noc19-ae08-lec17

[Music] let us start this lecture with a thought process from Mahatma Gandhi who is the father of nation he says that you must not lose faith in humanity humanity is an ocean if a few drops of ocean are dirty the ocean doesn't become dirty it's a very important and profound statement when Humanity is at stake at this moment we need to repose the faith in humanity and work for the humanity in ourselves and also that among other Philippines and we can now recall what we learned in the last lecture if you look at we started the discussion on looking at various structures in both over expanded nozzle and under expanded nozzle and then we also derived an expression for the thrust coefficient and if you remember well the thrust coefficient is a very important performance parameter particularly for judging how good the nozzle is and it is a function of pressure ratio that is PE by P t2 and it is also a function of PA by P t2 and area ratio and gamma and by knowing all these parameters one can evaluate theoretically but however one can also evaluate experimentally by measuring the thrust and the cross section of the throat and chamber pressure that is PT 2 suppose you know the thrust coefficient or you assume certain thrust coefficient from for designing a rocket nozzle then you can very easily find out the thrust by knowing the thrust coefficient CF and the a T is your throat area and PT 2 in your chamber pressure if I know all these parameters I can very easily evaluate the thrust and let us discuss little bit about the thrust coefficient you can recognize there are two distinct terms over here on the right-hand side of this equation and this portion is basically momentum test and this portion is your pressure trust and keep in mind that this PE by PT 2 is a parameter which is will be dictating what is a by a star or a by a star will be dependent on the expansion what you want to have of course one can notice here when PE is becoming small for a particular chamber pressure this pressure first term or the pressure first coefficient due to the pressure thrust will be decreasing and the thrust coefficient due to the momentum change will be increasing because these terms will become a smaller values P by P t2 and then this first coefficient due to the momentum change will be increasing and of course there is a situation where you will get maximum thrust coefficient that is possible only when the nozzle is expanded fully that means the exit pressure must be same as that of the ambient pressure or the back pressure pressure so that means we can achieve the maximum thrust when nozzle is fully expanded then as a result these terms will be 0 and then the optimum thrust coefficient becomes C F naught we are saying this is the optimum values is equal to root over 2 by gamma plus 1 power to the gamma plus 1 divided by gamma minus 1 to gamma square divided by gamma minus 1 in the bracket 1 minus PE by PT 2 power to the gamma minus 1 keep in mind that these optimum thrust coefficient this is your basically this optimum thrust coefficient is function of gamma and it is also a function of PE by PT let us consider three gamez and plot that CF naught versus the PE by PD 2 you can see that for a particular gamma let us say 1.4 you can see see F value attains a very higher value when the it is maybe around something 0.001 that is PE by PT 2 means the exit pressure is very very low as compared to the chamber pressure and this situation can occur at higher altitude and of course it decreases at a very faster rate very high slope over here and then after the slope decreases and it decreases slowly and the similar trend you can see for the gamma is equal to 1 point 2 and gamma for gamma is equal to 1 point 1 however the increase in CF value or the optimum thrust coefficient can be seen here interestingly you can see that higher PE by PT 2 this value let us say at zero point 1 there is not perceptible changes in the value of C of naught that means it is not really changing much the effect of the gamma at higher pressure on the optimum thrust coefficient he is negligibly small however when it the pressure ratio decreases then this effect goes on increasing and it becomes quite substantial at a very low pressure let us say PE by PT 2 is equal to zero point zero zero one at the axis origin that because this is increase is exponential of course this is not very much easily observed in this figure because it is all merging to the axis however if you plot this figure of the respect to log of the CF naught versus the log of P by PT 2 you will find there is a substantial changes in shear value with the change in gamma and another things you can see from this expression that when PE is decreasing then this value will be basically having very less and this very small values and this 1 minus of that is becoming Cal increasing this term is tending towards 1 that is the maximum value one can think of when the PE divided by PT 2 is equal to 0 or tending towards that means if P decreases maximum thrust increases when P is equal to 0 then C F attains a maximum value for a fixed specific heat ratio so if you look at this term is goes toward 0 so you will get C F naught is equal to root over 2 by gamma plus 1 power to the gamma plus 1 tau minus 1 and true gamma square minus one this is the expression one can get is the maximum C AB not so you can see that basically it is a function of gamma only and another interesting feature one can see from this that for maximum thrust coefficient we should have very higher chamber pressure and lower values of the gamma of course the higher chamber pressure means you should have a very thick wall kind of things and this because the temperature of the temperature in the test chamber will be very high and you should use very good materials that can withstand not only the stress due to the pressure but also due to the temperature so therefore one has to choose the chamber pressure properly although it is always desirable to a higher thrust coefficient as I told earlier that thrust coefficient plays a very important role in so far the performance of the nozzle is concerned because there is no term temperature over here and temperature generally will be generated in the thrust chamber and of course with due to the combustion of the propellants and it also increases the pressure in the thrust chamber and of course let me repeat little bit that you can estimate very easily by measuring the thrust F and also the chamber pressure P t2 and the throat area of the nozzle of the rocket engines so therefore you can evaluate experimentally CF values and using the right hand side expression you can very easily also find out the vertical that will be using and the experimental values need not to be same as that the theoretical value for the same pressure ratio across the nozzle or in other words for a particular area ratio let us look at this expression little further and see what is happening if you look at this expression that CF is basically function of PE by PT 2 and P a by PT 2 and a a by a star and gamma specific heat ratio and AE by a star we know this expansion ratio and if you look at this a by a star is dependent on P by PT 2 so therefore will be now looking at how does this CF that is thrust coefficient is varying with the area ratio and with also with P a by PT 2 for a particular gamma that we will see how does these plots will be looking like that and this is your plot which I have shown here that is your first coefficient CF he is plotted on the vertical axis on the horizontal axis the area ratio epsilon which is equal to a divided by 8080 is basically choked throat area sometimes we call that a t is basically a star because the choked conditions we are looking at and this if you can see that it is being plotted for PT by P a and starting from the 2 that is the pressure ratio across the nozzle 2 to the something around thousand and keep in mind that as I told earlier that a a T is a function of P by PT 2 so therefore there is no need to really look at the P by PT 2 already we have considered that in terms of this area ratio so you can see that let there are lot of lines and this is being plot in the semi log plot you can say this is the log scale on the horizontal axis let us consider one of the pressure cases for example PT 2 divided by PA is 5 you can see that this first coefficient goes on increasing with the area ratio 1 and if as the area ratio increases it can go like that which I have not shown here and keep in mind that at the area ratio of 1 this attains maximum value that is the thrust coefficient around something 1.09 something here and this is area ratio is equal to 1.5 and when the area ratio decreases less than 1 you can see that the thrust coefficient decreases and because of fact that this is area is being lower than the for which it is particularly designed so that the pressure will be expanded less that means PE at the exit will be higher than the back pressure or the ambient pressure at which for which it is designed as a result there will be expansion of gas downstream of the nozzle exit and and this we call it as a under expansion as further expansion needs to be undergone all the pressure PE is greater than the back pressure PE and we have already learned how the structures of the four structure at the downstream of the nozzle exit will be for any expansion and there will be expansion fan emanated from the leaves of the nozzle exit and then it will be interacting with the boundary layer and then reflected back as a series of compression wave which Qualis into south and again the shock will be impinging into the the jet boundary and then it will be again reflected back as a expansion wave of course before that there will be Mac disk at the centerline of the nozzle and this sock cell will be goes on formed it the sock cells will be formed as long as the strength of the soft cells will be in decreasing and this pattern is quite complex in nature as a result there will be decreasing the thrust coefficient and and if you go just a positive direction that means when the area is increases that became let us say 2 then there will be also decreasing the thrust coefficient because of fact that this pressure PE he will be less than the ambient pressure and it is basically over expanded as a result that it has to undergo the oblique shock and sometimes the normal shock depending on the pressure ratio what it will be there and as a result there will be of course the shock will be emanated from the lip of the nozzle exit and then it will be reflected back as expansion wave then the mag disk will be formed formed and then there will be again coalesces into the shock wave and further downstream this shock cell will be continuing to be formed till its strength is decrease or diminishes but however if you go a little beyond certain limit like here that point then there will be flow will be separated in the conical nozzle even in the bells of nozzle and as a result you cannot operate beyond that because the losses will be very high and the thrust coefficient may decrease drastically and this is the point where the maximum thrust coefficient you can obtain for a particular PT 2 by P a of 5 and of course if we decrease the pressure through three point three there will be decrease in the thrust coefficient if you look at this is the maximum values and if you go to two point five there will be decrease in the peak thrust coefficient value of the maximum has lower values and it goes on but if you goes on increasing to the pressure PT two by P a then you can see that there is a substance you will increase in this here values and if you go to the twenty again there will be substantial values of the peak CF coefficients and it goes to the if you go to thirty three point that is also increase but if you go beyond thirty three point pressure ratio that is PT two by P a then the increase in the peak thrust coefficient decreases or diminishes out as you go on this increasing the PT two by P a therefore the practically the people always try to restrict this pressure ratio the PD to by PA of thirty three point three because you won't gain much from the increasing the pressure ratio across the nozzle because if you increase this pressure then naturally the thrust the area ratio will have to increase to get a peak values of the thrust coefficient and then with increase in area ratio the length of the nozzle for a particular angle it will be increasing so also the thrust so as the weight of the engines will be also increasing and if the chamber pressure increases then you will have to pay the penalty for using very good quality material and beyond certainly it it may not be possible to have within the constraint of material what we are having at this moment so however theoretical when you look at this pressure can go up to very higher level you can look at the two here this CF that is thrust coefficient value of two you can get which is not shown in this figure around area ratio of 200 and this pressure ratio of 10,000 and when this PT 2 by PA goes to infinity that means chamber pressure is very high and P a is tending toward zero right then it became infinity values and at this condition the seer value happens to be two point two four six per gamma is equal to one point two and of course which is not shown in this figure and keep in mind that when you join these peak values of each pressure you will get the line of the optimum thrust coefficient that this is the line and of course for each pressure ratio the point beyond which the flow separation will be occurring and this is known as the region of flow separation of the conical model so as I have already discussed that this is a very important one one should understand let me repeat little bit about the under expansion when it is occurring that is if you look at a convergent divergent nozzle let us say with the area ratio of a ii and this is your a star and for that it is being plotted the pressure and p is decreasing as you can see along with the x direction and this is your x direction right and you can see that this is the pressure at the throat region when it is choked conditions keep in mind that this is very smooth or it is decreasing monotonically but however in practical situation there might be little bit kink over here because of fact that the sonic wave will be on that will be occurring at at the throat so therefore there will be but in we are considering the we are considering the theoretical one therefore there is no need to have this form and keep in mind that if it is expanded fully that is P is equal to P a you will get but if the you are using a throttle shorted nozzle then you will get that P II and this PE is basically greater than the P a therefore there is a need for expansion to be taking place at beyond the nozzle exit so we call it basically under expanded and if let's say there is a you are now operating in such a way that a the pressure ratio for which you are basically having a increase in area so that you will say that this will be the expansion will be taking place beyond that that means this PE is less than the P a for which it is designed so therefore this portion is basically is the over expanded portions and one has to avoid that one as I told earlier when PE is greater than PE it is under expansion and when P is less than P a is the over expansion this is basically over expansion and keep in mind that this pressure ratio has to be proper so that the obliques of formation leading to the flow separation can occur at a P by P LS than zero point four and this is we have already discussed this is basically the Sommerfeld criteria it must be used for designing a rocket engine so that the flow would be separated out as the shock is being formed oblique shock then the pressure will be very high over here at the downstream of the shock and then the flow is less is likely to be separated from the wall surface of course there are various reasons why separation will occur but this is the important point that must be kept in mind and otherwise the due to the flow separation the thrust level will be reduced drastically and then of course who can reduce the thrust coefficient and with this we'll stop over here and we'll be discussing further about various aspects of the nozzle in the next lecture thank you very much [Music] [Music] [Music]