 In the last class we had seen that how propulsive efficiency plays a role in determining what kind of systems we need to use and what conditions in this class let us look at what is the role of overall efficiencies okay. Now towards the end of the last class we derived expression for the thermal efficiency of the system for both air breathing as well as non air breathing systems and this is the expression that we had derived now from this we can derive an expression for the overall efficiency what is overall efficiency overall efficiency is this divided by this right now we have done this divided by this we have done this divided by this so the overall efficiency we can express it in terms of propulsive efficiency and thermal efficiency we will see how to do that a little later as I can write this overall efficiency as ? propulsive into ? thermal right now ? propulsive is what is ? propulsive fva by fva plus m dot a ve-va whole square by 2 into ? thermal is fva plus right this is the expression that we get so I can cancel these two out right and we will be left with this is nothing but this quantity divided by this quantity right the propulsive power divided by the input power is what you have as the overall efficiency now this expression is valid for both air breathing as well as non air breathing propulsion now let us look at the first case that is air breathing propulsion for air breathing propulsion we can write this expression we as ? overall is equal to m dot a ve-va this is the thrust part into va divided by okay now I can divide the denominator by numerator and denominator by m dot a I will get ve-va into va where f is nothing but m dot f by m dot a okay what is I will take again the definition of R R is nothing but va by ve if I use this definition of R and rewrite this expression I will get ? overall is equal to ve square right if I divide both the numerator and the denominator by ve square I will get this expression this is ve let me call let me define a parameter called e which is nothing but f into q divided by ve square by 2 okay now this ranges between 4 to 10 for air breathing engines okay and FR square if you look at the other quantity that is FR square FR square is typically between 0.01 to varies between 0.01 to 04 so this is very small compared to e so therefore we will neglect this portion okay for air breathing propulsion we will neglect FR square is very much less than e therefore neglecting FR square we can write the expression for overall efficiency as 2R into 1-R divided by right when this whole quantity here is nothing but this is nothing but e right so we will get this expression for overall efficiency when is this maxima again we need to differentiate this is very simple you will get 2R-2R square right or R is equal to half okay R is equal to half gives you the maximum value and what is the value of overall efficiency at this point R equal to half you put half here you will get 2-1-1-2 that is 0.5 so that is 1 by 2e now e value we know ranges between 4 to 10 so overall efficiency what is the highest value 1 by 20 is 5% so overall efficiency for air breathing propulsion varies between 5 to 5% to 12.5% which is a very very low number okay so please remember air breathing propulsion air breathing propulsion has very low efficiencies associated with it and it is typically in the range of maximum as around 12.5% okay now let us do the same exercise for rocket engines if we do the same exercise for non-air breathing or rocket engines again overall efficiency is FVA divided by m.fq right here there is no m.f right for rockets there is you need to carry both fuel and oxidizer so this is m. no m.f plus m.vx2 by 2 okay so we know that F is equal to m.ve right for rocket engines and if you substitute that you will get ? overall is equal to m.ve va okay fine I can cancel out m. here and again using r is equal to va by ve I get 2r or if I divide both the numerator and denominator by ve2 I get overall efficiency as I get the expression for overall efficiency dividing by ve2 as and in the denominator I have okay now using the definition of r I can write this as 2r divided by I will call this quantity as er I will define er as equal to q by ve2 by 2r indicates here rockets so I will get this as er plus r square okay now typical value of q for the rocket the heat of formation or heat of reaction what is it q for kerosene 42 Mg what do you think will be the q for rockets higher than that lower than that what do you expect it to be higher sure actually the q would be something between 4.2 to 6 Mg per kg is lower than that right we think we use such a bad propellant as rocket in rockets or is there a catch here if you look at rockets rockets need to carry both fuel and oxidizer right this is per kg now this includes both fuel and oxidizer whereas if you looked at kerosene it was only for fuel you did not account for the oxidizer because here you are looking at both fuel and oxidizer this will have to come down right this will have to be lower than what you get with kerosene okay so e q is something like this and therefore er will range between something like 1.5 to 3 okay so compared to that r square you cannot neglect if especially rockets can operate in a regime where r is greater than 1 so you cannot neglect r square compared to er the thing that we did in terms of aircraft engines or everything engines is not valid here so again we have to find the maximum value we will take the derivative so taking the derivative of overall efficiency with respect to r here I get 2 er plus r square minus er plus r square 2 whole square right and this simplifies to er minus 2r square divided by er plus r square the whole square so when will the derivative go to 0 in this case either when this goes to infinity or when the numerator goes to 0 in this case the denominator cannot go to infinity so the numerator has to go to 0 that is or r is equal to okay fine so you get the maximum value when r is under root of er and so r will be typically greater than 1 because er ranges from 1.5 to 3 so r will be greater than 1 in this case and it a overall maximum would be if you substitute this into this expression here so you get under root 2 under root er 2 er square so you get 1 by under root er okay so if it is 1.5 efficiencies can go very high actually in for rockets right and let us plot this if we plot overall efficiency for both air breathing as well as non-air breathing engines and on x axis we have VA in kilometers per second this curve is for air breathing engines this is for rockets or non-air breathing engines you see that overall efficiency is much higher for rockets compared to aircraft engines aircraft engines this was around 12.5% right whereas rockets it can go up to 85% fine now before we go there firstly I forgot to tell you that if you look at in this case also there is an optimal value for VE with respect to overall efficiency for air breathing engines one value and there is another value that you can choose in with regards to propulsive efficiency okay there are two efficiencies propulsive and overall efficiency for air breathing engines you usually the value of R that gives you maximum overall efficiency does not give you higher propulsive efficiency but you go for the maximum overall efficiency you choose R such that your overall efficiency is maximum for everything in okay for air breathing engines such that overall efficiency is maximum maximized right now if you see this graph you will find that I have to say what kind of fuel and other things Q is equal to 5 megajoules per kg V is 2.82 kilometers per second and here for everything Q is equal to and F fuel a ratio is somewhere around 0.02 okay so if you take a look at this graph it feels that overall efficiency of rockets is way better than overall efficiency of air breathing engines notice that V is 2.82 kilometers in the maxima occurs somewhere here which is what I had said earlier R is greater than 1 for maximum value it goes as under root ER okay so why not use rocket engines everywhere overall efficiencies are very high so we should be using rocket engines instead of air breathing engines everywhere is that a valid statement right efficiencies if you look at it says they are very efficient machines why not we use rocket engines everywhere what is the H ISP in a sense yes you are saying something but if you look at aircraft engines and rocket engines or air breathing and non air breathing engines there is an important thing that distinguishes them that is a rocket engine or a non air breathing engine will have to carry both fuel and oxidizer on board right because they carry fuel and oxidizer on board and rocket engines typically operate at very very high pressures okay so they add heat at very high pressures therefore they can expand to low pressures there is greater availability and hence they are more efficient in that sense but efficiency per se doesn't have any meaning here because you have to carry a lot of propellants in order to achieve what you want to achieve to give you an example if you take PSL is weight at lift off it is something like 294 tons it is payload for to low earth orbit would be something like 3250 kgs almost one-tenth of its overall weight okay so the useful thing that you can carry in a rocket mode rocket propulsion is very small compared to the overall weight although it might be efficient you end up having to carry more of fuel and oxidizer there because you have to carry both and the useful payload that you can carry is very very small typically what is this one hundredth is it 300 yeah one hundredth of the overall weight right now let us take a look at what are similar numbers for Boeing 747 its empty weight is around 178 tons yeah its fuel weight is there again around 173 tons and its maximum takeoff weight that is what is the weight of the aircraft maximum weight of the aircraft with which it can take off 397 tons so the payload weight should be the difference of this minus these two put together right so the payload weight comes to be something like if you compare it with the overall weight or the maximum takeoff weight it has increased from 1 x 100 to something like 1 x 10 which is far better right so if you have to have a mission wherein you are looking at going beyond the sensible atmosphere then you have to use rocket motors there is no other go but if you have to look at a mission wherein you are traveling within the sensible atmosphere then it makes sense to use at briefing propulsion again there is a restriction if you are looking at a very fast response system then again you need to go in for rocket engines even within the atmosphere okay the other point is what you made earlier that is for rocket engines it is more meaningful to look at what is the ISP ISP tells you what is it that you need to carry what is the propellant if larger the ISP then smaller is the propellant weight that you need to carry on board right but if ISP becomes smaller then you need to carry more propellant on board okay so ISP is a better indicator of performance and rockets and not overall efficiency whereas for everything a propulsion it is overall efficiency is a very good indicator of what is the performance okay now let us look at how this overall efficiency impacts the range of the aircraft okay now before we go there we need to make a certain set of assumptions in order to derive this expression for range of an aircraft and how overall efficiency impacts it okay what is range firstly range is the distance that the aircraft can travel without refueling right so okay that is if you are given a particular mass fuel what is the distance that the aircraft can travel now if you look at long range aircraft typically long range aircrafts there is firstly the aircrafts need to taxi to the runway then take off right climb to a particular altitude typically around 11 km and then it will cruise at that altitude right and further upon reaching the destination or getting close to the destination it has to climb down descend and then land and then taxi to its docking position right all these operations are there but most of the flying is done in the cruise range okay so we will only consider that part when we are looking at what is the range that it can do okay so most flying is done and level flight category hence we ignore climb descent take off and landing we also need to make another assumption which is quite valid for civilian aircrafts that is the mass of the aircraft changes only because of your expelling out fuel okay fine which is not true in case of military aircrafts especially if they have to drop bombs or drop fuel tanks and things like that or fire missiles then the mass of the aircraft changes because of other things when in case of civilian aircraft the entire mass of the aircraft only changes because of fuel being expelled out so aircraft changes only due to now so we made these assumptions now let us look at what is a level flight what are the things that are true in level flight thrust must be equal to drag and lift must be equal to weight so for level flight T is equal to D and lift is equal to weight okay and let me call M as the instantaneous mass of the aircraft then I can write this expression for thrust as thrust must be equal to lift into D by L okay and I can again rewrite this as lift is nothing but weight is M into G M is the instantaneous mass of the aircraft divided by L by D okay L by D is a aerodynamic parameter okay and can be defined as such for a overall aircraft any idea what are the typical values for L by D for an aircraft L by D for aircrafts no idea it is something like for Boeing Boeing 747 it is somewhere around 17 during cruise on quad it is somewhere around 7 during its cruise at Mach number 2 for Concord it is lower because drag is higher because it is going at supersonic speeds right so which is better L by D of a larger value of L by D is better or a smaller value of L by D is preferable which is preferable larger or smaller value of L by D yes a larger value of L by D is preferable because then you would have to spend less fuel okay so which is why Concord are very expensive right L by D is small L by D is large here so you can afford it and you something like how sparrow one that flutters very quickly and flies around I mean it is a short distance flights mostly the L by D for that is around 4 whereas albatross which is more of a long distance flyer and mostly sails through has a L by D of around 20 so L by D also indicates what you can do if you have a smaller L by D then you are mostly restricted to short flights they have a large L by D then you can think of longer flights okay coming back here so from here we can define thrust power as T x VA and T x VA is given by in this case MG VA by L by D okay and and we know from our efficiency calculations and other things what is thrust into VA we had called it F x VA in that it is nothing but it overall into M.F x Q right okay so now we combine these two we can write M.F is equal to MG VA x Q okay now this is mass flow rate of fuel this is a part of the entire mass of the airplane right so using that fuel to be part of aircraft weight I can write M.F is nothing but D M by DT and because this is going to decrease right it should be with a minus sign okay now I will change these variables from D M by DT to DS because we know that V DT is nothing but DS SS the incremental range so I get from this DT is nothing but DS by VA and if you substitute for that here you will get M.F DM by DS okay now SS nothing but the distance along the flight path SS okay so using this and this fine M.F I know that expression so I can write an expression for D M by DS as – MG divided by ? overall Q x L by D now in this case if we assume the overall efficiency to be constant and Q to be constant and L by D to be constant we can integrate this expression so assuming to be constant then DS is nothing but how we can integrate this integrating I will get S is equal to ? overall into log M1 by M2 M1 is the or I will call it MI and MF MF now I will retain it as M1 M2 M1 and M2 are initial and final mass of aircraft okay so in this expression you see that if the overall efficiency is high you get a longer range or also if your L by D is better you get a longer range right we will stop here and look at in the next class on the cycle analysis for various air breathing engines thank you.