 Good morning, I welcome you to the session of fluid machines. Today, we will first discussed the concept of specific speed in a fluid machine. Well, in the last class, we have seen that by the application of principle of similarity, we derived the different dimensionless terms representing the similarity criteria for a flow in a fluid machine. What are those terms? If you recall, the terms are like this pi 1, the different pi terms was q by n d q. The second pi term, pi 2 is g h by n square d square. The third pi term was pi 3. If you recollect rho n d square by mu. Fourth pi term, let us write here pi 4 is equal to p by rho n cube d 5. The fifth pi term is e by n cube d 5. The fifth pi term is e by n cube d 5 by rho n square d square. Now, these pi terms represented the similarity parameters. That means to maintain the similar flow situations in fluid machines, we will have to make these terms fixed. Now, we have also recognized that in case of fluid machines handling incompressible fluids, where the compressibility effect is not important, then the pi term does not come into picture. Again, we recognized that in most cases, the influence of viscosity is very negligible. We have considered to be secondary in respect of flow through fluid machines, so that pi 3 times terms can also be disregarded. Therefore, we are left with these 3 pi terms pi 1, pi 2, pi 4 and the flow in a fluid machines can be described by these 3 dimensions less terms that q by n d cube g h by n square d square and p by rho n cube d 5. These are pi 1 terms, pi 2 terms and pi 4 terms. Therefore, we see for a fluid machine handling incompressible fluid that is the liquid, these 3 non-dimensional terms represent the similarity criteria of flow through fluid machines. Now, at this juncture, I like to tell you one very important thing that the similarity is short or the similar conditions are short in a problem of same physics, problem described by the same physics. For an example, similarity cannot be achieved between the problem of a fully developed flow through a pipe and flow through a channel and a flow past a flat surface. Why? Flow through a pipe, fully developed flow through a pipe is governed by the pressure force and viscous forces. While the flow through a channel is dominated by the gravity forces, gravity force is the dominating force. Again, flow past a flat surface is a typical boundary layer flow, where the flow is dominated by inertia force and the viscous force. So, they are distinct classes of problems governed by the different physics. Therefore, similarities cannot be maintained or similar situations cannot be obtained between these different classes of flow. So, we can have a similarity in a particular class of flow. For example, in the case of flow through a pipe, in that case, we can consider different situations under different operating conditions. Similarly, flow through channels, we can consider different conditions at different different flow situations at different operating conditions, which are similar in nature. Another important thing in this respect is that, even for a particular class of flow. For example, we consider the flow with an open boundary, flow past bodies. Again, you see that the shape of the bodies or the geometry of the bodies is a very important thing. That means, it is an important thing. That means, flow past a sphere or flow past a cylinder are different. That means, similar conditions cannot be maintained between flow past a sphere and flow past a cylinder. We can maintain the similarity conditions for flows past a sphere, different situations of flow past a sphere or different situations of flow past a cylinder. cylinder. So, therefore, it is also very important that the shape or size of the system or the flow geometry is very important in define a particular category of flow. So, therefore, the similarities are short in same physics of flow in the same category of flow at different situations. Now, here also if you see the fluid machines well you see the different fluid machines are of different shape and size. Now, we have seen earlier that the geometric similarity or geometrical similarity between different condition or different conditions or between different system represents the similarity of shape and size. So, they vary in their geometrical dimensions and geometrical dimensions are proportionate to each other. That means, one system is a enlargement is an enlargement or a reduction of the other system, but shape is not changed. Now, in similar in similar way we can tell that fluid machines may be of different shapes. For example, the impulse machine is having a different geometrical shape shape means shape of their rotor and stator. Similarly, reaction machines are also having different shapes even for a reaction machine the radial flow machines and axial flow machines vary in their different in varying their geometrical shapes. So, therefore, the similarity conditions between the impulse machine and reaction machines and different types of reaction machines having different geometrical shapes cannot be maintained. So, what we can do that similar conditions can be maintained of a particular kind of fluid machines. For example, impulse machine of a particular type radial machine is sorry reaction machines of a particular type radial flow reaction machines. So, in that case we can tell the different situations may be there referring to a particular kind of machines of a particular geometric shape which operate under different operating conditions. And also the geometrical size or dimensions of the system may be varying one will be an enlargement or a reduction of the other. So, therefore, the machines in which the physical similarities are short belong to a particular category of the machine and all such machines of a particular category form a series known as homologous series this very important terminology homologous series. That means homologous series refers to a particular category of machine or a particular type or shape of machines where there may be different machines which are only varying in their geometrical dimensions working under different operating conditions and they form a particular series known as homologous series. Now, you come to another situation usually the performance of a fluid machine is expressed as or expressed by the rotational speed power head developed and the flow rate. These are the three four quantities that rotational speed power transferred between the fluid and the rotor head across the machine and the flow rate through the machines. These four quantities are the performance parameters of a machine that machine are usually specified by these four performance parameters for a turbine usually for a turbine for a turbine usually which develops power n p and a these three quantities are referred to as the performance parameters. That means performance of a turbine is expressed by these three parameters rotational speed the power which is delivered by the turbine and head which is given away by the fluid. Similarly, for a compressor or for a pump that means for a pump or for a compressor the parameters of sorry the performance parameters of the parameters by which the performance of a pump or compressor is expressed are rotational speed the flow rate that is the capacity of the compressor or pump and the head developed by the pump. So, therefore, we try to find out a parameter preferably a dimensionless parameter combining these performance parameters n p h and q depending upon whether it is the turbine or compressor and this parameter will be independent of this height of the machine rotor d. So, therefore, we search for such parameter how we can get a combination of n p h or n q h as a dimensionless parameters from the existing pi terms we see. Now, here for a turbine you see for a turbine what we can make for a turbine if we find out a term which is equal to pi 2 to the power pi 4 rather pi 4 to the power half divided by pi 2 to the power 5 by 4. This can be done by these manipulations in case of turbine then what it becomes it becomes what is pi 4 pi 4 is p by rho n q d to the power 5 to the power half divided by pi 2 that is g h by n square d square to the power 5 by 4. Then you get a term where d will be eliminated d to the power 5 by 2 will be cancelled we get a term like that n p to the power half by rho to the power half and g h to the power 5 by 4. If you calculate this that if I make pi 4 to the power half divided by pi 2 to the power 5 by 4 we get a term like that. So, pi 4 and pi 2 are dimensionless. So, therefore, they are combination in this fashion also a dimensionless parameter. So, therefore, we see that for a turbine we get a dimensionless parameter containing n p and h of course, rho and g are there and this dimensionless parameter we defined as k s this is the usual nomenclature t t suffix for turbine which is known as dimensionless dimensionless dimensionless specific speed for turbine. So, a name is given to this dimensionless term obtained from some typical combinations of the pi terms namely the pi 4 and pi 2 terms in this fashion and we get a combination of the performance parameters that is n p and h in this fashion n p to the power half rho to the power half and g h to the power 3 by 4. And that is also obtained a similar dimensionless parameters for a pump 5 by 4 sorry sorry sorry sorry sorry good 5 by 4. Sir, I have your suggestion this combination. Why you have taken this combination because this combinations will give a parameter which is free from diameter. You can see a step forward combination you have to make which will be a parameter independent on the diameter. I have just explained that we search for in dimensionless terms which will be free from the rotor size this is because the performance of a machine is expressed by rotational speed for example, a turbine power developed and head where the size of the machine does not come into picture. So, therefore, we seek for a dimensionless parameter which include only the performance parameters n p and h. So, we take this typical combination to get the n p h in the similar fashion if we follow for a pump and compressor with the idea that in or with the idea all right that in case of a pump or a compressor n q h are the three parameters or three quantities which are usually referred to as the performance parameters of the compressors. Then we can do that in case of a pump or a compressor how we can make a combination of a pump combination of the pi terms to eliminate the d and to get the term which combine only n q and h this is done in this way pi 1 to the power half by pi 2 to the power 3 by 4 and it gives you see that pi 1 is q by n d q and make it half that means d to the power 3 by 2 and pi 2 to the power 3 by 4 that means g h by n square d square and make it 3 by 4 d to the power 3 by 2 which will be canceling. So, therefore, we obtain a term like this n q to the power half divided by g h to the power 3 by 4 in the similar fashion we define this term is a dimensionless term because pi 1 and pi 2 are dimensionless which combine n q and h along with g of course the name is the k s is the dimensionless specific speed for pump or compressor we give a nomenclature p for pump. So, we write the same thing specific speed specific speed for pump or compressor also is equal to n q to the power half g h to the power 3 by 4. Now, therefore, we see that we have come or we have arrived at 2 dimensionless terms which are known as specific speed of turbine that is n p to the power half I am writing it again rho to the power half g h to the power 5 by 4 and specific speed for pump k s p n q to the power half g h to the power 3 by 4. Now, these two terms are the dimensionless terms this is because they have been derived from the dimensionless pi terms. So, therefore, these two terms also represent the similarity conditions in fluid machines that means for example, in turbine this term represent the similarity conditions in the flow to the fluid machines under altered conditions or n p and h which means the unique value of a k s t represents a similar conditions of flow through the fluid machines under altered values of n p and h. That means this can give a number of combinations of n p and h to yield a unique value of k s t under which the flows are similar in turbines and same is the case for pump or compressor that this quantity represents a typical combination of n q and h. So, a fixed value of k s p which refers to a similar conditions to the flow in pumps that means which can give a number of combinations of n q and h. So, that the conditions of flow becomes similar in the pumps. Now, we see that usually we are interested to know that when you develop a machines what are the ranges of this performance parameters n p h in case of turbine or n q h in case of a compressor a machine can cover. So, this is very interesting to know for which from which we can find out the particular machine a particular shape of the machine or a particular type of the machine which is best fit for a particular operating conditions. So, now you know that when a machine is designed it is always specified by its operating parameters in a way that at this operating condition it runs at a maximum efficiency. That means I develop a turbine and I tell if this be its revolutionary speed or rotational speed n if it develops this power p under a head of h this much it will be running at its maximum efficiency. So, with some tolerance limit from that rated conditions both the sides almost it will run at very high efficiency or if you run this machine at a condition much altered from this specified condition the machine will not be running at a higher efficiency or at maximum efficiency. So, it will be running at a very low efficiency that is not desirable to run the machine at those operating condition. So, therefore, you see here again that which means that for a particular machine of a particular design there is a unique set of values of n p and h which give a unique value of k s t for its running at a maximum efficiency. And this and since this is a dimensionless parameter and criterion of similarity we can tell that for a same homologous series or a same shape or same size or same type of machines we can consider a different machines of different sizes of the same series. That means of same shape working under altered conditions that means altered conditions of operating parameters which means the rotational speed power developed n h, but if we have to run the machine at the maximum efficiency you will have to make the operating condition such that the same value of this dimensionless parameter k s t has to be obtained. So, therefore, this value of k s t we get an unique value of k s t a unique value of k s t which refers to maximum efficiency. So, we get therefore, the conclusion is that we get an unique value of k s t for a particular shape and particular category of fluid machines which gives which unique value of that at the where the efficiency is maximum. Similarly, for a particular category of pump that means for pumps of a particular or a of a homologous series we get a unique value of this k s t corresponding to the maximum efficiency eta max of the pump. For example, a particular machine the set n p h or n q h either is a turbine or compressor or fixed, but it can vary, but the combination should yield to a fixed value of k s t or k s p for all machines of a particular homologous series. So, that if we operate at those that the parameter operating parameters the machines of that homologous series will be running at maximum efficiency. Now, I give you the actual concept in design. Now, for example, just a practical concept we know that n p and h now how the type of the turbine is first fixed. Let us consider that we know some operating conditions are there that we have a limitations on the rotational speed. We know that this much power has to be developed by a turbine and the turbine will be operating under this head that means we have got some available head h and we know that this much power we have to develop and we know that a rotational speed of the turbine will be like that. Then we find out this k s t now we see that this value of k s t has obtained from this n p and h which are specified as the performance parameters as the output parameters that have to be made. We will see that which class of turbines gives the maximum efficiency at that value of k s t then we will select that class of turbine because that class of turbine will give the maximum efficiency for these operations. That means from practical operating data that means we if we are told by our customers that we want this much power to be delivered. We have this much amount of head available with us and rotational speed is restricted for this r p n. What first of all we will have to think that what kind of turbine will be suitable for this purpose. Then we find out these dimensionless parameters the specific speed then we see from a list from a figure that which type of turbine that which homologous series corresponds to the maximum efficiency for this specific speed. Usually this specific speed for different shapes of fluid machines for different turbines or different category of compression or pumps that means for a different homologous series are coated for maximum efficiency. That means specific speed for axial flow turbine this much specific speed or radial flow turbine this much specific speed for an impulse tangential flows turbine this much that means that when the specific values are quoted for a particular homologous series referring to a particular kind or category of machine means that it refers to its maximum efficiency. So, we will match the specific speed calculated from the operating parameters given to us for its design to the specific speed quoted for a particular class that means which refers to this particular specific speed of a class when it runs at its maximum efficiency. We will choose that particular class and we will tell that this class of machines will be the best fit for this operation and then we will go for its geometrical dimensions for its design to find out the geometrical dimensions. So, this is the well the precise concept of the specific speed. Now, one thing we can derive also from here that since the acceleration due to gravity does not vary in our practical operations and if we consider the fluid handling and incompressible fluid for whose density is not varying much. For example, if we consider only the hydraulic turbines handling water then we can get rid of this row and g terms and we can define the specific speed for turbine n s t this is the dimensional specific speed which is the combination of only n p and h. So, what we do similarly for the pump this is n q to the power half and h to the power 3 by 4. So, what we do we eliminate the unnecessary terms g and row when we see that acceleration due to gravity does not vary from pump to pump or turbines to turbines and also the turbines or pumps handling water where the row remains virtually constant then we can get rid of this g and row because equality of k s t means the equality of n s t. Similarly, equality of k s p means the equality of n s p that means we come across through only the combination of n p h and n q h, but the only difference is that these terms are not dimensionless these are dimensional term because it is dimensionless when row and g were there in this fashion similarly here g was there. So, these are known as dimensional specific speed of turbine or dimensional specific speed or pump never the less they refer to similarity condition because the constancy of k s t which refers to similarity condition or constancy in k s p which refers to similarity condition is not violated since row and g are constant that means constancy in k s t and constancy n s t constancy in k s p means the constancy in n s p, but they are the dimensional specific speed that means they have their dimension depending upon the dimensions of n dimensions of p and dimensions of h here dimension of n dimension of q and dimension of h. So, therefore, whenever the specific speed are referred in usual practice or in usual problems when it is told the specific speed then it is the dimensional specific speed for our practical use when the specific speeds are quoted as dimensionless specific speed then we will refer to these expressions ok. Now, we will come after this to the a typical or hydraulic turbines now we will start our discussion on rotodynamic machines rotodynamic turbines first rotodynamic turbines now we will start the discussions on rotodynamic after this general discussions after this general discussions on principle of similarity applied to fluid machines the efficiency of fluid machines what we have done so far the basic principles of operation of a fluid machines how the force exerted by the fluid to the rotor and rotor to the fluid while the fluid flows to a rotor vane and then how the energies transfer what is the basic equation of energy transfer then the definitions of efficiencies in general for turbines and pumps and then the principle of similarity applied in general to a fluid machines we will come to the description of different fluid machines separately. So, first we come to rotodynamic turbines impulse turbine impulse turbine hydraulic turbine rather hydraulic so the impulse hydraulic turbine that uses water was first developed by the person is an American engineer known as Pelton. So, therefore, an impulse hydraulic turbine is referred as an impulse and impulse hydraulic turbine and impulse hydraulic turbine is referred to as Pelton turbine Pelton by the name of the scientist or engineer American engineer Pelton turbine or Pelton wheel turbine rotor is a typical wheel Pelton wheel. So, we will come to the description of an impulse hydraulic turbine you know the definition of an impulse turbine. So, it is an impulse hydraulic turbine that uses water and it was named after the name of the inventor was an American engineer as Pelton turbine or Pelton wheel. Now, let us come to this what is a Pelton turbine. So, this is a typical Pelton wheel here you see a Pelton turbine consists of a rotor which is a wheel or a disc on the periphery of which a number of spoon shaped buckets are attached these are all spoon shaped back buckets which are attached to the periphery of a disc or wheel which rotates let this is the angular speed omega rotate. Now, there are number of nozzles one such nozzle is shown which directs water at high velocity. So, what happens is that the water enters to the nozzles the nozzles are fixed nozzles water at high pressure enters to the nozzle and nozzle converts the pressure of the water into a high velocity water jet and the water jet is directed by the fixed nozzle to the spoon shaped buckets these are known as spoon shaped buckets or you can consider these as the vents usually the terminology buckets are used. So, that the jet strikes the bucket at its center tangentially which means the water jet strikes in such a direction which becomes the tangent to a circle drawn at the through the center of the bucket. That means, if you see a plan view of the bucket here that means this cross section if you see this cross sectional view here come this bucket is a spoon shaped bucket which is this sectional view sectional plan looks like this which has got two symmetry equal parts like this the jet comes and strikes the bucket and it smoothly glides along the two parts of the bucket these are this is the spoon shaped bucket. Now, you see here therefore, the jet strikes here a tangent to this bucket that means the direction of the jet velocity incoming jet velocity is in the direction of the tangent to the circle through the center of bucket the tangential direction. Now, if we look a side view of the bucket you see this is the bucket. So, at the middle here this one is the splitter reach there is a splitter reach which divides the flow equally into two half now you see if we consider the bucket a wheel and the buckets in a vertical plane the jet strikes bucket at each and every bucket tangentially that means this flow is in a horizontal plane which is in a tangential direction to this bucket. So, therefore, we see the inlet of the jet or inlet section of the jet and the outlet section of the jet is at the same radial location from the axis of rotation they do not vary in the radial location. So, they strikes the blade tangentially and comes out tangentially that means this is the plan view where your concept will be more clear the strikes here and goes out like this. Now, you see here the inlet velocity v 1 and the bucket speed u which is the tangential velocity at the center of the bucket that means that depends upon the rotational speed and the radius from the axis of rotation to the center of the buckets. So, this is the tangential velocity u shown for a typical bucket section the plan of a single bucket where this jet velocity is v 1 and they are in line. So, the jet is being deflected by the bucket now you see for a maximum change in the momentum of the liquid and hence accordingly the maximum force to be exerted on the bucket this change should be exactly 180 degree that means this relative velocity or the jet velocity with relative to vane should come exactly opposite to the v 1 that means the jet should be deflected 180 degree by the bucket, but usually this deflection theta in practice is made at 165 degree not exactly opposite to v 1 this is because that the jet coming from one vane or one bucket should not heat the back of the following bucket. So, that this can be made 165 this is the optimization. So, that we can get maximum change in the momentum. So, that maximum force can be exerted by the fluid while flowing through the bucket at the same time the fluid jet should not strike the back of the buckets following it. So, this is the arrangement of a pelton wheel now the number of wheels depends upon the different designs a type of operations it starts from number of wheels you write number of wheels usually greater than 15 number of buckets I am sorry sorry sorry number of buckets I am sorry number of buckets and number of nozzles which is fixed number of nozzles this varies from 2 to 6 depending upon the design. So, I will come again about this numbers and the diameter when the design aspects I will discuss, but usually the number of nozzles vary from 2 to 6. So, therefore, the fixed nozzles at this stator of the machine they are the fixed where the pressure of the fluid is converted to velocity here. So, when the fluid comes and strikes the bucket it partly fills the buckets wound shaped bucket. So, bucket is the rotor which is moving part and fluid when comes in contact with the bucket this is a free jet and this is open to atmosphere. So, therefore, the pressure of the fluid all along through the rotor that means through the bucket is atmospheric pressure and is constant. Therefore, this is an impulse machine that means while the fluid flows through the machine the relative velocity at inlet and outlet remains virtually same there is a little reduction because of the friction through the bucket, but there is no change in pressure. So, conversion from pressure to velocity does not take place whereas the entire pressure energy at inlet to the machine that means at the inlet to the stator of the machine that fixed part nozzle is totally converted into kinetic energy that means total expansion from a high pressure liquid at its inlet is made in the nozzle to exploit the maximum velocity available in the form of the water jet. So, thank you this is the basic introduction that means the type of the pelton wheel next class I will discuss the force analysis and the power generation by a pelton wheel. Thank you.