 So we were here where we found that if I now take into account the fact that the reaction zone in this flame thickness is only a fraction of the entire flame thickness and that is where the reactions are happening and therefore the heat release rate is going to actually come from a volume that is corresponding to only this and I scale the reaction zone thickness over the flame thickness by a factor given by beta which is a Zellowitz scaling factor then I can actually estimate a u0 which has a 1 over beta term inside the square root and so we now say we now notice that the flame speed is about that much smaller right because we are now factoring in that reactions are happening in only part of the flame and not in the entire flame we can now do further corrections in noting some some more things the question is first of all what we had done was to assume that W is a constant across this region and now what we have said is W is actually a constant across this region approximately now that is not very difficult to imagine because it is kind of rising very steeply within this region and going to value peak value and then draw dropping steeply back in this region so it is okay but the question is what is the W so if you now think about what the W is W is the chemical reaction rate and strictly speaking the way we have to look at so for example in a single step reaction we should be able to write W as I mean I am just not going to say equal to but this is going to say goes as because I am going to write this this is W in terms of mass right so this is actually mass produced per unit time per unit volume and I am going to write this as something like 8-E over RuT let us say yf and yo maybe you can you can have yf power something and yo power something as far as these this particular argument is concerned I am just assuming those powers to be one it is not very important what those powers are at this moment or you can keep a power you can say yf power nf yo power no that is also fine we can go ahead with this but not very important for the kind of order of magnitude analysis that we are doing here at this stage. So let us say that we have this right the question is this W is actually a function of T yf and yo right and the T varies like this but in the reaction zone the T variation is actually very small it is going from a value that is very close to TF to a value TF and looking at the way the variation happens it is actually very weak variation within the reaction zone. So it is it is reasonable to assume that this TF T can be plugged in as TF all right but what would be the values of yf and y yo that we have to use as a first good estimate what you would immediately think about is these are actually the mass fractions of the reactants okay this is FS the fuel oxidizer O is oxidizer and therefore these are the mass flow rate mass fractions of the reactants in the incoming stream and therefore we should actually if you have a premixed mixture you have a certain mixed mixture ratio in which they are they are mixed right. So this should be a certain mass fraction for the yf and certain mass fraction for the yo and that is what we need to use right. So that would be given by some value somewhere here let us suppose that this is yf and yo which are far upstream ahead of the flame and for us the flame is starting here with the preheat zone right and what we are talking about is to actually estimate a W that is going to be evaluated here we find that the W is actually lying dormant at nearly very low values up until here because of the temperature sensitivity to E right but there is not because yf and yo are 0 that happens only here you are going to have either yf or yo or both going to 0 over here. So what does that mean for yf and yo how are they going to vary can I just plug this up to this value and then now say well it is supposed to go to 0 well I think since I said tf is not going to vary a whole lot I should not expect yf to vary a whole lot and then it suddenly goes to 0 is that the way it is going to behave if this is the way it is going to behave then I would rather take the far upstream value like with subscript knots. So if in the Rankine-Hugonew formulation we had subscript 0s or knots for the upstream conditions and infinity for the downstream conditions tf corresponds to corresponds to an infinity condition and if I were to plug in yf as yf0 and yo as yo0 then this would actually mean that w0 infinity then goes as e to the minus e over Ru tf yf0 and yo0 right. Now this is actually a gross over estimate of what the w is so previously we said whatever is the w it is constant across the entire thing that was a gross over estimate then we said you now constrain this to a smaller region and got this factor and now we find how do you actually find out the w and if you now use the flame temperature in the Arrhenius expression and the far upstream reactant concentrations for the mass law of mass action then that is an over estimate as well because we cannot expect the mass fractions to drop so steeply in this region what happens how would the mass fractions change mass fractions of the reactants vary across the flame that is essentially the question that we have to worry about so the question that we have to ask is okay so we had reactants come in getting heated they get heated up and then they react to do what so what happens when the reactions happen they produce products right so the products are supposed to go that way right so this is the direction in which we are having the flow so the products are supposed to go that way but would they just do that would they just do that that is really the question right so if you now say that you have a product concentration that sort of is the opposite of what we anticipate without thinking about a particular thing that I am coming to then I would expect the product concentrations to probably rise like this right and then go all the way up to a maximum value let us say yp infinity all right but if that were the big the if that were the case then you can see that oh well the products are supposed to be get formed and then flow out but they now sense that you have a huge concentration gradient of the products over here so what happens when you have a concentration gradient you will have diffusion anytime you have a concentration gradient you are expecting to have a diffusion and the diffusion flux is in the direction of a region of higher concentration to a region of lower concentration so as the products are actually getting formed you have the fresh reactants coming and pushing them towards the other side but as they are getting formed they are looking around and finding hey I am not there so let me try to go there it is only that the reactant flow is trying to push them here but they have a tendency to mix up there right so you now have a competition between convection of the flow of the mixture on the whole and typically you are always thinking about convection for a mixture okay and upstream diffusion of the products. So sure enough in this place where you do not have reactants happening you have a lot of products that are produced here now or witnessing a incursion of products because of an upstream diffusion against the convection and there must be a balance between these two which is exactly the same as what we talked about for the energy balance which is a convective diffusive balance for the product species mass right so effectively we should now begin to expect a product concentration profile that is similar to the temperature profile itself because the way it happened was you now had a lot of temperature here that was produced code and code temperature being produced that is essentially heat is being produced so the temperature is locally getting raised and then it gets conducted that is not very different from species mass getting produced and getting diffused upstream right even in the case of energy balance you had an enthalpy flux that was trying to wash this this this hot region away from here and this heat conduction was like a convective diffusive balance it is trying to conduct heat upstream despite the flow wanting to go to ground downstream right exactly the same thing is happening for product mass right so this convective diffusive balance is happening for the product species mass as well and if you were to actually now invoke a unity Lewis number for the products right so the Lewis number is the one that is actually trying to tell you how well a species can diffuse versus how well it can conduct heat right so if you had a unity Lewis number for the products then what it means is it is going to go through the same extent of convective diffusive balance in the preheat zone for its species mass balance as the mixture went through for the energy balance therefore we should expect that the product concentration will be having pretty much the same profile as the temperature profile for a unity Lewis number assumption on the products okay. So the further correction that we have to make is is to take in account upstream product diffusion upstream product diffusion and with unity Lewis number for the products the product mass fraction profile can be expected to coincide with the temperature profile what has been by coincide how can you actually have mass fraction that is going from 0 to 1 coincide with the temperature profile that is going from let us say something like 300 Kelvin to about 2500 Kelvin the answer is of course we normalize okay so you take the maximum and the minimum and then try to fit them from 0 to 1 okay for both cases then you should actually expect that the profiles will coincide well if that means you had a product upstream diffusion what is that saying as far as the reactant concentrations is concerned concentration really means concentration how much of what you how much of what you have is there in relationship with what else is there right mass fractions are all about the density of this species divided by the density of the mixture right so that means if you now have products over here you have other things that are coming in and contaminating your reactants so the reactant concentrations are going to dip right so effectively then you do not have either this profile for the products that you would anticipate naively without taking the count of streamed product diffusion and correspondingly we have to now we have now to alter what the anticipated reactant concentrations are going to be so what you could expect is the reactant concentrations should now be the inverse of the product concentrations so if you now start with something like a normalized value with respect to the initial concentrations as here then you should expect your reactant concentrations should go like this now this is obviously for the deficient deficient reaction and let us suppose that we are considering a fuel lean situation okay if you now consider the fuel lean situation then that species concentration mass fraction is going to go all the way to 0 and that is the reason why your W is also going to 0 at the end of the reaction zone right so now look at what happens to the species a mass fraction variation within the reaction zone instead of actually having a value that is about that high you are now looking at a value that is about so small okay there is hardly any variation of this away from 0 so we are now talking about fairly small values so correspondingly you should anticipate that if you are if your W were to be calculated with GSTF which is alright you should probably actually now have a factor beta thrown in on your YF okay we have to now decrease the amount of YF relative to the initial or the upstream far upstream value and therefore we have to bring that in so in fact we could we could just for the sake of complete minutes go back and then say let us suppose that this is M and N and we want to say M and N so correspondingly the reactant concentrations decrease with the deficient react reactant with the deficient reactant going to very low values in in mass fraction in the reaction zone right now if you think about so that means we should basically say that is YF for example should now actually be more like YF divided by beta so correspondingly our W should be going as what we would call as W not infinity divided by beta power M so you would now say YF is YF not divided by beta so beta power M should show up for your denominator with W not infinity in the numerator to give you a W so question then is what happens to the excess species okay so the excess species will could actually come come out like that to whatever value let us say y o infinity alright now that is an appreciable fraction of the original value and so if you from an order of magnitude basis it is probably a factor of about 2 or 3 that is not a factor of 10 so it is not really an order of magnitude different right so without worrying about the non-deficient or excess species a mass concentration reduction we are still within an order of magnitude okay within the order of magnitude in our estimate so that that is reasonable so you do not have to worry about this that there is essentially how we should want to look at on the other hand if you think about like a stoichiometric mixture then both of them are going to be equally deficient right so you have to make this correction so keep that in mind so that that is an additional correction that we want to do we want to go back and say this W is not going to be just W not infinity right this beta here was to make sure that this W whatever it is W not infinity or not is going to be confined to a small region when compared to the flame thickness but on top of it if you now say W is actually W not infinity then that correction basically says no it is not it is W not infinity divided by beta power m this is what that you have to plug in there all right can we do more right can we anticipate something more than this or are we really pushing our luck trying to get insights into how things are supposed to happen right without doing the mathematics okay we are hardly doing much mathematics here we are just doing mostly order of magnitude anticipating things how they are supposed to happen and so on right so in fact I specifically say expected to coincide that that that is the crux there to get you this and this is not robust to say that this is going to be like this is not robust it is just in order of magnitude estimate good then how what else can we do what have we made an assumption here on we have made the assumption about unity Lewis numbers right we have made a unity Lewis number assumption on the product to say that it is going to coincide with the temperature profile all right and correspondingly we if you now say that if the product concentration profile is like this the reactant concentration profile should follow like this this is how we have actually built up our argument now that implicitly then assumes that the reactant constant reactant Lewis number is also one right because effectively whatever we talked about for the product happens for the reactant species mass balance as well that means as far as the reactant species mass is concerned you have a convective diffusive balance in this region for the reactant species mass right and if you now assume that the length scale of this particular balance is the same as the temperature profile then we are implicitly assuming that the Lewis number for the reactant species is also equal to one right that means the species the reactant species can diffuse just as well as it can conduct heat is what we are basically assuming that may not be the case and that is why are we talking specifically about the reactant Lewis number and just forgetting about or abandoning the product Lewis number that we originally constructed our argument based on is because the reactant concentrations the ones that are showing up in the law of mass action that is what we are beginning to correct and we want to now see if a reactant Lewis number is not going to be one is it going to be this way or that way right is this is this profile going to actually change right so how do you how do you take this into account there are there is a quick way of doing this so for the correction consider so consider non-unit loss number of deficient reactant so how do you expect the reactants profile to behave and how do you expect the temperature to behave so in the convective diffusive diffusive zone which is the preheat zone we could apply the Schwab-Zeldovich formulation and what would you have let us suppose that you had a velocity you not which which which did not change significantly then you have a convection balanced by the species diffusion you can write right which is assuming a fixed law formula fixed law kind of diffusion going on and a u0 dt over dx equals alpha d2 t over dx2 now what you are going to do is just focus only on this this particular region and as far as this region is concerned you are going to simply say let me not worry about the difference between this and this okay let me not worry about the difference between this and 0 right so it is as if like it started from this value and went to 0 here you start from this value and went to tf there within this region okay and so with these kinds of boundary conditions that you want to apply in fact you should be able to also fix your coordinate system and say that these values are actually in at a negative infinity okay so if you now apply these kinds of boundary conditions you can get solutions I am not rigorously solving this step by step for you just going to write the solution here and trying to explain the adoption of these boundary conditions intuitively it is not very important for us to go through the steps of the stage well let us go back and write df to be taking into account the diffusivity should be if the fuel particularly which is which is not necessarily the same as the other diffusivities of species and here you are going to have t equals t0 plus tf- t0 times e to the u0 u0x divided by alpha so let us see what is going on it is easier to see this first is what we have been considering first the thermal profile you see what is going on we have a temperature that is now going to actually increase in excess of t0 by a difference of tf- t0 times a exponential factor right which obviously is going to actually increase exponentially beyond the preheat zone. So the temperature as far as this profile is concerned is going to give you something that keeps going like that okay disregard that that is not part of our domain as far as this this equation is concerned this equation is valid only in this domain okay so once you understand that then this profile is quite reasonable for you to think about right and correspondingly you will now have your yf starting from yf0 and decreasing by this factor okay why are we doing this we want to be able to see this in both the cases right so as x increases within the preheat zone you are now actually looking at a exponential of x divided by df divided by u0 right so the length scale for reactant diffusion is I should say characteristic length scale right characteristic length scale for species diffusion is df over u0 are we okay this is like meter squared per second this is meters per second so meter squared per second divided by meter per second should give you meters right so always have to you come across these kinds of things you have to quickly check your dimensions right then the characteristic length scale for heat balance right this is this is in the preheat zone for the convective diffusive balance right is alpha over u0 that is a characteristic length scale so the ratio of the characteristic of the length scales of fuel concentration to temperature profile temperature profiles is then df over u0 divided by alpha over u0 which is equal to 1 over lef because lef is alpha over df okay so what do we have if you lose number of the fuel is greater than 1 alright then the fuel concentration length scale is going to be less than the temperature profile length scale that means if I go back and use a different color chalk piece and say forget about that that is a that is a x species so what you are talking about is we are saying that this length scale for these species is now going to be delta f lef greater than 1 if I had a loose number for the fuel greater than 1 then 1 over something greater than 1 is less than 1 therefore df over u0 is less than alpha over u0 and therefore this distance should actually be less than that distance so instead of just talking about a single delta we will now talk about a thermal flame thickness you and then we will now talk about a mass flame thickness right so what then happens is how do you now think about the concentration profiles the concentration profile is now going to actually begin to drop only here not here right and considering that it is continuing to be a deficient reactant let us say for the fuel is what we are talking about you have to get back to this as you come down so you are now going to drop a line that goes like this well let us try to make this a little bit better so what has happened here what we find is instead of having a variation that was like only so small within this region right you are now having a larger variation within the thermal region right so that means your concentrations have to be taken to be higher than what we did here so this beta was to take into account the reaction zone based on the thermal flame thickness right scaling the thermal flame thickness for the reaction zone thickness but we find that as the lowest number is greater than 1 the concentrations in the reaction zone based on the thermal flame thickness will be higher than this so correspondingly we have to say right yf then will become yf not divided by beta times Lef because Lef is greater than 1 and we need to now multiply by this excess Lef then unity to factor in that your concentration is going to be more now of course we hope that this factor of multiplying Lef should now be the same when you now think about a Lef that is less than 1 right so how would that work so maybe take a different color chalk piece let us try pink when you now say that Lef is less than 1 then the mass if you ma the species mass balance length scale is going to be greater than the thermal thermal balance length scale right so that means we should now be thinking about a a a flame thickness based on this to be ?f Lef less than 1 and how would you draw your profiles would start from here but you would start dropping your concentration right here and then reach up to this point right so that would be something like this what you then find is the concentrations are even lower than the situation corresponding to a unity lowest number so the lowest number was actually less than 1 for you so you just multiply by that factor to get a concentration that is less than that so sure enough so you need to have this kind of a multiplication going on therefore then your w should go as w0 infinity times Lef divided by ? power m so this is what we could actually plug in back here to now not only take into account upstream diffusion of products and correspondingly downstream diffusion of reactants in addition to downstream convection of the reactants so that the concentrations of the reactants are actually coming down in the reaction zone we could also now look for preferential diffusion that means you now say we really have a non unity lowest number the reactants actually diffuse preferentially depending upon their lowest number relative to the thermal balance right and then we find that the concentrations of the deficient reactant which is more critical in the reaction zone can vary with the lowest number itself in a way that we can see so this is amazing in my opinion because we just think about what is going on okay and then start expecting how these variations should be and the interesting thing is if you have to do the rigorous analysis as we will momentarily commence to do number one we will be constrained to make assumptions particularly about unity lowest number like for example the Schwab-Zelda which formulation essentially assumes one of the one of the assumptions is a unity lowest number so if you now adopt the Schwab-Zelda which formulation to solve the problem of a one dimensional laminar flame premix flame propagation then we will be constrained to do this for a unity lowest number and then we will have to relax the assumption of unity lowest number and then expand the scope of the analysis and then take the burden of non unity lowest number upon ourselves and do a fresh analysis for that what you will find at every stage of doing the mathematically rigorous analysis with a unity lowest number assumption that is up to the stage right or taken into account a non-unity lowest number and reaching up to this stage you will find that the mathematical analysis is different from this only by a factor of square root of 2 right otherwise you get all the all the factors that you want in there in the way we have done this square root of 2 is like 1.4 and 4 so you are your order of magnitude estimate is actually okay within about 41% right that is essentially the power of the order of magnitude analysis the additional bonus is we started actually using some of our gray cells instead of just you know plugging and chugging equations and so on which we will now do.