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 m3/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 [1].

[1] 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.