 We discussed the theoretical characteristics of centrifugal pumps in the previous lecture. And I have indicated the characteristic of a pump with a forward curved blade and a pump with a radial blade in red color. This is done to distinguish these two designs or characteristics from the backward curved blade characteristics. Primarily because these designs are inherently unstable even to the smallest disturbance in flow rate or input power. We will demonstrate this a little bit later but these are shown in a separate color to indicate this fact explicitly. So let us now look at the actual characteristic of a centrifugal pump. So here we have the actual head versus discharge characteristic of a centrifugal pump and the variational BHP with flow rate of the centrifugal pump. This is a backward curved design and backward curved blade design and you can see how it compares with the actual characteristic. The actual characteristic and the head versus Q is a falling characteristic throughout and the BHP curve or the power curve increases which is a maximum and begins to come down. Whereas here we notice that in contrast to the falling character, the theoretical characteristic here it increases then falls and BHP continues to increase monotonically. We have also plotted the efficiency. You may be recall that the efficiency was for the pump was defined as follows. The efficiency for the pump was defined as rho g Qh the hydraulic power divided by BHP and this increases reaches a maximum then it falls down as we can see here. So this point where the efficiency is a maximum is the design point. So this is the operating condition for which the pump impeller and dimensions are designed. So for a given omega and impeller diameter this is the design operating point at which you get the maximum efficiency. Efficiency falls off on both sides of the design point. Now why does the actual characteristic depart so much from theoretical characteristic? The reason for this or as follows at low flow rate some of the fluid in the that enters the impeller tends to recycle it within the impeller itself. In fact the flow rate is so low that the fluid pretty much stagnates in the impeller and doesn't really go through. So that is one reason why we see reduced flow rate at low heads. We see the flow rate beginning to come down at low values of flow rate. Number two, friction and leakage are also important factors. So the friction between the fluid and the impeller surfaces the no-slip surface actually can be quite high in the case of the centrifugal impeller because for the velocities that we are looking at the flow is likely to be very turbulent. So fluid friction is quite high and that has been neglected while deriving the theoretical characteristic. Perhaps the most important reason for the difference between the actual characteristic and ideal characteristic is due to the mismatch of the relative velocity vector between the impeller tangent at inlet and exit during off design operating condition. And this is called a shock loss and as we discussed earlier we assume V theta 1 to be 0 even for off design conditions and derive this theoretical characteristic utilizing that assumption. However it was made quite clear based on these velocity triangles that when the radial velocity changes magnitude at the inlet or at the outlet then CR both CR and VR change and the triangle will no longer be the same and C1 will no longer be tangential to the blade profile at inlet and C2 will also not be tangential to the blade profile at the outlet. Consequently the loss associated with this mismatch namely the shock loss can be quite high and that causes a significant departure of the actual flow characteristic from the theoretical flow characteristic. Now two portions of the characteristic are also highlighted here for special attention that is this rising portion of the characteristic which has been termed unstable and a part of the characteristic at very high flow rates which is also highlighted and shown indicated like this. Let us take this up first. Now cavitation is a phenomenon that occurs when the pressure at the inlet to the impeller goes below the vapor pressure of the liquid. So as we can see when the flow rate keeps increasing the overall head rise goes down and the dynamic head which is the static head plus the velocity head at the pump inlet also keeps coming down. So as a result there is a possibility that at very high flow rate this dynamic and this total head at the pump inlet could actually become equal to or maybe even become less than the vapor pressure of the liquid and when this happens then the liquid begins to boil and vapor droplets of the liquid begin to appear at the inlet. These droplets are then carried into the impeller where as the pressure increases the droplets collapse and the collapsing droplets can result in instantaneous values of pressure which are very high and if this collapse occurs in the vicinity of blade surfaces then this can cause a lot of pitting damage to the blade surface and also cause instability in the operation of the pump. So this cavitation is highly undesirable and must be avoided at all costs. So we will look at a way of quantifying the likelihood of cavitation occurring and then make sure that we know whether cavitation is going to occur or not and then adjust the operating point of the pump accordingly so that there is no danger of cavitation. This we will do next. The other portion of the characteristic that has been identified for special attention is the rising portion of the characteristic here. This has been termed or this has been indicated as being unstable for the following reason. If we take an operating point like this, let's say that this is an operating point and the pump is operating in a stable fashion at this operating point and there is a small disturbance in the flow rate. Let's say that a valve upstream of the pump is closed slightly causing a momentary reduction in the flow rate as indicated here by the arrow going to the left. The power to the pump, electrical power to the pump does not change and that remains constant because we have not changed that. So when the flow rate decreases momentarily like this, we know from the definition of power which is rho g q h that if q decreases slightly then the head is going to increase slightly with the power remaining constant. So that is indicated by this going up arrow by the arrow which is going up. So you can see that the pump when as a result of this disturbance seeks a new operating location where the flow rate is slightly less and the head is slightly more when compared to this point. But the actual characteristic looks like this, so there is no point in the actual characteristic that accommodates this requirement, namely slightly reduced flow rate and slightly higher head because when the flow rate is reduced because the characteristic looks like this, the head actually is less than this value. So there is no characteristic that the pump can latch onto. So the pump then enters a state of instability. Even the tiniest disturbance causes the pump to become unstable and the flow rate through the pump and the rpm everything becomes unstable and the pump struggles to operate under these conditions unless or until the flow rate is brought back to its original value or the input power is adjusted to accommodate the new operating or operating point which is the reduced flow rate. So if you reduce the power then you can actually bring the operating point down to a value here and try to get it going but this is also another unstable operating point. Now on the other hand if a valve upstream of the pump is open more let us say slightly then the flow rate through the pump actually increases slightly and by the same argument as we used to before the power supplied to the pump remaining constant causes the head to decrease because of the slight increase in flow rate. So when the head decreases the pump looks for a new operating point which matches these requirements namely slightly higher flow rate and slightly reduced head but the actual characteristic is in the opposite direction. So once again the pump enters into a state of instability and struggles to operate the operation is highly unsteady and can actually cause a lot of damage to the bearings to the shaft and to the impeller itself if it is left unchecked. So the only way to make it operate in a stable manner is to either reduce the flow rate bring it back to the previous operating point or change the power requirement so that we can get a new operating point which looks like this. So increase the power slightly so that for the increased flow rate the increased power will take us to an operating point which is here. So the rising characteristic as can be seen any part of the rising characteristic is or any operating point on the rising characteristic is an unstable operating point in the tiniest disturbance will cause it to become unstable and this is the reason why we marked these two characteristics in red colour and said that you know these are both unstable operating unstable characteristics and any operating point on this characteristic is an unstable operating point even this one here. So I leave it as an exercise to the student to convince themselves that both these characteristics are unstable using arguments similar to what we just made. So it is always desirable to operate the pump in the falling portion of the characteristic. The next concept that we are going to discuss is the so-called net positive suction head and the net positive suction head is defined as follows. It is equal to p1 over rho g plus v1 square over 2g minus pv over rho g where pv is the saturation or vapor pressure of the liquid corresponding to its temperature at the inlet. Notice that if the liquid is actually hotter or if you are pumping a hot liquid then the vapor pressure increases with temperature so pv will be higher than the value or the corresponding value at room temperature. So this is the net positive suction head. Now in the connotation of cavitation we talked about cavitation so in this connotation the net positive suction head is the minimum head that is required at the pump inlet in order to prevent the liquid from cavitating. So pump manufacturers usually provide the net positive suction head required which is usually denoted an additional R at the end. So the net positive suction head required is usually provided by the pump manufacturer as part of the pump characteristic. We can also evaluate net positive suction head if we know a few more details about the operating condition of the pump. And let us do an example to illustrate how we calculate the net available net positive suction head. Let us work out this example taken from Professor Coppola's book. A pump draws water at 20 degrees Celsius in density 1000 kilogram per meter cube at the rate of 20 liter per second from a reservoir whose surface is open to atmosphere at 101.3 kilopascal. The pump inlet is situated at the height of 4 meters above the surface of the reservoir. The pipe diameter is 7.6 centimeters and the suction pipe is 10 meter in length. The friction factor in the pipe may be taken to be 0.0187 calculate the net positive suction head available in PSHA. If we identify a streamline starting from any location on the surface of the reservoir and going up to the inlet of the pump and apply Bernoulli's equation along that streamline. And then we end up with this expression PS over rho where S denotes the surface of the reservoir PS over rho plus V square over 2 plus GZS equal to P1 over rho plus V1 square over 2 plus GZ1 plus and they had lost due to friction. So the subscript P here refers to the inlet pipe. Now if we take our datum to be the surface of the reservoir I can set the ZS to be equal to 0 and we assume the reservoir to be sufficiently large that its level does not change. So consequently the velocity of the fluid at the surface of the reservoir is very very small so we can neglect this term and the static pressure at the surface of the reservoir is of course equal to the atmospheric pressure. So now the net positive suction head is nothing but this term minus the vapor pressure divided by rho G. So if you recall it is that term of course divided by G minus the vapor pressure divided by rho G. So if we rearrange this expression then we can write the net positive suction head available as V atmosphere minus PV over rho G minus Z1 minus head loss due to friction. Now since the flow rate is given we can evaluate the velocity in the pipe as Q over AP this comes out to be 4.41 meter per second and the saturation pressure at 20 degree Celsius may be retrieved as 2.34 kilo Pascal from steam tables and so the NPSHA works out to 3.65 meters. Now this has to be compared with the net positive suction head required that is specified by the pump manufacturer to make sure that the available head is adequate in compared to the required NPSH.