 So, let us see, now we are going to do some derivations, we are going to derive some condition. The first important derivation is for how much is the power required for an aircraft when you are in the level study flight or unexerrated level flight, okay. So, power required is basically the thrust required into velocity and the thrust required is correctly shown here now as W over C L by C D, okay. So, thrust required W by C L by C D and V infinity is the velocity. Now, can you derive an expression for V infinity in terms of W for level flight? How can you do that? Now, do not look at the screen please. Now, you have to derive some expression. So, please copy down this expression first. P R is equal to T R into V infinity which is equal to W upon C L by C D into V infinity and V infinity now has to be replaced by an expression that incorporates W and C L for level flight. So, here we have used the simple expression that we derived last time. So, can you replace V infinity with something who will tell me what is the replacement for study level flight, what is V infinity equal to? Yes. Root over U 2W by rho SCL, okay. So, you have used the condition that in level flight lift is equal to weight. So, therefore, therefore L will be equal to W, it will be equal to half rho V square SCL. So, she is right V infinity will be root of 2W by rho SCL. So, now you replace this quantity V infinity inside the circle by root of 2W by rho SCL. What do you get? You will get an expression for power required as W upon C L by C D into V infinity will now convert. So, it is very straight forward. This V infinity which is root of 2W by rho SCL will go in that particular place and the W outside will come inside the square root sign as W square. So, it becomes W cube. The C L by C D outside to come inside the square root will become C L square by C D square. So, the C L square will multiply with the C L already in the denominator to become C L cube and the C D square which came inside to make it go up it will become C D square. So, in other words the expression will be P required will be under root of 2W cube C D square by rho infinity SCL cube. Now for a moment let us not worry about aircraft weight because that is equal to lift and it is a constant number. Rho infinity and S are also 6 values for a given altitude and for the given aircraft. So, in other words P required will be directly proportional to under root of C D square by C L cube or C L power 3 by 2 by C D. So, if the value of C L power 3 by 2 by C D. So, C L root C L by C D. If this ratio is large then the power required is going to be less. Now please tell me is this expression valid for only a piston-trop aircraft or a jet engine aircraft or both of them? What do you think? It cannot be none of the above it has to be either only for piston-trop. So, when I say piston-trop I mean piston-trop and turboprop. When I say turbojets I mean turbojets and turbofans. So, jets and props these are the two words we will use. So, is it applicable for only jets only props or both? What do you think? Yeah. valid for prop. Why only for prop? Because as you said earlier that thrust will be constant for jet. Does not matter. Does the whether the thrust is constant or variable does that affect the expression? The expression comes from basic physics. Yeah, maybe thrust is constant. Maybe it is constant. So, this argument is not acceptable to knock off turbojets just because TR is constant. Just because T produced is constant. This is about T required not T produced. So, what do you think? Is there some other compelling argument because of which this expression is not applicable for turbojets? Anyone? There is no compelling argument because when we when we derive the expression did we say turbojet ends this turboprop? No, we did not. So, this is true for any aircraft. For any aircraft the power required for study level flight is minimum when CL root CL by CD is maximum. It is independent of the aircraft type. The interesting thing is that the velocity at which this condition occurs may be different but the expression is the same. Okay, so now comes the interesting part. The interesting part is let us now go to thrust required. Power required we have already seen. Thrust required is actually actually much more straightforward because thrust required will be W by CL by CD into root of 2W by rho as CL. Actually, sorry this is not thrust required. This is basically the same expression which I have repeated. Okay, so let us go ahead and see this thing graphically. Graphically what do we see? Graphically we see that okay, one more interesting expression. Now it is not enough to say that CL power 3 by 2 by CD should be maximum. It is also important to find out the actual operating condition under which it happens. So, what do we do? We say that PR is equal to TR into V infinity for any aircraft and because thrust required is equal to drag, you can also call it as a drag into velocity. Drag is QSCD. This I have covered in the capsule on drag estimation, okay, where I defined the drag coefficient. So, D will be QSCD and CD itself has 2 components CD0 and lift dependent component CDi or induced drag component. In other words, the power required will be equal to Q infinity into S CD0 V infinity plus Q infinity into S into VD0 into CDi which is CL square by pi EAR. So, now we have an interesting task to do and that task is we have to now find the velocity to fly at which the power required is minimum. So, how will you get that? You have an expression which says power required is something plus something, 2 terms. The first term has V infinity, the second term also has V infinity but the second term also has CL square. Our task is to find out what would be the condition at which or what should be the V infinity at which the PR is minimum. So, how will you do that? Anyone can help me to proceed? Yes, Sushil, what do you want to suggest? But with differentiation, total differentiation or partial differentiation? Why total? Does this velocity have a time component? So, partially sufficient. If there was a, remember our discussion about compressible flow where there were 2 components changing with time, here it is steady, so it does not change with time. So, if we take a partial derivative of PR with V infinity and put it equal to 0, you get the condition for either minima or maxima, then you have to go for the second, third, etc. You have to find the first non-negative. So, the first component of this term is Q infinity S c d naught V, which is the power required to overcome the zero lift drag. So, we call it as zero lift power required. The second term will be the lift induced power required. So, there are 2 terms. Let us see how they are varying. So, the first one is a function of Q infinity S c d naught. So, Q infinity itself contains V infinity square. What is Q infinity? Half rho infinity into V infinity. Multiply by S c d naught V infinity. So, basically this term is proportional to V infinity cube. It is a cubic term. So, therefore, it is going to increase like this. You can see there is one dotted line here and the dotted line shows you how it is increasing. The second term again contains Q infinity which is V infinity square, but it contains C L square. Now, C L as we have already seen in level flight L equal to W. So, as she rightly said, V infinity will be root of 2 W by rho S C L. So, therefore, C L will be, what will be C L? Proportional to V square. 1 by V correct. So, now, what will happen is this expression, it will be proportional to 1 by, tell me how is it collected with V infinity? 1 by V or 1 by V infinity square. So, therefore, this term is going to actually reduce as the V infinity increases because you are dividing it by the term. So, this is the second term. It is a large value when you have a low value of velocity and it comes down. There will be some point which is marked as point number 1 in this figure where these two powers are going to be equal. So, that is a very interesting intersection point about which we will talk later. Right now, when you add these two terms up because they come from different expressions, they will be having a minimum value at a velocity at which C L power 3 by 2 by C D is going to be the maximum. So, our job now is to get that value of velocity. We are going to derive an expression for that value of velocity at which C L power 3 by 2 by C D is maximized. So, to do that, so this is what I was telling you that the first term is proportional to V cube. Second term is proportional to V square upon V power 3, that will be V. So, this is our job finding out the value at which this value is 0. So, for your convenience, I have repeated the expressions now. So, now you put pen to paper and now try and derive for me the condition for which the partial derivative of PR with the respective velocity is 0. So, you must tell me what is the condition? What is the condition at which this happens? So, when I say condition, I must also tell you what condition it is. So, what can you say about point number 1? At point number 1, what is the same? What is the same? Yeah, at point number 1, the two components of power required that is the induced power and the power due to 0 lift are same. They are equal. But now we are looking at this point where we have this value minimum. So, can you derive the expression now? So, for this what you have to do is you have to take the partial derivative or take the derivative of this expression with respect to velocity and remember that this term W square already has inside it velocity because W equal to L is equal to half rho V square SCL. So, you have to now derive the expression. This will need some time. So, what I am basically interested in is, see if you have some difficulty then you please ask me, do not just look at the screen, you should be now doing your calculations. I do not want people to stare at me, I do not want to stare at the board. I want you to do the calculation. What I am interested in actually is the relationship between CDO and CDI where CDO is the parasite drag coefficient and CDI is the induced drag coefficient. CDI is basically CL square by pi AE. So, I want the link between CDO and CL square by pi AE at which this particular condition is met. So, if you have got the answer then you have to raise your hands. Yes, what is the answer you have got? CD naught equals to one third KCL square. So, what is KCL square? K is one by pi AE. Correct. So, CL square by pi AE is CDI. So, you are saying that CD naught should be one third of CDI. Right? Okay. Anybody else? Yes. CL square is equal to, it is the same thing. Actually now you have to go further. You have to, because CL square by pi AE is CDI. So, you will get the same condition if you probe further, you have to probe little bit further. So, essentially what I got is this, if I differentiate it and if I take all the terms common then I get an expression which says 3 by 2 rho infinity v infinity square into S into CDO minus 1 by 3 CDI equal to 0 and hence the condition is that CDO equal to 1 by 3 CDI. So, the parasite power or the parasite drag coefficient CDO should be much lower than the induced drag coefficient. That is the condition at which you have the minimum power required. Once again this is aircraft independent or engine independent. Okay. Let us see. So, the interesting point is that the velocity at which this particular thing occurs is in general lower than the velocity at which the two powers are equal. Okay. Now, when you look at this point at which the two powers are equal that means Q into S into CDO into v is equal to Q into S into CDI into v which means CDO equal to CDI. So, the condition at which the induced power equal to the profile power or the zero lift power equal to the net equal to the induced lift induced power is one condition which will be useful little bit in the future. The other condition is when it is one third. Now, if I now look at the thrust required to be minimum not power required and thrust is now defined as power by v just because power is equal to T into v. So, then interesting thing is that the value at which minimum. Now, this is something I want to leave to you for homework. It is the same thing now. If you now go for T equal to D and if you want to derive now the condition for D into v that is power. So, if you say if I want to have a condition where the P required is T required is minimum that means D is minimum that means half rho v square Sd naught plus half rho square Sdi is minimum you can take partial derivative of again with v and you will get Cdo equal to Cdi. So, interestingly the operating condition at which power required is minimum is different from the condition at which the thrust required is minimum.