Introduction to Gas Dynamics and Basic Thermodynamics Review

[00:00:01][Music] [Music] okay so welcome to the advanced gas dyamics uh class uh so as far as the name is concerned it seems like we're concerned only about gases now this course could have been easily been called as uh compressible flows so um some of the questions I always request uh students to ask and also instructors to answer is that why are we doing this course beside obvious obviously the obvious answers to this so uh should we be concerned about gas Dynamics is that important do we see that around uh examples of this around us all the time is this a new topic or people have enough knowledge about this from uh the previous researchers Etc and uh do we know a lot about it are we still learning about it and so on and so forth right so like I said uh this course could have been easily called as compressible flows theories of compressible flows so does that mean that um compressible flows or compressible properties are only a phenomena of gases or others fluids for that matter can be compressible as well well we'll try to answer some of these questions like we said you know the purpose of of this uh course the importance of this course a little bit about history and uh what exactly we should be interested in and of course if we can find novel ways of you know uh implementing the knowledge that we acquire as we go along in this of course so um what we'll do start by starting is a little bit of history and try to understand how uh the study became the study of uh you know compressible nature of fluids became important and what exactly uh you know triggered studies or you know that today we are in a place we actually doing a whole course on uh gas dyamics okay now this is an example which you're probably familiar with so this is a wheel as you can see this is a wheel and these uh small little things out here are uh you know blades buckets if you will and essentially the wheel is is going to rotate but as you can see there is no mechanical mechanism to uh move this wheel instead we have these uh four four things here right well these things are called as nozzles okay let's call them just ducts okay and let's see let's uh sort of try and look at what is so special about uh you know that so essentially what we see is that there is uh so we have a large slightly wide entrance and a slightly narrow so we have a large entrance and a narrow uh exit right so let's complete the story first okay so what you see uh in on the wheel is that we have nozzles like this four His Kind right and um the blue here is essentially steam being injected into these nozzles right steam is injected into these nozzles it enters through this wider entrance and comes out through this um narrow uh exit and it comes out and hits these uh buckets or blades with such an impact that it is able to rotate this wheel at 30,000 RPM right and this ingenious way of doing this was first time um accomplished by the Swedish engineer called called Daval and this is also called Laval's wheel in 1893 so this was the first time he was actually you know used something like this so essentially what is it that this is causing it so we have a stream uh a steam coming in jet of steam coming in and producing a lot of impact right so essentially what is happening is steam comes in and it is choked you know that's the technical term to use so essentially it comes here it accelerates there's a corresponding pressure drop and when it uh hits the buckets is able to you know rotate it with as high speeds of 30,000 RPM and that was pretty large at that point of time right so this is essentially a convergent nozzle okay now I can we can have another type of uh nozzle as well like that where the end entrance is narrow and the exit is uh large right so now this is the entrance and this is the exit and this is a Divergent nozzle okay this is a Divergent nozzle okay now what we can do is combine these two so what we can do is essentially combine these two right so what you know as probably something like this so this is nothing but a convergent Divergent nozzle right this is the modern day convergent diversion nuzzle so it works on the same principle of the compressible flow which was used by Laval the first time round okay so this was the first of its kind way back in uh 1893 now I'll go over one of these um uh examples you know jump over to to 1947 right you're probably familiar with this uh picture out here so this is the B xx1 right so and this was the pilot chaker and this was the first time in 1947 that uh this plane and Jagger was able to break the sound barrier which means that he was able to fly at at a speed Which is higher than the speed of sound which is Mark 1 so basically he was able to break the sound barrier and why is this important why am I bringing this up uh in terms of uh you know Laval's experiment well the reason being that the thrust in the Bell xx1 the thrust in this plane was provided by four convergent Divergent nozzles and that was the one which was able to create speeds or um you know move this B xx1 at speeds for which he was able uh to reach you know Mark so essentially from a very simple experiment which Laval used it was basically a Swedish engineer uh to a rotat a wheel he used essentially the principles of compressible flow and we've been able to catapult that and use it for something as sophisticated as um supersonic flow okay now there are so similarly so uh we have been able to see um application of compressible flow theories in uh you know in literature so far in so some of those applications could be uh internal flow through rockets and gas turbines high-speed flow through uh wind tunnels external flow over airplanes for Mark 3 reentry Vehicles Etc flow inside and IC engines so what I mean by this is that um all people so from mechanical background or Aerospace background or you know so and so on and so forth Are all uh you know are interested in flows like this because uh compressible flow theories are applicable to a wide range of uh disciplines okay now I said something about external flow over airplanes for mark3 it seems like a thumb rule that if the flow over airplanes is uh less than 0.03 then the compressible nature of the fluid may be overlooked so we'll um I try to see uh what that means okay so we'll try to see that with uh say an example for example let us uh let us take an airplane okay and um say at this point say a right Which is far off from the airplane and um we have a the speed here the speed here is 100 mph which is around 45 m/s right so we have so let me call this as a free stream the free stream Upstream of this plane is 100 m hour now this comes over here and it accelerates over the plane right it accelerates and uh at some point here say the speed out here is 150 mph right which is around 67 m/s right now as we saw in the example which I showed you just uh earlier over here that the flow accelerated right there was a corresponding pressure drop right so here to what we would like to know is that if the flow is moving say from is being accelerated from 100 m hour to 150 M hour then what is the corresponding pressure drop right and uh what we should I mean how we we going go go about finding this okay now first things first let's do this let's just say that this flow out here is incompressible it's not compressible in nature okay so if that happens and am I sure of that no I'm not so I'm going to just assume it is incompressible and then go and uh verify that whether my assumption is correct or not so we'll just say it is incompressible for the time being and we'll use the B knowledge principle between A and B right so so this is the uh Bales principle so we'll apply that between A and B so what we will get so I get a uh relative pressure drop of pressure change of uh 1.5% okay and now we want to see uh whether you know the I used an incompressible uh Theory over here if I was correct in doing that if not then what we will do okay so let's move a little forward now I keep saying that it is compressible in nature now how do we quantify that how do we put math into this okay I understand that there is a flow over here there is a corresponding pressure drop the question is how much how do I relate the uh this pressure drop to the speed or do I have to do that can I do that so I need some sort of quantification right to say how much a gas or a fluid is compressible right and only based on that can we say that uh this is an incompressible fluid or this is a compressible fluid let's so let's try and find out some sort of uh quantification for this let's so let us say we have um a fluid element like this okay and uh this is the uh pressure and this is the uh volume okay so we have okay now say there is an incre increase in the pressure I increase the pressure so there is an increase which is um DP and there is a corresponding change in the volume right so I will write this as so this is a corresponding change in volume now uh if there is an increase in pressure the volume will decrease and that we know from our you know known uh principles of gases Etc so I will note that when I write change here I actually mean there is a decrease so I will denote that by a negative sign so if I have a positive pressure change there is a negative volume change right so negative change basically indicates there is a decrease in the volume now we shall Define what I understand by compressibility right so let me write this here so compressibility is defined so let we call that a sta so then I shall write that as every time I write an equation I always like to say that in words so that I understand what it means so that I don't have to remember the uh expression itself if I know what that means I can always develop the expression so what this means is that fractional change in volume which is DV by V per unit change in the pressure so that's what compressibility means there's a fractional change in the volume volum per your change in pressure and I can write this as okay let me call this as I like to name my equations for further reference so that's how I Define compressibility okay now uh the next thing we can see here that if there is is a change in the pressure and there is a corresponding change in volume what happens to the temperature if there's a change in pressure there is change in volume there should be a corresponding change in temperature as well right now if that is so what is happening to that over here are we taking care of the temperature or no now let us there therefore Define this compressibility for a few cases where we do that we say something about the temperature right how do we do that now first thing we'll see is that we'll say is that the temperature is constant right and how do we hold keep the temperature constant that that there is a um positive heat transfer that there is a nonzero heat transfer right and as a result of which the temperature is uh constant okay so then I can Define then I will write my um compressibility in this form okay and we can call this I think you should have guessed by now that this is isothermal compressibility so this is this is isothermal compressibility okay so in this case the temperature is uh constant okay now uh some things to notice here is that when I write these uh subscripts out here right so I write T over here this is a Deno and then when I write here I am just writing I'm just bracketing this part of the derivative I didn't include this is it the right thing to do or a wrong thing or I should have included that anyways well the point here is that the compressibility when I'm defining right the process the process under which the volume is changing based on this pressure now this process is happening under an isothermal condition right so when I increase the pressure there is a change in volume now that is the process during which the temperature is uh constant right and that this process I am comparing with what the volume was before that process started right so this is B basically the state volume right this is a state parameter or uh it is the it is a representation of the state right and then we started a process in which say there is an increase in the pressure and a corresponding decrease in volume now that is the process which is happening under isothermal conditions so therefore I'm going to write it in this fashion okay so one condition is where I keep the temperature constant now what if I don't keep the pr temperature constant there is no heat transfer happening there are no other dissipative forms but there is no heat transfer happening then what happens do we have a condition for that or uh what do we do now we will um Define something else okay now we will Define a an isentropic compressibility okay we will define an isentropic compressibility and I will write that as okay now what is this uh s that I'm uh writing here okay now this is entropy this is the the property called entropy and for an isentropic uh compressibility so if I may write it this way so this is uh isentropic compressibility right so here for this the entropy is constant right in the isothermal case if the temperature was constant for an is isentropic case the entropy is constant okay now what is this uh s now entropy I guess you're familiar with with this but um what I will do in the next lecture is do a basic review of thermodynamics because this entropy is a thermodynamic variable right so let me not gel on that too much but we will go ahead and do a brief review of thermodynamics and so we'll come back to what we mean by uh isentropic and entropy and so on and so forth fourth okay all right so now uh let me give you some numbers numbers always help right to get uh feel for you know how compressible or less compressible and so on and so forth for example um this is the isothermal compressibility for water right this is for water and this is for air and both these are at one atmospheres so what do you notice from here what what you see these are the numbers that we have so that's for water and this is for air so what do you notice what exactly can you infer from here what you can see is that the compressibility of air is much more than that of water which is essentially what it is that for gases the compressibility is much more than water right so we kind of beginning to see a few things about the name of this course on the first slide as to why we are doing this course or why do we call it Advanced uh gas Dynamics gas Dynamics as it were okay so um so essentially that's a thumb rule gases have more compressibility okay now um so let's find out so we've defined pressure volume density temperature etc etc so what is the relationship between them if we do have uh you know changes in them how they're related or if at all right so let us uh do something uh simple and develop a few Expressions right so say we have a mass of gas a unit mass of gas so we Define a specific volume right so therefore we have density right or we can write this as okay then let's do a little bit of math and do this right so basically what I'm you know this is simple here what all we're trying to do is trying to get a relationship uh in terms of the compressibility right so this is the equation one that we wrote about so this was we need a derivative something like this right so when I have this expression over here so DV by DP so if you can do this math right so this is uh simple over here so I will do this right so this is the uh derivative right so there therefore now I will try and write the compressibility so now if I write the compressibility how does that look now the compressibility is nothing but 1 by V DV by DP and we got 1 by v as row into DV by d uh DP which is over here so we inut that in there which is - 1 by row s d d row DP right so what we get over here so this is uh what we get right so what I will do is write this out this way so essentially D row is equal to row tow DP right we'll write that and or we can also write this as okay so let's say this is uh okay so what what we say over here is that um in here what you can see now we have just now said that for a given uh for a gas the compressibility is much higher than that of a liquid right so therefore for a given pressure the density changes in a gas is much higher than a liquid you know because this compressibility is much higher for a gas right so therefore to bring about um for a given uh pressure you can you can see here this is the a fractional change in density right so for a given pressure for air for example for a given pressure change there is going to be a corres a much uh larger uh density change than compared to say water essentially because this compressibility Factor here is much larger for air than for uh liquid having said that now let's go back to your previous problem so therefore we have this uh problem over here right oh this here and we said that the pressure change is 1.5% right now you can do the rest of the math yourself and relate the uh densities relate the densities Etc if you want in this case it was uh incompressible so there was we said there is no change in the density right with with when we consider no change in density what we found is that the pressure change is 1.5% but what I can say here is that since the corresponding relationship sh is for a compress if I were to consider this as compressible then the density change is essentially given by this relationship right now in here since this pressure change is very small it is just 1.5% so I can say that we have infin dismal or very small changes in density and hence I can it's okay for me to uh ignore that and that is so I can say say more or less safely say that it's okay to say that this is an incompressible uh fluid so therefore changes in density here are not very dominant so therefore it's okay to use uh the incompressible theory over here okay so hence I was fine doing this then of course the new uh you know the immediate question which we should be asking that okay so if we had a flow which was going from 100 m hour and it reached up to 150 miles hour right and that caused changes in uh density which we were able to ignore but then how how far would we be able to do that or for how larger speeds or how smaller speeds can we do that if I had a velocity which is going from say 1,000 M hour and going up to say 1500 m per R what I still see changes in uh pressure which was very small and causing changes in density which I could ignore would that be a possibility so therefore I need to relate this now to the speed here and that is why that is where when I first said that for uh flow over airplanes for uh Mark numbers less than three it is incompressible so now over a period of time with experiments Etc been able to see that more or less it's like a thumb rule that um okay so I me just right over here that for Mark numbers less than3 right for Mark numbers uh less than3 the changes in um uh changes in density can be ignored so therefore as a more or less uh thumb rules for this we can consider the flow to be incompressible right so to answer the question that whether we know a lot about this topic from you know history and literature Etc well we know enough to know that these thumb rules right so people have worked on this over a period of time so we kind of know these sort of things okay now um the next thing uh like I said the next thing to do would be to do a brief uh review of uh thermodynamics okay now what is this uh connection of thermodynamics with this okay I said something about entropy when we uh defined isentropic compressibility but what exactly why do we need to study the thermodynamics when we are dealing with compressible flows okay now we saw that there is when we have say higher speed flows okay or essentially to generalize when we uh we have changes in pressure which is causing changes in density right and uh these are changes in density which are not small enough to be ignored now there should be also changes in temperature right now when we um consider those changes in temperature that's when we need to uh study a little bit of thermodynamics before we proceed okay so I'll do a brief review of that now let's just say let's just uh you know begin with uh something like this okay let's just begin with now this P here is pressure p here is pressure V here is uh specific volume which is volume per unit Mass okay okay so I'll just write the unit here okay and uh R is the specific gas constant and T is the uh temperature okay now this is something that we are familiar with okay so let me write here what what exactly is this I hope you can recognize this right this is the equation of State so this is an equation of a state for a perfect gas there are two things over here one is equation of state and the other is perfect gas what do what do these things mean equation of State perfect gas what do we mean by a gas which is perfect right now what all we have done here is taking the pressure the volume temperature Etc connected them by some sort of uh constant Etc and we're saying that this is the equation of State it's some sort of equation right and uh we are saying this is for a perfect gas okay now what do we understand by this uh perfect gas okay now let's just uh let's just Define that at first okay now um a fluid essentially a fluid essentially is consists of molecules I think we all aware of that and these molecules are very widely spread about spread about for gases and more closely packed for a liquid right and these fluids these particles or molecules are in constant random motion um they're in random motion right so now if I um take say a volume like that say if I take a volume like that right so there is a constant uh inflow and outflow of molecules across this boundary so the number of molecules out here is constantly changing and we have you know free flow of molecules on either side which are constant motion now when we have these particles right it's by Nature there is intermolecular forces we know that right now if you have two particles which are like really far off we have particles which are very far off then they exhibit very small or very weak attractive force and if they close say like that then they have a strong repelling Force so these are the intermolecular forces right so therefore these um intermolecular forces kind of Define or kind of uh control for that matter the movement of these particles right I think it's kind of obvious to see from here now the thing is what is this perfect per gas the perfect gas is one when we ignore these when I say these I mean these intermolecular forces for a perfect gas we ignore the these forces and when we do that there is a certain relationship between the pressure volume and temperature density Etc of that uh gas and that relationship is what we Define as the equation of State so which means that if I have a set of you know some mass of gas then that will obey this sort of a relationship provided we ignore the intermolecular forces right now the next question is when I Define this pressure volume uh temperature Etc right do I Define it for a are we considering say the pressure of every single particle are we considering the uh temperatures of every single particle when I take a mass of gas or what am I doing right now it's important to say that all these parameters here are all defined on a Continuum these are all defined on a Continuum right now what do we mean by that now now what we mean by this is that let me take these a boundary like this right and I said that the number of particles is changing Etc you know and this is just a imaginary boundary so what I'm going to do here is take a collection of particles take a collection of these particles more or less I take such a boundary that these number of particles is not changing okay the number of particles more or less Remains the Same okay so then what happens is that the okay let me redraw this so I have a boundary like this right I have a boundary like this which is big enough right which is which is big enough so that I do not consider individual movements of properties of the particles right instead however we are concerned about the behavior of this whole collection of particles this collection of particles as a whole right so individual uh behavior of each particle can be ignored at the same time now this is not so large enough that it's such a big space that you know there is aeration of particles here here here and the particles are not uniformly distributed then as a collection of this we we may not be able to say that there is one particular temperature pressure volume because for example if I have something like this the temperature here will probably much larger than the temperature over here right so then it is not in so it should not be so large as well instead if I have a volume like this this would be a good Continuum right so that this I can just consider as uh you know a group of particles so more or less around here the the number of particles remains same and I can consider at this particular area for these group of particles a single uh value of the pressure temperature density volume Etc so that that's what we Define as a Continuum so all these parameters all these parameters pressure volume temperature Etc are all defined on a uh Continuum okay now we can write this expression now in uh some more uh ways for example okay so here we wrote this in terms of specific volume right so we said but which which means uh the volume per unit Mass right all I'm done here is because this here is the volume this is volume okay this is volume and this is gas constant let me differentiate between this and this so let me call this as a star this one is the specific gas constant and this is the gas constant so basically what we're doing is multiplying it by the mass out here the whole equation okay now what is this n here so the N is the number of moles in the gas which essentially this is doing nothing but giving us an idea of the number of molecules in the gas in the group of gas that we considering and I think you will remember that one mole of gas consists of 6 10 26 molecules they have AAL number of molecules right so what we're essentially saying here is that for a given number of molecules of that particular gas this is the relationship that it holds okay this is also for a perfect gas just trying to look at it a little differently okay now um again what is one mole what exactly is one mole what is the concept of a mole it is nothing but the molecular mass of a gas for example chlorine is 17 I think right so that is the molecular mass of a gas okay so that is uh essentially the um uh you know the equation of State for a perfect gas and I'll just give you an example of not not example just give you some numbers right just to remember some of these numbers to be familiar with it and we'll do just a small example and uh we'll wind it up for today okay so for example this is the the universal gas constant so this is equal to so this is our star basically how did I write it yes star so this is for example 8314 uh JS per kg mole Kelvin right and say specific gas constant is per Mass this is the molecular weight and for air for example so for air so this is equal to 287 Jew per kg Kelvin okay so the these are some numbers which you should be just familiar with what we'll do now is just quickly do this and uh wind up for today okay now all I'm saying is that so air is given okay so I have air and the pressure is at one atmosphere right then that I can write as calculate the isothermal compressibility right what is the isothermal compressibility so compressibility is nothing but fractional change in volume per unit change in pressure so this I can write as right so isothermal compressibility so how do I uh find this out okay now let us go back to the first equation of state so PV is equal to RT so if I do this then I will write the volume as RT by P right now if I do that okay so if I uh do that then all I need to do is DV by DP right so I do DV by DP and what I do here is that right this is what we get so then if I input that into this expression the DV by DP into this expression what I get is this hopefully you should be able to uh so 1 by P and the p is given right so the p is given okay now what you can see here is that I said isothermal uh compressibility right so what I mean by uh saying that over here is that you remember this expression right so essentially what we mean out here is that this change in volume and pressure is happening over a over a process which is isothermal so the temperature is constant so therefore when I did this right when I did this over here I kept the temperature constant and that's when that's how I calculated the uh derivative over here okay now uh temperature is a constant meaning that we are not considering any internal forces obviously because then there will be changes in its internal energy and I cannot say that the temperature is uh constant so therefore I can only calculate the isothermal compressibility from here right for example I have uh the isentropic compressibility as well it is the same formula and we have the same uh you know you know Etc is available so I can actually use that but the idea is that when you use such a formula you need to understand that what exactly are you using using it for so if you if I said find out the isentropic uh compressibility Etc you cannot use a process in which temperature is constant okay so we will have to you need still need to think uh about that so I think we'll uh stop over here what we shall continue to do uh next is that when we I talked about that so far whatever we did temperature is constant so why are we going to do a review of thermodynamics because when temperature is not constant then I need more variables just the pressure volume temperature is not going to be sufficient to um you know to study my gas or any process what I will need further is is an expression as to how these variables relate with temperature and to do that we will need more variables than that and that is what thermodynamics will provide right the concept of entropy internal energy and so and so forth we will continue to do that in the next lecture okay thanks [Music] [Music]