What is up guys Karma Medic here and welcome back to another dose. This video is going to be part two of the ethical dilemma series I'm doing for the MMI. We're going to be looking at the second half of the kidney transplant ethical dilemma that we looked at in part one. If you guys haven't seen that video already do check out the card over here or find it somewhere on my channel and watch that video first. Alright without further ado let's continue on to part two. So generally in the interview for ethical MMI scenario questions you're going to get the main question you're going to give your answer and then they'll ask you follow-up questions to sort of see how you would change your answer based on new information. Something else that's important to think about is the age of the patient and a lot of the time in MMI interview questions they will ask you the scenario and then they'll say what if the person was 39 instead of 29 or what if the person was 29 instead of 39 and they will always try and catch you out on this they always ask this age angle and the answer to that question is that age has absolutely no factor in determining who should get these organs. Remember that list I told you five seconds ago age was not on it. Age, gender, beauty, income, religion all these factors are not important when deciding who gets a kidney. Everything is based on clinical evaluation, logical decision-making, rational decision-making. For example if you're deciding to give a kidney to a 15 year old or a 70 year old it might be easy for you to say well obviously give it to the 15 year old because they have their whole life ahead of them but the truth is you don't know any of that. The 15 year old could overdose from drugs at the age of 20 the 70 year old could go on to live until they were 100. So in that case if you give the kidney to the child it would only last it for five years whereas if you gave it to the elderly person they could have lasted for 30 so you never know and age just is not taken into account in these decisions. Like I said they often try and catch you out with these questions in the MMI so do be careful. Another thing that they could potentially act is would you feel differently if the patient had a live donor who is willing to donate a kidney to them? So this is a very good question because it brings into light two of the four medical ethics pillars autonomy and non-melevalence. So autonomy is the idea that each person is their own individual and they should decide what they want to do with their own body and non-melevalence is the idea of doing no harm is that doctors should not inflict harm on any of their patients. So let's talk about autonomy first. If the person who wants to donate the kidney is a fully autonomous person that meaning that they have full competency full mental stability and they also have informed consent so they know the situation they understand the risks and they still want to donate the kidney they are still fully autonomous then that is within their right even if doctors might not think it's in their best interests however we need to take into account the idea of non-melevalence. So if doctors are going to remove a kidney from this patient that patient is going to have one less kidney and with that patient having one less kidney their life is going to change they're not going to be able to do things the same way they used to be and that also has inherent risks with the procedure and following the surgery. So really this is a cost-benefit ratio situation we have to weigh up all the pros and cons of allowing this donor to donate their kidney. Now regardless of whether this person is able to donate that kidney like we said before the woman has a GFR of 28 and we would need to wait until the patient gets down to a GFR of less than 20 before we can put them on the transplant list so that we can maximize the amount of lifespan that they have after the procedure. Okay so now in the case I'm going to give you some more information about this potential donor and then we're going to reassess if we think this specific donor should be able to donate their kidney. So let me give you guys a little bit more information. She's a 65 year old woman and you take a history which reveals she is a hypertensive smoker on one medication for successful blood pressure control. So successful meaning that her blood pressure is below 140 over 80. She has a family history of type 2 diabetes and her daughter's cause of kidney failure is Anca vasculitis. There's no other significant family history or kidney problems or significant medical history. Blood tests show no cause for concern. Okay so do we think that this woman should be allowed to donate a kidney and if yes or no what are some of the factors that we would take into account when making this decision. So first things first let's break down this woman's history. First of all she's a 65 year old woman. So as we've established one of the things we take into account when deciding whether someone should receive a kidney is the age difference between the donor and the recipient and the reason for that is that you don't want to give a 20 year old a 70 year old's kidney and that's because over time of course the 70 year old's kidney is going to have degraded infunction and if they have other risk modifying factors like smoking, hypertension, diabetes that's also going to affect that person's kidney function.  Second thing we know that she's a smoker and smoking as I'm sure you know has many harmful effects on the body one of which is definitely to also affect the kidneys. Smoking can restrict blood flow to the kidneys it of course can have many effects on the vasculature of the body. Smoking can also affect the blood pressure medication that she might be taking to control her hypertension so smoking overall is definitely bad and it's probably having an effect on her kidney function and in addition to that we know that this person is already hypertensive they already have high blood pressure which as we've said before is one of the biggest risk factors for developing kidney disease even though now it's controlled it could have been really high in the past and it will have probably done damage to this woman's kidneys. So these three points definitely don't make this person an ideal candidate for kidney donation there's no point in donating this kidney which is not at its optimal to somebody who's 30 years old if five or ten years down the line they're going to develop some major complications and maybe need a new kidney. Now let's talk about not doing harm or non malevolence to the donor so if we're going to remove this kidney from the donor that donor is going to have one less kidney for the rest of their life and that's obviously going to affect them in many ways if they're already at risk for developing any kind of kidney problems we don't want to remove one of their kidneys because then they're much more likely to end up having kidney disease and kidney problems in the future however if this person is fully autonomous has full mental capacity and is mentally stable and is able to with informed consent still know that they want to go through this procedure then they are an autonomous human and the answer to this question of whether they should or should not be able to donate their kidney is really really tough and I don't know basically you have to weigh up all the pros and cons of the conditions that we talked about that the person has and their ability to want to give up that kidney and this is something for an ethics board to decide but this is why we have an entire medical ethics board an entire team that meets up to discuss a single patient all the different factors that are relevant to help them make that decision on whether they should receive a kidney or not and who they should receive that kidney from and hopefully what you've taken away from this is that even if you don't have a concrete conclusion or answer at the end of the video showing your thought process your decision making and how you reached your conclusion or I guess all the factors that you took into account is what is really important ethical decisions are rarely going to be easy to make even though we have so many guidelines to follow and so many people to refer to and get expert opinions from it's usually down to a case by case basis and it can be very very difficult and that's something that I think you should definitely acknowledge in your MMI interviews you know just acknowledging that these are very very difficult decisions and that it takes a lot of people to be able to make them is something that's important that you should mention in interview so another follow-up question that they might ask you is that you tell all this information to the potential donor who we've said we don't think is a good candidate to donate their kidney and they come back to you and say you know what I really want to donate this kidney to my daughter I would do anything for her I just want to improve her life I want to save her I want to make her feel better please please please let me donate my kidney so how would that affect your decision and obviously the answer at the end of this is that it would not affect your decision you've made your original decision based on clinical evaluation very rational logical factors and you're not going to change that decision based on an emotional plea or a social desire but what you definitely should mention and talk about is that this is obviously very difficult you want to do what's best for your patient and you can definitely mention in interview how this is a very difficult decision how you want to do the best thing for both of your patients you want to make the mother happy because obviously this is affecting her not being able to help her daughter and you want to do the best thing for the daughter by giving her the best possible kidney that she can get and one last follow-up question that they might ask you is they might say what if the donor had a gene that made them at high risk for having hypertension or high blood pressure and how would that affect your decision so in this case of course if the donor has a very high risk of developing high blood pressure later in their life we want to make sure that they keep both of their kidneys because if they were going to donate one of those kidneys to the recipient and now only be left with one kidney and they developed high blood pressure and that was going to cause them kidney damage that's obviously going to lead to a lot of problems for them in the future so if they did have this gene that would increase their risk of blood pressure in the future you would want to make sure that they do not donate as they need both of their kidneys all right guys and I think that's it for this video this has been a really long case to cover quite a big ethical dilemma I hope you've enjoyed learning about the different factors that we might consider or take into account when looking at an ethical dilemma like this organ transplantation is something that does often come up in MMI interviews so it's something you probably want to prepare for if you guys have any questions do feel free to leave them in a comment down below do like this video if you enjoyed it and do consider subscribing to my channel if you want to see more videos like this from me in the future until next time I'll see you guys later peace


So, back many, many years ago, when I was first working with The Course in Miracles, I just really had a very deep, profound, insightful experience that, this big, thick, blue book is really just about cause and effect. And once you kind of get the gist of it, that only the mind is causative, which is very important, that can't be missed or skipped over, but then comes the transfer of training. How willing am I to practice that with everything that's floating through my consciousness? How willing am I, on a daily basis, or more of an hourly or moment by moment basis, how willing am I to reel it back off the screen? Any types of upset, disappointment, sadness, anything that's not supremely happy, how willing am I to, for a moment, bring it back to my mind and hand it over to the Holy Spirit and say, I was wrong. Oh, there I was wrong again. I was wrong-minded again. I thought there was some causation there. Something was happening to me. Something in the world was coming at me that was out of my hands, had nothing to do with my mind when actually it all has everything to do with my mind. So, to me that's what this experience, earthly experience has been. It's just been about practical application, just being humble enough to do the watching of the mind and being humbled by that. The first time I read that, and of course your mind is much too tolerant of mind wandering. I was like, whoa! Yeah, I want to come into alignment. I don't want to use my mind or my time for wandering, seeking and searching in vain. I would rather seek and find. So, to me it's all into, if we're called to forgive, then what's the attraction of that? It's just this glorious state of non-judgment. It's worth everything. That piece is its own reward. Non-judgment is so natural. It's almost like when we begin to trust and surrender and relax. We almost like fall into a bed of feathers and the bed of feathers is non-judgment. It's always been there, so soft, ready for us to just fall back. Some of you might remember from years ago, the Nesti commercials. Anybody remember Nesti? They would always say, take the plunge and the slow motion of people falling back into like a swimming pool or something. Refreshing, a very strong image, falling backwards and splashing into refreshing water. In one sense, Jesus is saying, take the atonement plunge, just fall back, fall back into me. Fall back into this soft, warm light that's who you are. If it's so simple as just being in alignment with the miracle, then it must be that when he says trust, development of trust, trust is the basis on which all the characteristics of a teacher of God rest. Then basically trust in spirit or the divine, Holy Spirit, Jesus, higher self-intuition. Call it whatever you want. That trust is the key and why is that trust so important other than how else are you going to experience that everything without exception is all involuntary? The miracle is involuntary and all the conditioning of the ego and all the conditioning of the world and all of our educational systems are all saying there's things that you can control and there's things that you can't control. Again, what is all involuntary, which is in the miracle perception is involuntary, the ego always splits everything up into opposites, the involuntary and the voluntary. Are you doing this involuntarily or voluntarily? You see how it's just more opposites and presence doesn't know those opposites. Presence transcends all opposites, all possibility of opposites. So for me, when I plunged into the course and immersed into it, Jesus let me study it for a while but he had a lot of commentary when I was studying it. He started in all kinds of commentary, pointing things out, go back, look at this and so on and so forth. But it was really, I would say for me, the study of A Course in Miracles was a very brief period until it was like you got it now, you got the gist of it, do you have the gist? Now we're going to practice. Practice, practice, practice, practice, make no exceptions, practice, practice, practice, practice. That's where the transformation comes in, that's where the experience comes in, is through making no exceptions to the principles. So it's not just another book to be studied or memorized or to do scholarly papers on or so forth, it's an actual conversion device for the mind and then the only way you make the conversion is by giving yourself over to it. And for me, after about five years it was like Jesus was like, okay, enough of the study, five years was plenty. It was, now I'm taking you on a road trip, we're going out for some practical experience and that was just another device, you know, you don't have to take road trips but in my case that was a sense of when pulling you out of your familiar perception, familiar friends, family, city, location and going to take you out on the road so we can go about this rather quickly, learn to transfer these principles very rapidly. And then we all know those points when you're just in this gloriously happy experience and you can't really say how you got there because there is no formula and you are just in so much happiness and joy but you do feel a bit like a twig in the river like you're being carried in the vast experience, you're in that vast experience, you are that vast experience but you can't break it apart into pieces, you can't analyze it, you can't figure it out, you can't say, well I did this and this and this and then it happened, it's just another story but that experience of everything flowing without conscious control or conscious effort, that's the flow that we were talking about early on, that's the flow. The ease of that is everything and there's no sense of a personal identity in that and there's no sense of a story with that. It has no goals, it has no direction in the world, in a worldly sense. It's not pointing in a certain direction.


Hello and welcome to lecture number 39 of this lecture series on introduction to aerospace propulsion. So, after series of lectures, we have now come towards the end of this lecture series and this is the last lecture in this lecture series, which will be jointly taken by Professor Roy as well as me. I will be taking the first half of this lecture and Professor Roy will continue with the second half of this lecture. So, over the last several lectures, we have been discussing and interacting about different aspects of aerospace propulsion. We have also had a chance to look at some of the fundamental principles behind some of these devices, basically thermodynamics of aerospace propulsion. So, we have spent quite a few lectures on understanding the thermodynamics and thermodynamic principles of aerospace propulsion devices. And also, we have discussed about some of the basic cycles involved in aerospace propulsion, basically used for aircraft propulsion. And we have carried out cycle analysis, the ideal cycle analysis of these devices. We have also had Professor Roy has been discussing about some of the aspects of piston propeller engines as well as rocket propulsion, elements of rocket propulsion. So, in today's lecture, which is the last lecture of this lecture series, we shall be discussing some of the aspects, which are related to advanced propulsion systems. Some of those concepts, which are as of now still a concept, but which we feel shall be used in future propulsion systems. So, let us take a look at what we are going to discuss in first half of this lecture. In the first half of this in today's lecture, we shall be discussing about what are known as variable cycle engines, which are part of the advanced propulsion concepts, everything propulsion concepts. And we shall discuss in brief about the supersonic transport propulsion. We will discuss two concepts, which seem to be promising, the mid-tandem fan concept and the mixed flow turbofan ejector concept. Subsequently, we will also discuss in brief about short takeoff vertical landing engines. What is it that makes them different from conventional engines? And towards the end of the first half, we shall also discuss about engine with intercooling, reheating and regeneration. We have had some discussion of this during the thermodynamics of the engine cycle. We will discuss a little bit more on intercooling, reheating and regeneration as well. So, these are some of the topics that we shall be very briefly discussing. It is just what we are going to have today is a very introductory discussion on some of these topics. I am sure you will be able to find more reference material on these topics. Now, one of the first concepts or topics that we shall be discussing is the what is known as variable cycle engine. And the significance of these engines are there are different applications of these concepts. One of the main applications that probably would use a variable cycle engine is the supersonic transport in the aircraft engine. As you are perhaps aware that we have had a supersonic transport aircraft, which was known as Concord which was operational for quite some time. Subsequently, due to several operational reasons, it was withdrawn and its service was stopped. So, at the moment, we do not really have a supersonic transport aircraft. Those supersonic engines are commonly used in military engine, military aircraft. They have not been used in civil aviation or for transport. And it is obvious that supersonic travel is going to substantially reduce travel time. And therefore, there is definitely a need to take a relook at supersonic transport and try and revamp the whole system altogether. But, this definitely requires us to take a relook at what happened really with Concord and why is it that Concord could not be sustained. And so, there are different concepts which are being proposed. And today, we will discuss two such concepts which probably might be used sometime in the future for supersonic transport. So, in supersonic transport, the main advantage as compared to what we have now is the travel time can be substantially reduced. And that is definitely a premium because everybody wants to travel and reach their destination as early as possible. And so, supersonic transport is one of the ways of doing that. And so, Concord had a certain period of time during which it was operational and lot of people got the benefit of traveling at supersonic speeds and reaching their destination much earlier than what they would have if traveled by conventional subsonic aircraft. But Concord had lot of problems due to which it was discontinued. Some of them had to do with safety which was an accident which took place subsequent to which Concord was discontinued. Then, economic viability was another issue because the cost of traveling at supersonic speed was substantially more than what it would be for subsonic travel. And so, it was not economically very viable. And the third problem was related to noise and of course, there are many other issues as well. But these are three main aspects due to which supersonic transport aircraft which was Concord had to be discontinued. And this is the reason why we do not really have a supersonic aircraft which is currently operational even though supersonic aircraft is used very commonly in military engines or military aircraft they have not really been used in civil applications. So, it is time that we take a relook at how we can revamp supersonic travel and what is it that needs to be done so that supersonic travel can again become a reality. So, there are few aspects that will need to be taken care of the key technologies which will need to be developed for achieving supersonic travel for successfully though it has been demonstrated it needs to be now done in a successful and economically viable manner. So, to revisit supersonic travel or supersonic aircraft technology there are different key challenges that need to be resolved. Some of them are basically to do with fuel economy the engine needs to have better fuel economy because that is the key to economic viability of the engine. Then there needs to be increased safety obviously because safety happens to be the key to all these technologies. And the third aspect is to do with the emissions both in terms of the pollutants as well as the noise. These are two parameters which keep getting stringent every few years. And so that is something which such an airplane aircraft will definitely have to meet because emission as well as noise restrictions and norms are getting stringent every few years. And so any supersonic transport aircraft that needs to be developed will have to keep in mind all these norms if the concept has to be successful which means that if we take a closer look at some of these requirements we immediately see that there are conflicting design requirements in the sense that if we look at supersonic cruise because that is what a supersonic transport aircraft needs to be doing. During supersonic cruise there is a requirement for high specific thrust whereas for subsonic cruise which is what would be happening or subsonic flight during take off and climb to the required altitude. We are not really worried about high specific thrust but we need a high propulsive efficiency and a lower noise. So here we have two requirements to design requirements which are conflicting. One of them which is for supersonic cruise needs high specific thrust whereas for subsonic cruise we need low specific thrust which is basically high propulsive efficiency and lower noise. And so conventional turbojet engines which can operate at supersonic speeds can develop high specific thrust but they suffer from lower propulsive efficiency and higher noise and which means that if the same engine were to operate on a in a subsonic Mach number the second requirement for subsonic cruise is not really met that is you end up having a relatively lower propulsive efficiency and higher noise which is not acceptable. And so we need to look at is there a possibility of modifying the existing engines either the turbojet or the turbofan engines in such a way that it can meet both these requirements. So there are there is a need for developing what is known as a flow multiplier device which can basically operate an engine in two different modes that is in one mode which is let us say the supersonic cruise wherein you require a high specific thrust which means you basically need a lower bypass and the high jet exhaust velocity so that you can get a high specific thrust. And the second mode of operation of the same engine would be wherein you would like to operate the engine on a high bypass mode so that the effective jet velocity is lower. So you end up getting lower specific thrust at the same time it also leads to lower noise. So there needs to be a certain method of being able to convert the same engine to be able to operate in both these different modes so that on both modes in supersonic flight as well as subsonic flight it satisfies the design requirements. So there are different concepts which seem to which have been proposed over the years and some of the two of these concepts which seem promising and have been are being researched upon are the mid tandem fan concept and the mixed flow turbofan with ejector. So both these concepts will in some sense try to satisfy the design requirements basically they try to adjust the flow rates during supersonic cruise and therefore the same engine operates in two different cycles one would be an engine without bypass or a very low bypass and in the other mode it operates with very high bypass so which is why it is in some sense a variable cycle engine. So let us take a look at the first concept that is the mid tandem fan concept. Well this concept will require using variable components like compressors with inlet guide winds which are variable it will require auxiliary intakes and variable geometry mixers and turbines and so on. So there are a lot of complexities which are being introduced in order that the engine can operate in different modes and so all these complexities are basically meant to adjust the specific thrust to the required values that is for high supersonic flight you would need a higher specific thrust for subsonic flight you would need a lower specific thrust. So all these variable components are meant to enable the engine to operate in these different modes and so this means that key to success of this concept would require us to design these components which are which not only can be adjusted for different thrust levels but also are aerodynamically efficient at all these different operating conditions. So that is the key challenge for this technology and so there is one schematic that I have here for you which is in some sense explaining what this concept is all about. So in this particular engine concept what is being proposed is to install a fan which is midway between the LP compressor and the HP compressor and so this fan is of a higher bypass than the core engine. So it looks like a turbo fan but the fan is placed in between the compressor stages and there is also an auxiliary intake which can be adjusted to adjust which can be altered to adjust the mass flow through the bypass duct. And after the turbine exhaust you can see that there is a variable mixer as well as a variable nozzle. So the variable mixer also can adjust itself according to the bypass ratio. So for example if this engine was to operate in supersonic cruise mode then you need a high specific thrust and the high jet exhaust velocity which means that we really do not need a very high bypass. So this auxiliary intake is partially closed and so this means that only a small fraction of air passes through the bypass duct and majority of the air passes through the core engine. So we have here an engine which is which resembles a turbojet with a very small bypass and therefore it is possible for achieving very high values of specific thrust because the jet exhaust velocity can be quite high. Now if the same engine was to operate in subsonic mode then the intake can be opened fully enabling more mass flow to pass through the bypass duct and therefore this engine now operates like more or less like a conventional turbofan engine with a high bypass and therefore the effective jet exhaust velocity is lower, specific thrust is lowered and therefore you also end up getting lower noise. So this is one of the means of achieving this design requirements conflicting design requirements for supersonic as well as subsonic flight and so by adjusting the auxiliary intake and also the other components which means that simply adjusting the intake alone would not suffice one would also need to adjust the mixer and the compressor guide vanes and so on. So that the performance can be matched and that is why the whole concept requires a very clever design of these different turbo machine components. So the second concept that is being looked upon is what is known as a mixed flow turbofan with ejector and it basically comprises of a long mixer ejector nozzle and basically this nozzle would entry in outside free stream air and it is mixed with the core flow leading to either leading to basically a cooler and slower exhaust jet which reduces noise substantially and also the specific thrust and so obviously it means that this ejector is required only during subsonic or low altitude operation when the aircraft is operating in a supersonic cruise the ejector can actually be switched off resulting in a high jet exhaust velocity and therefore higher specific thrust. So the second concept is simpler than what we discussed for the first one because here we do not really have complicated turbo machine mechanisms and all it requires is a longer jet pipe which is integrated with the nozzle and also has a provision for entraining ambient air and so as you entrained ambient air and mix it with the core stream it reduces the jet exhaust velocities and therefore noise and specific thrust. So this is the mode in which it would operate during subsonic flight during a supersonic flight the ejector can be switched off and so you it operates like a pure turbo jet with a very high jet exhaust velocity and therefore specific thrust. So one schematic of this concept is shown here in this engine that you can see it basically resembles a turbo jet engine. So this the initial part of this engine is exactly a turbo jet. So what has been proposed is to put a duct around this that is after the turbine exhaust a longer duct here before the nozzle and depending upon the operating requirements the duct can well the ejector can either be switched on or off leading to entrained flow. So this entrained flow from the surroundings will mix with the core flow and the jet exhaust velocities can be altered and therefore this engine can operate in different modes as required. So this is a mixed flow turbofan with an ejector which is also another contender for supersonic transport aircraft. Well an analysis of these two concepts have shown that the first one that we discussed which was a tandem fan concept seem to be in terms of performance better than the second one. But in terms of geometric complexity obviously the second concept is simpler because it does not require too many complicated turbo machine geometries involved there. So these are two different concepts which are being looked at of course there are several other concepts also which have been proposed for supersonic transport and it needs to be seen where which of these will eventually make it in actual prototype. Now the other application that I mentioned where one would need to look at such variable cycle engines is the short or take off and vertical landing or the short take off and landing or vertical take off and landing. These are different modes of operation of an engine which are being proposed and some of them are actually being flown. And that is another application where a variable cycle concept will be required that is if your engine needs to if the aircraft needs to take off with a very short runway length and land vertically which is what is known as short take off and vertical landing aircraft. The engine needs to again operate under two different modes depending upon the operating condition. So in a short take off and vertical landing engine or aircraft we would need a relatively low bypass at high flight speed and high bypass at lower flight speed basically close to ground. And here what is done is that the thrust is vectored depending upon the mode of operation of the aircraft that is if it is if the aircraft has to take off during a short runway length and the aircraft does not really have that required speed so that the wings can generate enough lift. Then the nozzles can be vector downwards to support the weight of the aircraft and so it is basically partly by adjusting the nozzle exhaust and also during different modes of operation like supersonic cruise one would need to operate the aircraft with a lower or higher specific thrust and therefore the engine will need to be operated in different modes like what we discussed for supersonic transport. So this again there are different concepts which are being proposed like a tandem fan concept very similar to what we had discussed wherein we could have two fans which are either used in series or parallel depending upon the application or design requirement. If the fans are operating in series then the whole flow passes through the fan front fan as well as the second fan and it operates more or less like a turbo fan engine with two fans and therefore it has a higher core exhaust velocity and therefore higher specific thrust. In a parallel fan configuration the flow from the first fan is vector downwards and the second fan takes air through the auxiliary intakes giving a higher bypass giving a higher bypass which is what is required for low altitude subsonic flights. And besides the short takeoff and vertical landing there are other concepts which some of them we have already discussed when we discussed about Brayton cycle with modifications that is intercooling reheating regeneration and so on. Out of these three the reheating is one which is very commonly used especially in military aircraft engines and we refer to reheating as after burning in such engines that is after burning is when turbine exhaust already has enough air present wherein you can inject additional fuel and raise the temperature to much higher levels than the turbine air temperature and therefore gain additional thrust. This is commonly used in supersonic military aircraft where the engine has to be accelerated to supersonic speeds and also cruise at supersonic speeds whereas other concepts like intercooling and reheating or regeneration are not necessarily being used or not really been used in aircraft engines but they are in some form being used in land based or in marine applications. The basic reason why they are not being used in aircraft engines is the associated increase in weight and complexity and so the benefits that we get out of using intercooling or regeneration is not being is not higher than the penalty that we need to pay for increase in weight and its complexity. So reheating is one thing which has been commonly used because it does not really require any additional complexities it is just a tail pipe that is attached to the turbine exhaust before the nozzle and so in aircraft engines reheating is usually referred to as after burning and they are used in land based engines for power generation where you have different stages of turbine with reheating in between. In aircraft engines in military aircraft engines it is used at the exit of the turbine. Now this is one cycle diagram which if you recall we had discussed during our discussion on Brayton cycle where we have intercooling which is between the compression stages that is after we split the compression stages into different stages and in between remove air remove heat from the system and therefore we achieve intercooling and reheating is we split the expansion from the turbine into multiple stages and add heat during the intermediate stages and regeneration is when we transfer or store heat during one part of the cycle and transfer it back to the cycle during another part of the cycle. Now intercooling basically reduces the work required for compression and in turn it reduces the turbine work as well and as I mentioned intercooling is not really being used in aircraft engines as of now but it is a topic of extensive research interest because there are lot of performance benefits that one can gain by employing intercooling and therefore there is lot of research currently being undertaken on trying to use intercooling or apply intercooling between compression stages in an aircraft engine application. Reheating on the other hand it has been used for a long time now in both land based as well as in aircraft engines and so reheating is something that can is already being used and its benefits are being already seen. The third aspect that one would like to use is regeneration where in energy storage can be done in the case of aircraft engine in the fuel where fuel can be used as a coolant and you can preheat the fuel by cooling some of the other components and therefore regeneration in some form can be implemented by using fuel as a regenerating fluid. And so these are some of the concepts which probably will like intercooling and regeneration can probably find applications in the future aircraft applications besides the other concepts like what we discussed for supersonic transport and also for short takeoff vertical landing and other such engines. So these are some of the concepts simple concepts that we might get to see in future everything propulsion systems. And there would be many more concepts such concepts which will be discussed in the remainder of this lecture. So professor Roy will now discuss about some of the other concepts which are being actively pursued for and would probably been employed in future aircraft propulsion system. So I will now hand over professor Roy to take up the rest of the last lecture on introduction to aerospace propulsion. We are looking back and looking forward. We will look at what we have discussed over the course of this lecture series and then look forward to what is in store for us in years to come in the field of aerospace propulsion. We have seen that all the propulsion units that are used for various aviation crafts they are basically heat engines and hence they are fundamentally governed by the various concepts of thermodynamics various concepts of gas dynamics. And as a result of which if you need to make fundamental change in the way the engines work you need to go back to the fundamental thermodynamics and fundamental gas dynamics to affect those changes. Now many of these changes that has happened over a period of last 100 years is basically dependent on various thermodynamic concepts and then of course those concepts would need to be converted to technology. The technology that is available at that moment of time the state of art of the technology that is available to the engine designer at that moment of time. So quite often conceptually thermodynamic concepts are available for quite some time before they are indeed converted to an useful technology. Now this is something which has been known to us for a long time many of the engines that have been devised essentially came out of the various concepts that fundamental thermodynamics had thrown to us quite some time back. And let us take a look at some of these concepts that are in store for us in future and we shall see how these concepts are indeed born out of fundamental understanding of how these propulsion units work the fundamental thermodynamics the fundamental concepts of basic aircraft and rocket propulsion that we have been talking about over the course of these lecture series. Now we know that a heat engine the kind of heat engine that is normally used in aircraft basically have higher efficiency if you have higher pressure ratio and higher pressure ratio directly translates to higher thermal efficiency. Now the kind of engines that have been used over the years have tried to enhance this pressure ratio more and more and more by using various technology that is available. And at various points of time there are limits within which these engines need to be created or configured. So there are technological limits within which the engines would need to be configured. And sometimes the concept that if you get higher pressure ratio you get better thermal efficiency and everybody wants engines with higher thermal efficiency. So you try to push the technology to accommodate higher and higher pressure ratio. And we will have a look at today what are the other concepts that contribute to better efficiency. And in terms of utility value of aircraft engine and the rocket engine how the fundamental concepts contribute to development of some of these engines. Now as far as the piston engines are concerned which we discussed in the course of this lecture series they were actually powering the aircraft for 50 years before the jet engines came in. And they continue to power many of the aircraft that are flying around the world especially the small aircraft. And they are still competitive they are still quite good in that particular range of application. The point is everybody wants to have even better engine, everybody wants to have more efficient engine. Now if you take a piston engine it is pretty much known that if you have again have higher pressure ratio you can get more and more efficiency thermal efficiency out of those engines. The problem is that if you go for higher pressure ratio you end up having engines that are very heavy because to withstand that high pressure inside the cylinder the engines need to be made stronger and made of thicker material and hence they become heavy and non viable for aircraft application. One of the candidates which have been around for a long long time and has never been considered very seriously earlier is the diesel engine or more specifically the compression ignition engine C I engine. Fundamentally the C I engine is heavier and as a result of which it was never considered very seriously for aircraft application. However in the recent years the C I engines have again been considered and one of the reasons the C I engines are being considered is because the C I engines can now be made of modern light and strong material which can withstand the high pressure that I created inside the cylinder. And as a result of that you can use the C I engines and hence the diesel engines. And now these are known to be thermally more efficient devices than the normal aviation kerosene aviation fuel that uses that are used in normal piston engines of aircraft usage. Now if these are technologically possible the C I engines would provide better fuel efficiency and that is always a sort after thing when it comes to aircraft applications because you need to carry then less amount of fuel with you. Now this is one kind of engine that is being developed and if the new materials that are new alloys that are being created the titanium alloys or the various kinds of aluminum alloys if they are strong enough to withstand the high pressure that is inside a C I engine they become a very good candidate for application in aircraft and the thrust to be created by the propeller propellers of course have been around for more than 100 years. So hooking up a propeller with a diesel engine is something which people are working at and it looks entirely possible that in years to come we may have these kind of engines powering aircraft propellers and flying them with greater efficiency. The other thing that is looking up is the small gas turbine engines which are basically used for jet engines powering propellers to create thrust. Now over the years what had happened was gas turbine engines when they became smaller and smaller they lost out in efficiency very small components of gas turbine engine do not promise to have same kind of aerodynamic or gas dynamic efficiency as the larger ones. And as a result of which for a long time small gas turbines were not considered very efficient devices and indeed even the piston engines were considered more efficient than small gas turbine engines. Now over the years because of the advancement of manufacturing technology the fabrication technology the material science and metallurgy it is now possible to create very small gas turbine engines which are also competitive in terms of overall thermal efficiency. Now this then makes another candidate for small aircraft for powering the propellers and creating thrust for small aircraft. So small gas turbine engines are becoming more and more competitive and years to come we will see that more and more gas turbine engines would be used. One of the reasons the gas turbine engines would be considered a good candidate if they can be made in small sizes is because gas turbine engine is intrinsically fundamentally a more efficient heat engine and that is the reason why people are still looking at gas turbine engine as small ones which can power aircraft power plants. The other device which we have touched upon is the prop fans which we have introduced once before let us look at it little more closely. We have seen that the fuel efficiency of an aircraft actually is enhanced if you introduce bypass devices. Now once the bypass is introduced the specific fuel consumption is vastly improved. Now this is a known fact for more than half a century and that is the reason why we have so many turbofan engines. Now if you stretch that concept a little more and introduce prop fans which are indeed devices which are somewhere in between a fan and a propeller. Now if you consider propeller is that where the entire mass flow bypasses the engine and creates thrust prop fan is something in between where a very large amount of mass is activated by the prop fan and if you talk in terms of bypass ratio we are talking in terms of bypass ratio of the order of 20 or 30. You see the normal bypass ratios of the big fan engines that we see today powering the commercial aircraft like jumbo aircraft or have a bypass ratio of the order of 5 or 6. Even the modern airbus or the modern Boeing aircraft 7, 7, 7 for example they have engines with bypass ratios of the order of 6 to 7. And of course people are talking about bypass ratios of turbo fans of the order of 9 to 10. Now we are looking at these prop fans where the bypass ratios are indeed of the order of 20 to 30 and the fuel efficiency that accrues out of this high bypass ratio very high ultra high bypass ratio is huge and this is being developed for quite some time now. You can see the shapes that have been developed and some of these have been around for 10, 15 years. These are becoming more and more efficient the designers are walking on them and they are getting more and more exotic shapes to create more efficiency. But the fundamental fact that the high bypass ratio very high bypass ratio gives a very high fuel efficiency is the biggest attraction and that attraction is what has kept them alive and now it looks entirely feasible that in very short while now we shall see these prop fans operative and flying aircraft specially the subsonic civil aircraft all over the world. They are extremely efficient devices and as I said indeed actually more efficient than the big fan engines that you see today flying around. It is simply a matter of time before these prop fans indeed take over flying the big aircraft. One version of prop fan that indeed again is being developed very seriously is known as counter rotating prop fan and we had a glimpse of it once before. Let us take a closer look again in this configuration you have two prop fans really one which is the front one and one which is a rear prop fan which is rotating in opposite direction and there is a technological device by which the counter rotation is affected. The two prop fans are driven by something like a big gas turbine engine and the shaft that comes out has a gear box which creates the opposite rotation of the rear fan and as a result of this you have two prop fans rotating together creating thrust. Now this prop fan compared to the earlier prop fan that we saw indeed would have much more thrust making capability. So it retains the efficiency of a prop fan and then nearly doubles it by creating more mass flow and more thrust making capability and indeed retains the huge fuel efficiency that accrues due to the very high bypass ratio that it can create. So this kind of counter rotating prop fans are indeed on the anvil and it is very likely that over a period of next few years we shall see many of them flying around the world powering very large aircraft. The design of these prop fans are indeed created using a combination of propeller theory which we have done in the course of our lecture and the compressor blade design methods which have been around for close to 50 years. So many of these theories that we are talking about or have talked about in the course of our lectures are combined together to create these prop fans and indeed the counter rotating prop fans. So you have the theory, you have the fundamental concept that high bypass ratio gives you better fuel efficiency and as a result of which you can create better thrust making devices. Now one of the issues or problems of these prop fans have been the noise. They make a lot of noise and some of the noise is indeed aerodynamic noise and the aerodynamic designers of these prop fans are now working on these machines or these blade shapes to reduce the noise and conform to the present noise regulations. So that the huge fuel efficiency that it promises can actually be made use of in actual operation. So noise is one of the issues that needs to be sorted out and once that that is sorted out we shall see many of these prop fans flying around the world. So the issues that indeed the modern engines would have to deal with at the stage of its concept and then at the stage of its technological feasibility study and finally making it the two concepts are one is the energy audit. How much energy the entire engine is making use of to create that amount of thrust? The thermal efficiency comes into picture, the entire energy usage of the fuel and other devices that power the aircraft would have to be taken into account and hence all aircraft propulsive devices would have to conform to very stringent energy audit from now onwards to be competitive in the market to be useful in the market for the operators to make reasonable economic profit out of it. The other conformity that is strongly required is the environmental audit. The environment audit that needs to be done is again to conform to the modern regulations environmental laws that have been put in place all over the world. Noise is one of the issues most of the aircraft engines over the years where indeed extremely noisy much of the noise have now been reduced to a very low level probably some more needs to be done and some of these noisy elements including the jet or including the fan and the big fans or the prop fans would have to be considerably redesigned to conform to the new noise regulations. The jet engines that are used would have to conform to pollution regulations more and more stringent pollution regulations are coming and have already been put in place and these have to be conformed to unless they are conforming to these energy audit and the environmental regulations these engines in future would not even be certificated. So, they would not even get a certificate to be operated and hence the new engines born out of the concept that we have talked about it may be thermodynamic concept, it may be concept due to the bypass engines or any other concept when the technology is finally applied one would need to conform to energy audit one would need to conform to environmental audit otherwise these engines would not be allowed to be flown they would not be certificated at all. Let us look at some of the more complicated engines that people are now looking at looking at very closely and some of them may have applications which are not necessarily civil applications these may have applications which are probably military applications. Now, what one of the concepts is we have talked about is the ramjet engine which takes a craft or a craft or a missile to hypersonic speeds. The point essentially is that ramjets do not operate at low speeds and hence you cannot take off a craft with the help of a ramjet. So, the concept is that if you combine a turbojet with a ramjet the turbojet will help it take off take it to some high altitude and at some high Mach number and from there onwards the ramjet would take over and take it to hypersonic speeds. So, these are concepts which are people are working on and it is simply referred to as a wrap around turbo ramjet where a ramjet is essentially wrapped around a turbojet. So, the outer annulus is essentially a ramjet and the inner core is a turbojet a normal turbojet when the craft is taking off it mainly works in turbojet mode and during take off and climb and climb to very high altitude it works in turbojet mode and once it is going to very high altitude and very high Mach number may be Mach number around Mach 3 then the ramjet takes over and takes it to even higher Mach number like Mach 6 or Mach 7. So, it is a combination of turbojet and ramjet and this allows the craft to take off from ground and even come back to the ground while landing at the same time going all the way to hypersonic speeds while flying at extremely high altitude above the earth surface. So, this is one concept that people are working on which will allow craft to go hypersonic from take off to hypersonic and then back to landing. This is a thermodynamic concept on which the turbo ramjet is being conceived you have done the cycles and there are hence possibilities which we can look into you have the normal cycle over here which is the turbojet cycle and then of course, you can have the afterburner on which will take it to Mach 3 you can have it operate without afterburner at low altitude and low Mach numbers then put the afterburner on which is a reheat engine and then take it to Mach number 3 or so and from there onwards you have the ramjet engine which then takes over and operates it at hypersonic speeds. So, it is basically again one can probably call it a variable cycle engine where during the flight itself it changes from turbojet with without afterburner then turbojet with afterburner and then a ramjet engine. So, it is a kind of convertible engine one can say which allows the craft to take off, climb, go hypersonic and come back and land. This is another concept which again allows people to fly very high supersonic to hypersonic or let us say from low hypersonic to high hypersonic. At low hypersonic flight conditions a ramjet engine is known to be quite useful and very good and provides thrust whereas, once you think in terms of hypersonic that is Mach 8 or above one would have to think in terms of scramjet engine. Now, scramjet engine is something we have done before and this is a concept where you have a ram and a scram together in one engine in which part of the combustion is done subsonically and part of the combustion is done supersonically allowing the craft to operate both under low hypersonic as well as under high hypersonic conditions. So, that one can operate reasonably efficiently under a wide range of hypersonic flight Mach numbers. So, these are devices which take craft to very high Mach numbers to hypersonic Mach numbers. This is a missile configuration. Now, this missile is basically finally is going to fly supersonic. So, if you look at the nose of the missile you will find a supersonic spike over there and that has been created essentially because some part of the missile flight is going to be indeed supersonic and during which this nose over here will indeed create shocks and the nose is then created to negotiate those shocks create minimum drag and fly the missile efficiently through those supersonic flight conditions. Now, as far as the propulsion device is concerned you can see there are essentially three stages the big first stage over here and then the nozzle of which is shown over there and then you have the second stage the nozzle of which is seen over here and then you have the third stage and the nozzle of which is seen over there. So, this is a very simple way of trying to show how a missile can actually be powered by a three stage propulsive device and many of these missiles depending on their flight range depending on their mission quite often have combinations of various kinds of propellants. We may have a combination of solid propellant first stage, liquid propellant second stage and may be solid propellant third stage again. So, combination of solids and liquids are often used in multi stage rockets. So, that they are optimally utilized and you have a optimal thrust making capability at various stages as you are probably aware that in such rocket flights once the first stage is used up the entire first stage is indeed ejected and it falls off. Similarly, once the second stage is used up it actually falls away and only the third stage then flies the main payload which in military applications could be warhead to its destination and a little before it reaches the destination the third stage also falls away. So, these rockets are essentially stages are essentially ejectable and they get ejected at various stages in a very planned manner and this plan is done by the designer. He designs very accurately at exactly at what point of time the first stage or the second stage or the third stage would fall off. So, these are designs which are created and it depends on the missile it depends as I said you could have a single stage missile you could have a two stage missile or you could have three stage missiles and some of the intercontinental ballistic missiles even may be four stage missiles. Some of the missiles do indeed use hypersonic go to hypersonic flights and they often use ramjets or scramjets for their missile powering. So, missiles are often powered by rocket, but in some cases the missiles may be powered by ram or scramjet engines also especially when they have long crews going over hundreds or may be even thousands of kilometers. Some of the rockets that we have discussed we may look at into the future of what we are moving into you are all aware that in India we have already launched Chandrayan which is our first step towards going to moon and some first launch has already been successfully done we have already retrieved some material and that experience has been loaded by all over the world and it is a successful experience. Now, this particular Chandrayan-1 was launched by PSLV what is known as PSLV or Polar Satellite Launch Vehicle and that rocket was used to launch the Chandrayan-1 it is a four stage combination of solid and liquid propulsion systems alternately and the first stage is the one of largest solid field rocket booster and it is indeed one of the largest solid field rocket boosters in the world and this has been used for launching Chandrayan-1. So, it is a combination of solid and liquid propulsion system. What we are looking into is Chandrayan-2 it will be launched with the help of GSLV which is Geosynchronous Satellite Launch Vehicle and this uses a more complicated more involved rocket propulsion system. It has four liquid strap on boosters if you look at the picture here you will see these boosters which are wrapped around the main core of the rocket. So, there are four of them you can see two of them on this side. So, there are four of them these are liquid propellant rockets and they use hypergolic propellants which are essentially the propellants which once they come into contact with each other they immediately ignite. So, you do not need a igniter to ignite them moment they come into contact with each other they get ignited or the combustion process starts. The propellants in use are UDMH which is unidirectional dimethyl hydrazine and nitrogen tetroxide. So, these are hypergolic propellants and they have been used in this liquid strap on boosters which you see around here which are on the outer side of the main rocket core. The first stage this is again three stage a first stage is of solid fuel which is just inside here. So, outside we have the four liquid boosters and inside we have the first stage which is a solid fuel rocket and then the second stage again is a liquid again of UDMH and nitrogen tetroxide which is a second stage and then you have the third stage which is a cryogenic rocket which we have talked about a little and this uses liquid oxygen and liquid hydrogen and this is a new technology which has been used in India comparatively recently and this is going to be used in the GSLV3 D3 which will power Chandra and 2 which we are hoping will take some Indian to moon. So, this is the technology that we are looking at for the future a very mature technology a combination of liquid and solid propellants being used and cryogenic rocket propulsion being used to power our flight to moon. So, this is a coverage that we have done in this course covering some ground on aircraft propulsion a little ground on rocket propulsion trying to bring to you the fundamental concepts that are required in making of these crafts and how the crafts are developing over the years and what we can look into in the few years to come. I would very strongly urge you to read books not just look at the internet websites, but look at the books because that is where you get the depth of the material that is where you get the theories that actually conform to the various things that we have discussed and theories that are indeed used for making this various kinds of engines. I hope we have been able to bring to you some of the fundamental concepts and I hope it enthuses you to read more books and more material so that you get more and more knowledgeable about how aerospace propulsion works.


This lecture is part of an online algebraic geometry course on schemes and will be a review of the operation of localization so We recall that we defined the spectrum of a ring and Stated it was a locally ring space, but we haven't yet got around to actually showing that it's a locally ring space Which mainly requires checking the sheaf axiom in order to do this. We need to know a fair amount about localization of rings So we're going to review that this lecture. So we recall that suppose R is a commutative ring and S is a subset and We want to invert all elements of s so for example When you're constructing the rationals you take the integers and you want to invert all non-zero integers Well, if you can invert Two numbers you can invert their product. So we're going to assume that s is closed under Multiplication and you can certainly insert one. So we may as well assume that s contains one So the subset with these properties is sometimes called a multiplicative subset We don't really have to assume that but it's convenient and if you've got any subset you can just make it multiplicative by adding one and Making it closed under product. So so these are harmless assumptions So we're going to construct a ring R s minus one With the following properties first of all, there's a homomorphism from R s minus one and the images of S should be invertible and it should be universal with these properties So if we've got any other map from R to a ring T with the image of S being invertible Then we want to have a unique map from the localization to T and You can check this property actually Characterizes this ring here. So how do we construct this ring? Well, it's very easy All we do is we forcibly add inverses. So we can write R s minus one Is equal to R and now we just add variables t1 t2 t3 and so on and Quotient out by the ideal S1 t1 minus one is equal to zero s2 t2 minus one is equal to zero and so on So this this just forces Ti si to be equal to one So so we've sort of forcibly added an Inversed each element si Well, that's fine But there's a bit of a problem that it's rather hard to see how big this ring is So we can ask how big is R s minus one Well, it's quite easy to check you can write every element of this in the form R Times R over S for some S in S So that sort of gives an offer bound on it. The problem is When is an element of this form zero? So we've got a homomorphism from R to R s minus one rather obviously We can ask what is the kernel so the kernel is going to be some ideal and sometimes I can be zero so if R s equals naught for some S in S then Then R must be in the kernel I Because if s is invertible and R times s is zero then R must be invertible So it's certainly sometimes possible for this to have a non-zero kernel And the obvious guess is that the kernel consists exactly of elements killed by some element of S So let's put I to be the set of R Such that R s was not for some S in S And you can check this I is an ideal so not is an I because Nought times one is equal to zero This is where we use the fact that S contains one and if R one R two are in I And R one S one equals zero R two S two equals zero so R one plus R two Times S one S two equals zero and this is in S because S is multiplicative So I is closed under addition So this is why it's convenient to assume S is multiplicative It makes it easier to define this ideal I Um So we want to show That I is the kernel of the map from R To the localization R s to the minus one in other words an element Maps to zero in here only if it's really obvious that it maps to zero Because it must be killed by some S And we're going to do this by constructing this Localization in a slightly different way and we're going to do this in two steps So the first step is first assume that S has no zero divisors In other words, I is equal to zero and then we can just copy the construction of Q from Z. So you remember That when you construct the rational numbers from the integers or from that matter any quotient fields of an integral domain you Do it as follows So the elements of R s to minus one are going to be given by ordered pairs R s And we're going to write the ordered pair R s as R over s because if we write it as R comma s it's impossible to remember what it means Um modulo the equivalence relation where we say R one over s one equals R two over s two Different only if R one s two equals R two s one So we take all ordered pairs quotient out by this equivalence relation And now we can define additional multiplication in the obvious way So R one over s one plus R one two over s two R one s two plus R two s one over s one s two and R one over s one plus times R two over s two is defined to be R one R two over s one s two remember this this is really a definition of the sum on the product and these things are abbreviations for the ordered pair Consisting of that element and that element or rather the equivalence class of that ordered pair And now to show this is a ring we have to check a large number of things. So we want to check first of all that this is an Equivalence relation Secondly, we want to check that addition and multiplication and subtraction are well defined I guess we should have defined zero and one but it's obvious how to define those and three we want to check the axioms Or a ring Okay, well, that's a dozen or so things to check and I'm not going to go through and check them all because Almost all of them are completely routine. There's one subtle point so You may say where in all this do we have to use the fact that s is no zero divisors well everything goes through No matter what s is except for the following question is the equivalence relation transitive So this turns out to be the one Routine check that isn't quite as routine as you might think So suppose R one over s one is equivalent to R two over s two and R two over s two is equivalent to R three over s three Well, this means R one s two equals R two s one R two s three equals R three s two And now if you can multiply this one by s three and this one by s one and subtract you get R one s two s three Is equal to R three? S two s one so s two but R one s three minus R three s one is equal to zero So R one s three minus R three s one equals zero if S two is not the zero divisor So this is the point where we need to use the fact that S is no zero divisors and of course this implies that R one over s one is equivalent to R three over s three and this proof really does break down if If s has zero divisors and if s has zero divisors this construction does not actually work the equivalence relation isn't transitive and We don't get a ring So how do we fix this? So let's now do the case suppose S has zero divisors well now we just replace R by R modulo i where i is as as above i is a set of R such that R s equals north for some s and s And now the image of S in R over i Has no zero divisors so we can form so we can form R over i And then invert all the elements of s or rather the images of the elements of s Well in the previous construction we noticed that if i is equal to naught the map from R to R over i Sorry to R s minus one is actually injective R is actually a subring of this And you can check this directly from the equivalence relation. So we see what is happening is we're getting a map from R to R over i And this is a subset of R over i s to minus one and We've also got the ring R s to minus one and you can check that these are actually the same In fact, you can define that ring to be that ring if you like so So I is the kernel of the map from R to R s to minus one So this just says that R over s is equal to naught if and only if Rt equals naught for some t in s So so this gives us control over the localization We can tell whether something is zero and of course if we can tell whether something is zero We can tell whether or not two things are equal just by subtracting them There is actually one step construction Which is the usual one you'll find in books This is where you define R1 over s1 to be equivalent to R2 over s2 if R1 s2 minus R2 s1 times s equals nought for some s Well, that sort of works, but it's a rather unintuitive definition of the equivalence relation It's not at all clear why you suddenly put s in there or why it works So I think it's clearer to construct localization in two steps You first kill off this ideal i and you then do the construction for when s has no zero divisors So an important special case is when s is the complement of a prime ideal P you remember a prime ideal has the property that if x y is in P Then x is in P or y is in P and if you think about it This just means the complement of P is multiplicative if x and y are not in P Then their product is not in P And in this case the localization r s minus one is often denoted by rp That's often a gothic letter, but I can't draw gothic with with these pens um so um, it's easy to confuse rp with the quotient r over p and in fact these are Completely different operations. They're sort of almost complimentary operations And to see this let's look at a few examples And so first of all, let's just look at the example when r is equal to c of x Then and then we have two different sorts of prime ideals p. We could take p to be zero Or p to be x minus alpha for some alpha We don't really care what it is And what I want to do is I want to draw the spectrum of the quotient And the spectrum of the localization so you can see what the difference is And so first of all the quotient by zero is just the ring r So it's spectrum sort of looks like this. It's got a generic point And it's got a lot of points on it and it's just the affine line um and If we're quotient out by this ideal, then we've just got the field c So we've just got a point here, which might be say this point So that's the quotient If we take a localization what we're doing If we localize at naught, then we've just got the field of rational functions and this only has one point So we just end up with a single point, which i'm drawing as a line because it's a very big point and If you localize at x minus alpha it means you're taking all rational functions Whose pole is not at this point here So this is the point alpha And as we saw earlier the spectrum is two points. It's got a generic point And uh one point there And if you look at the dimensions of these you find this is dimension one. This is dimension zero This is dimension zero and this is dimension one This point sort of looks one dimensional, but it's really zero dimensional because it's got nothing inside it um so um It's not quite obvious what's going on yet, but it may be clear if we look at a more complicated example So now let's look at r equals c x y We have three sorts of ideals. We can take p equals naught or p equals multiples of an irreducible polynomial or p equals So x minus alpha y minus beta Let's draw pictures of all these well, um, let's draw pictures of r over p And the localization r p in order to see what the difference is well as usual r over p It's just the spectrum of the whole space which you remember has um Some it has a generic point it has a lot of curves on it And it has some closed maximal points um, and as before if we localize At zero we just get the field of rational functions, which now only has one point And now we see this is dimension two and this is dimension zero If we quotient out by f then we're just getting the spectrum of the coordinate ring of this curve So the spectrum just looks like a curve look like whatever curve we The curve f equals zero and and and it still has these maximal points on it And finally if we Quotient out by this ideal then we're quotient out by some that then we're just Uh quotient is just a field so we just get a single point So so this point might be this point and it might be say this point So You can now see these have dimensions these quotients of dimension two one and zero Now let's take the localization. Well, we worked out this earlier and the localization here kind of Looks like this. It's got a point and it's got a lot of ghosts of one dimensional lines And it's got a generic point And it's sort of two dimensional And so remember dimension means it means the length of the biggest chain So we can have points in a line and uh and a plane. So we've got a chain of Of three prime ideals, which is length two um this one you remember we've worked out its spectrum and its spectrum has two points so it has the The maximal ideal p and it has a zero ideal Which sort of looks like this And now you may think this yellow bit is two dimensional the blue bit is one dimensional In fact the blue bit is zero dimensional and the yellow bit is one dimensional And it's got stretched out a bit so You see in each case the dimension of the quotient Plus the dimension of the localization is equal to two Which is the dimension of the thing you're started with and what is happening is If you've got some closed sub variety given by a prime ideal The quotient is kind of telling you what's going on inside this sub variety. For instance, if you take this curve Then taking the quotient of the coordinate ring is telling you what's going on inside this curve Or is taking the localization is telling you what's going on outside in the sort of neighborhood of the curve So and again if you pick a point here this localization is telling you what's going on outside this point nearby As I mentioned earlier, this is the reason for the name localize Localization you're sort of picking a point here and just looking locally at it So you might be looking locally at this line and if you look locally at this line You only see this bit here and if you look locally at this point, you only see this bit there. So this is a sort of Magnificent these are magnifications of neighborhoods of these points in some sense The other thing to be aware of is that when you go from ideals to Algebraic sets you invert the order so bigger ideals correspond to smaller algebraic sets. So here the um, what you're doing is when you Take a quotient You obtain you're left with all ideals that are bigger than this ideal But when you look at it geometrically You're looking at all sub varieties that are smaller than the sub variety you started with so so here for instance the localization will that's on the quote the prime ideals of the quotient the all prime ideals bigger than this whereas the spectrum contains all Irreducible sub varieties that are smaller than this blue one On the other hand when you localize It's kind of the other way around so here When we localize we only keep the ideals that are smaller than this And geometrically that this means we only keep the sub varieties containing the point where we localized So it's it's kind of all rather confusing because everything gets turned upside down when you switch between ideals and sub varieties um So you you can think of Taking a quotient as making P minimal In fact, you're not just making minimal you're making it the zero ideal whereas localizing is making P maximal You invert everything not in p which obviously forces p to be a maximal ideal and similarly When you localize at a sub variety you're kind of making that sub variety minimal So if we localize here, we're making it a minimal sub variety and somehow getting rid of all the points at it When you take a quotient you're making the sub variety maximal by throwing away everything outside it Okay, that's enough about localization and next lecture we will be Discussing why the spectrum of a ring is indeed a ring space