hello everyone welcome to this class on gas dynamics we will first start with the discussion on what gas dynamics is and how is it different from fluid dynamics what will be the basic difference if it is fluid dynamics versus gas dynamics what will you tell will be the difference in simple fluid dynamics which we talk about density remains constant roughly and when it goes to gas dynamics we will talk about density changing in a sense we are talking about compressible gases we're talking about compressible flow of gases is a better way to put it we are interested in flow of gases and now we are thinking compressible flow so first thing we need to understand is what is compressibility that is the first thing we need to think about compressibility so when we think about compressibility how will we define compressibility something to do with the density gradient should be related to density how is it density changing how we will density change and what are the various ways by which density of a gas element can change you're comfortable at gas element rate like fluid element we can say now it is gas element we will say specifically gasses okay so how can a fluid density change or a gas density how can it change various processes what all can it be change in pressure anything else change in temperature I could heat or cool a gas that will change at the density or I could think about changing its pressure that is I can compress the gas or expand a gas that is all the things I can do to the gas so now we will go write something on the word we will say I will use this notation for specific volume mass specific volume that is volume per unit mass and I put this cut for V to tell that it is volume versus if there is no cut I will use it for velocity I will keep it that way we will write this as a function of temperature and pressure because we know that these two can change it now we want to use simple calculus and tell differential change in this specific volume and make it D we will just write a DV by V what will this be equal to you guys know it already so I will just keep this one by V separate and then we will just write DV in terms of derivatives with respect to each of the terms next time I will be more careful about the corner line okay I will have these two terms of course you have to tell that this derivative keeping pressure constant and this derivative is keeping temperature constant so this is your expansion it is I heat the gas at constant pressure and that is expanding and this is your isothermal compressibility these are the two terms we are looking at so if there is any fractional change in volume specific volume here then it could be because of a small change in pressure or because of a small change in temperature now we are more interested in thinking about this term if we are talking compressible fluid flow it may also be changing because of heating or cooling we are more interested in this term mainly so we are looking for a term which is something like this this is the coefficient for that DP if this is very high then the fractional change in specific volume will be very high for a given change in pressure when that happens we call this highly compressible gas that is the way we are going to look at it so this is related to compressibility this is what we will call as compressibility of the gas it can be written in terms of density also what is specific volume it is volume per unit mass density will be mass per unit volume just reciprocal of it I can write it in terms of density compressibility can be defined as this this is also probably called compressibility whichever way you look at it okay in most of gas dynamics we will think about this this is easier to work with so we will typically think about density of a gas compared to a specific volume of a gas it is equivalent just 1 is a reciprocal of the other we will stick to this now this is one way of looking at the compressibility but if you go look at books there are so many books for compressibility they will tell you different different definitions for compressible flows one of the definitions happens to be it is related to compressibility where we will tell if the compressibility is very high then we will call it it is a compressible flow other definitions exist what are the others I did not get that yes in a way the bulk modulus is related to this term this is related to bulk modulus yes we can express this in terms of bulk modulus that is one way of doing it there are other forms as in in terms of velocities there is one way of defining how will I call a flow compressible flow Mach number related we will keep that also in mind we'll come back to it after some time and I will just tell you roughly what it means I am going to define something called a Mach number which is ratio of the fluid flow fluid element velocity divided by the speed of sound at that local point in the flow okay so that will be the local Mach number of that fluid element why are we and we are going to say that it is if it is very very low compared to 1 then we will call it incompressible and if it is pretty high compared to 1 or even higher than 1 we will call it compressible flow that is what we are looking at should there be a negative sign I guess the term will come out to be oh if you want to call it as a compressibility if you want to call it as compressibility and you want to say I want to keep my compressibility positive then I have to put a negative sign or else what will happen is this density - pressure relation will be opposite to each other right no nonce do not density to pressure volume to pressure will be opposite to each other I will have to put a minus sign if I want to think about it as bulk modulus if I want to leave it like this it should not affect in here it will not be a negative we will use only the density waster if we are looking at the density term we are looking at the density phases now if you are looking at specific volume increase if pressure increases typically volume decreases at constant temperature so DV by DP will be negative but we want to call bulk modulus as a positive modulus quantity right so if that is the case then we will put them a negative sign in front of it but if you are using in density it should not affect if it is density it should not affect at all we will keep it the same now we said something called Mach number and we said if it is far less than one it is incompressible otherwise it is a compressible fluid why are we comparing with a which is speed of sound of that particular fluid element at that point why are we comparing with speed of sound it is not just any reference I could have picked speed of light if you want as a reference but that is not what we picked sound travels by compression and rarefaction okay how is that related to what I want to tell compressibility we won't worry about obstacles right now in the flow even if it is flow with no obstacles just clean flow if it is mark three flow why are we comparing it with speed of sound one answer is something like Soundwave was related to compression and expansion of the gas fluid element locally if I am speaking to you I am sending compression expansion waves to you right that compression and expansion how fast can I compress and expand the gas will tell you how fast the fluid will go towards a quick way of talking about compressibility I always like giving this example is imagine a metal rod and imagine air column exactly the same volume out there now what we are thinking is I am going to take a hammer and hit the metal rod here and I put my finger on the other end to see when I will first feel the wave that there is something happening on this corner when will I feel it on the other side if I do the same thing on air which one will I feel quicker same length metal will be quicker why density is higher is not the answer molecules are far more closer but that is not the answer but that is related to that compressibility sitting there quick answer is speed of sound any information that needs to travel from one point to another point goes through collision of molecules think about it I hit with the hammer on this corner layer of the metal rod those set of molecules or atoms sitting there iron atoms let's say it's iron rod and atom sitting there are going to be displaced by this hammer when they move they collide with the next layer and then that will call later the next layer next layer next layer and then how are they moving and how does the last layer know that the first layer had something by collision this is the only process and after the first few layers the molecules do not know that it was because of the hammer that the molecules are moving only the first two layers may be knowing that it is because of the hammer other layers will just know that the first layer moved that is the only thing it knows if there is the only thing it knows is just that there is a wave that is coming from that end into the metal medium right just a wave that is going through this lattice of your medium that metal rod you know lattice is just arrangement of metal inside the metal rod or metal atoms inside the metal rod so that is being displaced and the lattice vibrations are going to send that wave in a particular speed only it won't send it at any speed it feels length for the particular material for that particular temperature there is just one particular speed at which it will go that happens to be speed of sound and that medium if I think about air same concept I have a let us say a tube which has air inside and I hit with a hammer here what will happen that molecules are more random they are going to move and push the next set of molecules they are going to crash into the next layer of molecules now those molecules should move away to give way for this now these will go and crash into the next layer of molecules like that that crashing process one into the other is slowly transferred from one end to the other right if I assume the molecules are sitting idle and in a particular matrix formation like in a lattice then life is simple I can tell first layer goes and collides with the second layer but that is not the case with gases it is a little more flexible they find gap between them so when one molecule comes it need not hit this one it may just move off in which case that wave did not really matter to the second layer there is more gap so it will not effectively go that fast it will go a little slower so speed of sound and gas medium will be less than speed of sound in lattice based medium like solids you know speed of sound of a speed of sound and air at room temperature rough number 340 meter per second 340 for our kind of room temperatures not in AC rooms and stuff and if you think about metal rod say steel of the order of 5000 to 6000 meter per second now we know that if I hit the wave travels faster in ion steel compared to air that is what we just found out right it is moving in speed of sound where did sound come in here we are talking about a wave of molecules moving with that information that hammer hit it suddenly I am talking sound how did I link it with sound what exactly happened there was one layer being displaced let us say this is one layer this is the next layer this is the layer that was hit by the hammer this comes in and now there is a short while when two layers are close by what does that mean density of molecules here is higher that means the gas is compressed locally gas or it does not matter even if it is a steel rod matrix is getting compressed locally it is getting compressed and what happens when this molecule layer moves to the next layer this region gets expanded so there is compression expansion waves going through from one end to the other when I hit with a hammer that is what is happening right I am sending in compression expansion waves which is what we call as sound waves right a loose definition for sound waves of course you go talk to acoustics people they will tell different definition for a sound wave we will keep it as this any wave which is a bunch of waves produced due to collision of molecules that is what we will call as a compression expansion wave related to speed of sound actually it is related to sound and the speed of which is what we will call as speed of son now we are linking things with this Mach number expression what do we want to say we want to compare the velocity of the fluid with the speed of sound in that medium so we are telling there is a fluid element now we do not have any steel or anything else there is just air and we are looking at air column there is one fluid element which wants to move fast now as it is moving it is trying to push the layer in front of it there will be a sound wave going if that sound wave reaches a head and then tells that this fluid element here that this is moving then my speed here is less than the speed of sound right then my Mach number is less than one if I had the opposite case that is fluid is going faster much ahead of the collision waves traveling then what will happen by the time the wave goes and hits it that fluid element the fluid element here will also go and hit it if that is the case this fluid element does not know that this flow was happening so what will happen to all the fluid elements in between they get crushed we will see that that is the flow behind the shock later we won't get that far right now that gets crushed we look at that we will give you more physical feel as time goes but that is the idea now I will get back to this name mark there was eminent scientist Ernst Mach who did pioneering work in explaining movement of sound waves in gases explaining shockwaves blast waves he did most of his work in blast waves most of his major contributions are in blast waves how blast wave propagates in time if there is an explosion of a bomb how far will I feel the pressure wave that kind of work he did a lot but he explained the Mach wave Mach reflection Mach wave all that it all became his contribution there is a lot of it so we just give it is a very apt name I would say I support that so we will follow that Mach number now we will go back to the original definition we got from you people which is if Mach number is far less than one we will call it incompressible and if it is higher than that low limit whatever if it is anywhere comparable to one higher than one far higher than one all that is compressible flow will keep it that way this is another definition for it we are given two definitions we will try and link these two definitions at a later stage not right now we will find that they are all the same eventually we will see that now we will get little more into the course this course is predominantly designed as starting from basics as if you are starting at undergraduate level this is the first compressible course you are ever having after your high school that is how we are starting with but I want to build up to a point where you can do graduate analysis with it full of such level analysis you should be able to do with this I am just trying to give you a lot of physical intuition in understanding this concepts then mathematical alone of course I am going to give you a lot of mathematics I will write a lot of differential equations on the board and all that and give you a lot of mathematical derivations but finally I want you to have this physical feel for things if something is happening you should start visualizing the wave traveling from one end to the other and because of that this is what should happen should be able to tell that that is the way I am thinking I will design this course if you are not getting any physical field for at anytime in the middle just stop me and ask then I will give you even better physical feel for it of course the any scientific analysis should start with laws of physics and it uses a lot of math then only we can understand anything engineering always starts with laws of thermodynamics and then laws of mechanics which is again subset of laws of physics of course we will ignore currently loss of gravity electromagnetism nuclear physics whatever will ignore all those other laws we will stick to only laws of thermodynamics laws of mechanics we will start deriving all kinds of governing differential equations which are essential for understanding this particular flow field system as incompressible flows any type of compressible flows once we have the differential equation remaining thing is all about putting special boundary conditions and you will solve different flow properties which overflows you want you can solve for it that is the basic idea so now I had to start with basic thermodynamics this is just a review so hopefully you'll understand this and we will go through this quickly but I will spend some time on laws of thermodynamics but before that we have to define what a system is what is the system any part of the universe is a better definition any part of the universe which we are currently interested in we are interested in looking at some property of say this small volume of gas then this becomes my system then what is a surrounding everything else around this in the whole universe everything around this in the whole universe is now considered surroundings to this whatever we are interested in is called system I may be interested in two three systems together may be system a system B system C and then everything around as surroundings that is also a possibility what is a closed system there is no closed system is just a system where there is no mass interaction with the surroundings we won't call it just there is only energy interaction as of now we leave it we can come back to it if you want and we will say closed system is a system where there is no mass interaction mass exchange across the boundaries of the system an open system is where mass exchange is allowed an isolated system is no exchange of any interaction possible nothing nothing is interacting completely isolated system so in that specific case we do not need to look at the surroundings if there is no interaction with the surroundings thermodynamics is simpler will consider only this unit nothing else next thing we need to define as a state what is a state state of a system we are trying to define the system we are going to tell it is applying so much force on the walls of my box it is having so much temperature it is having so much energy inside it is having so many molecules inside all that is a description of the system completely if I can define the system completely then all the defining statements I made put together forms your state for our simple systems we are talking about if I want to call a state say I pick an example I can tell this volume of gas here is at a particular pressure particular temperature and it is having a volume that will tell me that will enable me to tell you any property of this volume of the gas if you want me to give any specific property I will be able to give it so I have defined it completely then I have defined the state what all did I use pressure temperature and volume like that how many such properties should I have to define a state completely to at least two okay this needs a little more so I will wait a little bit and come back to this what is the process it's a combination of states not exactly a correct order to use some change some change from one one state it is going to another state that change is what we call as a process now what is a path it is a sequence of states that the system goes through in a process from beginning to end of the process it went from this state to this state to the state to the state and then the whole connection of all the states together becomes your path of the system path for the particular process which the system undergoes now I need it the next definition before I go back and answer the previous statement witches how many variables do I need to define a state for that I need to define intensive and extensive properties what is an intensive property those properties that depend on mass that are independent of mass or size or number of molecules in the mass whatever if it depends on all these quantities then that is an extensive property extensive mass volume or size whichever way you look at it intensive will be it does not depend on those intensive properties it does not depend on all that how will I tell whether a property is intensive or extensive say I am giving you some crazy property which is gibbs free energy per number of molecules does it depend on mass or not or does it is it an intensive or extensive property how will I tell ok say I am going to tell you an example which is enthalpy per unit volume it does not contain mass so what is it ok that is not a easy way of doing things so I will give you a clean definition or a simple experiment which you can do to find out whether it is extensive or intensive let us say we will do a I have a box this is one system let us say system 1 and it has that particular property which we want to explore whether it is extensive or intensive we want to explore so I have some property here I will create exactly identical system one more exactly identical system one more I created I put them next to each other remove the boundary between them now I will call the new new system as these two put together if my property value doubled then it is extensive if it remains the same then it will be intense a quick check now we can do the analysis let us say I pick volume volume doubled right if I have same volume one more one meter cube and one meter cube became 2 meter cube it doubled so volume is an extensive property if I think about pressure same pressure here and here and I remove the wall in between it's still the same pressure inside so that will be intensive property if I think about that specific case we picked enthalpy per unit volume let us say enthalpy per unit volume here is 1 Joule per meter cube same 1 Joule per meter cube here now I want to find enthalpy per volume for the whole unit for whole full box this will be 2 joules per 2 meter cube which will be again 1 Joule per meter cube it did not change so that is an intensive property okay you guys comfortable with this kind of easy check analysis if you do this you will never make a mistake now we will go back and again ask the same question to define a state of a system how many properties do I need one from extensive and one from intensive I hear to intensive is enough if we are interested in intensive properties only if you are interested in intensive properties only for our whole analysis I need only two intensive properties if I want some extensive quantity then I need to give one of them at least as extensive I could have it as two intensive one or one extensive or I could give you a special case which will work for some simple situations one intensive one extensive that is also fine so that is what we typically need to define a system now we will enter into laws of thermodynamics we will go for zeroth law of thermodynamics what is zeroth law of thermodynamics typically you would have heard first law and second law what is zeroth law of thermodynamics thermal equilibrium okay if mechanical engineering heat transfer people taught you weak thermodynamics yes it is defined as equilibrium thermal equilibrium but we are beyond them we are going to say not just heat transfer happens in my fluid it can also have pressure variations can also have chemical reactions if I want we will look at that one by one it depends on how you define things thermal equilibrium does not contain everything thermodynamic equilibrium defines everything we'll get back to that so first we have to give the statement of zeroth law of thermodynamics which will be if I have a system a I have a system V and I have a system C if I look at system a and system B and I say that they are in equilibrium I have not defined the word equilibrium yet if I say that these two are in equilibrium and I also say that these two are in equilibrium a and C are in equilibrium then zeroth law of thermodynamics states that these two are in equilibrium okay this is a law which is needed for comparing systems we are comparing one system with another system and so I saying what can be different between them so in this analysis we came to a point where we need a definition for equilibrium how will I tell that system a and system B are in equilibrium now we have to talk about different properties and tell system a and system B are in equilibrium if something is equal to that if I tell that temperature a is equal to temperature of B then we call this thermal equilibrium what did I introduce here a new variable called temperature this is our temperature is introduced we have we have introduced a concept called equilibrium we have introduced a variable called temperature to tell that these two systems have the same degree of hotness that is your temperature if temperature a and temperature B are same and temperature a and temperature C are same then temperature of B and C are same that is what is the actual statement we are looking for okay what about mechanical equilibrium now we have to think about how can I have system a and system B in mechanical equilibrium with between each other forces should be the same and we are going to make an assumption about our gas here we are going to make an assumption saying it is a simple compressible substance a simple compressible substance is a substance where the only form of work that can be done mechanically on it happens to be compression work compression work and nothing else only compression work which means it is just that P Delta V kind of work is the only form of work possible of course you can still have heat transfer to it that is another way of interaction with energy that is thermal interaction this is mechanical interaction if I define that then I ask for mechanical equilibrium now I can tell these two will be in equilibrium if the pressure of a and pressure of B are equal then they will not apply force on each other okay you can see that it does not matter what the volume is I may have a case where a is a small system B is a big system I put them together they do not interact mechanically here as long as the pressures are equal they won't push against each other or they are pushing equally nothing happens okay pressure into area is your force forces on both sides are equal that is your mechanical equilibrium right so now we have introduced the concept of pressure we introduced equilibrium concept while we are defining equilibrium we define thermal equilibrium where we said temperature is needed we had a defined temperature then we said we have to define pressure for mechanical equilibrium now I will tell you one more thing chemical equilibrium where we will say we need to know the composition of the gases if the composition here and composition here are exactly the same then they will not start doing any crazy exchanges across or when we go to chemical equilibrium of a given system instead of comparing systems if I just look at one system they think about it even otherwise they will say I look at the system now 10 seconds from now 10 hours from now 10 days from now whatever if the composition of the gases inside do not change then the system is in equilibrium even if it changes after 10,000 years the system is no more in equilibrium it may be changing very very very slowly but if it changes it is not considered to be in equilibrium that is the idea of equilibrium in theory the same thing here temperature of a temperature of B are equal and I put them together they will not exchange energy they will just keep the same temperature in both of them if that is the case then they are in equilibrium if they exchange energy then they are not in equilibrium if I have a hot metal and a cold metal I put them together hot metal will give energy to the cold metal so they are not in equilibrium at that moment if I wait long enough they both will become same temperature then it has reached equilibrium after that if I wait for ten thousand years it may not change that is the basic idea so we have defined if the pressures of the two metal blocks have become the same then it has reached thermal equilibrium if the metal blocks are still having different temperatures and the hot metal is giving energy to the cold metal then there is exchange of energy which means it is not in equilibrium yet if I wait long enough it may exchange enough that they will become equal at that point I can tell that the system is in equilibrium not till that point and from that point on if I wait how much ever time no change in temperature but now I will take a practical example I take a mental block 50 degree Celsius I take another metal block in this row another 50 degree Celsius I put it here what will happen the room temperature is 25 degree Celsius both the temperatures will lose equally they will lose energy equally what is happening here exchanging with surroundings when we talked about systems here we did not consider anything else here we just considered system a system B we did not have anything called a surroundings which means we considered isolated systems we did not have a surrounding if we have a surrounding of course I could take that into account by putting a big circle around this and call that region as D if there is exchange between a and D then a and D are not in equilibrium with each other okay may be TA and TB are still equal these two are in equilibrium with each other but they are not in equilibrium with the surroundings that was the example I just gave you 50 degree Celsius two metal blocks they will not exchange heat within themselves but they will give heat to all the air around if you just be very clear about all this that is zeroth law of thermodynamics where we introduced the concept of equilibrium introduced a variable called temperature for thermal equilibrium introduced a variable called pressure for MEC equilibrium introduced a variable called composition which is actually a whole bunch of variables right mole fraction of each of the species present inside say if it is air nitrogen oxygen carbon dioxide water vapor everything mole fraction of everything I have to give if I want to define the state very correctly okay now if I go for first law of thermodynamics what does that say energy remains constant energy can never be created energy cannot be destroyed such statements exist of course we are neglecting nuclear reactions and stuff in this particular case if you are taking into account nuclear reaction then we will have energy to mass conversion should be taken into account we will ignore that for now does not affect us the primary statement will be simply put energy is conserved what did I do with this law of thermodynamics I introduced a variable called energy we did not define anything very well okay we just said there is something called energy how do we define it we haven't defined it we just told energy is a fundamental quantity which defines the state we just said that we give expressions for it later and then we are going to say systems can have exchange between one and the other system a and system B can exchange energy but the overall energy of my complete system does not change overall energy of my universe never changes that is the law we are going to keep it does not change at all okay of course in a simple compressible substance the only kind of energy exchanged with my system happens to be through heat exchange or through compression or expansion work nothing else we won't consider the case like I put a fan inside my system and rotate it that is stirring of my gas which we are ignoring as a simple compressible substance we won't have stirring case if I stir a gas for a long time its temperature will increase but we will neglect that kind of energy transfer now we go for second law of thermodynamics what does that say direction of energy transfer okay and I heard something called entropy there okay so they are linked the actual statement is something like there exists a quantity called entropy we define the word entropy suddenly we introduce the word entropy in second law of thermodynamics okay and then we say if there is a process that is naturally occurring in nature then entropy must be increasing entropy of what must be increasing entropy of the universe must be increasing a quick example is refrigerator okay think about a refrigerator if I cool the system entropy decreases you would have already had a little bit of thermodynamics I am just using it like this loosely statement okay if I make the temperature go very very low molecules cannot move around there is no energy for X so they will just sit idle in one place that is the most restricted condition entropy will be the lowest so if I take my gas to say one Kelvin or Oh point zero one Kelvin then I will be in a situation where my entropy will be very very low but there are physicists doing such experiments what are they doing really they are using a very big refrigerator and they are having this small gas controlled to that point zero one Kelvin while all the other gases around in the whole room may be heated up if you go fill your refrigerator back side it will be very hot it is taking the heat from inside and sending it outside right net energy is still conserved but this process is naturally occurring why I am cooling a small volume and heating a big volume that is the idea and cooling a small volume and heating a big volume okay that is what I am thinking about so overall what happens is I have given heat to a large number of molecules removed it from a small number of molecules they are unhappy but the large number of molecules which I gave heat to they are all happy so it's in my opinion it's low entropy is like how happy are the molecules in your universe they want to be more and more disordered if you give it lot more energy they will move in random directions at whatever speeds so they are more happy in a way right we'll come back to this happy and entropy relation later come back to it again okay but when we think about isolated systems only of course isolated system does not have a surrounding right it is only system there only we will say D s is greater than zero if you want a system to occur spontaneously a process to occur spontaneously that is the idea if for an imaginary process which i think about if d s is less than zero that process will never occur in nature ok is this a law put by man on nature or is it something else it is just law of nature we do not know who put it maybe it is the god but nature obeys this it is called law of nature for that reason it is just law of nature all the thermodynamic laws we defined are obeyed by nature laws of mechanics are also weighed by nature but part of them are used to define variables momentum has to be defined by first law like that you have to define various laws even here zeroth law was used to define equilibrium first law was used to think about energy second law was used to think about entropy entropy so we said that if I have these variables I am sufficiently equipped who introduced this it was introduced somewhere around 1800 by I do not remember the exact set of people but Gibbs was one of them and then Boltzmann came and defined or gave numbers to it and gave a universal scale by which we can measure entropy and all that ok so that is where we are right now we will go to next class and we will more talk more about other thermodynamic variables other thermodynamic variables we still haven't gone to gas yet perfect gas we will have to define a gas then we will define perfect gas then we will go further along that direction and then we will enter thermodynamic processes then we will switch to laws of mechanics we are still in review mode first three classes are devoted to review of what you should know by the time you enter gas dynamics course and see you guys next time you