such as Eq. (5.48), provided for the DPAT term. Equations (5.48), (5.50), and (5.52) are presented again for this review of the Bernoulli equation. 

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. 

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 ... 

If the fluid flows into the pipe through a bell-shaped inlet section as shown in Figure 4.25, the losses in this inlet section will be small. In this case, if po is taken as the pressure ahead of the inlet as shown in Fig. 4.26 and p, is the pressure on the inlet plane then Bernoulli s equation applied across the inlet gives ... 

If a balance of forces is made on an element of incompressible fluid and these forces are set equal to the change in momentum of the fluid element, the Bernoulli equation for flow along a streamline results ... 

Remember from fluid mechanics that the Bernoulli equation is an equation for frictionless flow along a streamline. The flow through the screen is similar to the flow through an orifice, and it is standard in the derivation of the flow through an orifice to assume that the flow is frictionless by applying the Bernoulli equation. To consider the friction that obviously is present, an orifice coefficient is simply prefix to the derived equation. 

Given fluid conditions (pressure, flow rate, velocity, elevation) at the inlet and outlet of an open system and values of friction loss and shaft work within the system, substitute known quantities into the mechanical energy balance (or the Bernoulli equation if friction loss and shaft work can be neglected) and solve the equation for whichever variable is unknown. 

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