 So we have been looking at how this flame stabilization happens and essentially what basically means is you have a linearly increasing velocity profile but you also now have a flame speed that continues to decrease from the value of SL0 and it decreases because of the heat loss to the burner as one of the effects and the other effect is the dilution from the ambient for like let us say we are looking the fuel rich sorry I am sorry fuel lean flame and therefore if you now have a air that is coming in and diluting this it is going to decrease the SL further and you have to keep in mind that as you now get closer to the burner the decrease in the flame speed is going to be more because of the heat loss and less because of the dilution as you go further away from the burner if the flame goes further away from the burner the decrease in the SL is going to be more due to dilution and less due to heat loss right and that that is actually the effect along this direction the other way to look at it is also the radial direction as you now go closer to the as the flame stands closer to the burner the heat loss is more and the mixing field is actually quite narrow between the premix mix reactants and the diluting ambient gas whereas as you go further out you can now actually have the premix mixture going further out into the into the mixing zone and the flame can actually establish itself further and further out and what we are basically looking for is like a leading edge of the flame which is having a slow perpendicular to the flow where the SL has to match the velocity directly and what we are basically looking for is a condition where the SL has to the local SL has to match the local you since the you is also increasing linearly what we should be looking for is a condition where the local SL matches the local you and so the flame stabilization condition is locally we need to have this matching to be satisfied then what we look for is to two occurrences what happens when this condition cannot be satisfied in two ways one if you now progressively keep on decreasing your flow velocity right or progressively keep on increasing the reactivity may be by changing the mixture mixture stoichiometry towards the stoichiometric mixture so that the flame speed keeps increasing for the same flow velocity so in one one case what we are looking for is a relative decrease of flow velocity with respect to the flame speed or relative increase of the flame speed with respect to the flow velocity in which case the flame has a progressive tendency towards establishing itself closer to the burner and ultimately you are going to get to a situation where it cannot be held at the burner room anymore it wants to propagate even further in and that is actually situation corresponding to flashback the second situation is where we now do the opposite we just we now try to progressively increase the velocity relative to the flame speed or progressively decrease the flame speed by maybe going to leaner and leaner mixtures for the same flow velocity and that is going to have a tendency for the flame to actually get established further and further out stabilized get further and further out and at some point it cannot be stabilized anymore and then it starts blowing blowing off right so you have two such limits of flame stabilization two limits of flame stabilization or one is flashback the other one is blow off right so here what is happening is progressively we say progressively for a reason that I will explain soon SL becomes less than UL and here progressively U becomes greater than SL right so let us look at the flashback situation so what we are saying here is let us suppose that you had a you had a velocity contour that is a velocity profile that is going like this with respect to x which is r-r that is the x coordinate that is measured radially inward from the burner rim and what we want to think about is a match between a limit limit match between you so this is you and SL so the SL basically you see what is going on the U increases linearly and the SL for most part of the radius of the burner is nearly constant at SL not and then it drastically decreases within a within a short region and then goes to 0 ultimately when you when you when you're needing when you're reaching the what's called as a flame edge right so we are looking for a match that would that would give rise to a situation when we now have U that is at the most equal to SL now what happens is if you know if you now try to decrease your velocity you are now going to not be able to actually satisfy this match anymore and you're you're you're now going to have a condition where that is supposed to SL and now going to have a condition where the U decreases and that leads to your flashback right on the other hand if you now try to actually have a higher velocity then this is actually the condition for flame stabilization at the rim at the most and any decrease in velocity is now going to actually induce a flashback but if you now try to increase the velocity further it's now going to progressively starting to have a flame stabilization little bit away from the flame effort from the burner right and so increase so decrease in velocity below the value for the U slope to be tendential to SL leads to flashback any increase in velocity on the other hand leads to flame stabilization above the burner rim that means to say at this particular condition the flame is stabilized at the rim and that's the most you can do to prevent flashback then what happens if you keep on increasing the velocity further and further right if you keep on increasing the velocity further and further we have seen what's what is likely to happen there the flame shapes itself in a way to adjust to the increasing velocity situation all right we have found that with a artificially simplified condition that zeta is equal to 0 at r equals capital R you could now say that the flame is going to be a conical shape of any angle depending upon the SL over U ratio so for a fairly wide range of use given a certain SL you can you can actually have a flame that is held pretty much near the rim without worrying about the details of how it is attached to the rim if you have to worry about it then you start seeing that it starts moving a little bit further away the edge of the flame starts moving away from the rim a little bit and progressively so how does it really adjust so if you now think about the situation there so this is your U so and this is like the jet boundary and let us suppose that we now look at three locations where the flame is stabilized progressively so you can say the flame is stabilized here here and here of course when you when you go further and farther the flame gets curved like the way we are talking about but what you talk what what you basically saying is let us look at these three conditions the way the adjustment happens is at a lower location you now have intense heat loss so the flame edge is actually more towards the the center when compared to the other way when you now have less heat loss and further location the flame can be prevalent more closer to the rim when come I mean radially when compared to if it is much closer to the rim axially therefore what happens is for the condition a your SL actually is 0 for quite a while and then increases and then you now have a flame stabilization condition that matches matches this if you now try to increase your velocity to a let us say a higher value the linearly increasing profile would increase this way and the flame now begins to actually so the flame speed now begins to increase in such a way that you now have the match happening over here what this basically means is of course looking at how this curve looks like maybe you should have drawn it with a slightly greater slope than what or lesser slope than what I have done but essentially what this means is when you now increase your velocity the flame is getting pushed up therefore the heat loss is less so what it basically means is if you now think about a situ a distance going a little bit away from the rim right you can see that the flame speed is beginning to rise sooner than the previous case and the reason for that was a decreased heat loss situation all right and then you could still continue to have that happen if you now think about another velocity where we are now increased but then you are really getting pushed to the corner beyond a point as you now get the flames flame speed to increase because of less heat loss you are progressively so this is actually a this is B this is C progressively you are going further and further away and then the other mechanism of the dilution trying to decrease the flame speed is going to come in so the rate of increase of this flame cannot actually happen so soon from the rim and there is a cap on how much it can increase that is the SL knot so beyond this point you cannot really have a further increase in velocity admitted by the flame right and this is a mechanism of destabilization of the flame leading to a blow off condition right so effectively the flame speed because of these two mechanisms of heat loss and dilution they combine together to adjust in order to match the flow speed locally as it is being increased over a fairly wide range but not beyond a point right and that is when you are going to have a blow off so in effect the flame stabilization condition the flame stabilization condition is that you need to have a SL equals SL equals U at G equals 0 minus that is to say just upstream of the flame the notation used previously and if you use now let us say say y is the distance the distance of the flame base above the burner then partial SL over partial U sorry partial y now should be greater than partial U over partial y G equals 0 minus again just upstream right so this is typically the condition that we are looking for when we have an equality so when dou SL over dou y equals dou U over dou y G equals 0 minus this leads to a blow off right what I would like to point out is one thing here blow off is not the same as extinction of the flame let us see what is going on when you are when you are having a blow off when you are having a blow off the SL is actually more quickly rising to its high value closer to the flame base okay so the flame base is shifted over to this point and the SL as we now go from the flame base sorry flame sorry burner room you are now having a more drastic increase in the SL this basically shows that when you are now getting closer to a blow off condition which is corresponding to the curve C the SL is actually pretty high and you are having a fairly intense burning of the flame base at blow off okay so blow off is a situation of intense burning of the flame base at blow off extinction is the opposite of this extinction is where you are progressively starving the flame of reactant mixture right so effectively the reactivity is coming down further and further and since you now have a heat loss situation and the dilution situation that further decreases the SL at the flame base so the flame base is the place where you are going to actually have the biggest hit on the intensity of burning for a for a extinction situation where on the whole the flame is being stopped of reactants and the reactivity is coming down the burning intensity is coming down it is going to actually be the worst at the flame base because of the heat loss and the dilution so burning so should use the right word there so opposite effect least intense flame base during extinction so putting these two together what we can see is there is a role for the velocity gradient to do in this flame stabilization so one way of looking at this is I look at it in both ways one is to actually look at the velocity itself because you have a condition here another way is to actually look at the velocity gradient you can look at it in both ways they give you slightly different pictures but not very different okay so they convey the same idea one you have a limit like this is this percentage fuel and let us suppose that this is d du star over dy where du star over dy corresponds to velocity gradient what at a flashback or blow off and so obviously we are looking at two limits one for flashback and another for for blow off so this is this is the flashback curve right and this is the blow off curve and in between the two you are going to have a stable flame right and see what is happening here the flashback curve is non-monotonic so it is it is increasing and then going back again that means you should now be able to actually look at two points for the same gradient and this is now a different percentage fuel at these two points right that simply means that you could rest you could restabilize the flame from a flashback situation either by going towards this region or that region flashback is all about having a higher flame speed when compared to the flow so if you now are in a below a certain velocity gradient for the flow then if you are now within this stoichiometric range you are going to have a higher flame speed and therefore you are going to flashback so you now enter a unstable region so you now can restabilize by either going in the direction of fuel rich or fuel lean because on both sides the flame flame speed is going to come down right so it is possible for us to restabilize either by going the fuel rich direction or the fuel rate fuel lean direction that is not the case with blow off the blow off does not look like a a non-monotonic curve it does not really curve back right so it is it keeps increasing what that basically means is as you are going richer and richer the flame is going to be relatively indefinitely stable against blow off that means it is not going to blow off it is only the lean flames there are actually susceptible to this why is it the case naturally because of what we pointed out last last time the dilution from the oxidizing ambience if you now have a fuel rich flame and the flame is getting further and further lifted away from the room because of a higher and higher velocity and a higher and higher velocity gradient consequently then you are going to have a mixing coming from the oxidizing ambience which is going to cause a locally towards stoichiometric kind of mixture and therefore the flame the burning is going to be more and more intense and therefore it is not really going to try to blow off you see so what this means is blow off is progressively difficult with the increasing fuel concentration right and because burning intensity burning intensity increases should particularly say at the flame base at the flame base increases there is another way of looking at this of course the question that you have to ask immediately is what if I actually burn this in an inert environment is it okay if I now have like let us say a nitrogen ambience in which I have a reactant mixture that is coming out through the burner right what happens then I am going to allow for a nitrogen to dilute through the sides regardless of whether it is fuel rich or fuel lean should I expect the blow off to actually be happening for fuel rich mixtures as well yes sure enough correct so there is a there is a different way by which we can look at that which is a broader stability map so to speak so here we just look at it in terms of velocity directly instead of velocity gradient the picture does not change significantly at least in a qualitative way so if you now plot fuel concentration on the x axis or the horizontal axis as opposed to the vertical axis the period earlier then let us now say we are somewhere in the stoichiometric somewhere here right that means all this is fuel rich then our flashback curve would be somewhere there right and maybe I should locate this a little bit further out relative to this curve and then you have a blow off curve that keeps going like that that is for an air ambience or an oxidizing ambience but if you had an inert ambience you will have a curve that goes back like this so this is a blow off in inert ambience so it is sort of like reproducing this picture but then I would like to point out one more thing which is very which is beginning to look very interesting for us now and I would imagine you should stop thinking about it right away we are now beginning to look at a space which is significantly fuel rich right so if you are somewhere there what you should be looking at is you are now having like a lot of fuel that is coming in and only a little bit of oxidizer that is coming along with it right in and oxidizing ambience right what if I did not have any oxidizer at all here it is how it is highly fuel rich to the point of being only fuel in an oxidizing ambient what would I get right I am now progressively departing from a premixed flame so it is not a premixed flame anymore it is actually now beginning to look like close to a diffusion flame all right so what was happening all the while when you still had some oxidizer and we really had a premixed fuel rich mixture okay it could be flammable why we are not talking about getting past the flammability limit even right so you could have a fuel rich premixed flame all right but what are the products going to be the products are going to be fuel rich right they are still going to have some fuel that is remaining which would can mix with the air downstream of the flame not just at the base not not only at the base not only through the gap between the base and the burner rim what we have been talking about is this mixing effect that is happening between the flame base and the rim so as to focus on the flame stabilization but on the whole you are going to have a excess fuel that is coming out of the flame which would mix with the ambient and form a diffusion flame envelope that is around around the premixed flame this is typical and so long as you have this diffusion flame envelope it is lot harder to destabilize this flame because the fusion flame is going to actually be there with the fuel and oxidizer or the excess fuel and oxidizer oxidizer mixing all the time and it does not really have to propagate against a make a flowing mixture a flowing reactant mixture therefore the stability seems to be significantly protected for us in the case of fuel rich flames that is that is what is essentially is happening right however there is another interesting thing that begins to happen which is you now have to worry about something called a lifted flame so somewhere somewhere in here you now have another envelope that is coming up in the stability diagram where you now have a lifted flame as opposed to which what we have been talking about so far in this region as well as here is what is called as an attached flame that is to say whatever we have been talking about as a flame standoff here is still corresponding to what is called as an attached flame we are magnifying the region in the vicinity of the burner rim corner particularly and trying to decipher this standoff but this is still an attached flame what we mean by a delifted flame is when now the flame gets significantly lifted away from the burner because now it can get very very reactive as you now have a fuel rich mixture that is coming in and mixing with the ambient the flame can actually get significantly reactive right so it can get stabilized further out away completely away from the rim not even not anywhere near the vicinity of the rim so that is actually a lifted flame and it has its envelope as well and this is actually the blow out of the blow out of the lifted flame this means this is this is an edge where the lifted flame also blows out so that is distinguished between blow off and blow out blow off is a terminology that is used for a attached flame and blow out is for the lifted flame to just go out okay and this curve then becomes basically the lift transition from an attached flame to a lifted flame this curve becomes like a transition that that causes the lifting so it means if you now are fixing your fuel concentration and kept on increasing your velocity you should now lift at this particular point right and interestingly this curve corresponds to what is now called to say drop off or drop back what does that mean if you now have a difference between these two right what that basically tells you is if I now have a burner and I keep on increasing my velocity and I now have the attached flame trying to desperately hold on to the burner by giving way it is called of like elastic thing right so that the flame is trying to actually get pushed a little bit further and further but still attached and then it lifts okay and then it is actually burning there and you are like oops let me just bring bring back the velocity so that it can attach now it won't it would stay lifted until you decrease the velocity significantly lower than the point when you cause this lifting to happen that is what is called as a hysteresis right so what is going to happen now at that velocity is not going to actually cause a drop back right there we need to come down to a much lower velocity for the drop back to happen because now the flame is found a new home where it is got stabilized and it is beginning to get comfortable there it is going to stay there even if the velocity were lower and would allow a attached flame to happen so the question of whether you are going to have a lifted flame or an attached flame now begins to depend on which way are you approaching in your velocity direction for example right if you are now going in the increasing direction of velocity you would stay attached until you until you you went there if you had a lifted flame to start with at a very high velocity and you keep on decreasing you will stay lifted until you get to the drop back limit so this is a region corresponding to both lifted and attached flames so attached or lifted right and so if you want to actually shade this somewhat you see this region is now going to take shadings from both and is a common point for both would like to point out one thing that we have not talked in detail at all or as a matter of fact not not considered when we did the tip right we talked about flame stretch effects which is to take into account the flame curvature and flow divergence effects when we talked about the flame base where the stabilization happens we have not really bothered about that at all okay so considering flame stretch effects on flame stabilization alters the concept somewhat so we could do this we could do this we know how the SL changes with flame curvature and flow divergence as we have seen yesterday and we could take into account a diluting environment and a progressively changing field concentration or reactant concentration effect on SL so you can you can take into account a spatially varying SL because of spatially varying concentrations of reactants and all these things can be taken into account in like a computation right in a in a in a way we want to calculate this and take this into account and that changes things somewhat the other thing that I would like to point out is what we have been primarily talking about is flame stabilization over a burner rim right but in general we could be talking about flame propagation in a duct so for example when you now think about like a flashback it will say okay fine let it flashback and see what happens right so how does the flame propagate further down into the duct so there are some things that I would like to just say about flame propagation through the duct before we get to the next step right now here what we what we have to think about is how is the duct okay so far we have been thinking about like a Bunsen burner where typically we are looking at a Bunsen burner that is vertical and you now have a flame that is stabilized about the rim of the Bunsen burner and then the products go out up right now implicitly in our minds we have been adopting a gravitational field that's acting on these things and therefore there is a effect of buoyancy on these on these gases where the lighter gases would actually go up and the and the denser gases would stay down and the lighter gases are made lighter because of a higher temperature mainly for the products when compared to the colder gases being the reactants and therefore denser and so long as you now have a flame that is demarcating denser gases being down here and the lighter gases being above that is where they belong so nothing nothing is difficult or different right so what we would like to think about is a horizontal duct for example and you now may fill this with a reactant mixture that is supposed methane methane and methane air right keep it open on both sides for your safety you don't want to have transition to detonation here right so you now ignite from one end and now allow for the flame to propagate all right of course there are lots of nice interesting studies about flame propagations and tubes that are progressively of smaller dimensions and then you will find how the flame convolutes and so on right and you see that the gas gases behind have to flow away from the flame for a deflagration and then they begin to actually have a velocity profile as the flame propagates and all those things okay so but that is because of no slip boundary condition all this is pretty interesting so now talk about flame propagation in a duct rather than being stabilized on the other rim of a rim outside a duct that is vertical so far what we have been imagining is like having a burner and then having a flame that is stabilized above it but implicitly in our minds we have a gravitational field that is acting on this and you see what is going on you have a gravitational field that is acting downwards and you now have a flame reactants are coming up here and products are going up there and this gravitational field is now going to actually impose a buoyancy effect in which the lighter gases are going to go upwards and the denser gases are going to stay downwards and lighter gases are made lighter because of a higher temperature which is mainly for the products and denser gases are actually for the cold reactants and in a flame that we have that is stabilized the cold reactants are below and the hot products are above and then you have a fairly stationary flame but that is not necessarily the case all the time so the way to actually think about this to begin with is start looking at flame propagation in a duct which is horizontal so we now consider a duct which is horizontal and fix it up and fill it up with a reactant mixture let us say methane and air and you now ignite it at a 1n and keep the ducts to ends open so that you do not get into transition to detonation and those kinds of issues then you expect a flame propagation to happen right and there are lots of nice studies about how this flame propagation happens in this tube and for progressively smaller tubes how the flame actually curves because the float curves the flow attains a velocity profile because of the no slip boundary condition and this is actually the flow that is moving away from the flame and so on those are all nice studies that we will not really get into the question that we are asking now is if I ignite at this end you have a flame propagation from left to right if I ignited it from the other end would the would the scenario be different except that the flame propagation is going to be from right to left in a gravitational field that is acting only one way we do not expect a change on the other hand if you now consider a vertical duct and you are filled up with a methane air mixture and you now ignite it at the top and you now have a flame propagation that is happening downward from the top you see you have a cold reactant mixture at the bottom and the hot products are going to be at the top and they are going to go they are going to flow up right convectively they are they are flowing up the buoyancy driving is also going to make them flow up right and then you now have a propagation that happens further down and you get a certain flame speed but what happens when you now have the same vertical duct and you take the same mixture that is in it but ignite it from the bottom you now begin to have a flame that is actually beginning to propagate upward the reactants are cold and they are above and the products are hot and buoyancy is actually trying to drive them to go up against the deflagration expecting deflagration causing an expansion that is making it flow down right so if you now look at its momentum balance the product gas is now going to have a tendency to flow through the flame upwards and contaminate or dilute the reactant gases and therefore you are now going sorry yeah that is right therefore the flame speed that you are going to get is different on top of this we will also have to factor in the fate of the thermal diffusive instability that is like perturbation to a planar flame and how much heat that is going to get conducted versus how much reactant gases are going to get diffused right versus now product diffusion as well and so on on the one hand and also the hydrodynamic instability of what is the fate of a flame a flame that is perturbed from a planar condition with hot gases on one side and cold gases on the other side with the density jump across this you now have an instability because of that all these things eventually lead to the formation of things like cellular flames that means you do not really no longer have a flat flame at all the flame gets highly wrinkled and then you get into what is called a cellular instability these are things that are not going to be covered at all but I wanted to point out a very interesting phenomenon here where you just have to take a duct and then hold it horizontally and ignited at either end or hold it vertically and ignited it where either end you are going to get significantly different results point out something about the flammability limits is again like a informative thing rather than an analytical thing. So we had this kind of a curve right big right at the beginning to look at the dependence of SL on physical and chemical parameters and what we found was the flame speed actually increases and decreases about a point of 5 equal to 1 the equivalence ratio one slightly picking towards fuel rich side and in fact this is actually something that could be obtained by doing measurements on a Bunsen burner at the shoulder of the flame to look at what its angle is with respect to the incoming flow or you could actually look at a glass tube apparatus measured setup like this and then ignited at this end and then see what happens or you could also have a spherically propagating flame there are many apparatus that are used for measuring flame speed and each of the each of them has its own uncertainties because of additional effects that are not considered typically in the flame speed x from for example in a spherical flame you have a flame curvature effect that is coming into picture that is altering the flame the flame speed measurement and so on. Now what we are talking about here in the context of flammability limits is something that we found the other day that is we have the flame speed decrease on either side and from the maximum and then stop that means this does not really go to 0 you do not have any flame at all propagating on either side here and this is what is called as a fuel lean limit and the this is what is called as a fuel rich limit and at these limits you have a significant role for chemical kinetics coming in what is the effect of chemical what are the set of reactions that are happening and you are now starving the system of one set of reactants right or one reactant to start with and then corresponding set of radicals that are there okay so the chemistry can significantly pay a role play a role over here and for this reason the the fuel the the flammability limits or somewhat empirical I mean the knowledge on flammability limits is quite empirical even to this day you have to significantly work on the theory to be able to explain this correctly there are some theories which do a good job like for example a heat loss theory where you now basically say that there is lot of heat loss to the surroundings when you are now looking at these limits and so on and keep in mind what I meant by heat loss yesterday which is heat loss to non participating reactants right so you could have that situation but we are not going to get into all those details I am just going to factually state that you have these limits and what does it depend on see effectively the controls that we have in any any experiment or the temperature and pressure the pressure is kept constant and we are talking about reactant temperatures how do we actually change these with the reactant temperatures the effect of temperature is lot more straightforward you typically have a increase in the envelope of the flammability when you now increase your initial temperature Ti and the flammability limits expand linearly expand linearly with initial temperature with initial temperature and this linear so that means you can think about like a coefficient in the sense what is the rate at which the fuel lean limit expands as you increase the temperature and how does the fuel rich limit expand as you increase increase the temperature per unit degree for example right so what you will find is the slope if you now try to actually look at how the slope of the limit change is with temperature you will find that the coefficient or the slope for this relationship is higher for the fuel rich limit that means the fuel rich limit changes most deeply with with with increase in temperature right so that is to say basically you know that this gap should be quite less when compared to that gap although this picture does not reflect that significantly okay and keep in mind when we say expand we expect the fuel lean limit to decrease in the field rich limit to increase in Phi the corresponding Phi should actually decrease for lean and increase for rich okay so if you if you are asked to do some kind of calculations given what is this linear dependence you should factor in that lean is going to decrease and which is going to increase to the pressure dependence so pressure dependence is more significant than the temperature dependence usually you find a dependence that sort of looks like this if you now have a P versus let us say Phi you have a relatively weaker dependence relatively weaker for the rich for the lean limit I am sorry still it is sort of linear okay so over a range of pressures it is weakly linear and then almost remains constant over a smaller range if you think about it and then flattens out beyond it says at a certain low pressure beyond which you just have no flammability at all but then it is more sensitive at the rich limit so this is the rich limit and this is the lean limit if you now look at how this works this is actually controlled by like a second-order mechanism at low pressure and this is controlled by a first-order mechanism at high pressure high pressure or low pressure could be six subs substantially on either side of atmospheric right so we are no looking at tens of bars over here this is sub atmospheric pressure over here okay and one more thing I would like to point out in these kinds of things is the duct diameter plays a role plays a role in determining the flammability limits typically for small ducts so if you now look at small ducts as you know as now as this diameter actually decreases then you have you get into two problems one you could actually extinguish rapidly or or at a or within the limit of what was actually deduced with the higher diameter or harder to ignite as you now go to smaller and smaller diameters may not be even be able to ignite the mixture right and this is typically true at lower and lower pressures so you now have a fairly good sensitivity of the flammability limits to that diameter at lower pressure essentially because of the difficulty to ignite or even if you ignite you could have an extinction happen within a within a narrow range of equivalence ratios or giving a fairly narrow range of flammability limits correspondingly and I would like to point out that this this leads to something called quenching which we will discuss next.