 So today we will talk about spray combustion this is actually like a follow-up to the droplet combustion that we did where we were basically considering only a single droplet at a time and at the time I pointed out that it may or may not be exactly the way it happens in reality because in reality you may have a cloud of droplets that burn or you may have all the droplets evaporate by the time they reach a gas phase frame and therefore we do not necessarily have a droplet combustion and so on. So there are lots of different possibilities and it also depends on whether the spray is dense or dilute and typically we think about a single droplet combustion mode in a spray in reality most of the droplet combustion happens only in sprays right so spray is the more realistic situation that we need to think about and the single droplet combustion mode probably happens only in the in situations where you have dilute sprays and so on. So let us now see if we can take take the idea of the single droplet combustion further and see when that happens in a spray so basically we will now look at spray combustion now there are a couple of ways by which you can think about spray combustion similar to the way we have been dealing with the gas phase flames so in the case of gas phase flames we had on the one hand premixed flames on the other hand we had the diffusion flames right so we could think about these things as we could still think about like analogies to spray combustion similarly but however we have to think about in reality it is again in between it is not it is neither premixed nor diffusion so what is really meant by a premixed spray combustion or what is meant by a diffusion flame kind of spray combustion on the one hand you could now think about droplets that are now laden in an oxidizing gaseous flow gaseous oxidizing flow so if it is possible for us to mix liquid droplets of fuel in a gaseous oxidizing flow and of course the gaseous oxidizing flow could also have inert gases like for example if you are looking at a gasoline air mixture or something like that right so air has the oxidizing gaseous species as well as inert gaseous species and then you are now throwing in liquid droplets of gasoline and then let us suppose that you now have the flow up here flow in a tube and then you now stabilize the flame in between or you could also think about more like a Bunsen burner where you have the approach flow is actually made of the a mixture of liquid droplets in a gaseous oxidizing flow so that is that is one one configuration that you can think about on the other hand let us suppose that you now just have an orifice in which through which you just issue a liquid fuel and because the orifice is so small or the pressure injection pressure is high or a combination of both you now have a spray that is coming out so what is basically coming out is essentially a liquid fuel spray and the oxygen has to get in the oxidizer is in the ambience and it has to get entrained into this and as it gets entrained it mixes vaporizes the liquid and burns the liquid fuel and so on so that that would be like a diffusion flame counterpart in reality as I said you would have a combination of these two things it is neither this nor that and of course you can think about like a partially premixed flame that is established in a in what could be nominally a non premixed injection of liquid fuel but you since you have like a standoff distance it allows for some mixing and so on so many such possibilities exist in reality so what we will first consider is a premixed mixture force for the sake of simplicity mixture of liquid fuel droplets and oxidizing gish is oxidizing gish is oxidizer plus inert or diluent right. So as I said you could have like a possibilities or let us say you have a tube and then you have now the reactants and then you have a flame and then you have products or you could have a burner in which you now send the reactants and then you have a flame and then of course you have products downstream of the flame. So these are different possibilities in this case for example you now can talk about talk about a SL and what is really opposing it is the normal component of the flow right. So that is equivalent to saying you now have a reactant flow at the speed of SL that is approaching a stationary flame or in a flame fixed coordinate system. So in this case what we are basically talking about is let us suppose that in if you now look at this locally or in this region what this basically amounts to is if you now have a flame right what you are essentially saying is you now have a droplet that is approaching the flame along with oxidizing gas with inert and then it begins to evaporate because it is now in the influence of the flame as far as the heating is concerned and as the heating happens you now sorry as the evaporating happens it not the fuel vapor is now evaporating out and then also flowing along with the oxidizing gas and mixes and you now reach this is the flame and when this liquid droplet goes past it you now still have a liquid droplet with a flame behind around it as an envelope flame and this now decreases further into a smaller droplet and then flame and so on. So this is typically the kind of picture that we are looking at in these situations alright so if it is possible for us to now set up something like a Bunsen burner with liquid fuel droplets we should now expect the larger ones particularly and that is what I am just next going to talk about the larger ones will actually get you start burning as they flow through the flame and then begin to go as individual droplets that are burning not like in single droplet combustion that we have seen but the smaller ones will now have evaporated by the time they reach the flame therefore you would not have a single droplet combustion right. So the question that we have to ask is what is the limiting droplet size below which below which you will not have a single droplet combustion mode okay so if this droplet is going to actually shrink to 0 size by the time it reaches the flame that means starting size should have been small enough if it is large then it keeps on shrinking but when it gets past it begins to burn in a single droplet mode right. So the question basically is what is the limiting size below which the droplets do not burn in a single droplet mode the answer to that question can be done by a simple analysis. So the thing that we look at is what is the residence time of the droplet in the flame right so the so what we are basically saying is sufficiently small droplets or completely vaporized vaporized in the preheat zone the preheat zone of the flame is where the heat conduction is happening so the characteristic time the characteristic time of traversing through the preheat zone preheat zone is let us say Tau D is LD which is like the flame thickness divided by SL and that would be K divided by CP rho G divided by we should have actually at LD is K divided by CP rho G SL and we also have another SL before you get a SL squared and then what we have to compare this time is with the droplet evaporation time how long does it take for the droplet to evaporate versus how long is it going to be within the preheat zone right so the droplet vaporization time time is Tau V is Dl0 square divided by 4k that is essentially is into talking about the radius divided by the in terms of surface area divided by the evaporation constant it is essentially coming from the D square law and so this is this is as if this is D square law for evaporation we are not basically saying that it is going to burn in a single droplet mode but we talked about when we did see single droplets we also said that the same approach can hold good for evaporation so then we use the expression for the evaporation constant that we had obtained to get natural logarithm 1 plus B where B is now K CP T infinity minus T B the boiling point divided by the latent heat of vaporization so if you now say that these two times are comparable then we begin to get a diameter of the droplet below which you would not have a single droplet combustion that means you would not have anything that is burning and coming out of the flame right so for Tau D approximately equal to Tau V or the same order Dl0 min square equals 8 rho G by rho L K over CP divided by rho G SL the whole square natural logarithm 1 plus B the reason why we write rho G SL together is this is actually the mass burning rate or the burning mass burning mass flux okay so now for typical values of things like if you now assume approximately stoichiometric mixture for the fuel and air where the combustion happens and then you now throw in like row L over rho G is of the order of 1000 and then and so on so you do all these things then so this gives about 10 microns so Dl minimum Dl0 that is the original droplet size the minimum original droplet size is of the order of 10 microns 10 micrometers for typical values for other quantities right now we can also see the dependency here so for rho G SL squared of the all going as going a speed of the N you know it is coming from how the the burning velocity depends on the pressure and we are looking at rho G SL the whole square so we remember that and we say it is going a speed of the N then then D L0 minimum squared goes as p to the 1-n right and this is purely coming so what this means is the pressure dependence purely comes from the preheat zone length okay and we do not have to worry about the dependence of the evaporation time on the pressure so the evaporation time is not dependent on pressure it is only the preheat same length that changes with pressure and therefore the residence time of the droplet within the preheat zone is going to change like this right so that is mainly the effect on of pressure what we should actually look for and you now have these bunch of droplets here so we the picture that we drew is only for one droplet right but that is not that is not reality so this is assuming like a dilute spray essentially and then talking about a single droplet combustion event in the case of droplets of size larger than the D minimum right so question is what is the interaction effect okay so the droplet interaction now assume assume for for this purpose equal size of droplets right then what you essentially saying is let us suppose that these droplets are located in space at a particular time with some distance and what we would like to think is the distances are about the same okay for the sake of what we are doing not we are not very difficult we are basically looking for something like an average inter droplet distance so what you could then do is now draw spheres around these and so on and if you now say this is this this the sphere around which you around the droplet that you have drawn is having a diameter that is equal to the that is approximately equal to the inter droplet distance okay and then if you now take that sphere worth of mass of oxidizer neglecting the presence of the droplet okay and then take the mass of the droplet you can now form a fuel mixture based on that alright now we are there are voids here and those volumes of the oxidizer we are not considering so it kind of compensates between taking this volume and not taking that volume right so approximately speaking the the fueler ratio the fueler ratio is F over a is approximately 1 over 6 pi DL cubed rho L it is essentially 4 over 3 pi RL cubed but RL DL is RL divided by 2 sorry RL is DL divided by 2 so if you now plug that in there you get the 1 over 6 DG cubed rho G so the DG that we are talking about is the inter droplet distance average okay so DG average inter droplet distance right so what does that mean this is like row row L over rho G times DL over DG the whole cubed so what this means is of course if you have for nearly stoichiometric mixture stoichiometric combustion for typical hydrocarbons let us say typical liquid hydrocarbons your F over a should be around 0.05 all right you can you can evaluate that for typical so you look at the stoichiometric chemical reaction and the pins balanced and then you can look at the mass ratios for that if you now plug that in and row L over rho G is about 10 to the 3 right then we can we can find that DG over DL is of the order of 25 right so 25 is like more than 10 so we can say that the inter droplet distance is about is quite larger when compared to the droplet diameter itself so this does not mean too much interaction but if you now start thinking about the effective pressure okay so at high pressures rho G increases okay because as pressure increases rho G increases so if row L over rho G so if if rho G increases low L over rho G decreases if row L over rho G decreases then DL over DG DG will increase or DG will decrease effectively right so DG over DL then DL decreases for example in automobile engines if you now look at the typical kind of pressures you will find that those pressures DG by DL would be as lower like about 15 or something is still around greater than 10 but you are now hitting a limit where you have to consider the interactions what we should then start to think about as we started looking at a premixed mixture but let us now look at a jet jet a spray jet and a spray jet flame so this is now getting closer to reality so what is going on here is the two things one as you now have a jet that is coming out this is jet of liquid fuel it is also beginning to entrain air and then of course we have ignited it then there is a flame the question is are we having individual droplets that are burning in this flame or is it like an envelope of the all the droplets that is burning together that is essentially the question that we have to ask right so the answer to that is of course the droplets are evaporating from the heat of the flames right so the competition now is between how much is the entrainment right because you need the oxidizer for the flame to happen and so therefore you need to have the entrainment to happen so how fast is the entrainment versus how fast is the evaporation so the evaporation is going to give rise to the fuel the gaseous fuel the entrainment is going to bring in the great gaseous oxidizer and you need to have both of these for the combustion to happen right so typically when you are now looking at entrainment the entrainment is happening from the sides and then it starts penetrating to the core so there is a there is a core of liquid which does not see the ambient air significantly right for some distance may be on which the ambience penetrates so you could now think about the inner droplets not exactly going through a single droplet combustion right whereas the outer droplets are going through this because there is access to air so effectively fuel is sprayed and the air entrains so droplet at the core droplets at the core may have insufficient oxidizing ambience now so what you have to look at is the competition between entrainment and evaporation right so if entrainment is fast and or or okay it is a combination vaporization slow so entrainment is faster vaporization is slower than what do we expect air is now beginning to get available to most droplets and therefore most droplets can burn by themselves they are harder to evaporate therefore they are going to be waiting for the flame to come near them okay oxidizer to come near them and the flame to happen near them so this actually leads to single droplet burning more towards single droplet or so or droplet burning singly or in groups right now once you have a single droplet mode of combustion that is established then it is difficult for the flame flame to coagulate and then have a outer envelope flame because once you have the flame you do not have fuel penetrating outside of the flame so the oxidizer has to come to this to the flame in order to consume the fuel right so once you have this mode of burning this persists it is kind of like a stable solution that the system approaches on the other hand if entrainment is slow right and or or vaporization fast so this now leads to a flame that is actually enveloping a what is called as a spray sheet so now you can look at a spray you now have a cover around this and that is exactly why you expect the flame to happen so this is envelope flame at the spray sheet now once you have a envelope flame then that is not going to let the oxidizer get it okay so the droplets will now be waiting for the oxidizer but no oxidizer is it coming because it is all getting consumed in this flame so they just begin to get evaporated because of the heat from this flame and they will have to undergo continued and undergo combustion in the envelope flame so once you have an envelope combustion mode it is it sets in and it persists you it is hard for it to transition into a drop that combustion mode right so this persists now typical examples of these are for example if you now look in at a situation where the vaporization is slow that means we have to look for materials that are less volatile okay so typical example of this would be even solids for example look at coal particles okay so if you now look at pulverized coal so cold or cold dust okay example coal dust flame now the example of this would be the second one where the evaporation is fast is essentially oils so oil flame right where where the where where the oil is typically quite volatile and so the oxidizer spreads the fuel the gaseous fuel spreads out and meets the oxidizer even before it can enter in train significantly so then what we have to look for is how does how does this what we call as a cloud combustion or a dense spray combustion happen we define something called a group combustion number or we can you can call it a group group combustion index or number this would be called g and this is defined as n capital N dl not divided by L where n is the total number of droplets total number of droplets that can be now written as I should say goes as nl cubed where n is the number density that means number of droplets per unit volume and L is characteristic dimensions right so essentially what we are saying is if you now have a total number of droplets times the diameter divided by the characteristic dimensions let us say for example if you are looking at a spray flame or a swell flame where you are sending in the liquid in a swell flow and so on you have like a burner diameter so within this burner diameter or of the order of the burner diameter how many droplets can you space is essentially what is in what is indicative of the group group combustion number and what it basically means is if you have a large g it indicates a group group combustion if you have a small g it indicates single droplet combustion group combustion and small g single droplet combustion now in reality you could have also things that are in between that means you have small clouds of droplets that are burning with a flame enveloping it but many such clouds separately right as opposed to when you say group combustion it is like all droplets that are there or having a single flame so there is a reality that is in between so it is like these are extreme cases right so what would and then the other thing that you have to point out is this is purely a geometric parameter right so we are talking about what is like the spacing between the droplets if you will so g is purely a geometric parameter and it does not take into account these things like we are not looking at how well is the entrainment happening versus how well the vaporization is happening okay so these two will keep this in mind but but before before we proceed with taking into account those in a in a spray jet flame but what we will think about is is it possible for us to think about like a spherical cloud of droplets right what could happen if you now have a spherical cloud of droplets as a function of g so if you now start from a large g towards a small g what happens to a spherical cloud of droplets now there are four possibilities that that can that can happen with with the spherical cloud of so consider a spherical cloud of droplets right so what we are basically saying is can we have a large g situation where we are looking at a group combustion and what that means is you now have lots of droplets that are in the core and they are non-evaporating right so if you now have like a slow vaporization situation then let us suppose that you now have a spherical boundary imaginary boundary spherical imaginary spherical boundary of non-evaporating non-vaporizing droplets and out of which you now have let us say smaller ones which are evaporating droplets so they are obviously much smaller and I am just representing them by a dot and you can now talk about this as the spray boundary so in all the pictures that I am going to draw off the four possibilities the broken line refers to what is called as a spray boundary okay so there is another way of thinking about this it is you can look at like a spray that is coming out of the board right in a cylindrical manner it does not have to be spherical but you can look at things happening in a circular manner as a cross section right so it is like you now have a non-evaporating core and a evaporating surrounding that forms the spray so this is the broken line basically means this jet the spray boundary so this is a vaporizing droplets and the broken line is the spray boundary and you now have the actual flame around this right now as you now increase your sorry decrease your g as you now go in this direction of decreasing g the other possibility is you do not have a core of any non-vaporizing droplets that means this diameter basically shrinks to 0 you are now beginning to think about droplets that can vaporize faster but still not fast enough for flame to eat in okay so you know you now have a single spray boundary without having to distinguish between vaporizing and non-vaporizing droplets because all of them are vaporizing so that is vaporizing droplets and then you have a flame around right now the next possibility is for the flame to move in okay into the spray so as I said the broken line is the spray boundary so the moment the flame moves into the spray boundary you now begin to have single droplets in the periphery single droplet burning in the periphery and then you have group combustion in the middle so that gets us to a situation where let us suppose that you now have this picture of vaporizing droplets in the middle and still it is not the spray boundary you have actually have the main flame now we have to distinguish between main flame and individual droplet flame let us for those that are burning outside this so those are individual droplet burning right and this is what constitutes your your spray boundary and finally the flame now eats into each and every droplet there okay so that means you now have a spray boundary in which all droplets are burning individually and so on right so this is say limit of low G and this is individual burning and this is this is now OG right the question is what happens so this is as if like you had these droplets kind of put together in isolation as if they were not part of a flow right and then we were beginning to play with the G so if it if there is a high G then there is more group interaction when there is low G there is less group interaction you have more individual droplet burning and so on but what happens in reality for something like a spray jet flame the answer is the spray jet flame you now have a flow field with a variable G so you start with something's like a very dense spray that is just beginning to atomize and then spread out and then you now have more and more entrainment that allows for individual droplet burning for the route so it is like a variable G that starts with a high value and then keeps proceeding along the low value so this is where it is not just purely a geometric parameter that we can rely on we have to bring in the flow effects therefore in a in a spray jet flame G is variable so you now get into a situation where let us suppose this is a centerline and let us say you have lip of the nozzle and which issues out the spray and you can now think about like a potential core and you now have near the near the centerline you have bigger droplets but away from the centerline let us say you have smaller droplets because they are evaporating now you have a the main flame or the internal group combustion flame right and outside of which as you now keep going the the G actually decreases and you now have lots of cloud drop cloud combustion going on right forming the spray boundary so this is like the maybe keeping with the convention we should use broken line spray boundary and this is multi-droplet combustion and there is a overall turbulent flame brush that we will have that that encompasses the entire flame and so that is that is essentially a turbulent flame brush we will be we might look at it when we do turbulent flames that is it is an envelope of all the fluctuations right so that is let us say that is a turbulent flame brush how do you formulate this proud this kind of problem right so the way we have to do it is obviously to deal it deal it in a deal with it in a statistical manner so we have to look at how to do spray statistics spray statistics what you do is you now have what is called the spray distribution function function f of f of r rl which is the droplet diameter position x velocity u u vector and time such that this is like a density distribution that means f of rl u x to say x plus u and t d rl dx vector du vector so this is basically a seven dimensional space in 3d right so you have three components of velocity and three directions of location and then the radius so this quantity now right indicates the number of droplets of radius between let us say in range rl to rl plus d rl velocity position in range x to x plus dx that is within this bin and velocity in range u to u plus du right at any time t it is time dependent all right so we have to now let it let this fluctuate in time so when we now use the word rl sorry the the quantity rl we are implicitly assuming a spherical symmetry so zoomed there are ways in which we can relax this but we have to see when this is this is relaxable so the now typically small droplets for example if you are looking at things like 10 microns and so on 10 to 100 microns like they look at that range for most typical fields you have a significant surface tension effect that spherodizes the droplet okay so the sphericity is maintained because of the surface tension therefore we have to actually look at what else is causing the droplet to be non-spherical as competing against the surface tension so what typically allows for the droplet be non-spherical is when it now tries to flow you now have a shear that is happening at the surface as well as a hydrodynamic instability because you now have high density and low density interface which becomes unstable and so on these things are obviously proportional to the relative velocity between the two the droplet and the surroundings okay so the droplet were to be stationary you do not have these effects right so therefore the Weber number Weber number describes the effect of surface tension relative to that of the dynamic force on the droplet so we that is a simple symbol for Weber number is to RL that is for the diameter gaseous density times modulus of u vector – v vector the whole square that is a relative velocity between between the droplet and the ambient gas divided by sigma which is the surface tension so u is the droplet velocity v is the gas velocity right now for we greater than or greater than or equal to about 20 droplets tend to deform and break up so if you are looking at atomization situation you are looking forward to deformation of liquid sheets into ligaments and ligaments to droplets so you are looking at a high Weber number situation okay but once you get to fairly small droplets the Weber number becomes small and then the surface tension takes over so for a Weber number greater than 20 droplets tend to deform and break up so given this we can now write what is called as a spray equation for f on how this is actually going to go spray equation is a dynamic equation for f RL dot f plus now a divergence with respect to x of the droplet velocity carrying the f and a dynamic for a divergence with respect to u of the body force all equal to a s where now we will now explain these terms RL dot is something that we are familiar with which is DRL over DT right so this basically tells how this f is going to change because you are having a evaporation or burning that changes the R at a certain rate with respect to time and for example if the D squared law holds right we know that RL dot is goes is 1 over RL right that is something that we have seen before G is as I said as a force per unit mass whatever forces are acting that that change the droplet the spray the distribution function and s is actually the most important thing and also the most difficult thing is essentially the source term and essentially you are to look at two sets of quantities one is creation and destruction so creation and destruction so it is essentially a net source term of droplets right so creation is like for example by atomization and destruction is by coalescence and let us say impingement with the wall right so wall collision etc so it is kind of like what is called as a population equation that that keeps track of the droplet population given by f and what you are looking for is how do the droplets get created and destroyed as a source of this that is getting dynamically going in a flow field now these are difficult to model and the fidelity of this equation lies in actually how we model this and that is where all complexities remain