 ణార్టి రియానోయతోమ్ఒచేనితాడి కెన్రందాటాంచి. క్రర్మోఖేగితోదా. విలాపనాసి బరర్వసి నీంకుకాయారోాసి ఇలార్నోటి . .Also, we developed an expression for specific speed of Francis runner mainly in terms of the angles of the guide vents at the exit and the angle inlet angle of the runner blade. The usual values or the typical values that cover the actual case for a Francis turbine are like that. The alpha 1, which we already noted that it is the outlet angle of the guide blades usually varies from 10 degree to 40 degree. We must know these values in practice. The beta 1 that is the inlet blade angle which varies from 45 degree to 120 degree with respect to a typical blade of a Francis runner. So, this is beta 1. So, this angle is beta 1. Well, the ratio of width to diameter at the inlet b by d which is very important in determining the flow velocity v f, determining the flow velocity v f which remains constant throughout the flow through the runner blades which varies from a value of 120 to two-third. These are the typical ranges between which the pertinent geometrical dimensions are kept and the combinations of these covers a wide range of specific speed n s for the turbines from 40 to 500. Very wide range of specific speeds covered by a Francis turbine radial flow reaction turbines. Now, as the specific speed increases mainly beyond 500, what happens is that the head under which the runner is working is getting reduced. So, therefore, essential feature of a turbo machine at a higher specific speed that means that a lower head is that it should admit a relatively large amount of flow to get some definite work output. So, therefore, the runner should be capable of handling or allowing a more amount of flow to get a very definite work output when it is operating at a very lower head at a higher specific speed. And this for to meet this requirement the shape of the runner blades have to be changed and the type of the runner also changes. For a maximum flow through the runner the flow velocity has to be axial that means if the flow rate through the runner blade has to be made very high the flow velocity has to be axial that means the direction of the flow velocity should be parallel to the direction of the parallel to the axis of rotation. And to do this and to accommodate the flow and to extract the work from the fluid flowing through the runner the shape of the runner blades have to be changed. So, this is accomplished in a machine known as axial flow turbine axial flow turbine and a machine of this type axial flow turbine was first turbine sorry axial flow turbine was first developed by an Austrian engineer Victor Kaplan and that is why the name of this turbine is Kaplan turbine according to the name of the scientist Kaplan turbine. So, Kaplan turbine is specifically an axial flow reaction turbine which is used for a very high specific speed range beyond this 500 at max higher efficiency. So, let us look what looks like a typical axial flow turbine or the Kaplan turbine which is sometimes known as a propeller a propeller or Kaplan turbine you see these are the guide vanes. So, fluid while after passing through the guide vanes the pipe vanes are such the ducts are such it is bent right angles in axial direction. So, it first enters to the guide vanes almost in a radial direction as it happens also in case of a Francis turbine then it is turned to right angle to the axial direction. So, these are the rotor blades or the runner of the turbine. So, this is the shaft usually those machines have vertical shaft. So, if we look a plan view. So, the runner blade looks like this the numbers of runner blades are usually small. So, here what happens the fluid flows parallel to the axis of rotation throughout the runner which means the entry is axial the exit is also axial. Now, I have mentioned earlier also the purpose of the guide vanes is to direct the fluid accordingly to the runner blades and also to input a little amount of pre-wheel that means a tangential component of velocity to the fluid. Now, the tangential velocity in the fluid approaching the runner is such that it can be approximated by a free vortex motion why if we neglect the friction in this duct then we can say that in absence of friction and when no work is done or extracted from the fluid then the tangential velocity there is any follows a free vortex type of motion which means the tangential velocity decreases with the increase in radius, but what happens to the blade velocities the blade velocities increases with the increase in radius because it is a solid body rotation. So, to take care of this different relationship between the fluid velocity and the blade velocities typically the tangential velocities the tangential velocity of the fluid approaching the blade and the blade velocity linear blade velocity blades are made twisted twisted means the blade angles from inlet that means the root not inlet I will tell root to the tip is varying this is known as twisted blades these blade are made twisted twisted blades twisted blades blades, the blades are made twisted. So, the flow takes place axially, almost axial through the blade, a typical peripheral section of the blade is like this. Now, if you draw the velocity triangle here, you see that the velocity triangle is like this usually. The most important feature is that flow velocity, that means this is the rotor speed u, this is the v r 1 and this is the inlet velocity. This is the flow velocity v f, which is axial here. That means in reaction turbine, in radial flow we have seen the inlet, that means either with respect to blade or the absolute velocity was in radial and tangential plane. Then it has both radial component and tangential component. In case of an axial flow turbine, it is other way, it has got tangential component. This is the tangential component v w 1 and an axial component, which is the flow velocity v f 1. The outlet velocity triangle, if you draw, will be the same as that of a radial flow reaction turbine. This v f 2 is equal to v f 1, which is equal to v a, that means the axial velocity of flow. This is typically, this is this direction, this is the u and this is the v r 2. One important thing is that the blade velocity is same at both inlet and outlet. In my diagram also, it does not look like that. Their lengths are not same, because they are same, because they are in the same axial, they are in the same radial location. That means the flow is in axial direction. So, radial location at inlet and radial location and outlet is this same. So, this is the principle feature of a axial flow turbine. The blades are twisted and the flow is throughout the axial. The main feature is that, in this case the flow velocity is much higher as compared to that in the radial flow reaction turbine. The flow velocity is here in the axial direction, because the main flow is in the axial direction. So, this way this turbine is capable of handling a very high flow rate and therefore, it is suitable for a lower head or a higher specific speed at higher specific speed, higher specific speed. The runner runs full, completely filled with the liquid, so that the reaction is imparted on the liquid and moreover the degree of reaction in this type of machine is higher than that in case of a radial flow Francis turbine. Well, after this now I will go to the discussion on draft tube, application of draft tube. Now, I have already told earlier that a draft tube is attached always to the outlet of the runner of a reaction turbine to minimize the energy loss at the outlet. Now, at the outlet end of the runner of a reaction turbine, the kinetic energy which is coming out is loss. So, if the fluid comes with a very high velocity that means, the loss in energy in terms of the kinetic energy of the fluid is very high. So, therefore, what happens is that a divergent duct is attached to the outlet end of the runner of the reaction turbine, so that the velocity of the fluid is reduced that means, a decelerating flow is caused in the divergent tube as you know in the fluid flows through a divergent duct as the area of cross section increases in velocity decreases. So, therefore, at the outlet end the velocity of the fluid gets reduced and therefore, the rejection of energy at the extreme outlet of the machine is very less. In another look we can see we can see from another point that we have already discussed earlier as seen that if you write the energy equation of the Bernoulli's equation between the inlet to the draft tube that means, the exit of the runner and the final exit of the draft tube we see that since the discharge from the draft tube is at atmospheric pressure and more over the flow through the draft tube is a decelerating flow that means, velocity is decreased. So, therefore, the pressure at the upstream part that means, for example, at the inlet of the draft tube is will lower than the atmospheric pressure because pressure has to increase for decelerating the flow which means in other way the outlet end of the runner is running at a lower is running with a lower or a suction pressure in comparison to that of a runner without a draft tube. So, therefore, the effective head across the runner is increased by attaching the draft tube this can be looked from this angle also that draft tube reduces the pressure at the runner outlet. So, this is the precise principle of a draft tube how it increase the head across the runner. Now, therefore, we see that the purpose of the draft tube is to reduce the energy at the outlet and thus to increase the head across the runner. Now, two things have to be kept in mind while designing the draft tube. Now, you see draft tube is a divergent duct first thing is that when we extract more energy in the draft tube that means, how do you extract energy that means, we reduce the loss at the outlet or we reduce the pressure at the outlet end of the runner to do that we will have to keep in mind that the loss of energy while flowing through the loss of energy of the liquid while flowing to the draft tube should be as small as possible. So, what are the sources of error one source of error is the usual friction loss that is the friction between the fluid and the solid wall or between the fluid to fluid another source of error comes while a fluid flows through a divergent duct is the loss due to boundary layer separation do you know what is that the loss due to boundary layer separation do you know it well. Now, what happens when the fluid flows through a duct of uniform cross sectional area or fluid flows through a duct of converging cross sectional area that means, either fluid flows in uniform velocity or fluid flows with increasing velocity in accelerating flow loss which is incurred is only due to friction that means, between fluid to fluid that is fluid viscosity and fluid to solid friction, but when the fluid flows through a divergent duct then another additional loss comes loss in head the loss in total energy is known as boundary layer separation what happens is that when the fluid flows through a divergent duct you see the flow is a decelerating type that means, according to continuity when the cross sectional area increases the flow velocity decreases in compliance with the Bernoulli's equation the pressure increases when the velocity decreases the pressure increases that means fluid flows against an adverse pressure gradient that means fluid flows from a lower pressure to a higher pressure. So, why the fluid flows yes the fluid does not flow from a higher pressure to lower pressure that is not the nature's law nature's law is that fluid flows from higher energy to lower energy. So, fluid is capable of flowing from a lower pressure to higher pressure because its energy at the lower pressure condition that means that the upstream condition is more than the downstream condition then only the fluid is capable to overcome this adverse pressure gradient, but what happens what the fluid which is very near to the solid wall as you know the influence of solid to the fluid gives rise to the formation of a boundary layer or what happens is that due to the frictional interaction between the solid and the fluid it is the consequence of the fluid viscosity that the fluid particles near the wall loses its velocity and in fact at the solid wall fluid velocity is zero if the solid surface is at rest that means no solid condition you know from your basic knowledge in fluid mechanics that the relative velocity between this fluid particle and the solid surface at the wall at the solid surface is zero. So, therefore, for a static solid surface static that so fluid velocity at the wall is zero. So, therefore, very near to the wall adjacent to the wall fluid velocities are very small. So, the small velocity fluid particles do not have sufficient kinetic energy to make their total energy compatible for flowing from upstream to downstream section. So, therefore, they become unable to surmount the adverse pressure instead what they do they follow the favorable pressure gradient that means they follow the path from a higher pressure to lower pressure. That means in a direction opposite to the direction of the bulk flow which is away from the solid wall. So, therefore, you will see that the fluid particles near the wall goes on flowing in opposite direction a flow reversal takes place. So, this localized flow reversal makes a recirculatory flow and forms in terms of fluid eddies and this fluid eddies a recirculatory loop along with the main bulk flow causes a loss of energy. This loss of energy mechanism is like that a pressure energy a part of mechanical energy is converted into intermolecular energy which we call as a loss of energy from the few point of mechanical energy. As you know when you deal with the mechanical energy you have already heard this term loss of energy in Bernoulli's equation energy can never be lost from the conservation of energy principle which means a part of mechanical energy which is converted to other form of energy which is not the mechanical energy we call it as a loss of energy. That means this is a loss of mechanical energy due to friction because of the formation of eddies due to a recirculatory reverse flow at the solid wall why the fluid particles cannot surmount the adverse pressure in because of their low velocity. This is a very important phenomena of separation wherever you come across flow through divergent duct or desolarating flow this type of phenomena comes known as boundary layer separation. To avoid this or to keep this to a minimum value the most important factor is that the angle of divergence or rather the rate of diffusion that is change in pressure or the change in velocity has to be made very low. Usual recommendation is that the angle should be within 8 degrees 8 to 10 degrees to avoid the boundary layer separation as much as possible. So, therefore, we see that in designing a any divergent duct in any application it is not always for a fluid machines that there are two criteria comes into picture one is the loss due to boundary layer separation loss due to boundary layer separation loss due to boundary layer separation loss due to boundary layer separation loss due to boundary layer separation another d another is friction loss frictional loss frictional loss and one thing you must know at this level that these two losses are the consequence of viscosity if the fluid is ideal neither of these losses will take place. So, you must clear your basic concept in fluid mechanics along with the additional information in fluid machine that losses due to boundary layer separation and friction loss takes place because of the fluid viscosity why because the phenomena may occur, but the boundary layer separation will not take place even if there is an adverse pressure in because the fluid particle having a low velocity at wall depends upon the interaction between the fluid and the solid surface through the viscosity of the fluid. So, these are contributed by the viscosity of the fluid. Now, the basic intention of designing any draft tube is to keep this loss to losses to a minimum value. So, that the energy is not lost where we want to retain the energy in the form of mechanical energy. Similar will be the considerations for using or providing draft tubes in a reaction turbine. Well, let us see what are the different types of reaction turbines usually incorporated in practice. Number one is straight type you can read it this is straight type this is a simple conical the frustrum of a cone a simple divergent duct straight divergent duct the angle included angle is limited to 8 degree the same angle is 4 degree this type of conical draft tube this is a vertical one it is directly attached to the outlet end of the runner it is very efficient and its efficiency lies between almost equal to 80 percent in this connection I must tell what is efficiency the efficiency you can define of a draft tube or any divergent duct in transforming the pressure energy to velocity energy rather velocity energy to pressure energy like this the inlet head inlet head that means total energy at inlet per unit weight minus head lost in the flow divided by the inlet head for an ideal fluid this efficiency is 100 percent that means it is an index of the head loss the efficiency is high means the loss of head by this two mechanisms will be low. So, therefore, this type of machine gives almost 80 percent 80 or I feel 80 to 90 percent efficiency and these are used for small specific speed low relatively low n s t specific speed turbines with vertical shaft now another type of draft tube these two types are elbow types now these elbow types draft tubes are used where sometimes in certain places sometimes we see that to reduce the cost of excavation particularly in rocks we use this elbow type draft tube where if we go for a vertical draft tube from a from the place where the runner is installed at a height from the tail raise level we see that we have to go for excavation in rock. So, to keep the cost of excavation minimum the draft tube is bent in the horizontal direction to keep its length it is very very it can be seen very clearly in this figure. So, it is this is known as elbow type that is from vertical to horizontal because of the shaft change in the direction of the flow the efficiency of this type of draft tube is in the order of 60 percent. Sometimes the cross sectional area is changed from a circular at the inlet to a rectangular rectangular at the outlet to minimize the frictional losses. So, that the efficiency is little higher. So, in practice several types of draft tubes are used. So, you see the angle of the draft tube is limited that is diverges angle by the flow separation loss and the length of the draft tube is being compromised within the friction between the frictional loss in the draft tube and providing the length suitably either in the vertical direction or horizontal direction depending upon the places of application whether to reduce the cost of excavation in rocks we have to place a horizontal part you have to incorporate a horizontal part where the vertical part of the inlet portion has to be bent with horizontal part in sacrificing certain amount of loss by the change in direction. Now, after this I will come to another very important phenomena cavitation in a fluid machines. Now, cavitation this word as applied to the fluid flow is not necessarily restricted to fluid machines or a reaction machines this is applied to any fluid flow problem or to any hydraulic circuit probably you have heard the name cavitation while reading the siphon in your basic fluid mechanics class while you have read the siphon you have seen that the pressure becomes low at some points in the siphon where the problem of cavitation comes what is cavitation now let us define the cavitation in this way that if there is an hydraulic circuit that means the circuit which contains the flow of a liquid and if there is any chance of having pressure lower than the atmospheric pressure at some part or during some region of the flow of that hydraulic circuit. So, one in that case in that case one has to be very careful that this pressure that the minimum pressure which is below the atmospheric pressure should not fall to the vapour pressure below the vapour pressure or equals to the vapour pressure of the liquid at the working temperature why if the minimum pressure in the hydraulic circuit falls to the vapour pressure of the liquid at the working temperature liquid starts boiling at that pressure. So, when liquid starts boiling so pockets of vapour liquid vapour will be formed which will simply lock the flow or stop the flow this is precisely what is known as cavitation what happens there after is that when the vapours are formed where the pressure is sufficiently low they are carried away with the liquid to a high pressure region where the vapour collapses or busts forming into cavities and the liquid from surrounding zone rushed to fill up the cavities and if this phenomena bursting of the vapours takes place very near to the wall and liquid from the surrounding region rushes to fill up the cavities they causes an erosion effect to the wall of the tube or wall of the duct causing damage the damage is found in the form of pores in the wall of the duct or wall of the tube. So, therefore, it is dangerous to allow such phenomena to occur not only the flow will be stopped within few minutes the wall of the duct or the duct itself or the tube itself of the hydraulic circuit that part of the tube will be damage to the own out. So, this phenomena is known as cavitation. So, therefore, we will have to have a check at the minimum pressure section or the point of the hydraulic circuit. So, that the pressure at that point should not should be at least more than the vapour pressure of the liquid corresponding to its existing temperature similar is the case for a draft tube. Now, if you write the Bernoulli's equation as I discussed earlier at the inlet to the draft tube and the outlet from the draft tube. Let us consider the inlet where the pressure is p minimum because we know that in a draft tube this minimum pressure will occur at the inlet to the draft tube that means at the outlet of the runner. Then I can write the equation like that p minimum by rho g let v is the velocity there that means this v is the velocity coming out from the runner outlet or this v is the velocity at the inlet to the draft tube. Let z is the vertical height from a reference datum which is usually taken as the tail raise that means this is the height from the tail raise to the inlet of the draft tube that means this is the height at which the runner is placed. This I can write is equal to p a by rho g because the outlet pressure of the draft tube is the atmospheric pressure draft tube discharges liquid into atmosphere. Now, if we neglect the velocity at the outlet end of the draft tube we can neglect it if we consider the draft tube sectional area is such that its velocity at the outlet is negligibly small compared to its inlet. Then the datum it is zero because we have taken the reference datum at the tail raise level itself then another term is this head loss h f that is the loss in energy due to friction and boundary layer separation or due to the change in direction. So, it consists of all the losses in course of flow through the draft tube. So, here we see that p minimum by rho g is equal to p a by rho g minus v square by 2 g plus z plus h f. So, we see that if the velocity at discharge from the runner or at inlet to the draft tube is very high or the height at which the runner is placed from the tail raise level is very high. There is every chance that the minimum pressure at the suction may fall below the vapour pressure because the minimum pressure will depend upon this quantity more is the velocity at the inlet to the draft tube more is the height of the runner from the tail raise level less is the pressure at the inlet to the draft tube. Of course, the friction makes an advantageous case in advantageous case in this situation for the p minimum, but the frictional loss is very less as compared to the total energy v square by 2 g plus z. The friction puts a short of resistance you can understand physically this way sir why you may ask that friction increases the pressure because friction puts a short of resistance to the flow frictional loss. So, that the upstream pressure is somewhat increased. So, the influence of friction here is favourable as far as the reduction in the p minimum is required. Now, the basic consideration for cavitation not to occur is p minimum should be greater than p v that means, the vapour pressure of the liquid used. Now, what happens in practice is like that if I write this equation that equation little here itself in a different manner we can write with a little other form with an another form rearrangement v square by 2 g is p a by rho g minus p mean by rho g well plus h f or rather v square by 2 g minus h f is written p a little rearrangement is made minus p mean by rho g well minus z. Now, this part is expressed in practice as a function of the head across the runner that function is sigma c. So, let us express this way you can see this then we can write that sigma c becomes equal to becomes equal to p a by rho g minus p minimum by rho g minus z by h this sigma c is known as critical cavitation parameter critical cavitation parameter critical cavitation parameter critical cavitation parameter and it is an operating condition of the turbine which depends upon the minimum pressure at the runner outlet the height at which the runner has to be installed and the head across the machine. Another parameter sigma is defined according to a German scientist Thoma is known as Thoma's cavitation parameter which is used as a design criteria or a design parameter to determine whether the cavitation will occur in a particular situation or not what is this this is simply the same thing almost, but instead of p minimum we substitute the criteria that is the vapour pressure it is the limiting value by h that means in under all situations cavitation to avoid this p minimum has to be more than p v which means sigma has to be greater than sigma c to avoid cavitation well to avoid cavitation. What is done in practice that this sigma is calculated Thoma's cavitation parameter with the vapour pressure of the liquid and under the operating conditions sigma c is calculated and it is checked that whether sigma is greater than sigma c or not. Now you see the head of the runner is increased or the height of the runner from the tail less level is increased sigma is reduced and there is a chance of cavitation to occur. Usually what is done in practice that from this equation the maximum value of z is calculated that means let us write the z max that is the maximum height of the runner that means the maximum height at which the turbine runner can be placed will be sigma rather I can write this way p a by rho g when the p minimum will reach p v p v by rho g minus sigma c into a that means this equation gives z is equal to p a by rho g minus p minimum by rho g minus sigma c h. So, in p minimum will reach p v that is the vapour pressure which will give the maximum z. So, the values of sigma c the operating parameters are known to us in case of design while we design the turbine. So, knowing the value of sigma c we find out what is the maximum z that is required now sometime it appears that when h is very high that means turbine operating under a very very high head that means the low specific speed z max is reduced. Sometimes it appears that z max may be negative that is turbine has to be placed below the tail water level the situation is not so this is because the sigma c here is a function of is a function of specific speed of the turbine. I give you an example that if we plot the values of sigma c which are available to the design engineers in practice with n s t the values are like this for a francis turbine francis for a kaplan turbine it is like this kaplan turbine. Now, it is found for a francis turbine the values of sigma c is a direct function of the specific speed. So, when the head is high specific speed equation n p to the power half by h to the power five by four that means at a higher head the specific speed is low. So, therefore, we do not go for a kaplan turbine rather we go for a francis turbine where we find at a lower specific speed these all specific speed range for the reaction turbines only the lower specific speed the values of sigma c is also low. So, that if we employ the value of sigma c even at a higher head we get a value of z max which may not be negative. So, the difficulties can be avoided that turbine may not be set at a level below the tail less water level, but the height above the tail less water level must be very small not very high to avoid the cavitation well. So, this is about all this is all about the cavitation in a reaction turbine. So, if you have any questions you can ask me. So, today up to this only well please any question this cavitation principle of draft tube and the cavitation it is very important parameter cavitation is a very important parameter in designing any hydraulic circuits where there is a chance of pressure within the circuit to fall below the atmospheric pressure. So, you must know this thing any question ok. Thank you.