Bernoulli flow equation

As the potential energy term has an essential meaning in hydromechanics, the static head is selected as a comparison quantity. When the energy equation (4.32) is divided by g and integrated, it gives the Bernoulli flow tube equation... 

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

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... 

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. 

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 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 ... 

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. 

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

If the velocity had the uniform value u, the momentum flow rate would be mfpu2. Thus for laminar flow of a Newtonian fluid in a pipe the momentum flow rate is greater by a factor of 4/3 than it would be if the same fluid with the same mass flow rate had a uniform velocity. This difference is analogous to the different values of a in Bernoulli s equation (equation 1.14). 

Bernoulli s equation for this horizontal, turbulent flow is... 

Section 1.5). In flowing round the sphere, the fluid has to accelerate and therefore, by Bernoulli s equation, the pressure falls towards the midpoint of the sphere s surface. 

In practice the resistance of the exit pipe of the tank shown in Figure 10.1 will be sufficiendy large that the flow rate will be relatively low and consequendy conditions in the tank, in particular the fluid head causing the flow, will change only slowly. In these circumstances the emptying operation can be treated as quasi-steady. In view of this, Bernoulli s equation, which is valid only for steady flow, may be used. 

This is a statement of Bernoulli s theorem the quantity v2l2+Plp+gh is constant throughout the fluid for steady, irrotational flow. Equation A.33 is the same as equation 1.11. It will be recalled that, for rotational flow with friction, the engineering form of Bernoulli s equation applies only along a streamline and allowance must be made for frictional losses. 

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. 

Equation (5-14) is combined with Bernoulli s equation. Assuming flow on a horizontal axis and using a coefficient of discharge to account for friction at the orifice, the mass flow rate of an ideal gas through a thin hole in the containment wall is ... 

Proportional Element First, consider the outflow through the exit valve on the tank. If the flow through the line is turbulent, then Bernoulli s equation can be used to relate the flow rate through the valve to the pressure drop across the valve as... 

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 ... 

Bernoulli s Equation gives us the total energy in a flowing fluid and sums the energy due to fluid pressure, kinetic energy and potential energy. 

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 ... 

To obtain an equation for calculating the work of conpression, first apply Bernoulli s equation, Equation 5.1, across the compressor. The first term, the kinetic energy term, is small compared to the other terms in the balance. The second term is the change in potential energy, and it is also small. The last two terms are the work done by the system and the friction loss. First, we consider frictionless flow. Thus, the compressor work. 

The most important relationship in designing flow systems is the macroscopic mechanical-energy balance, or Bernoulli s equation. Not only is it required for calculating the pump work, but it is also used to derive formulas for sizing valves and flow meters. Bird, et al. [6] derived this equation by integrating the microscopic mechanical-energy balance over the volume of the system. The balance is given by... 

The problem that we must consider next is to relate the pressure drop across the valve to flow rate and valve size. After applying Bernoulli s equation. Equation 8.2, across the valve we obtain, for an incompressible fluid. 

To size a variable-head meter, we must calculate the orifice, venturi throat or nozzle diameter. Using Bernoulli s equation we can derive a relationship between the flow rate, the pressure drop across the meter, and the orifice diameter. 

To size a rotameter requires calculating the volumetric flow rate of a standard fluid at standard conditions. Most manufacturers calibrate rotameters using a stainless-steel float and water at a standard tenperature for liquids and air at a standard tenperature and pressure for gases. For other fluids, float materials, and operating conditions, the flow rate must be converted to an equivalent flow rate of water or air. To derive a formula for making this conversion, Bernoulli s equation is applied across the float shown in Figure 8.15 to give Equation 8.9. 

Apply Bernoulli s equation over the whole flow system to develop an expression for the punp head. After rearranging Equation 8.2, to obtain the suction and discharge heads we find that... 

When velocity components at the inlet boundary are not known, it is necessary to specify the pressure at the inlet boundary. Simplified equations can then be used (such as Bernoulli s equation) to calculate velocity at the inlet boundary (Fig. 2.3). For incompressible flow, if the specified total pressure at the inlet boundary is pq, the... 

---