 So, the last class we were looking at the effect of pressure on the flame speed and what we found was for a nth order reaction for let us say a global reaction the flame speed goes as p to the n over 2 n-2 over 2 and this basically gives a weak pressure dependence for typical hydrocarbon oxidation combustion reactions with a order close to 2 a global order to close to 2 so for typically second order reactions then there are lots of other parameters that we have to think about or variables course when we talk about pressure the next thing we obviously think about is the effect of initial temperature so or rather than saying initial temperature as if it is an unsteady problem and in a steady state situation we should be thinking about like a effectively temperatures of the unburnt reactants or far upstream reactants so temperature effective temperature reactant temperature T0 and of course these things are more like information so you can actually gather these things from most textbooks we will just go through this rather quickly to highlight what the most important aspects are and it is also important to think about this in physical terms other than just to look at explanations so what we are saying here is as far as T0 is concerned okay the gross effect is like if you now say SL does goes that goes as T0 to the M let us say where M is around 1.5 to 2 so if you think about this if you now look at SL versus T0 you get graphs that go kind of like that right so why is this this is mainly a preheating effect okay see because what you are thinking about is we have the reactants that are approaching the flame in a flame fixed coordinate system and then you have the preheat zone and the temperature has to now raise from there up to the reaction temperature in the reaction zone so this this effect is primarily a preheating effect all right so if you were to be thinking about like a PHT qualifiers question the question will not be what happens to the flame speed with the reactant temperature the question would be what happens to the flame thickness with reaction time the reactant temperature right so we will we will ask this question differently than what is normally done in the class so that is for you to think about okay so we say this is essentially a preheating effect now the other thing that you naturally have to think about when you are looking at the effect of anything on flame speed is what the effect on the reaction rate because you know you see that the flame speed is directly proportional to square root of the reaction rate right so as a matter of fact we got this n-2 over 2 for the pressure because the reaction rate was going a speed of the n and then since that the flame speed is proportional to square root of w you have n over 2 and then of course you had a 1 over density in it in as a factor out so which was linearly in pressure therefore you had this n over 2-1 that is how you got this so the first thing that you have to think about is the reaction rate what is the effect of whatever parameter you are thinking about on the reaction rate and therefore on the flame speed this is the first step approach the next thing that you have to think about is what is the reaction rate itself depend on and it primarily depends on the flame temperature right so what is the effect of anything on the flame temperature and through that on the reaction rate and through that on the flame speed this is how we have to look at this this relationship that is very important most of the time but in this case t0 of course affects tf directly and then tf affects w and w affects s l but in this case you know the tf effect is significant right when compared to the tf effect the the influence the t0 has on the tf and therefore on the s l is is marginal this is so this this rise is primarily directly because of the preheatings effect and much less through the effect through tf so the the the influence through tf is is less here if you now look at how the tf itself affects s l that is the next thing that we should look at effect of flame temperature tf on s l of course is significant so here what you will find is these curves are much steeper so if you now look at tf this is of course like we are essentially looking at different mixtures having different tf's and what what happens to the s l corresponding to that keeping lots of other things constant so this is a bit difficult to actually think about this experiment but if you now think about this this this this curve is much steeper when compared to this okay so a slight change in tf can cause a significant change in s l through through w that is a reaction rate so now if you if you go back and think about this the first thing that we did was when we wanted to look at the structure of the premix flame we said the reactants get in and then the temperature rises and then goes to the reaction zone but if you want to just do the gross order of magnitude balance you would say the the reaction rate is evaluated at the flame temperature right we do not have to worry about the variation of the reaction rate with temperature we will just evaluate the reaction rate at the flame temperature directly and this justifies that because as tf changes w changes and as w changes s l changes there is a direct link between these these things so we can we can see this justified experimentally then the most important thing that we should be looking at most of the time effect of mixture ratio so when you say mixture we are always looking at the reactant mixture okay in the context of preheat sorry premix flames we are always thinking about the reactant mixture that means the fuel oxidizer mixture right so here what happens is s l versus let us say equivalence ratio and of course you can think about equivalence ratio 1 as stoichiometric and then less than 1 is fuel lean greater than 1 is fuel rich you should typically look forward to curves that are like that with the peak a little bit off in the fuel rich region okay so peak nearly at stoichiometric condition or slightly fuel rich right now just to also give you some numbers we are talking about laminar flame speed so we are looking at these things measured typically in centimeters per second and these are about let us say 10 20 30 40 these numbers would probably work for methane lots of other fuels will give you these kinds of curves at much higher values maybe not much higher maybe twice as much or maybe three times as much or so around around that so we are talking about a few to several tens of meters per second sorry if you few to several tens of centimeters per second for the laminar flame speed okay so this is something that you got to keep in your mind and you know to get a feel for things when you do let us say experiments or even numerical work and so on now the other thing that we have to worry about is why are we having a peak that is slightly shifted off the reason for this is actually again through TF so look at how the variation happens for the TF with respect to equivalence ratio right you will find that it picks up near the stoichiometric condition or maybe slightly fuel rich again and that is because of the CP effect that is the CP if you now look at how the CP of the mixture varies with Phi right you will you will find that the CP affects the TF and then on top of it think about this TF is actually sitting in W and on top of and then there is an explicit CP dependence directly on for SL okay so the CP influences TF and CP directly also influences SL therefore you will find a further slide shift into the fuel rich region because of this region this is the reason so this is again again mimicking mimicking TF trend and a further CP effect we will look at the CP effect directly right away that means we want to keep the TF constant and see if we can vary the CP or so that that kind of thing so let us let us just see how that can be done so effective effective alpha and CP right I want to just go back and point out something here in this picture so this this curve stops on either side beyond a point as you go from go away from equivalence ratio 1 and this would be the fuel lean flammability limit and this would be the fuel rich flammability limit that means if you now have too much of either fuel or oxidizer the flame is not going to propagate and that is called the flammability limit and we will talk about flammability limits separately later on but we have to keep this in mind that this curve is going to actually hold only over a range of Phi around one significantly around one it is not like just around one but not too far away on either side okay so going back to the effective alpha and CP alpha is of course the thermal diffusivity so the way we can actually think about this is also pretty interesting so you can now think about SL and versus Phi if you now look at let us say CH4 in air right right consider various oxidizer inert mixtures right so that is like O2 N2 as an air and O2 organ and O2 helium all in the same proportions just to keep the proportions the same you should now get pictures that look like that and this is for N2 with oxygen this is for organ and this is for helium so question is why why are we getting this we can think about it in in multiple steps the first thing that we want to think about is this this combination both of them are inert gases and monatomic okay so when you say they are monatomic then their CP's are nearly the same right and the only thing that is different then is the density therefore the alpha changes so CP does not change but the alpha changes so if you look at how the SL behaves SL is like 1 over rho 0 square root of Kw divided by CP these are these are the dependencies so k over rho CP is what we are looking for as as alpha so we should now be able to write k over a CP explicitly in terms of alpha and then you should be able to look at an alpha effect versus a CP effect separately that is what we are trying to do so what you are saying here is because the helium is lighter than organ for the same CP alpha of helium is much larger when than for organ then for AR and then what happens is you know you can you can fix your flame temperature alright that means you can get your flame temperatures to be the same because the CP is the same so flame temperature depends primarily on CP as far as this is concerned so for the flame same flame temperature higher alpha higher alpha means higher SL all right so that is the reason why you are going to get a helium diluted SL to be having a higher value in compared to organ diluted flame speed now then what happens to this combination what you have to think about is here the CP's are different okay so between organ and helium nitrogen this is a monatomic gas but this is a diatomic gas so therefore the CP is different now N2 has a higher CP being diatomic okay relative to organ organ is a good comparison with N2 because the alphas are nearly the same case for for same alpha so that way you can fix your alpha somewhat and and then say look at the CP effect then what happens is SL SL with N2 then becomes lower than SL with organ right but then when you are thinking about change in CP right we start thinking about change in CP even here when we were talking about the SL variation with Phi following a TF variation with Phi trend and then we were thinking about why was the TF actually varying like that and that we argued it through CP of the mixture as the Phi is varied and therefore there was a strength that is shifting towards the fuel rich side so similarly CP is now going to directly affect the flame temperature all right so here what happens is TF also depends on CP so here in the case of SL SL varies through W and TF and therefore CP effect there and then an explicit CP dependence in the denominator therefore you now have a double effect both of them actually adding up together so what happens is typically this gap is larger when compared to this gap because you now have both the CP influencing TF and then CP influencing directly okay so TF also depends on CP and sorry and it is it also affects a cell so keep that in mind and then there are some more interesting things that we should be thinking about oh before we proceed this is a very interesting thing as an experiment for you to do and we will shortly talk about like a Bunsen burner flame stabilization all right so we will look at what is the condition for flame stabilization in a Bunsen burner for example and then we will say okay now I am going to have like a methane air mixture that is burning in a Bunsen flame and then I slowly shut off my nitrogen in the air and then try to introduce organ and then I progressively shut off the organ and then try to introduce helium okay so what happens the flame stabilization so this is like a trick question that is typically asked in like qualifiers or exams right so we don't necessarily ask you directly about this that's why kind of reproducing what's there in the textbook right but we also try to couple this with the flame stabilization issue rig up an experiment where we can vary the deluent type from nitrogen to organ to helium and ask progressively what happens to the flame shape in a Bunsen flame and look at it look at its stabilization and so on so effectively we are playing with the SL while you are trying to do that in relation to like the flames with the flow speed we will talk about that later so keep this in mind proceeding with chemical effects so effect of molecular structure of fuel typically fuel is what is of interest the oxidizer oxidizer is always air most of the time in what you are talking about or at least oxygen with any diluent now what we find is two things we have to think about one more unsaturated unsaturated saturated hydrocarbon higher is SL and saturated means you have like double bonds and triple bonds right so it's not like alkanes we are now talking about alkenes alkines those kinds of things they would they would have typically much higher SL when compared to when you have only only have single bonds then the other thing that we should think about is substituting methyl groups for hydrogen right leads to SL decrease that means like for example when you now have methane you have four hydrogen atoms right so instead of that you now say take take one of the hydrogens and then you have CH3 CH3 that becomes ethane right and then you can now have like CHCl3 or CCL sorry CCH CH3 twice and then CCH3 twice so you can progressively replace your hydrogen atoms by methyl groups and then form larger molecules with more carbon atoms and progress and in correspondingly hydrogen atoms as well in the methyl groups what this can lead to is two things one it implicitly also increases the number of carbon atoms okay so if you now directly look at what happens to the SL as a function of number of carbon atoms right what you have to do is look at straight chain hydrocarbons like methane ethane propane butane pentane and so on all of them N from propane onwards you have to think about like N propane and or N butane N pentane and so on right from from butane onwards you have to think about like straight chains the other way is like if you now take a methane and then start attaching methyl groups you know progressively you do not you do not you know begin to get like for the butane it does not have to be N butane it could be like a isobutane that means like you have a branched methyl group even if you have straight chains as the number of carbon atoms increase the flame speed decreases provided we are talking about alkenes and alkynes with the case of in the case of alkanes there is hardly much of an effect so if you know if you now look at the number of carbon atoms you can look at like a curve that looks like this for alkanes a curve that looks like this for alkenes the curve that looks like that for alkanes and these numbers are pretty interesting so this is like centimeters per second like for example you would have something like 50 100 150 200 and 250 and so on keeps you see this now of course when you now talking about alkines you are looking at a triple bond so obviously you need to have at least three carbon atoms for a triple bond to be formed in between so this is like acetylene and you can see that acetylene has a very very high flame speed for two reasons one the number of carbon atoms is less and it is unsaturated it is highly unsaturated so both of them contribute to a high flame speed so three is number of carbon atoms leads to SL decrease more with an unsaturated and one more thing that you have to think about is when you now have a increase in the number of carbon atoms there is a molecular weight effect that is coming in okay so additional effect is fuel molecular weight fuel molecular weight right and so this actually influences your alpha for the mixture and through that you also so one thing is the reactivity right so that means it affects the reaction rates the other thing is the molecules are getting heavier and heavier so it influences your alpha and from there you are beginning to get an effect on the SL so these two these two things compound and mix for example acetylene to be highly reactive so that is the reason why people say if you are working with something like an oxy acetylene torch like in welding applications and so on you got to be a bit careful should I should say bit careful maybe quite careful okay on safety considerations because it is highly reactive and in fact I should point out that here we are not looking at a TF effect TF note TF and as a matter of fact even the activation energies for like let us say global reactions of oxidation of most of these fuels are comparable right so this is not necessarily like a TF effect at all so then the other last point that I would like to make is about chemical reactivity so a good comparison that is made is for example if you now have a silicon hydrogen bond versus silicon carbon bond like Si H4 similar to C H4 right Si H4 would probably have like a large reaction what do you call flame speed okay so this because this this bond is actually quite reactive when compared to let us say for example if you now progressively replaced the carbon atoms in methane with silicon right and you go all the way to having Si H4 versus you can say I am sorry Si CH3 H2 and so on so if you now have some such thing that would be less reactive so if you now have like a silicon carbon bond that is going to be less reactive when compared to silicon hydrogen bond so these are things that you can pick up from textbooks not very difficult but what you have to keep in mind is to highlight two or three things one have a feel for numbers okay we are talking about tens or few tens to several tens typically for most flame speeds in some in terms of centimeters per second you can also look at some of these having a few hundreds of centimeters per second that is that is that is one thing the second thing always look for the effect of any of these things through TF okay because that is a big effect there are only a few exceptions for example T0 there is a direct effect of preheating and then there is a direct CP effect that you can think about and TF does not really influence significantly when you are thinking about different fuels with different carbon chains so there is a molecular effect and so on that that is coming the picture so some of these are exceptional but most of the time whatever TF does with whatever parameter they are thinking about SL would try to follow that mostly mostly so and then keep this shape in mind to this and then of course what happens when you know progressively diluted with different gases of different monatomic or diatomic gases and then also when you have different molecular weights or densities and so on so this is pretty interesting as well now with this what we should now try to do is a slightly different framework once we now know how to deal with how to get the flame speed then how do we deal with premixed flames so then we talk about what is called as the G equation so what happens when you are thinking about a G equation is when the flame is much thinner much thinner than the hydrodynamic length scale of the flow right we can treat this flame as like a surface surface that is propagating in the flow so it is essentially like a surface of discontinuity the flame can be treated as a surface of discontinuity right propagating in the flow that means the flame has a certain propagation speed which we now kind of take for granted we have gone through enough of what is happening across the flame within the flame how the temperatures rise the concentrations fall for the reactants lots of things and then we now have the preheat zone the reaction zone we could have like multiple preheat zones for non unity lowest numbers depending upon diffusion length scale versus conduction length scale lots of such things we have think thought about right and then we have also further thought about what are the effects of pressure temperature flame temperature all these things but now let us say flame speed is given right if the flame speed is given and you are now talking about a situation where the characteristic length scale of your flow is much larger when compared to the flame thickness okay and the flame is not very very thick many times a laminar flame for example is very very thin right it is about a few millimeters thick of course it is much it is quite thicker than a shock like if you are thinking about gas dynamic shock that is that is only a few mean free parts thick okay you have a sudden change in properties across a very very small distance where even continuum assumptions are broken down whereas that is not the case with flames but still it is quite thin when compared to most applications where you come across most applications therefore if you now think about this you now say that is the flame which is now beginning we are just trying to propagate into a flow right now where was all the preheat zone and all those things it is kind of like you now have to take a magnifying glass hold it against this right and then that is going to kind of look like and so on all the stuff that we drew and so it is almost like locally one dimensional so when you now say locally one dimensional that means the flame is trying to propagate with its flame speed normal to itself right so the flame could be curved that refers to the flame shape whereas what we have been talking about so far is what is called as the flame structure the flame shape is like a much global picture a flame structure is something that happens almost like at a point in the flame along the length of the flame right where you now try to spatially resolve temperature and concentration profiles reaction rate profiles all those things across the thickness of the flame so what we are talking about is locally normal to itself it is going to have a flame speed which is now of course going to be in different directions flame speed is a scalar right and it depends mainly on reactant temperatures pressure reactant temperature pressure and the mixture ratio alright and the all the other parameters like what is the alpha what is the CP and what are the kinetic constants what is the order of the reaction all these things right so once all those things are given if you are given a certain mixture at a particular ratio and temperature and pressure SL is fixed as a scalar the flame speed acquires a direction locally normal to the flame shape so dependent upon the shape of the flame is how the flame propagation direction is fixed and many times you would think wait a minute the flame is probably going to shape itself depending upon how it can propagate right and I will have to try to find the shape in the first place depending upon how it can propagate if I do not know how it is going to propagate how can I figure out what its shape is and how can I figure out the propagation direction so it is a loop that we have to think about so how do you do this right so the way we deal with this situation is we now say this is actually treated as a surface of discontinuity and we want to mathematically describe the surface by a by an equation called g of x vector, t is equal to 0 right so the geometry of the surface can be represented as g of x vector, t equal to 0 x vector referring to the position vector for that particular position there now if you are trying to do this analytically you could just deal with this surface having this equation right as it is or the other possibility is you can now think of g as a scalar like enthalpy or temperature or any anything okay just to just like any scalar so it is sort of like if you now have and particularly what is called as a passive scalar that means a scalar that does not influence the flow okay so if you now kind of have a scalar quantity that is actually dropped in the flow it just goes with the flow right unless it has a propagation speed that it wants to compete with the flow against right so that is the kind of thing that we want to see so if you want to look at it as a scalar that means it is a scalar field that means it should be this it should be defined or then described everywhere in your flow field right so if you want to do that then there is a way of dealing with this that is you can say g less than 0 could refer to the unburned state and g greater than 0 could refer to the burn state that means all reactants have a value the reactant field now should be assigned a value of g that is less than 0 and the product field should be assigned a value of g that is greater than 0 and what happens in the entire field as you now solve for g is the g rapidly rises across 0 within a very very thin narrow region mathematically speaking it is almost instantaneously happening that is why it is a surface of discontinuity at the flame alright that means it is having a value that is less than 0 up to the flame it is now suddenly jumping to a value that is greater than 0 on the other side numerically like if you want to now look do like computation like right okay discretized approximate numerical calculations you could now think about some values from let us say minus 1 to plus 1 and look for a contour or a surface where typically when you discretize your space over maybe about 2 or 3 grid points you now have a sudden rise in g right so that is like a approximate way of doing the flame but effectively we are looking at something like this so what you want to do is as I said SL itself is scalar so it acquires the direction normal to the flame therefore you need to actually define a unit normal vector unit normal vector we are of course assuming the flame surface to be continuous and smooth alright that means it is important to actually have a continuous and smooth flame as an assumption because at every point you need to be able to define a unit normal so if it is not smooth and it is kind of having like a cusp then you are going to have multiple unit normals at that at the point which is ill-defined right so you cannot really allow for cusps to happen so that is like saying it is really not smooth but it is still continuous but there are situations when actually the flame gets cut right but even when the flame gets cut it actually tries to coil around and then make like a circle and then start consuming reactants that are entrapped within this zone so momentarily you might you might find some discontinuities but pretty soon you are going to get something to be to be made continuous so it is it is not terribly bad assumption so what you are saying is under these kinds of assumptions we now say we can define a local normal like minus grad G divided by mod grad G and we take n hat as positive positive when pointed upstream we need to have a convention on this and n hat is taken as positive and pointed upstream okay so then what you do is we want to now have an evolution equation for G how does the G equation evolve or how does the the C the surface described by G equal to 0 evolve right so the evolution equation is obtained by adopting a Lagrangian system that means you now sit on the flame or you ride the flame course it is going to be pretty hot out there okay if you are able to bear that okay so you are going to have some fun sitting on the flame right what happens as you sit on the flame and it is doing whatever it is doing you are there I mean you are not nothing is nothing is happening because you are always sitting on the flame right all the fun can be seen only if you now view from outside like in a laboratory fixed coordinate system you can see the flame movement so the Lagrangian frame of reference says that on the flame surface right DG over DT equal to 0 the reason why we use a ordinary derivative is your space is fixed by the location of the flame you are on you are on G equal to 0 and the only thing that is varying is time right and then we are saying that the G is going to be invariant in time because you are sitting there nothing is happening while you are sitting there this in fact is what is typically written as capital G over capital DT in an Eulerian frame of reference then with respect to an Eulerian frame of reference if you now pull yourself out right you will now call this the material derivative right so this would then mean in an Eulerian frame of reference we have to write this as partial G with respect to time because G partially changes with respect to time besides there is an apparent change with respect to time because of its motion right therefore we have to say capital VF vector dot grad G equal to 0 this is essentially partial D partial D over DT plus V dot del of G so this is an operator that is acting on G that essentially is basically trying to say so what is what is VF now VF is nothing but DX over DG right that is essentially the position vector corresponding to where the G is and it is change in time so that means it is essentially a displacement of the points along the flame that taken a derivative with respect to time to get the flame flame motion so this is actually the local so this is just like local propagation velocity right now when you say local propagation velocity it is sort of relative to something that is very local it is not thinking about a flow field that is around it it is trying to have a relative motion of the flame with respect to the flow around right so this then by definition of the way SL is right SL is nothing but the VF minus the flow velocity V vector evaluated at G equals 0 minus this is corresponds to the upstream reactant velocity at the flame dot n hat right so this is to say that you are going to have a relative velocity between the flame movement and the and the flow that is manifesting as a cell if your flow were quiescent right or if your reactants were quiescent then you will identically have the flame propagation velocity that we talked about as the flames flame speed okay except we have to now consider the dot product because you have to look at the local normal propagation otherwise you would not get a scalar out of this vector or you could not have made a vector out of the scalar either way you can look at this right so if you now say so we then is the local flow velocity in lab frame right that is the Eulerian frame and from here we can get if you now plug this back here you can get in this equation you can get plus V obtained at G equals 0 minus dot grad G equal to SL mod grad G there is substitute for n as minus grad G divided by mod grad G and you should be able to get this right so this is what is called as the G equation the G equation essentially is an evolution equation for the flame surface as if it is a surface of discontinuity across which the temperature changes from the unburned temperature to the burn unburned reactant temperature to the burn product temperature and reactant concentrations become 0 for particularly the the deficient reactant and product concentrations go from 0 to whatever is the final product value and so on okay we will stop here this at this stage I will see you at 3 o'clock tomorrow.