 Welcome you to this session. Today we will be discussing the matching of pump and system characteristics. In last class we have discussed the pump characteristics which is typically the relationship between head developed by the pump with the flow rate at a given rotational speed. That means at a given speed of the pump what is the relationship between head developed by the pump and the flow rate through the pump. So, let us look on the diagram again that if we see this diagram that hq the typical hq plot for a pump considering a backward curve vein or whatever it may be that this typical hq plot. This is the characteristic curve along with that if we plot the efficiency well the efficiency variation we will see that the efficiency curve goes like this. So, that means if we also plot the efficiency versus this is the maximum efficiency point. So, this is the this is sometimes known as eta efficiency flow rate characteristics and this is the head discharge characteristics characteristics of pump well and this is the head discharge characteristics. So, characteristic characteristic curve characteristic of pump. So, the characteristic curve of the pump also describes the efficiency flow relation these two sets two curves one is the eta versus efficiency versus q and one is the head versus q describe the characteristics of the pump one is the efficiency flow characteristic another is the head flow characteristics and this is valid for a given r p m. So, a n is fixed now you see definitely the point where the efficiency is maximum is the design point. That means the pump is running at its maximum efficiency condition and corresponding to that the flow rate corresponding to that is the design flow rate if you write q here. So, this will be the design flow rate q d. So, therefore, this is the point known as the design point. That means this is the design point of the pump design point now pump is rated for its at its design point. That means it will develop this much amount of head and this much amount of flow at its design point means when the efficiency of the pump overall efficiency of the pump will be maximum. But in actual operation what will be the operating point of the pump depends upon the system resistance. So, pump is not in isolation pump does not run in isolation. So, whenever there is a pump there is a system to the pump that means what is that system that is suction line and the delivery line. So, therefore, the pump operating point depends not only on the pump characteristics, but also the system characteristics and the matching between the two. Now, let us see what is a system. You look back to this figure now you see here this is the pump. So, this suction pipeline with all the bends strainer the inlet intake inlet of the suction pipe delivery pipes this is the system to the pump. That means pump is attached to this system that is the suction and delivery pipes. So, therefore, when the flow takes place through this pipe along with the pump the operating point of the pump will be decided by the system characteristics of the we will depend on the system characteristics also. Well, so what is the system characteristics let us find out system characteristic means that what is the relationship between these head loss through the system and the flow rate which gives the head to be developed by the pump. We know that head to be developed by the pump is given by what let us see here that head that has to be developed by the pump is that means the energy that has to be imparted on the fluid while it flows from c to d. That means total head h d minus h c. So, this is the head developed by the pump this must be equal to this potential head difference between the sump and the upper reservoir because the fluid has to be put from the point a to the point a along with all the losses that it incur along its flow. That means the losses through the system system means the suction pipe and the delivery pipe not only on the pump also have to be taken account. So, that we can find out the total head developed by the pump. That means this is the difference between the elevation level of this two water surface that is h s known as static head plus all the losses. If we write the Bernoulli's equation at this point between this point and this point and between this f and d and from this two equations we have shown that the total head developed is equal to this difference in elevation head between this two surfaces that is the static head known as static head plus all forms of hydraulic losses in the suction and delivery pipes. Let us find out the mathematical expressions for that. Now, let us write the head loss in the suction pipe as h 1 suffix 1 is the head loss in the suction pipe which is the loss in total loss in the energy per unit wave due to flow through the suction pipe. This comprises two distinct part one is the loss due to fluid friction that is the friction between the fluid and the solid wall which can be expressed as a friction coefficient f 1. That means this is the Darcy's friction coefficient which can be expressed in terms of the friction coefficient l 1 where l 1 is the length of the suction d 1 is the diameter of the suction pipe l 1 is the length of the suction pipe one is the suffix suffix one used for the suction side v 1 square by 2 g where v 1 is the velocity of flow through the suction pipe. So, the typical fluid friction loss can be expressed in terms of a Darcy's friction coefficient f 1 times the l 1 by d 1 that is the length to diameter ratio and the velocity plus another loss take place due to the bends and bars. See in this pipe when the fluid flows to there are bends sometimes a valve may be there in the pipe line usually to control the flow a valve is given at the delivery side valve is not usually given is usually not given in the suction side because of the cavitation restriction because we want to minimize the losses in the suction side that I will discuss afterward usually a valve is placed in the delivery side. So, that to control the flow through the pump we want a less flow rate. So, we operate with the valve so that it gives the less a opening through the flow so that the flow is controlled. So, to control the flow through the delivery line we insert a delivery at a valve in the delivery line different types of valves are inserted. So, the when the fluid flows through the valves there is a loss of a similarly the while fluid flowing through this bend due to the change of direction change in the direction of flow there will be a loss. So, all these losses as you know are termed as minor losses in fluid mechanics. So, this minor loss is the second kind of loss that takes place in course of flow through this pipe this can be expressed as some constant time the v 1 square by 2 as you know all the losses can be expressed in terms of a constant loss coefficient k let it k 1 for the suction pipe times the velocity here all right I think. Similarly, we can think of the head loss in the delivery side to consist of this two distinct part one is the usual friction loss the friction between the fluid and the solid wall l 2 d 2 into v 2 square by 2 g plus similarly k 2 v 2 square by 2. That means, this h 1 and h 2 please k 1 is due to bend losses again I am telling this is the usual friction loss. So, second part takes care of all the minor losses that means losses due to bends and losses due to valves there are so many things in suction pipe there are strainers non return valves there is no such this type of valve gate valve or glove valve delivery line, but there are strainers there is another valve we are known as non return valve known as non return valve. So, that the fluid does not come to this sum from the pump. So, this is the non return valve you understand that is the non return valve. So, this there is a non return valve this is a strainer. So, all these things are there more over there is a pipe bend. So, therefore, this first term in this two equations are the usual friction loss that is between a friction between the fluid and the solid wall while the second term in both the equations represent the minor losses that means losses incurred due to the flow of fluid in the pipe because of their flow takes place through the pipe bends because of their change in direction of flow through the pipe bends through the valves through strainers at the intake and valve at the intake all these things are taken account in this if there is no valve no bend no minor losses then this will be zero all of course. So, therefore, we see this h 1 and h 2 sum of the h 1 plus h 2 is the total head loss in the system system. Now, you see that v 1 can be expressed in terms of the flow rate at q by pi d 1 square 4. Similarly, v 2 can be expressed and similarly, 4 d by pi d 2 square that means our intention is to replace the velocity of flow in terms of the flow rate. So, therefore, this h 1 will be if I just substitute it what will be the values of h 1 h 1 will be 8 8 by g into f 1 l 1 by d 5 d 1 5 d 1 by d 2 by d 2 by q square plus what will be this value if I put this there then 2 k 1 by g 2 k 1 by g pi here the pi will be there pi square will be there well. So, here also pi square will be there divided into q square. Similarly, h 2 will be 8 pi square g please check it very simple thing, but I can write the wrong expression q square what it will be 8 very good 8 because the same thing 8 pi square by g q 1 here there also d 2 to the power 4 please tell me it is very simple d 2 to the power sorry it is d 1. So, here also d 1 to the power 4. So, here also d 2 to the power 4 all right d 1 d 1 h 1 very good is there anything wrong k 2 here oh this is k 2 is already ok very simple thing, but these are not the important things important thing is that all these things this is constant. That means, these are constant these things constant. So, therefore, you tell that h 1 plus h 2 is equal to some constant into q square some constant into q square that is the sole intention this constant includes the friction factor definitely one thing you have to understand this friction factor does not vary with the flow velocity in the turbulent flow region. So, these are constants apart from them apart from them other parameters are the length diameter that is the geometry. So, for a given system of a given length and diameter and for given values of loss coefficients they are constant. So, constant into q square now the head to be developed by the palm head to be developed by the palm is what head to be developed by the palm by the palm is what is equal to the static head plus this loss h 1 plus h 2 which we have derived earlier. That means, this is h s plus this constant let is expressed by c c q. That means, this is the total head that pump has to develop this can be thought of as a resistance. That means, as if this h s plus c q square is appearing as a as an opposing head that a fluid has to overcome to go from the sum to the upper reservoir. That means, this is the total resistance head given by the system. So, if this we have write as the h system that system develop this head as a resistance that is an opposing head that has to be overcome by the fluid which is to be pumped from the sum to the upper reservoir. So, if we now draw this in the figure of h q h q plane where we already draw drew the pump character is this and it appears like this. That means, this is the system resistance curve this is h s let this is h s. So, this is again a parabola sorry this is like this. That means, this is h q characteristics of system h q characteristics of the system that is h s static head sometimes it is known as static lift. So, h s plus constant into q square. Now, therefore, we see this is the system characteristics and this is the pump characteristics h q characteristics. So, they intersect at this point. So, this must be the operating point. So, therefore, this is the operating where they will intersect that will be the operating point. That means, if the system is attached if the pump is attached to this system if the pump is attached to this system then this will be the operating point you understand that this will be the operating point. That means, this the pump will develop this amount of head and will develop this amount of fluid this is the system characteristic curve. So, the system characteristic curve value is valid for example, the pump characteristic curve is described for a particular pump. That means, the pump geometry is fixed and moving with a constant speed constant rotational speed. Similarly, the system characteristic curve is valid for a particular system. That means, pipes of given diameter and length and bends fixed bends and if there are valves the valve settings are fixed. Because, if you change the valve settings the loss coefficients will change. That means, you can constant different system resistance curves by changing the any of these parameters the most easier way by changing the valve settings. That means, if you change the valve settings the system characteristic curve will change. Because, the loss coefficients will change that means, the constant defining the system characteristic curve will change. You look here that means, we can draw different system characteristic curve. So, different these are all system characteristics curve. These are parameter the parametric variations are the different settings of the valves or you can compute the different dimensions of the pipe. These are the different settings. So, that means, if you change the valve position that means, if you create the system resistance different system resistance then the operating point may shift from here to there. That means, this is the system resistance where you close the valve. That means, this is creating more resistance to the system for a given flow the opposing head will be more. So, therefore, the pump has to develop that head at steady condition. So, the operating point will be shifted to this point. So, therefore, operating point is decided by the intersection of the system characteristics curve. That means, the system resistance characteristics that the opposing head which has to be developed by the system. How does it vary with the flow rate and the intersection of the system characteristic curve with the pump characteristic curve. So, this point may not be the design point. This is the design point. For example, here you see any of these three points for the three system characteristics curves are not the design point, but here you see if the system if you take these as the system characteristic curve the operating point is very close to the design point. Now, the closeness of the operating point to the design point depends upon the fact that how good an estimate is made about the system resistance while design of the pump was made clear. Now, after this I will discuss the effect of speed diameter effect of speed and diameter on pump characteristics effect of speed and diameter on pump characteristics. Now, we have seen that this h q characteristics is valid for a given speed that means n is equal to constant for a constant value of this speed. So, we can draw a family of these characteristic curves at different speeds. Similarly, we can draw a family of curves for different diameters. So, not only the speed the diameter of the impeller is fixed, because for a fixed geometry of the pump and moving with a fixed rotational speed this is the h q characteristic. That means in one two dimensional plane we can show a family of curves with different parametric values of either n and d which means that what is the influence of n and d on the pump characteristic curve. How to find it? Very simple very simple thing let we have a h q curve mathematically we have to relate the h and q with n. Now, if you recall that we know from similarity analysis pi 1 term is q by n d q and pi 2 term is j by n square d. So, this gives us the clue to find this mathematically. We know that we are we can only show the family of curves at different rotational speed for different diameters for a pump of the same homologous series. We cannot show these families of curves where one curve pertains to centrifugal pump another curve pertains to axial flow pump that will not do that I discussed at length earlier. That means it will represent the family of curves or different altered conditions of for example, the rotational speed the diameter in the same homologous series. That means for that series the conditions have to be similar provided conditions will be similar provided the pi 1 and pi 2 term remain the same. That means for two such machines that q by n d q will be same and g h by n square by d square will be same. So, therefore, if we consider only the influence of rotational speed influence of n for example. That means we keep d constant influence of n if we want to find then we can simply write q 2 by q 1 is n 2 by n 1. Well that means to find out the altered flow from a given flow rate we will have to multiply only by this ratio because they are directly proportional to their corresponding rotational speed. Similarly what is h h 2 by h 1 similar way for the same diameter h by n square is constant. That means is equal to n 2 by n 1 whole square. That means h 2 is equal to h 1 n 2 by n 1 whole square. That means h 2 is equal to n 2 by n 1 whole square. That means we can find out that means if we have point a different points graphically say primary school level job. So, we can find out the corresponding point a dash b dash c dash like this. So, that I can construct the curves. So, this is n 1 this is n 2 by this relation the most interesting thing is that if you look if you see these two relations that q is proportional to n and h is proportional to n square we can say from this h is proportional to q square which means that the locus of such similar points if you join this will give a series of parabolas this series of parabolas that similar points. Similar points means at different speeds that means this is n 3. That means c is c dash c double dash b b dash b double dash a a dash a double dash. That means all the similar points that means if c is the point corresponding to that the similar point is c dash. That means corresponding to c the flow if it is scaled down to another a rotational speed and h is scaled down. So, this will come to this point. So, the similar point corresponding to c or c dash to another rotational speed n 3 c double dash. So, all these similar points pass through a parabola the locus is a parabola and in fact this is the system resistance curve when h s is 0. Let me tell you again that I now here if you see that this is a system resistance curve starts from h s why the head developed has to be developed by the pump or the opposing head created by this system is h s plus some constant into q square which we have seen earlier which we have seen earlier this is the h s plus h 1 plus h 2 that is constant into q square. Now, in case when there is no static lift it is 0 that means the this is the pump delivers fluid in the same horizontal plane. That means this is the pump outlet this is the pump inlet the fluid is coming in the same horizontal plane here the total head is h 1 here the total head is h 2. So, h 2 is greater than h 1. So, therefore, the opposing head that has to be developed by the pump in pumping the fluid from this place to this place here the what pumping does not mean that there will be a change in the elevation. So, in that case the static head is 0. So, it is only to overcome the losses this loss include the exit loss also there also this losses include that exit loss that means this h 2 in h 2 the exit loss is also there. That means we takes care of the velocity head generated by the pump that means the static head is 0 in that case h system is simply c into q square that means in that case the system curve goes through the origin. That means in this case therefore, we can conclude that the locus of the similar points at different speeds in h q characteristics lie on a parabola which passes through origin. And these are in fact the system resistances and these are in fact the system resistance or the system character system resistance or system characteristic both the word got system characteristic. So, these are on the parabola h proportional to q square which gives a very interesting thing that means if we have a operating point here. Now, if you think in terms of an operating point that this is the system resistance and this is the pump characteristic curve system resistance means system characteristic curve pump characteristic curve this is the intersection point. Now, if the pump speed is altered to n 2 we can find out the corresponding similar point which is again is nothing, but the operating point. That means this operating point that the speed n 2 can be found out by direct application of the similarity principle similarity law which means the corresponding similar points of c c dash which also lies or which is also the intersection of the system characteristics and the pump characteristics. Because the locus of these operating similar operating points at different speeds lie on the system resistance curve. If this is another system resistance curve where this pump is set to that system. So, this will be operating point now without altering the system resistance that means without altering any pipeline without altering the valve setting if the pump is set to another rotational speed. Then we can find out this operating point by the direct application of the similarity principles because the point the similar points lie on the system resistance well understood. Now, the effect of diameter variation is very simple I will not go into that detail effect of diameter variation is again if we write again that pi 1 is again is a primary school job I feel that pi 2 is g h by n square d square then it is simple when the influence of d we consider only the influence of d. Then n is constant that means q by q 1 by d 1 q is q 2 by d 2 q that means q 2 is found out as q 1 into what d 2 q by d 1 h same way that h 2 for a fixed n value of n h 2 will be h 1 into d 2 square here we see that h is proportional to q to the power 1 h is proportional to q to the power 2 by 3. So, this is the locus h q locus for the similar points. So, therefore, for a alteration of d the operating points may not be found out by application of the similarity laws because the locus of all similar points in h q plane is not the system resistance curve, but they follows a curve where h is constant into q to the power 2 by 3. However, we can construct the curve at different values of d by changing the q that means to find out the new q with the new diameter d 2 by application of this formula and new h with the application to the for due to the change in new diameter by the application of this formula. So, that we can construct the curve for different that means if we have an h q curve like this for one diameter d 1 we can construct this for another diameter d 2 where n is fixed what happened this is n is fixed and here d was fixed d is fixed well any query today I will finish here please any question any question. Thank you.