*Chapter 3. The energy equation and the continuity
equation for liquids.*

**Introduction**

Mechanical engineers are involved with fluids (ie liquids and gases) that flow through, over and round solid boundaries as in pipes, weirs and aerofoils. We have seen how the pressure varies through a liquid when it is at rest and now we want to find a way of deciding how a fluid might behave when it moves and, for preference, find a simple way to make calculations about flow to an accuracy that is sufficient for our needs as engineers.

We would not have to do much to get the water in a plastic cup to start to move, just make a hole in the cup and let gravity do the rest. In many cases the flow of liquid in particular is initiated and sustained by gravity but the flow of liquids and gases can be maintained by a pressure exerted on them in some way. We have to find a way to predict the behaviour of flowing fluids and once more the methods that we use have evolved as a result of contributions by others who have gone before us and which can now be seen to be coherent and mostly adequate for our task. The accumulation of methods that has evolved is based on two simple expressions, the continuity equation and the energy equation and we must come to understand both of these because they will appear in several forms throughout this study.

Let me make a start. I have now used the word energy twice without offering an explanation of it. I find it to be a difficult concept to pin down to a few words for use by "the man in the street", indeed I think that it is used in so many ways in the wake of global warming that everyone ought to stop and decide what we do mean when we use the word. The root of our problem is that energy is a word that has a specific meaning in physics and a much earlier meaning in common parlance, eg a person might be described as being energetic. It is the special meaning that energy has been given by physicists that interests us.

Physicists do not define energy
as such but give us a test to decide whether something "contains"
energy. The test for energy is that it must be possible to conceive of it being
stored by the lifting of a weight in a gravitational field. This leads us
directly back to the model of the gravitational field of the Earth referred to
in Chapter 2. There we said that the gravitational field could be modelled as
many concentric spherical shells each having the same potential all over it.
Why do we use the word potential? In order to answer this we have to
contemplate some practical tests. Suppose that you hold a mass of perhaps 2 kg
in your hand. If you hold it still, you have to exert an upward force of 2.*g* Newtons, ie 19.62 N. You could lift
this mass and, if you do, you do work because the force you exert has moved
through a distance. Where has this work gone? There is absolutely no evidence
in the mass for the work having been done. Your arm may be tired but that is
not relevant. The only idea that fits the facts is that work can be done
against the gravitational field. If it can be done in this way the next
question must be, can we have it back? The answer is yes. You can lower the
mass and work is done on you and this work is exactly equal to the work that
you did initially. It seems to be inescapable that work can be stored in a
gravitational field and that in order to store work in this way a mass must be
lifted[1].
It also seems that a mass in a gravitational field has the potential to do work
if it is lowered. Hence the idea of each surface of the model of gravity having
the same potential and of course the idea that the higher the level the greater
the potential.

This test is sometimes easy to perform but sometimes it is not so easy. Consider a litre of petrol. Everyone would say, in common parlance, that it contains energy but showing that it can be stored in the gravitational field is quite impossible. The energy in the petrol is stored in its chemical structure and can only be released by combustion. It appears as heat and we cannot store heat in the gravitational field and we know of no way to convert all of the heat energy released to work[2].. However by dint of other relationships that have been established we can decide how much work would be done if only we could make the conversion

It is now important to distinguish between energy and power. If something contains energy in a form that can be used to do work it can be made to deliver power. Power is another word that has be taken from long established common parlance and given a special meaning. It simply means the rate of doing work, ie the rate at which energy is used to do work.

With this preamble we can look at a fluid on the move but I shall have to build up the ideas slowly from special cases to get to the general.

**The concept of pressure
energy**

Figure 3-1 shows a liquid in a large container that has a horizontal tube of small diameter fitted to it with a close fitting piston in the tube. There will obviously be a pressure on the piston and therefore a force on it. If the piston is allowed to move out of the tube, work will be done by this force. Liquid will flow to fill the space created by the movement of the piston.

The
container is filled with liquid of density to a depth above the axis of the tube. This produces an
absolute pressure on the piston equal to where is the atmospheric pressure. If the piston
has area the force exerted on it by the liquid is and, if the piston is not to move, this must
be resisted by an equal external force acting in the opposite direction. If now
the piston is allowed to move and the movement is slow, the piston will move in
equilibrium and, as the pressures remain sensibly unchanged, the force on the
piston remains equal to and if the piston moves a distance ,
as shown in figure 3-2, the work done against the external force is *.* This could be written as and as where and is the volume swept out by the movement
of the piston. However the external force is partly provided by the atmospheric
pressure acting on the piston and, as the piston moves, work equal to .
will be done on the atmosphere which will be lifted by a minute amount. The net
work done is:-

This work has been done at the expense of the potential energy of the liquid as a whole because the mass centre of the liquid has moved downwards by a very small amount.

Now suppose that a plate could be inserted to block the tube at the original position of the face of the piston so separating the volume of the liquid in the tube from the rest. No further work can be extracted at this level because any further movement of the piston will cause the pressure to fall to zero instantly because the liquid is not able to expand.

Then you can see that the
liquid in the container can be restored to the original condition by the
addition of a volume of liquid of *V* to the surface. This liquid will have
potential energy relative to the piston equal to its weight times its height
above the piston that is which of course numerically equal to the net
work done on the piston .
It looks as though the work that was done on the piston has come from the
lowering of the water in the tank but more correctly from the lowering of just
a volume of liquid at the surface through the height .
This is an extraordinary result. The work has been extracted from the
gravitational field and the work has been transferred from the liquid at the
top through all the liquid between it and the piston just as surely as if a
lever had been used

The system of pipe, piston and cut-off plate described above, is the basis of the three-cylinder hydraulic motor. The essential mechanical arrangement of the motor is shown in figure 3-3. Three cylinders are arranged at 120° to each other and their pistons act through connecting rods on to a common crankpin. When the crankshaft rotates each piston, in sequence, will move the stroke length and then back again. Each cylinder is connected to a valve that is driven by the crankshaft so that water from a high-pressure source can be admitted to each cylinder in turn as it moves out of its cylinder and then cut off from the source at the end of the stroke. At this point the cylinder is opened to drain by the valve and the water is expelled during the return stroke. The use of three cylinders ensures that there is a torque on the crankshaft at every position of the crank and effectively the hydraulic motor is a device that can operate continuously with power being extracted at the crankshaft.

Figure 3-4 shows how the motor might be supplied with high-pressure water from a reservoir. I have chosen this because it is clearly derived from the system in figure 3-1. If there were to be no friction effects in the water, no mechanical friction and no waste of water caused by valve action the system could operate steadily seeming to convert all the potential energy supplied steadily at the free surface directly to energy at the rotating shaft. It is a continuously operating version of the system in figure 3-1. It is only possible because the water can flow and maintain the pressure at inlet to the motor.

Let me continue with this hydraulic motor. In reality a hydraulic motor will have both hydraulic and mechanical losses and it would not be unreasonable to want to measure its efficiency. As it might in fact be supplied with high-pressure water through a pipe from a pump working in conjunction with an accumulator, any loss in the supply system should not be attributed to the motor, and some method is required of assessing the efficiency of the motor independently of the supply system. We would be faced with a motor on the floor with a pipe feeding it with water under pressure and another pipe carrying water away to drain and a need to test the motor.

Figure 3-5 shows the usual way of thinking about such problems. The motor is represented diagrammatically by the rectangle and the inlet and the drain pipes are shown with steady flow through them and the crank shaft is shown transmitting power to some external load. A closed imaginary boundary is then drawn to cut the pipes and the crankshaft. This done, the axiomatic law of conservation of energy can be applied to enable us to say that, if the energy inside the boundary is not to change, the rate at which energy is being supplied to the system must equal the rate at which it is leaving the system. We know how to measure the output power at the shaft by measuring torque and angular velocity and all the friction losses will go to heat the water and be carried away with it. This leaves us needing a means of measuring the rates at which energy is entering and leaving with the water. As it turns out this requires the recognition of a new form of energy that is special to a flowing liquid (or a flowing gas).

We have already met potential energy in this text and you will no doubt have met kinetic energy. They are well-recognised forms of mechanical energy but they are forms of energy that are normally associated with the centre of mass of a solid. We must recognise one special feature of potential energy. There can be no absolute value of the potential energy possessed by a mass in a particular position in the gravitational field. We must reckon the potential energy relative to some arbitrary datum and we choose that datum to suit the application. So potential energy is the energy possessed by a solid in a gravitational field as a consequence of the height of its centre of mass above some datum. Kinetic energy is the energy possessed by a solid as a consequence of the velocity of its centre of mass relative to some frame of reference usually the Earth. There are no reasons for not using these concepts for a liquid.

If a quantity of liquid has a
mass and its centre of mass is at a height * *above
some arbitrary datum where potential energy is taken to be zero, it has
potential energy of relative to that datum. For convenience, in
calculations in connection with the flow of liquids, it is usual to express
mechanical energy in terms of energy/unit weight. Then the potential
energy/unit weight is *.*

If the same mass of liquid has a velocity of it has kinetic energy of and for this mass, the kinetic energy/unit weight is .

In the case of the hydraulic motor the water arrives and leaves at the same level and at the same speed so the output power is not extracted from either the kinetic energy or the potential energy of the water crossing the boundary.

Let us suppose that the steady
rate of flow of the water into and out of the motor is , (in
S.I.units this would be in m^{3}/s). The water entering the motor at
the boundary has a high absolute pressure and, every second, must do work equal to just as if it was working directly against a
piston. At exit the pressure of the water is atmospheric and, as it emerges, it
pushes the atmosphere out of the way doing work every second equal to to increase the potential energy of the
atmosphere. It follows that the net rate at which energy (power) is being given
to the motor by the water is .

Now is equal to where is the mass flow and is the volume/unit mass of the water. It follows that and this, following the way that potential energy and kinetic energies are expressed, suggests that every unit mass of the water carries with it at the entry energy equal to . This is only possible because some means is used to maintain the pressure at inlet and the water, by virtue of its liquid state, can flow. The product is called the pressure energy per unit mass and there is nothing in the argument above that precludes its use for a gas. Indeed, in dealing with the flow of gases pressure energy is always expressed in this form but for liquids it is usually rewritten as energy per unit weight and in terms of density instead of volume per unit mass. As the pressure energy per unit weight becomes [3].

**Pressure energy in a
liquid at rest**

We have in a roundabout way
come to recognise pressure energy but we have done it for a liquid that is
flowing steadily. Now we must decide whether the idea can be extended to liquid
that is at rest. Figure 3-6 shows a tank filled with liquid that is at
rest. An element of liquid having volume and having the shape of a thin disc forms
part of the free surface. The depth of liquid in the tank is and so the
potential energy of the element relative to the bottom of the tank is where *r* is the
density of the liquid.

The element could be moved very slowly to a new position at a depth of and, as the movement involves no net force outside the liquid, no work would be done during the movement. This means that the element has lost potential energy equal to without doing any work nor apparently gaining any other form of mechanical energy in exchange. As this statement is contrary to the law of the conservation of energy we must again face the need to look for, and try to recognise, a form of mechanical energy other than potential energy and kinetic energy. As the only observable change in the water in the element is the rise in its pressure, we must expect this new form of energy to be associated with pressure.

The absolute pressure at the
surface is the atmospheric pressure and so, for the element when it is on the
surface, the product of pressure and volume is *.* At depth the absolute pressure is and the product of pressure and volume is *.*
Clearly, for the element, the product of pressure and volume has increased by which is numerically equal to the loss of
potential energy. It is also equal to the mass of the element times the same
expression as we used to define pressure energy/unit mass. So it seems that the
concept of pressure energy, which has been devised to permit us to quantify the
energy entering or leaving a closed system, can also be used to quantify the
energy in store in a liquid at rest. However it is still only true because the
liquid is continuous and able to flow in the gravitational field.

When the element is at depth its potential energy relative to the datum is . Our interest then lies in the fact that the sum of this potential energy and the pressure energy possessed by the element is:-

and that this is the same
wherever the element may be in the tank. We can deduce that in a liquid at rest
the sum of the potential energy/unit weight and the pressure energy/unit weight
at every point is the same and equal to *H*.
These two forms of energy are interchangeable.

**The continuity equation**

A fluid, that is, a liquid or a gas, can flow steadily along a pipe and that pipe can be made up of lengths of pipe of different diameters. It would be no surprise to learn that the fluid does not flow with uniform velocity at any point and nor is it surprising that finding out how the velocity actually varies with, say, radius is not easy. For engineering purposes we would look for some way to avoid this complication. The fact is that the only constant for all the different pipes is the mass flow . Then, for any given point in any one of the pipes:-

where is the density of the fluid at the given point, is the area of cross-section of the pipe and is the mean velocity of the flow.

For a gas the mean velocity of flow and the density change along the pipe but, for most liquids the density does not change.

There is a consensus view that the best way to proceed for most purposes is to treat the flow as having physical properties that vary along the pipe but not across it. Then it is called one-dimensional flow and, having stated that the flow is to be treated as one-dimensional we change the average velocity to and write:-

for any fluid and:-

for a liquid where is the volume flow.

This is called the continuity equation.

**Energy exchanges in liquids
in motion.**

We have seen that in a liquid that is at rest, the sum of the potential energy/unit weight and the pressure energy/weight is uniform throughout the liquid. The only other known form of energy that is relevant to fluid flow is kinetic energy and it becomes evident that pressure energy and potential energy can also be exchanged for kinetic energy. Of course all three forms of mechanical energy can be exchanged for random energy stored in the internal structure of the liquid but that is not mechanical energy and is normally an unwanted exchange that we try to minimise.

This leads to the idea that, if the water could flow without loss of energy to friction, the sum of the potential energy/unit weight, the pressure energy/unit weight and the kinetic energy/unit weight at every point in the water would be same. If we call the sum of the three forms of energy the total energy/unit weight we could simply say that, if a liquid could flow without friction, the total energy/unit weight would be the same for every point in the liquid. Then, in symbols,:-

= a constant

This is attributed to Daniel Bernoulli.

However it is not in its most suitable form for use in engineering. We could write that when the Bernoulli equation is applied to any two points, 1 and 2, in a continuum of liquid:-

For engineering we must take into account that there will be a loss of energy when the liquid moves from 1 to 2. A more general statement is that, for steady, one-dimensional flow, the total energy/unit weight at point 1 in the flow equals the total energy/unit weight at point 2 down stream of 1 plus the energy/unit weight that has been lost between 1 and 2. This can be written in symbols:-

+ the loss between 1 and 2.

**This is probably the most widely used equation in the application of
physics to the flow of liquids. It is the energy equation for steady flow of a
liquid in the form that is most suitable for use with liquids.**

This equation has been used to construct an empirical science. It is difficult to pin down the precise meaning of those two words because no science can exist without reference to the physical world. We need to know densities of ordinary things, electrical properties of common and uncommon materials, the speeds of light and sound and so on. Data on physical properties fill scores of books and it has all been gathered by experiment. Yet somehow this data is respectable but, it is my experience that, if you experiment to find out how the friction loss in a pipe varies with flow, it becomes “applied” and is no longer respectable and labelled empirical. Froude, with his timeless work on ship resistance, did not qualify for elevation to the peerage like other equally gifted men of his era. We must soldier on knowing that we are dealing with an empirical science and no amount of fudging will alter the fact. (We can console ourselves with the fact that whilst “slaves do arithmetic” some slaves like Brunel built Victorian England and the Wright brothers learnt to fly by using their intellect and gathering data by experiment.)

The energy equation is the
cornerstone and, before we go on, it is pertinent to think more deeply about
it. I have chosen to argue for this equation in words from starting premises
and I am certain that it came into existence in this way. It was not the work
of one man nor did it appear at one moment of time. Had it not turned out to be
useful as the basis of many of the methods used in engineering to deal with
common applications it would have disappeared. Its validity lies not in the
supposedly rigorous mathematical “proofs” of the equation that have been
constructed since but in the fact that it works with lots of other useful
expressions. It is important to decide just why it works. The equation involves
just three terms and each of them has the same dimensions and each term reduces
to a length. Then each term contains one important relevant dimension that can
be measured, the elevation, the pressure and the average velocity. If we go on
to apply the energy equation to things to mechanical devices that we might wish
to make we can deduce more expressions that involve measurable quantities on
which to base coefficients that are independent of the system of units and
capable of being determined by experiment. This gives us a foundation for an
empirical science. Such a science, just like physical science generates a mass
of data and that produces problem of storage and retrieval that shapes the way
that we create our science. We shall see that the energy equation lends itself
to very successful storage and retrieval systems. Engineers should get on and
use it and concentrate on *how* it is
used for the job in hand rather than trying to refine it.

** **

**The concept of energy
head.**

The unit of energy is the same as that of work and, in the S.I. system, it is the Joule. One Joule is equal to one Newton metre that is the work done when a force of one Newton moves through one metre. Each term of the energy equation is in energy/unit weight so they all have units of Joules/Newton. However these units can also be expressed as Newton metres/Newton which reduces to metres. This should not be unexpected as the potential energy/unit weight is in metres.

There are many systems in engineering where the flow starts from a free surface and the energy available in the system depends on the elevation of the free surface above some relevant datum level. This elevation is called the head and is quite clearly equal to the potential energy/unit weight. As potential energy/unit weight can be exchanged for kinetic energy/unit weight or for pressure energy/unit weight it is just a short step to talking of kinetic head and pressure head. The concept of head is convenient because it is shorter to write. Unhappily, when calculations of power are involved, it is easy to forget that head is really Joules/Newton and when I make such calculations I revert to the basic units rather than metres of head.

[1] This is the underlying principle of the hydroelectric power station. The sun operating through the atmosphere lifts water vapour which may fall as rain on high ground. There it can be stored and, under controlled conditions, the energy stored in the water high up in the gravitational field can be extracted by the turbines to drive alternators and produce power.

[2] Politicians, and others who should know better, make silly mistakes because they do not understand this problem.

[3] This system of a reservoir and a motor has an exact parallel in the steam engine. There the reservoir is replaced by a steadily operating boiler that produces a steady flow of steam under pressure to supply the “engine” that could be exactly the same as the hydraulic motor. The steam pushes on the pistons just as the water might. This is the basis of most model steam plant but in locomotives and marine steam engines the steam is admitted for only a part of the stroke and expands doing more work for the rest of the stroke.

In Cornwall I saw a water wheel that was arranged to drive a single acting water pump. The pump was about 40 feet above the wheel and the two were joined with a wire rope that was about 200 feet long. The wheel operated through a crank and had a counter balance that was raised during the suction stroke of the pump and the energy so stored was used with the energy coming from the wheel to make the delivery stroke. Judging by the deep groove in the stone surround of a window through which the rope operated it had run for a very long time. That rope was doing exactly what the water does except that it was pulling and not pushing.