 So, in this module, we will look at compressible flow through nozzles. Now, compressible flows are encountered in many applications mostly in aerospace and mechanical engineering. For example, flow through nozzles, flow through turbomissionary blade passages such as compressors or turbines and diffusers and compressible flow is also encountered in external aerodynamics like for example, flow over an airfoil or aircraft wing and so on, flow around aircraft and also flow through rocket engines. So, these are all examples of compressible flows in mechanical and aerospace engineering. Of course, for aerospace engineers, the working substance is always air, but for mechanical engineers, the working substance is not only air but also steam because mechanical engineers have to typically work with steam turbines and steam nozzles. So, steam is working substance that mechanical engineers encounter quite frequently. In addition, gas dynamics of refrigerants is also important because as I mentioned earlier, in domestic refrigerators, capillary tubes are used as a throttling valve. And so, the gas dynamics of flow in the capillary tubes is very important in the design of such vapor compression refrigeration systems. So, mechanical engineers will encounter air, steam as well as refrigerant as working substance whereas aerospace engineers will encounter only air as the working substance. And the scope of the lecture in this module will be confined to flow through nozzles because flow through as I said, terminationally blade passages or flow through nozzles or diffusers all can be categorized into a simple quasi one dimensional flow category. Now, of course, detailed calculations can be made, but once we learn quasi one dimensional flows or rather flow through varying area passages, then we would be able to apply this to most of the applications that we have in mind for a first cut analysis which is usually very adequate. So, that is what we are going to do. Our scope is to look at flow through nozzles or passages of varying area of cross section. Now, so we will learn whatever is required to achieve this objective. So, if air is the working substance, then we would look at normal shocks because normal shocks are encountered in nozzles whereas we will not really do normal shocks with steam as the working substance because it is a highly non equilibrium phenomenon and well outside the scope of this work. So, we will look at the basics in irrespective of the working substance, then we will develop the theory of normal shocks which is applicable for air and then we will look at flow of air as well as steam through nozzles. We will not really look at any particular application of involving refrigerants, but the theory is very straightforward. So, we will develop the theory both for a calorically perfect gas such as air as well as a substance like steam which is far from being calorically perfect. Let us first introduce formally introduce the notion of compressibility. So, compressibility of a fluid is defined as minus 1 over v partial v partial p where v is the specific volume or if you write it in terms of density we may write it as 1 over rho partial rho partial p. The negative sign in the front here ensures that the compressibility is a positive quantity. Now, the value of the compressibility will depend on how this is done. For example, any change in specific volume due to a change in pressure can be accomplished in many different ways. For example, let us say I have a piston cylinder mechanism and I have some compressible substance inside let us say air. Now, when I compress this substance there is a change in specific volume as a result of a change in pressure. But the magnitude of the change in specific volume due to the change in pressure will depend on whether the process takes place isothermally or whether the process takes place adiabatically or something else. So, the process is also important. So, based on that we may have something like an isentropic compressibility or isothermal compressibility and so on. Now, let us take a slightly broader look on this notion. We know that specific volume v is a function of two properties. The two property rule applies in this case, there are no other means or modes of energy storage. So, the two property rule applies and I may write the expression for change in specific volume due to a change in pressure as well as a change in temperature. So, when specific volume of a compressible substance changes by a certain amount let us say dv, part of it is due to change in pressure, part of it is due to change in temperature that is what each one of this term here represents. Notice that because it is a function of two variables when you take the partial derivative automatically this quantity is evaluated at constant temperature and this quantity is evaluated at constant pressure. The first term here may be identified as compressibility as we just defined it. So, any change in specific volume due to a change in pressure alone is called compressibility. Remember, here it is due to a change in pressure alone because temperature is being kept constant. Now, here any change in specific volume due to a change in temperature alone, remember p is being held constant is simply volumetric expansion. So, we have let us say air in a vessel or a room and we add heat. So, we heat the air then the air expands as a result of heating it becomes lighter and the lighter air will go up because the air is undergoing volumetric expansion. So, that is what this term signifies change in specific volume due to a change in temperature alone. Now, notice that any change in specific volume due to a change in temperature alone is not an indication of compressibility that is simply an indication of volumetric expansion. Any change in specific volume due to a change in pressure alone is considered to be an indication of compressibility. So, when you have a flow and you start talking about compressible flow there will be changes in density or specific volume. Now, the change in density or specific volume due to a change in pressure alone is considered to be an indication of compressibility. So, that is why we are starting with the notion of compressibility because that will naturally lead to compressible flow where compressibility effects are presumably significant. So, in a compressible flow where compressibility effects are significant changes in specific volume would be largely due to change in pressure. There could be a change in specific volume due to change in temperature also, but it will be largely due to change in pressure. Now, the isothermal compressibility of water is 5 times 10 raise to minus 10 and the isothermal compressibility of air at both at 298 Kelvin for air it is 10 raise to minus 5. So, it is clear that the isothermal compressibility of air is 5 orders of magnitude more than that of water not 5 times but 5 orders of magnitude more than that of water. So, what are we to infer from this? Does this mean that any flow of air should be considered compressible? Will compressibility effects be dominant in such a flow? For example, I may look at a flow of air over an automobile, I may be looking at flow of air over an airfoil or a wing, I may be looking at flow of air in a nozzle or flow of air in a turbomissionary blade passage. Do we classify all these flows as compressible? Because the isothermal compressibility of air is so high. Remember, this is isothermal compressibility of air. So, this is isothermal compressibility of air. So, because it is so high, do we classify all this as compressible flows? So, we need I mean it would be helpful to have a criterion by which we can actually make this sort of determination based on some expectations of what we are going to see in the flow. We have some ideas about the flow field. So, based on that we should it would be helpful to have some criterion which would tell us whether compressibility effects are significant in this flow field or not. So, what we are the distinction that we are trying to make here is this. The fluid itself, see it is clear from this that water is incompressible, no question about it. But it is clear that air is compressible. So, what we are trying to distinguish here is a compressible fluid versus compressible flow. The question is if the fluid is compressible, is the flow of such a fluid always compressible or is compressible flow slightly different or somewhat different from a compressible fluid or can we possibly have flow of a compressible fluid in which compressibility effects are not significant. That is the distinction that we are trying to make here. Now, you all know from your high school physics that sound which is nothing but propagation of pressure waves in any medium. So, sound travels in any medium with the speed which depends on the bulk compressibility. The less compressible the medium the higher the speed of sound. For example, speed of sound in air, if you go back to the same example as before, speed of sound in air is at room temperature is 330 meter per second. Speed of sound in water at this temperature is of the order of about 1 to 1.2 kilometer per second. So, you can see that it is about 4 times higher in water than in air. So, it depends on the bulk compressibility. So, speed of sound is a convenient reference speed when flow is involved. How? Let us say you know let us revisit the examples that I mentioned flow over an automobile. Let us say the automobile is moving at 120 kilometer per hour which is fairly high speed as you would know. Now, what we do is we try to calculate the speed with reference to the speed of sound. So, when the automobile moves with at such speed, let us say let us say the automobile is moving at a speed of 120 kilometers per hour which would roughly work out to about 30 to 40, about 35 to 40 meter per second roughly. Now, when I compare this with the speed of sound in air under these conditions, normal atmospheric conditions, it would be 330 meter per second. So, we can see that the maximum speed that we are likely to see in the flow around this vehicle, let us let me just draw this with a slightly different color. So, we are trying to let us say look at flow around this automobile. So, let us say this is the ground. Now, the maximum speed that we are going to that we are likely to see in this flow field is going to be 40 meter per second which is considerably less than the speed of sound which is 330 meter per second. So, the speed of sound serves as a useful reference speed in compressible flows. Compressibility effect becomes more pronounced as the flow speed becomes comparable to the speed of sound. Now, the flow speed in this case, remember we will encounter a flow speed when we switch to a frame of reference where the automobile is stationary and the air is flowing around the automobile like for instance, if the automobile had to be kept in a wind tunnel. So, in such cases when the flow speed, maximum flow speed is comparable to a speed of sound then compressibility effects will be significant in the flow field otherwise it will not be. So, good reference for this good reference speed to categorize the flow or to distinguish whether compressibility effects are going to be significant or not is the speed of sound. Now, I also gave some other examples like flow over an airfoil. So, the aircraft may be flying at a speed of let us say 800 to 900 kilometers per hour which would easily work out to at a very high altitude where the temperature may be of the order of about 220 or 230 Kelvin, 240 Kelvin something like that. So, this is at an altitude of may be 35,000 feet. So, speed of about 800 kilometer per hour or 900 kilometer per hour would work out to a Mach number of roughly about 0.9 or so. So, compressibility effects would be very much significant. So, in the same manner we can make a determination whether compressibility effects are going to be substantial in the flow field in the blade passage of a compressor or a turbine by having an idea about the maximum speed that we are likely to encounter. It need not be accurate. We want to see whether the maximum speed is anywhere close to the speed of sound. Even if it is half of the speed of sound we would then say that yes probably compressibility effects may be significant. We will in fact give a number for doing this also. But you get the idea that as we get as a maximum speed in the flow field keeps getting closer to the speed of sound compressibility effects are significant. So, basically what we are going to do is to define a ratio called the Mach number which is the actual speed in the flow field divided by the speed of sound at any point. So, basically in a flow field speed of sound will vary from one location to another depending on the temperature. And flow velocity will also vary from one point to another depending on the flow field. So, the Mach number is actually a local quantity. But what we are doing is we are actually using the Mach number sort of like a global quantity. We make a guess for example in the case of the flow over the automobile we make an educated guess that the velocity anywhere in the flow field is going to be less than 120 kilometer per second or 40 meter per second. So, the Mach number even for that case is going to be less than 0.1 or 0.1 or so. And in the same manner we said the aircraft is flying at 900 kilometers per hour. So, the maximum speed is going to be in that order of magnitude. So, for that the Mach number is 0.9. So, definitely compressibility effects will be significant in this flow field. So, we are actually using a reference speed to calculate the Mach number. But what we must keep in mind we will come to this little bit later what we must keep in mind is that Mach number is not constant for an entire flow field it varies from point to point. That is why we have emphasized it here V is the velocity magnitude at any point or location and A is the speed of sound at that location. We can try to get you know I mentioned just a little bit earlier that for Mach number 0.5 we can say let us say give the benefit of a doubt and say that you know compressibility effects are probably going to be significant or not insignificant. But at what value for Mach number would we say that compressibility effects are insignificant. It would be nice to have a number or a criterion which is what we will try to work out based on some simple order of magnitude arguments. So, if you go back to this example now as the streamline approaches the vehicle if you look at the stagnation streamline which would be this one. So, the pressure at this point is the stagnation pressure you know from Bernoulli s equation that the stagnation pressure dynamic pressure in this case is going to be rho times V ref square. Now you may say that it is actually V ref square over 2 and so on but that is not what we are interested in. We are interested in an order of magnitude estimate. Numerical factors like this unless they are very large do not make a difference to the order of magnitude analysis. If there is a coefficient in front which is like 10 or 100 or 1000 then we have to be concerned. If it is 1 or 2 or 3 we do not have to be concerned because we are doing order of magnitude not factors of magnitude order of magnitude. So, you can see that the pressure at the point that I indicated in the case of the automobile this is where the pressure is going to be maximum and the maximum pressure is going to be roughly of this order over the background atmospheric pressure. So, remember we are talking about delta P that is change in pressure. So, the maximum change in pressure is going to be at the front stagnation point and the value of the change is going to likely to be of the order of rho times V ref square. V ref here is a characteristic speed and we estimate it for the maximum value. What is the highest possible delta P which means we substitute 40 meter per second for V ref. Now, you also know from your high school physics that speed of sound is given by this expression square root of delta P over delta rho with entropy remaining constant. So, if I take now I can take a quantity like delta rho over rho and write it like this delta over rho over rho equal to 1 over rho times delta rho over delta P times delta P where I evaluate this quantity in an isentropic process. Now, delta P itself is equal to rho times V ref square and this is equal to 1 over a square where a is the speed of sound. So, we end up with this quantity so, we end up with delta rho over rho as being approximately remember this is an order of magnitude analysis. So, we say there it is of the order of M square where M is the Mach number based on the reference quantity not at each point we are calculating that for the reference value. So, basically notice that this quantity so, if you look at so, from our earlier expression of delta rho or isentropic compressibility we may also write delta rho over rho as equal to tau s times delta P which means that this is equal to tau s and which we knew already. So, speed of sound is thus related to compressibility that is why we are actually using speed of sound as the reference speed because it is related to the bulk compressibility of the medium through this expression. So, in fact you can show that tau s is equal to if you compare these two you can show that tau s is equal to 1 over times 1 over a square. So, a is the speed of sound that is that is the so, the so, you have velocity there because tau s is compressibility. So, the speed of sound links tau s and compressibility which is why the sound propagation of sound in a medium is always or the speed of sound in a medium is an indication of its bulk compressibility. Now, we are saying that delta rho over rho scales as M square. So, the guideline that we generally have for accounting for or neglecting compressibility effect is to say that if the change in density is less than 10 percent of the mean value which means I am sorry which means this quantity delta rho over rho is equal to about 0.1 or less 10 percent is 0.1. So, if it is 0.1 or less then we can generally neglect compressibility effects. So, if delta rho over rho is less than 0.1 equal to 0.1 or less then that generally implies that M is less than or equal to roughly 0.3 M is less than equal to 0.3. So, compressibility effects as a guideline we can say compressibility effects may be expected to be significant only when the Mach number exceeds 0.3. This is again just a guideline. And bear in mind that this is based on general estimation. So, you may have a flow field let us say where Mach number let us say we have flow over an airfoil or a wing or something like this. So, let us say M is equal to 0.3. So, it is quite possible that as the flow goes over the airfoil there may be some locations in the flow field where the Mach number is actually more than 0.3. That is quite likely. So, you have to take this guideline value with caution. You have to be very careful you should have an idea about the flow field that you are looking at. If it is too close to 0.3 then it is suddenly possible that there may be regions here where compressibility effects may be locally be significant, but not everywhere. In that case you have to treat the entire flow as compressible because if it is significant in some parts of the flow field, but not in other parts in order to account for parts where it is significant you have to treat the entire flow as compressible. So, this is the guideline value should not be used blindly. You need to have an idea about the flow field and then evaluate this criterion with respect to the flow field and then judge whether the compressibility effects are significant in this flow field or not.