 In the last class we had looked at cycle analysis of air breathing engines now let us look at rocket engines in the next few classes here we will be trying to derive expressions for specific impulse and how mass flow rate varies through a choke nozzle how do we get thrust how do we get exit velocity what are the assumptions that we make and as a consequence of these assumptions what do we have to deal with okay or what is the error of these assumptions we will look at it a little later in the course okay fine. So now let us look at rocket engines now rocket engines all the three kinds of rocket engines that we discussed in the earlier classes that is solid liquid and hybrid all three of them have the following that is they have propellant storage then they have a thrust chamber and lastly a convergent divergent nozzle in the solid rocket both these two are in the same physical location whereas in liquids and hybrids all three of them are separate but one thing common to all of them is this part that is the nozzle part okay. In the nozzle what you have is the thermal energy because of release because of chemical reactions is now converted into kinetic energy so in the nozzle you have thermal to kinetic energy conversion okay and as this is common to all three kinds of rockets we can study them exclusively okay and without having to pay any need to pay any attention to what kind of rocket it is whether it is a solid rocket or a liquid rocket or a hybrid rocket so we will do that now we look at the conversion divergent nozzle now before we go there what we are going to look at is quasi-steady one-dimensional analysis that is we are going to assume that all changes happen along the axis only there are no changes that are happening in the radial or the azimuth direction r and ? directions we do not assume any changes to be happening so all changes are happening along the axis so it is one dimension which is not strictly true for a rocket engine nozzle okay we will see what error this bring about a little later first is we are going to make this assumption then we are going to assume that the chamber pressure temperature are given to us that is let us say if this is the rocket motor we will assume that we know chamber pressure which is called PC and chamber temperature PC as known we will also take these to be stagnation quantities that is if you look at velocities elsewhere in the nozzle and if you look at the velocity at the entry of the nozzle these velocities are very small compared to the velocities as you go the nozzle so you can take them to be stagnation conditions at the entry of the nozzle okay and we are also going to assume that you are also going to assume that I said these are known to us firstly and we are also going to assume that fluid is an ideal gas with constant thermodynamic properties that is the CP gamma thermal conductivity all these do not change as we go from this portion of the nozzle to the exit of the nozzle okay then we will also assume that the flow is inviscid that is viscosities are very small and we also assume fluid flow is isentropic so with these assumptions let us see what we can derive and based on this we will also evaluate what are the shortcomings of some of these assumptions are these really valid and then we will try to estimate what is the error okay now from a basic gas dynamics you know that for a nozzle for a convergent divergent nozzle I can write du by u that is change in velocity as you move along the nozzle is equal to now we know that if m is less than 1 what should be the da by a if this has to fluid has to accelerate if m is less than 1 da by a has to be negative if fluid has to accelerate and if m is greater than 1 da by a has to be positive if fluid has to accelerate okay and we know that if at the throat at the throat of the CD nozzle Mach number is 1 then we call it choke nozzle let us now find out what is the pressure ratio if you have PC here what should be the PA or what should be this ratio of PC by PA for the nozzle to be choked okay so let us find the critical ratio of this pressure for choking in order to do this we will solve the energy equation from here to here okay from the entry to the throat section and I can write one dimensional energy equation as one dimensional steady state and for in visit flow with constant thermodynamic properties I can write the energy equation as HT plus is equal to UT where C where C there indicates our chamber conditions and T indicates throat conditions C indicates and T indicates that is we are looking at this to be C this to be T okay now there is no heat addition that is taking place from here to here okay and the composition is the same so if the composition is the same then CP has to be the same so C to T that is from chamber to throat which means that CP is same and there is only change in sensible enthalpy and it also means that only there is change in enthalpy because there are no reactions that are taking place so if all the change from C to T is only because of sensible enthalpy change then I can rewrite this equation as that is I have replaced HT by CP TT and it see by CP TC we also know that if the throat is if the flow is choked at the throat then at throat M is equal to 1 and therefore UT is equal to ET that is the speed of sound which is given by here RS nothing but R is equal to RU universal gas constant divided by molecular weight okay now using all this we can simplify this equation further and write it as because we know UT is equal to AT and AT is this I can write it that like this is equal to we know that CP by CV is equal to ? and CP- CV is equal to R so dividing the entire equation by CP TT that is dividing by CP TT we will get 1 plus ? R TT divided by 2 okay now we know that R and CP we can get through this so let me write that down this will then become 1 plus ? x 1 by 1- ? this cancels out so you get 2 is equal to TC by TT or simplifying I can write TC by TT is equal to ? plus 1 by 2 okay from this and knowing the connection between CP to temperatures to pressure for an isentropic flow I can write PC by PT is equal to so now we can get the critical ratio that is required for choking okay and if you substitute for ? equal to 1.2 which is usually the value of ? for burn gases you get PC by PT to be 1.7 that is if the chamber ratio to the exit pressure or the pressure outside is greater than this then the nozzle will be choked okay so for any pressure ratio PC by or for any pressure ratio that is from PC to ambient wherein it is greater than or equal to 1.7 then the flow at flow is called choke flow as I had said in the previous class the advantage of choke flow is now it becomes independent of downstream conditions and it is only a function of upstream conditions okay so let us see how we can use this to calculate what is the mass flow rate through a choke nozzle okay you have calculated what is the ratio that is required for the flow to be choked let us now calculate what is the mass flow rate through a choke nozzle now we know that m. Which is the mass flow rate is given by ? T ET into VT where T as I said earlier indicates throat conditions so this is the density into area into velocity okay now let me normalize ? T and VT with respect to chamber conditions why because I know for a choke flow that it is only dependent on the upstream conditions upstream of the throat of the chamber so if I normalize it with chamber conditions I will be able to derive some useful equations ? T and VT with ? C which is the density at in the chamber and ET which is the acoustic speed for the speed of sound at the throat so we will get m. is equal to ? T by ? C into ET into VT L by ET into what is ET I know that ET is equal to right so I will normalize this also with the chamber conditions and multiply by under root TC what is this ratio is the ratio of local speed fluid speed to acoustic speed which is nothing but Mach number Mach number at throat for choke condition is 1 so this is 1 so we get m. is equal to ? T by ? C ET so now we need to find this ratio and this ratio we already know for choke flow what is this ratio we also know pressure ratio so density ratio is nothing but ? T by ? C is nothing but PT by RTT into RTC by PC so R cancels out you get PT by PC into TC by TT you have already derived them that is nothing but into ? plus 1 by 2 okay this is the temperature ratio this is the pressure ratio so now if we because these two are the same we can add the powers and we can simplify it as ? T by ? C I will get it as I also know TT by TC which is nothing but I need under root so this would be – half now I know all the ratios that I wanted these two I can substitute them and get the relation for mass flow rate as m. is equal to I am sorry you did not remind me I needed to multiply by if I divide this by ? C I need to multiply by ? C again so this is ? C into ? T by ? C into this so I will get here ? C okay now I can deal with them because they are same base but different powers I can add the powers so I will get and further this – I can replace it as plus and this ? C I can write it in terms of pressure so I will get PC by RTC into AT ? RTC because I have a minus sign here I can write this as 2 by ? plus 1 into ? plus 1 now I have RTC here and RTC here so I can re simplify all of this and write it as this is very similar to something that you have already derived in gas dynamics wherein you will not use PC you will instead use P0 A star by T0 right this whole function this whole thing is a function of only ? so normally in rocket literature this is denoted as a ? of ? function okay so you get m. is equal to PC AT by C star where C star is called the characteristic velocity and C star is given by 1 by ? of ? RU by M is nothing but R so C star is given by this where ? of ? is only a function of function of ratio of specific heats that is so this is the typical expression that we use in rockets when we are discussing rockets for mass flow rate through a choke nozzle okay this ? of ? is also called as a Vanden-Kerchoff function varies from 0.64 to 0.667 for ? variation of 1.3 now we know how to get the mass flow rate through a choke nozzle now if you recollect back we had derived expression for the thrust of the rocket motor when we were discussing about the thrust of the turbojet engine so or an air breathing engine so we know that rocket engine F is equal to m.UE plus AE PE-PA where E indicates exit conditions A indicates ambient condition so you have here UE which is the exit velocity exit area exit pressure and ambient pressure okay now we in the thrust equation we already know how to calculate m. For a choke nozzle so this part we know we need to now find out how to get the exit velocity and then we need to get something about area ratios and other things we will look at it a little later firstly let us get an expression for the exit velocity how do we get this expression we know that the energy equation is valid from the entry of the nozzle to the exit of the nozzle so from that we know we can write from energy equation HE plus UE square by 2 is equal to HC we had seen a little earlier with respect to the throat this equation now we applied to the exit of the nozzle okay so I can rewrite this as UE is equal to under root 2 HC-HE since we know that there are no reactions that are taking place inside the nozzle and the thermodynamic properties are constant from the entry of the nozzle to the exit of the nozzle all this must be sensible enthalpy so therefore we can write this in terms of UE is equal to 2 CP TC x 1-TE by TC and TE by TC we can express it in terms of pressure as equal to under root ?-1 this part CP TC part I can use the UE CP TC I know is equal to CP by R x RTC okay now this RTC what is C star C star is nothing but RUTC by molecular weight or RTC right this is nothing but is equal to 1 by ? of ? x under root RTC so this part here RTC must be equal to C star square ? square okay so I can put this what is CP by R this I can write it as ? by ?-1 into so if I substitute back this into the expression for UE I will get UE is equal to okay because this has square here and this is in this route I can take it out and write it like this now what this tells us is that if we know the characteristic velocity and the pressure ratio we can get the exit velocity right so if PE goes to 0 what happens if this will UE will be a maximum when PE goes to 0 so when PE is equal to 0 UE is maximum and is called limiting exhaust velocity that is it will only become a function of the chamber temperature PE becoming 0 here means in this case if PE goes to 0 TE also has to go to 0 so the UE would this then become under root that is it becomes only a function of the chamber temperature but in reality we cannot get a condition where in the exit velocity this is not the ambient velocity ambient pressure this is the exit of the nozzle being at 0 pressure this is not possible this will be some finite number therefore you cannot get this condition but this is the maximum that one can get with a convergent-divergent nozzle okay now we need to also get one more parameter that is how does we still do not know how AE and PE are connected that is if we know the area ratio there from the throat to the exit how does how does one obtain PE from if knowing PC how can we obtain PE so let us do that so let us relate PE by PC to AE by AT to start this what we know is we know that mass flow rate once the nozzle is choked and the flow is steady the mass flow rate at any cross section is the same it cannot vary so we know m dot is equal to PC AT by C star and is also equal to rho e ue AE okay where e indicates exit conditions okay so from this I can get an expression for AE by AT as PC now I know rho e is nothing but rho e is PE by RTE and if I substitute for that here I will I also know the expression for ue which we just derived okay so if we substitute for both of them here rho e and ue we will get an expression for AE by AT okay this is the expression that we get once we substitute for ue and rho e now here what is again C star into if we take a ? of ? here so you get C star into ? of ? square which is nothing but RTC right so we get PC by PE into RTE divided by RTC that is we need to have a ? of ? in the numerator to account for this into 2 ? by ?-1 I can cancel the Rs and TE by TC I can again express in terms of pressures I already have a PC by PE so I know that TE by TC is nothing but PE by PC to the power of ?-1 by ? so if I substitute for all this for this here and simplify or I can because I can put them together and rewrite this as if you notice this is a relation connecting area ratios to pressure ratios mostly we will know what is the geometric area ratio of the nozzle that we are taking and to get the pressure ratio from knowing the nozzle ratio nozzle area ratio using this equation is very difficult so therefore we have gas dynamic tables which will give you this or there are plots that will tell you how PE by PC varies if you vary AE by AT we look at it in the next class thank you.