 In this module, we are going to discuss positive displacement machines. As I mentioned during the introduction, positive displacement machines are not turbo machines. Nonetheless, they are used quite extensively in many real-life applications. So for the sake of completeness, we will discuss positive displacement machines in the series of lectures, but not in a very detailed manner, somewhat brief manner. So the general mode of operation in a possible displacement machine is that a chunk of fluid is taken into a space in the rotor and this chunk of fluid isn't physically moved or displaced or carried by the rotor from the inlet to the exit. Hence the name positive, hence the name displacement machine. During this displacement, the space occupied by the fluid decreases. The rotor is designed in such a way that the space occupied by the fluid decreases causing the chunk of fluid to be physically squeezed or compressed thereby increasing its pressure. Hence the term positive. So positive displacement refers to the fact that the pressure increases while the chunk of fluid is physically displaced from the inlet to the exit by the rotor. So consequently, since the chunk of fluid is taken, compressed and then sent out, the flow rate from the positive displacement machines generally tends to be oscillatory. But the oscillations may be smoothed out using reservoir of suitable size on the exit side. That is usually done. Now, as already mentioned, positive displacement machines are not turbo machines since the pressure rise of the fluid is not due to rotodynamics but rather by physical compression of the fluid. Positive displacement machines are widely used for pumping liquids although there are designs that are available for compressing gases also. We will take a look at that as well. Here is an example of a positive displacement machine. This is a gear pump. And as the name suggests, this consists of two gears here. Four gears. They can also be helical gear or any other type of gear. Here we are looking at a design that utilizes a pair of four gears. The gear on the top is connected to a source of power. So that is the driving gear. And the one in the bottom is actually a driven or idler gear. This is actually idle and this is the driven one. So what normally happens is the following. If you take a cross-sectional view of this pump, you can see the driving gear on the top which is supplied with power and the idler or driven gear on the bottom. So when the gears come up, so the direction of rotation is in the clockwise direction here. So when a pair of feet come up to this point, the gears are completely unmatched. And so the volume available between the gears is a maximum. So a certain chunk of fluid is then taken into this volume. And as the gear moves up like this and as this moves down like this, half of the chunk of fluid is carried along in the passage on the top and the remaining half is carried along in the passage below. So you can see why we call it a displacement pump because this fluid is physically carried by the gear on the top and the gear on the bottom from the inner to the outlet side. Now as this pair of feet approach the outlet side, the gear begins to mesh. And so the fluid that is contained in the space between the teeth is subjected to an enormous increase in pressure because it is literally being squeezed. And the fluid is then sent out through the outlet side. So when the gear is unmatched, a certain chunk of fluid is taken in. This chunk is then transported from the inner to the outside outlet on the top and the bottom. And the gears then begin to mesh thereby reducing the volume of space that is available between the teeth and consequently increasing the pressure of the liquid, which is then sent out through the outlet. Now several observations can be made regarding the operating characteristic or H versus Q, flow rate characteristic of such a device. The first point is this. If no power is supplied, then this also becomes an idler gear. So as the gears unmatched, certain amount of fluid is taken here and this amount of fluid is moved from the inlet to the outlet side and it then goes out. So whatever is taken in is sent out and there is no increase in pressure of the fluid because no power is supplied. So the volume of fluid that is transported or displaced from the inlet to the outlet in this case is usually called a displacement volume. So displacement volume is the volume that is moved from inlet to outlet when the pressure rises to zero. Now the displacement volume is fixed because the geometry of the gears is fixed and so the amount of fluid that can be moved per revolution from the inlet to the per revolution of the gear from inlet to the outlet is fixed. Now when we start supplying power to the driving gear, the only change that happens is that the fluid gets squeezed and its pressure begins to increase. As we increase the pressure, the pressure increases even more. But notice that since the geometry is fixed, the amount of fluid that is delivered from the inlet to the outlet side more or less remains the same even at higher pressure. The only change that we will notice is that the pressure on the outlet side is much higher than the pressure on the inlet side, which suggests that the H versus Q characteristic of such a pump will more or less be a vertical line. There is no change in flow rate but the pressure increases with increasing power. The second observation is that since the displacement volume which is the amount of fluid that is moved from inlet to outlet per revolution of the gear is fixed because the geometry of the gear is fixed. The only way to change the flow rate from such a pump is to either increase or decrease the revolutions per minute of the driving gear. So if we increase the revolutions per minute, then the fluid that is delivered per minute will be more although the fluid delivered per revolution is the same because we have more revolutions per minute, more fluid will be delivered on the outlet side and vice versa that may reduce the RPM. So that is the only way to change the flow rate of such a positive displacement. So with this in mind, let us now look at the actual characteristic of a gear pump. So here we have characteristics of a gear pump which is reproduced from the fluid mechanics textbook by Fox and McDonald and as we had already said, it can be seen that for a given RPM, the characteristic H versus Q or pressure versus Q is almost a vertical line with a slight inclination to the left. So there is a slight inclination to the left from bottom to top and the only way to increase the flow rate is to increase the RPM of the device, RPM of the driving gear. What is that? The displacement volume is actually given to be 97 milliliters per revolution but this may also be obtained by extrapolating this characteristic all the way down to H equal to zero and then calculating the flow rate. Remember, we already said that the displacement volume is the amount of fluid that is delivered from inlet to outlet with no power is supplied and hence no head is developed. So when we extrapolate this all the way down to the x-axis and determine the flow rate there, that will be the displacement volume of this particular design. Now two sets of curves are plotted here. The dashed one is the so-called volumetric efficiency and you can see that for a given RPM the volumetric efficiency decreases as the head increases and the volumetric efficiency is nothing but the actual volume delivered divided by the pump displacement. Since the characteristic is slightly inclined to the left, the actual volume delivered will be somewhat less than the displacement volume in an actual application and that ratio is called the volumetric efficiency of the pump. If the characteristic that will be perfectly vertical then the volumetric efficiency will be one. Since it is slightly inclined to the left, it actually decreases from one to reasonably high value. These are not very low. Now the solid line indicates the overall efficiency of the machine and this is defined as the hydraulic power which is nothing but rho q g times h divided by the input power that is provided. So and as can be seen here the overall efficiency keeps increasing for a given RPM. The overall efficiency keeps increasing up to a certain pressure beyond which it begins to decrease just like what we saw for centrifugal pump also. Next let us discuss the effect of fluid viscosity on the performance of the gear pump. In other words what we would like to know is how these characteristics change if instead of pumping liquid like water we start pumping a liquid like a 10W oil or similar heavy oil using the same gear pump. So if it turns out that in the case of a gear pump the characteristics are shown here so it turns out that in the case of a gear pump there is a very little change in the performance characteristic as a result of a change in viscosity. And this is because of the manner in which the fluid is actually transported in the gear pump. So if we go back and take a look at the nature of operation you can see that the fluid is literally transported in the space between the gear from the inlet to the outlet. So whether we carry fluid of higher viscosity or lower viscosity makes a little or no different to the transport mechanism or increasing pressure of this fluid as we go from gear to gear. The only effect though that we notice here is that as a result of higher viscosity you can see that the volumetric efficiency goes up slightly which is sort of counterintuitive one would expect the performance to degrade usually with an increase in viscosity but here the volumetric efficiency increases slightly and again that is inherent in the nature in which the fluid is transported. So in an actual case the volumetric efficiency is not 100% due to leakage of flow in the gap between the crown of the gear team and inside the surface of the casing. So the oil keeps leaking back from the high pressure side to the low pressure side which then reduces the volumetric efficiency. Now whether increasing the viscosity this leakage flow through such a small gap is reduced significantly resulting in an increase in the volumetric efficiency at higher viscosity. So apart from this small change that we see here the viscosity of the fluid plays little if no role on the performance characteristic of the air pump. Now in contrast in the case of centrifugal pump or a rotodynamic pump the effect of viscosity on the performance is quite dramatic because with increased viscosity there is increased shear stress between the fluid and the solid surfaces such as blade surface and impeller surface and so more of the input power has to be used to overcome this frictional shear stress between the fluid and the solid surfaces and so lesser amount of the input power goes towards increasing the head of the fluid that is being pumped. So we consequently see for a given power and rpm an increase in viscosity of the fluid that considerably degrades the performance of the machine. The head drops, head develops, decreases dramatically. The flow rate also decreases dramatically in contrast to the case of a positive displacement pump in which the change in viscosity increasing viscosity or decrease has a little or no effect on the overall performance and as we said this is due to the manner in which the fluid is actually transported and the head increases achieve in a positive displacement pump. Okay let us now work out an example involving the gear pump. The problem statement reads like this a positive displacement pump with characteristics as shown before is required to deliver a fluid at a 10 ampere while running at 2000 rpm determine the volume flow rate volumetric efficiency and the input power. So it is required to deliver a fluid at 10 ampere so we are given the characteristic so 2000 rpm the characteristic looks like this. So 10 ampere assuming that this is the pressure increase across the rotor corresponding to delta P of 10 ampere if we go into this characteristic we can actually work out the volume flow rate by going in like this up to the characteristic and then dropping down to determine the volume flow rate. If you do that you get the volume flow rate to be approximately 178 litre per minute. So the volumetric efficiency is the actual volume delivered divided by the displacement volume. The displacement volume is given to be 97 millilitre per revolution. So if we employ the proper conversion the volume flow rate or volumetric efficiency puts out to be 0.9175 which actually corresponding to a pressure of 10 ampere and 2000 rpm and this appears to be consistent with the figures that are shown here of the values that are shown here. Now we are asked to calculate the input power. The hydraulic power may be calculated as rho qgh and notice that the product of rho gh is nothing but delta P so this may be written as 2 times delta P and it works out to be 29.7 kilowatts and if we go back into the characteristic corresponding to a delta P of 10 ampere and speed of 2000 rpm the efficiency in this case is roughly about 84 or so. So taking the efficiency to be 84 or 0.84 we can evaluate the input power of 29.7 divided by 0.84 which is 35.37 kilowatts. So this concludes our discussion on positive displacement pump for pumping liquids. As I mentioned earlier it is also possible to use a positive displacement pump for pumping gases and that would look something like this. So this is a reciprocating compressor which is used for compressing refrigerant. In the early refrigerated designs domestic refrigerators reciprocating pump was quite widely used. These reciprocating pumps can also be used for compressing air and other gases as well. So here the air is taken in on the intake straw through an intake valve so the piston moves from the top red center to the bottom red center so the displacement volume is known. So a certain amount of air is taken in so this is the displacement volume that is taken in. The valve the intake valve is then closed and the piston moves up compressing the gas that is here in the displacement volume. Once the pressure reaches a certain value the exhaust valve is pushed open by this gas and the gases then flow off. Once again it can be understood that the flow rate from such a reciprocating compressor will be oscillatory. So usually high pressure reservoir is used on this side to smooth out the oscillations and the actual flow is taken from the reservoir to the desired application. This concludes our discussion on positive displacement machine.