 Let us go to question number 2, this is on range and endurance. As I mentioned to you, if you look at the standard equation for range, it is V by C L by D integral 0 to 1 dW by W or sorry it will be integral 0 to 1 of V by C L by D log of 0 by W and now depending on which of them are constant, you take them out. So we will look at an example where the height remains constant and the C L remains constant and you are cruising okay. So this is also a live example of a turbojet engine aircraft. First there is some data about the aircraft. So wing loading is given, the thrust SFC is given, cruise fuel weight fraction is given, aspect ratio, possible efficiency induced, sorry CD0 per side drive coefficient and the cruising speed is all given to you. So these numbers you can note down because you will need them now for your calculation. And remember the aircraft is a turbojet engine. So normally the T SFC is given in terms of amount of fuel consumed per thrust produced per time. But if the amount of fuel consumed and thrust are in the same units, if let us say both are in kgs, you can cancel them. So units become only per time and in this case they have given the units in per hour okay. So this is the question. It is flying at 9 kilometers at 800 kmph and the pilot now decides to go for the range with C L constant and H cruise constant. So now as the aircraft travels the aircraft weight will go down because fuel is consumed right. So what will happen? If I want to maintain the same altitude, what will happen? What does the pilot have to do? Decrease the, when you decrease the speed, will the C L remain the same? Depends on angle of attack. So does the pilot have to change the angle of attack during this flight? Remember we want C L to be same and H to be same. The moment you change angle of attack, C L will change because for every given angle of attack there is a fixed C L value as per the alpha C L curve. So alpha cannot change and H cannot change. So interestingly you will change the velocity to maintain level flight but you will continue to fly at the same angle because the aircraft weight requirements are reduced alright. So for your convenience the value of sigma is given at 9 kilometers so that the density at 9 kilometers will be 0.3813 into 1.2256 kg per meter cube assuming ISA conditions which we assume unless it is mentioned. So now can you solve, did you remember the equation for constant, for constant height constant C L? For your information this is the equation. So the range will be 2 upon SFC into V L by D into 1 minus root W1 by W0 but W0 is actually the maximum takeoff weight which is W1 plus Wf or W1 is W0 minus Wf. So and it is given that the cruise fuel fraction is 0.3 that means Wf upon W0 is 0.3. So that is why you can rearrange the equation in terms of the known information. So V is given to you, it is given as 800 meter per second, kilometers per hour so convert it into meter per second by dividing by 3.6. So A is the induced drag coefficient 1 by pi into A into E. So A is given, E is given, you get the value. C L will be 2W by S by sigma rho 0 V square, sigma rho 0 is the density by definition and L by D will be C L by C D which will be C L by C D0 plus K C L square. You know C D0 given as 0.016, you know K now is calculated. So L by D is 15.62, notice this is not L by D max. In fact it will be 0.866 times L by D max. Then R is equal to 2 into V upon C now to convert it into the units, this is where most people goof up to convert it into per second because if you notice the units have to be maintained on both sides. So range is in meters, so you have velocity in meter per second therefore SFC must be per second not per hour, everything else is dimensionless, L by D dimensionless, weight ratio dimensionless. So therefore by doing this you get the range as 5102, I have made your job easy, you do not have to calculate anything, I have done the calculations for you, okay. So the range is 5102.7 kilometers but now I want you to calculate the endurance, that I will not tell you. So what is the formula for endurance for a jet engine aircraft under this condition? Any condition, same formula for 1 by C L by D log of, so do it, all information is available in this slide. So 1 by C L by D log of weight ratio, natural log of the weight ratio, so C is the same 0.8 by 3600, L by D is the same 15.62, it will become log of 1 upon 1 minus fuel weight ratio. So the number to calculate is 1 by C L by D natural log of 1 upon 1 minus 0.3, yes. We are confused about how the equation for the range came up, not like the L and C log. This was given to you as homework, which many people have not bothered to do. In the class I said that the range equation is integral of V by C L by D log of weight ratio or dW by W actually. So now what you do is you calculate this for various conditions because there are three variables, there is velocity, there is Cl and there is L by D. So there are three possible ranges, in one the height will remain constant and Cl will remain constant. The other one is, we will see that in the next slide, the velocity will remain constant and Cl will remain constant, but height will change, that is the cruise climb. And there is a third one in which the height remains constant and V remains constant, the angle changes or the Cl changes. That is more complicated because L by D also keeps changing, as Cl changes, L by D will change. So if you are really interested in deriving this formula, there is a very, very good book on performance by Professor SK Oja. So he has actually derived the expressions for all three ranges. And there you will get, so I have just put the equation here for your information because this is something which I wanted you to derive yourself. You can still do it. Okay, any more doubts? Yes. You have not derived the formula, so you do not know. This V is V at the beginning of the machine. Yes, V will change. But when you derive this formula, you derive it in terms of the initial velocity. So you are right, the value of V will change. But when you derive this formula, I should have put it as V0 or V initial. You are right, V will change. All right, so what is the value of endurance? 1 by C L by D log of 1 upon 1 minus 0.3, 7 hours. So it is 6.9 something hours, correct? Yeah, that is right. That is right. Okay, so let us see. 1 by C L by D log of weight ratio which is 1 by C L by D log of 1 upon 1 by W by F, 6.964 hours. That is right. Okay, so now let us look at question number 3. This is now height is changing but V is constant and C L is constant. So this is what normally the pilots fly. This is called as a cruise climb. If they are permitted to fly, actually this is what they are, the one that I showed you is what they are forced to fly because the ATC says sorry, maintain the altitude. Okay, but in a scenario where the ATC permits change in altitude, the pilots will love to fly this one. We will see the reason why. So we repeat the calculations for the cruise climb flight, C L constant V. In this case, the formulae are very simple. This is what you are normally used to. So it is just V by C L by D log of weight ratio and range upon V will be endurance. So 1 by C L by D log. So E will be the same as last time because everything is same. Just calculate this value now. Once again, W0 upon W1 will become 1 upon 1 minus Wf by W0. This is what I got, 5571 kilometers. Endurance is the same as last time. So if we sum up, we find that if you have a constant H and constant C L flight, then you get 5102 kilometers and you can fly in the air for around 7 hours. Whereas if you go for constant V, constant C L flight, you can actually travel approximately 470 kilometers more on the same amount of fuel. This is the reason why pilots beg for this particular flight profile, allow us to climb. And the ATCOs permit if there is less traffic and if there is operating conditions or such that it can be permitted. But in a very busy traffic where the ATCOs have to constantly monitor, they will say sorry, you may be little bit inefficient by following constant H, constant C L, okay. The third option is constant H, constant V, C L changing. But that pilots do not refer because it gives you more inferior value and it does not help anybody. So that one, it is there as one possible scenario, but it is not normally used. So pilots are forced to fly the first one, they love to fly the second one, okay.