 We have been doing propeller theories, we did some propeller fundamentals and we tried to understand how the propeller actually creates thrust. In the process of creating thrust, the propellers need to be supplied with power for which of course, you need an engine which could be a piston engine or a gas turbine engine or any other engine, any other device that produces power and propellers main job is to make use of that power and create thrust for flying an aircraft. Typically the propellers are made up of three or more number of blades, number of blades could be even two and these blades are made up of aerofoil sections which put together create the blade shape and these blades when in rotation in a pre-determined manner help producing thrust. In the last class, we did a theory of propellers in which we looked at propellers as an actuator disk which means physically and mathematically the propeller was replaced by an actuator disk and this actuator disk essentially is modeled to be a thrust creator or energizer which then produces the thrust. So, this is what we did in the last class, of course we have been introduced to the fact that propellers are made up of aerofoil sections, we had a look at one or two of these aerofoil sections. Today, we shall look at a propeller theory which again makes use of the propeller aerofoil sections and the proper propeller blade shapes and how this theory then uses the aerofoil sections to actually create thrust and how this thrust is determined with the help of simple mathematical formulations and as a result of which the designer can make use of this theory to design a propeller and then of course create a model for prediction of the propeller performance. We shall see how the prediction of the propeller performance is actually created which are called propeller characteristics and these characteristics essentially as the name suggests characterizes the propeller, this particular propeller which has been designed. So, every propeller that is designed needs to be immediately accompanied with a propeller characteristics which determine the entire propeller operation and its capability. Only then the aircraft designer can make use of the propeller because he has to match the aircraft characteristics with the propeller characteristics. On the other hand the matching the propeller with the engine requires that the propeller characteristics is matched with the engine characteristics. So, propeller as we see is the interface again between the engine and the aircraft, its characteristics must match with the aircraft characteristics, its characteristics must match with the engine characteristics and only then we have a propeller that can fly an aircraft with the help of the engine providing the power. So, let us take a look at the propeller theory that uses the propeller blade shape as it is actually is and the aerofoil sections which actually provides the conversion of lift and drag to thrust and matching of the torque that is provided by the engine. The theory that uses the propeller blade shape is simply known as blade element theory. There are of course various versions of the blade element theory, we will be doing the elementary version, there are more advanced versions of blade element theory which are beyond the scope of this course and we shall not be doing them, but the fundamental blade element theory that we will be doing provides sufficient backdrop for understanding of how the propeller blades are designed, how the propeller blades finally go on to create thrust and how the aerofoil shapes with characteristic values of C L and C D actually help in creation of thrust. The blade elements which create the propeller blade shape are assumed to be made up of aerofoil shapes. Now, these aerofoil shapes can actually vary from root to the tip of the blade. So, along the length of the blade, the aerofoil shapes can vary quite a lot really and as a result of which their C L and C D characteristics would also vary substantially from the root of the blade to the tip of the blade. Now, this is something which you need to understand very quickly because when the propeller is in rotation the as we have seen before in the simple velocity triangle that you can construct on a blade section, the incident velocity on a blade section would vary from root to tip. Now, one of the reasons is that near the root your rotational speed is rather low and near the tip the rotational speed is quite high and as a result of which the incident velocity at the root is likely to be low and the incident velocity at the tip is likely to be quite high and the difference is quite substantial. Near the roots you could have velocities which we could say are low subsonic velocities and the aerofoil sections therefore, you should be using there are essentially meant for low subsonic applications. On the other hand near the tips the incident velocity which we called we are would be quite high they could be high subsonic in the more modern aerofoils they are in fact going almost transonic and some of these would then correspondingly require aerofoils which are either high subsonic aerofoils or in the modern propellers transonic aerofoils. So, you see from the root to the tip of the blade the aerofoil sections would vary quite a lot from somewhat thicker aerofoil sections which are good for low subsonic aerofoil applications to thin aerofoils which are good for high subsonic or transonic applications. So, the aerofoils you would be using all the way from the root to the tip of a single blade varies substantially in its shape the in its camber and of course, its basic characteristics normally denoted in terms of C L and C D. The other difference of course, is these aerofoils would also have different kind of incident angle range. The C L C D that they use actually automatically would have the aerofoil characteristics built into it and these aerofoil characteristics would show the C L C D applicable over a certain incidence range. Now, this incidence range also varies from low subsonic aerofoil to high subsonic or transonic aerofoils typically the low subsonic aerofoils would have a somewhat higher incidence range of the order of 15 16 degrees of incidence range of operation. On the other hand near the tip where the incidence range is going to be very low because they are thin aerofoils meant for high subsonic to transonic applications and those incidence ranges are often of the order of 4 5 6 7 degrees. So, that is the range that you have of variability of the incidence which means your incident flow velocity V R could change its direction by that much depending on the aerofoil that is deployed at that particular section. So, this is the difference that happens from root to tip of a single blade and this is what the designer has to be very careful about when choosing the aerofoil sections and blending them together into one blade shape because they are actual sectional properties very substantially. So, a large number of different aerofoils are used to make up one propeller blade could be something like 10 15 20 different aerofoil sections with having each having its own CLCD and incidence range characteristics and all of them will have to be blended together into one single blade shape. And each of these elements they have their CLCD characteristics and in addition they have their as I just mentioned their incidence range which is built into the shape of the propeller that is being deployed over here. Now let us look at the various flow geometry and the flow parameters that every blade section can be say to be dealing with. On the left side we have let us say a three blade propeller which is let us say in rotation with an angular velocity omega and if we take one blade section here which has let us say a radial depth of d r and this is the blade section which we say let us say is representative of the propeller. So, typically a propeller is quite often represented by a blade section which is somewhere near about around 60 to 75 percent of the blade most of the subsonic propellers would have their representative blade section at around 75 percent of the blade length some of the transonic propellers may be even higher than that. So, it is somewhere between 50 to 75 percent that you have what can be called a representative blade section. This representative blade section is the first section that is typically designed by the designer and the designer creates its own characteristics which is the average characteristics of that particular propeller. So, every propeller is first designated with an average sectional characteristics or sectional property that is required to create thrust and this average sectional property is built into the reference section of the blade which is as I mentioned normally somewhere near 75 percent of the blade length and this section is the first section that is often designed and then the rest of the sections are designed from there onwards from root to the tip of the blade. So, let us take a look at this representative section over here in which you have let us say a blade which is expected to rotate with angular velocity omega and as a result of which it acquires a rotational speed of omega r which can be written in terms of twice pi n r n being the r p m and this propeller let us say along with the aircraft which it is flying is moving with a forward velocity and width of which is of the order of v infinity. So, we say that this is the velocity with which the air is coming and meeting the propeller. So, that is this is the relative forward velocity with which the air is meeting the propeller and this is the rotational velocity of the propeller section. So, the two of them together make up this resultant velocity v r which is now said to be actually incident on the propeller at this particular section. So, this particular section now has to be aligned as close to this direction of the resultant velocity. Now, this resultant flow is at an angle phi with respect to the peripheral direction or tangential direction of the propeller and then you need to set the blade this particular blade section which here is shown as a blade setting angle of beta and if you set it at an angle beta that means the chord or the zero lift line of the particular blade section is set at this angle beta with respect to the rotational direction then what you have is beta minus phi is alpha which is then the angle of attack of this particular blade section. Now, this is the angle of attack or sometimes called angle of incidence which characterizes this particular airfoils C L C D characteristics. So, one needs to be very careful what is the angle at which this blade section is being set with reference to the angle at which the flow is expected to come into the propeller. So, the angle of v r and the angle at which the blade section is set need to be very close to each other. So, that you have angle of attack alpha which is within the operating range of this particular airfoil as characterized by its C L C D characteristics. So, you have here the propeller blade section being typically harnessed on the one hand using the C L C D characteristics on the other hand aligning it to the actual propeller operation in terms of rotation and its forward velocity. Now, let us say that the blade has been aligned the blade has been set and the flow is coming into the propeller at an angle alpha to the blade section with a velocity v r. Then it creates a small elemental lift of D L born out of the lift characteristics C L. So, this is the elemental lift that is created born out of C L correspondingly this section would experience a small drag of D D which is born out of its drag characteristics C D and these are then the characteristic values of this particular blade element. Now, these are as per the definitions of lift and drag are parallel and perpendicular to the flow direction and as a result of which the lift is in this direction which is perpendicular to the flow direction and drag is in this direction which is parallel to the flow direction. Now, if you decompose the elemental lift D L and the elemental drag D D into two other direction which is the direction of motion of the propeller and the aircraft together and perpendicular to that you get two forces the one which is in the direction of motion of the propeller and the aircraft together gives you the thrust. So, the component of D L in the forward direction or what can we call the direction which is parallel to the axis of the propeller would give you the thrust component which we call D T which is the elemental thrust of this particular element. So, as a result of which you get a thrust now born out of the lift component of the propeller of course, here you would get a very small negative thrust in the actual direction. So, the net thrust would be composed out of positive contribution from the lift and a small negative contribution from the drag. On the other hand if you take perpendicular to the thrust the in the peripheral direction the two components of the lift and drag are actually additive and they together create a force. Now, this force is what needs to be countered or matched by the torque that is supplied by the engine. So, torque which is elemental torque which is shown here as D Q divided by r gives you the peripheral force that is coming in from the engine on this particular section and that needs to be matched with the peripheral component of D L and D D and if they are matched if the torque supplied by the engine matches with this torque requirement of the propeller of this particular section then we have a situation where we get the thrust that we want. So, the lift and the drag are the characteristics of the propeller the torque is what is supplied by the engine and if all these things are matched together properly we get a elemental thrust D T which is what you would require to fly the aircraft when all the elements are put together of a particular propeller. So, the thrust that is created by an element at let us say radial length D R is created with the contributions from the aerofoil which are characterized by D L and D D of this particular element and of course the torque which is supplied by the engine. So, this is how the thrust is elementally created by any particular element of a particular propeller and all the elements put together then create the total thrust of a blade which is in rotation. If you now look at this elemental lift and drag characteristics which then create the working capacity of the propeller. So, we have a blade element which is in work which is in rotation and in rotation it is doing work and in the process of doing work it is actually creating thrust. So, the thrust that is produced by this particular element can be now written down in terms of D T which is shown in the diagram and they can be written down in terms of the elemental lift D L elemental drag D D as I have stated the drag component actually gives you a slight negative component and as a result of which you can now write down the entire thrust equation. The first part that is half rho v r square c actually is the dynamic head which is created and C L is the characteristics of the propeller blade section. C D also is the characteristics of the propeller blade section which is under consideration. Phi is the angle at which the flow is coming into the blade and C is the chord of the particular blade section and as we have seen in the earlier diagram the propeller blade section could vary from quite a lot. The actual value of the chord could vary quite a lot from the root to the tip of the blade. So, chord of the blade would vary from one section to another. So, in this thrust creating equation rho is the density of the air which is an operation which let us say is invariant from root to the tip of the blade. On the other hand as we have seen from this diagram v r would vary from root to the tip of the blade c would vary from root to the tip of the blade d r the elemental length could be same for the for each and every element. Let us say that you take the C L value would vary from root to the tip of the blade depending on the air fall section you are using same with C D its value would vary from root to the tip of the blade of any particular blade and then the value of phi also would vary from root to the tip of the blade depending on the rotational speed which depends on r omega angular velocity being constant. So, phi would vary also from root to the tip of the blade. So, in this thrust creating equation as you can see the except for rho all the other parameters actually are varying from root to the tip of the blade. Now, supposing if you want to have the thrust created by each and every section of the same order you would probably have to manipulate these values in terms of lift a lifting capacity the drag penalty the resultant velocity that is coming in and the chord dimension of the chord to get the value of d t constant from root to the tip of the blade. Now, quite often that may not always be possible. So, quite often the actual elemental thrust produced by each section could indeed vary from root to the tip of the blade. Now, let us look at the torque that is to be supplied. Now, this is to be matched by the engine supply. So, torque is required for the propeller to be operated and this is to be supplied by the engine. So, again using the same diagram we can write down the torque equation here in terms of the elemental lift and the elemental drag and in terms of the local flow angle phi. Again if we write down in terms of the fundamental propeller blade characteristics the air fall characteristics we can write down half rho v r square c again is the chord d r is the elemental length of the elemental blade c l is again characteristics and phi is the flow angle c d is the characteristics and fly is the flow angle. So, here we can see that the components from c l or lift and drag are additive they are on the same side of the axis and as a result of which they are additive and they add up together to create a force and torque which needs to be matched by the engine supply and this when multiplied by r gives you the torque. Now, this torque has to be supplied by the engine without which of course, the propeller would not operate at all. So, this is how you create thrust this is how the torque is needed to be supplied by the engine is and is matched by the propeller requirement and if this requirement is matched and if we have a proper airfoil deployed there with proper c l c d characteristics then you have a thrust production that is required for flying an aircraft. Now, if you proceed along those lines and if you say that the from the flow geometry that is we have created the result in inflow velocity is a line to the blade element you know and if we write down v r as v infinity which is the forward velocity of the incoming axial velocity by sin phi which is the angle which it subtends and the incoming flow dynamic head based on the forward velocity of the particular element if we make these two substitutions in the equation that we are written down for thrust and torque from the flow geometry that is available over here the elemental thrust can now be written down in terms of q which is now the dynamic head c is the core d r is the elemental length and phi is the flow angle and again in terms of c l and c d. So the thrust elemental thrust can now be written down in a slightly different form but using the same characteristic values of c l c d and flow angle correspondingly the elemental torque can be written down again in terms of dynamic head q the chord c r of course is the length of the element from the axis and d r is the elemental length of the particular element and phi is the flow angle c l c d are the characteristics of the aerofoil being deployed at that particular section. So the elemental thrust and the elemental torque can be now written down in terms of the characteristics of the aerofoil and the geometry of the aerofoil and the flow which is coming in at an angle phi. So these are the things that are built into the equations that create the elemental thrust and the elemental torque. If we now try to put all together as we have seen the blade is a blended version of all the aerofoils put together and when all the aerofoils are put together you get total propeller thrust and the total propeller torque. So the elemental thrust and torque we were talking about are now to be integrated from the root to the tip of the blade and let us say we have a number of blades which is b which as we have seen could be 2, 3, 4 or 5 or 6 or 8 that is typically the number of blades normally uses these days and if you put them all together you get the total thrust of the propeller and the total torque of the propeller that needs to be supplied by the engine. So these two are to be very accurately estimated only when the total torque is properly supplied we get the thrust that is required to fly the aircraft. So this is the thrust you would require to fly the aircraft in a predetermined manner to do that you need to be supplied with that kind of a torque from the engine and of course the propeller power needs to be matched with the engine power to be supplied by the engine. So these are the parameters that you require finally for the propeller to create thrust based on the aircraft on which it is mounted and based on the engine to which it is attached. Hence we see that the net thrust and the torque are seen to be directly proportional to the number of blades b and the chord c as given in the final thrust and torque relationship. Now that gives an impression that if you keep on increasing the number of blades let us look at the thrust equation quickly you see here the thrust and torque are directly related directly proportional to the number of blades b and the chord c both of them. Now that gives a immediate impression that if you simply increase the number of blades you get more thrust of course you would require more torque or if you simply increase the size of the chord let us say all the way from root to the tip of the blade you would actually get more thrust and of course you would need to be supplied with more torque. So if you are supplied with more torque you would simply get more thrust. Now that is the impression one would get from the blade element theory and the torque and the thrust equation that we have put together in practice that is not quite true. What happens is if you increase the number of blades or the size of the blades by increasing the chord it shall result in more surface area of the blades if you have more blades around it will create more flow blockage and as a result of which very high aerodynamic losses. So the efficiency of the propeller blades would start falling. So if you increase the number of blades and the size of the blades or the chord the efficiency of the propeller would be affected the propellers would create more and more blockage to the flow and the thrust create is directly proportional to the amount of mass flow that it actually activates. So by increasing the surface area you are decreasing the efficiency of the propeller by increasing the number of blades you are increasing the blockage of the propeller and two of them together actually reduce the aerodynamic efficiency of the propeller. So just by increasing number of blades or the size of the blade shapes you would actually be reaching a situation where you would not get more thrust there would be what can be called a point of diminishing return at which you have to stop your increase of number of blades and that is how the number of blades are decided quite often and that is how you see today even today the number of blades are of the order of 3 or 4 or 6 or 8 and very rarely more than that. And the reason is here that if you just simply increase the number of blades you are not going to get more thrust. So the optimum number of blades that need to be decided would need to be found separately and does not directly come from the blade element theory. So blade element theory is not a correct indicator of the number of blades that need to be deployed for reaching a certain value of thrust. The next thing we need to talk about of course is the blade element efficiency. Every element is creating thrust as we have seen they use the airfoil characteristics C L and C D and as a result fundamentally they are aerodynamic entities and any aerodynamic entity would have a certain aerodynamic efficiency. So these blade elements would have certain basic elemental efficiency which when blended together would give us of course the total propeller efficiency. So the elemental efficiency typically as per known efficiency definition is the thrust power that is produced by the particular element and the torque power that is supplied by the engine. So the elemental airfoil characteristics that we have seen before now can be made use of and if you do that the elemental efficiency comes out in the form of C L and C D. So the elemental efficiency of any particular blade element of a propeller can be written down in terms of C L and C D and phi which is the flow angle into that particular element and these three together finally give you the elemental efficiency of the propeller which stands to reason really because the three of them together actually decide the aerodynamic working capability of that particular element which as you know is an airfoil section. So this airfoil section is operative in a particular situation which is incident flow at an angle of phi and this of course creates the local aerodynamic flow angle which creates lift and the drag and they together then finally decide what is the efficiency with which this element is going to perform. So we can directly calculate the elemental efficiency from the airfoil characteristics and the local flow condition. Now what can be done is you can find that the maximum efficiency of this element from the earlier equation that we have written down it can be shown by simple algebraic derivation that the elemental maximum efficiency occurs at this value phi by 4 minus C D by twice C L and this is how you can quickly calculate what could possibly be the maximum elemental efficiency. Now of course you know the elemental airfoil is characterized by the C L and the C D and hence it stands to reason that they together decide what the maximum efficiency is likely to be. It is generally found that all the things that we have been finding the thrust, the torque, the efficiency and the maximum elemental efficiency all these can be found with the reasonable engineering approximation and which normally gives us of the order of 10 percent approximation which is a fair approximation to begin with given the simplicity of the theory and it allows us to design the propeller it allows us to predict the propeller performance with reasonable approximate accuracy. And if we do that we can create a propeller which then can be deployed in a aircraft flight. So, these are the some of the simple things that can be derived out of this elemental propeller efficiency and the thrust and the torque that we can calculate. If we can now put together all of it into typical blade every element is now created with a lifting characteristic and let us say that we get a value of variation of lift variation of thrust and torque in the direction from root to tip that is designated as x from root to the tip of the blade. And if we can show that there is a gradient of thrust coefficient and torque coefficient which can be found and if they are plotted again from root to tip we would get a characteristic curve like this for any particular propeller. So, the elemental thrust variation would look like this and you would probably get a maximum thrust somewhere over here and the elemental torque characteristics would look something like this and you would get a maximum torque somewhere over here. What you can see here is that the maximum thrust and the maximum torque may not occur at the same element. So, the same element may not be giving you maximum thrust just because maximum torque is being supplied the thrust there could be less than maximum and you could get maximum thrust not necessarily at the maximum torque may be something a little less than maximum torque. And of course, as you can see the maximum thrust and torque typically occur on the outer half of the propeller blade and the lower half actually are less contributory to the thrust. And of course, they also consume less of torque and towards the root of the propeller you could see that it could actually be creating negative thrust because that is the portion which is often not properly aerodynamically shaped they may not have very good aerofoil shapes over there. In fact, they may not have aerofoil shapes there at all because that portion is structurally strengthened and hence they may be creating actually negative thrust around the root area. So, most of the thrust need to be created on the outer half of the propeller. However, near the tip of the propeller again as we can see the torque and the thrust dips very fast. So, very near the tip of the propeller the tip flow actually highly influences the propeller elemental behavior there and you are unlikely to get much thrust contribution from the tip area. So, one needs to be very careful in designing the propeller in which the thrust needs to be distributed in a manner so that together the blended propeller would give the necessary thrust that is required for flying the aircraft. So, this is the variation that you need to build into a propeller shape to get a propeller that is useful for flying an aircraft. A typical low speed aircraft propeller characteristics is being shown here which is shown in terms of C T and C P the C T of course as we have seen is the thrust coefficient and C P is the power coefficient and as we can see here and the efficiency of the propeller is being shown over here which shows that the efficiency of the propeller maximizes somewhere at one point this is plotted with reference to advance ratio V by N D and this shows that the efficiency of the propeller could be maximum at one point whereas, the thrust and the power could be maximum at some other point of advance ratio. Now, advance ratio as you know would vary with the forward speed of the propeller which is the flying speed of the aircraft and of course, the rotational speed. So, the ratio of the two of course gives the advance ratio. So, typically the higher the aircraft flying speed higher would be the advance ratio. Now, this is a low speed aircraft. So, advance ratio values we are looking at are somewhat on the lower side. So, typically that is the variation one would probably get from a low speed aircraft propeller and first thing we can see here is that the maximum efficiency operation if you if the cruise or the longest deployment of the propeller which is during cruise is somewhere near the maximum efficiency that is not the point where you of course use maximum thrust or maximum power you would probably be using less than the maximum power and you would probably using less than the maximum thrust coefficient and still creating sufficient thrust to fly the aircraft. On the other hand when you are taking off you would be somewhere near 0 of advance ratio and you would be using maximum near about maximum power to create near about maximum thrust. You would probably need to create a good thrust during the climb operation of the aircraft. So, you would need to create good thrust and then of course, during cruise you come down to lower thrust and lower power requirements and where you have very high efficiency of the propeller. So, good propeller efficiency is often obtained near the cruise whereas, as you can see here new during the takeoff the propeller efficiency is indeed quite low actually, but you operate there for a very short period to create high thrust of course, using high power. We can have a quick look at the high speed aircraft where the again the cruise is somewhere near the maximum propeller efficiency to get the best fuel efficiency of the power plant. It is it does not use the maximum power or maximum thrust coefficient which are typically high the highest near the takeoff and that is where you need to create maximum thrust for the takeoff and the climb operation and this is where you can see the advance ratios are little on the higher side compared to the earlier one and the advance ratio here shown is of the order of 3.5 maximum and you are probably operating at advance ratio somewhere near 2.6. So, this is a comparatively high speed aircraft on which a propeller has been deployed for providing thrust to the aircraft. Now, this is what you would probably get of a propeller characteristics which is a variable pitch propeller. Now, you see we have discussed the fact that propellers can be fixed pitch or variable pitch most of the modern propellers used in most of the aircraft today are variable pitch propellers. Now, these variable pitch propellers then would need to have their thrust versus advance ratio and efficiency versus advance ratio characterized with different pitch angles and this is the blade angle that is beta that we have seen before and each value of beta then would create one characteristic graph like this. So, if you have a variable pitch propeller one of the job of the propeller designer is to create variable pitch propeller characteristics within which then the aircraft would need to be flown the engine would need to be operated. So, that all the time the propeller is matched on one hand with the engine on the other hand with the aircraft. Now, what we see here is a probable cruise point selection which is where you are likely to operate at a comparatively high advance ratio and you would probably get at a you are working at a high pitch angle. Now, during the cruise your pitch angle is likely to be higher order during the takeoff your pitch angle of the propeller would likely to be rather low or as it is often called they are finely set whereas, during cruise it is going to be a core setting of the propeller and so during the cruise the efficiency of the propeller can be used for a high pitch angle and you would get a good efficiency there during the cruise operation of the aircraft. So, this is the kind of variable pitch propeller characteristics that you would get of thrust coefficient and efficiency. We can get a similar characteristics for the power coefficient and this is of course, a thrust power characteristics. So, you have the thrust variation here or on the y axis you have the power variation and what you would see here is a thrust power characteristics. The cruise point is likely to be somewhere over here where you would use modest amount of power and create modest amount of thrust to sufficient for the aircraft to fly. On the other hand near the takeoff you would need to create more thrust and hence you would probably using more power and as a result of which you would be operating somewhere over here which is at near 0 advance ratio whereas, over here you are operating at high advance ratio during high forward flight speed of the aircraft. So, these are the typical characteristics that characterizes the propeller every propeller once designed and created would need to have a characteristic plots like these for the engine designer and the aircraft designer to match to and this is absolutely necessary for matching the propeller with the aircraft and the engine. There is a fourth characteristics which is often used the first three being the thrust coefficient, the power coefficient and the efficiency. A fourth characteristic which is often used for propeller design or selection is simply called speed power coefficient and this is defined as C s and is defined as rho v to the power 5 divided by power p into n square n being the rotational speed all of it together power of 1 by 5 that is 0.2 and this is often used for designer selection of propeller. Now, this speed power coefficient can be related to the power coefficient simply with the advance ratio J and is related as C s equal to J by C p to the power 1 by 5. Now, why this speed power coefficient has been created this is of course, used only by the designers and the propeller selectors. This is because in the process of creating this definition the diameter of the propeller has been eliminated as a result of which you have a parameter normalized parameter in which the size of the propeller has been taken out of the equation and as a result of which you can create a propeller characteristics conceptually without to begin with a priori bothering about the size of the propeller which can be then factored in a little later in the design process. So, this speed power coefficient allows you to create a conceptual propeller without fixing the size right away and the fixing of the size can be slightly deferred to a later date. So, that you have a propeller conceptually already created and its characteristics also already created and this is the advantage of the speed power coefficient which is defined in many of the literature. This is a typical speed power coefficient characteristics of a propeller again a variable pitch propeller and this shows that at under various pitching angle operation the speed power characteristics would vary. This is the advance ratio this is the speed power characteristics on the x axis it is plotted against advance ratio j and the efficiency eta and these are the efficiency curves and these are the advance ratio versus c s curves and these are plotted at various pitch angle. So, we are characterizing typically again a variable pitch propeller. So, this is the kind of characteristic plot which helps selection or design of a propeller for a particular operation in which thrust needs to be created and the power needs to be supplied by the engine and as a result of this we get propellers which can be matched to a craft and engine. Some of the modern propellers we can take a quick look at have shapes which are which have sweeps and these swept propellers are being used in the modern propellers which have gone transonic. The airfoil shapes of these are of transonic airfoil shapes some of the new propellers that are coming up are counter rotating that means you have two propellers one behind the other and the second one is rotating in the opposite direction to the front one. So, these are called counter rotating propeller. The propellers can be fundamentally two types one is known as a tractor type in which the propeller is deployed somewhere at the front of the aircraft and it is pulling the aircraft. So, to say and that is why it is called tractor type sometimes they are deployed in the rear of the aircraft and quite often they are referred to as pusher type as if they are pushing the aircraft from behind. So, the tractor type is what is deployed somewhere in the front of the aircraft the pusher type is deployed somewhere near the rear of the aircraft. So, these are the various kinds of propellers which are typically used in aircraft. In typical aircraft application we can say that the propeller power will be equal to the engine power multiplied by the shaft efficiency multiplied by the propeller efficiency. So, we have to keep an eye on the propeller efficiency which is a composite of all the elemental efficiencies. The propeller torque we will have to be matched exactly with the engine torque that is being supplied with the engine and till you do that you are not going to get the proper thrust that is required. Typically during takeoff the torque requirement is low the blade setting angle or the pitch angle is low the power required is high and the rpm is high because during takeoff you require very high thrust normally you require very high thrust. During cruise on the other hand the torque requirement is very high that is because the blade pitch setting angle is very high and as a result of which the torque requirement is very high, but on the other hand the power requirement is rather low and the rpm is also rather low compared to the takeoff rpm. So, during the cruise and during the takeoff the propellers operate at quite different operating conditions to create thrust one for takeoff which is very high one for cruise where the thrust requirement is indeed rather low, but the torque requirement is rather on the higher side. So, these are the different two at least two different operating conditions which a propeller has to cater to to ensure that the aircraft flies properly. So, these are some of the fundamental issues that comes out of propeller theories and you need to ensure that you have a propeller which supplies sufficient thrust during takeoff meeting all these requirements and you have a propeller which supplies thrust during takeoff again meeting all the requirements that are shown over here. So, only then you have a propeller which is worthy of putting on an aircraft for flying the aircraft. In the next class we will be trying to make use of all these propeller theories that we have done the propeller fundamental definition that we have used the C p and the C t and when you use all these fundamental theories it should be possible for us to solve some very simple problems. So, that you get a feel of the numbers I will be bringing along a problem and I will solve a problem for you may be a variable pitch problem. So, that you get a idea of what happens when these theories are used to solve real life problems and so that is what we will be doing in the next class trying to solve problems making use of the theories that we have done over the last three lectures.