 I welcome you all to this course on fluid machines. The number of the course is ME263004. Before coming to the course contents, I first like to introduce the subject before you. Well, now first we start from the basic introduction to the subject of fluid machines. What is a fluid machine? Now you can define the fluid machine as a system or a contravence where the stored energy in fluid is converted to mechanical energy or vice versa. That means the mechanical energy is converted to stored energy in fluid. Now from this very basic definition the importance of fluid machine in our engineering field is very obvious because you know the need of mechanical energy today which is largely used for generating electrical energy and is partly used as mechanical energy for different engineering applications. So the importance of fluid machine lies in the importance of mechanical energy as we need today. Now next question comes about the nature of the stored energy in fluid and what is the nature of the mechanical energy that the conversion from one to other is taking place through this system fluid machine. Now stored energy in a fluid now I must say that if you recall your basic thermodynamics in your basic thermodynamics class you have read that the stored energy in any system the system may be a fluid is known as the internal energy that is the energy stored in a system. So if we look we can divide it into parts that when the fluid at rest that fluid the system is at rest then the only way the energy stored are the intermolecular energy and potential energy. Intermolecular energy is the energy due to molecular motions and the potential energy of the molecules that is the kinetic energy and potential energy of the molecules. Usually it is the kinetic energy of the molecules for ideal gases you know the kinetic energy of the molecules are due to the temperature. So for all system or fluid at rest the intermolecular energy is there by virtue of its temperature all system always at a temperature more than the absolute zero. So therefore they have the kinetic energy of the molecules and the potential energy of the molecules which give rise to intermolecular energy. Another form of energy which is stored in any system even at rest is the potential energy do you know the definition of potential energy very simple that it is the energy that its system possesses by virtue of its position or placement in a conservative force field. When all other conservative force fields are absent then obviously the gravity is the usual conservative body force field by virtue of which a system possesses potential energy. Now when the fluid is in motion that means if a system is in motion is a specific case we consider here a fluid what are the difference form of the stored energy. In other words I can tell that if there is a stream of fluid what are the difference forms the energy stored in the stream of the fluid. Now along with the intermolecular and potential energy as we have discussed in case of fluid at rest a fluid stream or a fluid in motion also possesses kinetic energy is very simple by virtue of its velocity know whose magnitude is phi square by 2 per unit mass. And another type of energy a stream of fluid possesses is the pressure energy or simply we tell flow work probably you know these things but I again tell you that a pressure what is the pressure energy you know that when a fluid is in motion any layer of fluid in a fluid stream pushes the neighboring layer to make its way through in the flow which means that all the time a layer does work on the adjacent neighboring layer downstream to it. So, this work done by the fluid layer to its adjacent neighboring downstream layer is known as the flow work and the energy by which it is capable of doing that work is known as the pressure energy we use both the words flow work or pressure energy. And this is by virtue of the pressure of the fluid stream and you know the magnitude of these energies pressure times the specific volume per unit mass of the fluid. So, these are the different forms in which the stored energy appear in a fluid mass whether it is at rest or it is in motion. Now comes the mechanical energy. So, mechanical energy is usually obtained or transmitted through a rotating shaft against a load through a rotating shaft against a load. So, this load depends upon the use for example, a large part of the mechanical energy which is obtained from a fluid machine is used for electrical purpose that means to convert it into electrical energy. In that case the load is an electrical load that means the rotating shaft is coupled with an alternated drives an alternator. So, load depends upon our use. So, therefore the mechanical energy is obtained from the fluid machines or given to the fluid machines in the form of rotation of a shaft against a load. So, therefore, we can now recognize the different types of energies that means stored energy and mechanical energy during their conversion through the fluid machines. So, therefore, next we will come to the classification of fluid machines which is very important. Now, fluid machines as such is very broad or very broad in their classification. Now, classifications therefore are made on certain basis or we can say that fluid machines are put into different categories on under certain basis. So, that is why it is basis of classifications are there. So, they are broadly divided into three basis one is direction of energy transfer one other is the fluid use another is the principle of operation. Now, direction of energy transfer as I have told earlier that in fluid machines either the stored energy is converted into mechanical energy or the mechanical energy is converted into stored energy of fluid. So, the machines in which the stored energy in fluid is converted to mechanical energy that means the output of the machine is the mechanical energy and input to the machine is the stored energy in fluid those are termed as turbines. And the machines where the mechanical energy is the input that means the mechanical energy is converted to stored energy in the fluid as the output are termed as pumps, compressors, fans and blowers there are different categories. So, this will be clear in the next category of definition where the fluid used. Now, based on the fluid used the fluid machines are classified into different forms. Now, you know the fluid comprises both liquid and gas one is the incompressible fluid the liquid another is the compressible fluid gas. Now, when the machine converts the stored energy to mechanical energy and uses liquid as the fluid then these are termed as hydraulic turbines the adjective hydraulic comes because this is in general turbines which convert the stored energy to mechanical energy. But if they use liquid the adjective hydraulic comes in almost all the practical purposes the liquid use this water for this turbines. So, hydraulic almost substitute the water in its adjective sense that is the hydraulic turbines sometimes water turbines the hydraulic is the more general terminology used as an adjective to that liquid which is usually water that hydraulic turbines. Similarly, for those machines which give mechanical energy from stored energy use gas this may be air or any other gas that is the compressible fluid are termed as air or gas turbines that means the adjective is air or gas that means the name automatically signifies when it uses air as the fluid is air turbines or it uses gas as the fluid for example, the combustion products as you know in a gas turbine the turbine fluid is the combustion products after burning the fuel with the air. So, therefore, it is a gas. So, gas air or gas turbine similarly for machines which convert the mechanical energy to stored energy, but uses liquid are known as pumps. So, pumps are those machines where mechanical energy is the input and stored energy in the fluid is the output, but the fluid is an incompressible fluid water or any other liquid that these are known as pumps whereas the same machines when they use gas air or gas they are termed as compressors fans and blowers these three names are there. Now, here in this context I like to tell you that there are differences between compressors fans and blowers which we will see afterwards that these machines uses these machines use gas and give the stored energy in the fluid. Now, the stored energy in the fluid are obtained either in the form of pressure high pressure in the fluid or in the form of high velocity in the fluid that means either the pressure energy of the fluid is raised or the velocity energy of the fluid is raised here you must know one thing that well we have already read that intermolecular energy is there, but usually by spending mechanical energy it is not advisable from thermodynamic point of view to increase the intermolecular energy of the fluid why can you tell why it is not advisable that you spend mechanical energy as the input through any device to increase this stored energy in the form of intermolecular energy why can you guess well this is not recoverable partly true, but I think it should be told in a more direct sense that intermolecular energy is the low grade energy whereas the mechanical energy is the high grade energy. So, we can get work from the intermolecular energy, but not 100 percent conversion is possible according to the second law of thermo so it is not advisable that we spend and high grade energy to convert the this to a low grade energy. So, therefore, the stored energy in the fluid which is being generated or being converted to from the mechanical energy as either the pressure energy or the kinetic energy when the stored energy is in the form of pressure energy that means the pressure of the fluid is raised by virtue of the mechanical energy in the fluid machines they are termed as compressors. So, that you know probably from your general knowledge that a compressor always provides high pressure here. So, flow or the velocity of the air is less whereas fans in fans and blowers the mechanical energy converts the kinetic energy the mechanical energy is converted to the kinetic energy of the fluid. So, therefore, fans and blowers provide mostly the kinetic energy of the fluid that means the stored energy in the fluid is in the form of kinetic energy where for compressors the stored energy in the fluid which is being obtained from the machine is in the form of pressure energy. Now, the third one is the most important one is the principle of operation is the most important one the prince depending upon the principle of operation the fluid machines are classified into two categories. So, one category is known as positive displacement machines where the principle of operation is based on the static action of the fluid. The other one is the rotodynamic machines where the principle of operation is based on the dynamic action of the fluid. Now, in the positive displacement machine the static action of the fluid means what is done here certain amount of fluid is entrapped within a given volume or in a enclosed chamber of the machine. And here the fluid mass behaves as a closed system thermodynamically a closed system then what happens is that one of the boundaries of the system then physically displaced to change the volume of the fluid entrapped either the volume is reduced or the volume is increased. And by virtue of this change in volume the energy transfer takes place between the fluid and the machine and the pressure of the fluid is increased or the pressure of the fluid is decreased. A very common example is a reciprocating motion of a piston in a cylinder that in a piston cylinder machines you see certain amount of fluid for example, air or it may be water even is entrapped. And then it is isolated from the inlet and exit of the machines during certain interval of time where it behaves as a thermodynamically closed system during that interval of time there occurs the displacement of the system boundary. For example, in case of a reciprocating motion of a piston within a cylinder piston moves either it moves so that the volume of the fluid is decreased or volume of the fluid is increased by virtue of which either the work is developed by the machine or work is being imposed done on the fluid system to increase the stored energy. So, these kinds of machines are known as positive displacement machines because of the positive displacement of the system boundary in a closed system that fluid mass itself behaves as a closed system changes or makes it possible for the conversion of energy. These categories of machines are known as positive displacement machines on the other hand there are machines which are in fact in large use for our engineering applications are known as rotodynamic machines they are based on the dynamic action of the fluid. What happens in those machines there occurs a continuous motion of the fluid and also of a part of the machines that means there occurs continuously a relative motion between the fluid and the machine both the machine part moves and the fluid and based on the hydrodynamic principle because of the change in momentum due to this continuous motion the conversion of energy takes place from mechanical to stored energy or stored energy to mechanical energy in the fluid. So, these are the basic classifications of fluid machines now after this classifications I think we will be in a better position to identify our course content which probably you cannot see here clearly I will give you this thing the course contents individually copy of this. Now as such as we have seen in the definition of fluid machines it is very broad and a single semester course cannot cover the entire fluid machines. So, these are termed as turbo machines fluid machines gas machines air machines a part of it is covered in different semesters for you also turbo machine course will be there afterwards which deal with gas machines usually the machines dealing handling gas and air are known as turbo machines that compressors air or gas turbines. In this course we will be dealing only with the hydraulic machines that means machines which deal with the liquid as the working system that means hydraulic turbines and pumps on the other end well. So, accordingly the course content is like this probably I tell you you can just first this principles of fluid machines introduction then classification of fluid machines which we have already covered. Then we start first with the rotodynamic machines. Now the basic equation of energy transfer in rotodynamic machines basic equation of energy transfer in rotodynamic machines this is in general for any rotodynamic machines whether uses liquid or gas. Then principles of similarity and dimensional analysis in rotodynamic machines this is also in general for any rotodynamic machines using liquid and gas. Then we come to different types of rotodynamic machines here only we concentrate only on hydraulic machines for this class for this course rather. The impulse hydraulic turbine known as pelton wheel now in this pelton wheel analysis of force on the bucket and power generation specific speed and wheel geometry governing of pelton turbine limitation of a pelton turbine. Then we will switch over to Francis reaction turbine turbines are of two types that we will come across during our course the impulse turbine and reaction turbine the Francis reaction turbine. Here we will cover net head across a reaction turbine runner of a Francis turbine specific speed development of kaplan turbine draft tubes in reaction turbines cavitation in reaction turbines performance characteristics of reaction turbines comparison of specific speeds of hydraulic turbines governing of reaction turbines this is up to turbines. Then we will go to rotodynamic pumps centrifugal pumps general pumping system and head developed by a pump impeller of a centrifugal pump characteristics of a centrifugal pump flow through volute chambers vane diffuser cavitation in centrifugal pump matching of pump and system characteristics pumps in series and parallel specific speed of centrifugal pump. Then come reciprocating pump this reciprocating pump is a positive displacement pump suction and delivery accelerating heads rate of delivery multi cylinder pumps air vessel well in this particular course at your IIT on this course subject named as M E 26 three zero zero four named as fluid machines a part of basic fluid problems are covered this is a particular or you can say typical design of our course curriculum at your IIT that though the course title is fluid machines we cover a part of the basic fluid problems these are theory of ideal flow which I believe you have read some extent that your basic fluid mechanics course. So, I will make a hurry recapitulation of all these things then superimposition of elementary flows flow past a cylinder with and without circulation concept of lift and drag aerofoil theory these are extremely important topics in fluid mechanics. So, a part of it probably have been covered in your basic fluid mechanics course there may be a little recapitulation in this class. Then you come to a new topic which you have not read earlier on the basic fluid problem these are compressible flow these are pressure field due to moving source one dimensional isentropic flow choking in a converging nozzle isentropic flow in a convergent divergent nozzle normal shocks this is our course content for this course. Now, before coming to this basic principle of operation of a general rotodynamic machine I like to recapitulate the flow of fluid through a moving curved vane the basis for which is like this now I will start the rotodynamic machines its general principle. Now, as I have told earlier the rotodynamic machines there is a continuous motion of fluid as well as a part of the machine. Now, this part of the machine is known as rotor which is a rotating element which usually consists of a disc rotating disc a number of vans are mounted on the disc at the periphery of the disc and the disc itself is mounted on a shaft where the actual rotation is imparted. So, therefore, it is the principle by which a moving fluid stream along a curved vane transfers the energy between the fluid and the vane which is a part of the machine part of the machine part of the rotor of the machine is the basic underlying or basic principle of fluid machines. So, therefore, we try to recapitulate the principle by which the energy is transferred or the force is imparted by a moving fluid through a moving curved vane. So, let us see that let us have you drawn this I will not start before you complete your figure I think you can very well see this now let me explain this which will help you in drawing the figure that there is a vane which is moving with a velocity in this direction let us make this direction as a coordinate direction x positive coordinate direction in which it is moving with a velocity u a fluid stream approaches this moving blade with a velocity v 1 this is the fluid stream approaches and it is discharged after flowing through the vane with a velocity v 2 this is the velocity v 2 this is the velocity v 2. Now, you see since the vane is moving so, what will happen with respect to the fluid will approach with an different velocity which is nothing but the relative velocity of the fluid with respect to the vane which we can get from a vector diagram by subtracting the velocity vectorially the from the fluid velocity the vane velocity for which we get a typical velocity triangle we know that that is the inlet velocity triangle where this is the vane velocity and this is the relative velocity of the fluid at the inlet that means this is the velocity with which the fluid strikes the vane with respect to the vane. Similarly, we make the vector subtraction of this vane velocity from the absolute velocity to get the relative velocity this is the relative velocity of the fluid with which the fluid is discharged from the vane well now we see that here as the fluid passes through the vane there occurs a change in velocity and also the change in the momentum of the fluid which gives rise to force exerted on the fluid or on the vane. Now, to analyze this type of problem we take the help of momentum theorem now we take a control volume if you recall in your basic fluid mechanics class we take a fluid control volume like this this is the fluid control volume this control volume is moving with a velocity u which is the movement or the velocity of the curve vane. So, now we make use of the momentum theorem applied to this control volume momentum theorem applied to the control volume momentum theorem applied to the control volume I feel the time is up for you today. So, I like to end it today up to this we will do this in the next class to analyze the force that is being exerted on the fluid because of its motion to the curve vane and the energy transfer between the fluid and the moving vane. So, I think I will stop today to this because time is up for you for the next class. Thank you I have told earlier the rotodynamic machines are those machines where there is a continuous motion of fluid and a part of the machine known as rotor and because of this continuous relative motions between the fluid and the rotor of the machine it is possible for an energy transfer to take place between the fluid and the rotor. So, therefore, the basic principle of this machine is based on the fluid dynamic principle fluid dynamic principle which is basically the utilization of useful work due to the force exerted by a fluid striking on a series of curved vane which is mounted on the periphery of a disk that is rotating the periphery of a disk that is attached to a rotating shaft. So, therefore, to understand the basic principle of a rotodynamic machines we should understand clearly the force interaction and the energy transfer that takes place while a stream of fluid passes through a curved vane. So, this is a little recapitulation of what you have already studied at your basic fluid mechanics course that we studied here the interaction of force and energy in the flow of fluid along a curved vane. Now, see here this is a curved vane which is moving with a velocity u and a jet of fluid is striking the vane with a velocity v 1 is the velocity absolute velocity with which the fluid strikes the vane and the fluid after flowing through this vane comes out with a velocity v 2 this is the velocity v 2. Since the vane is moving with a velocity u so the jet appears to strike the vane that means with respect to the vane the jet strikes it with a velocity v r 1 which is the velocity of the jet relative to the vane. Similarly, it is going out with a velocity v r 2 that is the relative velocity of the fluid with respect to the vane. Now, this relative velocity is at inlet and outlet are determined just by vectorial subtraction from v 1 the velocity u of the vane and from v 2 the velocity u of the vane. So, this vectorial subtraction is shown in terms of the velocity triangles as we have already read at the inlet and outlet. Now, let the suffix 1 refers to inlet condition and suffix 2 refers to outlet condition now we see in this triangle this is v 1 that inlet velocity of the fluid this is the u the vane velocity and this is the v r 1 that is the relative velocity of fluid with respect to vane at inlet this component this perpendicular component to the motion of the vane is denoted as v f 1 and is usually known as flow velocity. Similarly, the component of the fluid velocity absolute fluid present in the direction of the vane motion is conventionally symbolized as v w the suffix 1 is at the inlet. So, this v w I tell you this is a conventional symbol w stands for whirl whirling component this is because in actual case this velocity of the vane is in the tangential direction because the motion of the vane mounted on the periphery is a rotating motion. So, therefore, the linear velocity of the vane is in the tangential direction and that is why this component is known as tangential component or whirling component for which a conventional symbol w is given as the suffix similar is the case in case of an outlet velocity triangle this is the vane velocity this is the relative velocity of the fluid with respect to vane and this component is the whirling component or the component of the flow velocity in the direction of the vane velocity and this is the flow velocity that is the direction of the that is the sorry the component of the fluid velocity in the direction perpendicular to the vane velocity. Now, our basic purpose in this case is to analyze what is the force exerted by the fluid on the vane or vice versa vane on the fluid and by virtue of the vane motion which is the what is the amount of energy that is being transferred or developed due to this force due to this action of the fluid on the vane. So, to analyze this as you know we apply the momentum theorem. Now, to apply the momentum theorem we have to take a control volume of the fluid like this which is just adjacent to the vane now you see that this type of analysis can be done on the basis of both system approach and the control volume approach in a system approach what is done the new terms law is applied in a sense that you consider a particular mass of fluid and consider its change of momentum as it flows along the vane find out the change of momentum in a specific direction and in control volume the same thing is done, but the version is different we find the momentum efflux net momentum efflux in a particular direction and equates this with the force in that particular direction. So, if you if we apply this theorem for a steady state situation the situation is steady then we find that if f x is the force acting on the control volume in the direction x then it will be the net rate of momentum efflux from the control volume in that direction x because we are interested in the direction x that is the direction of the vane velocity the force in that direction. So, the expression in the right hand side is either the net rate of momentum efflux x momentum efflux from the control volume or from a system approach it is the change of momentum change of momentum in the x direction of a fluid mass taken as a system in either way you can see it and that becomes equal to the force is equal to the change of momentum times the mass flow rate. Now, you see the velocity at the outlet is v r 2 now here we have to consider the relative velocities because in this case the control volume is moving with a velocity use since the vane is moving with the velocity this is an inertial control volume. So, the coordinate axis will be fixed to this control volume. So, therefore the velocities which we have to take are the relative velocities. So, you see the component of the velocity in the direction of vane velocity here the if beta 2 is the angle made by v r 2 with the direction of vane velocity it will be minus v r 2 cos beta 2 because this direction is opposite to that of the vane velocity or to that of the positive direction of the specified axis x this is the momentum efflux minus the momentum efflux that means v r 1 cos beta 1 beta 1 is the angle made by the relative velocity with the vane direction. So, this component is in the direction of the vane velocity or in the positive direction of x. So, minus sign is that because it is the efflux minus efflux. So, both the terms are with a minus sign. So, it comes out. So, minus m dot now this v r 1 cos beta 1 or v r 2 cos beta 2 if you see from this triangle. So, this comes out to be v w 1 and v w 2. So, therefore, we say that force on the vane is equal to minus efflux that means this is the force that is being acted on what that is being acted on the control volume. So, the force acting on the vane is in the opposite direction that means if this is the efflux it is the in opposite direction of efflux minus efflux. Now, power developed due to the motion of the vane is then force into the velocity that is m dot v w 1 plus v w 2 into u, u is the vane velocity. So, this way we can develop an expression for the power developed due to the action of the fluid passing over a curved vane. I think you have understood it. So, from this two triangles you can get from here triangular relationships geometry that it is v w 1 plus v w 2 and this minus sign is because the force which is acting on the fluid element or the control volume is in the opposite direction to the specified axis that means in a direction opposite to the vane motion. So, therefore, the force on the vane is in the direction of the vane motion. However, the expression for power developed is written as the multiplication of efflux and u. They are in the same direction. So, their absolute values are taken well ok. Now, yes please v r 2 cos beta 2 is not v w 2 plus u and here it is v w 1 minus u that cancels out actually. So, ultimately you get v w 1 plus v w 2 yes correct v r 2 cos beta 2 is not v w 2 it is v w 2 plus u on the other hand v r 1 cos beta 1 is also not v w 1 it is v w 1 minus u. If you substitute that automatically it cancels out and becomes because I felt that you have already done it at your basic fluid mechanics course. So, this thing you know well ok very good I am happy that you are asking questions. Now, we come to the basic equation of energy transfer in rotodynamic machines. Now, in a rotodynamic machines what happens is that the rotor of the machine is a rotating wheel on which the vans are mounted and the wheel is mounted on a shaft where the rotation is imparted. So, in this case the same principle is applied here and we analyze this in the similar fashion with the help of a diagram here which is the general representation of a rotor or the representation of a rotor of a generalized fluid machines. Now, components of flow velocity in a generalized fluid machines here the most general sense we consider the rotor where the fluid enters at a velocity v 1 at any point whose radius of rotation from the axis is r. Now, before that I like to mentioning that there are few assumptions for this analysis one assumption is that the flow is steady. So, there is no mass accumulation no mass depletion anywhere in the system and number two assumption is that flow is uniform over any cross section normal to the flow velocity which is very important and that means that the velocity vector at a point is the representative of the flow over a finite area. That means we analyze with respect to a velocity vector at a point and we assume that this is uniform over the entire flow area that is an area normal to the flow velocity. So, that this is the representative of the entire flow through the fluid machines well. So, with these assumptions now we consider that at any point the velocity vector is v 1 that is the inlet point a very general case whose radius of rotation from the axis of rotation is r 1. Similarly, the fluid goes out or discharges at a point from the rotor whose radius of rotation from the whose radius sorry whose radius from the axis of rotation is r 2. Now, the velocities v 1 and v 2 can be resolved into three components there may be an arbitrary angle at which the velocity flow velocity strikes the rotor which can be resolved into three reference directions. One is in the direction of tangential one in in the direction of tangent a tangential direction which is the tangent to the rotor at that point another is the direction which is the axial direction. That means it is parallel to the axis of the shaft and another is the radial direction which is perpendicular to the axial direction. So, these three mutually perpendicular direction the velocities are resolved one is the tangential direction another is the axial direction another is the radial direction. So, these three perpendicular directions and accordingly symbolized as v w one is the suffix at inlet that is the tangential component whirling component that is why the suffix w is used the suffix a v a is the axial component that is component parallel to the axis of the rotation. And as I have told earlier the symbol f is used v f 1 for the inlet that is the flow velocity that is in the radial direction. Similar way the velocities are resolved in tangential direction as v w 2 at the outlet axial direction v a 2 and the flow direction v a 2. Now, the rotor is moving with an angular velocity omega is a constant angular velocity there is a steady state problem well now let us apply the momentum theorem or the new terms laws of motions either with respect to a system or a control volume here. Now, here the momentum which will be considered is the angular momentum this is because here the work transfer takes place due to the rotation of the shaft. So, we will be considering the angular momentum or moment of the momentum it is very simple if we consider a system approach our version will be that considering a fluid mass as it passes from the inlet to outlet what is its change in angular momentum. Or if we consider a control volume of a fluid then what will be the net rate of a flux of the angular momentum from the control volume. Now, here one thing is very important we are not bothered about the path in the rotor it is only the inlet and outlet that decides the change because if the inlet and outlet conditions are fixed kinematic conditions are fixed and the mass flow rate is steady. So, the change in momentum or the moment of the momentum whatever you say depends upon the inlet and outlet conditions well. Now, if we write the momentum moment of the momentum at the inlet for a unit mass at inlet what will be its value at inlet at inlet moment of the momentum moment of the momentum is equal to that is the moment of the tangential momentum. That means, v w 1 times the r 1 radius from the axis of rotation it is per unit mass per unit similarly the same thing at outlet at outlet the same thing at outlet is equal to v w 2 into r 2. So, therefore, per unit mass the change in the moment of the momentum of a fluid mass or the net rate of a flux of the moment of the momentum per unit mass from a control volume will be v w 2 and that multiplied by the mass flow rates. That means, this will be the net rate of angular momentum a flux or the rate of change of angular momentum rate of angular momentum net rate of angular momentum a flux when we refer it to a control volume that is a control volume approach control volume of the fluid or it is the net rate of change of angular momentum for a system as it passes from inlet to outlet. So, in both the cases that equals to the torque that is the angular momentum theorem that is the angular momentum theorem applied to a system or to a control volume that torque is equal to the rate of change of angular momentum of a system or torque is equal to the net rate of angular momentum a flux from a control volume at steady state. So, that is equal to the torque that is being imparted on the fluid by the rotating disk. Now, the energy rate of energy that is being imparted to the fluid will be nothing, but the torque into the angular velocity omega and that if we multiply the angular velocity and recognize that omega r 1 is the velocity of the linear velocity that tangential velocity of the rotor at inlet and omega r 2 is the linear or tangential velocity of the rotor at outlet and denoting them by the symbol u we can write v w 2 u 2 minus v w 1 u 1. So, therefore, we see that energy transfer per unique time the rate of energy transfer in the fluid as it passes from inlet to outlet becomes equal to the mass flow rate m dot flowing times v w 2 u 2 minus v w 1 u 1 where u 2 and u 1 are the tangential velocity that is the linear velocity of the rotor at the outlet point and u 1 is that at the inlet point because in a generalized case we have to consider that inlet and outlet are not in the same radius from the axis of rotation there is not at the same radial plane.