 Hello everyone. Today we are going to study the numericals on Pelton wheel that is Pelton turbine. I am Professor S.P. Mankani, Assistant Professor, Department of Mechanical Engineering Vulture Institute of Technology, Svalapur. So at the end of this session students will be able to analyze velocity diagram and the students will be able to solve the numericals on Pelton wheel. A Pelton wheel is working under a gross head of 400 meter. The water is supplied through penstock of a diameter 1 meter and length 4 kilometer from reservoir to Pelton wheel. The coefficient of friction for the penstock is given as 0.08. The jet of water diameter 150 millimeter strikes the bucket of the wheel and gets deflected through an angle of 165 degree. The relative velocity of water at the outlet is reduced by 15 percent due to friction between inside surface of the bucket and water. If the velocity of the turbine is 0.45 times the jet velocity at the inlet and the mechanical efficiency as 85 percent, determine power given to the runner, sharp to power, hydraulic efficiency and overall efficiency. So now we are going to write up the details given in the numerical. So gross head 400 meter, this is gross head 400 meter, diameter of penstock d 1 meter, diameter of penstock as representing as a d 1 meter. The length of penstock l is equal to 4 kilometer that converted into 4 into 1000 as 400 meters. Coefficient of friction f is given as 0.08, 0.008, diameter of jet small d representing as small d 150 millimeter that is 0.15 meter, angle of deflection is given as 165 degree, 165 degree, angle phi. So this angle I am going to represent in the next diagram, 180 degree minus 165 degree equal to 15 degree as a phi angle. Relative velocity at outlet V r2 equal to 0.85 that is 85 percent of V r1. Velocity of bucket u is equal to 0.45 the jet velocity. Mechanical efficiency nu m equal to 85 percent is given condition. So let we are going to consider this V star as the velocity of water in penstock and V1 is the velocity of jet of water. So now here it is a velocity diagram. So here is the inlet condition and this is the outlet condition. So the major information we required here as a V1 is not given in the problem. So to calculate V1 we are expected to utilize these all given values. So here this is a V1 as an inlet velocity, V1 is up to this particular point representing and Vr1 is the relative velocity and Vw1 is the wheel velocity at inlet condition. The corresponding values are at outlet condition. So V1 and Vw1 we are supposed to calculate it. Once we are going to calculate these two values we are going to calculate all the required values as power given to the runner, shaft power and hydraulic efficiency as well as overall efficiency. So here this is the angle phi. Phi is the angle here and this is 165 degree angle is given and this is remaining angle this is a 15 degree angle 180 minus 65. So this is 15 degree angle this is phi angle. So here we are going to see this one as the water storage. This is a total head Hg given in the problem. Hg value is given in the problem as 400 meter. This is a given condition and this hf is equal to 4fl V2 upon tri gd. So this hg is equal to hf plus whatever the head we are going to get it at the outlet of the nozzle. By using these equations we are going to calculate the value of V1 in the equation. So as far as the required values as power given to the runner. Power given to the runner is nothing but the outlet of the nozzle whatever the energy is given by this outlet of the nozzle is given to the blades. This blade is nothing but we are going to consider this as the energy given to the runner. So here from this water to this particularly blade is a hydraulic efficiency new edge and from the blade to the runner and runner to the shaft. Runner and shaft we are going to consider it as a mechanical efficiency and overall efficiency is considering it as shaft and this is outlet of the this nozzle. So here we are going to apply the continuity equation. Continuity equation from this particularly point and up to this particularly point. There is a frictional losses whatever it is going to be taken place in this particularly pipe. We are going to apply the continuity equation at the water level and at the exit of the nozzle. We are going to apply the continuity equation. So based on the continuity equation we are going to apply here area of the penstock, area of the penstock into V star. Area of the penstock into V star we are going to consider it as this is the cross sectional area at this particular as area. So pi by 4 d square d is the diameter of the penstock capital D and this V star is the velocity inside the pipe and that is equal to area of the jet into V1. Area of the jet is the outlet at the outlet of the nozzle at the outlet of the nozzle here. So this area we are going to consider this as a area of the water at the outlet. We are going to be applying here as a continuity equation. So based on this one pi by 4 d square into V1. So based on this particular equation we are going to calculate the V star. So V star is we are not going to get the value as a direct one number. So here it is in terms of V1 we are going to get it. So by using this value in the next step we are going to calculate the V1 value which is the required for the further calculation of the problem. So now we are going to applying the Bernoulli's equation to the surface of the water in the reservoir and outlet of the nozzle. We are going to apply the Bernoulli's equation to the surface of the water in the reservoir and outlet of the nozzle. So here we are going to apply from this particular surface the Bernoulli's equation. So Hg is equal to head loss due to friction plus V1 square. Here Hg that is a gross head. Hg is equal to Hf plus V1 square by 2g. So by using this value we are going to be calculating as the value of V1. So here Hg is given in the problem as 400 meters is equal to 4fl V square upon twice gd plus V1 square upon 2g. So this V1 value our intention is to calculate this value V star we are having in terms of V1. So by substituting this V1 sorry V star in this particular equation we are going to be getting this total value in terms of V1. So that we can calculate V1. So here 4f and this is l in terms of meter twice gd and this value is V1 square by 2g sorry this is V1 square by 2g and this value we are going to be taking it from this V star. So by substituting all these values we are going to get the V1 value as 85.83 meter per second. So now we are going to be getting the value of U. U1 is equal to 0.45 V1. So this condition is given in the problem. So based on that one we are going to be substituting the value of V1 and then we are going to get this as a value of 38.62. So once you know the value of V1 we can calculate Vr1. Vr1 is equal to V1 minus U1. So based on the velocity triangle we are going to get this as a value V1 minus U1. So this value is 47.21. So next Vw1 is equal to V1 so that we are with the reference of velocity triangle V1 is equal to Vw1 and U1 and this is Vr1. So based on this velocity triangle we are going to get the value of Vw1. Vw1 is equal to V1 then Vr2 is equal to 0.85 that is the 85 percentage of Vr1 so that we are going to get the value as 40.13. Vw2 is equal to Vr2 cos phi minus U2. This is from the reference of outlet triangle of the velocity. Vw2 is equal to Vw2 is equal to Vr2 cos phi minus U2. Vr2 cos phi minus U2. So based on this value we are going to calculate the value of Vw2. So with the reference of this value we are going to calculate the discharge through the nozzle. Discharge through the nozzle is area of the jet into velocity of the jet that is Av1. A is nothing but area of the jet pi by 4 d square V1. We are going to get this value as a 1.5 meter cube per second. With the reference of this particular values and Vw1 and Vw2 we are going to calculate the work done on the wheel per second. We are going to calculate this value as 5033540 Newton meter per second. So using this value we are going to calculate the power. Just dividing by 1000 we get the power as 5033.54 kilowatts in terms of watts. So now by using the mechanical efficiency equation we are going to substitute this value as a runner power here and that is a shaft power divided by runner power gives the mechanical efficiency. So based on this one we are going to calculate the value of shaft power. So once you know the shaft power that is a shaft power we can calculate the mechanical efficiency. So then we can go for the hydraulic efficiency. Here the requirement is Vw1 and Vw2. Both these values already are calculated and U value is calculated, V1 value is calculated so that we can calculate the value of hydraulic efficiency. So once you know the mechanical efficiency and hydraulic efficiency you can calculate the oral efficiency. Oral efficiency is equal to hydraulic efficiency into mechanical efficiency that gives the value as 76.62%. So now based on the value of V1, Vw1 we can calculate rest of all the values. The important thing is to calculate the V1 value we are going to be using the continuity equation as well as Bernoulli's equation. So based on that one you can calculate these two values then you can come to the overall efficiency of the given Pelton wheel. So for a further you can refer a textbook of fluid mechanics and hydraulic machines by Dr. R.K. Bansal. Thank you.