 good morning welcome you to this session today we will be discussing first the diffuser of a centrifugal pump and subsequently the cavitation in a centrifugal pump well as you know the purpose of diffuser in a centrifugal pump is to convert the kinetic energy of the fluid at the impeller outlet to the pressure energy at the pump outlet this is because at the pump outlet we want fluid at high pressure instead of high velocity so therefore the prime function of the diffuser is to convert the high velocity to high pressure in diffuser now if you see the different components of a compressed pump as we have already studied here you see that the pump this is the pump impeller after the impeller there are two types of diffusers one is the vaned diffuser where a number of static vanes are there which form a diverging passage while the fluid flow through this passage its pressure is increased while the velocity is decreased then after this vaned diffuser that means at the outlet of the vaned diffuser the fluid comes out and then enters in another casing that means there is another casing which is known as volute casing or spiral or scroll casing which in the fluid flows the cross sectional area increases in the direction of the flow so therefore an additional pressure rise or the conversion from velocity to pressure is a typical conversion process from kinetic energy to pressure energy takes place in this volute casing in some cases there is no static vanes or the diffuser vanes fluid from the pump impeller that means at the outlet of the pump impeller directly comes into the diffuser or scroll casing that means in that case the diffuser volute casing itself is the diffuser while in this case both the static vanes and the volute case act as the diffuser so that is why these are known as vaned diffusers and this diffuser is usually known as this volute casing is known as volute diffuser now let us first think of this diffuser volute casing so when the fluid comes out from the impeller if you neglect the frictional effect no additional torque or force is acted on the fluid so therefore in this passage either in the volute chamber or in the diffuser vanes the fluid angular momentum of the fluid remains constant that means we can write that the tangential velocity times the radius radius means the radius from the axis of rotation is constant which means that the fluid fluid so this free vortex motion that means a free vortex type motion exist if we disregard the frictional effect because no additional torque or force is exerted on the fluid moreover if we consider the fluid to be ideal that means the frictional effects if we disregard then the flow velocities with which the liquid is coming out from the impeller is also uniform that means the variation in radial velocity we can neglect as you know for a an ideal fluid the variation across at a section across the flow is uniform that means in that case we can consider the radial velocity is also constant so a combination of a uniform radial velocity and a free vortex motion gives rise to a spiral spiral vortex motion a spiral vortex motion for which the streamlines are spiral in shape that means this gives rise to a pattern of spiral streamlines this combinations of free vortex motion and the constant radial velocity gives rise to a spiral pattern of spiral streamlines and the most important fact is that or most important consideration is that the shape of the volume has to be made in matching with the streamlines pattern that means they have to be made spiral in matching with the pattern of the streamlines this is one of the most important considerations in designing a diffuser usually at maximum efficiency 10 percent of the total head is lost in the diffuser because frictional effect we cannot neglect of total head is lost that means in diffuser frictional effect cartels 10 percent of the total head in converting its kinetic energy into pressure energy so this is almost all in short about the volume casing as the diffuser then we come to the vane diffuser you see that how does a vane diffuser look like well a vane diffuser looks like this now as we have seen that in many circumstances there is a set of static vans act as vane diffuser these are provided when a shorter length of a pump is required to give a required pressure that means when the rise in pressure from the impeller outlet is required in a shorter length that means when the size of the pump is important there is an advantage of having static vans as the diffuser vans well that means when the size of the pump is reduced so the diffuser vans act in addition to the volume casing in changing the kinetic energy into pressure energy so that we can get higher pressure at the pump outlet so what is there you see that a ring of static vans surrounds the impeller outlet this is the impeller this is the impeller so these are the static vans so this creates the diverging passage this is the diffuser passage and here the process of diffusion that is conversion from kinetic to pressure energy takes place in a very efficient way within a shorter length the design criteria is that this angle of divergence should be limited to limited between eight degree to ten degree to avoid boundary layer separation another criteria is there that the number of vans should be compromise between two phenomena one is that if you increase the number of vans the process of diffusion will be better process of diffusion means the conversion from kinetic to pressure energy will be better but however if we increase the on the other hand if we increase the number of vans the frictional losses will be more so therefore the diffusion process and the frictional effect diffusion process and the frictional effect this two counter acting phenomena decides the number of vans that means if we have more number of diffuser vans the process of diffusion that is the conversion from kinetic to pressure energy will be better whereas the frictional losses will be more but if we have less number of vans the frictional losses will be less but the conversion will not be better that means though the total energy at the outlet will remain same but one of the major aims that means purpose to increase the pressure energy will not be much so that the compromise between these two decides the number of vans in this diffuser usually the number of vans another criteria is that the number of vans should not have any common factor with the number of vans in the impeller to avoid resonant vibration so these are the important criteria for vane diffusers so this is almost all in short the principle of diffusers in a centrifugal pump that to convert the kinetic energy into pressure energy we provide vane diffusers and also a valued chamber vane less diffuser known as valued chamber and scroll crescent so now I will come to the cavitation phenomena of cavitation in a centrifugal pump I think you all know what is the cavitation we have already studied it in case of turbine so here we will see that how the cavitation phenomena limits the operational or may put the constraint on the operational conditions that the height above which a pump has to be set from the sum that is the reservoir level from which the liquid has to be pumped now look the question of cavitation comes in any hydraulic circuit as you know when there is a chance of having a pressure lower than the atmospheric pressure in this hydraulic circuit now in a pump as you see that the liquid is usually taken from the sum which is at atmospheric pressure therefore in the suction line the pressure is below the atmosphere obviously otherwise it cannot draw the liquid from atmospheric pressure to a higher height so therefore while it is flowing to the suction pipe no energy is added from outside so fluid has to flow because of the pressure differences that means the atmospheric pressure should be higher than that in the suction pipe so therefore in the suction pipe pressure is below the atmosphere and this pressure is minimum at the end of the suction pipe obviously that means at the impeller inlet so therefore we see there should be a check for this pressure should that this pressure should not fall below the vapor pressure of the liquid at the working temperature this is the restriction for the cavitation so let us now see how we can develop the expression for this simply by writing the Bernoulli's equation now if we write the Bernoulli's equation between the two points one at impeller inlet another at the sum we can write this that at the impeller inlet if we describe the quantities with a suffix one then p 1 by rho g plus v 1 square by 2 g well plus z 1 plus head loss h f this head loss is head loss in the suction pipe is equal to the pressure at the sum at the liquid free surface of the liquid at the lower reservoir this will be atmospheric pressure p atmosphere rho g there is no velocity well there is well the data made is 0 because if we measure this elevation z 1 from the level of the free surface at the sum then the velocity is also 0 so therefore we see that this p 1 that is the minimum pressure let us now write the p 1 as the p minimum p 1 as the p minimum that is the minimum pressure which is at the impeller inlet it will be p atmosphere by rho g minus all these quantities v 1 square by 2 g plus z 1 plus h and it is obvious from the physical concept that if liquid has to be drawn to a height z 1 overcoming a frictional resistance h f and generating a velocity v 1 from an atmospheric pressure under static condition the pressure at the outlet end of the pipe has to be less than p atmospheric by this amount because this difference creates this velocity of the fluid it is elevated to a distance z 1 and over comes a frictional resistance loss loss due to frictional resistance h f so straight from the Bernoulli's equation we get this here one interesting thing is that in case of a turbine we have seen that in a draft tube h f made a favorable effect in this p minimum that means if the loss was more then the pressure at the runner outlet or at the draft tube inlet was higher but here it is just the reverse if the frictional losses are more so pressure at the impeller inlet is lower so therefore friction loss has to be made as minimum as possible this is the reason for which we try to avoid any additional restriction in the suction tube usually there is a very interesting thing we sometime ask that if a pump we see the two sides the pipes are of different diameters definitely you will install the pump with the it is obvious because if we cannot see the thing from outside everything is under the casing it is very difficult to know which one is the inlet and which one is the outlet it is from this sense you can guess that a pump of a pipe of higher diameter is the suction side because suction side losses has to be kept as more as possible because pipe with higher diameter give lower hydraulic losses similarly we avoid any extra bends or any extra valves if it is not needed in the suction line because we cannot help without giving strain in the suction line because in suction line we will have to get rid of the impurity in the liquid to be pumped otherwise it will cause bad effect or damage to the impeller blades similarly there is a valve known as non-return valve in the suction line this is there this is because the liquid should not go back to the sump again while in operation so these are the essential accessories or attachment to the suction line so we avoid any additional bends additional valves or preferably a relatively higher diameter in the suction line this is because to keep the hydraulic losses or losses due to friction minimum so that the cavitation is avoided so that the minimum pressure which takes place at the inlet to the impeller at the outlet of the suction pipe should be kept as high as possible so this is one of the important considerations in suction design in the suction pipe line now in a similar fashion if we define a cavitation parameter how did we define in case of turbine that if I write this velocity head then the velocity head will be p atmosphere by rho g simply algebraic manipulation minus p minimum now I write p1 as the p minimum the same thing that means I take it here p minimum by rho minus z 1 minus h so this is v1 square by 2 g and in the similar fashion as I did as we did in case of turbine if this is expressed in terms of sigma c into h where sigma c is known as critical cavitation parameter or simple cavitation parameter sigma c into h then we can write the same quantity that means the if this is write this is expressed as sigma c into h the straight way we can write sigma c is equal to p because I want to reduce another line by rho g minus p minimum by rho g minus z 1 minus h f by h so this is the value of the this is known as the critical cavitation parameter cavitation parameter which is definitely a operating parameter or a design parameter for selecting the pump for its operational conditions in a similar way as it was done in case of turbine we can define another cavitation parameter sigma known as thomas cavitation parameter where you see the minimum pressure will be the vapor pressure of the liquid at the working temperature that means if we just express or substitute this minimum pressure as the vapor pressure which should be the minimum limit the minimum pressure this minimum means this is the minimum pressure compared to the pressure in the entire hydraulic circuit of the pump. So, this minimum pressure should fall to a minimum value of the vapor pressure for the working of the pump to avoid cavitation now if I substitute this you know in the similar fashion as we did in case of turbine this is known as thomas cavitation parameter thomas cavitation thomas cavitation parameter the numerator of this is very important in testing a pump or designing a pump the numerator of this quantity which is known as net positive you write it is very important net positive net positive suction head in short we write n p s h net positive suction head this quantity this p atmospheric by rho g minus p v by rho g minus z 1 minus h this is the net positive suction head p v that p minimum should not fall below p v. So, therefore, we see for cavitation to avoid so the pressure at the impeller inlet should always be higher than this that means sigma should be greater than sigma c to avoid cavitation. So, this is a fixed design parameter this thomas cavitation parameter as you know earlier in case of turbine and sigma c is determined from the operating condition from the operating condition of the pump its specific speed and the operating condition now therefore, sigma c should always be less than sigma or sigma should be greater than sigma c to avoid cavitation. That means for a pump the design should be such that it should give a high value of the thomas cavitation parameter to do that or to maintain that this z 1 has to be as low as possible. So, depending upon the vapor pressure of the liquid at the working temperature and the head under which the pump is operating. So, therefore, sometimes we will see to make this sigma high to avoid cavitation z 1 has to be kept very low sometimes negative for which the pump may have to be said below the level of the some you know sometimes from your common experience if you have a deep tube well if you have to pump liquid from the tube well. So, if you place the pump at the top floor of the tube well then you can expect the cavitation. So, you can calculate it that simply the height of the water column could correspond to a pressure which is lower than the vapor pressure at the outlet end of the pipe. So, therefore, pump has to be said at a level below that floor level because if you give a very. So, therefore, it is a very much restriction you must know that for a centrifugal pump you should not allow a vertical height of the suction pipe to a level where it simply for the height itself without velocity head without considering the velocity head and losses gives a pressure which is very equal or very near to the vapor pressure of the liquid the water for example, at the working temperature. So, sometimes you will see in practice the pumps are said at a level very low the pumps are not said at the top floor of the deep well. Now, therefore, we see sometimes we may have to go for a negative jet that means sometimes below the some level pump has to be said to increase sigma to avoid cavitation. So, I feel at this juncture to understand cavitation better we should solve a problem immediately yes please yes no here the problem is that if we decrease jet you are correct sigma c is affected, but this affect cannot be seen altogether because jet and p minimum are not independent because if you decrease jet. So, p minimum will also change the minimum pressure at the inlet to the impeller or at the outlet of the pump. So, here you cannot say solely the effect of jet on sigma c yes if you decrease jet then p minimum will also increase. So, therefore, this value may not be increased. So, therefore, here you cannot see the effects solely that is why it is the sigma where z 1 is the only operational parameter well let us solve a problem good let us solve a problem. So, that we can immediately apply this concept how it is done just take one problem example when you can write it when a laboratory test was carried out when a laboratory test was carried out on a pump well it was found that for a pump total head of 36 meter at a discharge of 0.05 meter cube per second cavitation began when the sum of the static pressure and the velocity head at inlet was reduced to 3.5 meter that means the cavitation began the total head of the pump is 36 meter the discharge 0.05 meter cube per second these are the two important operational parameters operating conditions of a pump head and discharge cavitation began when the sum of the static pressure and the velocity head at inlet inlet means inlet to the pump which means inlet to the impeller because impeller is the first component which is reached by the fluid was reduced to 3.5 meter the atmospheric pressure well the atmospheric pressure was 750 millimeter of mercury this is the atmospheric pressure 750 millimeter of mercury the important data is that and the vapour pressure of water was 1.8 kilo Pascal that means this is the vapour pressure of water which means this is the saturation pressure corresponding to the working temperature of the water at this pressure water boils up at the working temperature well if the pump is to operate at a location where atmospheric pressure is reduced to 620 millimeter of mercury that means it is referred to another working condition where the atmospheric pressure is reduced to 620 millimeter of mercury that means reduced pressure you operate the pump at a higher altitude where the pressure is reduced and the temperature is so reduced and the temperature is so reduced that the vapour pressure of water is 830 Pascal that means you go to a region where the pressure atmospheric pressure is reduced to 620 millimeter of mercury and also the temperature is reduced such that the saturation pressure of water at that working temperature is 830 Pascal the temperature is so reduced that the vapour pressure of water is 830 Pascal what is the value of cavitation parameter when the pump develops the same total head and discharge that means in the second case the pump head and discharge remains the same well this is the clue to the problem the total head and discharge remains the same we have to find out what is the cavitation parameter that means this is sigma c is it necessary to reduce the height of the pump now in the second case is it necessary to reduce the height of the pump and if so by how much clear now let us well let us solve the problem well now let us solve the problem ok let us solve the problem now you see the initially the pressure head p 1 first case by rho g plus v 1 square by 2 g it is given as how much 3.5 meter now at this condition what happened the cavitation began well the cavitation began at this condition that means the p 1 is p v so at when the cavitation begins that p 1 is p v so therefore what is p v p v is 1.8 kilo Pascal so we can find out v 1 square by 2 g is equal to 3.5 minus p 1 is 1.8 kilo Pascal 10 to the power 3 Pascal and rho 9.81 so this gives well 3.32 meter so if you recall this is the net positive suction head this is the net positive suction head ok let us recall this what is net positive suction head this is net positive suction head and this is nothing but v 1 square by 2 g you know this is nothing but v 1 square by 2 g that we have already seen earlier this is the v 1 square by 2 g so therefore this is 3.32 net positive suction head v 1 square by 2 g so sigma the cavitation parameter sigma c is v 1 square by 2 g h so this will be 3.32 divided by what is the head that is 36 that becomes 0.09 now here one thing you will have to understand that this thomas cavitation sorry critical cavitation parameter is a dimensionless quantity and it depends upon the pump speed discharge and the head when the discharge head speed are constant so the this value of sigma c representing a dimensionless parameter representing the similarity criteria remains same for the second case also that means this value will not change until and unless the specific speed is changed understand the specific speed is changed because this dimensionless value is a function of specific speed so so long and n q and h remains same that means the pump is operating at the same specific speed well pump is operating at the same specific speed so this sigma c value will be unchanged this also I discussed while discussing the turbine so therefore sigma c in the second case as it is given in the problem in this way that means you calculate it in the first case and tell that this is same for the second case too now let us find out really that whether the height restriction that means whether the height has to be decreased or not if decreased by how much so let us find out from the Bernoulli's equation in the first condition p 1 by rho g they write the Bernoulli's equation between the pump impeller inlet and the sump inlet sump free surface at the sump inlet let us define by this prime at the superscript in the first case that v 1 square by 2 g well plus z 1 dash plus h f 1 h f dash is p atmosphere by rho g well so therefore we can write z 1 dash plus h f dash is p atmosphere by rho g minus p 1 dash by rho g that means the minimum pressure which is the vapour pressure in the first case rho g well vapour pressure in the first case second case I told that superscript for the second case or first case rho g minus p 1 dash by rho g that means the minimum pressure which is the vapor pressure in the first case rho g well vapor pressure in the first case second case I told that superscript for the second case or first case first case so this is the first case well p atmospheric minus v 1 square so velocity is same for both the cases because the critical cavitation parameter remains the same so in the first case what is the atmospheric pressure atmospheric pressure is 750 millimeter of mercury so it will be 0.75 into 13.6 so I rho is 10 to the power 3 in case of water so 0.75 into 13.6 into 10 to the power 3 9.81 9.81 will cancel what is p v dash p v dash in the second case is 1.8 into 10 to the power 3 pascals 10 to the power 3 into 9.81 and v 1 square by 2 g we have calculated v 1 square by 2 g that is 0.092 into 9.81 because 0.092 is v 1 square by 2 g h that is the sigma c into g so with consistent unit this is z 1 dash plus h m dash yes any problem p atmosphere well all right atmospheric pressure is 0.75 meter of mercury times is density 13.6 10 to the power 3 times the g that is in pascals so this divided by rho g of the water because this is all in head of the working liquid that is water so this becomes this this is p v dash by rho g this is v 1 square by 2 g this comes out to be if you calculate 6.7 meter that means the sum of the elevation head elevation head or elevation from the sum to the pump impeller inlet and the frictional losses comes out to be 6.7 meter well all right if I calculate it in the second case which I define without suffix p 1 by rho g v 1 remains same plus z 1 plus h f is equal to p atmosphere by rho g then what is the value of z 1 plus h f now you can find out only the application of Bernoulli's equation at the pump inlet that is impeller inlet and the free surface at the sum so this will be p atmosphere by rho g now I write the expressions what is the atmospheric pressure please tell me in that case 6 20 millimeter that means we can state right 0.62 into 13.6 rho g rho g will cancel minus what is the vapor pressure of water at that condition the vapor pressure of water 8 30 very good 8 30 Pascal's so it is already in Pascal's 9.81 into 10 to the power 3 this is p 1 by rho g minus v 1 square by 2 g this remains the same because this is again 9.81 into 0.092 that is sigma c which one which one which one which one which v 1 square by 2 g is 3.32 oh sure so why I am writing this v 1 square by 2 g h oh sorry very good 3.3 so this is not sigma c this is sigma c into h v 1 square by 2 g this is sigma c I am just putting the value of sigma c very good it will be 3.32 into 9.81 oh it will be simply 3.32 because it is rho g v 1 square by 2 g yes it is simply 3.3 you are correct so this value will be 6.7 as I am writing from this figure which is worked out problem so it will be 6.7 taking 3.32 I am sorry 3.32 yes because v 1 square by 2 g h is this 0.09 2 we have calculated so v 1 square by 2 g is equal to 36 into 0.090 you are correct because we already calculated v 1 square by 2 g is 3.32 then divided by the head we call this fine this is the head in terms of meter 2 g is already divided so 3.32 very good so here also 3.32 I think it is ok now if you calculate it it will be giving a value of 5.0 now here 1 assumption is there 1 assumption is there which is very important this is a very silly thing actually I just made a silly mistake the most important assumption is that we consider the h f to be constant here I have written that h f dash here here is because h f dash that means h f is equal to h f dash here 1 important assumption comes the next step before going to next step is that if we consider the frictional loss is same why this is justified that if the discharge is same and the velocity at the impeller inlet is same we can consider that the hydraulic losses that the losses due to friction also remain same. So, if we do that then we can tell that z 1 dash minus z 1 because h f will cancel will be 6 that means this is the amount by which so if we consider that h f in the two cases are same therefore by comparing the value we can tell that yes it has to be reduced and by what amount by this amount 6.7 minus 5.03 it becomes 1 point that means in the second case we have to reduce by this amount the height of the pump from the sum that means it has to be kept at this value otherwise the cavitation will occur simply because if you make this z 1 more than 5.03 this pressure will be lower than this vapor pressure 830 Pascal corresponding to this atmospheric pressure. So, you see how we use the cavitation parameter sigma c and by making use of the Bernoulli's theorem or Bernoulli's equation between the pump inlet impeller inlet and the inlet or the free surface of the sum we can determine the height above which from the sum free surface of the sum the pump has to be placed. So, this way you get a clear idea how the cavitation parameter is used in determining the height of the pump where it has to be set above the sum to avoid the cavitation that means the minimum pressure that is the pressure at the inlet to the impeller should be more than the vapor pressure of the working fluid at the working temperature. Well, thank you any question very simple I think very simple ok thank you.