 Consider the rotor of a turbo machine which is shown here like this. This rotates with an angular velocity omega in the clockwise direction. Fluid enters this rotor with an absolute velocity v1 and leaves an absolute velocity v2. The absolute velocity has components in three directions namely radial, axial and tangential also known as world component. As the name suggests the component of velocity in the radial direction moving outwards radially from the shaft is actually referred to as the radial velocity in this case as you can see this is the radial component of velocity in the radial direction and the axial component of velocity is along the axis of the shaft from inlet to exit. So as you can see this is the axial velocity of the fluid like this and the tangential component or world component is in the direction also confidential direction along the direction of rotation and as you can see this is the tangential component of velocity at this location. See that at this point this location entering location the tangential component of velocity looks like this and at the exit the tangential component of velocity because of the direction of the absolute velocity the tangential component looks like this. The tangential component is also usually referred to as the world velocity and if we actually look at the torque that is exerted on the fluid as a result of the passage of the fluid it is nothing but the change in angular momentum rate of change of angular momentum of the fluid. So m dot is the mass flow rate through the rotor and the rate of change of angular momentum of the fluid negative of that is the torque that is exerted on the rotor and that is what we have given here. So the rate of change of angular momentum is actually v theta 2 or 2 minus v theta 1 or 1 but since we are trying to evaluate the torque exerted on the rotor we change the sign so this is written as m dot times v theta 1 or 1 minus v theta 2 or 2. It should be noted that the tangential component of velocity at the inlet v theta 1 is in the same direction as the blade speed in the velocity triangle that is shown in the illustration. Also note that v theta 2 is actually in the direction opposite to the blade speed at least the black colored velocity triangle shows it that way. Now the red colored velocity triangle shows v theta 2 to be in the same direction as the blade speed. So the sign convention is that if v theta 2 is in the same direction as v theta 1 with respect to the blade speed then it is taken to be a positive number. If v theta 2 is in an opposite direction to the blade speed 2 v theta 1 with respect to the blade speed then it is taken to be a negative number. So this is something that we need to take into account when we actually do calculations of real applications. Now if we multiply both sides of this equation by omega t times omega is nothing but the rate at which work is done. So it is power so we can multiply both sides by omega and say that the power is equal to m dot v theta 1 omega r 1 minus v theta 2 omega r 2. What is that omega r 1 is nothing but the blade velocity at 0.1 and omega r 2 is the blade velocity at 0.2 or blade speed at 0.2. So we can write this as m dot v theta 1 u 1 minus m dot I am sorry minus v theta 2 u 2. This equation is called the Euler turbine equation and it is of fundamental importance in the theory of turbo machinery. So the product v theta u or the difference between the product v theta u at inlet and outlet determines whether the machine is work producing or work absorbing. Now the sign convention for the for the power or w dot here is the same as in thermodynamics which means if the right hand side comes out to be positive then the machine is a turbine right hand side is negative then it is the compressor or pump blower etc power work absorbing. So it is consistent with the sign convention for thermodynamic work. Now one point that we notice here in passing but we will take this up in detail later is that the Euler turbine equation involves only fluid mechanical quantities that namely velocities of a fluid and mass flow rate. No thermodynamic properties are seen in the Euler turbine equation as it is written here. So this is something that we will develop on at a later point. Now the next question that naturally arises is the following in case of an actual device let us say we take a device like this in case of an actual device if I look at let us say this set of blades the fluid enters the blade along the entire height of the blade and exits along the entire height of the blade. So how do we evaluate v theta one and v theta two or u one and u two what is that u one will vary along the height of the blade v theta one v x one and v r one all will vary along the height of the blade. So how do we evaluate v theta one v theta two u one and u two in this case or in the case of a centrifugal machine like this v theta one u one and the other components vary along the length of the impeller as the fluid flows through the impeller these components vary along the length of the impeller so how do we evaluate v theta one u one v theta two u two. Now for this purpose we introduce a concept known as blade element. Now in the case of an axial rotor like this we draw a circle at radius r typically the mid height of the blade and we take out an element of radial thickness dr at this location. So at the mid radius we take out an element of thickness dr which would be along the entire circumference so we take it out then we cut it open and then lay the elements flat. If you do that this is what it will look like in the case of a turbine. So we take out an element of radial thickness dr and remove the element and cut it open then lay it down flat this is what it would look like. If we do this for a compressor blade for instance then this is what the layout would look like basically the blades are all laid out along this so this is the circumference of the blade but laid out in a flat manner on this surface. Now in the case of a radial machine like the centrifugal pump what we do is we actually take an axial section of thickness dx at any axial location of the impeller. So we take an axial section along the vertical line like this for thickness dx and then we lay it out like this and this would be the blade element in this case so this would be the blade element in this case so the pump rotates like this so we have laid out a blade element of axial thickness dx so the axial thickness dx in this case would be perpendicular to the plane of the paper similarly here the radial thickness dr will be perpendicular to the plane of the paper in both these cases. So we can evaluate v theta 1 and u 1 at the entry to this blade passage and at exit to the blade passage in order to evaluate the power from the Euler-Termin equation. Note that one of the advantages of using the notion of blade element is that only two components of velocity are taken to be non-zero notice that only the axial component of velocity which in this case is perpendicular to the tangential direction and the angular component of velocity which is in this direction only these two components appear in this flow field. So the radial component of velocity which is perpendicular to the plane of the paper is taken to be zero similarly here the axial direction or axial component of velocity which is perpendicular to the blade speed and the tangential component of velocity alone appear in this and the other component of velocity namely the radial component which is perpendicular to this which is along the height of the blade is taken to be zero. In this case also notice that only two components of velocity namely the radial component of velocity and the tangential component of velocity or non-zero in a blade element the axial component of velocity which is along the width of the blade in this case is taken to be zero. For developing the basic theory this assumption is perfectly acceptable but in real life the flow will be three-dimensional so many such blade elements have to be taken and calculations may have to be done using many such blade elements. What is done in the basic theory is to take a blade element and assume that all blade elements develop the same power so that the analysis can be used in real life application with a great degree of accuracy. Small improvements on or any other improvement to this theory can always be made by doing three-dimensional calculations or measurements. So this is what we will do when we try to evaluate the right hand side of the Euler-Tangou missionary equation so v theta 1 u 1 v theta 2 u 2 or measure for the reference to this blade element. Now another important concept in the in the context of blade element is the notion of the relative velocity denoted here by c so we have talked about absolute velocity of the fluid what is the capital V denotes the absolute velocity of fluid by absolute velocity what we mean is a frame of reference where the rotor rotates and the fluid enters. Now relative velocity which is denoted by c is the velocity of the fluid measured in a frame of reference where the blade is stationary in other words the blade appears to be stationary and the relative velocity is the velocity measured in this frame of reference. This is important because it is used for designing the shape of the blade under design operating conditions. So under design operating condition the relative velocity of the fluid approaching the blade is such that it glides on and off the blade surface smoothly which means that the relative velocity vector is tangential to the blade profile at inlet and exit which is why the relative velocity is very important for designing the shape of the blade. Let us take a look at this what is that the relative velocity is tangential to the blade profile at inlet and exit in these situations and also in this situation so the relative velocity of the fluid is defined like this it is the absolute velocity of the fluid minus the blade velocity at that location what is that everything is done at a particular location so an observer sitting on the rotor at this radius or on this blade element would see the fluid approaching the rotor with the velocity c and they should actually be on this blade element would see the fluid approaching the rotor or leaving the rotor with velocity c. So if you draw the velocity triangle which is a graphical illustration of these three velocities you can see that velocity vector is the sum of the relative velocity vector plus the blade speed so v is equal to u plus c so you can see that the relative velocity vector I am sorry the absolute velocity vector v is the sum vector sum of the relative velocity vector c and the blade velocity vector u. In this case also the absolute velocity vector v is the sum of the relative velocity vector c at the blade velocity vector u. However, notice that the geometry of the velocity triangle is different in this case and different in this case.