 We are going to continue our discussion of droplet evaporation and lead into droplet combustion and spray combustion. So, we will look at the different physical processes underlying evaporation followed by combustion leading towards an understanding of combustion spray combustion today. At the end of the last class we had derived the famous d squared law where we showed how under quasi-static conditions the square of the diameter scales linearly with time decreases linearly with time and the rate constant is given by this KQ, this KQ which can be written explicitly in this form where By is a transfer number that is based on the difference between the mass fractions at saturation and mass fraction at infinite which is in the ambient and so essentially this ratio of the ratio of these mass fraction combinations occurs in this natural logarithm and that determines the rate of decay. There is one point I want to make here that the difference in the saturation concentrations or saturation mass fraction versus the mass fraction of water vapor at the infinite condition occurs inside a natural logarithm. So, for this quantity to be very large even if By is very large since it occurs inside a natural logarithm its effect is attenuated quite a bit. So, the point to make here is of all the properties that influence droplet evaporation it is the diffusivity that is the most important. The diffusivity of water vapor in air is going to significantly affect the value of KQ much more than the value of the saturation concentration versus the concentration of water vapor in the infinite ambient. This has a direct consequence to the lifetime of the drop. So, if Tl is the lifetime of the drop D0 squared divided by this KQ essentially determines the lifetime and if you take two drops one evaporate evaporating in a very very dry atmosphere and one evaporating under not so dry atmosphere. As long as By is small in comparison to 1 you will see that D0 squared is affected primarily by the diffusivity. Now and then we went on to include a correction to this correlation correction to this analytically derived D squared law to include effects of advection. So, if I have if I take a drop and flow past the drop at U infinity the drop itself is some diameter D0 I can define a Reynolds number given by that quantity and the rate of evaporation of this drop in this co-flowing air stream is also expected to follow a D squared law with a modified rate constant KU and KU is given by this relationship. Now this has no theoretical basis it is essentially an empirical finding while the finding that D squared decreases linearly with time for a quasi static in a quasi static evaporation case is entirely based on theoretical arguments in an advection case. The fact that D squared law seems to hold well is purely an empirical finding it has been validated in many many different instances that the square of the diameter decreases linearly with time even in cases where they you do have an advection. Now there has been other modifications for this recently there was a work by Wu et al in the international journal of heat and mass transfer I believe 2003 although I am not sure of the year where they provided a modification to the first link correlation in terms of a turbulent fluctuation the ratio of turbulent viscosity to molecular viscosity. So in other words if I went back to my advection problem I have a drop of some diameter D0 in a co stream of in a laminar flow of U infinity mean flow U infinity I will get a certain modification of the K value. Now if on top of this I had a U prime which is a turbulent fluctuation that would require an additional modification using the Damkohler number. Now why am I talking of all these in the context of sprays in the context of droplet evaporation primarily because evaporation is at the end of the day diffusion limited the process of mass being moved away from a liquid vapor interface into the bulk ambient is purely by a diffusive process and this diffusive process is accelerated if I have some kind of a turbulent fluctuation in the mean flow field if I have a purely laminar flow pass the drop I am only going to have to rely on molecular diffusive processes. So this ratio of the turbulent diffusivity or viscosity to the molecular viscosity or diffusivity is essentially the is another parameter that accelerates this diffusion process and therefore the D squared law rate constant has to be modified for the presence of turbulent diffusivity in the process. Now these are all still single droplet level phenomena at the end of the day all we discussed as far as phase change is concerned is with in the context of single droplets. Now is this how accurate is this in the context of sprays to understand that let us continue our discussion of the single droplet phenomena for into the case of combustion of single droplets. So we looked at evaporation of single droplets now we look at combustion of single droplets and the process essentially is where you take a liquid drop undergoes a phase change process to produce a vapor field and this vapor field uses oxidizer and there is a mixing process here follows is it precedes combustion which is essentially gives rise to heat release. Up until now we looked at this physical process now this physical process is significantly enhanced if I have if I am in the presence of a warm temperature field. So like for example if I have spray if I am spraying kerosene into a flame the flame has already created a high temperature field around these liquid droplets and that high temperature field is going to accelerate this process. Now that process involves three phases so if I now take this phase change itself it has three parts to it one it is called the initial heat up followed by vaporization followed by diffusion. Now if I take all these processes including heat up vaporization and diffusion if I purely include vaporization and diffusion in the in a hot field where the source of energy for this vaporization process is my is the temperature field around the droplet then I can still write a d squared law I will call this k combustion and that k combustion is given by so essentially in the context of combusting droplets one can write a modified d squared law again these are all the original d squared law is a theoretically derived version but it has been empirically validated in many many different situations where those theoretical arguments may not hold water rigorously and that is the beauty of this law. So now I can write a bc delta h combustion is the enthalpy of combustion or let us say delta h over nu is a specific enthalpy of combustion typically in units like kilo joules per kg cpg is the specific heat at constant pressure of the gas phase t infinity minus ts is a temperature difference between the ambient and at the droplet surface. So essentially if I look at this and of course I have to also define my lambda g here the thermal conductivity in the gas phase. So if I look at this expression for kc the only process that limits this evaporation the rate of evaporation is now controlled by lambda g where in the past we had the diffusivity. There is an important assumption underlying this called the unity Lewis number assumption. So Lewis number is essentially the ratio of thermal diffusivity also known as thermal conductivity to mass diffusivity. Notice how the units on all these are meter squared per second in SI terms. So the thermal diffusivity to mass diffusivity ratio is what is responsible for the rate of thermal diffusion or thermal transport from the ambient gas to the droplet which is also the same as the limiting factor of the limiting process in terms of the vapor diffusing away from the liquid into the ambient gas. So when we take this Lewis number and say Lewis number is order 1 essentially that thermal diffusion process and the mass diffusion process have are controlled by the same a parameter that is on the same order of magnitude. And therefore under this condition whether the process is limited by thermal diffusion lambda g lambda sub g thermal diffusion to the liquid drop or mass diffusion away from the liquid drop the 2 are the same order of magnitude. So you will see that in the rate constant kc the mass diffusivity which occurred in the previous arguments is now replaced by thermal diffusivity. It is actually not thermal conductivity but thermal diffusivity I think this is conductivity but if you go through the arguments we had a rho db dab that rho has been absorbed into giving us the diffusivity as a net result. So we now are at a point where I am able to introduce a droplet into a hot atmosphere and estimate its lifetime or the rate of release of vapor from that droplet. This is followed by the vapor phase reaction we are not going to discuss the reaction part as it is the purview or it is out of the purview of this current discussion. What we are going to do is understand that the reaction process where you have fuel let us say gasoline vapor reacting with oxygen in the air and to produce heat and the products CO2 and water vapor is essentially a gas phase reaction. So as far as the physics of that process is concerned it is no different from methane combustion where you have CH4 reacting with let us say oxygen in the air producing CO2 and water vapor and heat. So the process of methane combustion and process of gasoline vapor combustion are physically identical although the parameter space in which they operate could be different. So for the moment we are going to leave that out of the discussion. We want to understand the effect of the heat release on the droplets or the rate of evaporation of the droplets which is what we just posed in the form of a modified d2 law. Following this evaporation process you have a heat release and that heat release is going to now control the temperature field around this droplet. Now we have understood the process of vapor release and we said the vapor reacting with the oxygen is similar to any other gas phase combustion reaction. How does now various point sources of this vapor which is essentially what my sprays behave? So if I take a real spray I am going to start with dilute spray and then work my way towards the dense spray but how does the dilute spray look like and how does the dense spray look like as far as what are the differences between these two different kinds of sprays as far as my arguments for the combustion process is concerned. So if I think of a spray for a moment the lots of drops that are produced and these drops I am drawing a cartoon just to show the difference between the dilute and the more dense spray as you can see the one difference between the dilute spray and the dense spray is this distance to the droplet size. So if I take a ratio of ds to d0, d0 is like my mean drop size. If I take a ratio of this ds to d0 if I look at the limit where alpha is very large that is the droplet spacing is very large in comparison to the size of the droplet. I am going to create vapor field which is essentially superposition of evaporation processes happening from multiple drops. So each drop is evaporating in some mean temperature field completely unaware of the presence of the other drops. This is the limit when alpha is much greater than 1 as though they are unaware of the presence of the other drops. When alpha another way to think of this is if I take an individual drop of some diameter d0 I am going to create and I plot the mass fraction as a function of the radial distance. I know I am going to create an exponentially decreasing mass fraction and I am going to assume for a moment that this YA infinity is 0. So if I plot YA as a function of the radial distance I have an exponentially decaying distribution and that comes from my earlier relationship that I had derived which is given by this relationship. This is my YA as a function of the radial position. You can look at this there is an exponential decay of YA as a function of the radial position. It is actually not just an exponential decay it is a slightly complicated function where YA goes as e power minus m dot over r and m dot itself we found is determined by this parameter. So essentially for a given YA infinity and YA s I have m dot that scales as rs and this YA as a function of r scales as that rs over r. So if I now plot this I can think of a length scale over which I can think of a length scale over which the mass fraction goes down by a factor of half. So this is analogous to our half life in other typical decay processes. So I can think of a length over which the concentration gradient or the mass fraction gradient takes the mass fraction from the saturation value. So this is my saturation mass fraction value to a value that is half the saturation mass fraction. So this distance I will call this half the r sub H. r sub H also is important in terms of the ratio of r sub H to the droplet spacing itself I call that d sub s. So again if this r sub H over ds I will call this some a second ratio beta. If beta is much less than 1 that means r sub H is small in relation to ds that is each drop is again uninfluenced by the presence of the other drops. This is important because what where when alpha is very large and beta in this case defined as r sub H to ds when these are very small I have a regime in which I can treat individual droplet as though it is there by itself. So we have all the theory and empirical evidence that we have garnered from analytical studies as well as from all the empirical data we have gathered those can now be extended and applied to spray combustion situations if these ratios were in the right regimes. Now as far as dilute spray combustion is concerned this does hold true. So these alpha and beta do default to the appropriate limits in the dilute spray. So when the droplet spacing is in general very large you are quite okay. So in that regime I am going to take the vapor concentration field. So this YA so superposition when alpha is greater than 1 and you have to ensure that both of these are valid. You cannot just take one or the other and assume that what you find and assume that you can use single droplet studies superposed to study a spray. So if I want to look at evaporation in a spray or spray evaporation can be studied by superposition of single droplets when alpha is greater than 1 much greater than 1 and beta is much less than 1. So if this were not to be the case then I have to start to look at interference effects. So if I now look at a single droplet say of some diameter D0 thus the vapor surface is at concentration YAS so this is the saturation mass fraction lines of constant YA will now look like circles around a single droplet. This is not separate I mean this is essentially this is coming from the fact that YA is only a function of the radial distance and not anything else. If I take two droplets and do the same exact thing I take one droplet here the lines of constant YA are still circles around each drop and these are decreasing YA these are each circle is a contour of constant YA and of a decreasing value of YA away from the center. So if this if the droplet spacing is very large in comparison to the size of the droplet or this R sub H so this is where if you think of if this is a contour of YA equals YA,S and if this is the contour of YA,S over 2 both of these and if they do not intersect all I am saying is they do not intersect by a large distance what we are saying is that R sub H is much greater than is much smaller than D sub S the spacing between the drops and in this case I can use superposition of single droplets to create a vapor field around the drop. If I now look at multiple droplets coming together this may not necessarily hold true so if I take a situation where two droplets are spaced at some DS again but you can notice how DS is less than in the previous schematic and if I draw lines of constant YA. Now the two drops are not entirely independent of each other so the first drop the rate of evaporation of the first drop is affected by the presence of the second drop. So if I call this my first drop and this my second drop the 12 o'clock position to 4 o'clock position or 12 o'clock to you know 6 o'clock position on the first drop sees a different concentration gradient in relation to the concentration gradient that 6 o'clock to 12 o'clock the other half of the drop sees and so this causes a different rate of evaporation on the two drops which means that I still may be able to use like a D squared law but this is now a function of the droplet spacing like a mean droplet spacing and this is again an area of active research to extract information that relates to multiple droplet effects. In other words when I have drops that are overlapping in terms of their contours of constant mass fraction when I have significant overlap in those contours then each droplet is not evaporating independent of the rest which also means that I do need to think of multiple droplet interference effects or at least two droplet interference effects. Now this particular there have been several studies on this to think of to study evaporation effects or interference effects due to in the evaporation field. Now just to extract the effect of the droplet spacing on the rate of evaporation now again like we the where we need this information is where if we want to move away from our so called exact formulation in the multi phase models. If we want to give up the exact formulation and look at either a mixture or Eulerian multi phase or Lagrangian interaction we do need this kind of information especially in the Lagrangian models you are now not just limited by two droplets coming together spatially you are now limited by the sphere of influence of each droplet coming under the sphere of influence of the neighboring droplet. So the interference effects when you include evaporation in a Lagrangian model are significantly higher than in a non evaporating Lagrangian model. So if I want to use if I want to look at employing multi phase models to study sprays propagation and combustion I do need this level of information at the micro scale constitutive level. So at two droplet level I do need to know what the droplet spacing does. So at the larger level when I know the droplet spacing on average I can use this two droplet interference effect to get order epsilon information where order 0 is saying pure superposition no interference effects order epsilon is where you are saying that alpha is no longer much larger than 1 but or rather beta is no longer 0 but is some small quantity in relation to 1. We will then be able to estimate the effect of beta in relation to beta being 0 which is pure superposition of single droplet studies. So we have now looked at what dilute spray combustion spray evaporation looks like dilute spray evaporation looks purely like superposition of droplets then extending it more and more towards the nozzle we say when the droplet spacing becomes significant in comparison to that R sub H I do need to consider evaporation effects interference in the evaporation processes ok. And then when I come even closer to the nozzle is it possible that I have really no evaporation at all that is I have produced a very dense concentration of droplets and due to some process if the air interstitial air between the droplets if the interstitial air is completely saturated with vapor then my transfer number essentially becomes 0 in that case when the transfer number becomes 0 I no longer have any evaporation process possible ok. So we are now sort of moving from the dilute regime closer and closer to the nozzle dilute regime is essentially superposition of single droplets I am saying that very close to the nozzle it is possible that you really have no evaporation at all past a given point where you have saturated the interstitial air with vapor ok. So given this whole range of physical manifestations let us look at what spray combustion could look like ok. So we are now going to move on to our last point of argument which is to look at spray combustion this is adapted from a very important work by Chu et al in 1982 where they passed the different range different modes of combustion in among droplet clouds into 4 possible regimes. The first is essentially what we are familiar with as a single droplet combustion mode where you have droplets with individual flames. So for example I have a droplet and this droplet has a flame around it and multiple droplets each has a flame around it this is possible if the droplets are separated by a distance now the position away from the droplet at which this flame is stabilized is given by the point where the mixture ratio between the fuel vapor concentration and the air concentration is appropriate. So for example very close to the droplet I may have a very high concentration of fuel especially fuel vapor especially if I have a very volatile fuel and away from the very far away I may not have enough vapor diffused to create a zone which is ready for combustion. So where this the delta the standoff distance between the flame and the droplet is given by essentially the region where the mixture ratio is close to the equivalence ratio one could think of the equivalence ratio as being sort of the most appropriate concentration of fuel equivalence ratio being one is like the most appropriate concentration of vapor vapor and air for now we are only going to use qualitative arguments just to understand the different modes of spray combustion. So this is what we see individual droplets with the tiny flame around it in fact it is common observation that when I when we have a kerosene lamp you occasionally see a spark fly off and that spark is essentially either a burning droplet or an incompletely burnt droplet that still has some chemical energy remaining in it to cause a visible flame. So this kind of individual droplets flying off is what we are used to seeing in like a kerosene liquid fired flame when you may have an occasional spark fly off. The second regime is what is called internal group combustion inside the spray. So let us see what this looks like this is the case let us say where I have a bunch of drops and the drops are evaporating in the presence of a flame but the flame is sitting far away from the drops and this flame could be radiating heat or transporting heat somehow to the droplets which are in turn evaporating. So these droplets are evaporating because there is heat transport and the interstitial air is still unsaturated at that temperature. So you do have an evaporation process that is taking this vapor release process forward but the interstitial space is sufficiently small that I do not have a flame residing inside the spray ok. Now it is possible that some of these droplets which are being transported in this direction are incompletely evaporated. So I may end up with a tiny spray around it a tiny flame around it. So this is the case where we have gone from a pure dilute regime as we come as we make our way towards the nozzle you may have a visible flame but some droplets are incompletely evaporated and escape the flame boundary causing individual sparks which are these droplets with burning droplets essentially but the bulk of the vapor release is happening inside the flame regime inside the flame zone. The third is similar to this except I now have so I have the same U infinity process I have some transport of droplets but I can now clearly identify a region inside which the spray is too dense for evaporation. So this region is where I have a very high mass constant very high number concentration of droplets to the point where the droplets spacing is less than r sub h the distance required to avoid interference effects between droplets and that when you are in this too dense for evaporation regime you tend to have these are non vaporizing droplets that during the course of let us say a typical spray which has a diverging angle which has an angle of divergence to the cone encompassing all the droplets. So the cross sectional area available for the droplet flow increases and as a result the droplets spacing is increasing and in that increasing droplets spacing sense you tend to create vaporizing droplets. So I have a green boundary line that I have drawn in green color here now mind you these are schematics these are just a sort of a physical mode for our understanding mathematical modeling at this level involves a lot more is a lot more involved but this is essential before we go to that level. So this too dense for evaporation regime followed by vaporizing droplets followed by a flame zone that is completely separate and the flame zone is only supplied vapor. So the flame zone is purely sustained by fuel vapor transported into the into the flame with no droplets escaping the flame. This is our ideal situation in any kind of a spray combustion application. The fourth regime which is often called and by this is called this mode of spray combustion is called external group combustion and this is also an external group combustion the fourth mode is also an external group combustion without the too dense for evaporation region not going to draw a schematic essentially if I take the vaporizing droplets region and extend it to extend it all the way to the nozzle. So even very close to the nozzle I have not saturated the interstitial air if that be the situation we would essentially have vaporizing droplets emanating from the nozzle and this vapor these vaporizing droplets supply fuel vapor to the flame and the flame itself is completely separated physically separated from the droplets and all the droplets are completely evaporated by the time they get to the flame. So we will continue this discussion in the next class to look at how one would mathematically model these different regimes and then conclude our discussion with some applications of spray combustion.