 So, we need to actually look at this multi-component diffusion equation that means we need to look for something that is a substitute for fixed law that was applicable only to a binary mixture. So, we need to know what to do for a truly multi-component system. So, let us now just write out this equation it is going to go like gradient xi equal to sigma j equals 1 to n xi xj divided by dij vi vector minus vj vector plus yi minus xi divided by gradient p divided by p plus rho over p sigma j equals 1 to n yi yj times fi vector minus fj vector plus sigma j equals 1 to n xi xj divided by rho dij times dt, j minus dt, i divided by yj I am sorry this ought to be like this yj minus dt, i divided by yi times gradient t divided by t which will probably use a okay. So, does that look like fixed law at all could fix law so how what does it mean I do not understand I just wanted to go from two species to three species would life get so difficult yeah I guess when you get married you go going from 1% to 2% in life gets very difficult right. So, this is not terribly bad now that is here we can think a little bit more rationally than there so if I were to basically say can I reduce this to fix law situation that means if I now say my capital N is equal to 2 would I get fixed law back would I or would I not obviously not this is obviously a monster when compared to what we were talking about previously, but if I now said I have only two species would fix law be hidden in it could you see could you see thick peeping out of anywhere here remotely can you recognize him right yes we can first of all what did we get this okay well I just pull it out of my hat and that is it you just try to take it from me like it is like God given the answer is pretty much like we said that fixed law has to be from a molecular level basis instead of a continuum basis we can derive this from a kinetic theory consideration, but this is actually considering a lot more than Fick did empirically deduce for his set of conditions okay. So, we now are looking at about four different things out of which fixed law is contained only in this term first okay because how can I tell that because I started out looking for a relationship between the mass diffusion velocity of species I to its concentration gradient okay and if I were to looking if I were to look for a mass diffusion velocity related to concentration gradient I really wanted to look for a concentration gradient in terms of mass fraction rather than mole fraction, but this is what I have okay. So, any hope of getting fixed law is from here because that is what is first of all containing the mass diffusion velocity of species I unfortunately it is also containing the mass diffusion velocity of species J which is not equal to I okay if it were equal to hi I then you would have a 0 here V I- V I okay. So if you had two species this is actually a convoluted expression for the gradient the gradient of mole fraction of one of the species in terms of concentration sorry mass fraction mass diffusion velocities of the two species that are involved. So, if I can if I can now write this for gradient XA okay and then gradient XB what I am actually looking for is VA an expression for VA and an expression for VB. So now I have two equations I can I can now solve for one of them and then the problem is I will now have gradient XA and gradient XB which I did not have before but then try to convert that from XA to YA and so on okay. So, you start doing that then you can try to get that. So, whatever we were looking for is only in one corner of what the full picture is so what is what is going on so the first term denotes momentum losses due to collision of collision due to collision with other species this is the this is the mechanism that is actually in play in fix law as well okay and the second term so you now start looking at what are these different terms these are pressure related momentum changes pressure related momentum changes the third term we now have to explain what is small f small f is body force per unit mass of species i is fi vector fj vector is body force per unit mass of species j right. So, what is body force per unit mass it is kind of like acceleration okay. So, in a gravitational field this would simply be like g except you would be wondering why would g be different for species i then species j the answer is no gravitational field mostly you are not going to be looking at here this is more like things like electric fields magnetic fields and so on okay. So where dependent upon whether your species that you are looking at is neutral or positively charged negatively charged you are going to have different body forces on different species right. Therefore, this term is primarily coming from a differential body forces okay keep that in mind I am highlighting this because as we are looking at the second term there is a pressure related momentum changes go back and look at that you had a gradient p over p that basically says that this is driven by a pressure gradient that means a pressure gradient causes a concentration gradient is what you are essentially saying but also keep in mind there is a coefficient yi-xi okay we will be looking for situations where these are not important. So, for example here if the pressure gradient is not there then there is no contribution of that to concentration gradient okay or you could be looking for a situation when yi is equal to xi when would that be we talked about it right that is when all the species are having the same molecular weights here of course is a somewhat like a fictitious situation but for ease of analysis we could assume that if you want to consider pressure gradient otherwise okay but you do not want so we should be looking for when these different terms are important and so on. So one of the things here is you could get rid of this term if you do not have any body force to consider on any species in the first place that means Fi and Fj all are 0 or if you have to keep body forces for some of the consideration get rid of here you now take advantage of the fact that this is actually differential body forces therefore if the same body force is acting on this on all species then do not worry about it like gravity you want to take gravity into account okay but gravity is going to act on all of them okay then equally then you do not have to worry about this term. So this is what is going to happen so this is a differential body force term differential body force term and so here for example Fi vector is body force per unit mass of species I same holds for J and you are looking at the difference between the two this is a vector difference keep that in mind okay this is a vector equation on the whole so this actually has three components in three dimensions yeah and so it actually caters to your need we are looking for a Vi vector to be modeled in terms of concentration gradient that it does that this is a vector equation right away and finally the fourth term fourth term is called sorate effect this refers to thermal diffusion okay so sorate effect thermal diffusion that is that is diffusion caused by thermal gradients okay when we are looking at the energy balance we will look at the opposite possibility that is if you now have species of different concentrations that could give rise to a heat flux all right that and that we will recall we will now we will call that as a due for effect and this is a sorate effect these are named after these scientists so here this is again all these things all these things are reduced from molecular level considerations like in kinetic theory and what you would find is that if you had temperature gradients as you can see here if you had a temperature gradient that contributes to a concentration gradient okay so that is the effect that we are looking at okay and we will not derive this as I said we just state this okay now it can be shown it can be shown that that means you show okay shown that for a two component mixture for a two component mixture we can write we can write y1 v1 vector is equal to d12 gradient y1 minus y1 y2 divided by x1 x2 times y1 minus x1 times gradient p over p minus y1 y2 divided by x1 x2 the whole squared rho over p f1 vector minus f2 vector plus dt, 1 divided by rho d12 gradient t over t now dt, 1 or dt, I in general refers to what is called as a thermal diffusion coefficient okay so dt dt, I refers to a thermal diffusion coefficient that just shows up here or here now typically typically you can always see that okay let us just look at the units of these this equation here you are looking at gradient of x okay x does not have any units it is a mass it is a mole fraction okay so gradient will have like 1 over meter kind of thing so you are looking at 1 over length kind of units gradient t over t also has same units okay because the t units get cancelled and then you are only gradient of something non-dimensional therefore and of course you look at this x and y they do not have any units okay so what matters for you is d over rho dti or dtj divided by rho dij is what should be dimensionless okay I am sorry yeah that is right that is right so this is having the same units as this that means everything else should be dimensionless and since x and y is do not have any dimensions dt i over rho dj that is how typically we would compare things so dt i divided by rho dij is small all right typically so now just to get an idea of what is going on if you were to just look at the dij alone it is the highest for gases okay that means gases can diffuse very fast all right but lower for liquids and then still lower for solids you could be thinking about solids also going through a molecular level diffusion all right I can like an alloys and so on but that is a very slow process when compared to what happens in gases so this is fairly high so if you compare these they are like orders of magnitude different as you will see but if you now compare the thermal diffusion coefficient with the row of the mixture times dij for a pair of gases then this is obviously not for a pair of gases this is for one particular gas all right whereas the dij is a binary diffusion coefficient it is for a pair of gases okay the other thing that I would like to point out is when you now have a multi-component diffusion equation and you are looking at the same the same process that was there in fixed law showing up in this term you are still retaining the binary diffusion coefficient and then summing over all other species than what you are talking about okay we did not have to explicitly say that J is unequal to high over here because that would contribute to your 0 okay so you could just sum over but essentially we are looking at the interaction of all other species one at a time with species I okay so it is as if you now have a multi-component mixture and you are looking at what is the concentration gradient contribution due to mixing for a particular species it is coming from this species interacting with all other species individually in mixing with all other species individually as pairs of the species and the and each of those other species individually okay and as I said dij is equal to dij for any pair i and j and so it is like as species i is mixing with any species any of these species j other species j all other species j are equally mixing with species i in turn okay so it is kind of like if you now have a group of people and then you now let a bad guy in this bad guy actually interacts with each of the each of the guys in this group so every person in this group is equally responsible for entertaining this guy and contaminating the group you see so this is this is what is going on you are now still looking at a binary interaction and a collective effect of the binary interaction over here all right this this this is a little bit in direct because we would like to think of a diffusion coefficient that is a function of only one species at a time we do not want to this pairs of species business because when you think about a thermal conductivity or a specific heat or something of the sort we can easily think of those as being properties of only one particular species right but it is it is diffusion coefficient we have to think about a pair of species at a time that makes things complicated we will try to work around it by some kind of assumptions and simplifications a little later but at the moment I have to point out that is how the process is mixing always involves minimum of two species at a time and we always can look at them as a process that involves two species at a time even in a bunch of species together right. So that is the reason why the same binary diffusion coefficient that we saw for a fixed law shows up here and that is what exactly what it is here as well okay. Now we can as I was talking about we were trying to get rid of some of these these terms so that we can now begin to see the fixed law come up right and I also told you can we apply this can I can we apply this to a binary mixture and that is this result you can see that this is a little bit different from that that means some simplification has been made okay it is been made so that we actually have y1 v1 vector which is the diffusion mass flux that is what fixed law was looking for okay to get you something in terms of concentration gradient in terms of mass fraction constant mass fraction gradient right. So this is this is already rewritten in this form that we want now we have to look for what are the conditions under which you can make simplifications and retrieve fixed law out of here so under certain assumptions and that is what we are to see now certain assumptions under certain assumptions the fixed law is recovered okay that is to say you can say y1 v1 vector is equal to minus d12 gradient y1 did we see this well you only have to do is put in a row on either side if you now put in a row y1 becomes row 1 row 1 v1 vector becomes j1 vector and that is equal to minus row d12 gradient y1 that is exactly what we had before okay. So this is this is pretty much the same as what we so for the fixed law so the assumptions are the assumptions are one the mixtures binary of course we are looking at applying this to a binary mixture in the first place the thermal diffusion is negligible that is the sorate effect that is in whatever we do you know in our class we will always be pretty much assuming that the sorate effect is negligible because dti divided by rho dij is much less than 1 we would not worry about it for most of the most of the conditions okay. The body forces body forces are same on both species or negligible you have you have two choices of making assumptions okay so you can assume that the body forces the same even if they are not negligible or if you have to get rid of it you have to make an assumption that it is negligible the question basically is you know for example if you are now looking at buoyancy effects that means you cannot get rid of body force okay buoyancy buoyancy effects in flames and that is very typical and under gravitational conditions look for the reason why you see these candle flames looking elongated upwards is because we are in earth's gravity if you now go to a zero gravity environment you will not see this kind of a shape right. So if you want to actually think about that effect not exactly that effect mean there are some things that go associated with that shape for as far as see heat transfer back and all those things is concerned so from a quantitative point of view gravity does affect not just the shape of the flame but lots of other interesting parameters so you might want to keep gravity but fortunately for us the gravity would affect different species to the same extent so that is fine so it does not affect this equation it shows up in the mixture momentum equation okay so keep it there you do not have to worry about it here constant pressure or constant or uniform pressure constant or uniform pressure or identical molecular weights right either of them will sell if you had a grad P that is 0 that is okay approximately or if yi is approximately equal to xi that means both the molecular weights approximately the same then that would work as well now how good is this assumption or either any of these assumptions the answer is not terribly bad although it is not strictly valid in the most conditions you can think about so many times when you are looking at assumptions you have to ask yourself is it a reasonable assumption to make or is it completely unreasonable just to make our life simpler alright the answer is you will find we will go through this quite carefully under we will note that under typically low Mach number conditions alright and most combustion processes that we commonly encounter right from candle flames wounds and burner onwards kitchen gas stoves onwards to even highly turbulent flames and furnaces gas turbine combustors and power plans and all those things are all fairly low rockets okay are all fairly low Mach number how low any ideas how low is low 0.2 0.3 is low it is not even that high it typically is 0.1 or less many times it is like 0.01 alright now if you really think about what I am talking about okay now at these kinds of temperatures that we are looking at the speed of sound is so high so you are looking at something like close to temperatures close to the adiabatic flame temperature the speed of sound is like of the order of 1000 meters per second right and 0.1 times that is going to be like 100 meters per second that is going to be very fast it is not slow really strictly speaking but the Mach number is low because of high temperatures involved you see and typically we are looking at fairly low subsonic conditions for most of these and what we will show is under those conditions it is okay we will show that the pressure actually becomes constant more or less alright so this is not this is not terrible or if you now look at molecular weights if you think about molecular weights between for example nitrogen and oxygen there is not hardly any difference okay so like looking at between 28 and 32 carbon monoxide is what about right 16 plus 12 is again 28 carbon dioxide is a little bit of a high on the higher side but methane could be like about 14 water is like 18 so they are all in the same ballpark for for most part so it is not a terrible assumption under most conditions you do not you should not too much worry too much about the molecular weights being too disparate they are not for most point conditions so for analysis purposes it may be okay to make this assumption but not not not a lot of times for example you are looking at propulsion you have like the specific impulse or something like that is going as a square root of T divided by molecular weight and it is in the denominator so if you now have a small change in the molecular weight like let us say from if it changes from let us say 20 sorry 30 to 20 or like 28 to 18 it makes a difference it makes a significant difference in your ISP calculations and so on and people die for getting that kind of increase in ISPs from there so you got to be careful on the application and you might be able to get away with assuming equal molecular weights for the purpose of getting rid of this term or for the binary mixture this term so then with these we can we can get the multi component diffusion equation to reduce to gradient xi equal to sigma j equals 1 to n xi x j divided by d ij times vj vector minus vi vector right that is what we can get it to so if you want to now make sure that this is a binary mixture make this i is 1 and so j equals 1 to 2 we can now make this x1 xj d1 and so on and v1 what we are interested if you now go back and look at look at this this and this are there okay do not have to worry too much about d1 j because j goes from 1 to 2 and j is unequal to 1 so this is just going to be only one term right so we do not have to worry about a summation you can you can j is going to take only a value of 2 and therefore you can get rid of this and then start putting your j as 2 for you you can now have a j2 d12 and v v2 the problem that I am talking about is you now have v2 x2 and x1 as distractions from this expression see they did not and then of course you have x1 here you wanted a gradient y1 but this is an expression for gradient x1 okay so you know how to convert y's to x's okay there is a relationship between x and y for each species which involves all other species that unfortunately okay but this is a binary mixture therefore it should be possible to for you to figure that out and also you have to eliminate v2 that means you will have to consider the next equation and so on keep in mind in a binary mixture x1 x2 is equal to 1 and y1 y2 is equal to 1 you should use those additional equations and then you should now simplify this to something like this that is an exercise as well for you to do right okay I think we will stop here for the day we started too late.