 So we were talking about what happens to the Bunsen burner at the tip because what we found about the flame stabilization was largely speaking we are now talking about like the shoulder of the flame where you have the normal component of the flow velocity tries to balance the flame speed that is normal to the flame itself and therefore the flame shapes itself corresponding to that but that is true for most of the shoulder of the flame but if you now think about what happens to the tip no matter what the flame speed flow velocity balance is the tip also always has to actually propagate against the flow directly right so which means no matter what the flow speed is and how the flame speed is able to balance that the tip always has to adjust itself to match the flow speed to the extent it can right so question is how does that happen right so what we find is we now have to think about think about a locally highly curved flame there and then see what happens when you now have a flame that is so curved out there and therefore what we want to do is think about flame curvature and also another aspect of it that we need to worry about call the flow divergence effect in the context of a small perturbation to a plain flame so if you now begin to think about a nominally plain flame that is propagating against a propagating with a flame velocity SL superscript not all right which means in a flame fixed coordinate system you have your reactance unburnt reactance coming in into the flame at SL superscript not then let us now think about what happens when you now perturb the flame about this point okay and this particular picture here shows couple of things one is you have one part of the flame that is curved convex against the incoming flow and the other part of the flame is curved concave relative to the incoming flow we need to think about what happens to these two parts separately but we should probably come to the same conclusion either way so typically in the classroom if you explain what is happening to the concave convex part in the exam you can you can expect a question of what happens the concave part right so so you should be able to think about this just as well what we want to talk about here is actually two process to say two different kinds of processes that are that that come about when you have this kind of curvature one is a thermal so one is a thermal diffusive imbalance or balance or imbalance depending upon what really happens so one of the things that happens is when you now have a flame that is curved let us now take the concave part this is a little bit more intuitive and it is also kind of like what is happening in the Bunsen flame tip so that this is pretty instructive when you are now thinking about a curvature which is now beginning to be of the order of not exactly equal to but of the order of which means still a little greater than the flame thickness now so keep in mind what is meant by flame thickness the flame thickness includes the preheat zone and the reaction zone and the preheat zone is where the upstream heat conduction and downstream species diffusion happens along with convection superposed on for both the species mass as well as the enthalpy what we are now beginning to see is if you now have a curvature like this then beyond what is being heated up well for the cold reactants you now have a focused effect of heating the reactants so this conduction now becomes multidimensional the heat conduction becomes multidimensional and therefore you are now actually heating the reactants significantly more than if the flame were planar this should actually give rise to a tendency for the flame speed to increase all right now on the other hand look at what is happening to the species diffusion the species diffusion now is going to actually go radially outward so typically when you are looking at something like a fuel lean flame for example then the the deficient species is deficient reactant is fuel in a fuel lean situation and when you are now trying to have this speed that so that it is more critical to think about the deficient reactant so the deficient reactant now spreads thinner across in a radially outward manner so instead of actually feeding a planar flame straight it is actually spreading itself thin that means the flame is running out of the deficient reactant more than in the case of a planar flame if you have a concave curvature relative to the upstream right so this has a tendency to decrease the flame speed all right so you now have a possible balance between these two these two effects that will tell you whether the curvature is going to stay or grow or decay so when you now have to think about a balance between heat conduction and species diffusion what comes to our minds it is a Lewis number right so we have to think about what the Lewis number is and depending upon what the Lewis number is okay so if you now have a Lewis number that is greater than one right so if you now have a Lewis number that is greater than one that means the heat conduction upstream right is more than the mass diffusion downstream right that means the species depletion effect is not going to be as significant as the the focus sheet conduction which means that the flame is actually going to propagate more vigorously or there is going to be a net effect of increase in the flame speed all right so if you now think about what is happening at the tip of the flame for a greater than unity Lewis number right we expect a more intense burning at increased flame speed that can try to match the flow speed there all right for a greater than Lewis the greater than unity Lewis number but if you now have a for in the Bunsen burner tip right the Bunsen burner tip for early less than one you now have a a greater extent of the deficient species depletion that is happening when compared to the extent of upstream heat conduction so the flame is essentially running out of reactant even if it is trying hard to conduct conduct heat upstream to heat up whatever is remaining of the reactants so this leads to this could lead okay so could lead to a local flame extinction all right now many times I shouldn't say many times sometimes you do see flames which have a hole in the middle so that means you now have a flame that keeps going up like this instead of turning around it actually goes up like that and see see that I am actually drawing it with the lighter line that simply because the flame fades as it goes along that indicates like a lesser burning intensity progressively and the extinction there all right so you now have a hole in the flame it is not as if like it you need to have any less than one for this hole in the flame there is also one more thing that we have to think about so the first thing that we talked about was a thermal diffusive imbalance thermal hyphen diffusive imbalance the second thing that we have to worry about is the density jump okay so density the density jump across the flame so what is the consequence of having a decrease in density for the products relative to the reactants the flow gets past the flame right any ideas here what is going to happen if the flow now tries to go past so we try to still go through the same idea that is you now have a nominally flat flame with a SL not but then now you are beginning to think about the fact that the density of the products is going to be less than the density of the reactants and perturbed the flame to be non-planar all right so as a consequence of the density difference local mass balance dictates that the velocity should compensate for it all right so if you now think about a point here along along a curved flame what we are basically saying is you now have a let us suppose that your this is your you not and what you want to do is try to decompose this as parallel and perpendicular to the flame right so this is you n not and this is you t not and what happens is when you try to have a flow that is going past the flame it is mainly the normal component that suffers the expansion because that is how we are normally thinking about a flame so if you had a unperturbed flame with a with a with a with a flat profile then what you basically saying is the normal component of the flow is going to actually expand right so what we then what we then have to expect is we need to have the normal component increase significantly this is significant because what we are talking about is a constant pressure more or less a constant pressure system all right and you have to think about what is the density change because of the density changes because of the temperature change so the temperature rises therefore the density falls right so what is the temperature rise like if you take like a 300 K over here and let us say it is reasonable to expect something like a 2400 K there right so that is about a temperature rise of about eight fold can be 2100 2400 2700 K think in terms of factors of three 300 for the sake of simplicity right of 3000 so you are now talking about a factor that somewhere between 7 to 10 times more all right so this has to be significant and then so the density correspondingly decreases right so P equals rho RT and as T increases rho has to decrease and then rho 0 U 0 is equal to rho infinity U infinity or more specifically in this case rho 0 U N not it should be equal to rho infinity U N infinity right so if rho 0 over rho infinity is going to be about a factor of 7 or 8 or 9 then UN infinity is going to be that much higher when compared to UN not right so it would be okay if we extended this line even more like this particular vector curve that I drew is only about three times so this is exaggerated little bit more it is still not an exaggeration and so this should actually be your UT sorry UN infinity right and the UT not gets preserved as such when you now want to locate your U T infinity because that does not really suffer any expansion and then so what happens to the resultant so the resultant now obviously gets tilted more towards the normal of the flame all right so this is basically then your U infinity this is the picture you would like to see so what is it what essentially is happening it essentially means that if you now have a streamline that is coming like this it is going to get compressed right so if you have a streamline that layer that that goes like here it is going to go together but obviously streamlines can't hit each other right they try to turn and in trying to turn and accommodate each other they try to disturb the streamlines that are that are approaching the flame as well okay so you now get into a situation where your streamlines should actually turn out to be in reality somewhat like this it starts bending outward in anticipation of having to converge again further down right and correspondingly if you now look at this kind of a curvature which is what is pertinent for us in a Bunsen burner tip right so we expect that the flame should actually increase sorry the flow should actually increase or diverge and then converge back again I am sorry my domestic so look at what is going on here you have a situation where you are thinking that you want to have a flame that is stabilized at SL0 right now this is what is equal to you locally here that means you always this is very important I think I think I mentioned this earlier the way you always want to think about flames and I will reiterate this when we are dealing with triple flames after doing diffusion flames the way we always want to think about a flame propagation speed is always relative to the flow speed far upstream of the flame that means you don't want to worry about all this mess you have to say it all depends on it all boils down to how much do I have to open the valves for my flow to actually set in so that the flame is approximately stabilized what has been by approximators it is stationary there I can actually have a flame fixed coordinate system what it is doing within the flame fixed coordinate system is not of concern to me so long as it is there right and what is the flow speed at which it is there is what is the flame speed is what I would like to think about right so if I want to now say that this is what it is then what is the problem that is happening near the flame you now find that the flow is locally accelerated when you now have a flame that is curved concave relative to the far upstream reactants alright so the what has essentially happened is the flame has now induced the flow to go faster at it when it when it when it tries to have a concave curvature right and this means the flame needs to actually try to compensate for a higher level of increase in flame speed which should be by a mechanism that we talked about here if it has to survive if not it is going to actually cause a local extinction right so we see this actually happening in a in a in a Bunsen burner for example when you now talk about a streamline path that is approaching the flame you now know that know for a fact that you now have a streamline path that goes like this so you do have a a flow that is actually getting diverged for most part and then that continues to happen over here but locally it tries to actually squeeze in and then diverge so effectively what happens is you need to have to take into account the effect of flow divergence on the the planar flame speed so if you want to think about this effect you want to try to actually bring this into bring this into modifying the planar flame speed such that all this is like a black box and the if you now had a planar flame speed how is it going to be modified in a in a way that will take into account the local flow deceleration or acceleration all right so there is a way by which you can do this I am not going to derive this this little bit is quite beyond the scope of this particular course but we will just state these things so if you now assume constant properties constant properties the flame speed is modified as SL equals SL0 minus SL0 script L kappa plus script L n hat dot divergence or n hat dot gradient V vector dot n hat so here the SL0 is basically the unstretched flame it does not suffer from the flame curvature and flow divergence effects okay but SL is actually the flame speed that you would that you would have to use for a flame that is curved and also induces a flow divergence correspondingly and therefore alters the flame speed as well right so here kappa then is the flame curvature right so that is equal to divergence of n hat which is nothing but negative of divergence of grad G divided by mod grad G in the framework of the G equation and so of course you can now try to do a chain rule differentiation here so this is del square G divided by mod grad G plus grad mod grad G times grad G divided by mod grad G squared which is minus del square G plus n hat dot grad mod grad G divided by mod grad G keep in mind we are actually trying to find the flame shape okay and the flame shape is now contained in the flame speed okay so previously we were trying to find the flame shape based on the unstretched flame speed right but now we have to use a stretched flame speed which depends on the shape and the shape depends on the speed so you will have to do a iterative solution of this it is possible and there are solutions that can be made available for these things L here the script L is what is called as the Mark Steins length and recall I kept talking about a length scale that is of the order of the flame thickness but not exactly equal so it is somewhat greater gates is a little bit greater than the flame thickness because if you now have a flame curvature the heating effect is going to be felt a little bit further upstream than the original unstretched flame thickness okay so previously the unstretched flame thickness contained the preheat zone but now if you curve like let us say concave the heating is going to be felt further out right so that is kind of like the mark Mark Steins length we are talking about so it is a quantity of the order of the flame thickness flame thickness and of course we know that the flame thickness delta depends on the flame speed this we did long ago right so the first time we started talking about the structure of the premix flame the mass balance in the preheat zone resulted in something like this okay but here keep in mind this delta is actually for the stretched flame thickness the SL is actually the stretched flame speed okay and what we are now saying is this depends on L okay and so how does L depend on the delta so there is a derivation for this again which we would be if we won't go through but I will just state for large activation energy of of a one-step chemical reaction right so within the framework of assuming a one-step chemical reaction with the large activation energy for it L over delta script L over delta is 1 over gamma natural logarithm 1 minus sorry 1 over 1 minus gamma plus beta Lewis number minus 1 divided by 2 1 minus gamma divided by gamma integral 0 gamma divided by 1 minus gamma natural logarithm 1 plus x divided by x the x evidently x is a dummy variable of integration from 0 to gamma over 1 minus gamma gamma is now not the ratio of specific heats that you are used to in gas dynamics this is actually the ratio of temperatures which is basically a temperature difference divided by the initial temperature and beta is our Zeldovich number so it is EA times TF minus T not divided by Ru Tf squared so you can see that gamma is actually embedded in here you could say Tf minus T not divided by T not as gamma so E over Ru T or Ru Tf times gamma would be your beta now typically it sits found that typically L over delta is 2 to 6 so there are analysis so what what you what you will see see is now if you want to try to actually find a curved flame shape it is quite difficult it is more getting more involved your Mach Stain's length depends on the flame thickness the flame thickness depends on the flame speed the flame speed depends on the Mach Stain's length and so on so there is like a loop within a loop that is going on here which we have to solve iteratively there are analysis where we would just assume a constant Mach Stain's length let us not worry about so you pick a number between this because it is about a variable that varies within this range so let us not worry about it is the attitude for those analysis so then what happens as I said you see as the flame speed increases if you now go back to the tip of the Bunsen burner that that is what we were concerned with if you now keep on increasing the flow speed right as it is the flame speed in this case the flame speed for the unstitched flame is less than the flow speed okay and the flow speed or put another way the flow speed is actually greater than the unstretched flame speed and at the tip because you now have a concave curvature the flow tends to actually converge and actually increase faster the flow faster that means the flame flame speed has to increase much more to counter that and that would be through the thermal diffusive imbalance alright and that is obviously possible mainly when you have a lowest number greater than 1 but still since this effect is countering that even when you have a lowest number greater than 1 you could have conditions of local extension right so so as the as the as the flow rate is progressively increased right the flame shoulder adjusts its orientation but the tip can't the tip always has to stay as the tip it can't adjust to the adjust its orientation to the flow velocity therefore this results in local extinction regardless of of any whether it is whether it is greater than 1 or less than 1 and so on fine what next we talked about the Bunsen burner we realized that at the shoulder of the flame the the flow normal component of the flow velocity should match the flame speed for the flame propagating perpendicular to itself then we realized that that's not as simple a situation at the tip so we went through what we needed to go through to think about the tip and it we found that it is not so straightforward you have to worry about these effects two effects mainly the thermal diffusive imbalance and the density variation effect density change with temperature effect so all these things are involved in explaining what's going on at the tip but what about the base right so what's going on at the base when we did the G equation we made an assumption that over zeta should be equal to 0 at r equals capital R that was like a boundary condition that we had adopted okay basically saying that the flame is touching the rim of the burner of course within the framework of the G equation we don't bring into effect the thickness of the flame alright so it's as if like the flame is having a thickness and we are basically saying that the SL is a constant it doesn't change and therefore we don't have to also worry about a varying thickness of the flame near the near the base and or and or or a varying flame speed as we approach the base right but none of these things is true so we need to now focus a little bit more on what's happening near the base and see what is the fate of the flame speed as we approach the base as well as the flow isn't uniform either right so we tend to think that you have a uniform flow that's approaching the flame and you have a uniform flame a constant flame speed with which the flame is trying to attack the reactant flow and therefore you have a shape but that was not the case for the flow as well the flow wasn't uniform there it is coming out of a burner and it has to satisfy no slip boundary condition that means the flow velocity has to start from 0 right so we tend to begin to think wait a minute if you are going to have a 0 flow velocity there that means the flame should be able to propagate against the flow right there along the walls and never really be stabilized above the burner on the one hand it looks like the flame has to actually not be held at the burner at all and maybe get pushed up by the flow but if you now Bob begin to look at the flow right it looks like the flame has to go deep in what's going on right so let's so this idea of having to look at the base of the flame takes us to what the issue of what's called as flame stabilization flame stabilization so this is just refers to the fate of the flame base flame base or flame anchor point flame stabilization can also be referred to as flame anchoring because the base is the one that is actually holding the flame so correspondingly you can also use the term flame holding right so flame holding flame anchoring flame stabilization all these things refer to pretty much the same and it deals with the fate of the flame base because the base is where the flame is held right so we need to look at so what we what we just said was we need to look at SLU variation as you approach the burner rim approach the burner rim right so let's look at the flow velocity variation that's simpler so flow velocity you so what we expect is if you now have a burner we have a no-slip boundary condition all right and of course you could quibble saying we don't know exactly how the flow was formed in where and how is it developing as it reached a fully developed profile do we have an entrance length of associated with it and so on those things are not very very relevant okay so even in the in a worse case where you assume like a fully developed flow that is emerging out of the of the tube right and we are talking about laminar flames so we are looking at a laminar flow lamina fully developed flow that's coming out of the tube we expect a parabolic velocity profile but what matters to us is what's happening here right so this basically means that we are looking at now what we drew as a line has suddenly become not a thick block I am I am even exaggerating and going further into saying that this is almost like a semi semi semi infinite solid filling one quadrant and with the corner here corresponding to that burner rim and the flow velocity is locally linear that's your you so when we said that we wanted to look at the flame speed and the flow velocity variation as you approach the burner rim we picked the flow velocity because the flow velocity is always going to be somewhat linearly increasing from the rim inwards towards the center of the tube so that is more like a monotonic variation you increase the flow speed the slope is going to increase if you decrease the flow speed the slope is going to decrease that's all right so there is no there are no counter effects or competing effects here but that's not the problem that's not the case with the flame speed there are competing effects that we have to worry about for how the flame speed changes as it approaches the rim so it's not so straightforward so we consider that next so how does the flame speed behave right there are two factors in fact to two contravening factors that we have to think about one first and foremost as a matter of fact is the heat loss to the burner heat loss to the burner so when you now have a flame that is approaching the burner so think about what's happening if you now have a burner rim that's look looking like this when you are now trying to zoom in into the burner so much right you can't simply draw a flame like a single line anymore pretending that that is going to now consume contain both your preheat zone and the reaction zone and all those things right you now have to worry about how does the preheat zone look like how does a reaction zone look like what I know for a fact is far away from this burner rim I am going to have a certain thickness for the flame and the preheat zone is going to be thicker than the reaction zone larger the activation energy okay but what happens as you now progressively come closer to the rim is keep in mind that this is actually the region where the temperature actually rises from the initial temperature to the flame temperature as we go along and this is the region where most of the conduction is happening but as you now come down to the flame with the burner rim you now have a heat conduction that's going on inside the rim as well so it's suddenly like the the burner rim intruded into the preheat zone because the preheat zone essentially is part of where the conduction happens right but the preheat zone was supposed to actually heat up preheat right so preheat means it's supposed to heat up the reactants to react so there is no point in heating up the burner rim because the burner rim is not hopefully going to react right it's quite of like the joke about whether we have so how does the how does the combustor work the combustor burns it's not good news because the combustor burns means the combustor is not going to be there so similarly you don't want to have the burner be part of the reactions right so what this but what then happens is you know essentially you are thinking about a thickening progressively thickening preheat zone and the fact that we are talking about a heat loss is because the reactants are not the only ones that are getting heated up okay in an adiabatic flame there is a heat transfer going on it's not it's not as if like there is no heat transfer going on in an adiabatic flame but it is kind of like you know the father gives us money to the son and the son gives the money to the grandson and so on it's there the money is kept within the family right it's not going going away so it's like the reactants are of the ones that are actually getting the heat and reacting in the flame so so long as the reactants are the ones that are going to get all the heat and then react in the flame to release that heat it's an adiabatic system but the moment the reactants are not the ones that are going to get heated up anymore that's not an adiabatic system so obviously then the local flame speed should decrease right so the the flame essentially becomes a lot thicker and and the flow the flame speed progressively decreases to towards the towards the as the flame is held closer and closer so it's essentially the heat loss to the burner basically means that closer the flame base is located to the burner room right more the heat loss and SL decreases more right the decrease in the SL is more what we are basically thinking is as you now have a burner and you're now trying to increase the flame speed we are beginning to imagine that the base is not exactly touching the room anymore the base of the flame is now going to go up a little bit and try to adjust its position why would it adjust is something that we're going to see pretty soon right but when it with the moment we now begin to think that the flame is not going to be at the rim and and can move up or down relative to the rim what we're basically saying is closer it is to the rim more is the heat loss and therefore the flame speed near the rim is going to be progressively less so this is a a contribution that is going to decrease the flame speed okay the second decrease the flame speed more and more when your the flame is closer to the rim second is the mixing with the ambient mixing with the ambient right now for the purpose of what we are talking about let's fix the ambient as air right air as in what we find on earth here right that means you now have the oxidizing ambience right and then think about what happens as you as you now allow for mixing to happen so essentially what this means is you now have more and more standoff of the flame let's say the flame is now further out okay so if you now have a flame that is progressively further out and for this purpose it's sufficient for us to just draw a line to represent the flame let's not worry about its thickness right it is thicker and so on that's fine but what you're looking at is this distance more this distance greater is the mixing of the reactants with the ambient right what's the consequence of this for the flame speed that's really the question that we have to ask that depends on whether the flame is a fuel rich flame or a fuel lean flame right so if it were a fuel rich flame right so great so let's just record this greater the standoff distance right more is the mixing and the question is is that mixing going to be increasing the flame speed or decreasing the flame speed right so that depends on whether the flame is going to be fuel rich or fuel lean so fuel rich flame in an oxidizing ambience what happens now this region actually becomes closer to stoichiometric right because it is a fuel lean fuel rich flame and then you're now sending in more oxygen from the side and therefore it becomes closer to stoichiometric so this leads to a increase in flame speed right with increasing standoff and this goes in the same direction as what the heat loss effect was in the case of the heat loss what we understood was the closer the flame is to the burner lesser it is a lesser it is it lesser is its flame speed or conversely the farther it is from the burner it's going to have a higher flame speed because of less heat loss is exactly what we're saying here increase in the same flame standoff means a increase in the flame speed because it's fuel rich so in this case this is a little bit more complicated and we will talk about this later on what's important for us to is to look at a competing effect that means an opposing effect right and that is presented in the case of a fuel lean flame so in the case of a fuel lean flame your already fuel lean and a greater mixing with the ambient is going to dilute the flame further and further right that means the flame speed is going to decrease as the flame standoff distance increases the heat loss was less so the flame speed doesn't have to decrease that much more but it's going to decrease because of the dilution right so dilution decreases the SL with increase in standoff right with these two we should now be able to tell how the flame is stabilized we will do this now and we will worry about pushing the limits on this to lead to a flash back or a blow off later so essentially what happens is if you now have a let's now put these two together that means originally if you now had a flow velocity that was locally in linear right and I'm on let me now consider the flame the flow like this right that's a flow velocity you what we are looking for is if you now have a flame that is like this let's look at how its flame speed is going to be right as it is over here with respect to like the free stream velocity the flame speed is low but the flow speed is actually decreasing linearly and the flame speed is decreasing because of heat loss on the one hand as its proximity to the burner is determined and because of the mixing and we are now considering a fuel lean case the fuel lean case it is a more straightforward for us to think about so there is a position there is there is a condition when maybe the scales are in good here so let's consider the scale that looks like that so there is a condition when the local local velocity and the local flame speed match exactly where you have this curvature facing the flow right so this is where you now have a u equals SL I am not saying u bar equals SL not or you not equals SL not okay I am saying you is equal to SL so this is the stabilization condition right now what happens when you now try to actually have a flame that is perturbed inward is it now gets closer to the flame right so its SL actually becomes less than you because of a greater heat loss effect all right but the mixing is not much so it is not trying to decrease the SL a lot but the heat loss is actually trying to decrease the SL and therefore the flame gets pushed backwards to this point but if you now try to actually have a flame that is pushed further out it now becomes a lot more extended you can see how I am drawing these pictures you now see that it is actually stopping somewhere here it's stopping here it can extend further because your mixing fan is more you now have the reactants reaching a greater distance laterally outward right so you can you can have a flame over here but in this case the SL actually now becomes greater than you because you don't have too much of a heat loss effect and therefore it has a tendency to move upstream you now have a larger flame because if the reactants have actually fanned out and they have less heat loss and they have a tendency to move against a linearly decreasing velocity until you reach a matching point okay so effectively it now becomes a local match between a varying flame speed and a varying velocity the flame speed is not uniformly varying the the velocity is linearly varying and this match is what is going to actually dictate how the flame is stabilized locally there and therefore correspondingly a flame standoff distance stop here