 Good morning everyone. So, we are back again with the course on rocket propulsion. Till now what we have done is, we have talked about the vehicle dynamics, we have talked about the orbital mechanics, we have talked about the multi stage rockets and how they function, the performance of rockets. Then we have gone into the nozzle, the nozzle design and then the combustor design. So, that pretty much gives the basic fundamentals of rocket propulsion and every component. Now, what we do is for the remaining lectures, we will go to specific type of rockets and discuss how they work and what are the specific features or special features of those rockets. So, today we will start with solid propellant rockets. After that we will go on to liquid propellant rockets. That will finish our discussion on chemical rockets. Then the couple of lectures we will dedicate to electric propulsion. So, now coming to the solid propellant rockets. Solid propellant rocket motor essentially as the name suggests the fuel is solid. So, solid fuel rockets rely on controlled burning of a mixture of substance which are in their solid phase. So, the name solid propellant comes from this word that they are in their solid phase. And typically of course not always the entire solid propellant mixture is fairly homogeneous. So, a nearly homogeneous material is burned. The composition of this propellant is quite similar to gunpowder that is about 75 percent potassium, nitrate, 10 percent carbon and 15 percent sulphur. This is the composition of gunpowder by the way, but the solid propellant rockets the propellant will have slightly different composition or sometime even drastically different composition. But essentially all of them will be in solid stage. Now, what are the advantages of solid propellant rocket? First of all the thrust produced by these rockets are fairly high. The advantage is that since the propellant is solid. So, therefore, mass per unit volume is high and that is the mass the density of the propellant is high. So, therefore, the same mass can be packed into a mass smaller volume. So, a small size rocket can produce fairly high thrust. So, therefore, thrust to weight ratio is very high for the solid propellant rockets. Second advantage these are very simple or relatively simple to make they have much less components compared to a liquid propellant rocket. So, these are the two major issues apart from that the third issue is the storability. This rockets can be stored can be prefabricated and made ready and can be stored for a large period of time. And because of that they are very good devices as weapon or missiles because they can be kept ready and can be fired at ease. So, these are the basic advantages of the solid propellant rocket. However, as with any other device there will be some disadvantages as well. The main disadvantage is that the specific impulse compared to liquids of course is fairly low. If you recall towards the beginning of this course I discussed that what are the specific impulse for different type of rockets. Solid propellant rockets the specific impulse is in the range of 200 to 300 second. Whereas, a liquid propellant rocket let us say cryogenic rocket can be as high as 450 seconds. So, almost double almost double propellant specific impulse can be obtained by a liquid propellant rocket compared to a solid propellant rocket. Second disadvantage is throttling or changing the thrust at need is fairly complex actually it has to be pre programmed. So, that you can get the thrust variation as you go along, but online throttling is very very difficult because the solid grain is burning to change the burning rate as it flies is very very difficult almost impossible to attain. However, there are advanced rockets where specific materials are embedded which can be triggered at will to give some kind of throttling, but again this is very complex to do. So, the throttling is very fairly complex process. Third difficulty is it is very difficult to stop and restart. Since it is a solid propellant and you do not have anything else inhibitor in between once it starts to burn it is almost impossible to stop the burning. So, therefore, stopping the rocket from firing completely is very very difficult and even if by some means let us say we have pre programmed it to cut the combustion or the burning in between if you do that then the restart becomes a difficult issue because as we will discuss today that one of the component of solid propellant is the igniter and igniter is actually as one short devices. So, once it is used up you cannot relight it again. So, therefore, unless you have multiple igniter you cannot relight it on the go. So, that makes it difficult for the ignition of this rockets if it is stopped and safety because many times as we will go along we will see many time explosives are used as propellants. So, therefore, have controlled reaction with these explosives which are known to burn at a very rapid rate is a major challenge. So, therefore, unless they are handled properly safeties which are always present in solid propellant rockets. So, with this little bit of introduction let us look at some some applications of solid propellant rockets. As far as launch vehicles are concerned solid propellant rockets are most widely used in boosters and as you can see here a picture where the booster is lighted and it is taking it up. If you look at the picture of this rocket the solid propellant boosters are fairly small as you can see compared to the large rocket, but within this small boosters a large amount of energy is packed because of the high thrust to weight ratios of this vehicles or these devices as well as high storage density of these devices. So, primarily solid propellant rockets are used as booster rockets for rocketry application. However, they are very extensively used for missile applications particularly ballistic missiles because of the propellant storability which I have just discussed few minutes back sorry then the excellent aging process and quick response. So, they can be kept ready and they can be lighted very easily. So, unlike liquid propellant rocket where you require to have lot of preparation time just before the launch we will discuss those issues solid propellant rockets does not require any preparation time they are ready to fire once they are built they are ready to fire. So, that is why this because of this quick response they are the best option for missile applications and of course, the high weight to thrust to weight ratio will give you more payload carrying capability. So, these are couple of applications of solid propellant rocket. Now, let us come to various components of a solid propellant rocket here is a schematic of a solid propellant rocket. The green part here is the propellant grain this is the thing which will be burning. You can see there is a something like a tube going through this green part this is the igniter and you can see that there is some kind of charge here which will ignite this grain. So, like I was mentioning that this is a one short kind of application it can be a small rocket itself or some charge will be used I will discuss the igniters in detail. So, this is the igniter which will initiate the combustion process because this is a solid propellant. So, there are fairly closely packed and lot of initial ignition energy is required to start the ignition process once that starts then it will flow on its own, but the initial ignition requirement is quite high. Therefore, the igniter is a must to provide that initial ignition energy apart from that of course, this thrust has to be terminated somehow. So, you can see in this end there are some plates which work as the thrust a terminator then there is this is the casing. Now, when the rocket will burn lot of energy will be produced we do not want this energy to be lost laterally through the casing. So, that all the energy should flow out of the nozzle. So, in order to do that there is a thrust or the insulation between the motor case and the grain you can see here the internal insulation is present. So, this will stop the heat transfer to the casing and then if you recall when we talked about the wide the vehicle mechanics we said that in order to get lateral stability. So, that the lift is 0 you need to have the rocket nozzles slightly tilted right which is thrust factor. So, rockets use thrust has been using thrust vectoring for more than 100 years now from the very early beginning of modern rocket tree that thrust vectoring was used. So, therefore, that thrust vector as you can see here this is the thrust vector control at the nozzle thrust and this is the thrust vector actuator which will turn the nozzle about this point. So, essentially to give the lateral stability to the vehicle and then this is our nozzle this is the converging diverging nozzle this is a nozzle exit cone through which the burnt product will expand to the ambient. So, if I go back to what is written here we have a it is a simple solid rocket motor schematic is given here which consists of a casing a nozzle the propellant charge or grain and the igniter. So, these are the four main components of a solid propellant rocket the grain behaves like a solid mass which burns in a predictable fashion. So, therefore, the grain is designed in such a way that they will burn at a predictable fashion and thus producing the exhaust gases the nozzle dimensions on the other hand we have talked about nozzle design in detail the nozzle dimensions are calculated to maintain a design chamber pressure P c naught we will discuss this also while producing the thrust from the exhaust gases because the thrust will depend on how much chamber pressure is generated inside this rocket plus chamber pressure and temperature. So, the nozzle will ensure that that pressure and temperature is maintained. Now, let us come to the types of solid propellant use the propellant what I mean is the fuel and oxidant together. Now, since as I have been saying that it is like a homogeneous mixture completely. So, the both the propellant that is the fuel and oxidizers are mixed together in this propellants. So, it will be a single solid piece. So, primarily there are four type of solid propellants which are used one is a double base then a composite propellant then composite modified double base and nitramine propellant. So, these are the four types of propellant primarily used for different rockets solid propellant rockets. Now, let us discuss each of them one at a time. So, first let us start with double base propellants what are double base propellants typically these are nitro cellulose or nitroglycerin mixed together at the molecular level. So, this is a mixture of these two propellant nitro cellulose and nitroglycerin one of them is the fuel other is the oxidizer. So, they are mixed in the molecular level. So, therefore, a homogeneous mixture of these two are created. Now, when I have to be mixed in the molecular level you have to add some additives. One of the additives is a plasticizer which is typically diethylphate let and triacetine this essentially allows this two nitro cellulose or nitroglycerin to be compacted together. So, it gives like the binding agent it works like the binding agent to keep them together. At the same time you need some kind of stabilizer to control the burning rate the stabilizer controls the burning rate. So, it is like works like a damper. So, that you can get a required burning rate. So, diphenyl amine is used as the stabilizer with this propellant. So, the additives gives it better fluidity. So, it burns at a constant rate and also it is less sensitive from a to ignition from impact. Now, ignition as I have mentioned that if it is ignited it is almost impossible to stop it. So, therefore, ignition should not be starting on its own. So, this two will actually provide a threshold of ignition energy. So, that so that from the ignition system we have to provide at least more than that energy. So, that the ignition can start. So, therefore, essentially ignition cannot be started by accident that is the idea of putting these things. So, this all four of this will be mixed together and processed and a propellant rod will be created and then this rod will be put inside the case and when ignited this rod will burn depending on the exact design and the mixture ratios at a particular rate which will give us the required pressure and temperature. So, now the second type of propellant is a composite propellant. These propellants are typically heterogeneous propellant that is they do not have a smooth mixture everywhere mixture fraction everywhere, but there will be local non-homogeneity embedded in a homogeneous mixture. So, therefore, this is a heterogeneous propellant. Typically, oxidizer which is crystalline in nature like say ammonium per chloride which is a solid oxidizer will be embedded. So, this is an oxidizer, but it also works as a fuel which I will talk about that that it breaks into two components one ammonia and one hydrogen chloride. So, ammonia is the fuel, hydrogen chloride is the oxidizer, but it is predominantly works as a oxidizer. So, therefore, in the presence of this sorry in the presence of the ammonium per chloride the combustion process will be fuel lean because they are adding more oxidizer. So, in the vicinity of those propellants you have a fuel lean combustion going on. Then typically, polymeric fuels are used for this type of application that is a composite propellants which requires a binder. It consists of some hydrogen, carbon and oxygen atoms. Typically, the different type of propellant use are P-Ban which is polybutadiene acrylic acid acrylononite trial or CTPB, carboxyl terminated polybutadiene or HTPB which is hydroxyterminated polybutadiene. These are typically the propellant which are the polymeric fuel that are used as composite propellants. Apart from this, so the polymeric fuel the oxidizer sometime we also embed some metal powders into this propellant this this mixture. So, this metal powder typically aluminum is used as the metal powder. Aluminum and alloparticles will be used as the metal powder which will be mixed along with this composite. So, that you get actually more in non-humidinous heat, more heterogeneity rather. So, it will be like a grainy structure. Now, the advantage of this metal powder is that when the metal burn, they burn in a highly exothermic manner. So, therefore, the energy density increases and the energy release is enhanced. So, this metal powders will be burning giving additional energy. So, therefore, this composites will have a higher energy density compared to the double base propellants because of this structure. Then, the third type of propellant is composite modified dual base. Here in double base sorry here in addition to the ammonium percolate crystal sometime even some explosives are added. So, we add some ammonium percolate to double base propellants which we have already discussed. So, that changes this double base to something like a composite. On top of that sometime even some explosives are added. So, therefore, the reaction rate is even more. Reaction rate is enhanced and first of all the double base propellant typically burns like in a diffusion flame because although they are mixed in the molecular level this is not premised because it is solid propellant. So, it has to for vaporize and then the vapors will mix and then burn. The presence of ammonium percolate gives the local fuel leanness. So, as I said this is oxidizer. So, locally it will be fuel lean. So, therefore, the overall mixture will be less fuel rich. So, having the ammonium percolate mixed with a double base propellant reduces the fuel richness. So, moves it towards more complete combustion because of that the specific impulse improves. So, we get a better specific impulse with this type of propellants. It increases the rate of burning because of explosives are used or also the apex crystal will burn at a particular rate higher than the double base propellant burning rate. So, because of that there is a increased burning rate also and these are more rugged compared to composite propellants typically used in missile propulsion. So, this type of propellant which is a composite modified double base propellants are typically used in missile propulsion. So, this is the third type of propellant we talked about. Now, let us come to the fourth type of propellant which is the nitramine propellant. These actually contain high explosives like RDX or HMX high explosives are added to the fuel binder. Typical binder is HTPB. So, HTPB we have discussed in the case of composites hydroxy terminated polybutadene this is the binder. So, in this binder you have RDX or HMX added. So, these are explosives. So, therefore, that makes it more dangerous to handle, but if they are burned in a controlled manner they will provide much higher energy density. So, the hydrocarbon gases from the polymer combines with the fuel rich products of explosives to form low temperature products. So, the explosive will not have the freedom to burn on its own because of the presence of this binder. So, that kind of dams out the explosiveness of the explosive and gives a controlled burning which essentially helps in producing the required condition for our thrust generation. So, the carbon dioxide and water vapour are absent in this mixture. Typically as you know that most of the hydrocarbons when they burn because they have oxygen, hydrogen and carbon many times they produce carbon dioxide and water vapour, but in this type of propellant carbon dioxide and water vapour are absent. Now, what does carbon dioxide and water particularly water vapour does? It is a very good signature of IR infrared. So, typically the identification of missiles path is by the IR signature, infrared signature. So, if you do not have water vapour the IR signature will be drastically reduced. So, therefore, you have much lower IR radiation. So, therefore, it is very difficult to detect this missiles because they are coming at a high speed and if you do not have these detectors to detect the IR signature is very difficult to detect them. So, therefore, these are used for strategic applications. They have a fairly good stealth capability because of the low IR signature or low IR radiation. So, this is a propellant which is typically used for strategic applications in missiles. Now, here is a table giving the characteristic of some of the propellants. So, D B stands for double base, then D B A P A I is double base with aluminium percorate with aluminium percorate with alumina solid particles. So, I will just show you few features. First column gives the ISP. You see that typically the ISP for all of them range between about say 180 to about say 270. So, about 250 to 260 range and typically range of about 10 among all of them. So, this is fairly close. Flame temperature however, if you look at the double base is about 4100 Fahrenheit, but for the double the composite one it goes to as high as 6500 and then with the embedded HMX it goes to 6750 Fahrenheit. That is a fairly high temperature and because of that high temperature particularly if you look at the burning rate because of the presence of this aluminium percorate and alumina. If you compare the burning rate of a double base and a double base with aluminium percorate the composite you see that there is a fast difference in the burning rate. Burning rate is much higher here. So, these are the different propellants which are traditionally used and different conditions for all of them. Now, this pressure exponent n I will discuss this because it is part of the burning rate equation which will come up when I talk about the burning rate. So, these are various propellants which are used extensively and the properties of those propellants are listed here. Now, next let us talk about the mechanism of burning. How do the solid propellant rockets burn or the solid propellant burn? First let us start with a double base. So, here you can see a schematic of a double base propellant. So, is a nitroglycerin and the other propellant nitrocellulose and nitroglycerin. In this stands for nitroglycerin, in this stands for nitrocellulose plus the additive. So, this is our double base propellant which is fairly homogeneous. Now, in the combustion of this propellants like we have discussed earlier in various courses in combustion courses that any solid propellant goes through a particular process of burning. The first process is pyrolysis where because of the supply of certain energy the propellant starts to evaporate from the surface. But in pyrolysis like say pyrolysis of coal or wood it is a full hydrocarbon and the oxidizer is in the atmosphere. So, when it evaporates it mixes with the atmospheric oxygen and then the combustion takes place. But in this double base both the oxidizer and flue are together it is not reacting with the atmosphere. So, therefore, the combustion mixture will be formed right next to it without any outside help. So, first you see this is the propellant the first zone here is the foam zone. Here in the foam zone the propellant surface degrades exothermally that is because of when we start the ignition process the propellant surface absorbs this energy and then it starts to change its phase. It changes from solid phase to the combustible gas phase. So, the combustible gases are released in the foam zone, but combustion does not take place there. Combustion takes place slightly ahead of it which is called a Fies zone. In the Fies zone which is the zone of gas phase reaction. So, you can see here this is the zone of gas phase reaction or Fies zone. So, reaction takes place here because we have both the fuel and oxidizer present in the vapor the mix and burn. So, in the Fies zone the actual reaction takes place. So, if you look at this bottom plot the temperature is here same then close to the surface there is slight increase then in the Fies zone there is a substantial increase in the temperature and it goes to a temperature T 1. So, I will discuss this T 1 also later. So, there is a zone of gas phase reaction which increases the temperature as I just discussed and this zone is highly luminescent. So, most of the flame color comes from this zone. Now, luminescent means that most of the radicals are present in that zone right the reaction when they occur most of the radicals are present in that zone. After that after this luminescent zone there is one zone called dark zone. In the dark zone which is which will be typically present if the pressure is less than 10 mega Pascal there is not much significant increase in the temperature that takes place only there is some kind of convection of this propellants take place and more mixing will follow. So, after that after this dark zone where nothing much happens we have the secondary luminous zone or secondary combustion zone where this molecules as you can see here NOCO or NO NH 2 these are heavy molecules they further break down into more fundamental molecules like carbon dioxide water vapor etcetera. So, this is the secondary luminous zone here the chemical reactions further proceed and get completed. So, by that time we come to the end of this zone the chemical reactions are completed and all the heat is liberated because of that as you can see here it attains the final temperature at the end of this zone. So, this is the mechanism of burning for a double base propellant. Now, let us talk about the estimation of burn rate at what range it is burning because later we will see that the pressure and temperature will depend on this burning rate. So, the burn rate is typically estimated by energy balance at the propellant surface that is at in the phase zone mostly at the foam and phase zone phase zone will consider a one dimensional process. So, this is here you can see a propellant is given here at x equal to 0 this is my phase zone. Let us consider a small control volume in this phase zone at a distance x from the propellant surface having a width of d x and with the unit cross sectional area for this propellant. Now, this propellant is burning as the burning takes place there is some amount of heat that will be released. So, this heat release is given here by q dot cam this is the heat released by the combustion process. So, it is given by A p to the power m e to the power minus e by r naught t. So, this comes from Arrhenius law which represents the rate and then this represent the magnitude of the heat release. So, this is the total heat release rate that will come out from this expression t is the pressure this is one index which will be has to be estimated a is a constant again that needs to be estimated this e is the activation energy r naught is the universal gas constant and t is the temperature. So, this is the volumetric heat release rate for this propellant which is burning apart from that what are the other energy transfer taking place. First of all there is conduction of heat into this control volume and there is convection of enthalpy there is a change in enthalpy across this control volume right. So, now if I do a full energy balance of this process that is occurring then what we have is first of all let us identify three sources of energy one is other the three changes in energy not sources one is the net heat conducted into the control volume. So, the rate of heat conduction will be given by Fourier's law which is kg dt dx where kg is the kg is given here thermal conductivity. So, therefore the rate of conduction of heat will be given by the double derivative of that now you can derive it also if I go back here in the left surface energy the conduction is given by minus kg dt dx and what is leaving is this amount that is this plus a slight increase right. So, difference of this two that is this amount is the net rate of heat conduction into the control volume. So, that is given by this term then next what we have is the net heat generated in the control volume due to the exothermic chemical reaction that is taking place. So, that will be equal to q dot dx time 1 because the area is 1 in the depth right is a unit area. So, therefore the surface the total volume will be given by 1 times dx. So, q dot time that will be the net heat generated in the control volume. So, now this heat conducted plus the heat generated is the net change of enthalpy from energy balance. So, that is the net change of enthalpy of the gases flowing through the control volume. So, how do we get the net change of enthalpy? This term is the enthalpy of the gases flowing out where rho g is the density of the gases u g is the velocity of the gases c is specific heat and t plus d t is the temperature at the right surface. Similarly, rho g is the density of the incoming gases u g is the density of the incoming gases velocity of the incoming gases c is the specific heat and t is the temperature of the incoming gases. So, therefore this term in the right hand side is the net change in enthalpy. So, now this is our expression for enthalpy balance as you can see this is a differential equation. So, you have to solve this differential equation with certain boundary condition to get the details of the process. So, this is the formulation then this is our governing equation given here which is that is k g d square t dx square plus rho g u g c d t dx equal to q dot c. Now, this q dot will come from the chemistry which we have already discussed right. So, now we have we can solve for the temperature field from this. What are our boundary conditions? First of all the temperature at the surface is T s at x equal to 0 at the surface temperature is T s at a distance of L which is the edge of the phase zone the temperature is equal to T 1 right. So, therefore these are the two boundary conditions with these two boundary condition we need to solve this governing equation. So, I am not going into the details of solution just just look at few specific specific features of this formulation. First of all the gas phase velocity that is u g is a function of heat transfer to the propellant surface leading to vaporization right. So, the rate of the rate at which is vaporizing and the gases are moving is my u g. So, therefore is a function of the heat conduction to the surface right and from mass balance at the surface if I do a mass balance at the surface the propellant density times the burning rate gives the mass flow rate of the propellant from the surface. Now, this mass flow must be conserved therefore the mass flow rate of the gases moving out should have the same value and what is the value for the gases moving out this rho g u g because rho g is my density of the gases u g is the velocity. Therefore, this equation must be satisfied at the surface where r is my burning rate and this is what I want to find out. So, let us look at burn rate r first of all this r is a function of pressure temperature and activation energy E. So, therefore as you can see from here that q dot will depend on this or this depends on q dot vice versa right. So, now the solution of this entire process is quite involved I will not go to the details of that, but typically it has been seen that r that is the burning rate is given as this A p to the power n where A is a constant depending on the particular propellant we are using p is the pressure and n is the index that gives the burning rate. So, the table that I have let me go back to the table I was discussing this pressure exponent n see here this pressure exponent n is that n that I am talking about now. So, that is typically obtained experimentally. So, this is the expression for the burn rate now r equal to A p to the power n this relationship is called valley's law or saint law bus law this dictates the burning rate of solid propellant typically the double base propellants. Now, here A is a function of initial temperature composition and fuel properties and n is a function of pressure typically. Therefore, it depends on the combustion process this A and n are typically obtained by fitting experimental values of r at different pressure. So, how do we get it we conduct experiments at different pressure measure the burn rate or estimate the burn rate and then do a curve fit to get this relationship and from there we estimate A and n that is how typically the exponents are estimated. So, this is the burn rate of double base propellants next let us look at the composite propellants. So, burning mechanism first for the composite propellants is little more complex here you can see a schematic representation of the burning of a composite propellant A c is the hydrocarbon. So, this is the main fuel it can be double base fuel and all A p is the ammonium per flow rate which is embedded in it. Now, A p itself first of all because of this as you can see that the burn rate is not constant everywhere. So, locally some higher burning and then other places there are lower burning and finally, away from the surface we get fairly constant flame, but in between there is a lot of inhomogeneity in the entire process because of the heterogeneous nature of the propellant. So, since the propellant is heterogeneous we have distributed burning. Now, first of all let us look at the ammonium per flow rate. This ammonium per flow rate like I was mentioning sometime back is actually a mono propellant it can burn on its own. So, when it is burning it will decompose into ammonia and A c l o 4. So, this after burning of A p we produce this and now because of this being a mono propellant it can burn on its own. So, it is like a premixed flame right. So, when it is heated it will burn with releasing this vapor in a premixed flame and in this flame this is a fuel lean flame. So, therefore, it is a oxidizer rich gases will be released after it is burned and typically this temperature will be about 1300 Kelvin. Now, let us look at the ammonium per flow rate. As I have mentioned earlier that this is a mono propellant. Therefore, when it it can burn on its own and decompose into ammonia and A c l o 4. So, since it is mono propellant burns on its own it is like a premixed flame because it is mixed in the molecular level and burn on its own. So, this is a premixed flame is produced when burning the this propellant and this typically gives an oxidizer rich gas or a fuel lean rich gas fuel lean gas sorry act 1300 Kelvin. Now, this oxidizer mixes with the hydrocarbon vapor which is produced by evaporation of this hydrocarbon from the polymeric bond binder and a stoichiometric fuel air mixture is created slightly away from this surface and therefore, now at this surface a prime it burns in a diffusion mode. So, a primary diffusion flame is created and the combustion is controlled by diffusion and mixing process as it is seen here temperature will be increased to as much as 2800 Kelvin. And now the product of this diffusion combustion and this premixed combustion finally, mix to give us this final flame which is the final diffusion flame sitting slightly away from it downstream from this is the final diffusion flame is created where everything else whatever is remaining unburned will burn and the temperature will go up to as much as 3200 Kelvin. So, this entire burning zone is about 0.1 mm thick and the thickness decreases with increase in pressure. So, this is how composite propellant will burn. Now, the burning rate of a composite propellant is simplified here that represent a simplified manner here, but this is the surface of the propellant and this is where the flame is located the T x is the surface temperature T f is the final temperature this x is the distance of the flame from this surface and this is the variation of temperature. The burning rate is given by this expression again the burning rate is estimated by energy balance similar to the process we have discussed earlier. So, this is the burning rate is given by this expression. So, the all the three frames are modeled as a single flame surface with the final temperature T f. T i is the initial temperature of the ammonium chloride combustion which is appearing here and the energy balance once again as I was mentioning that conduction plus heat of reaction is equal to the change in enthalpy. So, for this also it can be shown that R is equal to A p to the power n. So, the terminal functional representation is present only this values of A and n will be different right for the different propellant or the composite propellant, but the basic mechanism is like this and therefore, it satisfies this basic functional relationship as well. Now, let us see that how do we use this information for actual rocket design. So, thrust produced by a rocket f is given by C f P c naught A star where C f is the thrust coefficient P c naught is the chamber cum stagnation pressure A star is the throat area mass flow rate through the throat is m dot is given by P c naught A star C star where C star is the characteristic velocity we have discussed this in detail. So, I am not going into it again now however, the mass flow rate due to the propellant burning is the propellant density times the surface area of the propellant times the burning rate of the propellant. So, surface area of the propellant is given by S p the burning rate of the propellant propellant R on the other hand as we have just discussed is equal to A p c naught to the power n because p c naught is my combustion chamber pressure. So, I can replace it here equating the mass flow through the nozzle and the mass flow due to propellant burning for steady rocket performance we can get this expression. By the way here this S b is the surface area of the propellant is same as this S p. So, these two are same S b and S b here I have written as S b because I just wanted to show that it is the propellant now this is the burning area S b is the burning area. So, these two areas are same. So, this is the expression just by equating this and this I get this equation. So, this is the chamber pressure as I like I was mentioning at the beginning the chamber pressure is dictated by the burning rate etcetera which now you can see here and where n is the exponent of this. So, this should be small for reduced influence of parameter on stable operation typically n should be less than this value that is 1 upon 1 minus rho g by rho p it should be less than that. So, now once we have this p c naught we can put it back here in this expression and get the expression for the thrust. So, now in this expression we have the burning rate properties or the characteristics also embedded included. So, this gives me the equation for thrust produced by a solid propellant rocket burning at a certain rate. The rate at which the propellant will burn depends on this surface area right. So, as we can let us go back to the previous slide this S p or S b they will dictate at which rate it will burn right. So, therefore, identifying the effect of this this surface is very important and that depends on the configuration of the grain that is why grain configuration plays an important role. So, thrust is a function of surface area we have just discussed. So, higher surface area means higher thrust and chamber pressure is also a function of surface area. Burning of the propellant take place normal to this surface. Now, we can have three type of burning we can have a neutral burning where the surface area is constant and because of that constant surface area we get constant chamber pressure and typically burning from the end is will give that because if surface area is constant then the chamber pressure is not going to change. So, this is the neutral burning. On the other hand we can have burning called progressive burning in which the surface area the burning surface area increases with time and as the surface area increases this increases the chamber pressure. So, chamber pressure becomes a function of time typically this type of burning is seen in hollow cylinder burning out wide from inside. So, the combustion starts from the inside of a hollow cylinder and burns outward the surface area will increase and that lead to an increase in chamber pressure. We can also have of course, is very rarely seen regressive burning where we have a decreasing surface area and because of that a decreasing pressure we know that we want to operate always at higher pressure. So, this is something very rarely used a decreasing pressure. This is if we have a radial inward burning we start to burn from the outer radius and burn inside that will give us a regressive burning. Now, with this background of gain for configuration let us look as some of the gain configurations which are typically used. So, here I have some schematics of certain gain configurations. This is for a neutral burn end grain completely filling up then we have a dog bone where we have a shape like this. This will also give us neutral burn this internal burning tube where we have a hollow cylinder going in this gives a progressive burning. So, it will start to burn from here the area will increase. So, this is progressive burning. Then we have slots and tubes also give neutral burning internal external burning tube neutral burning slotted tube also neutral burning rod and tube also neutral burning wagon wheel neutral burning internal burning star also neutral burning or multiple perforations. So, as you can see here more often than not we actually go for neutral burning why because then my chamber pressure remains constant and is chamber pressure remains constant my nozzle behaves properly. In the chamber pressure starts to change your nozzle will not be behaving or producing the required thrust because the nozzle conditions inlet conditions keep on changing. Therefore, it is always advisable to go for a neutral burning. So, that the chamber pressure remains constant. So, that is why you see that most of this configurations give us neutral burning. Some of the gain designs as seen shown here if you have tubular that variation of thrust as you can see with time will change if you have a progressive burning the thrust will increase with time. On the other hand if you have this configuration is a neutral burning for a quite a bit of period like you can see here in this schematic then you can have a rod and tube again a neutral burning for a quite a bit of period. Of course, in the initial and final stages will not be neutral is progressive and regressive, but otherwise it is neutral. Then sometime we need double thrust or dual thrust as I was saying the thrust can be programmed in the grain design. You can see here this multi fin gives us a dual thrust. Initially high thrust then a lower thrust which is because of this grain configuration and how does this vary because the chamber pressure varies. Then we have a double anchor configuration this gives us regressive burning and we can have a dual composition which gives us two-step thrust. So, by properly designing the grain we can actually program the thrust history required to attain a particular mission. We can get different type of thrust at different time instance. So, that is I can see here for different grain configuration will give us that. Next let us look at the variation of chamber pressure. The chamber pressure depends on initial surface area S B we have just discussed about the grain configuration, burning rate and propellant type. So, essentially the characteristic velocity will be dictating what will be the chamber pressure as well as the nozzle throat area. So, this expression we have already talked about gives us the chamber pressure variation with different parameters. Now, next let us look at the burn time. How much time it will take to complete the burning of a particular grain? This is obtained by integrating the burn rate R with respect to time. So, pressure term can be replaced in the burn rate equation the pressure term can be replaced by the expression for P c naught. So, for a hollow cylindrical grain where n naught equal to 0.5 we get this expression for the burning time. So, this is the time it will take for it to burn out starting from the initial diameter D i to the final diameter D f. This will the time that will be taken by this entire grain to burn out completely. So, this is the duration where the reaction is or the burning is taking place. This is not applicable to n greater than or equal to 1 and because of that stable operation is not possible if n is greater than 1. Because this gives us a stable operation for n equal to 0.5 this is the expression for the burning rate. Why it is different from this? Because if you look at this expression n equal to 0.5 will make this term 0. So, T b will be infinite. So, therefore, it is a singularity there. The singularity is removed and we get this expression for the burn time if n equal to 0.5. So, this is the burn time that will be essentially a function of P c naught also. Now, so this completes our discussion on the complete burning process. I mentioned that one of the component is the igniter. So, let us now discuss the igniter. Igniter essentially helps in achieving the ignition process which is achieved by transferring heat energy to the propellant surface to take the temperature of the surface to T i which is my ignition temperature. So, that requires additional source of heat energy. Now, this heat energy can be obtained by combustion of easily burnable mixture or it can have potassium nitrate, potassium nitrate, charcoal and sulfur like the black powder. We can also add some of the metals like aluminum, boron, copper into it to give further enhance the heat release. So, igniter is something that is a device which will initiate the ignition process or the combustion process or burning process. So, these are typically type of igniters which are used. One is a pyrotechnic igniter which is a long tube and on one end we have end is sealed with hot polyglu. Inside this we have this ignition mixture or ignition powder. It can be gunpowder and then 2 you can see this is a nichrome wire is placed here and it is connected to the power supply. When you supply electricity the nichrome wires get heated up and it starts to burn and then it burns at a rapid rate providing the energy to the propellant to burn. So, this is my igniter. So, combustion of ignition charge is initiated electrically which is essentially a nichrome wire. This is also called squib. So, it is a squib ignition. Squib is enclosed in another easily ignitable substance like KCL. Rapid combustion of this KCL initiates the burning of this pyrotechnic substance which is given here. Now, why we need this is this is a particular threshold current is required only then it is the ignition will start. So, it is also a safety device. It is not only provides the ignition requirement, it is safety device unless we have that current it will not ignite. So, this is the pyrotechnic igniter. It prevents accidental ignition also. This is most commonly used igniter. We can also have pyrogene igniter which is nothing, but a small solid propellant rocket in itself. Pyro technique is a separate it is not a rocket. Pyrogene ignition on the other hand is a small rocket in itself where we have everything all the components of a rocket. There is a small propellant charge as you can see the ignition powder is an this is a burst diaphragm it will burst when the ignition pressure which is a particular value and then it will start to ignite the entire process. So, it is used in large solid propellant rockets because it requires because those requires large amount of ignition charge. So, a small solid propellant rocket is used to ignite the to get the initial ignition and plume of the solid rocket initiates the ignition as this seen here and this small rocket is ignited by pyrotechnic igniter. This is a pyrotechnic igniter sitting here and this is a small rocket and the different things here present. So, this will be ignited by this and then this on the other hand will ignite the main rocket. So, that is the pyrogen igniter. Now, how does ignition takes place? This is a very complex process is a multiple physical and chemical processes take place. We have heat transfer to the propellant again grain by the conduction convection and radiation then heating of propellant to release fuel and the oxidizer vapour then pyrolysis of propellant and its decomposition then generation of heat at the surface due to chemical reaction. So, all this must be present to give the proper ignition essentially is a three step process. First we have a local ignition then the frame produced by this local ignition spreads. So, it is a frame spread and then finally, the entire chamber is filled to equilibrium pressure when the ignition process is complete we get equilibrium pressure in the entire chamber. Ignition time as you can see from here it depends on the flux heat flux. So, as the higher heat flux will give less ignition time also the chamber pressure at higher chamber pressure we get less ignition time. So, if a higher pressure means the reactions will be faster. So, we require less time for ignition. So, now what I have here is a complete schematic of an actual rocket. You can see here this is the propellant grain, this is the igniter, igniter fix in here is the propellant again and then everything is put together is a nozzle insert and arc enclosure makes it the complete solid propellant rocket. So, this is a complete solid propellant rocket in parts. Now, one of the most widely known solid propellant rocket is space shuttle solid rocket booster. So, this is a space shuttle solid rocket booster with all its components is perhaps one of the biggest rockets ever made SRBs is the largest solid propellant rotors ever flown. And it is a diameter is 14.17 feet length is about 150 feet is a huge rocket C level thrust it produces is 3.3 million pound and its weight is with of the with the propellant is about 1.3 million pound. So, this is a huge rocket it provides a 71 percent thrust at lift off of the space shuttle. So, you can imagine how big this rocket is the propellant mixture by oil it has ammonium perchlorate, aluminum, iron oxide, polymer, binder and epoxy cutting agent. So, it is a composite propellant and the propellant is a 11 point star shape in forward motor segment and double truncated cone in half segment of half segment. So, it is a two type of propellant grain that is used at two different locations. So, this is the full schematic of a space shuttle solid rocket booster. Then we have another one is a graphite epoxy motor which is give the schematic is given here again we have a graphite propellant and then the igniter this is a thrust vector nozzle and every component I will not go into the detail of that is you can see that is a different variant of the solid propellant rocket. Then another one which was used is Pegasus solid rocket. So, you can see the full Pegasus solid rocket picture here and it is Pegasus being launched from here. So, this is the solid propellant rocket Pegasus. Some performance data is tabulated here for various motor designation various chamber pressure and the burning time the thrust and ISP. You can see that for the solid propellants the maximum ISP is in this one Orbus 21 295.5 no sorry Orbus 60 303.8. On the other hand the maximum thrust is in this RSRM right 11,648 kilo Newton that is a huge thrust being produced. So, in general, but these are lower thrust and the chamber pressure as is given here. So, this is the full data data for most of the solid propellant rockets which are used widely and you can this is given just to give you an idea of what are the numbers numbers of thrust and specific impulse etcetera. So, this brings us to the end of this lecture I think I have over shoot the time, but so I have I have to acknowledge the first of all the book by Dr. Ram Moorthy and the some of the wave pages from which I got the material. So, with this I stop here and the next lecture we talk about the liquid propellant rocket which is another class of a chemical rocket and with that we complete our discussion on chemical rockets.