 Hello and welcome to lecture number 22 of this lecture series on Introduction to Aerospace Propulsion. Over the last several lectures, we have got introduced to several aspects of thermodynamics, thermodynamic principles and also laws of thermodynamics and how to use them in applications like in thermodynamic cycle analysis and so on. And what we are going to discuss today is a little bit different from what we have been discussing so far in the sense that this particular topic that we are going to take up for discussion today and also in the next lecture combines some of the thermodynamic principles with fluid mechanics in some sense. And so, what we are going to discuss about is on compressible flows and what are the different properties of compressible flows and also we shall look at compressible flow through some simple cross sectional geometries like convergent nozzles and converging divergent nozzles and so on. So, this is a slightly different topic from what we have discussed so far, but obviously this is going to also use many of the thermodynamic principles that we have been discussing in the last few lectures. So, what we shall discuss in today's lecture are the following. We shall be talking, we will begin our talk today with a discussion on what do we mean by one dimensional compressible flows. We shall talk about stagnation properties and then we shall talk about speed of sound and Mach number, how do you define Mach number and then we shall look at one dimensional isentropic flow and variation of fluid velocity with flow area will derive an equation which defines or governs the variation of fluid velocity with area. We will also be discussing towards the end of the lecture on isentropic flow through nozzles. We will discuss two types of nozzles, converging nozzles and converging diverging nozzles. So, these are some of the topics that we shall be discussing in today's lecture and so you might have got a feeling right now that this is going to be a discussion primarily on compressible flows and also on their significance in terms of analysis of compressible flows. Now, during our thermodynamic analysis that we have discussed and also we have solved several problems using the thermodynamic principles. We have always made an inherent assumption that at a given state of a system the density is a constant and so there is an inherent assumption that the flow is incompressible in the sense that we do not really consider variations of density which is not necessarily true in most of the in well in many of the engineering applications. In most of the day to day applications that we are familiar with the velocities of the fluid are very low and therefore, the inaccuracy that we achieve because of assuming incompressible flow is not really high. Whereas, if fluid velocities are very high then it is not really possible for us to make this assumption that the flow is incompressible. The flow no longer remains incompressible and therefore, we should be we should be taking into account the compressibility effects and that is some one of the aspects or that is one of the reasons why we are taking up this topic for discussion today that there are lot of engineering applications where the fluid velocities can be significantly high and therefore, we cannot really assume that the flow is incompressible changes in density are incompressible or changes in kinetic energy are incompressible for that matter. So, what we shall discuss to begin with is that in the basically the significance of the compressible flows and why do we need to discuss them basically because there are some fluid applications there are certain implications where it is important for us to understand or analyze the system in the form of considering the variations in density and also taking into account the variations or effect of kinetic energy though potential energy may still be negligible kinetic energy cannot really neglected. So, the flows that basically involve significant density variations are known as compressible flows though most of the analysis we have considered so far neglected density variations we shall also take a look at some applications where this effect can no longer be neglected. Basically, we shall be again making an assumption here that even the even when we consider the flow to be compressible we shall be analyzing it in a one dimensional sense for an ideal gas with constant specific heat. So, we are going to assume that the specific heat is a constant and the gas is ideal and therefore, we can assume the ideal gas behavior and so on. And where do we see such applications basically in devices that involve flow of gases at very high velocities like in nozzles and so on. And so, it is important that there are we have an understanding of how to analyze such systems which involve very high velocities therefore, changes in kinetic energies can no longer be neglected. Now, so in order that we analyze a system where kinetic energy cannot be neglected we need to define what is known as stagnation property of a system or stagnation property different types of stagnation properties. One of the stagnation properties to begin with we shall understand is the stagnation enthalpy. We already defined what is enthalpy we know now that enthalpy basically represents the total energy of a fluid in the absence of potential and kinetic energy. This comes from the first law of thermodynamics and when we take up this property of enthalpy at high speed flows potential energy being negligible may still be valid, but not kinetic energy. So, we have to take into account the changes in kinetic energy. So, what we do is we basically combine enthalpy with kinetic energy that is we add up enthalpy and part of the kinetic energy and we define a new property known as the stagnation enthalpy. And why is it known as stagnation enthalpy will be clear when we take up when we discuss further. So, this sum of the static enthalpy so called the static enthalpy which I will now be referring to as static enthalpy and the kinetic energy term put together is known as the stagnation enthalpy. So, if you look at the definition of stagnation enthalpy which I had defined stagnation enthalpy which is usually denoted by h subscript 0. 0 is applicable for most of the stagnation properties which we are going to define like temperature pressure and so on. So, stagnation enthalpy h naught is equal to h which is now the static enthalpy plus v square by 2 which is the kinetic energy. And you might notice that these are all per unit mass and therefore, it is specific enthalpy specific stagnation enthalpy is the sum of the specific static enthalpy plus v square by 2. So, the first term on the right hand side is the static enthalpy which does not have a subscript 0 and the second term is the kinetic energy term. And on the left hand side we have the stagnation enthalpy. So, this is how you would define a stagnation enthalpy which is primarily the sum of enthalpy as we had defined earlier which we are now calling as the static enthalpy. So, whenever we take up different properties at high speed cases that is incompressible flows we also define these properties as stagnation as well as static parameters. And so static parameter will become stagnation parameter if the velocities are 0 that is if kinetic energy is 0 then static and the stagnation properties are the same. So, we will now use this principle to define other properties in terms of stagnation parameters like stagnation pressure and stagnation temperature, stagnation density and so on. Now, if you consider a steady flow of through a duct let us say through some diffuser or a nozzle where there is no shaft work there is no heat transfer etcetera. Steady flow energy equation for this is something we have already defined that is h 1 plus v 1 square by 2 is equal to h 2 plus v 2 square by 2. So, the left hand side as we have now defined is this stagnation enthalpy at state 1 which is h 0 1 and on the right hand side we have stagnation enthalpy at state 2 that is h 0 2. So, this means that in the absence of any heat and work interactions if there are no heat transfers or there are no work interactions this stagnation enthalpy remains a constant during a steady flow process. So, this is a very significant property that we need to keep in mind that if there are no heat interactions or there are no work interactions or in a steady flow process for example, then the stagnation enthalpy of such a process remains the same it does not change if there are no heat transfers into the system or from the system stagnation enthalpy does not change. And that is something we have to keep in mind because many of the systems that we are going to consider will have this property to be used that is there is no heat and work interactions taking place like in nozzles and diffusers. And therefore, stagnation enthalpy has to be a constant, but this is not applicable for systems like turbines or compressors because there is a work interaction taking place in the case of compressor work done on the system in the case of turbine work done by the system. So, stagnation enthalpy obviously will not be a constant in such cases it is only applicable for those cases where there are no work or heat interactions. Now, we have now seen that if you look at this duct example that I was mentioning where we had h 1 plus v 1 square by 2 is equal to h 2 plus v 2 square by 2. Let us say that at state 2 the velocity is now equal to 0 that is by some means we bring the velocity at state 2 to be 0. Now, what we have is h 1 plus v 1 square by 2 is equal to h 2 which is also equal to h 0 2 because as we have discussed if velocity is 0, static and stagnation parameters are the same which means that at state 2 the static enthalpy and stagnation enthalpy are the same. So, what this means is that stagnation enthalpy also represents the case where the fluid velocity is isentropically or adiabatically brought to 0 brought to rest. So, at state 2 if you were to bring the fluid to rest we have h 1 plus v 1 square by 2 is equal to h 2 which is also equal to h 0 2 which means that stagnation enthalpy represents enthalpy of a fluid when it is brought to rest adiabatically that is because in this case there was no heat interaction or work interaction across the system boundaries. So, if you bring the fluid to rest adiabatically and in which case stagnation enthalpy basically represents the fluid enthalpy of a fluid when it is brought to rest adiabatically. So, what happens during this stagnation process basically and this is why it is basically called stagnation enthalpy that is the term stagnation comes because we are assuming that the fluid is adiabatically coming to rest that is how the stagnate the term stagnation enthalpy has arise. So, during a stagnation process kinetic energy of a fluid is converted to enthalpy that is internal energy and flow energy which results in increase in the fluid temperature and pressure that is if you were to bring a fluid to rest adiabatically it has to reach I mean that energy as part of conservation of energy principle that energy has to get transformed into some other form and what happens is basically that the fluid energy increases in the form of flow energy and so on. So, that ultimately leads to an increase in temperature and pressure of the fluid. So, based on this we shall now define what are known as stagnation pressure and stagnation temperature. So, on the left hand side if you recall we had h 1 plus v 1 square by 2 which was equal to h 2 and internal equal to h 0 2. So, in general we can write h naught that is stagnation enthalpy is equal to h plus v square by 2 which means that for an ideal gas which is where we began our discussion that we are going to assume that the gas is going to be ideal with constant specific heats. So, with constant specific heats for an ideal gas enthalpy is simply equal to c p times t therefore, we have c p t naught is equal to c p t plus v square by 2 or t naught is equal to t plus v square by 2 c p that is left hand side we have a temperature with a subscript 0 that is known as the stagnation temperature. Stagnation temperature is equal to the sum of the static temperature plus one term which we are going to define as the dynamic temperature because that changes with fluid velocity and that is why it is called dynamic temperature. So, if you write the equation for enthalpy in the form of a product of specific ratio of sorry specific heat at constant pressure to the temperature then we have c p t well c p t naught is equal to c p t plus v square by 2 or t naught is equal to t plus v square by 2 c p. So, here t naught is defined as this stagnation temperature and this represents the temperature an ideal gas will attain if it is brought to rest adiabatically that is similar to what we did for enthalpy if you bring a gas to rest adiabatically the temperature that the gas attains at the end of this process that is when it becomes when it comes to rest adiabatically is known as the stagnation temperature. And the second term that is v square by 2 c p corresponds to the temperature during such a process and because it is it can change with velocity it is called dynamic temperature. So, you can immediately see that for non-zero velocities stagnation temperature will always be greater than the static temperature which is something we have defined earlier as well that is as you bring a fluid to rest the energy gets transformed into internal energy and so on which ultimately leads to an increase in its stagnation well increase in its temperature and pressure which we have now defined as stagnation temperature and pressure. And this has to be higher than the static temperature because it has an additional energy term which is the dynamic temperature term. Similarly, we can also define pressure that is stagnation pressure which is basically the sum of the static pressure plus the dynamic pressure. So, stagnation pressure well pressure is equal to p naught is equal to p plus half rho v square I guess you might have learned this in fluid mechanics and this was basically as part of the Bernoulli equation. So, we can immediately see that there is a direct correlation between what you get in thermodynamics with what you feel or what you get in fluid mechanics. So, stagnation pressure is equal to static pressure plus this dynamic pressure and so what we are trying to say here is that there are parameters which we need to take into account when the fluid velocities cannot be neglected. So, kinetic energy needs to be accounted for in calculating parameters. So, similarly the way we have defined for temperature we can also define stagnation pressure. So, the pressure that a fluid will attain when it is brought to rest isentropically is known as the stagnation pressure. And so what we are saying here is that all these stagnation parameters whether it is temperature or pressure or enthalpy will have a value greater than that of the corresponding static parameters for all nonzero velocities. Now, from the ideal gas well we have already assumed that all these gases that we are discussing about will be for ideal gas. So, from the isentropic relations that we had discussed we can see that the ratio of we can will basically relate the stagnation pressure to the stagnation temperature in terms of the pressure and temperature ratios. So, if you were to relate the pressure ratios to the temperature ratios we have p naught by p is equal to t naught by t raise to gamma by gamma minus 1 where gamma is the ratio of specific heats which is equal to C p by C v. Similarly, we can define it for density which is rho naught by rho is equal to t naught by t raise to 1 wide gamma minus 1 where again gamma is equal to the well it is basically the ratio of specific heats. So, these are isentropic relations which we will be using very frequently in our thermodynamic analysis of different cycles and some of them which we have already discussed was not in the form of the ratio of stagnation and temperatures and pressures, but in the form of static pressures and temperature ratios. But in some of the later analysis that we will be doing for aircraft engines where the kinetic energy terms cannot be really neglected these equations will be used very frequently. Now, what we shall do now is to see what happens as if there is a change in let us say the stagnation pressure. So, stagnation pressure well we already seen that stagnation enthalpy does not change as long as there are no heat transfer or work transfer across the system boundaries which means that since stagnation temperature is directly related to stagnation enthalpy this also is applicable to the stagnation temperature and therefore, stagnation temperature also does not change as long as there are no heat and work interactions across the system boundaries. But is it applicable for pressure because pressure is a parameter for ideal gases it is not directly related to the enthalpy it is a temperature which is a direct function where which happens to be related to enthalpy. So, what about pressure? So, to understand that what we will do is to look at a process where we could have a loss in total pressure and we will see what correspondingly happens to the enthalpy. So, what we have done here is that on an enthalpy entropy scale we have h and s scale here and this is let us say a compression process where there is an increase in pressure or even if there is no compression let us look at a fluid which is just a having a certain velocity which means that it will have stagnation parameters which will have to be accounted for which is basically the sum of the static parameters plus the dynamic term. So, the actual state of the fluid is represented here at state one which is shown there and so this is the initial state where it has a certain enthalpy h correspondingly there is an entropy as well and this is on a constant pressure line. So, this pressure line that is seen here is p it is a constant pressure line. Now, what are the corresponding parameters for this particular fluid if you were to look at this stagnation parameters. So, we know that stagnation enthalpy h naught is equal to h plus v square by 2 and so we have h naught which is equal to h plus this dynamic term that is v square by 2. So, this is the isentropic stagnation state that is if the process were to be isentropic then we get the straight line because entropy is a constant. So, it is a straight line it is an isentropic stagnation state where the enthalpy at the end of this process is equal to h naught and the corresponding stagnation pressure is equal to p 0 or p naught, but if the process is not isentropic that is if we look at a process which has frictional losses and some such irreversibilities it cannot be any more isentropic. So, if you look at an actual stagnation state there has to be an increase in entropy. So, there will be a certain positive slope for that particular process, but there are no heat or work interactions which means that the enthalpy should not change because there are no heat or work interactions which are taking place and therefore, enthalpy cannot change in such a process, but because of irreversibilities it is non isentropic and there will also be some pressure loss because of irreversibilities. So, if there is a pressure loss then we have a certain slope for the process and so we have on the same enthalpy line because enthalpy does not change stagnation enthalpy line extended the process ultimately meets the stagnation enthalpy line and the corresponding pressure is the actual total pressure. So, p naught actual need not be equal to p naught isentropic because there could be a pressure loss total pressure loss taking place because of let us say friction, friction can cause decrease in velocities which means that at the end of the process you have a lower velocity and therefore, correspondingly a lower total pressure. Therefore, p naught actual can be less than the p naught ideal or p naught isentropic. So, it is not necessary that the total pressure remains the same at the end of such a process, but what should remain the same is the enthalpy, enthalpy does not change because there are no heat or work interactions taking place. So, this is a very important aspect that we need to keep in mind that in a process where there are no heat or work interactions the stagnation enthalpy cannot change. Therefore, stagnation temperature also does not change, but what is possible is that there could be a difference between the stagnation pressure ideal or isentropic to the stagnation pressure actual which could be because of frictional losses which could lead to non isentropic processes and there could be p naught actual which is less than p naught ideal and in the actual in the ideal case where the if you assume all irreversibility is to be 0, then the p naught actual will be equal to p naught ideal because there are no more pressure losses. So, this is a very important aspect that you definitely need to keep in mind because we will keep using this aspect in many of the analysis that we are going to do in some of the later lectures when we analyze ideal cycles and real cycles of gas turbine engines and so what we have discussed now are on stagnation parameters, stagnation enthalpy, stagnation pressure, stagnation temperature density and so on and the bottom line is that in the absence of heat and work interactions stagnation enthalpy and therefore, stagnation temperature cannot change, but what is possible is that you may have a change in stagnation pressure due to some irreversibilities and non isentropicity of the process. So, what we shall discuss now is a different aspect related to our compressible flows, we will be defining what is known as the Mach number. So, before we define that we need to define modismian by speed of sound. So, speed of sound is basically by definition the speed at which an infinitesimally small pressure wave travels through a medium because sound is a pressure wave which is basically a small pressure wave. So, speed at which the infinitesimally small pressure wave travels through a medium is basically the speed of sound. Now, if you assume an ideal gas speed of sound which is usually denoted by symbol c can be shown to be equal to c is equal to square root of gamma r t where gamma is the ratio of specific heats r is the gas constant for the medium and t is the temperature static temperature. So, c is basically a function direct function of temperature speed of sound is a direct function of temperature. And of course, it also depends upon the ratio of specific heats and the gas constant, but for a particular medium both of these are constant. And so of course, as long as we assume that ratio of specific heats does not change with temperature. So, we can see that speed of sound is a direct function of the static temperature. Now, based on this we are going to define a non-dimensional parameter which is basically taking the ratio of the velocity of the fluid or the object to the speed of sound and that is basically known as the Mach number. So, Mach number is basically the ratio of the actual velocity of an object or fluid to the speed of sound. And so in some cases the fluid may be moving and the object is stationary which is what we would do in a wind tunnel testing for example, the fluid is at a certain speed the object is stationary. Whereas, on the other hand an aircraft an actual aircraft moves at a certain speed in a medium where the air is relatively at 0 velocity. So, ratio of that speed to the speed of sound is known as the Mach number. So, Mach number is defined as V by C where V is the velocity of the object or fluid and C is the speed of sound which is equal to square root of gamma R T. So, Mach number is basically a function of the ambient temperature as we have seen which means that it is possible that an object which is moving at the same velocity in two different mediums of two different temperatures will have different Mach numbers. Because even though their velocities are same they are in different mediums which have different temperatures which means that the speed of sound will be different for different mediums depending upon their temperature. And therefore, it is perfectly possible that Mach number of two objects which are moving at the same velocity but in different mediums which have different temperatures the Mach number can certainly be different. And so it is not necessary that if the velocity is the same Mach number has to be the same. Now, if depending upon Mach number if we have a case where Mach number is equal to 1 then such flows are called are basically known as sonic flows. If Mach number is greater than 1 the flow is known as a supersonic flow. If the Mach number is less than 1 it is called a subsonic flow and if Mach number is much greater than 1 usually greater than 5 we refer to such flows as hypersonic flows. And Mach number approximately equal to 1 it is around 1 then we call such flows as transonic flows. So, these are different terms we use for flows depending upon their Mach numbers. So, Mach number less than 1 is subsonic greater than 1 is supersonic and greater than 5 is usually referred to as hypersonic and so on. And so what we will do now is to look at the variation of fluid velocity with the area and we will derive an expression which relates the area different ratios and fluid velocity with the Mach number. So, which means that for different Mach numbers we can see how area and velocities are related. So, for deriving an expression what we will do is basically consider a mass balance for a steady flow process. So, we know that mass flow rate for such processes are equal to the product of density area and velocity. So, rho a v which will be a constant for a steady flow process. So, if we differentiate this equation and divide this by the resultant mass flow rate we can write the above equation rewrite the above equation as d rho by rho plus d a by a plus d v by v is equal to 0. Now, from our steady flow energy equation which we had derived in earlier lectures if you assume work done heat transfer kinetic and potential energy to be more or less 0 then from the steady flow energy equation we get h plus v square by 2 is equal to 0 or d h plus v d v is equal to 0 that is if we differentiate this equation energy equation we get d h plus v d v is equal to 0. So, this is basically coming from the steady flow energy equation where we assume work done heat transfer and potential energy to be 0 and from the T D S equation that was the second Gibbs equation T D S equal to d h minus v d p. Now, for isentropic flows T D S will be equal to 0 and therefore, d h is equal to v d p which is d p by rho because specific volume is the inverse of density and therefore, d h is equal to d p by rho. Therefore, our previous equation becomes which was d h plus v d v is equal to 0 becomes d p by rho plus v d v is equal to 0. So, if we combine this equation with the mass balance equation we get and simplify basically we get d a by a is equal to d p by rho multiplied by 1 by v square minus d rho by d p. Now, it is also known that the ratio d rho by d p for constant entropy is equal to 1 by c square. So, if you were to apply this principle we get d a by a is equal to d p by rho v square into 1 minus m square. So, m square is coming because we will get a ratio of v square by c square which is basically equal to Mach number square. So, from the mass balance equation we get d a by a plus d p by rho v square into 1 minus m square and this again we can rearrange based on our earlier equation. So, if we rearrange that equation which is basically from d p by rho is equal to minus v d v and. So, if we rearrange that equation we get d a by a is equal to minus d v by v into 1 minus m square. So, this equation has a lot of significance in the sense that this equation governs the shape of a nozzle or a diffuser in subsonic or supersonic isentropic flows. Now, in this equation area and velocity obviously are positive quantities and if that is. So, Mach number obviously can. So, depending upon the Mach number whether it is greater than 1 or less than 1 area velocity changes can be inferred that is for subsonic flows where Mach number is less than 1 we have d a by d v less than 0. For supersonic flows where Mach number is greater than 1 the rate of change of area with velocity is greater than 0 and for sonic flows where Mach number is equal to 1 then we have d a by d v is equal to 0. So, we will we will try to understand what is the implication of the rate of change of area with velocity depending upon the Mach number. So, as Mach number changes there are changes in velocities with reference to changes in areas. So, let us look at what happens as you change the Mach number and what happens to velocity as you change areas. So, from these equation it basically follows that to accelerate a fluid in subsonic flows you need a converging area that is there has to be a decrease in area because for Mach numbers less than 1 d a by d v is less than 0 which means that if you have to accelerate a flow we have to have a corresponding decrease in area in subsonic flows. Whereas, in supersonic flows you will need an increase in area to accelerate a fluid a diverging nozzle is required at supersonic velocities and what we will also see a little later is that the highest velocity that you can achieve in a converging nozzle is the sonic velocity that is the maximum velocity that you would be able to achieve in a converging passage in subsonic flows will be basically that you would get only a sonic velocity at the end of the converging nozzle. To accelerate a fluid to supersonic velocities you will need a diverging section or diverging area or increase in area after the flow reaches a sonic velocity at the minimum area which is basically known as the throat. So, after the throat there needs to be a diverging section to accelerate a fluid to supersonic velocities and so what I was trying to say is that in subsonic flows if you look at Mach numbers less than 1 as you start accelerating the fluid the maximum velocity or Mach number that you can get at the end of the acceleration is Mach number equal to 1 which is basically known as a sonic velocity. So, a flow which has certain total pressure and temperature at the inlet of the nozzle. So, a nozzle section is shown here will accelerate the maximum that it can accelerate is Mach number equal to 1. So, what if you reduce the area below this that is at the end of this nozzle let us attach another nozzle hoping that this will accelerate it to supersonic speeds, but that is not going to happen what will basically happen is that the section where you get sonic velocity will shift and the section which had sonic velocity earlier will now have a subsonic Mach number. And so the sonic velocity will occur only at the exit of this convergent section instead of the exit of the original nozzle and basically the mass flow rate will now reduce because you are trying to force a certain amount of mass flow through a lower passage area and so you would get a decrease in mass flow. So, we will see later that as Mach number reaches 1 the mass flow is at its maximum and that is known as the choking of a nozzle that is if you were to force fluid through a nozzle till the point that at the exit of the nozzle Mach number is equal to 1 then the mass flow rate that can be passed through such a nozzle has reached its maximum level that is known as choking of the flow which means that if you add another convergent section to that you are not going to increase the Mach number anymore it will in fact lead to reduction in the mass flow rate because your choking area is different now. And so it is not possible for us to achieve supersonic Mach numbers in a convergent section. So in order to get a supersonic Mach number we need to have a divergent section at the exit of the convergent section and such nozzles are known as converging diverging nozzles. We will do some analysis of the variation in pressure across a convergent diverging nozzle with change in back pressure we will do that little later. Now so in summary from the area velocity Mach number relation what we have are the following. So if you look at the first case that is shown here we have a nozzle which has an inlet Mach number which is subsonic. So along the length of the nozzle the pressure and temperature the static pressure and temperature would decrease the velocity and Mach number increases. So this is basically a subsonic nozzle. So area reduces Mach number increases in subsonic flow reverse of that happens at if the area is increasing Mach number is less than one at the inlet the static pressure and temperature increases along the length of the nozzle velocity and Mach number decreases. So this is basically a subsonic diffuser. If you look at the supersonic version of this it is the exact opposite. So if Mach number at the inlet is greater than one if we have to have an increase in Mach number along the length of the nozzle area has to increase. So pressure and temperature decreases velocity and Mach number increases and this is known as a supersonic nozzle and if you look at a supersonic diffuser it is basically decreasing area. So pressure and temperature will increase and Mach number decreases. So a supersonic diffuser at least theoretically is basically a subsonic nozzle. So supersonic diffuser will act as a subs a nozzle in subsonic flow and a subsonic diffuser will act as a supersonic nozzle in at Mach numbers greater than one which is what should happen theoretically. So this what we have discussed now is an outcome of the area velocity Mach number relation which we had derived and so we know how the area will well how the area has to change given a certain Mach number so that you get an increase or decrease in velocity accordingly. Now what we will do is basically relate the stagnation properties that is stagnation pressure temperature and density to the corresponding static parameters like static temperature pressure and density through the Mach number. We have already seen the isentropic relations which relate ratio of temperatures and pressures through the ratio of specific heats that is T naught by T is equal to P naught by P raise to gamma minus one by gamma and so on. So now we will relate the stagnation parameters with their corresponding static parameters through the Mach numbers. Now to do that we have already seen this equation which was relating temperature stagnation temperature to static temperature and the velocity ratios. So T naught is equal to T plus V square by 2 C p therefore, T naught by T is equal to 1 plus V square by 2 C p T. Now since we know that C p is gamma r by gamma minus 1 and C square is gamma r T and also Mach number is V by C. So if you make the substitutions in the dynamic temperature term we get V square by 2 C p T is equal to V square by 2 into gamma r by gamma minus 1 into T which basically is gamma minus 1 by 2 into V square by C square which is gamma minus 1 by 2 M square. So if you substitute this in the first equation we get T naught by T is 1 plus gamma minus 1 by 2 M square. So this relates stagnation temperature to static temperature through the Mach number. Now this can also be extended to the corresponding pressure and density from the isentropic relations. We get P naught by P is 1 plus gamma minus 1 by 2 M square raise to gamma by gamma minus 1 and stagnation density ratios rho naught by rho is equal to 1 plus gamma minus 1 by 2 M square raise to 1 by gamma minus 1. So these are property relations which relate the stagnation parameters to the static parameters through Mach number. Now if Mach number is equal to 1 we have seen at the end of nozzle if Mach number is equal to 1 then if we equate M is equal to 1 in those equations then the properties that we get are known as critical properties and they are denoted by a star superscript star. So if we equate M equal to 1 in them we get T star by T naught is equal to 2 by gamma plus 1. Similarly P star by P naught is equal to 2 by gamma plus 1 raise to gamma by gamma minus 1 rho star by rho naught is equal to 2 by gamma plus 1 raise to 1 by gamma minus 1. So these are equations which relate the critical properties to the corresponding stagnation properties and you can see that it depends only on the ratio of specific heats. So this was about property relations for ideal gases inherently assuming that there is an isentropic flow. Now what we will do next is to analyze isentropic flow through nozzles and we will take two different types of nozzles a converging nozzle and a converging diverging nozzle. So a converging nozzle in a subsonic flow will have a decreasing area as we have seen. So let us look at what happens as you keep decreasing the area that or if you keep the area fixed as you decrease the exit pressure that is known as the back pressure how does it affect the flow parameters. So a converging isentropic flow through a converging nozzle will basically have involve a decreasing area along the flow direction. What we shall do is to consider the effect of back pressure on the mass flow rate and pressure distribution along the nozzle. We will assume that the flow enters the nozzle from a reservoir where the velocities can be assumed to be 0 and so this stagnation temperature and pressure will remain unchanged through the nozzle. So we will not assume any losses taking place in the nozzle. So that pressure is constant and obviously since there is no heat or work interactions stagnation temperature also remains a constant. So if that were the case this is the nozzle that we are talking about which has flow entering through a reservoir which is at pressure P naught temperature T naught exit pressure is P e and back pressure is the term which we can change depending on the back pressure the pressure across the nozzle also will change. So in the first case first scenario we have back pressure is equal to the reservoir pressure there is no flow taking place. So the variation of pressure ratio is a constant. So the pressure would vary in the format shown here. Now as we reduce the back pressure but back pressure is still greater than the critical pressure. Now there is a decrease in the pressure ratio P by P naught and so this is how the variation would be. So from state 1 to state 2 we have a case where back pressure is still greater than the critical pressure. Now if you reduce it further at state 3 where back pressure is equal to the critical pressure the pressure ratio continues to reduce and reaches state 3 let us say and if you were to decrease back pressure below this critical value what basically happens is that there is no change happening it will continue to have the pressure ratio which will continue to drop till the extent that if you reduce the back pressure even further the Mach number at the exit of the nozzle does not change it will remain the same. And so the point when we have the ratio or back pressure is equal to the critical pressure is known as the choked condition or the choked flow and the pressure ratio at that point is P star by P naught we have seen this depends only on the ratio of specific heats. And so for any other pressures which if we continue to drop the pressure back pressure lower than that we will continuously see a decrease in the pressure ratios till the point if we continue to reduce back pressure to 0 the Mach number at the exit is still going to be Mach number is equal to 1 which because basically because it is a choked flow there is no more change happening there. So if you look at Mach number and the pressure ratio plots so we started off at Mach number state 1 so if you look at the mass flow rate it was equal to 0 because the pressure ratio the back pressure was equal to stagnation pressure at the reservoir there was no mass flow mass flow was equal to 0. As you start reducing the back pressure there is an increase in Mach number till a point when it reaches state 3 which was the critical pressure ratio that is P star when Mach number reaches maximum mass flow that is its maximum value. If you reduce back pressure even below that Mach number does not change we get the same mass flow rate and which is basically the Mach number is equal to 1 at the throat mass flow rate remains the same throughout for state 4 and 5 as well from state 2 3 4 and 5 the mass flow rate is the same and if you look at the exit to stagnation pressure ratio after state 3 it remains the same basically because it is the choked condition and it has not changed after the flow reaches the critical state. So this is how the variation of a flow would be through converging diverging nozzle and from the above what we basically see is that when exit pressure is equal to back pressure well exit pressure P will be equal to back pressure for back pressure greater than or equal to the critical pressure and exit pressure will be equal to critical pressure for P be less than P star. So for all back pressures lower than the critical pressure the exit pressure will be equal to critical pressure the Mach number is unity and mass flow rate is maximum which is basically the choked flow a back pressure lower than the critical pressure cannot be sensed in the nozzle upstream flow and it basically does not affect the mass flow rate. So back pressure this is basically relating the back pressure and the exit pressure to the critical pressure as you change the back pressure values. So let us now look at the same scenario for a converging diverging nozzle because we have seen that the maximum Mach number that you can achieve in a converging nozzle is unity for achieving supersonic Mach number you need a diverging section after the throat. So a diverging section obviously alone will not guarantee a supersonic flow it will happen only as you change the back pressure accordingly. So for back pressures which are different from what it should be you may not really achieve a supersonic flow. So let us analyze a flow a supersonic flow as we change the back pressure. So this is a converging diverging nozzle there is a converging section a throat where the area is minimum and a diverging section P naught is the pressure in the reservoir and P e is the exit pressure P b is the back pressure. So let us say P b is pressure which is currently higher than the stagnation pressure P naught as you reduce it to P a there is a flow which takes place and as you increase the P b what basically or as you reduce back pressure how it affects the flow as you reduce back pressure which is still greater than the critical pressure then we get initially an increase the static pressure reduces velocity will increase and then in the diverging section the static pressure will increase. Now as you reach the critical pressure which is P star which happens when P is equal to P c or back pressure is equal to P c the flow reaches its minimum static pressure here which is equal to P star and you get Mach number is equal to 1, but it does not achieve a supersonic flow subsequently it becomes subsonic and the static pressure continues to rise. So if you reduce the back pressure even further what happens is after the throat the static pressure begins to continues to drop up to a point where there is a sudden increase in static pressure which is basically due to the occurrence of a shock. So at the in the diverging section of the nozzle there is a shock which causes after which the flow becomes subsonic we will discuss more about shocks in the next lecture and then exit of the shock the flow becomes subsonic and it take static pressure again rises like in a subsonic flow. So you have still not gotten a subsonic supersonic flow at the exit of the nozzle it is still subsonic flow because of the presence of a shock. If you reduce the flow back pressure further we get a supersonic flow because the shock which was there in the diverging section will continuously move outward as you reduce back pressure eventually the shock will be pushed out of the diffuser and the flow will become supersonic at the exit of the nozzle. So if you look at the Mach number plot for state A there was no Mach number because there was no flow for state B the Mach number increases and then it decreases in the diverging section state C it reaches Mach number equal to 1 because that is the critical condition and then of course it does not become supersonic it again decelerates and becomes subsonic. So we have subsonic flow all the way here at state D the flow is supersonic after the throat continues so till this point after which there is a shock and because of the presence of a shock the flow becomes subsonic and we still have a subsonic flow at the nozzle exit. If the back pressure is lower than what is happening at state D then the shock continuously moves towards the exit and for all other states which is E and F and G we will have continuously have a supersonic flow all the way to the exit of the nozzle. So we have now achieved a supersonic flow all the way up to the exit of the nozzle and this was possible only because the back pressure was adjusted to values which were lower than what we have seen here lower than pressure at D that is P D and P E and for pressures lower than that we get a supersonic flow continuously all the way up to the exit of the nozzle. And so in the Mach number plot you can see that it goes up to Mach 1 at the throat after which if the back pressure is not low enough it can again become subsonic or if it is low up to a point you may get a shock in the normal shock in the divergent section which means that you get a supersonic flow up to the shock and downstream of the shock it again becomes subsonic and if the back pressures are lower enough then we get supersonic flow all the way up to the exit of the nozzle. So this is the variation of supersonic flow or this is how you would achieve a supersonic flow by changing the exit back pressure and it is not just enough that you put a divergent section at the end of the from the throat and still get a supersonic flow that will happen only if the exit back pressure is low enough. So let me wind up today's lecture where we have discussed at least had a very quick introduction to some of these topics which are related to compressible flows and we had some discussion on one dimensional compressible flows on stagnation properties on the speed of sound and Mach number. And we have seen variation we have derived an equation which relates the fluid velocity and Mach number with the flow area and we have also seen isentropic flow through nozzles two types of nozzles converging nozzles converging diverging nozzles because our interaction for this was a very brief one and there are separate courses which are offered on compressible flows which are known as gas dynamics and so on. And so this is just to give you an idea of what is the importance or significance of compressible flows and why you need to understand stagnation properties and take due care in calculating properties which involve significant kinetic energies. So in the next lecture we will continue our discussion on compressible flows. We will discuss little bit more on shock waves and expansion. We will discuss about normal shocks oblique shocks and Prandtl mayor expansion waves. We will be discussing about two duct flow cases one is duct flow with heat transfer and negligible friction which is basically known as the Rayleigh flow. We will also discuss about duct flow with friction but without heat transfer known as the Fano flow. So these are some of the topics which are in continuation with our discussion on compressible flow. We will take up for discussion during our next lecture.