Bernoulli s equation

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

Bernoulli s equation (Equation 2-53), which accounis for static and dynamic pressure losses (due to changes in velocity), but does not account for frictional pressure losses, energ losses due to heat transfer, or work done in an engine. 

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... 

Equation 2.43 is known as Bernoulli s equation, which relates the pressure at a point in the fluid to its position and velocity. Each term in equation 2.43 represents energy per unit mass of fluid. Thus, if all the fluid is moving with a velocity u, the total energy per unit mass ijf is given by ... 

Bernoulli s equation on a center streamline ahead of and behind the flame and the momentum flux conservation across the flame front however, the steady-state, backpressure drive theory [29] used only the momentum flux balance across the flame front. These resulted in the -v/2 difference between Equation 4.2.10 and the first term of Equation 4.2.7. 

Thus, it should be noted that the flame propagation in combustible vortex rings is not steady, but "quasi-steady" in the strict sense of the word. This may explain why prediction 9, based on the momentum flux conservation can better describe the flame speed for large values of Vg than prediction 4, which adopts the Bernoulli s equation on the axis of rotation. 

The instantaneous velocity is then used to estimate the instantaneous local static pressure using Bernoulli s equation of the following form ... 

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... 

If the friction loss is negligible, the energy balance (Bernoulli s equation) becomes... 

The relationship between flow rate, pressure drop, and pipe diameter for water flowing at 60°F in Schedule 40 horizontal pipe is tabulated in Appendix G over a range of pipe velocities that cover the most likely conditions. For this special case, no iteration or other calculation procedures are required for any of the unknown driving force, unknown flow rate, or unknown diameter problems (although interpolation in the table is usually necessary). Note that the friction loss is tabulated in this table as pressure drop (in psi) per 100 ft of pipe, which is equivalent to 100pef/144L in Bernoulli s equation, where p is in lbm/ft3, ef is in ft lbf/lbm, and L is in ft. 

Evaluate the kinetic energy correction factor a in Bernoulli s equation for turbulent flow assuming that the 1/7 power law velocity profile [Eq. (6-36)] is valid. Repeat this for laminar flow of a Newtonian fluid in a tube, for which the velocity profile is parabolic. 

Also, the total driving force in a branch between any two nodes i and j is determined by Bernoulli s equation [Eq. (7-45)] as applied to this branch. If the driving force is expressed as the total head loss between nodes (where hi = i/pg), then... 

The head at both the entrance to the header (/q = 230.8 ft) and the exit from the branches (/q = 0) is known. If the head at node 2 were known, Bernoulli s equation [Eq. (7-67)] could be used to calculate the flow rate from 1 to 2 (g12) and the flow rate from 2 to 5 (g25) By continuity, the flow rate from 2 to 3 must be the difference between these (Q23 = Qn Q25)-This flow rate is then used in Eq. (7-67) to determine the total head at node 3... 

Bernoulli s equation applied between the suction and the discharge of a... 

The suction pressure Ps is determined by applying the Bernoulli equation to the suction line upstream of the pump. For example, if the pressure at the entrance to the upstream suction line is P1 the maximum distance above this point that the pump can be located without cavitating (i.e., the maximum suction lift) is determined by Bernoulli s equation from Px to Ps ... 

Bernoulli s equation applied across the valve relates the pressure drop and flow rate in terms of the valve loss coefficient. This equation can be rearranged to give the flow rate as follows ... 

Equation 1.13 is simply an energy balance written for convenience in terms of length, ie heads. The various forms of the energy balance, equations 1.10 to 1.13, are often called Bernoulli s equation but some people reserve this name for the case where the right hand side is zero, ie when there is no friction and no pump, and call the forms of the equation including the work terms the extended or engineering Bernoulli equation. 

The equations derived are valid for a particular element of fluid or, the conditions being steady, for any succession of elements flowing along the same streamline. Consequendy, Bernoulli s equation allows changes along a streamline to be calculated it does not determine how conditions, such as the pressure, vary in other directions. 

As an example of a simple application of Bernoulli s equation, consider the case of steady, fully developed flow of a liquid (incompressible) through an inclined pipe of constant diameter with no pump in the section considered. Bernoulli s equation for the section between planes 1 and 2 shown in Figure 1.5 can be written as... 

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. 

Applications of Bernoulli s equation are usually straightforward. Often there is a choice of the locations 1 and 2 between which the calculation is made it is important to choose these locations carefully. All conditions must be known at each location. The appropriate choice can sometimes make the calculation very simple. A rather extreme case is discussed in Example 1.1. 

An alternative is to choose locations 1 and 2 as shown. These points are in the bulk of the liquid where the liquid s speed is negligibly small. Applying Bernoulli s equation between points 1 and 2 gives the pump head as... 

It is appropriate here to define some pressure terms. Consider Bernoulli s equation for frictionless flow with no pump in the section ... 

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