 So, let us now look at something called a multi-component diffusion equation diffusion coefficient right, what we did originally was to get fixed law for applied to a binary mixture where we notice that you have something called a binary diffusion coefficient that is a diffusion coefficient that is valid for a pair of species and then we now switched into a multi-component mixture where we wanted to look at how the diffusion happens in a situation where we have lots of species and now we came up with a monstrous equation the other day which under some conditions might begin to look like fixed law let us say for example again for a binary mixture but what we noticed was even in the presence of multiple species the role of the binary diffusion coefficient was still there that is to say we were still looking at mixing happening between a pair of species at a time right. So if you now had like three species ABC then we were looking at a DAB, DBC and the DCA okay so we are now three diffusion binary diffusion coefficients for each pair of species that is involved there but many times when you are looking at let us say 10s or 100s of species then we now have to worry about binary diffusion coefficients of each of the species which with each other species right. So that becomes little bit more complicated and it is not quite amenable to further simplification for example when we now did the multi-component diffusion equation we could reduce it to fixed law for the case of a binary mixture okay by making lots of simplifications in trying to justify disregarding the three other terms and to look for only the first term that contributes to this and then we could see okay fixed laws is retrieved but the question is if you had a truly multi-component mixture could be even think about a something like a fixed law something as simple as a fixed law right and for this the binary diffusion coefficient is going to be a bit painful to deal with because we are now saying that you have to have a DAB times gradient YA DA DBC times gradient YB and so on and these binary diffusion coefficients are not going to be the same right. So therefore many times it is worthwhile to think about what is called as a multi-component diffusion coefficient this is not the reality okay the binary diffusion coefficients are the reality whereas we will now think about a multi-component diffusion coefficient artificially used artificially used to relate to relate diffusion velocities as a function of linear combination of concentration gradients. So the way we want to do this is so commonly approximate as let us say yi vi vector is equal to minus di gradient yi i equals 1 to n course or we should also be able to say xi vi vector is equal to di gradient xi right we have seen this before for a binary mixture like for example if I now multiplied both sides by row the mixed density then I get rho i vi vector is equal to minus rho di gradient yi previously we have done this only for a binary mixture so if you were to apply this to a species a then we would have a DAB so that has two subscripts to denote that this D is valid for a pair of species a and b but now we are looking at a multi-component mixture that means we have more than two species but we are having only one subscript that is to say we now want to tag the diffusion coefficient only to one species much like how it is CP or maybe thermal conductivity or any of those quantities would be that does not make sense honestly because as we said if some species has to mix into another species you need to have other species involved in this process right so DAB made sense because it was actually for a pair of species that way but di how can you have a diffusion coefficient that depends only on one species the answer in this situation is as I said it is a bit artificial but it is not too artificial okay because what we could now think about is how is this species how is one particular species in a mixture mixing relative to let us say a species that abundantly present in the mixture let us now begin to think about how each of the species in the mixture is going to mix with a one species that is the most abundant in the mixture right so typically in combustion applications we are now looking at like fuel air mixtures right in air you have a lot of nitrogen that is pretty much sitting there and that is like an abundant species around so if it is now possible for us to think about how each of the species like the fuel or the intermediates the products the oxidizer air the oxygen that is and so on mix with nitrogen okay and that is just present like a reservoir and it did not matter to nitrogen or the abundant species how well it mixed with the individual species it is not going to get affected because it is available in abundance right when that is the case then it is possible for us to still think about the diffusion happening were for a power of species as opposed to a tagging it to only one particular species but we achieve the goal of thinking about only one species you simply do not worry about the abundance species right so so the way to be typically do this is this is this is valid this is valid when one species is in large concentration large concentration and others are in trace now typically trace is a word that is used for something that is very very small okay so it is sort of like an exaggeration that we are trying to do here in saying that when we are now having most species that are present in nominal quantities not negligible not in abundance we cannot really say trace but you get the idea that this is strictly speaking valid when one of them is in large concentrations other in trace but we will not go ahead and say well we will identify an abandoned species the other ones are not going to be in trace but we will still go with this idea right okay. So then what happens generally if we use then we then use di is now considered identical to what is called as din okay that is the that is as if you are now looking at a binary diffusion coefficient of each of the species relative to the abandoned species okay and this you can now say is generally written as 1-xi divided by sigma j equals 1 to n and j unequal to i xj divided by dij so that means we are actually using the dij information for all the species okay in the in the mixture in evaluating the what we call now as the multi-component diffusion coefficient okay strictly speaking this the derivation can be possible only if you make an assumption that all the species are having equal velocities whereas we would ignore that if we have to apply this situation where there is not the case and that is in general not the case alright. So that there is an assumption that goes with it which we normally while it in addition there is also one more caveat that we have to use which is use this for n-1 species okay and obtain the dn okay as follows the point we could the point the reason why we cannot use this for all the n species is if you were to use something like this and therefore get something like this and then plug it in your species conservation equations and add strictly speaking we cannot set the summation of all these diffusion fluxes due diffusion mass fluxes to 0 okay so it is sort of like it sort of violates mass conservation. Therefore strictly speaking if you are going to use this kind of a multi-component diffusion coefficient and correspondingly your relationship between the diffusion velocity for the species and its concentration gradient can be expressed as simply like a fixed law okay that is what we are trying to do we are trying to get this form for relating your vi's to gradient yi's or vi stars I should say to gradient xi's right. So if you have to do this then do not do not do this for all the n species okay the n species has to be obtained from adopting sigma yi vi vector is equal to 0 i equals 1 to n and sigma i equals 1 to n yi equal to 1 okay. Use these two things your dn as sigma j unequal to n dj gradient yj divided by sigma j unequal to n gradient yj but many times again just as we ignore the assumption that is involved here in saying that the velocities of all the species is or equal we also many times ignore this that we just say let us not worry about the nth multi-component diffusion coefficient being obtained this way provided all the n-1 diffusion coefficients are obtained this way okay you see that these are actually the multi-component diffusion coefficients they are obtained so this is ignored many times okay let us now move on what we have done so far in trying to look at conservation equations in general for a reacting flow system of truly multi-component mixture is to derive the species conservation equation okay. We did this for a binary mixture we now expanded it for a multi-component mixture then we notice that we need to have a closure for capital VI vector for which we wrote the multi-component diffusion equation and then looked at what are all the situations by which you could simplify this and we have got to this point okay and when you also noticed that as you now add up the species conservation equation all the species conservation equations you will now get your overall continuity equation for the mixture as if it was a non-reacting single species case right we have not shown this in these lectures for the multi-component species conservation equation but that I would like to leave it as an exercise for you to go back look at the multi-component species conservation equation equations and then add them up to show that even for a truly multi-component system you should get the overall continuity equation to be like a single species non-reacting situation alright so that means we have in our kitty the species conservation equation and the overall continuity equation these are two things that we have okay and of course we also have the multi-component diffusion equation so far we could now count the number of unknowns in our minds density for example density of the mixture is one of the unknowns the corresponding equation that you can think about is like the overall continuity equation right and then you have a mixture velocity the mass average to mixture velocity that is showing up in the overall continuity equation that is an unknown for which typically we resolved the momentum equation for the mixture right which is what we would now go ahead and do and then we have the mass fraction is an unknown yi for i equals 1 to n for which we now have species conservation equations 1 to n and species conservation equations which in turn shows up capital VI for which we have the multi-component diffusion equation okay or its simplifications or many derivatives which ultimately leads something like a fixed law but if you do not want to do the simplifications you want to deal with the multi-component equation in all its complexity then keep in mind that you have a xi showing up there we need an equation for grad xi rather than for yi which is your unknown in your in your list therefore you need to now use n equations that relate xi and yi okay so as and when you now come across unknowns you have to start thinking about corresponding equations to solve okay so in this list so far of course wi contains temperature for which we will have to resort to the energy balance okay and then we will find that the momentum equation involves pressure for which we also have to look for the equation of state so that will completely close the system of equations so in what we have looked for so far we have not listed the mixture momentum mixture energy and mixture equation of state these these three are yet to be done so let us now do that so momentum equation I am just going to write it out I am not going to do anything to derive this and I am going to adopt the tensor index notation to do this in tensor index notation rho times vj dou vj by dou x sorry dou vi by dou xj plus dou vi over dou t is equal to minus dou p over dou xi and xj delta ij plus dou by dou xj of mu dou vi over dou xj plus dou vj divided by dou xi minus 2 by 3 mu dou vk over dou xk delta ij close the brackets there plus vi where vi is equal to rho times sigma k equals 1 to n yk fki I am going to say a lot of things now just say now I am going to write it out okay first in terms of index notation if you now have one subscript that means we are looking at a vector okay so for example we now say bi that is a vector it is i goes from 1 to 3 1 1 2 3 it will take index values 1 2 3 for 3 components in a three dimensional space okay we can probably go a little bit backwards bi is actually rho times sigma k equals 1 to n yk fki all right and then of course rho does not depend on the summation it could be taken in and if you now were to think about rho yk as rho k rho k fki fki obviously is a vector but k means that depends on species right so whenever we seen a f corresponding a small f corresponding to a species before we did not do we did this not too far back we probably did it like last class or yeah that is true this is nothing but the body force per unit mass okay so in this case it is going to be like the ith component of the kth species body force okay so ith component of the body force per unit mass of the species k here so it is sort of like acceleration we went through this sometime back times its density it is going to mean that this is actually done per unit volume okay this is a volumetric equation this equation is now valid at a particular point on a per unit volume basis that is the reason why we are multiplying by density because this is actually per unit mass times density gives you per unit volume all right so fki is basically body force per unit wall unit mass and therefore we now want to do this to plug it into this equation so bi then is like the net body force on the mixture all right this keeps in mind that the body force could be different for different species okay and this is because you are looking at like an electric field or magnetic field that could be acting on different species differently and so on okay then I want to also point out that we have a summation over repeated indices in place okay so if you now have like a vj and vj and xj then we have to sum over the repeated index the remaining index will be I which is the prevalent index in the entire equation so this is for the ith component of the vector equation as in this case for example you now see just that the summation is in place over here as well where ?ij is the Kronecker delta and Kronecker delta stands for a value of 1 when i is equal to j and 0 when i is not equal to j therefore this basically will give you a gradient p so what we use to write in vector equation vector notation is grad p needs to go through a complicated route over here and this is first of all assuming that we are looking at a Newtonian fluid for the mixture all right and or rather we should say Stokesian fluid okay because it also has in place the Stokes hypothesis about the relationship between the linear viscosity and the bulk viscosity and here ?vk over ?xk again has a repeated index summation over it so that actually indicates a divergence of velocity all right so the divergence indicates a dilatation and the – 2 by 3 mu is a coefficient that comes out because of the bulk viscosity so that is acting on the dilatation of the fluid and of course for an incompressible fluid ?vk over ?xk will be equal to 0 the divergence is 0 that is actually coming from the overall continuity equation if you sum up your overall continuity equation and recognize that the mixture is incompressible then the density does not change with time and your and the divergence of velocity will be equal to 0 so you should be able to get this right so in all these things we now are having to reckon with two more new things that have that we have not seen before the first one is pressure okay and the second one is mu everything else is supposed to be counted as either an unknown or given unknowns that we have already counted the row in the overall continuity equation and vi or vj whatever it is in the overall continuity equation again yk is an unknown that is counted in these species conservation equation fk I is supposed to be given to us okay on how individual species is being acted upon by body forces per unit mass of them therefore the thing that we do not know here is P that is an that is reckoned as an unknown okay whereas mu is a parameter in the problem it is reckoned as a fluid property the question is we now have a mixture of fluids okay what mu do we take all right so keep that in mind we have a little problem here okay keep that in mind and we will proceed in fact we are now continuing to our search for more equations because the species conservation equation had a wi on the right hand side which had a temperature in the Rhenius term okay so we want to now close for temperature and for this we want to now look for the energy equation so of course we now also have a pressure and we will now have to close for that with the equation of state we will do that next but the next thing that you are going to do right now is the energy equation the energy equation just like the momentum equation is also written for the mixture that means we are not worried about energy of this of a particular species and so on the only thing that we do that is species specific is the mass conservation for a particular species okay but rest of the things we do for the mixture as a whole so if the energy equation is also applied to the mixture now there are several ways in which you could write the energy equation so one of the maybe the mother of all equations would be to write out an equation for the stagnation enthalpy because the stagnation enthalpy will take into account the kinetic energy involved in the motion of the of the mixture and it will also involve the pressure work because we would now say H is equal to E plus P by rho okay where E is the internal energy and P by rho is the pressure and density you now look at a change in enthalpy it also takes into account not only the internal energy change but also the pressure work term and that is for the H the enthalpy alone okay if you are now looking at H not which is H plus V squared over 2 then it also takes into account kinetic energy changes so the mother of all equations probably should be to write in terms of the stagnation enthalpy now what I would like to point out here is that is overdoing things a little bit the reason is if you were to write your energy equation in terms of the stagnation enthalpy keep it with you and then you take a dot product of your momentum equation with velocity vector which is of course we are now looking at the mixture velocity that is the mass averaged mixed velocity of the mixture you take a dot product of this is a vector equation okay so you now take a dot product of this equation with velocity you now get this in dimensions of energy okay and that would be what is called as the mechanical energy equation that means this equation then will only deal with the mechanical energy there is a reason why in many Phc qualifying exams people may ask you something that you learnt in your undergraduate fluid mechanics is the Bernoulli's equation a statement of energy conservation or is it a statement of momentum conservation the answer is both right you could conserve momentum or you could conserve mechanical energy it is a matter of just taking a dot product of the momentum equation to get a mechanical energy right so in other words if you are conserving momentum it is amounting to conserving mechanical energy what it does not do is to conserve the thermal energy okay so if you now obtained your mechanical energy and subtracted that from the energy equation written in terms of the stagnation enthalpy then you get an energy equation that takes into account only the thermal energy conservation okay that is what we would do we would just write out the equation for the thermal energy conservation we will not worry about the stagnation enthalpy at all right. So here we are rho dh over dt equals minus gradient q vector plus q dot plus rho sigma capital k equals 1 to n yk fk vector dot vk vector plus phi plus dp over dt now I told you any time you now look at an equation that keeps going on and on okay do not worry about it you start looking at it term by term okay and then start making sense out of this capital d over dt capital d over capital dt refers to the substantial derivative or material derivative or total derivative right which in turn would be like dou by dou t plus v dot del that is operator that we have to replace for this to look at the this is this is essentially the inertial term in general which can now be looked at in an Eulerian frame of reference as a unsteady term and a convective term together right. Now if you were to if you were to derive this for yourself you would understand that you got the divergence because this was actually a surface term in a arbitrary control volume this is the way we actually got our species conservation equation derivation okay we had a surface mass flux across the control surface okay and then we applied the Gauss's divergence theorem to get a divergence so whenever you have a divergence you had immediately think that this is actually coming from like a surface or from the sides okay and we will expand on this q dot is actually the what do you think is q dot let us let us just tease ourselves a little bit for a minute what do you think is q dot sorry energy supplied by what external source damn you knew the answer what else could have been what are we looking for here what are we looking for why are we here huh put into the reactor did you say put you put into the why would I put into the reactor I am a combustion person right I want to get out something right I put in fuel and I want to get heat so how many of you think that this q dot is actually the heat that is coming out of the reactions like half hands up why or is the heat from the reactions hidden in the here anywhere this is the test of a combustion scientist or a student they should say right where is the heat released from chemical reactions present in this we should probably ask this question at the end of explaining all the terms but I cannot wait I mean that is what I am really looking for forget about everything else in fact I will forget about lots of those right so some of these are basically written once and then we will say we will forget about this but we cannot forget the heat released from chemical reactions because that is how that is our essence right where is it in this is it the q dot right the q dot is essentially an external energy release rate term so this is a is the heat released due to chemical reactions contained within your system or is it or is it to be externally put in right so you now take like a reactor or a combustor or whatever it is and then you now put in your air and you put in your fuel okay and do we have to put in heat no right where was the heat contained that is now released the H where is the where is it in the H standard what the standard heats of formation of what the products of this on the reactants right so this this H is the mixture enthalpy mixture specific enthalpy okay so what is going to happen here is H is the mixture specific the reason why we are multiplying by rho specific means per unit mass okay specific enthalpy and that is given by equal to sigma k capital K equals 1 to n yk hk where yk is the mass fraction still unknown okay in the system of equations but hk is the specific enthalpy of species k okay by the way notice that we have shifted from a and b or i and j and all those things for species to a capital K because if we now did this with the tensor index notation we had i's and j's competing with us okay so we now use capital K for species from now on yeah so hk is the specific enthalpy of species k that still does not show you the heat due to the chemical reactions because you now further have to identify that hk is equal to ? hf not k plus integral t ref to t cpk dt small cpk dt there are two things that I would like to point out in this this is what is contributing to your heat release this is not the full heat release of course this is only the standard heat of formation of the kth species if you now plug this in here you now go on to get a summation yk ? hfk that is like saying I want to add up all the heats of formation of all the species that are involved in this mixture weighted by their mass fraction okay and keep in mind it is a products that are going to have very high negative heats of formation the reactants are not going to have that high negative heats of formation they could have like small negative heats of formation or 0 or close to 0 or small positive or positive or more positive heats of formation all that stuff that algebraic sum is like the net heat that we are going to get by thinking about a hypothetical process by which we broke down the reactants into reference elements by putting in some heat and then got the reference elements to form the products right and then get lot more heat than what we put in right that is what is the heat release from the chemical reactions is hidden in there okay the other thing is we were actually writing this equation searching for temperature as an unknown and look at where it is that is in the sensible enthalpy term on the top of this integral that is the unknown right so okay let us look at a few other terms let us look at grad sorry divergence Q or what would skew first of all so I told you it is coming from the surface so this is heat flux okay so this is heat flux this is an energy rate equation okay so this is actually per unit time whatever we are writing here is actually per unit time things are happening as they happening you are looking at what is happening per instead per unit time right and this is like heat flux that means what is happening per unit time per square meter per unit area right so what is this heat flux this is now going to be like equal to minus k grad T plus rho sigma capital K equals 1 to N HK YK capital VK vector plus or ut sigma I equals 1 to N sigma J equals 1 to N XJ DT comma I divided by W capital W I DIJ times capital VI vector minus capital VJ vector plus QR vector clearly heat flux is a vector okay which is now going at an arbitrary direction say arbitrary it is going at a direction not exactly normal to a surface it is having like a N N N vector unit vector unit normal vector that we have to take a component with respect to and that is how you would get the Gauss's divergence theorem applied over there and then there are about 1 2 3 4 yeah 4 terms there okay how do you get 4 terms the first term minus k gradient T what is it that is a heat conduction the heat conduction is actually coming from this from the sides at from a particular point because you have a temperature gradient with respect to this point and the adjacent point right so if you if you had a harder region on the side you are now going to have heat into this point if you are going to have a cooler region on this side you are going to have heat away from this point right so that is the familiar Fourier law of heat conduction so effectively we are assuming that we are looking at a Fourier fluid that means a fluid that the heat conducting fluid that satisfies Fourier law all right but the fluid is actually a mixture of lots of species okay so what is this k it is a conductivity of the mixture right so similar to this mu being viscosity of the mixture so we need to have an idea of how to deduce the conductivity of the mixture and the viscosity of the mixture for a mixture whose composition is yet to be solved for okay so actually we do not know this and this is changing from point to point and time to time okay in this mixture because the composition is changing as we go from fuel air over here to lots of stuff and then finally some products and so on okay so I am not going to tell you how the mu and k depend on the species concentration instantaneously I am going to put this out in a website with that that the that this video will go with it has to go to kinetic theory okay so we now we cannot do this in the continuum framework we will write out equations that that give you this later on what is this term can we carefully look at this okay rho is out you can take this in then make it make this rho rho yk as rho k hk all right capital vk as a matter of fact so long as we had yk hk sigma you would get back your H okay but then you had a capital vk that prevents you from summing over only yk hk right yk is weighting the summation sorry vk is weighting the summation what is capital vk that is the diffusion velocity of species k right how did we get this term the answer is you see when we were looking at this we had a dou by dou t plus v dot del H the v there is the mixture average the is a mass average velocity of the mixture okay and then keep in mind that H is sigma yk hk the mass average velocity of the mixture is not species dependent it is now a mixture property so you could have pulled it out of a summation and kept your sigma yk hk as H that is how you got your H okay but in reality how is the enthalpy flowing into your point and getting out of that point this looks like it is convection right in convection of enthalpy but this is like convection of enthalpy of the mixture okay but in reality whenever any matter moves it carries with it mass momentum and energy that is the reason why we are able to do mass conservation momentum conservation and energy conservation here we now have a bunch of species each of which is carrying some mass for which we did a species conservation equation at any particular point okay and then we had a mixture momentum conservation all right but if you now thinking about the energy or the enthalpy that the species carries with it okay now it carries with it enthalpy at its velocity but what we have accounted for is like a overall enthalpy that the mixture carries at the mixture velocity that is what this accounts for then there is a deficit there is still enthalpy carried by individual species at its diffusion velocity with respect to a mixed affixed coordinate system right in reality it is carrying its enthalpy at its velocity but then now we have to taken like a mixture average mass average velocity of the mixture okay and then that does not depend on species anymore so we should now be able to add up a contribution of all the species put together in terms of enthalpy to see how the enthalpy of the mixture gets convicted right but then there was one more contribution that we forgot there is like a diffusion velocity with which the enthalpy is still going all right that is coming up if you now have a mixed affixed coordinate system that is what this is so you now have a vk times hk okay so that is actually the enthalpy of the mixture that is due to diffusion okay so you have to keep that in mind and then this is pretty interesting this is what you would call as the due for effect okay where this is the opposite of the sordid effect that we saw the other day okay where there what we saw was if you now had a temperature gradient it gave rise to a contribution to a concentration gradient here if you now have diffusion velocities okay which in turn are supposed to be functions of concentration gradients it gives rise to a heat flux okay so why you had a temperature gradient giving rise to a concentration gradient is called sordid effect where you have a concentration gradient giving rise to a heat flux is is called the due for effect okay and you can clearly see that you have a dt, i divided by dij showing up here as well all right so that is what is called as the due for effect and finally qr is a radiation heat flux so what we have had so far here is only the conductive heat flux therefore you also have to add a radiation heat flux the radiation heat flux will have to be model based on Stefan's law and if you are look at an enclosure or something of that sort you have to have these shape factors coming in and so on absorptivities and emissivities and all those things that is something that you want to do separately but at the moment you just had to plug in a keep in mind that it is going to depend on temperature okay so it has a t to the 4 expression and that could be an unknown because temperature is an unknown okay so you could plug in an expression for that if you want we will stop here for now and then pick it up from here next week we still have few more terms to go we want to know what this is what this is and so on okay the energy equation takes some amount of energy.