 Let us move on to the next topic which is one of these additional topics and this is what we call introduction to compressible flow, I am Balchandar Puranik, technically this is slightly different from your standard thermodynamics business I will say. Compressible flow as many of you would know is really part of fluid mechanics, however this is a special branch of fluid mechanics in the sense that it is a combination of fluid mechanics principles and thermodynamics principles and therefore in some situations, in some courses, some places some of this material is included in the thermodynamics course especially toward the end. In IIT as far as I know some instructors included in the thermodynamics course toward the end of their course some instructors do not and typically in fluid mechanics the standard undergraduate fluid mechanics class that we have at present does not include the compressible flow material. So, in that sense in case your fluid mechanics class does not have compressible flow included maybe it makes sense to include it in the thermodynamics part if it can be permitted. As somebody had said I think two days back in the initial session obviously there are places where a complete semester long course on compressible flow exists and in that case you can do a far more detailed but what we decided is that at least let us look at some of these basic ideas and in case there is any requirement in the final workshop we can expand these and include those. So, it is really the choices yours at the end of this I suppose you can decide whether some of this material can be included in the final workshop or not. So, what I have done is I have just put together a bunch of slides for two reasons one is that the time is about one hour or probably even less now. So, I thought let me just give you a brief overview of what we are going to do if this material is going to be included in the final workshop and this is typically what we include in a thermodynamics course. So, let me just go through this. So, what we will go through is a discussion on the following topics. So, conservation of momentum speed of sound in normal shock analysis and finally a discussion on steady quasi-1D isentropic flow analysis. We will discuss each of these in as much detail as time permits. But right at the beginning when I said that compressible flow is actually part of fluid mechanics and uses sufficient amount of thermodynamics ideas. The first point here will emphasize that the conservation of momentum. So, conservation of momentum is something that you normally do not see at all in thermodynamics. It is a concept that we normally utilize from fluid mechanics. However, when it comes to compressible flow analysis it is going to be required. So, I do not know really when at your institution when students go through a thermodynamics course whether they have completed a fluid mechanics class or they are simultaneously doing it or I do not know exactly what the situation is. But here for our purpose what I am going to do is I am just going to talk about what is the overall principle of conservation of momentum as employed or applied to an open system or what we will call in fluid mechanics a control volume rather than getting into the detailed mathematics of the conservation of momentum principle. I will write it in plain English so that hopefully we can understand what is going on there and then utilize it directly for a couple of situations immediately afterwards namely the determination of speed of sound and for the normal shock analysis. And finally, when we will come to the steady quasi-1D asymptotic analysis we will try to understand what that means. So, here is something this is a slide which I am sort of recycling from the CFD workshop that we had run several months back if anyone of you was part of that workshop the first part of that workshop included fluid mechanics as was required for CFD and then the later part was CFD. So, this was one of the slides which I had used in that fluid mechanics part to simply convey what is meant by conservation of momentum when we employed for an open system or equivalently what we call a control volume in fluid mechanics. I do not really expect that you will write all this down. What I am planning to do is for whatever it is worth I will put this set of slides on model so that in case you want to go back and download it and look through it you can do that. At present my opinion is that it is sufficient if you just look at the screen and simply go through whatever is written out as I discuss it. Of course you are free to write it if you want but there is no need. So like I said in fluid mechanics we call this a control volume I think in thermodynamics you will call it a an open system and a balance statement just like what you would write for a mass balance or an energy balance for an open system we write it for momentum. Now it is not that obvious actually when it comes to quantities such as momentum what this balance statement is going to be but try to couch it in the same form as what the mass balance statement you are aware of and what the energy balance statement that you are aware of. When we are talking about control volume or open system we are necessarily talking about fluid flow coming in and going out of the control volume and as the fluid flow comes in and goes out along with it it brings in certain rate of momentum and takes out a certain rate of momentum and those are the two first terms on the right hand side of this equation. Finally it is something like an account balance statement as you can think of and for a control volume or this open system what we will say is that the rate of change of momentum that is contained within the control volume is simply going to be the rate of inflow minus the rate of outflow plus in case there is a source term available within the control volume because of which there is a chance of increase of momentum within the control volume. This is exactly the same the overall statement is exactly the same as what you would write for a mass balance statement when it comes to control volume or open system. There you would write rate of change of mass contained within the control volume or open system is equal to the rate of inflow of mass minus the rate of outflow of mass plus the rate of increase of mass if there is a source. So now let me just take a minute to say that what we mean by rate of increase due to a source as far as mass is concerned. So in case you are dealing with a situation where your open system or control volume contains nuclear type of reactions where mass is actually created or annihilated through nuclear reactions using the famous Einstein's law. Then you would require a rate of source to be included for the mass balance. Normally in standard engineering applications we never are dealing with such situations and when we write the mass balance for an open system or a control volume customarily you will realize that the last term which is the rate of increase of let us say mass due to a source is never really brought into picture. We simply say that the rate of change of mass within the control volume is simply going to be equal to the rate of inflow minus the rate of outflow of mass that is it. That is the standard mass balance statement. So in the same way you can come up with a balance statement for control volume momentum situation as well and that is what I have written out here. So the only thing that is worthwhile pointing out in the momentum balance statement is what do we visualize and what can we convey to the students as a source term for momentum. And typically the easiest way to convey that is that what we are talking about here is an open system. So that there is a mass flow in and there is a mass flow out. So from instant to instant the mass content within the open system or control volume is going to change. That is fine but at any given instant if whatever mass content exists inside the control volume if there is a net force acting on the mass content or the material that is contained instantaneously within the control volume that is going to be treated as essentially a source for momentum. Because Newton's second law of motion tells us that if there is a net force acting on a certain fixed material its momentum is going to change. It will experience a rate of change of momentum and that is the way one can come up with a balance statement and that is what I have written here. The other thing which I would like to point out immediately is that just like what you have in mass balance statement if you are dealing with a steady flow situation there is no rate of change of mass within the control volume in which case what we say is that whatever is coming in is whatever is going out. I cannot resist it here but this is a typical situation where in class I will say that many times this is a situation with the students. You consider your head as the control volume and whatever information is coming in if it goes out at the same rate nothing is getting accumulated and that is not a good thing. So your head should not be a steady state system is what I jokingly point out. But yeah if we are dealing with steady flow situation there will not be any rate of change of mass momentum energy whichever one that you want to talk about within the control volume and in that sense you can simplify this balance statement. Now those who have background in fluid mechanics and those who have gone through fluid mechanics particularly can relate these terms and relate to these terms and can put these in mathematical form. I am not going to do that here the reason is because we are going to immediately employ this for a fairly straight forward situation where we have a one dimensional flow and we will see exactly how this statement is going to be employed. So then moving on it turns out that the speed of sound is one of the most important parameters that one has to deal with when it comes to compressible flow analysis. So what it really implies or entails is that we are talking about some sort of a wave propagation through the fluid medium and sound is just one type of wave. So what I have tried to point out here is that in general when it comes to compressible flow analysis we may have to deal with different types of wave propagations. So one type of wave is what many of you would know something called a shock wave. Some other people will be also aware those who have background in compressible flow analysis is that there is some other type of wave possible which is called as an expansion wave and so on. So what we mean by wave motion is essentially that let us say you introduce some disturbance at some location. For example right now when I am talking I am continuously introducing some sort of a pressure pulse or a energy pulse at this location and this increases the local molecular activity at this location. So what happens is that when you impart a certain amount of energy at location it will go into increasing the molecular activity at that location. Likewise what ends up happening is that these higher energetic molecules will move this energy pulse outward by bringing out more collisions away more and more away from the location where it all started and therefore the physical picture associated with wave propagation is that the molecular collisions is essentially responsible in transferring this pressure pulse from one location to the next location and so on. So that is what I have pointed out in the couple of slides, so couple of points on the slide at the top. While the molecular picture is fairly well known for these wave propagation we do not really have to bother with dealing with that when it comes to a course like thermodynamics or even fluid mechanics. What we end up using is a standard model where we say that let us represent this wave with what we call a wave front across which the fluid properties are going to change. So you can imagine and you can try to point out to the students that sound wave is something that you can visualize or represent using a model like a wave front across which there is a change in the properties from P to P plus delta P let us say pressure likewise temperature likewise density likewise velocity and then this wave front is moving with a certain speed which I am calling C here for a general wave propagation. So what I have shown is in the picture a standard one dimensional representation of a wave propagation picture where the wave is shown as a wave front which is moving with a speed of C in general. The region of fluid into which it is moving has properties pressure and temperature density etc. given by P rho T and just behind the wave front the properties have been changed by amounts delta P delta T delta rho etc. And if it turns out that these changes are infinitesimally small then that particular wave front is what we call a sound wave front. And this has actually experimental backing in the sense that people have measured what is involved in sound wave and it turns out that the sound waves as we know fairly well approximate what I have put it inside the box namely it can be actually shown that the sound waves do form wave fronts across which the fluid properties will change infinitesimally. So we will have P to P plus delta P where delta P is very very small. So this is what typically we would like to begin when we want to talk about what is meant by speed of sound and the information that you want to convey to the students. So formally we want to start with a reasonably general picture of wave propagation and bring about that molecular collision idea. Eventually when it comes to the analysis we say that the experimental evidence suggests that it can be modeled as a wave front across which the conditions will change by certain amount. If it so happens that those changes are infinitesimally small then we essentially have by definition what is called as a sound wave. So with this background let me just go through the formal derivation which is doable in this case to determine the speed of sound. So now the previous slide was sort of a general description of wave propagation. Now I am specializing to speed of sound and most authors prefer calling that speed of sound with the letter A. Again it is no hard and fast tool but typically at least in fluid mechanics you will see that people prefer using the letter A. C has also been used but A is something that I am using. So what we have here is this wave front traveling in air. We are necessarily restricting ourselves through dimensional situation. The wave front is traveling with a speed of A in two conditions which are P, T, rho, pressure, temperature, density etc. And just behind the wave front the conditions are changed to P plus delta P, T plus delta T and so on where again since we are talking about sound wave the changes are necessarily very very small. So infinitesimal changes. So for the analysis what we do is the top picture is something like you are sitting in the lab coordinate let us say and you create a sound wave and you see that it is propagating in front of you. The conditions in front of the sound wave are given by P, T, rho and the one behind given by the change amount. For the analysis what we do is we take the coordinate system which is ourselves let us say and attach it to the moving wave front itself. So then the bottom picture simply shows that the coordinate system has been attached to the moving wave front. In that case the wave becomes steady and what you will see is that the fluid which is ahead of the wave is approaching the wave front with a speed of A and will leave with a change speed which I am calling A minus some amount delta V. This is what formally mechanics people will call a Galilean transformation where all that I am doing is that I am making a coordinate system change to make a situation so called that was unsteady to what we will call a steady situation. So when we are sitting on the sound wave front you will see that you are not moving anywhere you are always there but the flow ahead of it is approaching you with a speed of A and leaving with a change of delta V. Now what we do is we come up with an open system or a control volume which is essentially surrounding the sound wave front just about hugging the sound wave front you can say and in that sense this control volume or the open system will be a one inlet one outlet open system. So the inlet is with respect to the flow that is coming into the sound wave front and the outlet is the part which is where you see that the flow is leaving the sound wave front. Once this is done what you can go ahead and do is you employ your standard conservation principle. So the first conservation principle that you can employ for this open system is conservation of mass which is shown on the left hand side. Since the coordinate system within which we are performing this analysis makes the flow steady when you are sitting on the sound wave front you will essentially see that there is no accumulation of mass within that control volume. So whatever mass flow rate that is coming in is the same that is going out and that is what the first line there says that m dot is simply the mass flow rate which is coming into this control volume or open system is the same that is leaving. What we do is since this is a one dimensional picture we perform the analysis on a per unit area which is normal to the direction of propagation. So you can imagine a normal sorry a unit area along the sound wave front and in that sense we perform the analysis. Since we are assuming essentially a uniform one dimensional flow picture the mass flow rate in on a per unit area basis is simply going to be equal to the density times the velocity with which the flow is coming into the control volume and with respect to our control sorry coordinate system that is simply going to be rho multiplied by A which is now seen as the flow velocity coming to the sound wave front. And mass flow rate that is going out is simply rho which is now changed by an amount delta rho times whatever is the change in the flow velocity which is simply going to be A times delta rho. Usually when you come to such calculations there is a little bit of algebra involved. So in this case for example what one will be able to do is you can open up those two brackets on the right hand side and perform the calculations. When you do that you will see that that rho times A on the left hand side and rho times A on the right hand side will drop out. In addition when it comes to the product of delta rho multiplied by delta V what we will claim is that we are necessarily working with the situation where each of these quantities delta rho delta V delta T delta P are very small. So their products are going to be even smaller. So those products delta rho times delta V etcetera will be neglected in comparison with the other terms. This is a simplification and it becomes more and more accurate as your delta rho and delta V are smaller and smaller. And by definition since we are dealing with a infinitesimal change in each of these properties it makes sense to do that. When you perform such simplification you will see that whatever I have boxed out there is the final form of the conservation of mass statement employed for this particular control volume or open system. On the right hand side I am writing the momentum balance. So let me go back for a second to my second slide where I had written the balance statement in a general fashion. So all that I am now going to do is I am going to come up with expressions for each of these terms and put it up there for that particular control volume. Since we are dealing with a steady flow situation the left hand side which is the rate of change of momentum within that control volume is going to be 0 and that is what you will see here. The equation which you see here p dot is something that I am using as a rate of momentum that is flowing out minus the rate of momentum that is coming into this control volume is simply going to be equal to the net force acting on the control volume. And the rate of momentum coming in or going out is simply going to be the mass flow rate multiplied by the corresponding velocity component. So the mass flow rates are already there for example if you look at m dot out it is that rho plus delta rho times a minus delta v that is the mass flow rate on a per unit area basis. You multiply that by again the velocity of the flow which is in that case a minus delta v and you will have the rate of momentum leaving this control volume and that is what the first term is likewise rho multiplied by a multiplied by a will be the rate of momentum that is coming in. And in this case we are basically neglecting any viscous forces altogether which are acting on the control volume. This is a fairly useful approximation to make and in particular you want to point out to the students that in general viscous forces will be present in fluid flow situations. However, if it turns out that the domain that you have selected for analysis if it is sufficiently far from solid walls etcetera the effect of viscosity in general can be neglected. So the only terms contributing to the net force acting on the control volume turn out to be the pressures which are acting on to the control volume. So if you see the pressure on one side is p on the other side it is p plus delta p and the signs have been taken into account such that I am essentially using the direction pointing to the left as the positive direction for the flow in that sense the signs have been taken into account. Again you perform a simplification of terms drop out the higher order terms as what we say and the equation at the bottom is something that is the left out part of the simplified momentum value. And what is then you can do is right here this term rho multiplied by delta v is something that will appear in the simplified momentum equation which you can eliminate using that conservation of mass statement. You can see that rho times delta v is available as A times delta rho. So you can replace that and if you simplified I am going slightly faster understand but I am just trying to give an overview of what is involved in such analysis. If it turns out that we are going to include these topics in the final workshop I will do a far detailed job of working out the derivations and pointing out how things are going. But right now hopefully I want to just give an overview. Finally what turns out after the simplification is that the square of the speed of sound turns out to be the ratio of the change of pressure across that wave front divided by the change of density across that wave front. Let me point out something which some authors prefer to do in addition to what we have done. It is up to you whether this approach is something that is appealing to you or you would like to do something else when we want to argue that the process across this wave front that is happening is essentially isentropic. So let us see how we can do it in a couple of different manners. At least some authors prefer to do it in this fashion. So in addition to that conservation of mass and momentum balance for the control volume one more conservation equation that you can come up with is the energy balance and I think after having gone through the open systems analysis earlier you would recall the so called one dimensional steady flow energy equation. We can employ that steady flow energy equation in one dimension for this control volume. You can note that there is no external heat transfer into or out of this control volume neither there is any sort of work transfer going in or out of this control volume in which case the neglecting of course changes in the elevation because we are talking about essentially one level where everything is happening. You can bring about the one dimensional steady flow energy equation to say that it is simply the specific enthalpy times the inflow velocity squared over 2 is going to be equal to the specific enthalpy on the outlet side plus the velocity at the outlet side square over 2. So this is simply the form of the one dimensional steady flow energy equation that will reduce to in this particular situation. Going ahead and neglecting higher order terms which will be in this case delta v times delta v or square of delta v what you can come up with is a reduced form of the one dimensional steady flow energy equation which is that the change in the specific enthalpy across this wave front is going to be simply equal to the speed A times delta v. These are fairly straightforward to work out. This algebra is really extremely straightforward. So I am skipping these right now. If required we will look at it later in the main workshop in detail but right now this is one. So this is some sort of an intermediate result that one can obtain from the energy equation employment. Finally you use your so called TDS relation from the thermodynamics that you have gone through and starting with that you are able to simplify step by step using the two previous expressions that we have obtained namely that the change in the specific enthalpy across this wave front is equal to A times delta v and A square is equal to delta p over delta rho. If these two are used in a sequential manner which I have outlined here on the bottom part of the slide what you can actually show is that the change in the entropy in the specific entropy in this particular case across the sound wave front can be shown to be equal to 0. In other words what we can argue in this case is that the process that is occurring across the sound wave front is essentially isentropic. So this is one way of doing it and as I said some authors especially from the compressible flow gas dynamic side prefer to do it in this fashion. Some others will simply say that let us go back to the control volume. Here is my control volume. What do I have? I have a control volume where there is no external heat transfer that is one good thing and the properties across the wave front or from the inlet of the control volume to the outlet of the control volume are changing by infinitesimally small amount. So simply by looking at these two factors that there is no external heat transfer and the properties are changing only infinitesimally. Right there we can argue that the process is essentially isentropic that is one way of looking at it. One way or the other finally when we want to close this discussion we realize that the sound wave propagation can be approximated as an isentropic process and therefore formally when we want to write this speed of sound we want to write it as the change in the pressure with respect to the change in the density all occurring at constant entropy and formally the mathematical way to write it is partial differential of pressure with respect to density at constant entropy. So this is what we will normally see as the expression written out for the speed of sound that the square of the speed of sound is given as the partial derivative of pressure with respect to density at constant entropy and hopefully I have tried to argue why we can say that the process which is involved across the sound wave front is isentropic. So that subscript s here is added to remind us that the process across the sound wave front is essentially isentropic. We can go ahead and include one more relation from your previous knowledge and that is isentropic relation which simply relates the pressure and density through that isentropic law which is that the pressure is equal to a constant times density raise to gamma. I am using gamma as the ratio of the specific heats Cp over Cv. I think some authors use k as well I am preferring to use gamma. So gamma here is the ratio of specific heat at constant pressure to that at constant volume. This note that this relation p equal to constant times rho raise to gamma is applicable only for isentropic flow. So if you go ahead and perform differentiation of this particular relation you can actually find out this dp d rho as we will say and substitute it back here and if you simplify that again this analysis we can do later in detail in the main workshop. I am simply pointing out what you can do here. If you do that then you will see that the dp d rho at constant entropy can be equivalently written as simply gamma times p over rho and then finally using ideal gas equation of state p over rho can be replaced in terms of the specific gas constant r times t and therefore finally we can arrive at probably the expression which many of you are definitely familiar for the speed of sound that is the square of the speed of sound is given by gamma which is the ratio of specific heat at constant pressure to that at constant volume times r which is the specific gas constant times t which is the temperature of the fluid medium into which this sound wave is propagating and that is the square of the speed of sound. So here you know normally what I do is once you have arrived at this you can say that let us estimate the speed of sound in standard conditions in air for example. So for air we have gamma is equal to 1.4 in standard conditions r is about 287 joules per kg Kelvin 287, 289 and let us assume a temperature of say 27 degree Celsius for the purpose of calculation which is 300 Kelvin. If you put it all in here you can show to the students that the speed of sound is roughly about 350 meters per second in these conditions. So this is something that is the first part that we want to do in these compressible flow discussions. It turns out that this speed of sound is one of the most important parameters that you will always come across within this portion of compressible flow and that is what we normally begin with. So just to summarize quickly what we have done we have gone ahead and utilized three conservation principles. The additional one that normally thermodynamics does not see is these momentum balances but that is something that we are borrowing it from fluid mechanics and then the conservation of mass, the conservation of momentum and finally as an additional exercise we do this conservation of energy as a steady flow energy equation for the control volume and arrive at certain important results one of which is the estimation of speed of sound. So is that roughly acceptable in the sense that has it been followed what I am doing? I know that I am going fast. Then let me move on and then define the other parameter which is of importance in compressible flow and that is something that we call the Mach number and that is simply the ratio of flow velocity to the sound speed which we have just calculated. And if you look at the fluid mechanics side of it normally we like to provide some sort of a physical meaning to many of these ratios. Those who are familiar with fluid mechanics will recall that the Reynolds number has a certain interpretation in terms of ratio of inertial forces to viscous forces and so on. Likewise there is a nice little interpretation that we can associated with Mach number and to do that although I have not done it here I will do that in the main workshop if it is taken up. We look at the square of the Mach number. So when we look at the square of the Mach number on the right hand side you will deal with v squared over a squared and with certain manipulations including the expressions for the specific heats and the gammas and so on. You can actually come up with an interpretation that the square of the Mach number can be used as a measure of the ratio of the bulk kinetic energy to the random thermodynamic energy within the flow. Why this is important is because in compressible flow typically eventually you will find out that the bulk kinetic energy contained within the flow on a specific basis which will be simply v squared over 2. m times v squared over 2 is what we will call the kinetic energy. So v squared over 2 is simply on a per mass basis, unit mass basis. That if you compare with the thermodynamic internal energy which can be roughly classified as cv times t let us say. You will see that when it comes to compressible flow that bulk kinetic energy is not negligible in comparison with the thermodynamic internal energy and that is what one of the most important facts is about compressible flow. If you want to point out to the students that what is difference, what is different between compressible flow and incompressible flow, one factor that you can point out is that the bulk kinetic energy incompressible flow is not negligible in comparison with the random thermodynamic energy. In case of incompressible flow which normally we are familiar with at least in mechanical engineering, the bulk kinetic energy usually is very, very small compared to the thermodynamic internal energy and that is why many times flows in mechanical engineering you will see can be safely approximated as isothermal flows. There is no change in the temperature within the flow unless there is an external heat transfer happening. But in compressible flow that cannot be done even though there is no external heat transfer occurring because the bulk kinetic energy is sufficient as a fraction of the internal energy. There is actually an interchange between those two happening and you will see that because of which the temperature within the flow will change from location to location even though there is no external heat transfer in or out of the flow. So that is something that I normally like to point out to the students. To be honest with you this is more on the fluid mechanics side than a thermodynamic side but something that you may want to keep in mind. Once we define the Mach number then we can say that if it is less than 1 we will classify the flow as subsonic flow. If it is equal to 1 we will call that as a sonic flow where the flow velocity is exactly equal to the speed of sound. That is v equal to a right here. And if it turns out that the flow velocity is higher than the speed of sound we call that situation a supersonic flow. There are many more slides obviously I knew that I will not be able to complete it but let me just quickly walk through what we can expect in the later part. Here you know rather than going through the isentropic flow discussion now I thought that while we are discussing the conservation equations for a control volume that is surrounding a wave front which we just did for a sound wave. One thing that you can do is immediately you can talk about what is a shock analysis. Fundamentally there is nothing different between what we did for the sound wave analysis and what one can do for shock analysis. It is exactly the same. The picture is the same. The only difference is that the fluid properties now are not changing by infinitesimally small amounts across the wave front but they are changing by finite amount. So a normal shock is something that I will just go through very fast. It is essentially a wave front across which fluid properties will change by finite amounts as opposed to infinitesimally small amounts in case of sound wave. So some people will want to classify sound wave as a special case of a shock wave where the changes across the wave front are infinitesimally small which is a special case of any change. Physical measurements have shown that the stream wise extent of the shock is very very small. Typically it is about fractions of micrometers and therefore for a primary analysis as what we would do in case of a typical undergraduate course we would like to treat this shock front as essentially a discontinuity. In that sense that sound wave front which we talked about can be considered as a weak discontinuity. So weak is supposed to convey to us that the changes that are occurring across the sound wave front are infinitesimally small that is about it. So here we have again the same picture. There is a upstream flow which is coming to the shock or approaching the shock with properties p1, t1, rho1, v1 and the mark number m1 and across the shock things have gotten changed to pressure 2, temperature 2, density 2, velocity 2 and mark number 2. So the formal purpose or objective of a normal shock analysis is that given all these upstream quantities can we determine the downstream quantities without really bothering about what is exactly happening through this shock front. So what we do is we again come up with that control volume which surrounds the shock front and we simply look at what is coming in and what is going out and simply relate those through the standard conservation equations which have already been discussed. So if you did that what you will see is the first equation where my cursor is pointing right now is your mass balance equation for that control volume again written on a per unit area normal to the flow direction. The second equation is that momentum equation which we had discussed earlier when it comes to a normal shock the momentum equation is written in the form that is written on the board right now. And finally the third equation is your 1D steady flow energy equation again noting that there is no external heat transfer and work transfer in or out of this control volume. And immediately I am writing the specific enthalpies in terms of the CTs and the temperatures. Again as was being discussed with in the previous session there is a certain datum that we define and let that datum be sufficiently low so that we do not bother about getting into any negative or whatever quantities. We are going to essentially look at only the changes with respect to that datum and therefore I can safely write that H is simply Cp times T and so on. Finally the closure relations for this set of equations as we would like to point out is provided by the fact that the ideal gas equation of state is valid everywhere including the front of the shock and the back of the shock. And therefore P equal to rho RT is applicable both in conditions 1 as well as say and conditions 2. So that is it this is really the normal shock analysis if you want to simply say that all that you need to do is employ the conservation equations of mass momentum and energy across that control volume and use the closure relations of P equal to rho RT to relate the properties behind the shock to the properties ahead of the shock the shock analysis is over. The remaining part is only algebra manipulations of these equations to couch them in certain form which I have not outlined here but we will look at it in detail if it is to be included in the final workshop. There are couple of different ways of doing it and it turns out to be slightly more tedious than the sound wave analysis but it is fine it is purely algebra is nothing more. So sometimes students get dissolution that you know what is going on. Every time I would like to point out to them that we are doing nothing but simply manipulating the same set of governing equations that we are familiar with. So that you need not lose the sight of what you are up to finally there is a little bit of mathematics and manipulations involved but finally we are simply looking at the governing equations and looking at how those can be transformed in a manner which we can use. So just for the sake of this purpose I have written that algebraic manipulations you can do and one way of obtaining the solution is in terms of what is called as the Rankine-Ugonio relations which are nothing but the governing equations represented in a special manner where we are representing everything as ratios of properties now. So we represent the density ratio, the velocity ratio, the pressure ratio and the temperature ratio across the shock. The solution of this problem turns out to be in the form of a single parameter which is that p2 over p1. So you can see that here rho2 over rho1 which is same as v1 over v2 through the mass balance equation can be written using the expression that is shown here which involves this p2 over p1 ratio. Similarly the t2 over t1 ratio can be written in terms of an expression which involves p2 over p1 and therefore to summarize what I will say is that this Rankine-Ugonio relations are essentially the normal shock solution in terms of the parameter p2 over p1 which has a special name in the gas dynamics or compressible flow terminology it is simply called as the shock strength. So far you point out that everything is algebra nothing more nothing less. Finally some physics has to be brought in and we say that what is it that makes the shock shock in some sense. So then you say that we go back to our control volume and we realize that conditions are changing from 1 to 2 across the shock no heat transfer no work transfer. So the only way this process is going to be permitted is by employing our second law of thermodynamics which must state that entropy in conditions 2 has to be equal or greater than condition 1. So finally you come up with that statement that the requirement of the second law for this shock process to be feasible or possible is that s2 minus s1 has to be greater than or equal to 0 and with much more algebra which again I am not doing here you can show that this s2 minus s1 can be greater than or equal to 0 only if that shock pressure ratio or the shock strength is greater than or equal to 1. So only if this condition is satisfied then you can show that we are essentially in a position to call a situation shock and to conclude this we then say that the pressure must increase likewise you show that the temperature and density must increase across the shock and then using the continuity equation or the mass balance equation you can show that the flow must decrease in its velocity. So this is one way of performing the normal shock analysis the same analysis rather than using that parameter p2 over p1 can be performed using an upstream mach number as the parameter. So everything can be couched all entire solution that we went through can be couched in terms of m1 which is the mach number of the upstream flow and again when you put in the condition that the second law requirement is that s2 minus s1 has to be greater than or equal to 0 for this process to be feasible you will come up to the conclusion that upstream mach number has to be supersonic. So the condition that you come up with then we will work out this in detail later is that m1 must be greater than or equal to 1 corresponding to which there is an associated conclusion that the mach number behind the shock has to be less than 1. So therefore then to conclude again the upstream flow with respect to the shock has to be supersonic and the downstream flow with respect to the shock has to be subsonic. I will just use three more minutes although I do not think I am going to be able to complete it but just to point out what else we are going to plan to do in the main workshop and that is this so called steady quasi-1D isentropic flow. So quasi-1D in plain English is supposed to mean almost 1D which basically is supposed to mean that we are talking about a flow inside a duct whose cross sectional area is varying very very slowly as you move along the stream wise direction. So that at any particular cross section the two dimensional effects are fairly negligible. That is all that means by the word quasi-1D which is almost 1D and we will look at adiabatic and frictionless flow which makes everything isentropic. Adiabatic and frictionless which essentially means that there are no irreversibilities involved within the flow will make the situation isentropic. In which case what we end up doing is we perform the same analysis everything is the same again and again in fluid mechanics nothing is different. All the time what you do is you select a certain control volume and you employ the conservation laws as what we have seen already the mass balance, the momentum balance, the energy balance and then come up with some more manipulations to see how things work out. So here in this particular case what we are going to do is we will identify a little section within your duct which is varying in cross sectional area. The length of the section is dx which is elemental length and things change from p, t, rho, v, m etc to p plus delta p, p plus delta t and so on across this length and area also changes from a to a plus dA. So now what I will do is I will just point out what is done here I am not going to get into the details because there is a little bit of algebra involved again. We employ the mass balance which is on the left, we employ the momentum balance which is on the right, we obtain certain intermediate relations which you combine to bring the so called very well known area velocity relation based on which then you can point out what happens in case of supersonic flow versus a subsonic flow. So if you are dealing with a subsonic flow which means that the m is less than 1 what you will see is that if you decrease the cross sectional area of the duct correspondingly there has to be an increase in the flow velocity as shown by this area velocity relation. On the other hand if you are dealing with supersonic flow it turns out that if you decrease the cross sectional area the velocity will also decrease. So that is something that is worth pointing out from this area velocity relation and further there are a few more additional things that you say that in case there has to be a Mach number 1 to be achieved it will be achieved only at one area minimum within that converging diverging duct and that area minimum is something what we are going to call as the throat. One thing that I like to point out at this point to the students though is that just because you have a throat does not necessarily mean that you are going to get m equal to 1. It depends on the operating conditions. However if m is equal to 1 is to be happening it has to occur at that area minimum. That is something that is worthwhile pointing out. Many times students go out with this concept that as long as there is an area minimum there is going to be a Mach number of 1. That is not true only if certain conditions exist Mach number of 1 will occur at that area minimum. On the other hand because physically we have an area minimum we will always have a maximum or a minimum velocity depending on whether you are dealing with a subsonic flow or a supersonic flow. But m need not be necessarily equal to 1 just because you have a throat. That is something that is worth pointing out to the students. And then there are more terminologies like the stagnation conditions and so on which I will skip right now. Finally what we are going to wrap up this is with a discussion on the operating characteristics of nozzles and here what I personally want to do and like to do in the class is first discuss the operating principle of a purely convergent nozzle. So that we have only a continuously decreasing area of cross section ending up with the so called throat let us say there is no further part to the nozzle. It is essential because with respect to that purely convergent nozzle I think you can reasonably well explain the phenomenon of what we call choking. Many times what I have found is that if you do not do the discussion on purely convergent nozzle and go directly to the converging diverging nozzle sometimes students have difficulty in figuring out exactly what is happening. On the other hand if you do the discussion on purely convergent nozzle first you can then say that well now I will simply add one more part which is a divergent part and make it a CD nozzle or a convergent divergent nozzle and it turns out to be at least in my experience that the explanation becomes slightly better I think and the students also appreciate it little bit more. So I am sorry I have used up almost one hour and again I am sorry that I have not been able to go through enough details here but roughly I thought that I have given I hope so what we will be incorporating in case this topic is to be included in the final workshop. So I think that is where I will stop. Thank you for your attention.