 Today, we will start a new chapter on propellers. Propellers have been providing thrust for flight of aircraft for little more than 100 years, 107 years to be exact and ever since the Wright brothers flew their first aircraft, it's a propeller which have been providing thrust for flight of aircraft. The creation of thrust by the propellers is done quite differently than let's say compared to the jet engines, which of course came much later, much much later. The propellers create thrust by virtue of rotation of blades and how the thrust is created by the propellers, by the rotating propellers is what we will discuss in today's lecture and in subsequent lectures. The propellers initially when they were flying their craft were made of wood, ordinary wood because that was the material that was most suitable for giving the complex shape of the blades that the propellers are made of. Later on over the years new materials have given in, materials like aluminum alloys, materials like titanium alloys and of late the composites have given rise to more intricate and more accurate shaping of the propeller blades and that of course has enhanced the efficiency of the propellers and of course allowed propellers now to go to high speed applications and in fact today some of the modern propellers are working under supersonic and transonic working conditions. That is the advancements that has happened over a period of almost 100 years. To begin with the propellers were made of very simple material as I said wood and as a result of which the shapes are also rather simple. Let us start with the fundamental issues that govern how a propeller works. To begin with a propeller is an interface between an aircraft engine and an aircraft. You see the engine which we have been discussing for example in the last few classes the piston engine IC engines they do not give you thrust. They actually provide only power and this power needs to be harnessed for creating thrust which only then makes the aircraft fly. So, the aircraft requires thrust to fly and the propeller is the intermediary between the engine and the aircraft for creation of thrust and of course as we have discussed this thrust creation needs to be done differently during different modes of the aircraft flight. For example, when the aircraft is taking off you need one kind of thrust or one amount of thrust when it is cruising you need another kind of thrust another amount of thrust. All this needs to be provided by the propeller during the various modes of the flight. So, the engine and the propeller need to be matched in a manner that will enable the aircraft to fly during its various flight modes. The creation of thrust for flying of an aircraft is what the propeller is mainly entrusted with. So let us see what kind of propellers people have been using over the years. For example, propellers create thrust essentially by harnessing the air which is available in the atmosphere. So, propeller works in the open atmosphere unlike an engine which works inside an enclosed atmosphere in an enclosed chamber of an engine whereas the propeller is out in the open and is working medium is the atmospheric air the open air. So the air is deflected or rather pushed backwards by the propeller and this push backwards of course creates the thrust. The propellers over the years have changed their shape size and number of propellers. We see here in the picture there are three bladed propellers, there are four bladed propellers, one could actually have two bladed propellers for very small aircraft one seater or two seater very small aircraft or you can have propellers which have up to eight blades. If the number of propellers blades is more than 10 in a propeller the modern designers would prefer to call it profan rather than propellers. There are of course propellers which are counter rotating propellers. So you have two rows of four bladed propeller let us say eight of them and the rear four is rotating in a direction rotating direction counter to the front four. So that is a counter rotating propeller and such very large propellers have been used in the past for flying very big aircraft in subsonic flight speeds. So some of these designs have been used over the years for creating massive amount of thrust. We have also seen that the propellers actually are very efficient means of creating thrust. It is actually a more efficient device for creating thrust than a jet engine and because of this basic inherent efficiency of the thrust creation that propellers have held their ground even today even after 100 years. In fact many of the modern very modern aircraft propulsive units that are coming up essentially are using the concept of propellers for creating what are known as profans and essentially they are modern versions of the earlier propellers. We will probably have a chance to look at those later on in these lectures. Let us take a look at what a propeller details are of a propeller blade. A typical propeller blade would have what would be called a leading edge and of course what would be called a trailing edge. The root of the propeller is shaped in such a manner that it blends with the hub over here and quite often the root may not have a proper shape for creating thrust whereas the tip of the propeller quite often is very thin and often may not make a very large contribution to the thrust creation. We shall see very soon that the main portion from let us say from here to here which actually are the main thrust creating part of the propeller are made up of airfoil sections. So propeller blades are essentially made up of airfoils which invariably then participate in the creation of thrust and in this lecture today we shall see how that actually happens. The propellers are finally rotating around central shaft which is the hub of the engine and this shaft is getting power from the engine as necessitated by the actual propeller rotation and this matching between the propeller and the engine in terms of power is extremely important and we shall see today that this matching is required for power as well as for torque which is also to be supplied by the engine. So the matching of the power and torque is important for creation of appropriate amount of thrust which of course then flies the aircraft. So these are the some of the details that we will be going into in today's lecture. Let us look at the fundamental concept of how the propellers actually start creating thrust. Propeller is fundamentally what can also be called a propulsor and a propulsor by definition creates thrust and the creation of thrust the fundamental notion of creation of thrust is by making a small change in the momentum of the propulsive fluid which in our case is air and that has to be in the direction of motion and direction of motion of the aircraft. So the direction in which the aircraft needs to move in that direction a net positive change in momentum needs to be created which then creates the thrust. Now this is what a propulsor is supposed to do the amount of thrust that is necessary is fundamentally dictated by the amount of drag that the aircraft is experiencing and that is the minimum amount of thrust that the propulsor must create. If the aircraft is climbing or doing any other maneuver then the thrust required could be more and the propulsor is supposed to provide that additional thrust that is required for such maneuvers. Now again fundamentally the propulsor creates thrust by changing the momentum. This is an equation which you probably have seen before in which the thrust is equated to the mass of working medium in our case air which is taken from a velocity V a which is the incoming velocity let us say and the exit velocity V e and the differential of these two velocities multiplied by the mass of working medium essentially creates thrust. Now in case of propulsors or propellers more specifically it works on the principle that the mass of air it is activating is very large very large and the amount of change in velocity is rather small compared to many of the jet propulsion units which you may have already encountered. So in a typical jet propulsion the mass of activation of air is comparatively much smaller whereas the change in velocity is of a much higher order and then thereby they create a sufficient amount of thrust for flying aircraft. In case of propeller the mass of activation of air is very large and it takes a very small change in velocity to create sufficient thrust for flight of aircraft. So there is a fundamental difference in the way propellers create thrust and how the jet propulsion devices create thrust using the basically same concept of creation of thrust. Let us take a look at the propeller blades the propellers modern propellers have many blades but it can have minimum of two blades or more and each blade is exactly similar to the other and they are set in a symmetrical fashion in an annular space. So you could have three or four or six or eight blades all arranged in a symmetrical fashion in an annular space and each of these blades is actually created by stacking of airfoil sections from root to the tip of the blade. So the entire length of the blade is actually made up of airfoil sections. What you see here in the picture is what can be called a plan form of the blade typical propeller blade in which typically from the root to the tip of the blade the size of the airfoil actually changes. The width of the propeller is approximately representative of the chord of the airfoil and this width quite often in more popular way changes from root to tip. It is somewhat large in the middle section of the propeller and becomes somewhat lesser towards the tip and quite often the tip is rounded essentially for reducing the losses related to the tip flow the flow around the tip and the root is quite often round kind of a section could be almost a circular section sacrificing all the airfoil shapes essentially to provide good structural strength to the blade. Because as you can see here the entire blade is a cantilever it is fixed only at the root and rest of the blade when in rotation is actually a more like a cantilever beam. So it experiences a lot of bending forces over this entire length and the entire force and the moment has to be borne by the root section which is fixed over here and is also rotating. So those sections are called often non aerodynamic sections. So they do not participate in the creation of thrust their basic job is to bear the stresses that are coming up due to the activation of the blades in rotation. So the shaping of the blades are done for various purposes as I mentioned the middle 60 to 70 to 80 percent of the blade actually is aerodynamically shaped for creation of thrust the root section essentially bears the load the stresses and the tip section is quite often rounded to ensure that the aerodynamic losses around the tip are minimum. So that the efficiency of the propeller is also maximized. So this is how normally a blade shape is created for propellers. We shall see later on in this lecture series that very modern blades often have somewhat different looking shapes to cater to the modern blade design and the modern aerodynamic thinking and we shall introduce those things as we go along in this lecture series. As I mentioned the blades which are used in thrust creation are actually made up of aerofoil shapes. If you take a sectional cut at any part of the blade you will probably see an aerofoil section. These aerofoils have certain properties as you probably know and these properties are harnessed essentially in a stacked up manner to create the blades and all the blades together are then composition of aerofoil sections which are then together create the thrust. We shall see today how actually the blades finally create thrust because as you all know blades fundamentally their aerodynamic properties create lift and drag and we shall see today how this lift and drag are finally harnessed for creation of thrust. What you see here today is a particular kind of aerofoil which has been created essentially as a basic aerofoil shape for propellers. These blades are often designated by the creators. For example, some of the blades are known as the NACA aerofoils. Those are created by the American organization NACA more than 60 years back. Some of the blades are known as Clark Y profile and some of them are known as Göttingen profile which were created in Germany and these are the profiles which have been made greatly used of in various propeller designs. The common feature in some of these profiles is that many of them actually have large part of one of the surface as a flat surface or what is often known as flat under surface. So, one of the surface is deliberately made rather flat and there is a rounding near around the leading edge of a typical aerofoil and then you have the cambered side of the back of the propeller which is then the main lift producing surface of this propeller. So, one of the surfaces is basically more of a non lift producing surface whereas the main lift production is left to one surface. This makes the job of the propeller designer a little easier and that is one of the reasons why these aerofoils were created. Of course, the aerofoil always ends with a trailing edge which is normally has a very small rounding over here much smaller than the leading edge rounding typically much smaller and as a result of which they create lift and the aerofoil surface as a result of the various frictional losses experience certain amount of drag. So, that is a property of the aerofoil and these properties are harnessed in the process of creation of thrust in a rotating blade. The coordinates of the NACA aerofoil are given here they are little small I hope you would be able to sit down and look at those profile coordinates. However, these profiles are available in the open sources and you may be able to get them from various books and other literature quite easily. The other profiles which I was talking about the Clark Y and the Gottingen aerofoil and the fourth one is RAF which is Royal Air Force aerofoil are given here I hope you would be able to look at these profiles and create your own aerofoils and these profiles are also easily available in open literature in many books and other literature. All of these profiles given here have a common feature that all of them are flat under surface aerofoils as shown here. So, those are the aerofoils which have been used for many many years for creation of propellers which are mainly subsonic propellers, but many of the modern propellers are transonic and very high subsonic and those aerofoils are likely to be quite different from these aerofoils which are given here. These are the standard aerofoils used for somewhat low speed applications and they have been used for little more than 100 years now. Now, we will look at the modern aerofoils later on in this lecture series. Let us see how aerofoils actually are stacked in a blade typically when a blade is operational the lift creation depends on the way the flow is locally incident on the aerofoil and it depends on the aerofoil characteristic which is often decided by the local angle of attack. So, what is shown here for example, in this is that at the root section of the blade it is subtending a very high angle of setting blade setting and this angle of setting is necessary for the root blade section and we shall have a look at the velocity diagrams in a few minutes now. On the other hand near the tip of the blade section it is set at a low angle. So, this is often called fine angle this is often called the coarse angle. As a result of this necessity of tip being at a fine angle and the root being at a somewhat coarse angle inevitably the blade gets a twisted shape and that is the amount of twist which the blade would invariably have. Now, this would depend on the particular propeller design and it will of course vary from one propeller to another it also depends on what it is being designed for and quite often the propellers may like to optimize the blade shape keeping in mind the various aircraft operations which are often given in terms of the so called design point and various off design points which cater to various aircraft light schedules or various aircraft maneuvers which are required for the aircraft. So, quite often the propeller blade shape could be some kind of a optimized blade shape and finally, it is given certain amount of twist to cater to all kinds of necessities that the propeller would have to bear during its operation for creation of thrust. So, a propeller not only is a composition of various aerofoils the aerofoils are stacked at various angles of setting and that gives rise to the twisted blades which are invariably the shape of each and every blade that are part of the propeller. So, what we see now is that a propeller is typically dependent on the local aerodynamics of the flow incident on the blade elements and these blade elements are stacked as we have just seen from root to the tip of the blade and this root to tip stacking essentially creates local flow incidents at various stations or various sections of the blade. The flow incident of the root as we have seen in the earlier picture is likely to be at this angle whereas, the local flow incident at the tip is likely to be at this angle. So, the blade setting is actually given to the particular blade section because the flow incoming flow or the relative flow angle would be actually around this angle at the root and around this angle near the tip. So, to meet the incoming flow incident on to the blade section the blade sections are provided with the particular blade setting. For efficient operation of each of these blade elements which are of course aerofoils as we have seen we need to create an appropriate angle of attack. As we know all aerofoils have a characterized value of angle of attack at which it is most efficient and it has a zone of angle of attack or a range of angle of attack over which the aerofoils are actually operative and they give good performance over this range. Beyond these range the aerofoils often likely to give very bad performance there is of course a maximum angle of attack where this aerofoil is likely to stall that means it will simply refuse to create any more lift and the drag would be too high. So, these ranges of angles of attack for each of these aerofoil sections would have to be factored in for each of these blade sections which are stacked in creation of the entire blade. We have to see that if possible most of these sections or as many of these sections as possible should operate somewhere near its maximum lift creating angle of attack or more appropriately the maximum elemental lift to drag ratio of each of these blade sections and that is what the designer has to do to stack the blades in a manner that most of the blade sections operate somewhere near their individual maximum elemental lift to drag ratio. Now, let us take a look at a typical propeller blade its sectional geometry and the local flow details. You see the sectional geometry as is shown here is fundamentally of an aerofoil section. What happens is you have a propeller which is rotating and the rotation provides a certain rotational speed of the particular blade section. So, each blade section is now rotating at a particular rotational speed or what may also be called a tangential velocity which is omega r omega being the angular velocity which can also be expressed in terms of twice pi n r n being the r p m of the r r p s of the blade that is rotating in a solid body r of course is the radius of the particular section from the axis of rotation of the propeller. So, this provides a sectional rotational speed or tangential velocity of that particular section with which it is now rotating. Now, if the aircraft is moving forward and if the propeller together with the aircraft is moving forward it has a forward velocity which we may now call v infinity which is the entire forward velocity of the aircraft propeller combined. Now, with which if the they are moving it creates an angle of phi between the two velocities the rotational velocity and the forward velocity and then two of them together create this resultant velocity v r which is now the relative incident flow onto this aerofoil section. But this incident flow now creates an angle of attack of alpha with reference to this blade section which is an aerofoil section. So, this angle of attack is what we were talking about and this needs to be something which is within the characteristic property of the particular aerofoil that we have chosen. So, this angle of alpha needs to be kept in mind while creating the propeller blade because this angle of attack has a maximum beyond which this aerofoil will refuse to do any aerodynamic action of any use to us and very low angles of attack or negative angles of attack also could start creating non aerodynamic effects of the propeller which are again of no use as far as creation of thrust is concerned. Now what happens is by virtue of the geometry that has been created these propellers have what can be called an ideal forward motion which is created or built into the geometry of the propeller. So, if this is the angle at which the blade has been set it is expected ideally or theoretically that the forward motion of the propeller would be so much per revolution of the blade. So, for one revolution of rotational motion of the blade the forward motion would be so much and that is called the geometric pitch. So, that is the ideal pitch or the motion of the propeller forward motion of the propeller. What happens quite often is that because of the fact that air is flowing over the blade the real flow that flows over the blade and certain amount of small angle of attack is created the angle of attack and the final angle that is subtended by the actual velocities creates an actual pitch which is a result of real flow aerodynamics over the blade and there is a small difference between this geometric pitch and an actual pitch and we shall be talking about this difference in a few minutes. So, when a blade is operational it has a certain ideal geometric feature which is built into the aerofoil shape of the particular blade section it is given a shape and a blade setting keeping in mind certain norms of its operation certain aerodynamic properties that it has and then of course, the real flow happens. So, there is always a small difference between what actually happens and what it was ideally designed for and that difference is in terms of the geometric pitch and the actual pitch. The geometric pitch of course, is given in terms of twice pi r into tan beta which is of course, the angle at which the blade has been set. So, that is the geometric pitch which the blade is supposed to encounter in its forward motion per revolution of the propeller. Now angle of attack which we are talking about and which is a property of the aerofoil section is a function of the blade element and the geometric pitch which is the blade setting and which is built into the design of the propeller and of course, the effective pitch angle. So, these two things together create the angle of attack which means the geometric pitch is something which is already designed and created and set and let us assume for the typing that it does not change. On the other hand the effective pitch angle the flow angle could change depending on the operation of the propeller. For example, if it is rotating at some other speed this tangential velocity a rotational speed would change. If the aircraft is moving with some other velocity forward velocity v infinity would change in which case this value of phi would change. Now you see the beta has already been fixed by design and for the timing we are assuming that beta is not changeable in which case the difference between beta and phi that is the angle of attack alpha would change. Now this change occurs during the actual flight or operation of the propeller and as a result of which the angle of attack is a variable quantity during the operation of the aircraft. So we have to keep an eye on this variation and see to it that it conforms to the aerofoil properties which has as I mentioned a range of angle of attack within which this aerofoil is good beyond that the aerofoil is really not good for use as an aerodynamic entity. The rotational speed often designated as u which is equal to omega r of each blade element is different it varies with r omega of course is fixed from for the entire blade but the r varies from root to tip and hence that rotational speed would vary from root to tip. And as the forward speed is let us say same from root to tip the pitch setting needs to be vary from hub to the tip so as to maintain the best angle of attack for each of the blade elements and this is what give rise to the twist and this is the reason why because of which we needed to have the blades at various angle of setting from root to tip because the local rotational speed at the root is much less the local rotational speed at the tip is much high and as a result of which the angle required here in fact the value of phi also here be very low so that we need to keep the angle of attack low over here whereas in this case for the root the angle of attack also needs to be kept within a certain value but the value of phi here is going to be high and as a result of which your blade needs to be given a twist and this is from the fundamental fact that the rotational speed omega is constant from root to tip but the actual rotational speed that is u varies from root to tip the tangential velocity varies from root to tip for each element of the blade. Thus the angle beta which is made and that is what we are talking about it is known often as the pitch angle or the blade setting angle and this is defined often with respect to either the zero lift line of an aerofoil or the chord line which is more and easily understood or in case of propeller quite often the flat under surface of the blade section as we have just seen many of the propeller blades are indeed made up of flat under surface aerofoil sections so for propeller making that is quite often a very convenient reference line to which the blades could be set. The pitch which we have just defined refers to the forward movement of the propeller for one revolution of the blade or the blade particular blade section. We shall see that theoretically each section of the propeller as per the definition we have just given may have its own pitch which is theoretically possible. However, you know that entire blade of the propeller is a solid body and hence the entire blade is moving together with all its blade sections and aerofoil sections as one solid body and hence they have all of them have only one single forward motion. As a result of which this difference between the geometric pitch and the actual pitch which is for the entire blade could be different for different sections and this difference arises as the propeller or the aircraft together starts moving. So, the difference between geometric pitch and the actual pitch for the same section arises when the propeller starts moving both rotating as well as in forward motion. So, this is basically the concept of pitch which needs to be understood with reference to the operation of the propeller. We shall see later on that this pitch as a concept has great importance in the operation of the propellers. We are using aerofoil sections for creation of the propeller blades. Aerofoils fundamentally create lift and drag. They are fundamentally shaped to create lift and drag. The lift is the positive component which we want. Drag is the penalty you pay in the process of creation of lift. So, these are the fundamental forces that are created by the blade shape and typically the lift is perpendicular to the chord of the blade and drag is parallel to the relative wind direction coming on the blade element. The relative wind direction is often described with reference to the chord of the blade. So, these are the fundamental properties of the aerofoil section which is the blade element of the propellers. What we shall see now is how these fundamental forces of lift and drag are harnessed towards creation of thrust and of course, the tangential force which is met by the supply of torque from the engine. If you look at this picture now, you see that an aerofoil is used here fundamentally for creation of lift, a particular blade element being shown here. So, we are showing the lift in the form of d l, a small elemental lift let us say. So, l is lift, d is drag, t is the thrust, q of course, is the torque, v infinity is the forward velocity, v r is the relative velocity incident on this aerofoil section, r is the radius at which this particular blade section is of the propeller and alpha is the angle of attack, more specifically the local angle of attack for this particular blade section. What we are showing here is the elemental values of lift and drag for this particular blade section and hence we are showing them as d l, d t etcetera to designate their elemental values for this particular blade section. So, this particular blade section is now creating a lift over here and it is experiencing a drag. As I mentioned drag is the penalty you pay in the process of creation of this lift and this is created by the shape of the propeller subtended at an angle alpha which is the local angle of attack. Now, if you decompose this lift and drag which are perpendicular and parallel to the chord, if you decompose them in the form of axial force and the tangential force which is d q by r, what you get are elemental contribution to the thrust. So, d t is the elemental thrust of this particular blade section, d q by r is the elemental tangential force which has to be met by the supply of the torque from the engine. So, this is the component that the engine needs to supply for creation of this thrust through the shape of this propeller blade section which is an airfoil section. So, this is the mechanism, this is the aerodynamic mechanism by which the thrust is created by the propellers where airfoil shape is the fundamental element in the creation of thrust. We had talked about various kinds of pitch settings from root to tip. However, the pitch settings of the propeller as a whole can actually be defined in different ways depending on the operation which the propeller is intended to have. The simplest of the pitch setting is what is known as fixed pitch propeller, where the propeller pitch from root to tip is fixed and it is not going to change during the flight of the aircraft. It is not changeable at all and this kind of propeller is what the Wright brothers used to begin with when they started flying and was being used in the early generation aircraft for many, many years. Those blades as I mentioned earlier used to be made of wood and these fixed pitch propellers were also needed to be optimized as I mentioned. So, that they catered not only to the aircraft takeoff, they catered to the crews and then later on to various maneuvers. So, these were the fixed pitch propellers which were being used in the early days of aircraft flight. The variable pitch propeller which came a little later actually had the capacity to vary the pitch manually through variable mechanisms which could be done on ground during the ground servicing or through some hydromechanical control system and this could be done also during flight by the pilot and he could fix a pitch depending on whether the aircraft is actually taking off or whether the aircraft is cruising and then it becomes a fixed pitch. So, quite often they have two or three or four settings. Sometimes it is a fine setting which is often used for takeoff and the other is often the core setting which is often used during the crews. So, that at these respective flight modes the propeller works at their maximum efficiency which translated to the blade sections means that the blade sections are now optimized at their best angle of attack at each of these sections during takeoff and crews. So, this variable pitch setting propellers allows the propeller to work close to their good efficiency operation and as a result of which the propeller efficiency is much better during each section or each segment of the aircraft flight. However, the most modern kind of pitch setting is referred to as constant speed propeller which is often controlled by a control mechanism and the propeller pitch has a built in control law which is often also referred to as floating pitch. So, as to maintain torque balance between what the propeller needs at a particular time of operation and what the engine is supplying and this torque balance is extremely important during the entire operation of the propeller and the speed of the propeller which is being supplied by the engine through the shaft is maintained constant during this operation. So, the pilot or the control system tries to maintain a constant speed which is good for the engine because the engine can supply a steady power supply at a constant speed and if you keep if you can keep it at a high speed the engine can continue to supply at that high speed a good quantity of power. On the other hand the pitch is now set on a floating mode the result of which is that it automatically sets itself to a good pitch setting. So, that the propeller efficiency is good or as high as possible and this is often a electro hydro mechanical control system and most of the modern day propellers do have these kind of control systems built into the aircraft control system which of course is often called simply called control constant speed propeller. So, these kind of propellers are being used in all the modern aircraft today which allows the propeller to set its own pitch during actual flight and it sets its own best pitch for the best possible efficiency of the propeller for that particular flight segment of the aircraft. Let us try to take a look at some of the propeller performance parameters that are often used as figures of merit for the propeller. These performance parameters are the fundamental parameters by which a propeller is often specified or designated. The first parameter that is typical of a propeller is known as advance ratio in most of the literature it is designated as G as equal to v infinity divided by N d. N is the rotational speed of the propeller often expressed in r p s and d is the propeller diameter which is the tip diameter of the propeller expressed often in meters. Now, this advance ratio is a non-dimensional parameter and is often used to characterize the propeller as you can see from the definition it will change with the forward speed it will change with the rotational speed. So, if you change any of the two parameters the diameter of the propeller let us say is fixed once the propeller is made if you change any of the two operating parameters the forward speed or the rotational speed the advance ratio J is going to change and hence the J is a fundamental parameter by which the propeller needs to be characterized. Now, the basic definition of J actually means that it is the forward motion of the propeller per unit rotational speed. Now, this of course brings in the fundamental concept that propeller is actually executing a screw motion through the working medium which is air and in some of the literature and many of the literature is often referred to as air screw and hence the propeller effectively is executing a screw motion as it is going through the air. This is what the fundamentally propeller does when it is creating thrust it is executing a screw motion through the air in the process of creation of thrust. The other important characteristic parameters of a propeller are of course the thrust which is what we need to fly the aircraft the torque which is needed to be supplied by the engine and the power needed to be supplied by the engine. So, power and torque are the two quantities that need to be supplied by the engine thrust is what you get from the propeller shape by the rotation of the propeller and is used for flying the aircraft. So, supply is p and q thrust is what you get by operation of the propeller. Now, these parameters are often defined in terms of certain coefficients which are as I mentioned figures of merit of a particular propeller or a propeller shape and these are defined in terms of the density at which it is actually operating the density of air, the rotational speed n, the diameter of the propeller d and c t c q and c p which are the three coefficients which are characteristic of a particular propeller. These are called thrust coefficient, torque coefficient and power coefficient. These three coefficient together with advance ratio characterize the entire propeller. We shall have a look at these characteristics later on in the course of these lectures. Now, these are what defined in terms of the fundamental parameters of the propeller which are built into the size of the propeller which is d, rho is the density of the air in which it is operating and n is the rotation and speed. So, we get the three parameters which we need p and q which are to be supplied by the engine thrust is what you need to fly the aircraft. How are these parameters of these particular coefficients created? They are created by using the dynamic similarity theories often popularly known as the pi theorem and these are created much the same way as the lift coefficient, the drag coefficient and aerofoil are defined. Also you would remember the coefficients like Mach number, Reynolds number. These were all created by the same dynamic similarity theories and the same theories have been used to create these coefficients which characterize the propeller. So, instead of C l and C d of an aerofoil, we now have C t, C q and C p of a propeller which define the activities of a propeller. So, these are the characteristic parameters of a propeller and we shall have a look at their characteristic nature later on in the course of these lectures. The propeller performance can also be given in terms of the efficiency of the propeller which is of course, also can be called the propulsive efficiency of this particular thrusting device and it is simply given in terms of the T v infinity which is of course, the thrust work being done by the propeller and p is the power that is being supplied by the engine. So, this is the efficiency of the propeller which as I mentioned can also be called the propulsive efficiency of this propulsive device. Now, using the definitions of the coefficient that we have just seen, the propulsive efficiency or propeller efficiency can also be shown in terms of J C t by C p J being the advance ratio and C p of course, can be written in terms of twice pi C q. So, we can see that the efficiency of the propeller can be very quickly determined from the propeller characteristics. Once we know what the characteristics are and these characteristics are often designated in terms of the advance ratio which is operating point which defines operating point of a particular propeller. So, these are the basic definitions of a propeller performance. We have one more definition which we can look at which is the propeller tip speed and that is given in terms of the diameter, the rotating speed of the tip and the forward velocity with which it is moving forward. So, this is the rotational velocity of the tip of the propeller and this is the forward velocity of the propeller and that gives the tip velocity which can also be called the helical velocity because as we just discussed that the propeller is executing a helical screw motion through the air and hence the tip actually executes a helical motion through the air and hence the v tip actually is a helical speed. Now, what happens is this speed v tip which is a combination of rotational speed and the forward velocity can experience compressibility and even shocks when the tip speeds go very high. The in the modern propellers this tip speeds actually go very near the sonic values and if it crosses the sonic speed the mark number one you could experience shock and once you have shock the shock losses come into the picture which you have studied in the earlier lectures and they will tell you that the shocks would reduce the efficiency of the propeller. So that is exactly what happens if the propellers start rotating at high speed the propellers would experience tip speeds which are supersonic and then you would have shocks and those shocks would then reduce the efficiency of the propellers and the propellers could become of lower efficiency which of course would tell on the fuel efficiency of the engine. These shock losses then are avoidable things as far as the conventional propellers are concerned and the conventional propellers which earlier were being made of metal or even earlier of wood had limitations of tip speed for the wood it was of the order of 0.75 and for the metal they were of the order of 0.85. However, the modern propellers are being made which are made of carbon composites these have actually crossed the sonic barrier I will have occasion to show you some of these blade shapes later in the course of these lectures where transonic airfoils have been used as airfoil profiles for the propeller blades and they are transonic propellers which are being used in the modern propellers and they are made of carbon composites where the shock losses by design can be kept to the minimum and you can still have a very good and very high efficiency of propeller operating at very high speeds. So, these are the various performance parameters that defines how a propeller actually operates during the course of its operation under various operating conditions. In the next lecture we will have a look at the various propeller theories that are used in the propeller design that are used for analyzing the propeller and creating the propeller characteristics which of the fundamental parameters which we discussed today. These parameters would need to be characterized by design and these characteristics are to be then given to the propeller manufacturer and the propeller user for use of the propeller during the actual aircraft flight. So, we will have a look at these theories over the course of next few lectures and we will see how these propeller fundamental parameters are characterized in what is known as characteristic plots of the propellers over the period of next two lectures.