 Let us put out an example involving actual characteristics of centrifugal pump next. So, the details of a centrifugal pump are given here, 37-diameter centrifugal pump running at 2140 rpm with water at 20 degree Celsius produces the following performance data Q, H and BHP are given at the various operating points including at Q equal to 0. The best efficiency point B, if the pump is to be used for pumping 0.4416 meter cube per second of kerosene at 20 degree Celsius at an input power of 400 kilowatts, what pumps feed an imperil size of needed, what will be the head developed and see if the pump is to be used for an application that requires 5 meters head and discharge of 5 meter cube per second of water, what would be the diameter of the impeller speed and BHP. Let us start with part A. So, we are asked to calculate the best efficiency point. It may be recalled that the efficiency of the pump at any operating point is defined as rho GQH, the hydraulic power divided by BHP. So, BHP is given here using these values for Q and H, we can evaluate the efficiency at each operating point. We have skipped the first operating point because Q is equal to 0 there. So, the efficiency really not meaningful. So, from this, we can see that the efficiency increases as we had said in our previous lecture. So, the efficiency increases, reaches a maximum and then begins to decrease and we see the same trend in the actual characteristics here. So, what we have done right now is that we are given different points along this characteristic, along the H versus Q characteristic and along the BHP versus Q characteristic. We are just calculating the efficiency curve for this pump. So, here is what the efficiency variation looks like and there is we can see that there is a steep drop in the efficiency towards the higher flow rate region. So, the best efficiency point comes out to be 0.92 to best efficiency is 0.92 to 7 and the best efficiency point is the corresponding operating condition which is 0.2 meter cube per second, 95 meters of head and BHP of 202 kilowatts. Now, before we can answer the next two questions, notice that the next question involves pumping kerosene whereas this data pertains to pumping water with a particular impeller at a particular speed. So, obviously, if you change the flow rate and you have different flow rate and input power then everything changes seemingly. And here we want to pump water but again the head and discharge are not within the range that you are seeing here. So, how do we use this information to select the pump of suitable size and RPM and so on for this because this is only for one speed, one RPM and one impeller diameter. So, these questions B and C particularly can be answered if we revisit a concept that is probably well known to you from your undergraduate fluid mechanics course namely similitude. So, let us kind of get an informal introduction to that. So, here we are actually looking at centrifugal pump characteristics but for different impeller diameters with the same angular speed. So, basically each one of this curve corresponds to the data that is given in this table. So, each one of this curve and each one of this line corresponds to this is the power equal to constant line. So, each one of this corresponds to a set of data like this particular diameter and a particular omega. So, the corresponding efficiency data is superimposed on each one of this curve. So, you can see that if you pick any diameter equal to constant and you see that the efficiency increases reaches a maximum and then begins to fall. So, this is increasing impeller diameter and each one of this is t equal to constant curve. So, as the impeller diameter increases the power required also increases and each one of this is h versus q for a constant diameter. So, as the diameter increases the head developed also increases. The net positive suction head required for each diameter is also indicated qualitatively here. So, as we can see this curve moves to the right as the diameter increases. And you can also see there for a given diameter the net positive suction head increases required in net positive suction head required increases with flow rate. So, when we did the calculation of net positive suction head available which was this example we calculated this to be 3.65 meters. So, what we need to do is compare this with the net positive suction head required for the corresponding operating point and make sure that the actual value is comfortably more than the required value. So, here we have actually shown the characteristics of a centrifugal pump of different impeller diameters. We can come up with similar characteristics for constant diameter but different angular velocities that would generally look very similar to this with lower angular velocities developing smaller heads and higher angular velocities developing higher values of heads. So, it looks similar but not quite the same. So, what we would like to do now is to figure out some way of combining the data. So, in addition to having data like this for different impeller diameters different angular velocities for let us say pumping water you can also generate similar data for pumping let us say kerosene or gasoline or some other liquid. So, we have a huge volume of data that is generally available in connection with pumping liquids using centrifugal pumps different impeller diameters different omega different fluids and so on. It would be nice if all this information can be combined into something like a universal characteristic for this centrifugal pump. By universal characteristic what we mean is head versus q that is universal for any diameter any rpm and any fluid. Similarly, power versus q characteristic that is applicable for any rpm any impeller diameter and any fluid. So, this objective may be accomplished by using what is known as similitude or dimensional similarity which we have studied in the previous course on fluid mechanics. Of course, in today's world such an exercise probably would be accomplished using say machine learning when so much data is available you can use machine learning techniques you know to actually come up with this sort of universal behavior and machine learning strategies ideally suited for something like this. But it must be borne in mind that machine learning is a brute force approach you know with a given so much data and with enough computing power it can come up with this combination for a given application. So, it can detect universal characteristics from the data but it is a brute force approach. Now the approach that we are going to take namely the similitude approach or dimensionless characteristics is a much more of a finished approach based on physics or dimensional analysis of the quantities that are likely to influence the quantities of interest to us. So, the quantities of interest to us are the head that the machine can develop and the power that is required. So, we are we are seeking dimensionless characteristics of head universal characteristics of head versus q and universal characteristics of php versus q. What is that we have taken the g term the acceleration due to gravity on the left hand side because that is a universal constant and well known. So, we do not wish to have it on the right hand side. So, both the head and the php can easily be seen to be functions of the flow rate, diameter of the impeller, angular velocity, density of the fluid being pumped, viscosity of the fluid being pumped and the roughness of the impeller or rotor surface. These are possibly the quantities which can affect this. So, now following Buckingham's Pi theorem we form dimensionless groups. So, we can form one dimensionless group for the head which looks like this. So, this is a dimensionless head denoted as c sub h. This is a dimensionless flow rate denoted c sub q. This quantity here is also dimensionless it may be recognized as the Reynolds number and this may be recognized as the relative roughness. So, you would have used the relative roughness to look up friction factors from the Moody's chart. So, this is the same relative roughness. So, this is the dimensionless php denoted cp. So, what we are saying now is that the dimensionless head is a function of the dimensionless flow rate Reynolds number and roughness. Similarly, the dimensionless power is a function, a known function of the dimensionless flow rate Reynolds number and epsilon over d. So, it is this functional form F3 and F4 that we are seeking now. So, this says that the head and the php are both dependent on three dimensionless parameters c q which is what we want and in addition Reynolds number and epsilon over d. But generally for the kind of flow rates that we are talking about in the context of centrifugal pumps, the flow is normally turbulent and the relative roughness value is such that the flow is generally fully in the fully rough regime. So, Reynolds number and the relative roughness hardly play any role in influencing the head or the php. So, we may actually drop these two terms and then say that the dimensionless head is a function only of the dimensionless flow rate and the dimensionless php is a function only of the dimensionless flow rate. So, we are looking for a universal relationship connecting the dimensionless head and the dimensionless q which is consolidation of data relating to many different flow rates, many different impeller diameters, many different angular velocities and pumping of several different fluids. So, this consolidates all the data and gives it to us in the form of a single curve. Note that we do not require such a curve for the efficiency because efficiency by definition is dimensionless and it is nothing but the product of c h, c q divided by product of c h and c q divided by c p. So, we do not need to do anything special for the efficiency. So, what we need to do now is how do we get this functional form. So, basically what is done to construct these two functional form is to go back to data like this for again many different impeller diameters, many different RPMs and while pumping many different fluids. So, for each case we have this set of data and from that we actually calculate the efficiency like this and what we then do is select the best efficiency point for each one of this case. So, we select the best efficiency point for each one of this case impeller diameter or PM and fluid and we calculate c p and c h for each one of this case c p, c h and c q for the best efficiency point and then plot all this data in one plot. So, notice that if we go ahead and do that this is the curve that we are going to get but let us see what we have done here. So, we have taken the best efficiency point for each case namely one impeller diameter running at a particular RPM pumping a particular fluid. So, we have calculated best efficiency point for that and then from the best efficiency point we have calculated c h for the best efficiency point, c q for the best efficiency point and c p for the best efficiency point and then plotted that here and of course efficiency also and plotted that here. So, when we do this for the entire volume of data that we have many different diameters, many different RPM, many different fluids all of them generally fall into curves like this universal curves like this. So, this is the universal or universal functional form that we have been looking for this is so this here is what we have denoted as this here is what we have denoted as f p and this curve here is what we have denoted as f h the universal relation connecting c h and c q. So, once we have something like this for any application this information will be used to select a particular impeller diameter and particular RPM for the application in hand. So, let us see how we do this for the given example although we have only one set of characteristics for this, we will use this to illustrate the procedure on how to select a particular pump characteristic. I am sorry on how to select a particular pump for a given application. So, let us go back and look at the second question. So, if the pump is to be used for pumping so much of kerosene with an input power what would be the impeller size and RPM that are needed and what will be the head that is developed. So, we follow the procedure that we laid out. So, we calculate c q, c h and c p and we have calculated for all the operating points, but remember the universal characteristics are actually constructed with the c q, c h and c p corresponding to the best efficiency point which is shown in bold here. So, the other data are discarded. So, the advantage of doing it with the best efficiency point is that whenever you select a pump from here for a particular requirement you are actually selecting the best possible pump among all the data that is actually given. So, that is guaranteed. So, we construct this curve with the best efficiency point, but we are illustrating it here for all the points. So, now in this particular case both the flow rate and the BHP are given. So, we are given the flow rate and we are given the input power and the density is known. So, since q and BHP are given, what we do is we can eliminate omega from the expressions for c q and c p. So, the expression for c q looks like this, the expression for c p looks like this. The other quantities are known diameter and omega alone are unknown. So, we can eliminate either omega between these two expressions or diameter between these two expressions both of. So, we eliminate omega between these two expressions and finally write the unknown diameter in terms of these quantities. What is that? Everything inside this bracket is known c p, c q from the best efficiency point. So, we take c p corresponding to the best efficiency point, we take c q corresponding to the best efficiency point and rho is 820 kg per meter cube, because this is in kilowatts, we have written this as 0.82 and the q is given to be 0.4416, BHP is given to be 400. So, using this data, we can select an impeller diameter of 53.76 centimeters for this particular application. So, once the diameter is known, we can go back to this and calculate omega. So, omega may be evaluated from the definition of c q. So, we use the value of c q and the given diameter and evaluate rpm to be 1540 rpm and the head developed may also be calculated from the known value of c h. So, c h is known, so we plug in the value. So, for the best efficiency point c h is 0.1356. So, we plug in the numbers and we get. So, given the flow rate BHP and the liquid to be pumped, we can get all the other quantities of interest namely the diameter rpm and the head developed. Notice that this is head in meters of kerosene column because that is the fluid that is being pumped. So, we can get all the other quantities that we want provided we have such dimensionless characteristics available with us. So, that is the power of constructing dimensionless characteristics like this. We can actually select a pump for any application because the data that goes into this actually spans all the diameter I mean the entire diameter range, the entire speed range and the sort of fluids that can possibly be pumped. So, it is a very extensive data and if using dimensional analysis, the power of dimension analysis also becomes clear from here. So, once we do the dimensional analysis, we know what dimensionless groups are relevant here and once we do this, this can be used for selecting suitable centrifugal pump for any application because we use the best efficiency point here, whichever point we select from here is the most efficient pump for that particular application that we are looking at. Now, let us now look at part c. In part c, we are actually given the head and the required head and the flow rate and we are pumping water. So, we are asked to select the diameter of the impeller speed and the and determine the BHP required. So, we proceed in the same manner since q and h are given, we eliminate omega from the expression for c h and c q and calculate the diameter of the impeller using this quantity and the diameter comes out to be 3.8626 meter which is huge and we can in a similar manner evaluate the rpm which comes out to be 47 ridiculously low and a BHP value of 265 kilowatts. What these numbers are trying to tell us is that a centrifugal pump is not the best choice for this. These curves will still tell us what a centrifugal pump would be required or what the centrifugal pump that would be required for this particular flow rate and head that is given, but it may not be the best choice as we can see from here. So, clearly the given set of head and flow rate or such that a centrifugal pump is not suitable for this particular application. So, the question that arises next is how would we then select a pump for a given application. So, if it is a centrifugal pump then we know how to select, but after selecting if it turns out that centrifugal pump is not the best, how do we actually go across different kinds of pumps and then make the best choice or selection for a given application. This is what we are going to take up next.