Bernoulli equation pump work

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) ... 

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 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... 

When a pump is in the control volume, hp = (ftshaft — ftFriction)p is often used where ftp is the actual head rise across the pump an is equal to the difference between the shaft work into the pump and the head loss within the pump. Notice that the ftpriction used for the turbine and the pump is the head loss within that unit only. When hs is used in the extended Bernoulli equation, ftpriction involves all losses including those within the turbine, pump or compressor. When /it or ftp is used for ftshaft, then ftpriction includes all losses except those associated with the turbine or pump flows. 

Also, the usefulness of the corrected Bernoulli equation in solving problems of flow of incompressible fluids is enhanced if provision is made in the equation for the work done on the fluid by a pump. 

If we write jthe head form of Bernoulli s equation, Eq. 5.11, between the free surface of the fluid (point 1) and the inside of the pump cylinder, there is no pump work oyer this section so... 

For incompressible fluids without friction or pump work at points upstream (1) and downstream (2) on a streamline in the flow, the Bernoulli equation becomes Eq. (9), where zi, Z2 are the levels at the two points, pi, pi are the pressures at the... 

Applying Bernoulli s equation between the free surface in tank A (point 1) and the free surface in tank B (point 2), we see that the velocities are negligible. Since there is no change in elevation and no pump or compressor work, we have... 

First we apply Bernoulli s equation from some upstream point in the canal to some downstream point in the canal. Since both points are open to the atmosphere, the pressures are the same. For steady flow of a constant-density fluid in a canal of constant cross-sectional area, the velocities at the two points are the same. There is no pump or turbine work. Therefore, the remaining terms are... 

If we write Bernoulli s equation (Eq. 5.7) from the inlet of this pump to the outlet and solve for the work input to the pump, we find... 

Now we apply Bernoulli s equation from the tip of the blades (point 2) to the outlet pipe (point 3). Again the change in elevation is negligible, and we neglect friction. The pump does no work on the fluid after it leaves the tip of the blades, so dW Jdm is zero. The outlet velocity is small and may be neglected. Thus... 

In defining the efficiency of a pump (Sec. 9.1), we compared the useful work to the total work. For an incompressible fluid, this is done most easily by means of Bernoulli s equation, which is restricted to constant-density fluids. The definition for a pump could be restated as... 

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... 

This is an example of Bernoulli s equation in action. A steam vacuum ejector (jet) works in the same way. Centrifugal pumps and centrifugal compressors also work by converting velocity to pressure. Steam turbines convert the steam s pressure and enthalpy to velocity, and then the high velocity steam is converted into work, or electricity. The pressure drop we measure across a flow orifice plate is caused by the increase of the kinetic energy of the flowing fluid as it rushes (or accelerates) through the hole in the orifice plate. 

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