The Bernoulli equation

1. Work and energy

We know that if we drop a ball it accelerates downward with an acceleration (neglecting the frictional resistance due to air). We can calculate the speed of the ball after falling a distance h by the formula (a = g and s = h). The equation could be applied to a falling droplet of water as the same laws of motion apply

A more general approach to obtaining the parameters of motion (of both solids and fluids) is to apply the principle of conservation of energy. When friction is negligible the

sum of kinetic energy and gravitational potential energy is constant.

Kinetic energy

Gravitational potential energy

(m is the mass, v is the velocity and h is the height above the datum).

To apply this to a falling droplet we have an initial velocity of zero, and it falls through a height of h.

Initial kinetic energy

Initial potential energy

Final kinetic energy

Final potential energy

We know that

kinetic energy + potential energy = constant

so

Initial kinetic energy + Initial potential energy = Final kinetic energy + Final potential energy

so


Although this is applied to a drop of liquid, a similar method can be applied to a continuous jet of liquid.

The Trajectory of a jet of water

We can consider the situation as in the figure above - a continuous jet of water coming from a pipe with velocity . One particle of the liquid with mass travels with the jet and falls from height to . The velocity also changes from to . The jet is travelling in air where the pressure is everywhere atmospheric so there is no force due to pressure acting on the fluid. The only force which is acting is that due to gravity. The sum of the kinetic and potential energies remains constant (as we neglect energy losses due to friction) so

As is constant this becomes

This will give a reasonably accurate result as long as the weight of the jet is large compared to the frictional forces. It is only applicable while the jet is whole - before it breaks up into droplets.

Flow from a reservoir

We can use a very similar application of the energy conservation concept to determine the velocity of flow along a pipe from a reservoir. Consider the 'idealised reservoir' in the figure below.

An idealised reservoir

The level of the water in the reservoir is . Considering the energy situation - there is no movement of water so kinetic energy is zero but the gravitational potential energy is .

If a pipe is attached at the bottom water flows along this pipe out of the tank to a level . A mass has flowed from the top of the reservoir to the nozzle and it has gained a velocity . The kinetic energy is now and the potential energy . Summarising

Initial kinetic energy

Initial potential energy

Final kinetic energy

Final potential energy

We know that

kinetic energy + potential energy = constant

so

so

We now have a expression for the velocity of the water as it flows from of a pipe nozzle at a height below the surface of the reservoir. (Neglecting friction losses in the pipe and the nozzle).

Now apply this to this example: A reservoir of water has the surface at 310m above the outlet nozzle of a pipe with diameter 15mm. What is the a) velocity, b) the discharge out of the nozzle and c) mass flow rate. (Neglect all friction in the nozzle and the pipe).

Volume flow rate is equal to the area of the nozzle multiplied by the velocity

The density of water is so the mass flow rate is

In the above examples the resultant pressure force was always zero as the pressure surrounding the fluid was the everywhere the same - atmospheric. If the pressures had been different there would have been an extra force acting and we would have to take into account the work done by this force when calculating the final velocity.

We have already seen in the hydrostatics section an example of pressure difference where the velocities are zero.

The pipe is filled with stationary fluid of density has pressures and at levels and respectively. What is the pressure difference in terms of these levels?

or

This applies when the pressure varies but the fluid is stationary.

Compare this to the equation derived for a moving fluid but constant pressure:

You can see that these are similar form. What would happen if both pressure and velocity varies?

2. Bernoulli's Equation

Bernoulli's equation is one of the most important/useful equations in fluid mechanics. It may be written,

We see that from applying equal pressure or zero velocities we get the two equations from the section above. They are both just special cases of Bernoulli's equation.

Bernoulli's equation has some restrictions in its applicability, they are:

All these conditions are impossible to satisfy at any instant in time! Fortunately for many real situations where the conditions are approximately satisfied, the equation gives very good results.

The derivation of Bernoulli's Equation:

An element of fluid, as that in the figure above, has potential energy due to its height z above a datum and kinetic energy due to its velocity u. If the element has weight mg then

potential energy =

potential energy per unit weight =

kinetic energy =

kinetic energy per unit weight =

At any cross-section the pressure generates a force, the fluid will flow, moving the cross-section, so work will be done. If the pressure at cross section AB is p and the area of the cross-section is a then

force on AB =

when the mass mg of fluid has passed AB, cross-section AB will have moved to A'B'

volume passing AB =

therefore

distance AA' =

work done = force distance AA'

=

work done per unit weight =

This term is know as the pressure energy of the flowing stream.

Summing all of these energy terms gives

or

As all of these elements of the equation have units of length, they are often referred to as the following:

pressure head =

velocity head =

potential head =

total head =

By the principle of conservation of energy the total energy in the system does not change, Thus the total head does not change. So the Bernoulli equation can be written

As stated above, the Bernoulli equation applies to conditions along a streamline. We can apply it between two points, 1 and 2, on the streamline in the figure below

Two points joined by a streamline

total energy per unit weight at 1 = total energy per unit weight at 2

or

total head at 1 = total head at 2

or

This equation assumes no energy losses (e.g. from friction) or energy gains (e.g. from a pump) along the streamline. It can be expanded to include these simply, by adding the appropriate energy terms:

3. An example of the use of the Bernoulli equation.

When the Bernoulli equation is combined with the continuity equation the two can be used to find velocities and pressures at points in the flow connected by a streamline.

Here is an example of using the Bernoulli equation to determine pressure and velocity at within a contracting and expanding pipe.

A contracting expanding pipe

A fluid of constant density = 960 is flowing steadily through the above tube. The diameters at the sections are . The gauge pressure at 1 is and the velocity here is . We want to know the gauge pressure at section 2.

We shall of course use the Bernoulli equation to do this and we apply it along a streamline joining section 1 with section 2.

The tube is horizontal, with z1 = z2 so Bernoulli gives us the following equation for pressure at section 2:

But we do not know the value of . We can calculate this from the continuity equation: Discharge into the tube is equal to the discharge out i.e.

So we can now calculate the pressure at section 2

Notice how the velocity has increased while the pressure has decreased. The phenomenon - that pressure decreases as velocity increases - sometimes comes in very useful in engineering. (It is on this principle that carburettor in many car engines work - pressure reduces in a contraction allowing a small amount of fuel to enter).

Here we have used both the Bernoulli equation and the Continuity principle together to solve the problem. Use of this combination is very common. We will be seeing this again frequently throughout the rest of the course.

4. Pressure Head, Velocity Head, Potential Head and Total Head.

By looking again at the example of the reservoir with which feeds a pipe we will see how these different heads relate to each other.

Consider the reservoir below feeding a pipe which changes diameter and rises (in reality it may have to pass over a hill) before falling to its final level.

Reservoir feeding a pipe

To analyses the flow in the pipe we apply the Bernoulli equation along a streamline from point 1 on the surface of the reservoir to point 2 at the outlet nozzle of the pipe. And we know that the total energy per unit weight or the total head does not change - it is constant - along a streamline. But what is this value of this constant? We have the Bernoulli equation

We can calculate the total head, H, at the reservoir, as this is atmospheric and atmospheric gauge pressure is zero, the surface is moving very slowly compared to that in the pipe so , so all we are left with is the elevation of the reservoir.

A useful method of analysing the flow is to show the pressures graphically on the same diagram as the pipe and reservoir. In the figure above the total head line is shown. If we attached piezometers at points along the pipe, what would be their levels when the pipe nozzle was closed? (Piezometers, as you will remember, are simply open ended vertical tubes filled with the same liquid whose pressure they are measuring).

Piezometer levels with zero velocity

As you can see in the above figure, with zero velocity all of the levels in the piezometers are equal and the same as the total head line. At each point on the line, when u = 0

The level in the piezometer is the pressure head and its value is given by .

What would happen to the levels in the piezometers (pressure heads) if the if water was flowing with velocity = u? We know from earlier examples that as velocity increases so pressure falls …

Piezometer levels when fluid is flowing

We see in this figure that the levels have reduced by an amount equal to the velocity head, . Now as the pipe is of constant diameter we know that the velocity is constant along the pipe so the velocity head is constant and represented graphically by the horizontal line shown. (this line is known as the hydraulic grade line).

What would happen if the pipe were not of constant diameter? Look at the figure below where the pipe from the example above is replaced be a pipe of three sections with the middle section of larger diameter

Piezometer levels and velocity heads with fluid flowing in varying diameter pipes

The velocity head at each point is now different. This is because the velocity is different at each point. By considering continuity we know that the velocity is different because the diameter of the pipe is different. Which pipe has the greatest diameter?

Pipe 2, because the velocity, and hence the velocity head, is the smallest.

This graphical representation has the advantage that we can see at a glance the pressures in the system. For example, where along the whole line is the lowest pressure head? It is where the hydraulic grade line is nearest to the pipe elevation i.e. at the highest point of the pipe.

Losses due to friction.

In a real pipe line there are energy losses due to friction - these must be taken into account as they can be very significant. How would the pressure and hydraulic grade lines change with friction? Going back to the constant diameter pipe, we would have a pressure situation like this shown below

Hydraulic Grade line and Total head lines for a constant diameter pipe with friction

How can the total head be changing? We have said that the total head - or total energy per unit weight - is constant. We are considering energy conservation, so if we allow for an amount of energy to be lost due to friction the total head will change. We have seen the equation for this before. But here it is again with the energy loss due to friction written as a head and given the symbol . This is often know as the head loss due to friction.