 to this session of the course. So, today our topic of discussion as it is shown is axial flow pumps, but before coming to this topic we like to discuss about the characteristics of multiple pumps used together or connected together in a system. So, first we have to know why multiple pumps that means more than one pump is used or are used in a particular application. When the head developed or the flow rate delivered by a single pump is not sufficient for a particular application a number of pumps that is more than one pumps are used. Pumps are used either in series or in parallel, when it is used in series then the heads are added that means when you require more head than that developed by a single pump we use the pumps in series and where we require more flow than that developed by or delivered by a pump we use pumps in parallel that means in case of pumps connected in parallel the heads the flow rates are added just like this you can see here the pumps in series that if you have number of pumps that is pump one it is simply geometrical series in series geometrically connected in series like this pump three pump three like this that means the discharge from one pump is connected to the inlet to the other pump and so on. So, therefore, you see if we consider the total head developed across the system of multiple pumps. So, it is some of the head developed by individual pump. Similarly, when the pumps are connected in parallels for example, pump one the system is like this pump two then pump three if they are used in parallel the arrangement is like this the inflow to the pump it means inlet to the pump inlet flow of water is divided into three parts for example, if three pumps are connected these are the common shaft. So, they are not connected this way this is the common shaft. So, therefore, the flow is divided and ultimately the flow meets like this. So, that the final discharge is the sum of the discharges from all the pumps this is the final discharge. So, here you see the q is divided like this in three parts q one q two q three. So, it automatically meets and gives q. So, the total flow which is taken up by the system sum of the flow rates through individual pumps. Now, we come to the characteristic of such a system of multiple pumps. Let us consider the characteristics of pumps in series and let us consider two pumps of identical design that means two identical pumps. If you recollect the head q characteristics the centrifugal pump you see the head q characteristics like this. This is the we give it by a dotted curve this is the head q characteristics of a single pump. Now, you see if we should use another same similar pump in series what will happen the head will become double at any given flow rate that means for a given flow the head will be doubled. So, therefore, we can draw the h q characteristics the relationship between head and discharge q for the two pumps connected in series to identical pumps like this. So, this so this meets here that means at any flow the heads are added up double that means if this is h one. So, this is h one that means the two such pumps in series are added. So, at any point the heads are being added double the same pump. So, this is h one this is h one. So, therefore, we can write this is the single pump characteristics single pump characteristics single pump characteristic well and this is the double pump or combined pump characteristics combined the two pumps connected in series combined pump characteristics characteristic. So, this is there is two pumps in series two pumps identical pumps here in series in series. Now, as we know the operating point depends upon the system resistance that means it is decided or determined by the point of intersection between the pump characteristics and the system characteristics. For example, as we have seen earlier that for a single pump that let it be the single pump characteristics we if we know the system characteristics for example, if these be the system characteristics what is system characteristics it is nothing but the head discharge system characteristics head discharge relationship for the system that is the pipelines and valve to which the pump is connected that means it is the head loss to the system for a given flow rate. So, if this characteristic curve is drawn we know that this is the point of intersection the single pump that means the single pump and with this system characteristics this is the operating point in case of multiple or combined pump here two pumps the operating point will be this is the operating point this is the operating point for combined pump system for combined pump operating point for combined pumps whereas this is the operating point for the single pump. So, one interesting thing is that though the characteristic curve for the combined pump when these two single pumps of identical shape size and design are used then the heads are summed up. So, head at any flow rate is doubled, but the operating point gives a head which is higher than the operating head when the single pump was in use, but it is not exactly double. So, the head will be increased, but will not be doubled that depends upon the system characteristics well similarly we can show we can draw rather the characteristics for combined pumps in parallel let us see the combined pump the same way we can draw the combined pump let us consider this as the single pump h q characteristic. Now, when the pumps are used in parallel as I have told you earlier so flow rates are added at a particular head. So, therefore the combined pump characteristics can be drawn like this that means at a given head the flow rates are simply doubled that means if this be the single pump flow rate q 1 for example. So, this part will be q 1 that means for a given head we can find out the point in the combined pump by adding the flow rate that means the same flow rate because the identical pumps are used in parallel. So, this q 1 this q 1 so it will be same for all heads. So, therefore this is the single pump characteristics as it was shown earlier in case of pumps used in series single pump characteristics and this is the combined pump characteristics well combined pump characteristics characteristic well when the pumps are in parallel. So, this is the case that pumps are in parallel similarly here also the operating point is decided by the intersection of the characteristic curves with the system resistance that means let this be the system resistance let this be the system resistance that is system characteristics or system resistance as you tell so system resistance curve. So, this is the operating point for the combined pump where we see the operating point does not give double the head as shown by the operating point in case of the single pump. So, the head is increased, but is not doubled. So, this is finally the sorry not head the flow rate operating point. So, flow rate is increased, but not double operating point for combined pump I repeat it again that as we have seen earlier in case of pumps in series that pumps in parallel. So, the operating point is here. So, the head that is ultimately delivered depends upon the operating point. So, this is increased from that delivered by single pump because single pump operating point is there, but it is not doubled, but characteristic curve itself shows that at any head the discharge is doubled, but here the discharge is not doubled this is the combined pumps in parallel when they are not identical. In a similar way in most general cases we can show we can show the pumps in series and parallel when they are not identical that means pumps are not necessarily to be identical when they are used in series and parallel. So, it is also very simple same thing we may have two pumps let this is one pump pump one characteristics of one pump and let this be the characteristics of another pump where this characteristics is steep steeper than the number. So, number this is one pump the two pumps are dissimilar pump. So, if we use the two in series it is nothing at one flow rate we have to add the heads that means we will have to start from here. Now, it will come like this and then it will follow the same because here the head is 0. So, where the head of the second pump is 0 it will follow the first pump. So, this is the characteristic for combined pumps for combined pumps in series. Similarly, we can draw for the pumps in parallel. So, up to this part there is no flow delivered by the pump one. So, therefore this will closely follow or totally follow the pump two then it will be like this because after that at any head the flow of the flow rate delivered by the two pumps will be added. So, it is very simple. So, therefore it is for characteristic for characteristic for combined pumps in parallel. So, therefore we see that the characteristics of multiple pumps whether parallel whether they are used in series or parallel can be drawn in this way by adding the heads at a given flow rate for pumps in series or by adding the flow rates for all the individual pumps at a given head when the pumps are connected in parallel. And the operating point will determine entirely by the for example, this will be the operating point if the system resistance is this for the combined pump in parallel and this is for the combined pump in series under this system operating characteristic. It will determine by the intersection of the system characteristics or system resistance with the characteristics of the combined pump system. So, now we have completed this section of centrifugal pump. Now, we will come to axial flow pump. Now, an axial flow pump is a pump where the flow of liquid that is the water is in the axial direction that means the flow is in the direction of the axis of rotation. Well, you have seen that in centrifugal pump the flow is in the radial direction it is a radial flow pump and the flow is radially outward as you know for a pump the for a radial flow pump the flow has to be radially outward this is because to gain pressure to gain pressure energy of the fluid from the centrifugal energy. So, that the fluid has to go outward so that at the outlet the centrifugal head is more than that at the inlet. So, therefore, we see that centrifugal pump is a radial flow pump. So, axial flow pump is a pump where the flow is almost in the axial direction that means the inlet and outlet of the fluid do not vary in their radial location from the axis of rotation. We have discussed this in axial flow turbine also this is the definition holds good as well for any axial flow machine. So, therefore, an axial flow pump can be thought of as an converse to an axial flow turbine or a propeller turbine and we can just have a look what how an axial flow pump looks like. So, you see basically the axial flow pump consists of a central boss on which a number of blades or veins are mounted this is basically the impeller and this impeller rotates within a cylindrical casing this is the cylindrical casing with fine clearance this is these are the clearance. So, therefore, this boss with the number of blades mounted on it consists the impeller this is rotating the cylindrical casing. So, these are the inlet. So, therefore, you see this is the impeller this is the impeller this is this you cannot show see clearly impeller. So, this is this one I write it here stationary stationary guide veins the purpose of this stationary guide veins is to direct the fluid in the correct way to this impeller blades. So, that they can enter the impeller blade without any shock the most important part of an axial flow machine is this stationary blades stationary outlet guide veins this is stationary outlet. Which is not there outlet guide veins in a radial flow machine the purpose of the stationary outlet guide veins is not to convert any energy from kinetic energy to pressure energy it simply is change the direction of motion. That means, it reduces the whirling component of velocity which the fluid possess the water possess from the outlet or at the outlet of the impeller. So, this whirling component is reduced. So, that it is directed in such a way the fluid is directed in such a while it passes through this stationary outlet guide veins. So, that the final discharge from the machine becomes almost axial that means, in the direction parallel to the axis of rotation this is the axis of rotation this is the rotation omega this is rotating like this. So, this is in general a schematic view of an axial flow pump. Let us see the velocity diagram of an axial flow pump now if we take a section of the blade like this if we see a section of the blades then the it looks like this that this is the well this is the blade impeller blade one impeller blade and this is the outlet guide blade. So, this is the impeller blade impeller blade one impeller blade the section is made like this if we take a section like this we will see this this is the inlet guide when this is the impeller impeller blade and this is the outlet guide. So, we are seeing the impeller blade and the outlet guide vane. So, this is the outlet. Now, usually what happens is that the inlet guide vane directs the water in such a way as you know earlier cases also that it glides that means it strikes smoothly with this that means this is the tangential direction so that the angle made by the relative velocity that is the in this direction the blade is moving that is the one so this makes the same angle with that the blade at the inlet so therefore this is the velocity triangle at the inlet this is the absolute velocity at the inlet and this is the whirling component or the blade velocity this is the blade velocity rather I will tell this is the blade velocity now another thing is very important the inlet and outlet of the fluid rather you see here takes place in such a way that the inlet and outlet do not vary in the radial location from the axis of rotation so therefore the velocity triangle is shown at a mean height so that inlet and outlet which is varying with their axial location not in the radial location so this is this diagram is made at the mean height of the pump that means at a mean radius at a mean radius that means this is the axis of rotation so the radius varies from this place this is the half radius that is the root of the impeller blade and this is the tip radius so this is made at the mean radius and all the velocities are therefore considered to be mean if there is any variation along the radial direction so therefore we consider this as the mean velocity or some average velocity in the average velocity where if there is any variation in the radial direction so therefore we see the outlet diagram here now when it comes out this is the absolute velocity coming out from the impeller blade this is the relative velocity so which is gliding out of the blade and we see this is the blade velocity at the outlet and since the inlet and outlet at the same radial location so u2 is equal to u1 u1 is equal to u2 so therefore we see this is the vw2 vw sorry vw1 the whirling component of velocity at the inlet and similarly this part is the whirling component of velocity at the outlet and this is the axial velocity because this is the axial direction va1 and va2 at the outlet and the design is made in such a way that va1 is va2 now we can write that the energy imparted to the fluid in the impeller blade per unit mass can be written as from our earlier discussion as written as vw1 u1 minus vw2 u2 rather this is with a negative sign with a negative sign with a negative sign because this is more than this or we can write is equal to this is the energy imparted to the fluid if we tell vw1 times the u1 or u2 simply we write u1 is equal to u2 is equal to so this is the amount of energy imparted to the fluid per unit mass so mass flow rate m dot can be expressed as the density times the average axial velocity which is known as the flow velocity is either va1 or va2 they are same in the design times the area that is pi r t square minus r h square where rt is the root diameter and root radius rather rh is the hub radius that means if you see that that this is the root radius and this is the tip radius so this is the hub radius this is the tip radius root rh that is the hub radius that is the tip radius so impeller tip so this is the tip radius and this is the root rh so this is the average axial velocity or flow velocity so this way we can find out the mass flow rate which when multiplied with that will give the power that is being transferred to the fluid minute passes through the impeller blade. Now, this is the outlet guide vane as I have told you earlier the purpose of this outlet guide vane is to reduce the whirling component of velocity here you see at the inlet to this blade is the velocity v 2 which is the absolute velocity from the moving impeller blade and it has got a whirling component of velocity of this much. So, this is being reduced that depends upon the shape of the blade so that at the outlet the fluid which is coming out with the velocity v 3 which is the discharge velocity where the whirling component is almost reduced. So, this velocity is almost axial this component is reduced. So, you understand very well this component is reduced. So, velocity is almost axial. So, this alpha 3 represents the angle with the axis which is almost 0. So, it is almost an axial discharge here alpha 2 represent the angle of the absolute velocity that makes with the tangential direction the direction of blade motion. Similarly, this is alpha 1 in the inlet velocity triangles this is beta 1 that is the velocity of relative velocity angle of the relative velocity with the tangential direction. Similarly, this is beta 2. So, beta 1 and beta 2 are the blade inlet and outlet angles for a smooth chocolate flow well. So, this is the blade diagram and we can find out the energy or power imparted to the fluid in a in the impeller blade of an axial pump. Now, obviously as you know in case of an axial flow machine for axial flow turbine the specific speed in case of an axial flow turbine was more that means axial flow machines the head developed will be less and the flow is more similar the case in similar the case is with axial flow pump. So, in axial flow pump the head develops a relatively lower or smaller rather the flow rate or flow delivered is higher as compared to a centrifugal pump centrifugal form develops more head the low flow whereas, an axial flow pump develops low head, but more flow in other word the specific speed as you know the specific speed the definition specific speed of pump if you recall the specific speed of pump dimensional specific speed n q to the power half h to the power 3 by 4. So, specific speed for an axial flow pump is high that means it compared to that of a centrifugal pump that means it handles more flow, but at a lower head relative to a centrifugal pump well now after this I will solve an interesting example before closing this lecture please see that example we will solve this problem is a very interesting problem calculate the least diameter of a centrifugal pump to just start delivering water to a height of 30 meter if the inside diameter of impeller is half of the outside diameter and the manometric efficiency is 0.8 the pump runs at 1000 rpm again I am reading calculate the least diameter of a centrifugal pump we have to find out the least diameter of a centrifugal pump to just start delivering water to a height of 30 meter this is the static head of the pump and the other condition is that the inside diameter of impeller is half the outside diameter and the manometric efficiency is 0.8 the pump runs at 1000 rpm ok well just before this problem I like to inform you another data which I have forgotten to tell you just I have remember here if you please excuse me that you must know this that number of impeller blades in an axial flow turbine usually lies between 2 to 8 the number of blades number of blades number of blades impeller blades number of here impeller blades and the ratio of the half to tip radius that means r h r t varies between 0.3 to 0.6 this two are very important design information well so now again coming back to the example of this problem example problem the pump runs at 1000 rpm that means if a pump has to start to lift a water to a head of 30 meter the pump cannot start without a minimum diameter calculate the minimum diameter that means the impeller diameter we will have to calculate if the pump runs if the rpm is fixed so pump diameter has to be there is has to be a minimum diameter below which the pump cannot start working that means this lift of 30 meter is not possible so how to solve this problem now as you know if you recollect that the head developed by the pump to the fluid in case of a pump that head developed by the pump to the fluid is given by v 1 that is head means energy developed per unit weight energy per unit weight to be given to the fluid can be written as v 2 square minus v 1 square by 2 g if you recollect plus u 2 square minus u 1 square by 2 g plus v r 1 square minus so for any turbo machine this is the total energy transfer between the machine and the fluid for a pump this is the energy per unit weight that is the head energy per unit weight that has to be developed by the pump or that has been imparted by the pump to the fluid now as you know if you can recall this is the kinetic head or kinetic energy this is the kinetic head developed or imparted to the fluid and this part is the pressure head pressure head so this part of the pressure head is due to the change in the centrifugal head because the fluid is displaced in its position in a centrifugal force field from one radial location to other radial location that is u 2 is the final one that is the outlet u 2 square minus u 1 square by 2 g u 2 is always more than u 1 because the flow is radially outward and this one is the change in the relative velocity usually in a pump the relative velocity v r at 1 at the inlet is more than that at the outlet so therefore the relative velocity of the liquid while flowing through the pump impeller is reduced so because of this reduction in the relative velocity there is an increase in the pressure so this part is the gain in the pressure energy because of a reduction in the relative velocity according to Bernoulli's theorem there is a gain in the pressure energy so these two part combined gives the pressure head now when a pump just starts working then at that moment we can neglect these two parts because the velocity has not yet been established so at the start at the onset of the start the pump has to develop only the centrifugal head so therefore in this problem we will equate that the pump dimension should be such and its rotational speed should be such that at the start the centrifugal head is the only head that is being developed while these two terms are zero that must be sufficient one to overcome the friction and the static lift that means this must be equal to the static lift hs divided by the manometric efficiency this is very important thing so in our problem so if we do so we see that u2 square minus u1 square by 2g according to the problem it is given that 30 meter is the static lift divided by point t now according to the problem the impeller diameter inner diameter is half of the outer diameter that means you can write that u2 by u1 is d2 by d1 diameter where d2 is the impeller diameter or the outside diameter of the impeller blade and d1 is the inside diameter of the impeller blade so d2 by d1 is 2 given so u2 by u1 is 2 u2 is the velocity blade velocity impeller blade velocity at the outlet impeller blade velocity at the inlet so therefore we can write u1 is half u2 so if you substitute this then we get u2 square minus half u2 whole square divided by 2g very simple 30 0.8 which from which you get 3 by 8 1 fourth 1 minus g u2 square is 30 by 0.8 which gives u2 is equal to root over 8 into g is 9.81 into 30 divided by 3 into 0.8 under root which will be if you calculate 31.32 meter per second now if we use this then we can find out the diameter we know that relationship between rotational speed with the linear speed of the impeller diameter which will be if you calculate 31.32 meter per second now if we use this then we can find out the diameter we know that relationship between rotational speed with the linear speed is like that where n is expressed in revolution per minute so by 60 per second pi d2 so this becomes u2 here if you use this pi into d2 into revolution n or in rpm it is given 1000 in the problem now next part is simple u2 which is equal to 31.32 meter rest part is simple which gives d2 is 0.6 meter so this is a very interesting problem that a pump therefore this is the most interesting information is this so therefore for a pump of given dimensions there should be a minimum rotational speed for pump to start against a static lift with a manometric efficiency or for a given rotational speed the pump must have a minimum diameter impeller diameter for it to start well so today I think this is all alright we have completed the axial flow pump and we have solved an interesting problem today next class we will start the reciprocating pump ok thank you