 So, let us look at what exactly determines how much you can travel on a given amount of fuel and that one of the important parameters is the SFC. So, for an IC engine it is the brake specific fuel consumption and for a jet engine it is TSFC. So, in the brake in the VSFC or in the IC engines we are concerned about the brake or the shaft horsepower, here we are concerned about the thrust produced directly. This shaft horsepower as you know is consumed by a propeller. So, depending on whether you are concerned about power or thrust, there are two different definitions for the specific fuel consumption. So, for jet engine aircraft we talk about TSFC or the thrust specific fuel consumption called as C subscript T which is the fuel flow W.F, fuel flow per unit thrust produced okay or amount of fuel consumed or amount of time per unit thrust. For propeller aircraft we are concerned about power. So, we define SFC C as the fuel flow per unit power produced. Now is there a way to connect these two together? Is there a way by which we can define a single value for both the aircraft types? So, power will be thrust into velocity because at least in level flight thrust is equal to drag. So, P will be T into V. So, you notice now but P also has now it is not just T into V because you have a efficiency also. So, eta P also will come. So, if you replace P with T V by eta P, so then you can get an expression for CT equivalent jet SFC for a piston prop aircraft okay. So, the equivalent jet SFC for piston prop is equal to the SFC for piston prop into V infinity by eta. So, this is used many a times in performance calculations and I will explain to you why when I do the derivations in that. So, let us now look at some mathematical formulations for arriving at how much range an aircraft can travel. So, just at the point of memory the distance travelled on the given amount of fuel is called as a range. We assume it is steady level flight. So, there is something called as a generalized range equation which is common to both the or all the aircraft type but at some point some deviations will come. So, for this purpose we divide the aircraft into two things. Everything other than fuel that is W1 weight and only fuel is Wf. So, the aircraft gross weight W is equal to W1 plus Wf, simple right. Other than fuel is W1 and Wf is only the fuel. This Wf consists of reserve fuel. So, all of it is not available to you for your mission planning but that distinction we will not worry right now. We will assume that Wf is basically is the fuel available to you. Now, we make one very fundamental assumption that during flight we do not catch a passenger and throw him out or take somebody's suitcase and throw it out or do not open the payload bay and drop off something. The only reason why an aircraft in this calculation reduces its weight is because fuel is consumed. That means W1 will remain constant then and only then we can apply this equation. So, before we go ahead, can you tell me which are the situations in which the aircraft fuel, the aircraft weight reduces not only because of fuel consumption but also because of other things. What are the scenarios in which, yes. Give me a scenario in operation of an aircraft. If we transport aircrafts, airdrop, vehicles, bombs. So, do not call them transport aircraft then. The word transport means nothing is dropped. But I know military cargo aircraft or military aircraft which drop things off. For them we cannot calculate the range by this formula, agree. But I will not use the word transport. The word transport basically means transporting people without dropping them. So, it is okay or cargo without dropping it. But cargo aircraft or military aircraft agree, bombers, fighters, if they lose anything we cannot use it. That is one reason. Okay, any other scenario in aviation where we cannot apply this equation because the aircraft weight during flight changes for reasons other than fuel consumption. Think about it. Yes. Baby, dive pullout. The weight is also changing. Why weight is changing? We have seen that MV square by R, that part is further reduced to the weight. Correct. But MV square by R my M remains constant. In flight even V remains constant. You can have a constant velocity dive pullout. So, the aircraft weight will not change during dive pullout. Aircraft weight will remain the same. Any other scenario? One class you talked about the second world war plane and that is landing gear and all will be dropped. Okay. That is right. That is just at takeoff. So, that is okay. Technically speaking yes, you drop the landing gear just at takeoff. That is one very special aircraft agreed, answer is not wrong. But there is one more mission that is regularly flown in which W1 changes, air to air refueling. When you acquire fuel from aircraft or when you give fuel to somebody, it is not consumed, it is only transferred. So, when we do air to air refueling or when we have bombing or drop of payload, you cannot use this equation. For all other applications, you can use it. So, we will assume that change in the W with time is equal to change in the WF with time. I am going to put this in the formula to derive the expression. Another assumption will be no headwind, no tailwind, so stationary atmosphere and no acceleration. Yeah. Sir, in some case, we can see that pilot used to dump the fuel. That is at the end of flight when you want to suddenly reduce the weight. So, in that case also WF changes while the fuel is not being. Exactly, but you have already reached the destination. So, you are not at a stage when that will affect the range. So, either when you drop landing gear in the beginning or when you drop fuel at the end, you are doing it not to increase or decrease the range but for some other reason. So, during flight, during study level flight, the basic reason would be only one of those two. I agree with you. You can also suddenly dump fuel at the end. So, if you look at study level flight and if you look at stationary atmosphere and if you look at a situation in which the change of the weight is only because of fuel consumption, then only then we can apply. So, let us start with the fundamental definition of the thrust specific fuel consumption. So, SFC or CT, now I am using CT because I want to talk about both the aircraft together. The reason is this is generalized range equation. It does not say applicable only to turboprop or pistonprop or turbojet or turbofan or a pulsejet. It is applicable to all aircraft type. So the thrust specific fuel consumption basically is rate of consumption of fuel divided by the thrust and the rate of consumption of fuel basically is nothing but at any particular instant it is actually DWF by DT. So, this gives us an expression that DT, in a small elemental time DT, you have a change in WF upon SFC into time and because the change is negative because we lose fuel we have put a minus sign. So this becomes our basic building block, DT is equal to minus DWF by CT. So let us stick it and copy and paste it here and now we start looking at the manipulation. So first manipulation that I will do will be to replace, let us see one by one, what do I do? You just see what I do. So first I say that the distance traveled, a small distance traveled in the elemental time t will be V infinity into t. The assumption here is that V infinity is constant, steady flight. So at least in that small element of time t, we do not assume the engine V and therefore the distance traveled in time DT is V infinity into DT, DT is already given there so I can replace DT with minus DWF by CT, how have I done this? Because minus DWF by CTT is defined as DT, in other words DS is equal to minus V infinity DWF by CT. Now DWF is basically DW, do you agree? Change in the fuel weight is change in the aircraft weight, so we replace it. So we get DS is equal to V infinity by CT, W by T, DW by W. Now what we have done here is we have multiplied and divided by W, is it permitted? Because W is non-zero, so it is permitted. And now the last thing that we do is because it is level flight, therefore T will be equal to D and L will be equal to W, so what I am going to do is because L equal to W, the W in the numerator I will replace with L with your kind permission and the T on the bottom I will replace by D, so that is it. This is the elemental equation or sorry generalized range equation. For any aircraft the distance travelled in a small elemental time DT is given by the product of the velocity of the aircraft during that time, assume to be constant, divide by the thrust specific fuel consumption into the lift over drag into DW by W. So now we start integrating this over a range, so the total range will be from the aircraft weight equal to W0, W1 to W0 from initial condition to the final condition. Now this minus sign I have consumed in changing the integration limits, so instead of putting minus W0 to W1 I have put W1 to W0. So from this point onwards now there are many, many paths available. So this particular equation has a very interesting name, it is called as a Brighay range equation. What is special about this equation? Number 1, it is valid only for study level flight. Number 2, it is valid for any engine type provided you use the equivalent thrust SFC. If the engine is an jet engine then you directly use CT, W.F by T, if it is a piston prop turbo prop you use a converted value. From this point onwards you will have different paths depending on what is assumed constant. So the most common assumption is a constant altitude flight, flight at which the value of L by D remains constant, so that means a constant angle of attack flight. You know that corresponding to any angle of attack the aircraft has a given value of CL, there is a alpha CL curve for the aircraft, so for any alpha value there is a CL and for that CL value you have CD is equal to CD0 plus KCL square, CL is constant, K is constant 1 by pi into A into E that does not change and CD0 is also a constant that is the function of the aircraft shape geometry etc. So as long as the angle of attack of a flight during flight is fixed it may not remain 6 but as long as the pilot flies at a given angle of attack you have a corresponding velocity. So level flight at a constant angle of attack one can assume that CT does not change, CT is a function of the engine behavior and the engine consumption of fuel does not change if you have the same density. So if V is constant, CT is constant, L by D is constant then all of them can come outside the integration sign then R will be only V infinity L, V infinity by CT L by D integral of W1 to W0 dL upon W and that will become log of weight ratio, so that is one very simple expression that is what it is. But remember you may not have this condition always for example there is one condition of flight in which velocity may remain constant but altitude may not remain constant, do you remember I talked about cruise climb condition in which as the aircraft is losing the weight you slowly increase the altitude so during that time this cannot be acceptable because CT will not remain same as density changes CT will change. So you may have a constant V constant L by D but CT changing so any combination of these 3 parameters is permitted and therefore 3 different types of ranges also are there and there are 3 different equations for those ranges but we will not go into that direction because this is not a class in aircraft performance where you want to go into detail. So we will look at only a fundamental expression okay, so let us see. So for a propeller aircraft we use that relationship CT is equal to C V infinity by eta p so R will be eta p R by C L by D log of weight ratio. So now if I ask you a question let us say I would like to maximize the range what should I do tell me it is right there in front of you. First thing is fly at a condition at which eta p is maximum. Second thing is minimize the SFC third thing is maximize the fuel capacity by increasing W0 by W1 and also fly at maximum L by D. There is one particular angle of attack at which the aircraft acquires maximum L by D that angle is approximately 3 to 4 degrees for most aircraft so the pilot will fly at that condition but when you fly at 3 or 4 degree angle of attack it is quite possible that the aircraft may be also inclined which is not convenient for passengers. So there are many years of doing this one is you set the incidence angle of the aircraft itself so that the aircraft is horizontal when you have the optimum L by D. So let us look at propeller driven aircraft so we can derive a condition. Now this is something that I would like you to derive yourself. If you just look at the board and say yes then you will be in a real soup. So it is a very simple expression which I would like you to do as homework. So I will just show you to you back again basically you have to derive the condition for L by D maximum what is the CL at L by D maximum. So that can be really calculated and if you know this L by D maximum you can put it there so the at what speed should you fly so that your range is maximum for turboprop you can estimate by this particular expression. So here K is the induced drag factor for a jet propelled aircraft the only influence is L by D so one can easily derive the expression. So it is very messy it is not a very simple expression so L by D alone is not important because V also is there so you have to actually optimize V L by D.