Welcome back. So, in the last class we talked about solid propellant rockets. We discussed various aspects of solid propellant rockets including the type of propellants, the ignition etcetera. Today, we are going to talk about the liquid propellant rockets. Now, as the name suggests the liquid propellant rocket essentially uses liquid as propellants. So, here we have a simple schematic of a liquid propellant rocket where liquid separate liquid fuel and oxidizers are stored in the fuel tanks. Then this fuels both fuel and oxidizer are fed into the thrust chamber through some pumps. There are valves in between to control the flow rates of this and the metering and atomization is done in an injector and then the combustion takes place in the combustion chamber and the gases are exhausted through the nozzle. So, that is the basic principle of operation of a liquid propellant rocket. So, if you see here in this picture we have two tanks fuel and oxidizer, then the pumps this pumps feed into this thrust chamber or combustion chamber, combustion takes place then we have this nozzle here and after burning or expansion through the nozzle the exhaust goes out with the velocity V e at the pressure P e and the exit area is P a e then the thrust produced from the thrust equation which we have derived at the beginning of this course is equal to m dot V e plus P e minus P naught a e where P naught is the ambient pressure P e is the pressure at this exit a is the exit area V e is the exit velocity and m dot is the mass flow rate. So, that is what in to it in a liquid propellant rocket, but it is not that simple in operation. There are a lot of engineering issues involved in liquid propellant rocket. So, before we go into the details first let us look at the little history of liquid propellant rocket. Liquid propellant rocket was first proposed in 1903 by Salski-Weski in his book the exploration of cosmic space by means of reaction devices in 1903 he first proposed the use of liquid propellant rocket. The first flight of the liquid propellant rocket dates as back as 1926 in past 16 March 1926 at Auburn Massachusetts by the father of modern rocket tree Robert Goodart. I have discussed at the beginning of this course about rocket Goodart and Robert Goodart and his contribution to rocket science. So, he essentially made the first liquid propellant rocket in 1926 on successfully using liquid oxygen and gasoline as the propellant. So, after Goodart's success the German engineers and scientists become installed with this technology and they started to design better liquid fuel rockets. So, from 1926 till we come to the first second world war there has been a rapid development in the liquid propellant rocket because this was identified as the most promising technology to make a rocket viable. So, therefore, the first practical rocket which was ever used was V 2 rocket developed by Germans. So, this was first actually utilized rocket which was the first ballistic missile short range ballistic missile a picture of the V 2 rocket is shown here. The propellant of this rocket were to the main combustion chamber we had liquid oxygen and alcohol which is alcohol was 75 percent ethyl alcohol and 25 percent water mixture was given to the main combustion chamber. To the gas generator which essentially allowed the compression of the gases to put into the combustion chamber or the turbo fan to run that sorry to run the turbo pump that used 80 percent H 2 O 2 and Na Mn O 4 H 2 O. This was able to take it the temperature to 685 Kelvin this run the pump which fed the propellant to the main gas chamber. So, I will talk about different type of liquid rockets then I will explain this more. So, essentially the propellant were liquid oxygen and alcohol. Now, this was successfully fired during the second world war after the second world war this German rocket technology was transferred to various allied countries like Soviet Union, United States and UK etcetera. So, R 1 was the Soviet version of V 2 rocket US a reassembled V 2 at White Sand in USA and similarly UK also reassembled U 2 rockets. So, this is a full picture of a schematic of a V 2 rocket that was used during second world war and the technology so far has not much changed. Ethyl remains almost similar technology where all the main ingredients are components are already there in V 2. So, there has not been much of technology advancement beyond this as far as the rocket part is concerned only this the component efficiencies were increased, but the basic schematic remains same. So, what are the advantages of a liquid propellant rocket? First of all its high specific impulse compared to the solid rocket as we have discussed in the previous class they have much higher specific impulse going up to as much as 450 seconds. On top of that they are fairly high thrust because the liquid propellants are pretty high energy density as well because of that we have a high thrust to weight ratio. The biggest advantage of a liquid propellant rocket is easy throttling or easy to control. We can control the propellant flow rate that is the both fuel and as oxidizer flow rate by using control valves which are used in line after the pump and the combustion chamber we can control the flow rates. So, that essentially gives us control over the entire energy your heat release rate because we can change the mass flow rate we can have control over the entire heat release rate or the heat flow rate produced by the combustion process. And because of this it is very easy to abort the mission also we just need to cut off the supply of supply of fuel and the combustion will stop. So, mission can be very easily aborted it can and another thing is that unlike liquid rocket solid propellant rocket once you create the grain if you fire it is gone you have to produce another grain to test the system. Whereas liquid propellant rocket can be tested before the fire also we can still refill the tanks and test the system. So, the testing of the system is possible which is not possible in solid propellant rocket solid propellant then you have to make a new grain and burn it again. Whereas liquid propellant rocket you can just have to need to change the fuel and oxidizer and test the system. So, that advantage even gives us flexibility to test different type of fuel oxidizer combination also. So, the development is much more scientific and it is very very reusable this technology these rockets can be used again and again and because of that space shuttle main engine uses a liquid propellant rocket. So, we can have the main thrust chamber everything pump and everything can be written except the tanks can be thrown out if needed otherwise you can written the tanks also, but typically tanks tanks are thrown off. So, just tanks need to be put in and the rocket remains the same. So, now different type of liquid propellant rockets are possible first and foremost is a mono propellant rocket where we have a single propellant which essentially is a mixture of both fuel and oxidizer. Typically this type of propellants are hydrogen and hydrogen peroxide and of course, some catalyst is required to start the initiate the ignition process. Typically it is a granular alumina coated with iridium is used as the catalyst. So, this is a mono propellant rocket which will mono propellant means the propellant is single is a the propellant itself will burn when energy is provided. So, the rocket system since it consists of fuel tank now unlike the bi propellant system we can have a single fuel tank right single fuel and oxidizer we do not need two tanks we need a single fuel tank, but since these are highly reactive you need to have some special lining. So, ethyl propylene rubber coating is given inside this or a surface tension propellant management device is used which is filled with the fuel and then this tank is specialized by an inert either helium or nitrogen which pushes the fuel out of the out to the motors and then the pipe leads from the tank to a valve and then to the thrust chamber of the rocket motor and then this propellant when fed into the rocket chamber it will burn on its own. So, this is a mono propellant system other system is a bi propellant most widely used liquid rocket are bi propellant. So, that is why I can see there are as you can see there are various rockets listed typically propellant likes a cryogenic propellant liquid oxygen liquid nitrogen is used or we can have liquid oxygen and kerosene which is semi cryo then we have liquid oxygen and alcohol nitric acid with di nitrogen trioxide and kerosene then arosine 50 then UDMH then MMH all kind of MMH and di nitrogen trioxide UDMH and di nitrogen trioxide all type of different combination of fuel and oxidizers are present. So, different countries this use different propellant for example, isro isrosine which is a propellant fuel developed by isro with liquid oxygen. So, you have a semi cryo engine liquid oxygen and isrosine burning together. So, therefore, different I have listed here different rockets which use this propellant as you can see it pretty much covers most of the rockets used over the world. So, again every depending on the mission requirement all this propellants have their own energy density own ISP. So, depending on the mission requirement you choose which one to use that is why you can see that I have a big list of different type of rockets and different type of propellants. Now, let us look at how this work in reality how this liquid propellant rockets work in reality. We talk about the engine cycles there are typically four different ways of powering the injection of the propellant into the chamber and that essentially dictate what type of cycle it will follow. One is a pressure feed cycle, then an expander cycle, then a gas generator cycle and stage combustion cycle. So, these are the four type of cycle cycles that are used to feed the injector into the combustion chamber alright. So, now let us look at one each of this one at a time first a pressure feed cycle. Here is the schematic of a pressure feed cycle this is our combustion chamber the fuel tank and oxidizer tank and we have the control valves here what we have is a pressurized gas it can be helium it can be nitrogen it is stored at a high pressure into a container and this is fed into the fuel and oxidizer to push it full push the fuel and oxidizer at a high pressure into the combustion chamber. So, the propellants are forced in form of in from pressurized tanks which are relatively heavy tanks and because of the heavy tanks relatively low pressure is optimum we need to have and typically the pressurization is done by helium due to is lack of reactivity because it is fairly inert. If the fuel and oxidizer are hypergolic they burn on contact immediately otherwise you need to have an igniter to burn the fuel and oxidizer. The space craft attitude controls small rockets used for space craft attitude control or orbital maneuvering orbital maneuvering thrusters are almost universally pressure fed designs because this pressure fed gas can be maintained up to a long time. So, it can be operated on the space also. Now, however there is a problem with this you must take enough care essentially during long burn hours or the long burn times to avoid excessive cooling of the specialization gas because as the gas comes out it expands and as it is expanding the temperature will go down. As the temperature goes down the cold helium will not liquefy but your oxidizer and fuel that may get frozen. It may freeze the fuel and oxidizer and if that happens the flow will be of course, impart right some flow will not be smooth it will not come the same flow rate will not be maintained. Similarly, it may damage the components like valves etcetera and this devices are not designed for low temperature application. Now, to be a system safe from this problem is to have a heat exchanger in between. As you can see this is the heat exchanger. So, it will be wrapped around the combustion chamber it absorbs some of the heat from the combustion chamber so that the helium temperature does not go to very low levels. In that case this problem of freezing will not occur but that precaution must be taken otherwise there is a practical probability or chance of having the fuel or oxidizer frozen then it is not going to work. So, this is as I said most widely used during in the spacecraft or attitude control or orbital maneuvering thrusters. Now, next let us look at the another cycle which is called expanded cycle. At the name studies there is some kind of expansion taking place in a expanded cycle. So, here what happens here is the schematic fuel comes through a valve and then it is channelized all around the combustion chamber and the nozzle. So, the fuel flows through this channels and absorbs some of the waste heat. Because of this absorption of this heat the fuel gets expanded and gets converted into gas. And there after that this gas is supplied across a turbine and because of that the turbine starts to rotate this turbine is connected by a shaft to this fuel pump. So, the pump starts to work and therefore, fresh fuel is sucked in. Now, the gas after it crosses the turbine is fed back into the combustion chamber. So, therefore, there is no wastage of fuel and everything is essentially burned. So, this is the cycle entire cycle of a gas expanded cycle. However, there is one problem with the cycle of this cycle because we require a phase change where the liquid fuel must be converted into gas in order for this to work otherwise the turbine is not going to work. So, because of this phase change requirement the operation is thrust limited. It cannot operate beyond a certain amount of thrust. Now, let us understand why that happens. How do we increase the thrust? For the same combustion chamber we do not change P c naught and T naught. So, only way we can increase the thrust is by increasing the nozzle right nozzle size. As we increase the nozzle size the nozzle surface area increases we need to have more volume to heat it up to rather more volume of fuel to absorb the heat produced by this right. So, the volumetric flow rate requirement increases and as the volumetric flow rate requirement increases the also the in order to produce more thrust you required to have more volumetric flow rate also. So, as the volumetric flow rate increases more energy has to be given out to convert it into gas, but that energy is no longer available right. So, because you have increased the amount of fuel which absorbs that energy. So, therefore, you do not have enough energy now to convert it to gas. So, beyond a particular point as we keep on increasing the thrust the conversion will stop rather the phase change will stop and as the phase change stops this cycle is not going to operate. Therefore, this limited thrust will be produced by this. So, there exist a maximum engine size of approximately 300 kilo Newton of thrust beyond which there is no longer enough nozzle area to heat enough fuel to drive the turbines and hence the fuel pump. So, this is the problem with the expander cycle. Next let us look at the third type of cycle which is a gas generator cycle. It is very similar to expander cycle only difference that we do not need it to gasify by heat transfer from cooling. Instead we have a separate burner a pre burner or combustor which is called gas generator. So, some of the propellant as you can see here some of the propellant and oxidizer is taken here from the oxidizer and fuel and burned in a pre burner. Now, the resulted hot gases from the pre burner or gas generator is used to run the turbine and then this turbine runs the fuel pump for fuel and oxidizer pump. So, that fuel and oxidizer come into and then they go into the chamber the heating is used to cool it and then, but it is directly fed into the chamber itself. So, this heated thing is not used to run the turbine. So, you do not need to completely converted into gas it can still remain in the liquid phase. So, pre burner actually eliminates the requirement of this thrust limitation. We can go to higher thrust by having a pre burner which is a separate combustion system. However, the gases after going through the turbine is exhausted directly into that ambient. So, that is why it is called an open cycle it is thrown away this gas is thrown away this is actually a combination of fuel and oxidizer. So, some of the you could have produced some of the thrust, but we are not utilizing that thrust. So, because of that the efficiency of the system is less than if you burn completely everything. Since, we are throwing out this which is not going through the nozzle we are using some of the efficiency. The gas generator turbine does not need to deal with the counter pressure of injecting the exhaust into the combustion chamber because that requires certain amount of pressure. This simplifies the plumbing and turbine design and results in less expensive and lighter engine. The main is disadvantage is the lost efficiency because we are throwing out some of this gas outside. Gas generator cycles tend to have lower specific impulse than stage combustion cycle. Next I will discuss stage combustion cycle primarily is less because this is the amount of propellant which is not participating in thrust generation right because of that the specific impulse is less. So, the fourth type of cycle is the stage combustion cycle. In stage combustion cycle which is again actually is combination of both the gas generator cycle as well as the previous cycle we talked about the stage combustion sorry what was that expanded cycle. This is the combination of expanded and gas generator cycle. So, here is the schematic. The liquid fuel comes in goes around the combustion chamber gets heated up, but it is fed into the pre burner. Oxidizer directly comes in here, part of the oxidizer is taken from the supply and put in the pre burner. Pre burner burns this fuel and complete all the fuel. Earlier in the gas generator it was not all the fuel which was going to the pre burner. Here all the fuel goes to the pre burner. So, as you see you can see this is a fairly fuel rich combustion going on and then it goes to the turbine and then the entire product is fed into the combustor again. So, nothing is lost and it is already a vapor phase now. So, your fuel is now coming as a vapor of product and fuel because some of the product is created here. So, this is the stage combustion cycle. Some of the propellant is burned in the pre burner and resulted hot gas is used to power the engine turbine and the pumps. The exhaust gas is then injected into the main combustor along with the rest of the propellant and combustion is completed. So, all the cycles gases are going through the combustion chamber therefore, overall efficiency does not suffer any loss. So, we have fairly high combustion efficiency. This combustion cycle is often called closed cycle because everything is going round and the cycle is closed as propellant produce products go through the chamber as opposed to the open cycle where some of these was discarded in the open cycle, some of the gas generator cycle some of it was discarded. Stage combustion gives an abundance of power which permits very high chamber pressure. We can go to very high chamber pressure by this schematic. Very high chamber pressure on the other hand means high expansion ratio nozzles are possible and because of this high expansion ratio nozzle we can get fairly good pressure at takeoff as well even at ambient pressure. We can get fairly high pressure in the combustion chamber to allow even takeoff because it is possible to operate at very high pressures and this nozzles give far better efficiency at low altitude because they are designed for high pressure operation. There are some research advantages with the stage combustion cycle. First of all the turbine conditions are fairly harsh because it is experiencing hot products coming from this combustion. So, turbine has to withstand these conditions. Secondly, the gases that is coming now to the combustion are hot gases. So, it requires more exotic plumbing it requires insulation and all to prevent heat from getting lost and also from thermally insulated materials so that it can withstand the high temperature because it is coming at high temperature right. So, therefore, the plumbing has to be more exotic and first last and another most important thing is that you require a very complex feedback control for proper operation of all these devices. This is a major problem with this stage combustion because everything has to be controlled in a proper way so that everything works in unison. So, that is the stage combustion cycle. So, I have discussed all four cycles. Now let us look at some other aspects of liquid combustion liquid propellants. One of the most important component of liquid propellant is the injectors. What are the injectors? The injectors actually atomize the liquid into the fine droplets. Today the injectors cost primarily of small number of small holes lot of holes but of very small diameter. This small holes aim the jet of fuel and oxidizer in such a way so that they collide at a point in the space a short distance away from the injector plate and because of this collision that the jets become unsteady and then they break into fine spray. So, this helps to break the flow into small droplets which are then easy to burn. They will evaporate faster, mix faster and burn faster. Actual performance of a rocket and its efficiency depends a lot on the injector performance because if you have poor atomization the droplets are big the combustion will be incomplete because then by the time the evaporation gets completed it will be almost outside the thrust chamber. So, therefore, the combustion will be incomplete. So, poor atomization will lead to incomplete combustion and that essentially means reduction in combustion efficiency. So, to have good atomization is very very important or essential for good performance of a rocket liquid rocket. Secondly, the injectors work in as a secondary application in reducing the thermal loads on the walls. The injectors actually spray the fuel and oxidizer in such a way particularly close to the wall to create a small thin film on the wall. So, the providing increased proportion of fuel around the edge of the chamber it reduces the chamber temperature the wall temperature of the chamber. So, that prevents the melting of the chamber material or the heat transfer out of the system the losses out of the system and then that is can get evaporated at burn. So, therefore, the wall the lower temperature on the walls of the nozzle can be obtained by essentially tailoring the spray in such a way. So, that we get a particular type of special distribution of temperature field injector can provide that. Now, what are the different types of injectors which are used? First is a shower head type of injector will I will discuss all of them in detail then an impinging injector which can be self impinging either fuel fuel or oxidizer oxidizer or cross impinging which is fuel oxidizer fuel. Then we can have swirling injector or pintal injector these are the different type of injectors used in rocket application.  This is there are some of the pictures of shower head injectors essentially it consists of various many many small holes which are oriented in a particular manner. This inject small streams of each propellant into the combustion chamber. The positioning of this streams across the face of the injector and the diameter of the streams will be such that the injectors mass should be evenly distributed across the face to reduce the horse powers. And secondly the stream should be small enough fast enough and at a particular angle best suited to force the atomization process. So, this hole design should take care of these two requirements how we align and locate these holes. The element testing shows that these elements are better for one type of propellant or other. That is we cannot take one of this injector element and use it for any type of propellant because atomization quality will depend on surface tension of the propellant or viscosity of the propellant or density of the propellant. So, therefore we cannot have a single shower head which will give best performance for all the propellant. So, therefore, this diameters or orientation need to be changed from propellant to propellant. Most important design criteria is that the element must maximize the propellant atomization. This is the most important thing. If the injected propellants are not automized fast enough then that will reduce lead to incomplete combustion, wasting of fuel and reduce efficiency which we do not want. And the orifice size is another issue here that the orifice size that is provided for this propellant we cannot have very large holes then the atomization quality is going to be poor. At the same time we cannot have very small holes either because in order to push the amount of flow rate through the small holes then you require a very large pressure drop. Now, the pressure is provided by your turbo pump. So, if you require a very high pressure drop you have to extract more power from the turbine which means that in their entire cycle the efficiency is going to go down. So, therefore, we cannot have very small holes either all the very small holes will give you better atomization, but at a cost. So, therefore, there must be a balance depending on the designer's choice there should be a balance between the size of the holes. It cannot be very small it cannot be very large. So, this balance must be maintained the velocity of the injected propellants should be fast enough. So, that the injector is not sensitive to chamber pressure oscillations. Now, this is very important the velocity at which or the delta p that is provided it should be such that the combustion chamber instability should not lead to feedline coupling I will discuss this in detail again. So, when there is a chamber pressure oscillation that should not be transmitted back and affect the fuel supply. So, that can be attained by having a high enough velocity of the flow of the fuel. Next, let us look at the impinging type of atomizer. This is a schematic of impinging type of atomizer which is also called doublet atomizer in this one. As you can see there is a pair of holes here 1 2 1 2 together you can see here 2 holes are placed side by side. It is a schematic here these are the injection holes. So, this is the most widely used injector here the atomization is due to the impingement of liquid jets. So, the fuel and oxidizer jet as you can see here are set at an angle. So, that they strike in space and then because of this collision they break into fine droplets. So, this is designed to provide best atomization and once they collide the resultant will flow is in this direction. So, it gives the axial flow also. So, it starts with angular flow then gets into the axial direction after the collision. So, this is the schematic diagram of that. So, you can see there is a picture of doublet atomizer working the 2 jets coming and colliding making the fine spray and then moving in the axial direction. So, this is the by far the most commonly used rocket injector and the fuel and oxidizer are supplied through manifolds in the inside the plate and then they come out and create the spray. The third type of atomizer is a swirling atomizer. This in this atomizer liquid is atomized by imparting a tangential motion. So, the tangential motion can be imparted by either providing a tangential inlet and then a settling chamber. So, that the liquid gets a tangential motion or passing the liquid through a helical passage as it is done here in this atomizer is a helical passage. So, the liquid passes through the helical passage gives a tangential velocity. So, when it comes out it has a tangential component of velocity. So, it spreads out like this. So, this is the spray typically a holocone spray is created. So, because of this spreading out of the liquid surface is an increase in surface area and this leads to a breaking of liquid sheet into droplets. Typically, holocone sprays are created, but this holocone spray is created after a particular pressure is given. After a particular pressure is applied across the injector that you create a holocone spray. At lower pressure the cone will not break up you will not get fine spray. So, fairly high amount of pressure is required to produce this type of a fine atomization from this type of atomizer. So, this is very similar to a pressure atomizer or pressure soil atomizer. Then the other type of atomizer is a spindle injector or spindle atomizer. This was first used on a flight verica during the Apollo program in lunar excursion module descent engine. It is a coaxial injector as it you can see here. The oxidizer comes here this is the oxidizer passage we have a spindle or plug sitting here. Now, this plug does not completely close the passage, but a small space is space remains and then the oxidizer fade under pressure goes out through this passage. The fuel on the other hand as you can see here is fed through a small passage directly into this. So, the fuel and oxidizer interact or collide with each other at this surface and because of this the atomization takes place. Both of them are coming at a fairly high pressure. So, therefore, they have a high momentum. So, they break in collision. The flow rate can be controlled by moving this spindle up and down. Now, if you have a spindle moved in the passage area decreases that increases the that it increases the velocity. So, atomization become finer. If you take it out the passage area increases the velocity decreases that atomization becomes coarse. So, you can have a control of the atomization property also. So, if the propellant is the B and B propellant B is the fuel and A is the oxidizer. The printel arrangement can be set up to get fuel flim cooling of the chamber as well. We can design in such a way that it can use the fuel flim cooling as well. The printel atomizer the biggest advantage is allowing deep throttling because we can just move the printel and get different flow rate also. So, it allows for deep throttling without large losses in combustion efficiency. This is the biggest advantage of this type of atomizer. So, this is the schematic and how it works. So, one component is the atomizer or injector. Now, let us look at some of the problems associated with liquid propellant rockets. One of the most recurring problem or very nagging problem is combustion instability. One type of combustion instability is chugging. This is a relatively low speed oscillation of the pressure in the combustion chamber. Now, the engine must be designed with enough pressure drop across the injector to render the flow largely independent of the chamber pressure because this chamber pressure oscillations always take place. So, normally it is achieved by using at least 20 percent of the chamber pressure across the injector. So, the pressure drop across the injector should be at least 20 percent of the chamber pressure that is the basically rule of thumb. Now, what happens here in combustion instability that due to the feedback between the pressure and heat release the pressure starts to oscillate. As the chamber pressure starts to oscillate your turbo pump is feeding at a constant rate. So, turbo pump is feeding at a constant rate, but a chamber pressure is increasing because of that there is a decrease in flow rate. As the flow rate decreases then what happens that since the flow rate decreases your PC naught will decrease. So, the more flow will go through the throat because it has a throat layer rather less flow will go through the throat. So, there is a as the PC naught increases the chamber pressure increases sorry decreases. So, first chamber pressure was increasing because of the increase in chamber pressure the flow rate decreases. As the flow rate decreases the chamber pressure starts to decrease. So, again it will as the chamber pressure start to decrease the flow rate there is a because your throat is choked right. So, therefore, the flow that will go through is increasing now that will lead to again a decreasing the flow that will go through is decreasing that will again lead to a rise in pressure. So, then the pressure will go to rise and fall rise and fall periodically. So, that is called that is what combustion instability is. Now, what is the effect of that that the fuel flow rate then directly gets affected. Now, if it is a limit cycle oscillation is fine it will operate, but it may be possible that the feedback is such a way that the pressure rise flies away it starts to increase. And it starts to increase then the fuel flow rate will be more and more affected and a point will time will come when the flow rate will be such that it goes beyond the flow ability limit. Then the combustion will stop. So, therefore, the feed line coupling can lead to the combustion being stopped. Therefore, combustion instability is a dangerous phenomena to occur in gas turbines sorry rockets. And it is a fairly common phenomena in liquid propellant rockets. So, that is why I mentioned in the previous case also that the injectors should be such that they are kind of isolated from this oscillations which will be possible if you provide high enough pressure drop. So, that the pressure does not get affected. So, this is one of the major problems with liquid propellants. Now, next let us look at the another issue is associated with liquid combustion is the cooling. Proper alignment of injector is required to provide proper cooling. A fuel rich layer is created in the combustion chamber wall otherwise the combustion chamber gets heated up. So, fuel rich layer is created in the combustion chamber wall this reduces the wall temperature downstream to the throat and even into the nozzle. Because of that it allows the combustion chamber to run at higher pressure. And because of that higher expansion ratio nozzles can be fitted which will give us higher specific impulse and better system performance. So, if you can get proper cooling of the combustion chamber wall we can get higher specific impulse and better efficiency of the system. Liquid rockets apart from this cooling also often employs regenerative cooling which uses the fuel or oxidizer to cool the chamber and the nozzle like in the expanded cycle we mentioned that it goes all along the liquid flow channels are made all along the chamber. So, that continuously keep on cooling maintaining the temperature to a lower value. So, that you can operate at higher pressure. So, cooling is a major requirement for this entire system, but again this cooling or heat transfer has to be done in a proper way because your fuel is supplying passing through this and many times we have hydrocarbon based fuels. So, if the heating rate is very high it will form suits or rather choking it will cook and then that will block the passages and that can be disastrous. If the passages are blocked you do not get enough fuel and rockets will fail. So, therefore, it has to be done in a proper way. I say again one of the major engineering problem to get proper heat transfer in cooling the chamber. Then another issue is the ignition. Now ignition here for the liquid propellant there are of course, the ignition system are going to be similar to that on the solid propellant which we have already discussed. The three major issues involved in the ignition part. First is the hard start. What is hard start? Typically when we start the process the quantity of propellant that enter the combustion chamber and prior to ignition may be very high or large, large amount of propellant come in. Now if that is the case then we can have an excess spiking pressure. The pressure can increase very high to a very large amount and because of this high spiking or increased pressure it can lead to a structural failure or even an explosion. So, this is typically a problem when the we start the engine where lot of unburned propellant and fuel and oxidizer come in and then once the ignition stabilizes then we get the proper flow rate. But initially still the ignition is process is started we keep on feeding and that gets accumulated and suddenly ignition starts we have a lot of propellant sitting there lot of fuel and oxidizer. So, when the combustion occurs it may lead to a mass flow rate which is more than the nozzle can handle nozzle throat can handle and. So, if the nozzle throat cannot handle the entire mass flow rate the chamber pressure is going to be higher. So, chamber pressure increases beyond our design limit it may lead to structural failure. So, this is one of the major problem that how much time it takes to ignite the process. So, that we do not get into this problem we have to have the ignition as soon as possible. So, that even before the pressure goes to this high value the ignition is started and then it is burning and flowing with the required flow rate. So, that brings up to the next point the delay in ignition. In some cases as small as a few tens of milliseconds is required or even the small delay can cause over pressure of the chamber due to excess propellant. So, therefore, this ignition timing is very critical if the ignition is not timed properly we can have catastrophic failure in rocket. So, that is why in the starting the combustion process once you have the say countdown is complete you have to keep your fingers crossed till you have proper ignition because that is where most catastrophic accidents can happen on the launch by itself. If we have this hard start problem another problem associated with this is that see what happens is that you keep on throttling till you have the ignition, but how do you know this successful ignition when to stop you do not know right. So, detection of the successful ignition of the igniter is very difficult some systems use thin wires that are cut by the flames when you have the flames this wires are cut. So, we know that there is ignition and then we give the throttling and the system works on its own or we use pressure sensors to see that the chamber pressure is rising. So, that can be used as detection, but it is very difficult to detect also. So, therefore, these are the problems associated with ignition in liquid propellant rocket it needs to be addressed. What are the different methods of ignition we have discussed in solid propellant similar methods are used pyrotechnic or we have electrical ignition like spark or hot wire we can have chemical ignition or a pilot flame we can use hypergolic propellants which has an advantage of self igniting it has a reliable and less chance of hard start because it is hypergolic propellant. Generally ignition system try to produce flames across the injector surface with a mass flow rate of approximately 1 percent of the full mass flow of the chamber and ignitions this then provides the ignition source before the main valves open. So, it is like a pilot flame. So, we have a pilot flame in the system before the main valves open hoping that as soon as the fuel and oxygen are coming because of the presence of this flame that will catch fire and the combustion will proceed without any problem. So, we have enough energy density sitting inside the rocket from the from this pilot flame. Now, let us look at some practical issues associated with liquid combustion sorry liquid propellant rockets. First of all the shift in center of gravity here the propellant is carried in large tanks and propellant is a very large portion of the mass of the vehicle. Now, as the propellant get used up the center of mass shift significantly rearward as the propellant is gets used up and it becomes difficult then to control the vehicle if the center of mass goes very close to the center of drag. So, this is one issue that how long we are use we are going to use it. So, the timing of a liquid propellant rocket is set by this beyond that the control becomes difficult. So, we do not want to use it for more than that time. Then pressurization of tank itself is a problem typically a very thin wall propellant tanks are used and must have positive gas pressure and all times otherwise they are going to collapse. So, if the propellant tank is at a pressure less than the atmospheric pressure is going to collapse and there is collapsing tanks then can have severe implications. Then sloshing the propellant that is stored in the tanks as they move it may slosh and because of that sloshing the fuel head is changing pressure difference or head is changing that may have a negative impact. It can have loss lead to loss of control of the vehicle this can be controlled. However, we can have slosh sloshed bluffles in the tank or judicious control law in the guidance system can be provide to prevent the sloshing. Then as I mentioned previously we have combustion instability they can suffer from pogo oscillation. Rocket suffers from uncommanded cycles of acceleration because of the combustion instability. Because as the instability side seen the thrust then starts to oscillate and that will lead to periodic acceleration field or acceleration change which is something that is not I would say advisable for rocket operation. And in the particularly in the case of a large motors 0 gravity or during stages it have to avoid sucking of gas into the engine during startup. So, this is something very important that gases should not be sucked into the engine because then you create an environment which is different from what you have designed for. Because your propellant has to come from your system suddenly you have additional gases sitting there which will not burn right it has to be pushed out then that will delay the ignition process. So, therefore, this problem that sucking of burn gases into the engine itself should be avoided as much as possible. Then vortexing within the tank particularly towards the end of the burn vortices are formed in the tank and that can result in gas being sucked in the engine pump and that will stop the operation. The gas being sucked will stop the operation of the engine. Then the propellant leak particularly if you are using hydrogen as a propellant this can lead to possible formation of explosive mixture which is very dangerous. Then the turbo pumps which we are using they are used to pump the liquid propellant they are very fairly complex in design and can suffer from serious failure modes particularly if you are going for if you run dry it may go to over speeding and that may completely destroy the blades or shedding fragments at high speed if metal particles from the manufacturing process enter the pump. You have some bolt or something red somewhere or small pieces if it goes into the pump it will shed all the blades. So, therefore, that is a very dangerous mode of failure. So, therefore, this turbo pump itself is a various significant issue and particularly this pumps have to come into full life within a very short time because the rockers the maximum operation time is about 40 seconds. Ignition should start in few microseconds. So, in this sorry few milliseconds in that time it has to go from 0 flow to full flow. So, the pump have to be have very fast response and that is what makes it more complex special materials are required because the loads are more. Then when we are dealing with cryogenic propellants the propellants are ok, but the atmospheric water vapour may freeze into very hard crystals and this water vapour may damage or block seals and valves because water vapours are present which may create problems. And to avoid this problem one often requires lengthy chilled down periods where we once we drain the system we have to give it chilled down for a long period so that all this water vapours and all are removed. And then ice can also form an outside of the tank which may fall and damage the vehicle. Then the external foam insulation can cause problem as happened in the special Columbia. Then non cryogenic propellant do not cause such problems, but cryogenic propellants propellants may be safe, but the water vapour around it will freeze on them and create problems for you. So, therefore, those needs to be taken care of. The biggest problem with the liquid propellants is the preparation time. Most of them are non storable particularly cryogenic propellants are not storable at all. So, they cannot be stored for long duration and they require considerable preparation immediately before the launch. So, therefore, they cannot be kept ready you have to prepare it for launch and that makes them less practical than solid rockets for most weapon systems, but they are good as say rocket vehicles or space varying vehicles. Because once you prepare you can fire and then it goes out, but if the mission is aborted then you have to go through this process to make it safe. So, this is what I wanted to discuss on the liquid propellant rockets. So, that brings us to an end of our discussion in chemical rockets. So, in the next class we will talk about the electric propulsion systems.


The Surface Navy Association hosted the second day of its 25th annual National Symposium in Arlington, Virginia, January 16th. Admiral William Gortney spoke about the fiscal changes in the fleet and the future mission of the Navy and Marine Corps. There's going to be uncertainty out there and the naval force over the next decade, the Navy and the Marine Corps team will be sailing to crisis because that's what we've done since we were founded. Gortney said that the fiscal challenges currently facing the military will improve over time. We won't get into a trough and stay into a trough forever. The nation will need us again and they're going to increase the DOD budget. You can see that over history. That's what's going to occur. The question is, there's tough choices that we have to make as we're in the downward trend to where we get in the upward trend. From the Defense Media Activity, I'm Petty Officer Brandy Wills.


So, just a quick review of what we have done last time we derived the formula for least square estimation we only through geometric arguments. So, we said that the error between the predicted and the known or measured values predicted by the model and the known values of the dependent variable that error should be perpendicular to the subspace spanned by columns of matrix A. So, we had this model which was u is equal to A theta plus e where e is the error vector. So, we just made geometric arguments something that you know from your school is this error vector u minus A theta this should be perpendicular this should be orthogonal to the column space of. So, e should be in the least square estimation e is perpendicular to the column space what is column space of A span of columns of A column space is nothing but span of columns of A. So, this error should be orthogonal to this just using this simple argument that is argument of projection we arrived at the basic formula that was theta least square is equal to A transpose A inverse A transpose u. So, we could arrive at this fundamental result which earlier we had arrived through algebra through geometry. So, today what I want to do is extend this beyond just finite dimensional spaces also give you some more insights geometric insights in the finite dimensional spaces. Next class after this we will start applying it for different purposes. So, again let us again stick to the interpretation part of it as I said there is a third way of interpreting all these things that is through statistics unfortunately we do not have time to get into that but I am going to just upload my notes about statistical interpretation. So, those of you who are interested should go and look at the notes but that is extra reading and this is a post graduate course. So, you should do extra reading it is not that just you end the things end here what we actually managed to cover in this course is the tip of the iceberg. So, we derived this formula and I want to generalize this to some extent to any Hilbert space approximation in any Hilbert space before that let us look at some geometric insights. Now I just now define column space of A probably in some other course you also have looked about other spaces that are associated with the matrix what are the other spaces that are associated with the matrix null space what about row space what is the row space span of rows span of rows okay or span of columns of A transpose okay. So, column space of A transpose is nothing but the row space. So, we just need to know little bit about these four spaces. So, given a matrix A span of so span of columns of A is called as column space of A then span of row space of A row space of A is nothing but span of A transpose or set of all possible linear combinations of rows okay what is null space what is null space of A A x equal to 0. So, null space of A this corresponds to set of all x such that A x equal to 0 vector. So, set of all x such that A x equal to 0 vector this is called as null space of A and there is one more space associated with matrix that is called as left null space okay. So, left null space left null space is nothing but null space of A transpose null space of A transpose null space of A transpose is called as left null space of A okay. So, there are four fundamental subspaces associated with a matrix okay one of them is the column space column space is all possible linear combinations of columns of A row space span of row vectors null space set of all x such that A x equal to 0 and left null space that is set of all vectors y such that A transpose y is equal to 0 okay. So, this is four fundamental subspaces and what you are going to see or what you will realize here when I go to this okay we split a vector u this vector u was split into two components one component was the projection was the projection p. So, we are able to split a vector by this approach geometrically what we are doing we are splitting the vector into two components what are the two components one is in the column space this is in the column space because A theta is nothing but linear combination of columns of A right A theta is linear combination of columns of A. So, this vector has to belong to column space of A okay. So, if I just draw this picture again here if you have a two dimensional if you have a matrix with three cross two matrix with two linearly independent columns let us say this is the first column A1 and this is A2 and this is my u vector this diagram I had drawn earlier this is my u vector. So, this is the projection this is the p vector this is the projection vector which is lying in the column space of A okay then there is a orthogonal component what do you mean by orthogonal component orthogonal component lies in the null space of this matrix the orthogonal component lies in the null space does it lie in the null space or does it lie in the left null space just think about it where does it lie where does the orthogonal component lie does it lie in the null space or left null space. So, we are able to split a vector into two components in the column space and orthogonal to the column space okay. Will a vector belonging to null space be orthogonal to the column space? What is the property of a vector that is orthogonal to the any vector that is orthogonal to column space? Yeah, a times that vector okay will give me 0 vector okay. Of course e is not in the null space, e is in the left null space. Let us think about it, e is in the left null space, e is in the left null space and p is in the column space okay. So there is a fundamental theorem of linear algebra okay. I am just going to mention it here without getting into the proof you can refer to Strang's book or you can refer to any linear algebra book for this. The column space of a is often referred as r of a, the column space is often referred as r of a and the null space is often referred as n of a, n stands for null space a null space of a. So there is one fundamental theorem of linear algebra which says that I think you know this result, I am just going to state it again. Number of linearly independent columns okay is equal to number of linearly independent rows. In any matrix number of linearly independent columns is always equal to number of linearly independent rows which is equal to rank okay which is equal to rank. So this is the algebraic result which you know which is also a geometric result because linearly independent columns okay. It relates rank which is a algebraic quantity with a geometric property that linear independence okay. So what is the dimension of the column space? Dimension of column space is always equal to rank okay. So the fundamental theorem of linear algebra, dimension of row space of a, dimension of column space of a, dimension of column space of a is equal to rank of a okay. I am looking at a matrix a which is n cross m, I have a n cross m matrix. So dimension of column space of a is equal to rank of a, dimension of null space is equal to n minus rank of a, then dimension of row space of a transpose, sorry column space of a transpose that is nothing but row space is equal to rank of a, dimension of null space of a transpose m minus rank a. Number of columns is m, if I put n cross m then it should be m minus rank a and then this should be n minus rank okay. This is just the background, this is a well known result I am just stating it again okay. Now with this background I am going to prove some nice algebraic properties of projection that we have looked at okay. So now actually what we have achieved is a way of splitting a vector into two components. See we found out the least square estimate right, we found out the least square estimate and how did we get the least square estimate? How did we get the least square estimate? So theta least square this was a transpose a inverse a transpose u correct, this is how we got the least square estimate okay. So what is the projection? See we are talking of two things right, one is this projection onto the column space, other is the vector error which is perpendicular to the column space okay. So how do I project given a matrix a, how do I project it onto its column space okay. So the projection p is given by a times theta hat Ls right, the projection we found out the theta which projects now projection vector itself, what is the projection vector a times theta okay.   So given a matrix a how do I project any vector onto its column space, I take matrix a, I find out this matrix this is called projection matrix, this is called projection matrix, this projection matrix projects any vector u onto the column space of a okay see because look at this theta here see theta happens to be this but what is theta, theta is a least square estimate is some number. So a times theta is nothing but linear combination of columns of a, so p has to lie in the column space of a okay. So actually what we have done is we have found out a way given a matrix a which is non square which is actually tall matrix more it has more rows than the columns for such a tall matrix we know how to project any vector u onto the column space okay, this is the component along the column space. What is the component which is orthogonal to the column space, so that is e, so how do you get e, e is u minus a theta least square okay that is u minus a a transpose a inverse a transpose u which is i minus projection matrix what is this i minus projection matrix everyone with me on this i minus projection matrix okay you can check that you can check that p is orthogonal to e take inner product of p and e okay you will get 0 these two are orthogonal components. So now given a vector u we know how to split this vector into two components one in the column space other orthogonal to the column space okay this matrix a into a transpose a inverse a transpose is called as a projection matrix there is something very nice about the projection matrix is projection matrix symmetric is it symmetric matrix if you take transpose of this matrix what will you get I am not going to do it on the board just think about it there is one very nice property of this projection matrix I want to do check it and I want you to understand what it is meaning this is a very funny matrix it is square of itself yes this matrix this matrix has a very nice property p r square is equal to p r can you check this just multiply p r into p r you will get back p r mind you this is not an identity matrix this is not an identity matrix and you have to explain me geometrically what is meaning of p r into p r is equal to p r first of all are you able to get this is this coming out just multiply what is a into a transpose a inverse a transpose a a transpose a inverse a transpose right this a transpose a inverse a transpose a this will become identity you will get a a transpose a inverse a transpose this is not a this is not a matrix which is this is not identity matrix yet it has this very nice property it is square of itself what is the geometric meaning of this yeah yes not matrix vector u plane same vector precisely very very nice so if you project the projection again you will get the same vector right see here this my u vector is split into two components projection and orthogonal to the projection now this projection if I project again on to the plane what we what should I get the same vector because there is no the best approximation of this vector inside this plane is this vector itself this vector itself so the geometric interpretation of projection square okay you can just go on you can just say projection cube is also equal to very very very very nice matrix okay vice versa any matrix that has this property is called as projection matrix that is square of itself is a projection matrix okay so this this is a this is a nice property of projection matrix you can go on multiplying it with itself and you will get back the same matrix what about transpose is a transpose of this matrix symmetric matrix okay here transpose it is a symmetric matrix okay the symmetric matrix it is square of itself okay what will be projection matrix if your u is perpendicular just without computing you should tell me what should be the projection matrix if u is perpendicular projection matrix should be 0 because the best approximation best approximation of u which is perpendicular to this plane is the 0 vector okay so projections these square approximations associate this geometric ideas okay then you will remember it better than just what happens if u is already inside we talked about it if u is already inside you will you will get back the same same vector so same matrix so it is projection matrix is and what happens but it is square if a matrix is square matrix then what is the projection matrix if a is a square invertible matrix if a is square invertible what will be the projection matrix a is square invertible so what is a transpose a inverse it is a into a inverse which is i into a transpose inverse into a transpose which is i identity matrix okay so if a is full rank and square okay projection matrix will turn out to be identity matrix there will be no error you are already in the full space okay so now I am going to generalize this to any Hilbert space this is very nice result and then we need to you know to extract it is full potential we should not just restrict ourselves to n dimensions till now I have been looking only at n dimensions why not go ahead and look at you know any Hilbert space any infinite dimensional space in fact what you will see is that soon what will pop out is Fourier series in not just the Fourier series but generalized Fourier series okay let us look at this very quickly because this is something which probably Fourier series haunts you from your second year of engineering and then this will probably explain you the basis to how it works why Fourier series why we look at orthogonal functions and so on okay so now what I want to do is to generalize this result into any Hilbert space so what we are going to look at now is see basically when you will study all these methods numerical methods for solving partial differential equations boundary value problems algebraic equations whatever whatever I wanted to remember all this geometric insights all this you know it is not it is not just the procedure it is these foundations which help you to modify to tweak to twist for a particular you know application because if you want to if you want to really concoct a solution for a given problem you cannot many times just rely on the standard tools you have to come up with some way of you know concocting the new solution procedure that is where you should understand all these fundamentals okay should know where it is arising from how whole thing was derived and then you can just derive another approach if you if you need to so that is critical okay so just like a let us look at what we have done till now what we have said that I have a way of projecting a vector okay so we looked at the three dimensions we looked at a vector which is outside the plane you know and then suppose this is the plane no I have a way of projecting this vector onto this plane and a vector component component which is perpendicular to this plane okay in general I said well why just a two dimensional plane if I have a vector in Rn and I have a subspace in Rn okay I have an m dimensional subspace I know how to find a component lying in the subspace and orthogonal to the subspace right we looked at m dimensions and n dimensional vectors being projected onto m dimensional subspace and a component which is orthogonal component which is orthogonal so we were able to split in in finite dimensional vector spaces we were able to extend our school geometry that you know the shortest distance of a point from a subspace or from a plane is drop of perpendicular I want to do the same thing in any Hilbert space why can I do this in Hilbert space I have in a product with me since I have in a product with me I have you know I have orthogonality I can talk about projections I can just talk about you know two vectors being orthogonal to each other okay so I can just talk about now you know taking a subspace in a Hilbert space okay or taking a subspace in a product space any space on which in a product is defined and then in that subspace okay and given a vector which is outside the space I can project this vector onto the subspace right I can find a component of the vector along the you know in the subspace and component perpendicular subspace okay so what was the trick the trick was that the component which is error okay our approximation error is orthogonal to the subspace okay so I find the error vector find it is orthogonal and you know that will help me to come up with approximations see for example just to motivate you know let us say I have this function ft a plus bt where a and b are some known values okay and then I want to approximate this function using I want to construct an approximation let us say g of t which is alpha sin t plus beta cos t is this a vector it is a vector let us say this is a vector in where t belongs to 0 to 2 pi is this a vector this function is a vector this is a continuous function over 0 to 2 pi I want to construct an approximation alpha sin t plus beta cos t for this vector so what do I mean by that the error what is the error function what is the error vector error vector that we call this et will be ft minus gt ft minus gt okay what is the subspace here what is the subspace that these 2 vectors span it is a 2 dimensional subspace right 2 dimensional subspace offset of continuous functions between 0 to 2 pi why 2 dimensional there are 2 linearly independent vectors sin t cos t okay there are 2 linearly independent vectors sin t cos t now just we extend our ideas of projection what should happen this error function this error function should be orthogonal to orthogonal to subspace which subspace sin by sin t and cos t 2 dimensional subspace sin which is span by all possible linear combinations of sin t and cos t so this should be perpendicular to span of sin t cos t this error this approximation error between the function original function and approximation okay see this a and b are known constant let us take as you know 1 plus 5 t let us take specific a and b are not known we have to find out a and b this square approximation okay I want to get a this square approximation so I want to get a this square approximation of this I am just taking very simple function in general you may have very complex function which you need to approximate for example you may have data which is actually coming from you know daily temperature variation which is cyclic right daily temperature variation is cyclic and you could approximate using sin and cos functions you could develop very nice approximations so I want to develop the approximation alpha and beta which is which what is the advantage of let us say I have data for temperature variation throughout the day throughout the day what is the approach what is the advantage of fitting see I what I mean is you know I collect temperature at every 5 minute and I plot it see let us say it is you know every 5 minutes I have plotted you have data for 5 minute for 24 hours how many data points 12 per hour into 24 so many data points if I have to save this data for years let us imagine how much data I have to save okay as against if I am able to fit sin and cos okay so I fit some sin and cos which is approximation which does not pass through every point okay nevertheless it sort of gives me a best fit least square fit okay for today suppose I have to save data for today for today's data okay I could find alpha and beta for today's data I have to save only two numbers you must have heard about data compression right you heard about data compression this is a data compression algorithm data for 5 minutes for temperature for the entire day can be compressed and written in terms of only four numbers or two numbers if you are not happy with sin t cos t add two more you know sin 2t cos 2t okay and then you will be able to get an approximation of temperature versus time and instead of saving all this 12 into 24 values for one day I could just save two approximation coefficients all that I have to know that these two approximation coefficient correspond to sin and cos I multiply them I can get a very good approximation of what happened on that day right it may not be accurate to the extent that it may not be passing through every point does not matter but I can get a trend I can get a trend so this is very very important very very important okay so let us leave this aside for a while let us look at this problem we want to get geometric insights and we want to extend them to okay so now what should happen is this error should be perpendicular to span of these two vectors now in general my model could be of this form okay let me write a general model my general model is ut is equal to this is my general model okay I will just make one small change here just to be consistent with our notation let us call this p because this is the projection this is the projection let us call it p okay and then here let me call this as ut okay now let us come back here so in general my ut okay is some linear combination of these known vectors a1t a2t a3t amt are known vectors just like sin t cos t okay you want to project this onto some known subspace okay you know the you know its basis vectors okay and linear combination of these basis vectors will define that subspace what is all possible linear combination of this a1 to am if they are linearly independent it will be m dimensional subspace right it will be m dimensional subspace now the question is if I give you u if I give you ut okay can you find out alpha 1 alpha 2 which are v square how do you find out alpha 1 alpha 2 which is v square so this part this part here is the projection this part here is the projection onto onto the subspace defined by span of a1 to am right so I want to find out projection project ut on okay okay now just for the sake of convenience of notation I am going to drop this tt tt I am just going to work with up a1 a2 so t is there I am just going to drop it for the sake of convenience otherwise the writing will become cluttered what I am saying here and what I told you here is not different just remember this this is a two dimensional example okay here so this is my a1 this is my a1t and this is my a2t this is my a1t this is my a2t right the subspace which is defined by linear combination of these two vectors is two dimensional in general you may have m such vectors and you are talking about distance of a point from the subspace same problem okay visualization will not be different from what you know in three dimensions but I am extending it to any m dimensional subspace of an infinite dimensional space okay so the projection theorem I am not the statement is formal statement is given here the projection theorem says that you know u-p that is equal to error is orthogonal the statement of the projection theorem classical projection theorem in Hilbert spaces is that error is orthogonal to the error the point which is at the least distance from the sub the point in the subspace which is at least distance from a point outside can be reached okay by dropping a perpendicular which you know from three dimensions same result holds here that error is perpendicular to the subspace okay that is the statement of this theorem formal statement is given here I am just compressing and writing here that this error vector is orthogonal to the span so this is my let us call this subspace s okay is orthogonal to the span now how am I going to use this to compute alpha 1 to alpha m okay so what I am going to do is very very simple okay I am going to start I know that this is orthogonal to a1 to am okay which means this error is orthogonal to a1 is also orthogonal to a2 is also orthogonal to a3 so one by one I am going to write these equations okay so I am going to say my first equation is that in a product see now I am just generalizing this result I am saying in a product of u- u-p into ai is equal to 0 u-p is this okay what is the meaning of this statement u-p this error vector is orthogonal for i is equal to 1 2 up to m how many equations I need how many unknowns are there there are m unknowns alpha 1 alpha 2 alpha 3 up to alpha m there are m unknowns okay I need to write m equations okay here are my m equations I am just going to use them okay so just look at this well I will just rearrange this and write it I will say that this means that in the product of uai uai is equal to these two are same equations right I am just writing them I am just expanding and writing this equation so I am going to use this now to create so what is this okay so this equation first equation is that you know alpha 1 a1 plus up to alpha m am in a product a1 is equal to u in a product a1 right so this equation will actually be can be written as alpha 1 a1 in a product a1 plus alpha 2 a2 in a product a1 alpha m am in a product a1 is equal to u in a product a1 right I am just expanding this equation I am just expanding this equation what is my second equation so second equation if I just write like this and expand what will I get I will get alpha 1 a2 in a product a1 in a product a2 plus alpha 2 a2 in a product a2 alpha m am in a product a2 is equal to u in a product a2 okay how many such equations I have I have m equations okay so alpha 1 a1 in a product am up to finally up to alpha m am in a product am is equal to u am is everyone with me on this how many equations you have got m equations in m unknowns okay do I know these vectors a1 a2 a3m I know these vectors I have chosen them sin cos okay sin 2 cos 2 and so on so I know these vectors so if I if I if I know these vectors okay this in a products can be computed okay whatever integral over 0 to 2 pi we can compute them we can compute them and then this equation can be solved this equation can be solved because you have m equations in m unknowns is it a linear equation it is a linear equation it is like matrix times vector is equal to some vector u is known to you a1 to am are known to you so right hand side is known to you all this coefficients are in a products are known to you alpha 1 to alpha m is not known okay now now just see what is the advantage of choosing these vectors to be orthogonal suppose a1 a2 a3 am or orthogonal what will be this in a product okay which in a product will not be 0 a11 a2 to a33 a44 yes if if you choose orthogonal vectors okay the solution is particularly very easy then you will get first equation will be alpha 1 times this is equal to actually what we have derived is nothing but generalized Fourier expansion I will just come to this in my next lecture very briefly this is if you if you take a1 to am to be orthogonal what you get are nothing but Fourier coefficients this is called as generalized Fourier expansion and then we will get Fourier coefficient Fourier expansion need not be only with sin cos tomorrow in some function you know you want to approximate some function over some domain say 0 to 1 using shifted general polynomials why shifted general polynomials shifted general polynomials are orthogonal polynomials okay and then this approximation will be very easy so we will continue on this particular equation is called as normal equation and this is like cornerstone of optimization projections and helps us in discretizing and so on okay so actually if you go back and look little bit carefully this is not different from A transpose A inverse or A transpose A into theta is equal to A transpose U because inner product of columns A transpose A if you take columns of A inner product of the same equation written in a generic form for any Hilbert space same equation not different okay so let us stop here and then let us continue with its interpretations in our next lecture.