 We have been looking at fixed law with the hope that we should be able to relate the diffusion mass flux to the concentration gradient so that we would now reckon the concentration gradient let us say in terms of a mass fraction as one of the primary unknowns then we should stop thinking about what would be its corresponding equation okay so when you now want to solve the combustion problem that means you have to actually find out how the composition of your mixture of reacting species changes so your reactant whether the species concentration is an unknown and therefore correspondingly you need to have an equation that determines it and the equation that we will be looking for is the equation of conservation of mass of species a this is this is a general mass conservation equation for a particular species as an extension of what we have already seen before in the in the case of the thermal thermochemical processes that we have looked at like constant volume constant pressure fixed mass reactors in the well stirred reactor and so on so there we had a very limited approach to conservation of species or species mass but here we will try to actually have the most general approach that is possible that would include convective effects as well as in the most general way as well as diffusion effects right so we want to now bring those into the account in addition to unsteady effects that there are possible and chemical reactions right so the way we want to do this is to now consider a arbitrary volume so we can now look at this is like a volume so shedding basically is to give a 3d effect there the surface as the surface is denoted by sigma and the volume itself is denoted by script V and so we consider a consider a an arbitrary control volume I would like to emphasize that the arbitrary arbitrariness of the control volume is pretty important for us okay and that is what actually tries to make it quite general for us and we will probably try to get the same equations in the end but we will invoke the arbitrariness of the control volume to do this pretty soon so then the question is how do you how do you conserve mass the answer is just like you consider anything else okay so how do you conserve anything the answer is so you now look at a situation like let us say a control volume and then you have to now keep in account the rate of rate of change of whatever you want inside the control volume as equal to whatever is coming from outside versus what is going out what is coming in okay so this is pretty general as I have said earlier you know this you know to do this from looking at your bank account on how much money you have like versus how much money is getting depleted versus what is coming in and what is going out and so on so this is a very intuitive idea nothing great greatly mathematical about it so what we will first do is write the conservation equation like a sentence okay like a verbal statement so that it just appeals to us appeals to a common sense without getting into mathematical notation so here rate of species a generated slash consumed by the by chemical reaction in volume V in volume V is equal to rate of species accumulated or depleted in volume V plus rate of species entering or leaving right entering or leaving the surface sigma okay so there are three things that are happening primarily one you have species getting generated or consumed within this control volume okay and you have species coming in and going out of the control surface so if you now look at how much of your species a so we are looking at a particular species a okay so whenever we say C species we should be looking at species a here right the amount of species a mass in this case okay when we say amount we are looking specifically at mass of species a that is accumulating or depleting the rate at which it is accumulating or depleting how much of it is growing or decreasing it depends on how much of it is generated or consumed within the control volume okay plus how much if it is entering or leaving in the control surface so in other words we will now see that there are these three things right generated or consumed accumulated or depleted entering or leaving and you have all these things with a slash in between to say this or that and you can always look at whatever is on the left hand side of the slash they should kind of go together that is if something is generated okay or entering it will get accumulated if something is getting consumed there is a disappears or it leaves it will get depleted okay they all go together so this is fairly straightforward for us okay the second thing that I would like to point out in this verbal equation is two of them are happening in the volume the other thing is happening across the surface okay so keep this in mind we will have to apply the Gauss's divergence theorem to convert the surface term to volume term that is that is what that is why I am basically looking at this okay so this can be now written as then a integral over the volume dou rho a over dou t dv plus integral over the surface m dot s vector dot n hat d sigma equal to integral over the volume wa dv all right here this is the first term on the right hand side of the verbal statement which is to say the rate of accumulation or depletion so how did you get this rho dv is the mass in a elemental volume that is inside this control volume so you now have a dv that you want to integrate over the entire volume ultimately so rho times rho a times dv so rho is the density the rho a is the density of species a times the elemental volume is the elemental mass there and dou by dou t of that is the rate of flux rate of mass accumulation or depletion right we all want to worry about the dv being changing in time because that is like fixed in time so the only thing that is accumulating that is changing is the density itself right there is a sentence like the mass concentration as I said so the concentration of the species is changing in time and so this is the rate of accumulation or depletion term that is a volume term we can see that that is a volume integral the rate of mass entering or leaving the control surface is a second term on the right hand side of the verbal statement which we now put on the left hand side of the second term here so this m dot m dot a m dot a is the mass flux okay so we have looked at this as a mass flux so essentially what happens is if you now were to look at a little window on the surface as a d sigma this has a m dot m dot a vector mass flux that is cutting across at an angle to let us say that is your n hat if this is your unit normal locally all right to this to this elemental area d sigma which we want to now integrate over to get the entire sigma this is the mass flux that we are looking at that is going at an angle so if you want to now look at the projected mass flux it is coming out that should be the dot product there right and this this now takes care that the dot product will actually take care when you now integrate over the entire surface whether it is going to leave or enter it will take care of that okay so the dot product could be negative or positive depending upon whether it is going to be entering or leaving okay so here WA as we know is the is a reaction this is the mass reaction rate this is not the molar reaction rate okay we are looking at the amount of mass that is produced per unit volume per unit time so this is actually going to be the units units is going to be kgs per meter cube second right so this is not this is not to be given by the law of mass law of mass action and Arrhenius law you need to you need to use that expression and multiply by the molecular weight of the species a in order to get this because this is mass all right keep that in mind all right now so we now notice that two of them are volume volume terms this is a surface term therefore we now use the Gauss's divergence theorem to convert your surface term to a volume term as well and so we now have a volume term for the first as before plus a volume term for the second as well now which is the origins of m dot a vector dv equal to integral sign over volume WA dv so what you can do is group all of them rho a dou rho a over dou t plus divergence m dot a I am sorry minus WA dv equal to 0 here is where we now invoke the arbitrariness of the shape of the control volume that we have considered okay now if this integral has to be 0 over this integral for the entire volume for any arbitrary volume in general that is possible only if the integrand is 0 right so we now get from here for any arbitrary control volume the above can be satisfied the above integral we should specifically say the above integral equation can be satisfied only if the integrand equal to 0 so that implies dou rho a over dou t plus divergence m dot a vector is equal to WA we are still not quite looking at the 3 fundamental processes of combustion showing up here although we see three terms okay this is a unsteady term but this actually embeds convection and diffusion together because we now notice that this is actually corresponding to a rho a VA vector where VA vector is the velocity of species a in a laboratory fixed coordinate system for a stationary frame of reference okay and then we will now have to split it into the mixture average the mass average velocity of the mixture and its diffusion velocity when you do that then we will get into a convective term and a diffusion term separately in addition to the reaction term that is showing up already okay then we will begin to see the three basic processes convection reaction diffusion reaction okay so that is where we are heading you can now do this for species B okay so let us let us say we now say similarly species B do we have to do this again shall we right so how would I do this all I am just going to do is wherever I see a I am just going to replace that by B are that that is that simple so I can write do rho B over do t plus divergence m dot B vector is equal to WB now let us just go back and call this equation 1 in this class and let us call this equation 2 okay let us now try to use fixed law okay using fixed law using fixed law for a binary mixture of species A and B okay as a reason why we wrote another equation for B it is not like we want to have fun with all the letters in the English alphabet and so on right so if you now have a fixed law then how do you how does this work out we can we can now say do rho A over do t plus divergence rho A V vector equal to divergence rho DAB gradient YA plus WA and do rho B over do t plus divergence rho B B equal to divergence rho DAB gradient YB plus WA W I am sorry WB all right is there a problem I guess not we had a m dot m dot B let us pick m dot B in this panel and m dot B is a row row B VB vector okay so VB VB vector could be now written as V vector plus capital VB vector okay so you now write split that and then you now get your row B V vector this is the mass averaged velocity of the mixture of A and B then you have the other part you have the row B capital VB vector and that is actually JB vector for JB we had a negative row DAB gradient YB all right and you now try to take it on the right hand side you now have a row DAB gradient YB we already had a divergence so we now have the same thing outside all right so this is how we have actually got these two equations now what happens when you try to add these two so here okay before we do that so we can now begin to see that this is the n study term okay that is because the process could be n steady we now have a rate of depletion or consumption of a so the depletion or generation of or accumulation of a in an n steady process so this is an n steady term okay so I can now say this is n steady term right this is a convection convection term right heat transfer people might probably use convection for natural convection okay but aerospace people are used to force convection most of the time so we and then they might call this advection okay so what we are like we are convective people so we we connect so all the time therefore this is we just call this convection and this is the diffusion term right if you want to now generalize most conservation equations well you would call you would have an n steady term all right let us say for example looking at a momentum conservation you may have an unsteady momentum term you have a n steady momentum convection sorry you may have a momentum convection term this would be diffusion that would be a viscous effect okay and then this is this is the reaction term in our case but in general this this this diffusion term could be referred to as a transport term okay or the reaction term could be referred to as a source term in the case of momentum for example a body force could be like a source but in this case the chemical reaction provide the source so these are these are typically what is going on so primarily what is happening is we have a convection diffusion or reaction that is coming up that is that constitutes your combustion process this is something that we have seen in the past all right now let us try to do some interesting thing here let us try to add up these two equations right at the above what would you get you have a partial derivative of rho a with respect to time plus partial derivative of rho b with respect to time that is dou by dou t of rho a plus rho b okay in our looking at a mixture which has only two species right so rho a plus rho b is nothing but sigma i equals 1 to 2 of rho i what is that that is just a mixed density rho so rho a plus rho b is equal to rho so effectively you now get a dou by dou t of rho dou rho by dou t right so you now get the rho by dou t plus you can do all the summation all that stuff through divergence and so on no problem so divergence of rho a times v vector plus divergence of rho b times v vector is nothing but divergence of rho a plus rho b times v vector so rho a plus rho b is again rho so this is divergence of rho times v vector right so this is divergence of rho v vector equal to let us look at what happens here right divergence of rho d a b well if you are if you are very picky we should probably put ba db a here right we just wanted to like a like a computer we wanted to wherever you got a here we put a b and vice versa to to construct this equation right so we should have put db a let us do that but then we notice that d a b is equal to db a we saw that the other day right and so these two are the same row is the same in both and then you have a gradient so all that this is going to happen is divergence of rho d a b times gradient of y a plus y b what is y a plus y b that is one okay you now have a mixture and y a and y b are mixed of fractions okay and this is making the mixture has only a and b so the two fractions together should become unity right and what is gradient of 1 0 okay so you now get a 0 what does that mean is it because gradient of 1 is equal to 0 that we get this addition right it actually tells us the same thing as what we saw the other day for d a b equal to db a if species a is mixing into species b then species b is mixing into species a they are just having fun with each other nothing else is happening outside of this so you do not have a extra so the notice that this is act this and this are actually coming out from surface terms right so it is a it is if like these two together are all happening inside there is no extra mass of the mixture as a whole that is entering or leaving because of the diffusion process if you now put things together right so that is exactly why the diffusion fluxes the diffusion mass fluxes of all the species together will equal to 0 it all looks like they are all interacting with each other but the sum effect that the net effect of this is nothing there is really no net mass exchange by diffusion of all the species put together in a mixture okay this is a very important idea that comes out of what we are doing here plus what do you get for w a plus w b can we just write that there if it were a n component mixture right where we were we had subscripts like I for the ith species equation we would have written like sigma i equals 1 to n wi would be should be can be we did it in the past we had these sigma i equals 1 to n the question is did we have a wi or did we have an omega i and does that matter Greek or English did that matter did not matter if you were to be looking at omega i that is a number of moles of species I produced per unit volume per unit time okay and you try to sum over all the species you might find that there is like a net mole production in this reaction that we are looking at okay but here you are looking at w which is the mass that is produced per unit volume per unit time right so if you now try to add up the mass that is produced or consumed okay keep in mind things have to get consumed if things have to be produced right if mass of species I produced per unit time produced or consume per unit time per unit volume plus mass of species be produced or consume per unit time per unit volume right what would that be that should be equal to 0 because if you cannot have a depleted or consumed you cannot have be produced or vice versa as first mass goes because mass is conserved in a chemical reaction because chemical reactions are only based on electron exchanges and nothing to do with nuclei right so you get a 0 again okay so does not look good to have an equation with two zeros right next to each other sufficient to write one and what do we have do we recognize this yeah so this is basically your continuity equation of the mixture as if there was no diffusion going on no reactions going on so when your aerodynamics professor writes this equation for flow past an air foil okay air flow past an air foil right you could strictly speaking argue with him or her that there could be chemical reactions and diffusion of species going on and you would not know would you right that is what that is as that is as first the mixture is concerned okay the mixture does not really feel these things as first the mass of the mixture mass mass conservation of the mixture you will find that the there are effects in the energy of the mixture and so on okay but as far as the mass the mixture is concerned it did not matter that you had a multi-component reacting flow is not that kind of interesting right so you get back your mass conservation of a mixture as if it is a non-reacting single component situation okay now for fun let us also have one more thing that we can do we could we could could write we could write conservation species conservation equation we will simply call the species conservation equation that there is a stuff that we have done here we will simply call this a species conservation equation okay so we could write species conservation equation on a molar basis okay we will pretty much get the same thing except that we will just change our notation we will not use row we will use C we will not use M we will use N we will not use W we will use omega right so simple right so we could write ? CA over ? T plus divergence NA dot vector is equal to omega A all right can you do what I said before what will happen if you now write the same thing for a for species B and you look at only a binary mixture okay what do you think will happen you will get CA plus CB is equal to C right and you can you can also get a similar similar expression over here we would we would not recognize this because we never really do a mass conservation kind of thing for mold conservation for a mixture or a non-reacting single component species flow right but then what is going to happen is this two terms the corresponding two terms for this we will have a XA and a C we will have a XB and a C right you now add these two you will still have zero because you are not going to get like gradient of XA plus XB which is one again right just like yA and yB right so this is going to be zero but ? A plus omega B is not going to be zero right that is the only difference okay now you can do a couple of you can do one more thing we will have a I am sorry we will have a V star we will have a CV star okay we keep that in mind so using Frick's law we get ? CA over duty plus divergence of CA V star which is the molar average mixture velocity equal to divergence of C DAB gradient XA plus ? A right now for a for a quiescent quiescent non-reacting binary mixture with meant by quiescent effectively we say V star is equal to zero right and what is non-reacting what is meant by non-reacting ? A equal to zero each of those ? A ? B would be zero right so we get the CA over duty is equal to DAB gradient CA sorry del square that is the plush in CA right this is what is called as Frick's second law I remember studying this in high school in a verbal as a verbal statement along with the Frick's first law of course but later on I found that students in subsequent years have not heard of Frick at all okay now what is the second law say it say something like if the first law were to mean that the diffusion mass flux is directly proportional to the concentration gradient okay the second law then says that the rate of change of the concentration is directly proportional to the second derivative of concentration okay so that the mass flux is proportional to the first derivative the rate of change of concentration is proportional to the second derivative this is how the fix laws were formulated but we can find that the first law that we started using cannot be derived from continuum point of view it is stated as a law okay we said we needed that law because we wanted a connection between the mass flux that the diffusion mass flux that is what else the species is doing other than going with the rest of the mixture we needed a expression for that in terms of a primary variable namely the concentration so that was a constitutive relationship that the fix first law provided to us and it can be obtained only if you get down to molecular level you need to get into at least kinetic theory or quantum statistical mechanics those kinds of approaches in order to be able to derive it from fundamental first principles right so you now go through like a course like a physical gas dynamics in order to explain how you can get fix law okay and when and the two not necessarily very satisfactorily alright so that is a that is a fundamental law whereas the second law is something that you can actually find out from molar conservation for a specific for a special situation of non-reacting quiescent binary mixture right so that this is not as special as the first one good now where we started talking about mixing there is diffusion we have been stuck with binary mixture binary mixture is boring can we need something more right so let us look at what to do for a multi-component system now a truly multi-component system in a truly multi-component system when you say truly that means it is not even binary that is what it means binary is multi-component okay but by is not multi enough okay so we want to have like at least three so you always look for the simplest situation three species becomes complicated okay for us so a truly multi-component system that is more than two species right the we can write the species conservation equation okay on a mass basis course that is to say an equivalent of this particularly something something intermediate between this and this that means we will now be we will now say let us let us open up m.a as row a row a times v plus row a capital VA we will do that but then we will start blinking because we do not know how to write row a capital VA which is JA in terms of the concentration gradient because Fick has not said this for us for more than a binary mixture right but we can go up to that point so we will write this equation by just opening up into a mixture average velocity and the and the diffusion velocity we will write up to that point and then we will start blinking okay so not yet at the moment okay so we can go ahead so we then say and unless otherwise stated it is always going to be mass basis from now on we will never really go back to molar basis okay so we will now write this as row yi for this is writing for the ith species right so row a can be written as row y a all right so similarly I am going to write row yi over here for the y ith species plus divergence again I am going to write row a or yeah row a as row y a or in this case for the ith species row yi times I am going to write this as the mass average to mixture velocity V plus capital VI right is equal to wi or can now try to do a few things pull out the row use chain rule so can write yi do row over do t plus yi divergence row V with a mixture average velocity plus row row V dot gradient yi plus divergence row yi times capital VI is equal to wi okay so the way it is written up is to group these two together right and then notice that this is just why I times this so therefore this can go out and therefore you have a do yi by do t plus V dot del yi plus one over row divergence row yi capital VI equal to one over row wi this is built this is good enough to begin with because we can get some insights into what is going on you can see that this together you say doh by do t plus V dot del of yi right what is doh by do t plus V dot del that is the material derivative capital D over dt capital D over capital DT of yi okay so in a Lagrangian frame of reference you would simply look at a rate of time rate of change of species mass fraction yi as far as this is concerned all right so this is actually together then call the inertial term okay this this together is called the inertial term inertial term of course you can in an Eulerian frame of reference we notice that there is a basic change in the concentration because of with respect to time as well as there is an apparent change because of its motion all right so this is this is this is the distinction between an Eulerian and Lagrangian frame of reference if you were to go with the particle you will not know that you are moving therefore you will see everything as only a rate of change of time that is that is what indicates your material derivative right but then you now step back and say wait a minute all the changes that I went through is because I was truly changing plus I was also going go experiencing the world around me as I was moving this is happening in our lives right so there is a lot of philosophy in fluid mechanics you see you see start thinking about this is very intuitive okay so if you now look at an Eulerian frame of reference you can now identify this as the unsteady term right before and this is your truly a convective term now the moment you see your capital VI that is your diffusion mass diffusion velocity okay that is the relative velocity of species I with respect to the mixture right so this is your diffusion term and as before this is your reaction term right so you always have to learn to read equations term by term and try to assign meanings those terms physically then a mathematical equations begin to look like sentences in English okay or your favorite language right and these are basically words okay it so happens that these words are composed of spellings that are very jumbled looking from a language point of view but that is all it is the equation tries to tell you something and it is a string of words that makes sense and you have to start looking at each of those terms like words that make sense right and of course you know you have to keep in mind these pluses and minuses they are like the verbs and all the things that there are thrown in between these meaningful words to convey the meaning for the sentence as a whole right so this is what we are essentially looking at for the species conservation equation we still have a problem we do not know what is capital VI vector alright but you have to keep in mind now that you are looking at looking at a truly multi-component system this is one equation that represents actually n equations and n could be pretty large okay could be 5 10 40 100 not more than that mostly okay so this is a equation that is actually consisting of large number of equations keep that in mind okay and then the next problem that we have is we have to start looking at a much more generalized version of fixed law for a truly multi-component system that tries to relate your V to why I VI to why I okay and you might be worried VI is a vector capital VI is a vector so it has three components okay so we are actually having three n equations that we should be looking for but fortunately if you are now going to be looking for VI in terms of gradient why I all the three components are buried into just one unknown why I because it is a matter of taking gradients in different directions for you to get your capital VI right we hope that we will now be able to get a fixed fix law to work for a truly multi-component system but unfortunately it is not going to be as simple okay it is going to take some more time for us to get there we will start doing what is called as the multi-component diffusion equation tomorrow.