 In the last few classes we have seen the different cooling techniques namely the film cooling radiative cooling and regenerative cooling these are primarily used in liquid rocket engines where you have liquid or if you look at radiation cooling it is primarily for satellite applications. Now let us look at how we cool a solid rocket motor okay if you look at a solid rocket motor as I said earlier the heat that is coming in to the nozzle is something that needs to be taken care of other than that the if it is a port burning configuration the propellant itself acts as a thick insulation layer so you do not need to worry about cooling the thrust chamber or the entire motor but only the nozzle portion now in the nozzle portion this is a more acute problem than in liquids because you essentially do not have any liquid on board okay and yet you need to have cooling okay and let us look at what are all the techniques that are available and how do they cool the solid rocket motor now I remember writing this in with regards to radiative cooling also there are a few books that say radiative cooling heatsink and radiative cooling okay because it just acts as a sink and then radiates on the other side okay but there is also this method that has been used to look at the throat region of a solid rocket motor now if you remember earlier lectures we had said that throat is a very critical region because the highest heat flux is at the throat so you need to cool it otherwise it is going to damage the throat region now one way of cooling it is to use high density graphite as shown in this figure this is the throat region right and you have a high density graphite here which acts as a heatsink now heatsink is something like this if you have a particular thickness of this material and if it is a high temperature resistant material right then if you look at how the temperature profile is for this material it is something like this shown in a here that is into the solid the temperature decreases and depending on the thickness this temperature is achieved right so you need to know what is the thermal profile thickness in other words right and if you look at this figure here you see that there is high density graphite and there is a backup material there is some phenolic asbestos phenolic and also a metal backup as such the graphite can take high temperatures but not high pressures you cannot use it to make the motor casing itself right so you need a metal there and this could withstand high temperatures and could be used above the metal portion. So what we need to know is how thick should we have this graphite to do that if you remember this is a problem of unsteady heat conduction okay and you can solve this and the thickness is given as thickness of the thermal profile is given us where alpha where alpha is nothing but thermal diffusivity that is in meters square per second and T is operational time right so depending on the one these two you will get a thickness right it also depends on if you look at this figure here it also depends on what is the allowable T0 you can have on the other side right on the other side I remember I said there is either going to be a metal or a asbestos phenolic so what is that temperature that you are looking to have on the other side is what determines what is this thickness going to be okay so depending on that you can choose the thickness of this throat region with high density graphite the next technique that has been used is ablative cooling okay now ablative material as such is a material that consists of two different things one is it has a base material and a resin okay now if you remember our discussions about solid propellant and how heat is transmitted in the solid right we had this equation for the heat flux at the surface that is right now if you are looking at this ablative cooling what happens here is there is this resin portion that vaporizes so it absorbs heat to change its state from solid to a gas phase okay as this heat is absorbed it cools the rest of the material and there are two kinds of heat resistant ablative cooling techniques that are available one is charring and non eroding as shown here in B and the other one is charring and eroding or ablating okay the basic difference here is in this case charring and non eroding it forms a char but the char as you see here even in a flow cross flow this good doesn't get eroded it stays as is okay so the if you look at how the heat is going to be transferred there is a new material that in a sense you are going to have which is going to resist the heat transfer right you have now when you started out you would have had only two materials one is the virgin material and the other one is the charring material as time progresses you are going to have three layers with the char layer increasing and the virgin decreasing okay which means that you can see here very well that the temperature profile through these decreases and on the other side you can have a very low temperature so the essential idea here is to have a high QS as possible right then if you have a high Q higher QS no matter what your TS is going to be if you have a high QS then this T0 is going to be lower and more so if you have different layers of these material right and the next set of materials that is used as a charring and eroding or ablating material in this case because of cross flow the material gets eroded and you are going to have a region which you are going to have a region which which is if this were the nozzle earlier indicated by the dotted line then that portion is going to move to this new line right so the area is changing right so area is increasing and therefore the gas phase if you see is coming here as the charring portion moves into the virgin material it still has three layers as in the other case but the thickness of the char layer is very small here okay and the charring layer is the same thickness and the virgin material after that so with this you can have a lower temperature T0 at the other end right well this is in some sense the same idea which people use to burn camphor on their hand right you have seen that people can hold camphor on their hand and have it burning and do the earthy right essentially what they do is camphor burns so the temperature profile in the camphor will be something like this right so you will have a fast burning here and because of which the on the other side the temperatures are really lower until the flame comes and hits the other portion but what they usually do is they will have a layer of oil right so that protects their hand at the time when the flame reaches the other end and if that is a very short time it will go off without any major damage to the hand so in a in a way one good way to protect the material is to have something burn off right in that sense it is taking away all the heat and that is what happens even inside a rocket motor as well as in ablative cooling the materials used for this are silica phenolic and carbon phenolic this is the resin part which upon heating absorbs the heat and evaporates okay so a combination of if you look back at this picture here if you look at this picture here this is the convergent portion this is the throat and this is the divergent portion this is the axis so we have shown only one side of the axis now if you see here the highest heat transfer is near the throat which is protected by two layers one is high density graphite and then a layer of phenolic resin and after which the heat transfer coefficient reduces as you move towards the exit so therefore you can use some kind of ablative material to cool it right so using a combination of both ablative material and heat sink a solid rocket motor throat is or the nozzle is cool this is the same cooling technique that can be used in a hybrid rocket also primarily because hybrid rocket as we will discuss a little later will have a fuel and it is also a port burning configuration so the fuel itself protects it in the combustion chamber and only the nozzle needs to be cooled which can be cooled using this kind of technique okay so we have discussed the various cooling techniques that can be used to cool both the nozzles and the thrust chamber now let us look at what are the penalties and where do we use what kind of cooling if you look at the different kinds of cooling one was heat sink then ablative and film cooling radiative and lastly we had regenerative now various considerations that we need to keep in mind is one is operation time then the limit on PC remember when I when we discussed the equations and how we can rewrite the heat transfer equation in terms of pressure right using m dot is equal to PC 80 by C star we found that if you go to higher pressures then the heat transfer coefficient will be higher so in a sense there is a limit on the chamber pressure and restart throttling pulsing and what are the penalties with these kind of cooling firstly regarding the operation time if you look at the heat sink if you have given a particular thickness of the material that you have to you can use there is a upper limit on the time that which you can allow it for operation okay because if you look at a steady state then the steady state would be at if it is a large time then the actual chamber temperature will reach the other side also if you are giving it a very large time and if you are not taking out heat on the other side so in a sense the operation time is limited if we use a heat sink and same as the case for ablative cooling here it is less than 20 minutes and then you have film cooling film cooling if you look at it you are constantly pushing out some liquid on the periphery therefore there is no limit as long as you have the liquid on board you can keep doing this and there is no limit to the time you operational time and this should be the same for regenerative cooling also what about radiative cooling you think it is going to have some kind of limit it should have a limit because it is also in some sense a heat sink and then on the other side you have radiation taking away the heat so this is something like typically one hour okay. Now limit on PC heatsink obviously cannot be unlimited so this is again limited this is also limited this would have no limit as long as you have the coolant on board this is also no limit this is restricted to something like less than 10 atmospheres so if you look at radiative cooling it is only applicable wherein it the operational time is also not very very large but the thrust of the motor is also very small now there are a few things that are possible only with a liquid rocket engine that is restart throttling and pulsing throttling is you can change the thrust level okay by adjusting the flow rate of the liquid pulsing is as we have discussed earlier with reference to monopropellant thrusters if you look at the thrust versus time curve it will have something like this so there are a short pulses of thrust and there are no also times where there is no thrust so what about heatsink can we have this with restart right it should be fine the only thing that will have a problem is regenerative cooling because if you look at regenerative cooling when the motor switches on the flow in the coolant pipes might not be that good because there is always some propellant that will be trapped right and that would have absorbed some heat and would have been a gas or something like that so you need to be careful in only the regenerative cooling so you cannot have too many restarts if you have regenerative cooling till the time it reaches some kind of steady state the flow will not be established and that might cause harm to the motor casing itself so you need to be careful while using regenerative cooling but otherwise the rest of the things it is fine to have as many restarts then what about throttling throttling other than these two right you are increasing the thrust level or decreasing the thrust level again there are issues of what is the flow rate through the coolant pipes or how quickly can you take off heat or through the coolant pipes is an issue right if you are going up in thrust the coolant pipes could be having a lower flow rate and then for it could be detrimental so in these two cases it is limited but otherwise for the rest of them throttling is no problem and pulsing in this case it is simply not possible okay and this is in some sense again limited because you have to have a flow on the surface of the chamber itself but for all others this is fine except of course ablative cooling because ablative cooling if you look at it if you are wanting to have a very sharp cut off and if you are having an ablative material that charge and erodes right there is some amount of gases that are going to be released so the thrust cut off will not be as sharp and if they could have a residual thrust okay so in this case so let us now look at what are the penalties involved in using each of these techniques these two it is only the weight that is a penalty because depending on the time of operation you need to have a certain thickness and depending on the density of the material that you are using it adds to the weight of the system so in both these cases weight is a penalty and in film cooling if you remember we are using one of the fluids that is fuel on board and this is not contributing a great deal towards ISP right so in a sense the penalty is here ISP right it is not contributing towards producing a thrust in that sense if you look at it you are going to operate it at a particular O by F it is not that it does not participate in the combustion or something like that but in the chamber right if you are wanting it to be used as a coolant you do not want it to be participating in combustion right ultimately it will happen but if you remember what you want to do is the length that is available is only up to the throat if it burns further beyond that is not going to contribute towards thrust if the pressures and temperatures are increased beyond that that is not going to contribute greatly towards thrust because we know from thermodynamics if you add heat at the highest pressure that is the one that is going to give you effectively a better efficiency right if you are going to add things if things are going to burn later on beyond the throat portion they are not going to contribute greatly towards thrust although I am not saying this will not burn right this does burn but it is only a matter of where it burns it could as well burn in the atmosphere beyond the nozzle also so in that sense the ISP the delivered ISP what I am talking about is how much of propellants you are carrying and what is the overall specific impulse that you get that will be lower then with regards to regenerative cooling if you remember when we talked about regenerative cooling I said this is mainly for high thrust and long burn duration engines and I spoke of the space shuttle main engine is a liquid engine and there I said the chamber pressures of the order of 200 bar right and I said somewhere in the pipeline it is going to be something of the order of 300 bar or above that if you look at the coolant pipes they are going to be very very small right and there is an enormous pressure drop in pushing liquids at a very high rate through these small pipes so if you are going to use regenerative cooling the penalty that you are going to pay is in terms of pressure loss okay pressure drop in other words is also going to effect if you are using a turbo pump to pressurize it there is some amount of energy that is expended which is not going to impact the ISP right so in that sense it comes back to a reduction in ISP but as you as we said earlier in this method you are pumping back the heat that is lost to the walls so you are going to take it closer towards the adiabatic condition that we had assumed while we were deriving equations so in that sense there is a gain in ISP because it is going to be adiabatic and there is some amount of loss because there is something that you need to pay for the pressure rise or the pressure drop across the coolant channels and radiative cooling as we discussed earlier I had clubbed it with heat sink because this also needs to have a certain amount of thickness right so again the penalty here is weight so we have discussed all the cooling techniques that are used on board a rocket motor and if you look at it if you have a liquid engine regenerative cooling is probably the best method of cooling but if you look at some of these aspects you need to have a certain amount of backup or redundancy if you are restarting and if you have throttling and other things so invariably people also tend to use film cooling to back it up or to add redundancy into the system so as to have a safer margin okay although it is not so efficient film cooling has been used and will continue to be used because it gives you that little bit extra that you can do even after designing everything right even after you design the motor everything is done you still have that margin to play around with because you have this film cooling so if we look at picture wherein what are the regimes that each one of them is going to be used in it is going to be something like this so if you have burnt I am on the y-axis in seconds and this is on a low log scale and thrust also on a log scale so you can divide this into various regions if you see that you have a smaller burn time and whatever thrust level that you want to have it is possible to use heat sink but as the burn time increases you also need to back it up with ablative cooling and for a very low thrust and large burn time it is better to use radiative cooling and the region for high thrust and high burn time it is regenerative cooling and have not indicated where film cooling is as I said if you are using a liquid engine it can be used everywhere in this domain it is to add some kind of redundancy and therefore can be used all across the domain okay so this completes our discussions on the various cooling techniques to cool the rocket motor nozzles now let us look at a we are discussing about a liquid rocket motor and we know that it has to have a feed system either it is going to be pressure fed or turbo pump fed let us try and look at the design of the feed system so firstly let us look at pressure fed systems if you look at the figure of a pressure fed system here if you look at this figure what we are essentially going to do is if we are given see when we are asked to design the pressure fed system what we are going to do is try and design this size of the gas bottle what is the size of the gas bottle that we need to carry and what pressure so as to have a chamber pressure of certain value okay fine and for a burn time so essentially you will be given for doing this going about doing this you will be given firstly the thrust time curve which means that you know what is the thrust F and you are also going to know what is the burn time TB right after knowing this let us say you are going to use some kind of propellants right if it is a bi propellant system or a mono propellant system you know what are the propellants that you are going to use and if you know the chamber pressure PC and you also know the area ratio of the nozzle and propellant combination so you are knowing from this propellant combination what you can get out of this is the chamber temperature and you know the area ratio and the chamber pressure and also the ambient pressure or its variation right so you can calculate what knowing all this you can calculate the specific impulse of the motor right so at the end of this you will get to know what is the specific impulse if you know the specific impulse and if you know the thrust what can you calculate mass flow rate through the motor because we know that m dot is nothing but F by ISP so you can calculate what is the m dot and after knowing what is the m dot you can get depending on the ratio fuel a ratio of oxidizer to fuel ratio right you know the overall mass flow rate and the you know the ratio of fuel to oxidizer that is let me call this as s okay then you can calculate the individual mass flow rates of oxidizer and fuel right so if you can calculate the individual flow rates of oxidizer and fuel and then you also will know the density of these materials right so you can get volume flow rate of oxidizer and fuel and now you know the burn time so you can calculate the overall volume of the fuel and oxidizer that you will need for the particular machine right but is that going to be enough if you just do this calculation and carry that much of liquid on board right you will know the burn time so you carry based on this whatever is the volume of the liquid is that going to be sufficient for your fulfilling the mission no why what kind of losses yes if you look at any kind of feed system there is some amount of fluid that you might not be able to expel out of the tank itself and obviously there is going to be some kind of fluid depending on what kind of cooling you have if you have regenerative cooling then a larger fraction of the coolant is going to be trapped in the pipes okay so that is not going to be useful so you need to carry a slightly higher volume than what you get from this typically of the order of 2 to 3% extra okay so if you then estimate what is the volume it is going to be 2 to 3% higher than this volume so now you know what is the volume of the fluids you need to carry in addition to what we discussed if you look at if you are using cryogenic propellants like liquid oxygen and liquid hydrogen after you store it in the tank right and keep the vehicle ready for takeoff as it continues with its operations because of the temperature difference there is going to be some amount of boiling of this propellants right so these are not going to be useful because it is already going to become a gas and therefore it is not going to be useful so you have to account for that also while calculating the overall volume that is required okay fine so once you calculate all this taking into account all this you are going to calculate some volume now if you look at this design here you are not going to fill the tanks completely because then there would be no space for the pressure into act so you are going to have a small amount of volume that is left unfilled through which the pressure and can act right so if we were to design what is the pressure and how much is the amount of pressure and we need to calculate carry on board we need to know the volume flow rate and from the volume flow rate calculate the overall volume then account for losses like boiling and some coolant trapped in the pipes and also you are going to have a small extra volume known as eulage volume so if you account for all this and then you know what is the tank volume right now this tank has to be filled at some pressure by the pressure and system at the end of burning if you look at it so the pressure inside the tank should be the same as what we started out in the beginning if you are not doing any thrust variation so you need to take into account what is the volume of the tank and what pressure it is going to be stored at the end and then calculate what is the volume you need to have and that will help you size the pressure and tank so if you look at doing that so the weight of the pressure and could be mg is equal to this is given by ideal gas law multiplied by a compressibility factor okay this Tg is the temperature of the pressure and gas then vt is the vt is the volume of the tank and z is the so using this you can calculate what is the mass of the pressure okay we will continue with this in the next class thank you.