 So in most of the spray applications one of the inputs that we have to give to the problem is what is called as the initial spray droplet size distribution. In fact when we look at these the straight spray distribution function it says we need to actually look at a bin of certain droplet radius and a bin of certain droplet velocities and a bin of certain droplet locations and so on. So typically for example you do something called a PDPA experiment in order to try to find out what is the droplet velocities and as well as droplet sizes simultaneously so it is possible for us to actually measure them together but you do that in a particular experiment and then you want to use that as a inlet condition to let us say a flow field in where for example this burner is now fitted into a combustor or a reactor and so on. So obviously the positions are not going to exactly be the same you are going to only get dry statistics out of it rather than exactly know what the positions of the droplets are in this experiment. So the best thing that normally people do is if you now want to fit a certain burner with a let us say oil spray you at least have to know what will be the droplet size distribution at the inlet to this combustor from the burner exit and therefore we have to look we need to know what is called as a initial droplet size distribution. This is usually obtained empirically of course there are recent efforts to actually try to computationally predict this as well by taking into account the shearing of liquid sheets or liquid rods, liquid tubes kind of thing by the aerodynamic as well as hydrodynamic effects and so on but of course there is a lot more that needs to be done computationally in order to be able to predict droplet size distributions accurately at the moment I think they are mostly used for design of atomizers from a qualitative viewpoint like for example if you now change some parameters how does it affect the droplet size distribution kind of thing but they are always benchmarked and calibrated against experimental data that is obtained. So this is basically like let us say used with a symbol G of RL with this you should actually be able to obtain several spray properties so the spray properties are based on the droplet size distribution or integral 0 to RL DRL DRL this actually is the total number of droplets of radio less than RL and we can also look at the volume so 4 over 3 integral 0 to RL RL cubed G RL DRL that is the total volume of droplets of radii less than RL right. Now many times we use this quantity called the sotamine diameter or SMD which can be written as in the form of integral it can be written as integral 0 to it is a certain maximum RL that you have in the distribution RL cubed G RL DRL divided by integral 0 to RL max RL squared G RL DRL so notice that you are going to get something like an average radius out of this but SMD is actually a diameter so you have a number 2 a factor 2 there to make that a diameter essentially or okay then there is one more way by which you can define this which could be more familiar that is twice of sigma over all I ni Rli cubed divided by sigma over all I ni Rli squared. Now this is actually something that is that is more familiarly used because typically what we do is we now have a bunch of droplets and then we now try to size them into bins of certain elemental radii radius bins and then you now count the number of droplets that are there in each bin that is the ni and so on so you essentially get what is called as a histogram and if the bin size is very small that means you are now counting the large number of droplets and then placing them over smaller resolutions or higher resolutions of radii then they just can actually become an integral right so the integral definitions are typically when when the histogram bin sizes are small and you can now try to fit a curve out of the histograms rather than keep them as histograms themselves but this would actually hold if you have histograms and essentially the idea of put the solid mean diameter in both these definitions is we are looking at basically a ratio of total volume to total surface area right so ratio of total volume to total surface area so you know when you now have atomization of a spray and if the atomization is very fine then the total surface area is actually very large so if you now have a larger and larger atomization done of course the volume corresponding to it is also preserved so typically if you have the same radius for all droplets then it did not matter you will get the same mean radius regardless of how you actually averaged you did a arithmetic average let us say so for example this is the total number of droplets and then if you now have a integral 0 to RL or RL max RL G RL D RL divided by integral 0 to RL max G RL D RL right that would actually be the arithmetic mean and this would be like the sort of mean diameter you could also have like a weight mean diameter and so on so all these things will actually amount to the same if the radius of all the droplets are the same okay but when you do not have the radio of all the droplets progressively higher and higher weightages is going to shrink the size the average size basically here in this case you can clearly see that you have a large surface area then that is going to actually pull down the mean right so typically what is done in order to get this the smooth curve that we talk about is from the histogram data you can now fit a what is called as a rosin rosin Rammler distribution so this is very popular so the generalized rosin Rammler distribution for sprays is a distribution function G G of RL is equal to B of RL to the power T exponential negative a of RL over S so here a, b, s and t or constants that are used to fit the data to this distribution so a, b, s, t or constants fitting parameters now there are special cases so t for example t equals s minus 4 is rosin Rammler distribution that is the original rosin Rammler t equals to corresponds to what is called as a Nuki Yama Tana Vasa distribution and s equals 1 is what is called as a chi square distribution and so on so you now have these four constants what we should be looking for in trying to fit curves is four conserved parameters so essentially the four characteristics that we should be looking for as four characteristics N the total number of droplets which is something that we saw but here we try to take all droplets so 0 to infinity G DRL the total number of droplets total number of all droplets then you have a mean which is defined as 1 over this is the arithmetic mean 0 to infinity RL G DRL which is the average droplet radius we can simply see size sigma so when somebody says size usually they talk about diameter that is a colloquial parlance but in this case we are actually referring to radius so keep that in mind sometimes these definitions might be different and then you are off by a factor of too many times so you know what to suspect so sigma equals equals 1 over N 0 to infinity RL minus the mean or the whole square G DRL which is the standard deviation and finally you have this skewness which is S times sigma cubed equal to 1 over N integral 0 to infinity RL minus mean of RL the whole cubed G DRL which is the skewness so if you are now able to actually get some data with which you know the mean but the total number and the standard deviation and the skewness you can fit the distribution that satisfies these things right and then try to obtain the constants and then now you have a curve that is a standard template that follows a standard template good so with this what we then do we can do is one once you have a good description of the Doppler size distribution we now look at the conservation equations for a spray flow so conservation equation for spray flow including combustion of spray so conservation equations here we want to now define row F as the mass of the gaseous mixture per unit volume of the two-phase mixture so you now have a two-phase mixture with droplets in there and in gas and the gases and so gas in turn is a mixture of gases all right so this is basically the mass of the gaseous mixture mixture mixture per unit volume of two-phase mixture the row G is actually the gas density so it is to say that row F is kind of like taking into account the two-phase part of it but row G is without taking into account the two-phase part of it right so if you want to distinguish these two then row F divided by row G is nothing but 1- integral should say actually double integral you want to go back to the spray distribution function taken into account different velocities for different droplets so then that would be 4 by 3 pi RL cubed divided at times f d RL du then the overall continued equation for the gas for the gas is partial derivative of row F with respect to time plus a divergence of divergence with respect to X of row F V vector is equal to negative double integral row L times 4 pi RL squared RL dot F d RL du so here what is going on is you know this is basically the rate at which the liquid evaporates or its surface regresses and then produces gas right so since this is a this is d RL by dt where the RL decreases as time increases you need to have a negative quantity to make sure that you are adding mass to the gas so this is basically the this is coming from reduction in the liquid mass liquid mass due to droplet evaporation causing gas phase density variation basically what you are saying is whatever mass that is well that is lost by the liquid is added to the gas because masses to be conserved right so the momentum momentum conservation is row F partial row F times partial derivative of V vector with respect to time plus row F V vector dot divergence with respect to X of V is equal to minus divergence with respect to X of the stress tensor including the normal stress namely the hydrodynamic pressure right plus row F sigma over K for species YK FK that is a body force so we will name these once we finish writing the full equation then you have another term which is coming from the evaporation 4 by 3 pi RL cubed or no this is this is the drag on the droplet this is the drag experienced by the droplets from the gas and in turn the droplets exert an equal opposite drag on the gas okay and we will talk about that plus the thing that I was just mentioning which is the momentum that is imported due to droplet evaporation so that comes from row L 4 by 3 pi RL cubed RL dot U minus V F DRL DU right so here FK is the force per unit mass of the Kth species that is basically the acceleration experienced by the Kth species this typically is different for different species if you have things like ions that are subjected to an electric field or magnetic field and so on so of Kth species G is the drag force that is equal to 3 over 8 rho G over rho L modulus of the relative velocity times the relative velocity to give you the direction divided by RL times CD now CD is a drag coefficient drag coefficient depending on depending on Reynolds number and mass transfer number so it will also depend because you now have an evaporating droplet so it depends on mass transfer number Reynolds number RE so Reynolds number is RE equal to 2 RL that gives you the diameter times density times the relative velocity magnitude divided by the gas to gas viscosity so that is taking care of the this term on the right hand side and finally this term is basically the momentum transferred to the gas by the vaporizing mass because when the gas comes out of the surface of droplets it now imparts a momentum to the gas around so this momentum transferred by the vaporizing mass to the gas similarly we could also write the energy equation partial derivative with respect to time of rho F times HF plus V squared by 2 plus divergence with respect to X of rho F V HF plus V squared over 2 this is the total energy equation that means it includes the kinetic energy also equals minus divergence with respect to X of Q vector minus the stress tensor dotted with gradient of V so this is basically a tensor gradient of a vector is a tensor the stress tensor is a second order tensor therefore this is actually a double dot product giving rights to a scalar finally as part of the energy equation plus the partial derivative of pressure with respect to time that is for the that is coming from the PV work and then you have the effect of body force over K YK FK vector dot V plus capital DK this is the diffusion velocity of the kth species keep in mind this is what the momentum equation is all about so this is the total energy equation if you got the thermal equation a thermal energy equation alone that would be by dotting that with V which is the mixture gaseous mixture average velocity so that this might go away but this would still stay right so the body force acts on the actual velocity of the species which includes the diffusion velocity as well as the average velocity of the mixture the gaseous mixture minus then you have a couple of terms that are specific to droplets so you now have a rho L 4 4 3 pi R L cubed g u this is the work done due to the drag force times F D R L du minus integral rho L 4 pi R L squared that is a surface area times R L dot H plus u modulus squared times F D R L du right we need to go back and check here I think this must be 4 this must be 4 or 4 pi R L squared because we are looking at the surface of surface evaporation okay then finally we have the species conservation which is rather straightforward with respect to time partial derivative of rho F yk plus divergence with respect to x of rho F V plus capital Vk yk so this takes care of the diffusion mass flux right equal to the chemical production minus we also have production of the kth species of the gas from droplet evaporation right so rho L 4 pi R squared R L squared R L dot omega k F D R L du where omega k is actually the radial mass flux of kth species from droplet droplet surface okay just could squeeze some space there for writing that so what then happens is if you now write out these equations is a sort of similar to the governing equations that we wrote for a homogeneous combustion that means all species being gaseous but here this is for heterogeneous combustion where you are now throwing in a bunch of liquid droplets typically fuel droplets and in each of these equations you can see that there is a source term that is coming out at least one or two source terms that are coming up because of the droplet that means the droplet would now add mass to the gases it could now impart momentum in couple of different ways energy in a couple of different ways and also throw in particular species into the gas and therefore bring comes in as source terms in the in the equations now what you have to imagine here is let us suppose that we try to solve this for a typical flow field which is quite turbulent there are some very nice research that have been done which now begin to look at what is the effect of the turbulence on these droplets and what is the effect of these droplets on the turbulence okay. So typically what happens when you now have a locally high Reynolds number situation based on the mean based on the relative velocity between the gas and the droplet velocity you could now have a wake there is sharing behind the droplet okay. So you now have a overall turbulent field into which now the droplet is actually moving following the turbulent flow around but as it is moving it is also producing a wake behind it and this week is obviously of length scale of the order of the droplet size rather than that of the characteristic dimensions of your combustor or your burner or something which the overall turbulent the prevailing turbulence is of a length scale of so what happens is you now have these multiple length scales of turbulence the larger length scale corresponding to the combustor geometry on the small length scales of the wake turbulence from these droplets so a snapshot for example if you were to now do something like a PIV of this and take like a planar data of what the flow field is you now have like a large scale motions captured over the larger area with like pockets of intense small scale turbulence here and there because the droplets have just gone past these planes here and there. So that is typically described as a Swiss cheese because if you now take like Swiss cheese you now all you have these lots of these holes that these holes now filled up with in this case the droplet turbulence this is pretty interesting the other thing now if you now begin to think about these droplets burning rather than just evaporating and let us suppose that we were talking about the group combustion number number which now allows for individual droplets to have flames associated with them we never really talked about what happens when you have droplets in motion right so the kind of even in even the kind of pictures that we drew were as if like the droplets were stationally and you have an envelope flame that is spherically symmetric about the drop that assuming the droplet itself is spherical to begin with that is all right but when you now have a droplet in motion because of the motion the flame now becomes a symmetric right so a simpler experiment for that would be if you now have a droplet and then you have a let us say a gas flow that is coming from the bottom you now have a flame that is now getting elongated along the flow direction right and you could still think about something like a envelope flame that is enveloping the droplet all around but getting elongated along the direction of the flow right but then as you now increase the relative velocity between the droplet and the gas you now get into a situation where you no longer have an envelope flame you now have what is called as a wake flame that means so if you now have a droplet and then you have a flow relative to the droplet airflow relative to the droplet coming up you now have a envelope flame that is enveloping this and elongating in the direction of the flow this is what we would call as an envelope flame and if you now have a greater relative velocity between the two you actually begin to have a only a wake flame that means the flow goes around and actually causes like a lift off of the flame similar to how you have lifted flames you now have the flame getting really confined only to the downstream part which is what is called as a wake flame so this is something that typically happens in these cases that is a droplet let us see that the flame there behind okay. I want to stop talking about droplet combustion but we are still in the subject of what is called as heterogeneous combustion that means we are talking about some of the reactants or some of the species chemical species in general or in different phase when compared to the other others right so far until we were talking about droplet combustion it was all homogeneous combustion that means we were typically talking about all the species being of the same phase and that is that is a gaseous phase but the moment you start talking about droplet combustion we are beginning to step into heterogeneous combustion and yesterday we mentioned the situation of let us say coal combustion where you have like a single single particle combustion rather than a cloud combustion that is because of lesser volatiles so just quickly talk about a coal combustion and other other solid combustion we also mentioned for example metal combustion where it is similar to droplet combustion because by the time the metals are actually burning they melt and so you have molten metal droplets that are burning and it is although it may seem like we are actually burning solids in reality you might end up burning liquids so if you now think about coal particles the problem mainly with the coal particles is it is essentially twofold one it is okay before before we in the context of talking about coal particles if you want to now go back to liquid fuels you can still talk about something called a single component liquid fuels versus multi-component liquid fuels okay so what is meant by single component versus multi-component is if you now take a fuel for example like methanol okay you are essentially talking about only one chemical composition or only one chemical substance there but if you are now looking at something like diesel it is actually made up of a large number of different hydrocarbons and each of those different hydrocarbons has different volatilities and you could now get into volatiles that are actually volatilizing from somewhere near the core through the less volatile substance and penetrating through that and then coming out as gas right so one of the dramatic examples for that is for example if you now have something called aluminized gelled kerosene kerosene itself is a multi-component fuel on top of it you now let us say you try to gel it with aluminum and aluminum is less volatile obviously when compared to kerosene so what happens in these kinds of situations where you have a mixture of materials that are of different volatility is the more volatile substance so that the less volatile substance will actually begin to accumulate on the surface as the surface is regressing right and as it now accumulates and the surface is regressing and shrinking the density of the less volatile substance actually increases and it actually begins to form more like a shell for example in the case of aluminum it could even get sintered together like a shell and it is a and when it is it sintered it is like a porous structure through which you now have the more volatile substance get evaporated right at some stage this shrinking continuously will now cause this and then of course you have more and more aluminum reaching the surface right from within and therefore you now have this shell actually kind of hardened you do not necessarily have it porous anymore and you now have a gas because the more volatile substances actually evaporated from within the shell and it now starts acting up with pressure right so when you now tries to pressurize the shell the shell breaks and then it now becomes smaller fragments. So similarly you know this is quite dramatic when compared to that if you now simply have multi-component liquid fuel droplets that means you have everything is liquid nothing is really concentrating and forming a hot shell there but if you know look still look at liquids the more volatile once can actually volatile from the core and bubble through the the less volatile once and can cause what is called a secondary atomization that means it now shatters the less volatile once into smaller droplets which which which from a from a single parent droplet so the single parent droplet itself was actually obtained by atomization but then you now have a secondary atomization because of this so such things are possible when you now look at coal particles you have a similar scenario where it is not really the coal particle is not a single component substance right it is a solid particle all right and the typically you pulverize it into very small sizes but you are essentially going to have a bunch of different things first thing you know you have a lot of moisture which is pretty bad because this is not going to burn right then you have what is called as the volatiles which is what is exactly going to burn right and burn fast and or evaporate fast or get released into the combustion zone quite quickly then you have the minerals which are bad because they are not going to actually aid the combustion but they are going to actually finally bring in ash and what is called a slag so this ash could in the combustion zone become like a molten region and then as droplets or liquid molten liquid form they could go and get splashed against the combustor walls and so on so this produces ash and slag and then of course you also have char so this goes through a solid phase combustion and gives rise to a residue and so effectively you are now looking at only two parts of this that are giving rise to energy the other ones could actually soak up energy all right and this is pretty important because in the case of Indian coals which is an India really relies a lot on coal and there is a lot of cold reserves here for quite some time that we can burn but it is all very high ash content okay so the ash content could be as high as somewhere like 30 to 40 percent it is almost like we have coal and ash rather than ash and coal okay so that is a kind of situation we have as for a school is concerned similarly you can also think about this sort of a composition for biomass okay so if you now think about biomass which essentially is let us say the most commonly encountered biomass is wood but then you can also come up with lots of all kinds of things like for example even rice husk is considered like biomass okay so you can burn any of these things a lot of agro wastes can be used in incineration or gasification there are lots of technologies that are getting developed through these things so typically again you have to characterize what is the moisture content what is the volatiles content what is the minerals content what is the char content so doing this kind of thing is what is called as approximate analysis trying to find out what are the different percentages of these different things and then in turn you also have to find out what is the elemental composition of the energy containing constituents and that is what is called as ultimate analysis right so typically one of the major problems with biomass and coal more with biomass because you have varied kinds of biomass in the first place but even for the same kind of biomass and for coal the problem is different locations where you mine them or where you what do you call harvest them you will have different compositions of these so you have to do the approximate and ultimate analysis for these first to get an idea of what kind of energy content they have and what kind of ash problems you are going to face and so on in these things but then once you are in a position to actually do like pulverized coal particles then the analysis is not very different from what we have done so far and it all takes us back to having diffusion flames or premixed flames and so on of course these things typically will be turbulent flames and we will talk about a little bit about turbulent flames but to but within a gaseous framework okay in the next few days I will stop here for the day.