 In the last class we had seen what is specific impulse what is vacuum specific impulse and density impulse and how CF varies with PC by PE and AE by AT we also noted at the end of the last class that thrust varies with altitude for a rocket motor let us look at how it varies with altitude and what we can do to derive some benefit out of it. Suppose I were to plot the C level thrust thrust at any altitude versus C level thrust okay wherein if we plot the thrust at any altitude versus by divided by the C level thrust for different altitudes and also find out how the AE by AT should vary if we are to have something known as adaptive nozzle so if you look at this plot what it tells you this dotted line here is for a nozzle that is adapted at C level if we use the same nozzle to fly to different altitudes then this is the thrust that one can get if we have a nozzle that is adapted at each and every altitude okay what we mean by adapted nozzle is when PE is equal to PA or when the flow is optimally expanded through the nozzle then we call it an adaptive nozzle if it is adapted to every altitude then this is the thrust by C level thrust so you can see that the thrust delivered by this is far superior to the thrust delivered by a nozzle that is adapted at C level because it this one accounts for the increase in area ratio that needs to be had because the ambient pressure is dropping and if you look at what we need to pay or how do we account for it in terms of area ratio for an adaptive nozzle this will go like this notice that beyond some altitude the increase is very very steep okay obviously we cannot have a nozzle that is experiment that is also increasing its area ratio as the vehicle moves up that would be very very difficult so let us look at if there are other ways of getting to this or if not improving this in some sense so that we get as close to the adaptive nozzle as possible okay we always have said this that when PE is equal to PA right we get the best performance that is when the nozzle is optimally expanded when the flow in the nozzle is we get the highest thrust is what we have said let us look at how that is possible if you look at the thrust equation the thrust equation that we have is F is equal to m dot ue plus Ae right this is our thrust equation and if you look at it we are saying when this term goes to 0 we get a best performance from this equation it looks like if we throw out a term that is causing a positive addition to it it is still giving us a better performance this looks a little paradoxical to start with let us look at how this happens okay how when PE is equal to PA we get the best performance now there are two approaches to do it one is you know you have studied in your earlier classes that if you differentiate and show that the derivative is 0 then it will be a maxima or a minima and if it has to be a maxima then the derivative second derivative has to be negative that is one approach and we will also look at whether we can explain the same physically okay now firstly in addition to the thrust equation we know that the momentum equation in one dimension is dp by ? plus u du is equal to 0 now let us multiply this equation by ? a okay and we will get you will get a dp plus ? u a okay fine now we know that for this to be a maxima or a minima the derivative should go to 0 so let us take the derivative of the thrust so we will get df is equal to u e dm dot plus m dot PE minus PA into dae plus taking the derivative of the terms inside ae dpe minus ae dpa fine now in this equation there are certain things that are constant that are if you look at doing this derivative we need to hold PA constant right and the mass flow rate through the nozzle constant right so for m dot and PA constant then what happens to these terms this goes to 0 and this goes to 0 so we will be left with three terms right now let us rearrange this a little differently and see what we can do so I will get df is equal to plus ae dpe okay now if you notice we had multiplied the momentum equation with rho into a and got this expression now we take this for the exit plane right I can rewrite this as ae dpe plus rho u a is nothing but mass flow rate right so I will get okay now if you notice the first two terms of the derivative of thrust as this which means these two will go to 0 so we will be left with df is equal to for it to be a maxima or a minima df has to go to 0 and that is possible when only PE is equal to PA so df goes to 0 now let us look at what happens to the second derivative right so we will make d2f is equal to we had already said that PA is constant and therefore this term goes to 0 so we left with only two terms right now if you look at these two terms for PE equal to PA the first term will go to 0 they are looking at what happens to the sign of d2f when PE is equal to PA right so at PE is equal to PA this term goes to 0 okay so we will be left with only this term now let us examine what is the sign of this term how is the in the supersonic portion of the nozzle convergent divergent nozzle what is DAE whether it is positive or negative in the supersonic portion that is in the divergent portion the area is always increasing right so it is positive and if you look at what happens to pressure in the nozzle it is decreasing so the derivative will be negative so you have one positive term and one negative term the product will always be negative and therefore this is a maxima so we have a maxima at PE is equal to PA now let us try to understand this physically also if you look at the convergent divergent nozzle let us say you have a rocket motor and then you have a convergent divergent nozzle you all know an aerodynamics that you can get the lift of an airfoil by integrating the pressure over the entire surface similarly you can get the thrust by integrating the pressure over the entire surface okay so if you do that if you look at this all this portion to the left of this portion is immaterial to us because that will not change with change in ambient pressure okay or change in exit pressure fine upstream of this will not have any bearing and also if you look at these two surfaces they anyway cancel each other out we are only looking for thrust in this direction right we are looking for thrust in this direction so it is good for us to only consider the divergent portion now if you look at the divergent portion alone I will take one section of the nozzle now on the inside what is happening to pressure as you move from the throat to the exit plane pressure is decreasing okay now let us take the case where the ambient pressure is equal to the exit pressure okay so then this is the exit pressure so it is constant on the other side throughout the okay so for PA equal to PA okay now if you look at this the thrust net thrust if you look at what is happening here this will produce a force in this direction right and you will have a component of the same in the direction that we want the other component will get cancelled each other out because of symmetry so we are only interested in one component that is in the x direction right so we are only interested in this component now let us say we add a little more nozzle what happens some extra portion of the nozzle such that if you add an extra portion what happens to the flow inside the pressure will drop even further right and you will get a situation wherein the inside pressure is lower than the outside pressure so in this very small portion you will have a component of force in this direction which will lead to something like this if you resolve it in these two directions so this is adding to a negative thrust you would not want that so therefore let us take this portion out and get back our earlier figure wherein we had PE is equal to PA now let us say the exit pressure is more than the ambient pressure right then what will happen we need to take out some portion of the nozzle what we have essentially done is we have taken out a portion which was giving us net positive thrust but we have taken that out so therefore we will get a reduced thrust so it works out that physically when PE is equal to PA only we will get the best thrust if you look at the thrust equation here right this term is increasing when as you reduce pressure the velocity is keeps on increasing as you reduce the pressure and when PE is equal to PA you will have this portion as the maxima and that is why you are able to get the highest thrust when PE is equal to PA okay now let us look back at some gas dynamics and try and understand what happens in the supersonic nozzle so now we know that in our thrust equation F is equal to m dot UE if PE is equal to PA then the flow is expanded optimally expanded and gives us the maximum thrust when PE is greater than PA what is this case this is a case when it is under expanded that is there was scope for this to expand up to PA but we have left it some portion unutilized okay and when PE is less than PA it is over expanded flow okay here we have taken it more than or lower than PA and therefore it is over expanded okay so now let us look at what happens inside the CD nozzle when all these three conditions take place I mean whether what will happen to the flow as we look at it when these things happen okay so to do that let us look at convergent divergent nozzle I will only take one half of it okay this up to this is the reservoir this is the throat and this is the supersonic portion we know from gas dynamics that if we have a reservoir pressure PC then depending on what is the ambient pressure outside we have derived that in the class that for gamma is equal to 1.2 this ratio should be around 1.7 for the nose throat to be choked let us say if the ambient pressure is somewhere here right ambient pressure and the pressure inside is the same okay then what will happen there is no flow right so there will be no flow and pressure is constant throughout now let us keep reducing this pressure on the outside okay so then there will be flow okay and if the fluid is viscous then we will not recover the actual pressure but otherwise we will recover the pressure okay and if we still keep reducing it further once it reaches a critical pressure the ratio here and then it will expand and it will reach Mach number 1 M is equal to 1 at the throat okay and then if the pressure on the outside is still lower it will expand further let us say this is the point where now there could be two solutions possible when PA is there if you after you have reduced the PA if you increase it further it will let us see what happens to the flow in the nozzle now we have gone to a case where in PE is equal to PA and the flow is optimally expanded now let us say if we increase the pressure in this direction what happens as we increase all that the flow knows is as soon you are confining it within the nozzle here right so it till it is in the nozzle it cannot experience what is the outside pressure and also this flow remember is supersonic so it does not have anything it does not know what is going to come ahead right so as long as you confine it it will not feel what is the outside pressure but the moment you release it outside it knows that it has to equilibrate with the surrounding fluid so it will try to process itself either through shocks or expansion fans okay and then equilibrate with the surroundings if the pressure here is slightly greater than PA let us say at this point it will go through this and it will process itself through a series of oblique shocks and expansion fans okay and then it will equilibrate over some length with the external pressure okay let us say we increase the pressure beyond this there is a point at which the fluid cannot process itself through these oblique shocks outside and you will get a oblique shock inside and the flow will separate and therefore you will get something like this further still you will get a normal shock and after that the flow will be subsonic and therefore you will get a pressure recovery right now let us look at what happens if we go below this okay if we go below this the flow is going to be the same up to this point okay and then it knows that the outside pressure is much lower than the exit pressure so it will expand further okay and then equilibrate with the surrounding fluid so it will process itself through a series of expansion fans and these will get reflected and we will see that here if you see this case here the first figure this is when PE is greater than PA so therefore you will have expansion fans right and the flow will process itself through these expansion fans these expansion fans when it hits the jet boundary will get reflected as compression waves okay these are Mach waves and they will get reflected as compression waves which will again get reflected from the free jet boundary as expansion fan so it will go through series of expansion fans and compression waves and process itself and finally it will equilibrate with the surrounding pressure now if you have a case where in PE is less than PA then it knows that it has to process itself through a oblique shock so you will have an oblique shock first and then this oblique shock when it hits the free jet boundary gets reflected as a expansion fan and the same here on the other side oblique shock gets reflected as an expansion fan and then these expansion fans will get reflected as Mach waves so it will process itself through a series of this till it equilibrates with the ambient pressure okay now how does this effect our thrust equation and what is its role is what we have to see now if you look at this figure here what you see plotted on the X axis is AE by AT and on the Y axis it is CF okay and this is plotted for PC by PA different values of PC by PA that is seen to increase in this direction and finally it reaches infinity okay and the dotted line here that you see is the locus of all this maximas that you have here for each AE by AT so if you connect them you will get this line this is actually the line if you say that an optimal that an adaptive nozzle will have the CF variation if you have an adaptive nozzle that is if you continuously keep on increasing the AE by AT an adaptive nozzle will give this kind of CF profile okay is that clear if you have an adaptive nozzle that is allowing the flow to be expanded optimally at each and every altitude then the CF of that will be something like this but let us say you have a nozzle that has a particular area ratio let us say you have a nozzle that has an area ratio of 5 then what happens to the CF we look at 5 is somewhere here right so if you look at having only area ratio of 5 then depending on PC by PA PA keeps on decreasing as you increase in altitude PC let us say we hold it constant so the what happens to this ratio this increases so you will go from smaller values to larger values and then there is a point at which it will be optimally expanded and then beyond which it will also drop right now there is if you look at this line here this dotted line here beyond this the flow will separate okay and you will have a recirculation zone in the expand in the supersonic portion of the nozzle right and the flow will separate here okay so if we want to have a single stage to orbit vehicle it is extremely difficult to design this it will not function optimally because what will happen is if you look at having a single nozzle right let us say you have a nozzle expand wherein which gives optimally expanded flow at sea level then if you continue to use it at higher altitudes it will perform badly as we go in altitude compared to an adaptive nozzle right so we are losing out on some thrust that we could have probably got but let us say we do the other thing we take the nozzle that is expanded at some altitude and try to use it from ground level to higher altitudes what happens is beyond a point the flow will separate and you will get to this kind of situation so either ways it is very difficult to have this which is why having a single stage to orbit is very difficult and also if you look at rockets as soon as you expel out some propellant the structure is a waste right the structure is not useful enough so therefore if you multi-stage it okay then at each stage you can have the optimally expanded flow for some altitude and therefore you will get a much better benefit right we will not probably be able to do if you looked at if you look back at the adaptive nozzle figure that is altitude we had seen that if we use a nozzle that is adapted at sea level and if we use it for higher altitudes also this would be some performance and if we were to use a continuously adapted nozzle then the performance was superior right if we do multi-staging we will probably get firstly it will be adapted at sea level let us say and you go to some altitude then you make sure that it is adapted at that point so it will probably go like this and again if you separate that stage it will go in this fashion so we in a sense we are trying to do the adaptive nozzle in stages okay that is what we do in multi-staging also if you look at PSLV right PSLV has six trap on motors okay they do not switch on all the six of them at ground level they will switch on four of them in the ground level and two of them at higher altitude the reason being if you switch it on at a higher altitude then you can have the nozzle adapted to a higher altitude and therefore it will perform better okay so that is in some sense some kind of multi-staging or this thing that is done in PSL okay so such things are possible in order to get a slightly better performance now there are also other issues that sometimes if you use the nozzle where in the flow is under expanded you will get into some kind of other difficult situations I will explain that with a small figure here let us firstly consider a booster or first stage then a second stage motor and then a third stage motor okay what would you use for a booster what kind of area ratios would you want to use for a booster is it going to be very large or is it going to be smaller number smaller so you will typically use something like six to eight let us take the case where it is six and for the second stage you will lose a larger area ratio and then also for the third stage you will lose use a still larger area ratio if you take the nozzle for this during flight the pressure if it does not go to a very high altitude that is the maximum altitude is something like five kilometers the nozzle flow nozzle will flow full slide under expansion and the ISP will be of the order of 2670 this is in or I will put it as 267 this is in seconds okay now if you test it at sea level conditions remember whenever we want to send a rocket up we need to also test it in a static test facility right so if we test it in a sea level static test facility this nozzle would probably be flowing full if you test it in sea level conditions there is not going to be too much of a variation here and you will probably get similar values for ISP now if you look at the second stage it has a fairly large area ratio and the flow will be under expanded here here again the altitude is something like 24 kilometers and the ISP that you get will be something like 312 seconds. So the flow is here under expanded okay what happens if you use this nozzle at sea level conditions it is in such a way that it is optimally expanded at some higher altitude now if you use the same nozzle at sea level conditions in a static thrust what kind of ISP will you get lesser because if you look from the graph the CF will reduce because there will be probably flow separation that takes place and the nozzle might not flow full right so therefore you are probably going to get something like ISP of 254 seconds okay and if you use a third stage motor wherein it is expanded for something like 100 kilometers altitude and it gives something like 334 seconds right then the flow beyond this will expand further right so it is under expanded again and if you use this at a sea level condition probably you are going to have a normal shock sitting somewhere in the supersonic portion of the nozzle and you might get a very low ISP because the flow beyond the supersonic after the shock will be after the shock the flow will be subsonic and therefore probably you might end up getting a very low ISP here okay. So one of the problems is if you are looking at this scenario wherein the flow expands even further you should be careful to avoid having any instrumentation in this region because at a higher altitude the flow might turn back and you know you might have hot gases impinging on this equipment which it might not be able to withstand so therefore it is either better to have some insulation there so as to prevent the heat heating of these equipment okay. Now till now we have looked at an analysis wherein it only looked at was a one-dimensional flow we had assumed the flow to be isentropic then we had also assumed the flow to be an ideal gas and then all the parameters that is the CP thermal conductivity viscosity whatever inviscid flow so with there is no viscosity the CP does not change with a flow in the nozzle right we had assumed that CP is constant and thermodynamic properties are also constant in the next class let us look at how changes in the real world will affect whatever we had derived in most cases in engineering it would be nice if we can get a closed form solution that is if you have a real situation and if we can get the entire solution for that particular situation otherwise the next best thing that we can do is let us say we can get the bounds for it right if you have an upper bound and a lower bound and if you say the solution might be somewhere in between these two that is also good for us because then we are we have some we know it is going to be within these two values so that is what we are going to do in the case of nozzle flow which we will discuss in the next class thank you.