 Good morning in the last class we had seen this that we had derived the thrust equation and we had also derived equations for mass flow rate the AE by AT relationship in terms of PE by PC and we had derived the expression for the exit velocity UE now if you look at the thrust equation we had derived expressions for m dot UE by AT and in terms of PE by PC right we can now go ahead and define parameter that is of prime importance and rocket propulsion that is the specific impulse specific impulse as the name itself suggests is nothing but impulse per unit mass okay if you look at the expression for impulse impulse is nothing but integral of FDT per unit mass of the propellant this is known as specific impulse or it is denoted by ISP okay and when you work this out you can get this expression for ISP as thrust per unit mass flow rate okay m dot is this so you get an expression as thrust per unit mass flow rate now we also know that if you take the m dot here you will get ISP is equal to UE plus AE by m dot okay now let us look at how to proceed on this further and get an expression for ISP in terms of we if you look at this equation we know the expression for UE we know ME m dot and we know a PE by PC in terms of AE by AT we will use that to get an expression for ISP now m dot I we had got this in the last class that is m dot is equal to PC AT by C star okay so I will use that and I will get now UE we also had derived an expression for in the last class that is UE the exit velocity and we had also derived an expression for AE by AT in terms of PE by PC okay now if we substitute back all these terms we will get an expression for ISP if you notice in this equation ISP the unit of this is meters per second and if you look at all the other terms here PA by PC PE by PC and AE by AT are non-dimensional and you have C star which is again meters per second but if you look at this definition you get thrust per unit mass flow rate which means that this is Newton right or in other words Newton second per kg right if you do expand Newton then you will get Newton is nothing but kg meter per second square so what do you get so you end up getting meter per second okay and also if you look at most of the literature in rocket technology most of the books will give you a unit for ISP in terms of seconds now if you divide this by the acceleration due to gravity right that the units of that is meter per second square then you will get the unit that usually rocket technology books will give you that is in terms of seconds so both of these are used Newton second per kg and seconds are the units that are commonly used if they are using an SI system they will use this otherwise they are going to use seconds okay. So now we have derived this expression let us substitute back I am not going to use this primarily because if we remember the discussions last time we would be knowing the geometric parameters that is AE by AT right and we would want to know what is the variation of PE by PC from this expression it is very difficult to extract out this so therefore what we said in the last classes we either use graphs or tables to get this value so I will not use this here but without this I can write okay this is the expression that we get so I have to have this brackets here now if you look at this looks like ISP is a function of two terms one is C star and there is another term in the brackets this entire term we will denote it as CF okay which is known as the thrust coefficient why it is called as a thrust coefficient I will be able to show it to you in a minute so ISP we can now write it as a function of now C star this depends on only the energy content and CF if you notice here it depends on the nozzle characteristic which is given by PE by PC okay and the nozzle geometry AE by AT so if we know the nozzle geometry and if we know the ambient pressure and the chamber pressure we can calculate the rest of the parameters so it is only a function of nozzle characteristics okay so the reason for dividing this into these two components will be obvious in a short while from now now C star if you remember is given as 1 by ? of ? into under root right so C star depends on TC and the molecular weight TC is nothing but the adiabatic flame temperature okay so this is determined by what is the energy content in the propellant okay and M is the molecular weight of the gases so if we divided like this we will be able to differentiate for a given nozzle which of the propellants gives us the best performance okay we will come to that in a little later we will now derive what are known as rocket equations based on all these things we now know that F is equal to M dot UE plus AE and we also know F is equal to M dot into ISP from the definition of specific impulse and if I use this ISP is nothing but C star into CF and M dot we know is nothing but PC 80 by C star so if we plug these two into that equation we will get F is equal to okay I had called this as the thrust coefficient now if you look at this equation it is obvious why it is called the thrust coefficient because these two terms gets multiplied by a coefficient CF to give us the thrust which is why the name thrust coefficient okay so these are known as rocket equations and we will be using them pretty frequently in this course as I had said if you look at ISP we have defined it as C star into CF now C star I said depends only on the energy content of the propellant now if we are to determine which of the propellants is better which of the propellant combinations is better then we can do the following exercise we can make it independent of CF that is if we fix certain nozzle parameters if we fix the chamber pressure at a particular value the area ratios at a particular value and then the ambient pressure as one atmosphere then we will be able to make the ISP independent of nozzle parameters right so that is independent of CF that is what is done and typically so we need to fix the value of CF and to do this we keep the chamber pressure that is PC at 70 atm then the ambient pressure is the mean sea level pressure that is one atmosphere and we ensure that the nozzle is such that it expands in a way where PE equal to PA which is also known as the case where the flow is optimally expanded okay and we choose an area ratio such that it gives this that is AE by AT is chosen such that okay now having done this we will get a value of CF for this to be around 1.6 for ? equal to 1.2 okay so having fixed this CF we can now evaluate various propellants and see how would each one of them is okay and if we do that okay we will now try and see various combinations of propellants and what is the TC molecular weight and C star that we get if we do this okay there are two kinds of solid propellants one is double base then there is storable liquid okay and lastly locks LH2 that is a cryogenic system now if you look at TC for this these are the values that you will get if you notice in this there are several interesting things the C star value for locks LH2 system is the highest okay and therefore the ISP for this system will also be highest and that will be around 400 plus seconds okay if we need to understand why this is happening we also need to look at this and this if you look at these two the temperatures are similar in fact in this it is a little higher right but yet the C star values of this is higher than this the answer to this lies in the fact that C star is not only a function of chamber temperature but also a molecular weight if the molecular weight is lower and the chamber temperature is high enough then you will get a very good C star and that is what is done in this LH2 locks cryogenic system okay you not only get a lower molecular weight but you also get a reasonably high temperature and therefore you will find that the ISP for this is the highest okay it is also true that any kind of propellant that we try to make we always try to make it fuel rich okay the reason being if you look at the molecular weights of field the typical fuel elements are carbon and hydrogen right carbon and hydrogen have molecular weights of 12 and 15 sorry H is 2 right so carbon as molecular weight of 12 and hydrogen 2 now if you combine this with oxygen if it completely burns this will give to CO2 and this will give rise to H2O okay if it completely burns in a liquid engine locks LH2 engine the reaction would be H2 plus half O2 to give rise to H2O if you look at the molecular weight of H2O molecular weight will be 18 right but what we have got here is lower than that that is around 12 to 15 the reason being we want to make it fuel rich while not severely decreasing this component that is the chamber temperature okay then we will be able to get a higher C star okay so in a sense of propellants we would want to make them fuel rich so that the C star values that we get are higher okay the other important thing that is present here is if you look at solid propellants to liquid propellants in a sense you will find that liquid propellants have a slightly better performance even in the case of storable liquid compared to solid propellants the reason is like this if you look at solid propellants solid propellants with the fuel and the oxidizer need to be present together and in the same chamber right so that imposes a restriction on what kind of fuel and oxidizer that you can choose they need to be compatible and they need to not start reacting as soon as they are mixed together right so that places a severe restriction on the choice of fuel and oxidizer but in a liquid propellant there is no such restriction because they are stored separately okay they are stored in different chambers and they are only made to come in contact with each other in the combustion chamber so therefore you can choose better liquids in this case whereas the choice is more restricted in the case of solid propellants and therefore you will find specific impulse of liquids will be superior to those of solid propellants we all know this that rockets also perform outside the sensible atmosphere the kind of things that we discussed here that is PC a by 80 such that it is optimally expanded flow is optimally expanded to a ambient pressure of one atmosphere all these things are useful for assessing engines operating inside the atmosphere that is the lower stages of launch vehicles okay and also certain tactical missiles now what do we do when we have to look at things operating beyond the sensible atmosphere now what happens to the thrust equation that we have derived in this equation if we are operating beyond the sensible atmosphere PA goes to 0 and therefore we will be left with an expression PA is approximately equal to 0 and therefore we will get F is equal to notice that in the equation here the term PA is with a negative sign so therefore if you remove that component your thrust will increase therefore you will find that the thrust that is delivered by a rocket motor at vacuum conditions is higher and correspondingly the specific impulse which is known as vacuum specific impulse will be higher than the specific impulse that you will obtain when operating within the sensible atmosphere typically vacuum specific impulse will be 10 to 15% higher than the specific impulse obtained within the sensible atmosphere or the C level ISP so we now learned what is vacuum specific impulse and specific impulse there is also another parameter called as density impulse density impulse is nothing but the specific impulse multiplied by the density of the propellants that is product of that is ISP into ropey ISP we know is F by m dot that is F by m dot I can write it in terms of mass flow in mass into volume flow rate right m into m is equal to rho V if I use that then I can write m dot as rho P into V dot so using this that is thrust per volume flow rate please remember the ropey that we are using there is not the density of the product gases but it is the density of the propellants okay so where do you think this would be useful solids this would be higher yes if you look at systems that operate within the sensible atmosphere you would want them to be as compact as possible primarily because then you will be able to reduce the drag or your net thrust will increase because net thrust is nothing but thrust of the system minus the drag so therefore if you want to have the net thrust to be higher you need to also have drag lower so if you have a system that is operating within the sensible atmosphere that is very bulky or a large volume then that will not produce the lowest drag so therefore you need to have this optimized for systems that operate within the sensible atmosphere that is for small boosters and tactical the reason for this is if you look at the lock cell H2 system what is the density of liquid oxygen any idea around 1100 right and if you look at liquid hydrogen the density of liquid hydrogen is around 70 kg per meter cube okay it is a very low density liquid if you look at the space shuttle right the large tank that is on the back side of the orbiter most of it is the liquid hydrogen tank because it is very low density fear okay if you have such a system for a tactical system then the volume will be very large although the ISP is better the volume is very large and therefore the drag will also be very large so in such cases it is better to go in for a solid system wherein the density will be higher if you compare the density of this to density of solids solids will be of the density will be of the order of 1600 to 1800 kg per meter cube which is very large so the systems will be much more compact so for tactical systems and for small boosters it is better to go in for solid and for large rocket motors that operate beyond the sensible atmosphere it is better to go for storeable liquid or cryogenic engines because then your ISP will be higher okay yes what should be the density that we should be taking is it the density of oxygen or is it the density of hydrogen it would be a combination of these two so you have typically as I said earlier we do not use the stoichiometric ratios we use it fuel rich that is typically oxygen will be of the order of 5 to 5.5 the oxidizer to fuel ratio so you need to take that and then calculate the density okay so we have now learnt what is it that we need to use if we are looking for a large system that is we need to use storeable liquid or a cryogenic system right when the operation is beyond the sensible atmosphere but when we are looking for tactical systems that operate within the sensible atmosphere and for small boosters it is better to go for solids now there is also if you look at the thrust equation the there are two things that are varying the CF will vary with respect to the area ratio right that we choose so if you look at this graph here right on the y axis you have CF variation and on the x axis you have PC by PE variation and the graph also shows for different area ratios of the nozzle and for different values of gamma fine notice that for a low PC by PE right and for a low area ratio things do not change too much all the values different values of gamma give rise to almost a single value of CF around 1.2 okay as we go to a higher and higher PC by PE the value of CF changes with values of gamma right and for large area ratios you see here that CF will change with the value of gamma and change is not very small if you notice somewhere around 1000 it varies between 1.7 to 2 okay so that variation is not small because CF and C star is what gives us ISP so the ISP could severely change depending on the value of gamma so also we need to keep in mind that the value of gamma that we have made use of in our all our calculations is like this that we have taken gamma to be constant and not varying beyond the nozzle entry point okay but as I said earlier also the value of gamma will change because these are reacting compounds and temperature and pressure are changing and therefore you will find that gamma will change during the expansion process okay and it is important to note that during the expansion process if the gamma changes then the there could be a significant change in CF depending on the change of gamma we look at all these things a little later in the course now we know that the thrust varies if we have a rocket system that is operating through the atmosphere at sea level there is a ambient pressure and the ambient pressure keeps on falling as you go higher in altitude so if you go with altitude PA will decrease so therefore if you look at the thrust thrust is nothing but m dot UE x AE PE – P let us say we have a fixed area nozzle okay which is typically the case then the PE is also fixed right only the PA keeps on decreasing as you go higher in altitude and therefore you will see with altitude the thrust increases because of this increase in thrust right people have been looking at what is known as adaptive nozzles which we will discuss in the next class thank you