 So we have been looking at these different models of combining chemical and thermal processes like the fixed mass reactors of constant pressure and constant volume and then we looked at the open flow systems of well stirred reactor and plug flow reactor the question is what can we do with these okay. So many times we can actually use these to solve complicated problems in more simplistic manner but something that is quite interesting is you could also actually use combinations of these models to set up like a network that deals with a more complicated system. So one thing is to take one system and try to simplify it into one of these but another way is to actually put a bunch of these together in a network that will mimic a more complicated system. So the model reactors can be used in a network. So for example so we have let us say a gas turbine take a gas turbine combustor. So we know that for example you now have a gas turbine outer casing that goes like this and then you now have a fuel inlet that comes in and then you now have a swirler so you have an atomizer here you have a swirler and then you have a liner and so you have some swirl cup and so on. So you now have the primary air that goes through this the secondary air goes like this and this is fuel this is air and then you have this liner having these holes for secondary air to come in so you now have a primary combustion zone here and a secondary combustion zone here. So this is of actually different fueler ratios so you now have like a close to stoichiometric mixture over here and then you now have some air that is coming in further into this and you now have a more fuel lean condition of combustion there and then you have a further situation of dilution because you now have to have the turbine blades withstand the temperature at the outlet so you need to have the turbine entry temperature reached at a particular value and not higher than that therefore you have these called dilution holes right. So this is like a flow that is a diluted flow a diluted not just in not in terms of air but also in terms of temperature that means you are relatively cooling this gas so this is the primary combustion zone secondary combustion zone and this is the dilution zone. So how do you deal with this in a they are actually highly complicated system in terms of the fluid mechanics that is involved here the chemistry the combustion all those things the multi phase flow all the stuff that happens here it is a highly complicated situation. So how do you try to use the simplified model reactors that we have had like the W the well stuff this is obviously an open flow system so we have to actually choose something like the WSR and the PFR that we have seen before so what you can actually think about is you now treat the primary combustion zone as a WSR one and then what happens is so you now have a fuel line and a airline that are feeding into these so you now have a inlet from the air and the fuel right and the exhaust of the WSR one gets into the gets into another WSR right that can be used to simulate the secondary combustion zone and you now have this airline can keep going you now take additional air in here and then the output of that into well it depends on whether you want to treat this is a PFR or so on but typically you do not have too much combustion going on and you are now a detail end of the combustor. So you could actually more easily treat this as a PFR and you can even keep track of how things evolve in space because PFR allows you to do that in a one-dimensional manner right so you could you could try to model this is a PFR with additional air and then you now have this coming out right in case you are thinking of having a additional fuel that is participating in the secondary combustion because if you now have incomplete combustion in the primary zone and it is not just the combustion products of the primary zone that feeds into the secondary zone but additional fuel you could have this as well just to just to be sure that you are using all the fuel that is in there inside the combustor so you see that this is now like a network of these these model reactors in tandem one after the other and also using the output of one as an input of the other and so on and taking air from a airline and fuel from the fuel line for these different parts differently and then finally you now get this these are the two inputs and that is the output there is a there are temperatures for these so that means you have a enthalpy for this and therefore correspondingly you will get the enthalpy for that and so on so something of this sort can be done. So I would like to conclude the section on these model reactors which combine chemical and thermal processes without getting too much into flow details except for the little detail that we did for the PFR and without getting into any mixing at all okay and justify trying to avoid mixing and deal with quite complicated situations in a simplistic manner but then we cannot avoid mixing completely so we have to step into getting into mixing as we go along so we let us now start doing the real combustion real combustion now involves you to be dealing with convection, diffusion and reaction all three together so we have so far been looking at chemical reactions and we did not really to do too much of convection or flow we would not do that much but we will have to ultimately get into some amount of mixing that we need to look at so it is what we would want to do. So we want to now look at some mass transfer definitions this is actually continuing from the set of definitions we had on mass concentration of species and molar concentration of species and mass fractions and molar fractions so we be more fractions we had made these definitions earlier on but now we have to actually get into a conceptual idea here so the first definition that I would like to make is again we will have two parts one is the mass averaged velocity of the mixture let us suppose you use the symbol v vector small v vector for the velocity of the mixture this would be defined as i equals 1 to n sigma rho i vi vector divided by sigma i equals 1 to n rho i what sigma i equals 1 to n rho i do you remember from past that is just the density of the mixture itself okay the density of the mixture itself is nothing but the sum of all the individual densities of species because the density is nothing but mass per unit volume so per unit volume of a mixture let us suppose that we have air around us and then air is a mixture of let us say nitrogen oxygen and a few other things in trace so you just pick a unit volume of this then you start counting the mass the amount of mass of nitrogen amount of mass of oxygen and so on in this amount the mass of nitrogen per unit volume is the density of nitrogen the mass of oxygen per unit volume is density of oxygen and so on you know add up all the mass that is there in that that is the mass of all the all the species there per unit volume is the density of the mixture so it is simply adding up all these so this is not this is nothing but row so you now begin to see that if you do not have a subscript right it now may it now big belongs to the mixture if it has a subscript then it belongs to a species alright first of all and then how does it work the density is summation of all the individual densities but the velocity is like an average it is a mass weighted average why is it why am I saying mass weighted because this is actually a mass averaged velocity right so I could have I could have simply said this is mass of individual species times its velocity per unit volume divided by mass of individual species per unit volume so I could have had like per unit volume on both sides so you got the density from the mass so essentially a mass weighted average or velocity of the mixture what does this mean is if you now have a mixture of gases what we are trying and then each of these each of these species is actually trying to go with its own velocity alright and since I am showing my thumbs that that that is to actually indicate the direction that means you now have the species a goes this way species be goes this way and so on at this particular location and time right out of this we are now trying to picture how the mixture is going to go on the whole without worrying about what the individual species are doing so how do you deduce how this mixture is going to go as a whole we now try to average the velocities weighted by their mass that means if you have a very small amount of mass of a particular species it does not really count too much for the mixtures velocity let us say you have a very small amount of mass going very fast okay you are still going to have the mixture on the whole going slower because lots of other things which are that are present in a more greater abundance are going slower right so as a mixture it is now going to go just as fast as this very small component right. So this is what we are saying we could have also done this instead of having a mass average what else could we have done moles right so we can use moles so the molar average velocity mixture we now call this small V star vector that is given by ? i equals 1 to n C i Vi vector divided by i equals ? i equals 1 to n C i of course this is also the same as C which is the concentration of the mixture the molar concentration of the mixture because it is also going through the same kind of arithmetic which is just summing over all the number of moles of all the species together per unit volume all right. So what is happening here is a mixture actually is a mixture traveling at a velocity V or is it traveling at a velocity V star both are not same okay the C i's and the rho i's are not exactly the same how are they related there is a molecular weight coming into picture okay so they would actually be the same if your molecular weights of all the species were the same but that is not necessarily true you could have hydrogen on one hand which is a very low molecular weight when compared to let us say butane okay or carbon dioxide all these things have much higher molecular weights. So these are not necessarily the same the question then is what is exactly the mixture doing is it going at this speed or is it that speed it depends on what you want how you how you want to deduce the mixture speed so the mixture speed is not an absolute quantity it is the species velocity that is that is more absolute the mixture velocity depends on the way we want to evaluate the mixture velocity right it is kind of like saying in a democracy you have a lot of people who have to get together to vote and select the leader okay so if they now select the president in one country or the prime minister in the other country who is bigger the president of the prime minister no it did not matter the people of the ones that are the most important that is what the democracy is all about right so here it is a species velocity that that is really the absolute how do you want to actually deduce the mixture velocity is up to you right this is now going to be very important the next step that we make right so the next thing that we want to do is a we now define a mass diffusion velocity of species I species I okay now this is actually for a individual species and the way we want to define this is capital VI vector is equal to small VI vector minus small V vector and let me just complete the next step B and then we will talk about this molar diffusion velocity of species I is now capital VI star is equal to VI vector minus V star vector you see here VI are the same because they are absolute they are always there as they are okay depending upon whether you are subtracting the mass average to velocity of the mixture or the molar average velocity of the mixture you now get apparent velocities of the individual species which are capital V or capital V star and why do we say apparent what is the going on it is like saying this is where we have to really listen okay it is like saying you now have a mixture that a ear okay right there in that picture okay so you are now into a combustor you are now peeping into a combustor and then you are finding lot of species going hither and thither right you want to now try to deal with this like a mixture so you now choose to pick a mixture velocity that is defined either this way or that way all right and then you now want to tag this mixture velocity to the mixture in reality all the species are going this way in that way then you got a mixture velocity out of this that is going that is saying that the mixture is going like this and then you go along with it that means these velocities are in the mixture fixed coordinate system we are essentially transforming our coordinates if you were in a ramp fixed coordinate system where you had your combustor running and then you saw your species going this way and that way and you could deduce a mixture velocity for it as whatever it is for the mixture that is going in whichever way you want it is all in the lab fixed coordinate system but now you decide to travel with the mixture when you are traveling with the mixture at the speed that you have determined for the mixture in a certain way right you begin to see that the species are going this way and that way it is not all going like this okay because you are going along okay it is going like that is even going backwards right because you subtracted out that mixture velocity right so it can now this is vector vector addition this is vector algebra okay so you look in at vectors so from your mixtures fixed coordinate system it looks like as if the mixture is stationary right that is very good because forget about the combustor forget about aerospace engineering and all those things they will now take a box okay keep the mixture on our desk table it is not going to go anywhere let the mixture mix right because the mixture is not going anywhere we are going with the mixture anyway right so while we are going with the mixture we are now in a mixed effects coordinate system so the mixture is now on the table you do not need to you do not need to go anywhere so whatever you are going to be looking at for the species velocity if the mixture we are not moving is what you are going to see with this or this right and that is how we would have actually tried to make experiments to try to find out what else is the mixture is doing other than going I am sorry what else is the species doing other than going with the rest of the mixture if you now have air in this room I suppose we do okay and it is a mixture it is a mixture of nitrogen and oxygen predominantly and then you want to now try to start looking at what the oxygen molecule is doing okay we do not really bother about things at the molecular level because we want to stick to a continuum framework so the particular point if you want to look at what is the oxygen doing what is the oxygen species velocity that is actually a bunch of molecules millions of oxygen molecules at this particular point collectively what are they doing when compared to the rest of the mixture the rest of the mixture is mostly nitrogen so you will now ask what is the nitrogen doing okay we will find that most of them most of the time in what we are thinking about the oxygen is pretty much going to do the same thing as what the nitrogen is going to do because it is all completely mixed when it is completely mixed it is now acting as if it is one species air then the means the species are going to do the same thing that means they are just going to go as one single species which is the mixture right so what do we understand if a species is going to do something other than going with the rest of the mixture then it is not completely mixed right and to quantify what it does other than go with the rest of the mixture is what you have these velocities for right so this is a and this is very important this is fundamental to anything that you do with mixing yeah so this is this is in the now mixture fixed coordinate system right this is lab fixed coordinate system so in general let us just make this point very clear in general small v and small v star okay and capital vi and capital vi star are different or different for species in a mixture with just similar molecular weights okay and this is going to come to haunt us after some time yeah so sometimes you will find combustion scientists analysts are making these seemingly reckless assumptions like equal molecular weights for all species what is a big deal I mean can't you even deal with more species that are of two different molecular weights the answer is yeah but we have lots of other equations to solve so can we get some reprieve here we will find okay so what do we do with this we can now define mass fluxes we have we are now stepping up our approach towards something particular so of course we know that mass flux of species i is m dot vector i equals rho i vi okay and so of course we have a and a b motor flux of species i let us call this ni dot vector equals ci vi vi that is it vi is absolute so this is in the lab fixed coordinate system right something that we are familiar with mass flux is nothing but density times velocity okay similarly motor flux is nothing but concentration times velocity all right then we should now be able to write for a relative mass flux relative mass flux or diffusion mass flux diffusion mass flux of species i is ji vector is equal to rho i capital vi which is rho i vi-v right and b go through the same thing replace the word mass by molar and then we will see how the symbols change a little bit relative molar flux or diffusion molar flux of species i how does it change ji star is equal to rho i I am sorry ci capital vi star vector that is equal to ci vi vector-v star vector right and obviously this is in the mixture fixed coordinate system right that is in the mixture fixed coordinate system what are we done the first thing that we have done is for the first time in our lives we have used the word diffusion okay welcome to the confusion the moment you have diffusion you are going to get into trouble okay and that is what that is what we will we will see for from now on and so obviously things get more interesting exciting and stuff right whatever we have been doing so far has been boring and then what you are saying is if you are now thinking about mass flux let us actually think about in a mixture fixed coordinate system right so in a mixture fixed coordinate system if I were to move with the mixture right what is the mass flux of my species that is going here and there it is all not going in together in one direction because I am going along then I am subtracting that so you could now also have some mass flux or molar flux depending upon how you want to look at it go backwards forwards either this way that way and so on right so why are we interested in this flux where could we have just talked with velocities right that is because that is how fixed law comes about okay so we now have to have what is called mass fixed law we are now ready to look at fixed law at all stick around 1858 stated that da r is equal to – C da B gradient xa for a binary mixture of species a and b the plural of species is species okay so we say species we were talking about individual species I you know talk about two species a and b all right okay we got it now it is very likely that Fick exactly did not say it in a mathematical way but we are now geared up to state fixed law in a mathematical way right and because we have now come from a mass average and molar average velocity all the way to a relative molar diffuse molar flux or a diffusion molar flux J I star for a particular species what it basically what did he say he found that the relative molar flux or the diffusion molar flux of a particular species is directly proportional to its mole factor it is basically he said it is directly proportional to its concentration gradient when would you have a concentration gradient if you now had a variation in the concentration in space okay that means you have more of it here more of a particular species here less of a particular species there then you now have a variation and a spatial variation would mean that you now have a gradient and that is what is actually driving the species to come from there to here right that simply means that you do not have a mixture in which all the species are uniformly mixed if you had all the species uniformly mixed like we think air is around us at the moment okay then we would not have any concentration gradients of oxygen here versus here versus here and so you do not have any gradients that means oxygen does not rush this way while nitrogen is rushing somewhere else or the mixture is going somewhere else alright of course then the oxygen rushing this way counts for the mixture going somewhere anyway okay so in a lab fixed coordinate system it will look like the oxygen is going like that but it does not do that other besides nitrogen going somewhere else they are all going together because you do not have any concentration gradients that means the mixing is perfect it is only when you do not have a perfect mixing right you have to worry about concentration gradients and when you have concentration gradients we now have these species go from a region of higher concentration to a region of lower concentration until the concentration gradient vanishes and then the species does not go anywhere it is all living happily ever after right so it is kind of like Robin Hood okay take the money from here from the rich give it to the poor until all of us are living happily ever after okay that is what transport processes always do okay so the DAB is a binary diffusion coefficient that appears as a constant of proportionality between the diffusion molar flux and the concentration gradient right and this is a transport transport property this is a mass transport property correspondingly the momentum transport property is your kinematic viscosity and the energy transport property is your thermal diffusivity all of them actually have the same units in Si it is meter squared per second right and what do they do their job is to take mass of a particular species from a place where it is more and put it in a place where it is less take momentum from a place where it is more that is what viscosity does okay so momentum from a place where it is more and put it in a place where it is less take heat from a place where it is more and put it where it is less that is heat conduction so that is what conduction means so this is like transport phenomena out of which we are essentially looking at only one part which is the species mixing in a mixture. We have to look at something a little bit more carefully we are now saying it is a binary mixture that means we are now looking at a mixture which has only two species how does it matter and how does it work out for does it does it make sense we are doing combustion and do we deal with mixtures that have only two species what combustion means you have to have reactions right so for reactions to happen you need to have two reactants yeah sure so we have two and then they will mix but what about products if two reactants are required to form a third product you do not have a binary mixture anymore you see so we are out in fic is not good enough for us right away unless you had like one of them as a reactant the other one is a product okay that is like a recombination reaction you now had a molecule that that just let us say you it was subjected to heat from the flame let us say okay and then it desire that to reorganize itself and become another species okay like like a bunch of atoms in one corner of the molecule started getting excited to dissociate and then go around and then form and then go back and of course you now find that you will now have radicals and all that stuff right because this is all got dissociated and so on so it is not a binary species is something that we do not really come across typically but you could if you if you want to simplify your situation you say a equals B is my global reaction I will deal with only two species and so okay you could do that Fick did not really have that much of a complication in his mind okay he was only looking at a isothermal maybe non-reactive okay mixture of two liquids as a matter of fact is what he was doing right so it is an incompressible situation and because I as I said J star is a mixture fixed coordinate system you did not have to worry about a mixture in motion in a lap fixed coordinate system you could think about a mixture that is right in front of you sitting there mixing as you speak or observe okay and then you can come up with this law and therefore the C is a concentration of the mixture just like an incompressible flow you say the density is constant you say the concentration is constant for the mixture as a whole can be pulled out of the gradient and then you now have a more fraction gradient rather than a concentration gradient right so this is the simplistic setting in which Fick really came up with this law okay and you could of course so we have a of course we could write in terms of in terms of diffusion mass flux let us say we are not chemists so we are not used to dealing with things in terms of molar flux we are more used to things in mass flux so so so let us say diffusion mass flux Ja is equal to – Rho DAB it is the same DAB as before you now have a mass fraction gradient rather than a mole fraction gradient and a density instead of concentration for the constant for the coefficient Ja star becomes Ja vector right. So what is DAB DAB DAB is the binary diffusion coefficient binary diffusion coefficient for a pair of species A and B so the binary diffusion coefficient is actually defined all ways regardless of how many species are there in your mixture okay you can still think about a binary diffusion coefficient that is defined for a pair of species in that mixture you are now beginning to think a little bit more okay is it possible for us to apply this to a mixture which has more than two species maybe okay when what would DAB be yes you can still use the DAB that is a binary diffusion coefficient for a power of species is that so it is always valid for a power of species well what if I have a mixture of B and A instead of A and B well I have a DBA there is a difference from DAB is it like a 10 0 or something like that no right so DAB is equal to DBA that is because we are now in a mixture fixed coordinate system right so if and this is very important because we are in a mixture fixed coordinate system effectively what it means is if species A is mixing into species B then correspondingly species B is mixing into species say just as well so it is like saying if you now had hydrogen and oxygen a thing that hydrogen is a much lighter gas when compared to oxygen so you now had a bunch of hydrogen over here and then a bunch of oxygen over here in a box with a diaphragm in the middle and then you now have like we have very magic wand that makes this there are from disappear a T equal to 0 and you start looking at this at T greater than 0 okay you expect that all the hydrogen is going to rush into oxygen the oxygen just sitting like that huh and then you say all the hydrogen was the one that did the job of mixing oxygen did not do the damn thing no the oxygen is going to say yeah of course I am mixing right it is going to mix just as well into the hydrogen as hydrogen mixes into oxygen because from the oxygen's point of view the hydrogen is doing is done you see this is very very important okay in a binary mixture there is nothing like this species is mixing more than that species why are we doing this why do we need the fixed law the fixed law states that the more diffusion flux okay or the mass diffusion flux are directly proportional to a concentration gradient in terms of it that a mole fraction gradient or a mass fraction great why did we need it why did we need thick why are we talking about right what are we achieving any guesses if you are you think a little bit more okay I told you that these are these are this is a transport process and then there was like there are correspond this is a mass transport case was correspondingly momentum transport and energy transport and you now had viscosity and conductivity come up in those as transport properties what do they connect okay. So if this is like a proportionality constant between a mass flux and a concentration gradient what was the viscosity a proposal it was viscosity a proportionality constant of anything it was a proportionality constant that was relating shear stress to velocity gradient okay keep in mind velocity is a vector a gradient of velocity could be a tensor and therefore you have additional problem and momentum transport of dealing with a gradient of a velocity that means your shear stress could be a tensor and so on okay but whereas here this is a vector gradient of a scalar that is a vector okay but when you get back to energy transport things are back to normal we are now looking at the thermal conductivity is showing up as a proportionality constant of heat flux there is relating heat flux to a temperature gradient right temperature is a scalar gradient of temperature is a vector heat flux is a vector right. So what is going on we are always trying to relate some of these quantities like diffusion mass flux or shear stress or heat flux to gradients spatial gradients in concentration or velocity or temperature these are all quantities that I can measure I can stick a probe and get my concentration velocity or temperature these are things that are actually showing up in my equations you have a control volume there is a heat that is coming in from the surface or mass that is coming in across the surface control surface and so on and I did not know how would I know I can only measure concentrations in two places to find out this is more than that then it is that you have a mass flux I now measure temperature at two places if this is more than that I now have a heat flux right. So in our system of equations we would like to keep the unknowns as essentially our primitive variables right but you now have these heat fluxes mass fluxes and shear stress coming in as extra items along the control surface like for example shear stress is essentially a surface force it is coming through a surface force okay to drive the momentum heat flux by conduction is trying to influence a enthalpy change right similarly mass flux by diffusion is going to now change your mass balance a continuity equation right and we would like to relate these to things that we want to keep us keep us our primary unknowns and we do not want to deal with any additional unknowns so this equation and it is like for momentum transport and energy transport or constitutive relationships that connect a secondary unknown to a primary unknown now plug this in an equation which treats xa as its unknown you are safe you are not reckoning additional unknowns you see so this is very very important to us unfortunately we will be grouping to see how to deal with this for a truly multi component system where you have a mixture of more than two species next class.