 Welcome back, we are going to continue our discussion of design of atomizers in the last two classes we looked at simplex or a precious swirl atomizer the different parameters that go into the design the degrees of freedom like we said the other day and the constraints that you have to have your design satisfy. So those are the that is essentially the objective of any design exercise that you take advantage of the degrees of freedom and deliver a certain performance constraint. We are going to continue that discussion today to look at drop size if you look at if you go back we really left off at a point where we were able to get a film thickness we did not go much further than that so we are going to today look at drop size as a constraint that you could place on the performance of an atomizer we will see how to make sense of it okay. So we are going to continue our discussion of design now the one of the ways in which drop size has been factored into spray equations is through the use of linear instability analysis we looked at that say for example in a simplex atomizer you may have let us say this is the exit part and if you did the reentrant trumpet face at the exit like we said you essentially have a liquid film that is exiting your spray nozzle and this liquid film itself could be swirling so if your liquid film is swirling and the swirling liquid film is subject to subject to some disturbances it could cause it could cause the liquid film to break down. So the way we suggested we could model this in one of the earlier classes is you can develop a dispersion relation governing the linear instability the growth of small instabilities on this system that dispersion relation gives you omega as a function of k k is your wave number we looked at at least 3 examples where we derived this omega the growth rate as a function of k and we find the value of k at which dfdk is 0 so that gives me k star this k star is the wave number for which the growth rate is maximum which means that in a very short distance away from the nozzle that particular wave number is going to dominate the structures the fluid structures in the liquid sheet or in the liquid film okay and so that gives me which also implies the 2 pi over k star is the characteristic size of the fluid structure that is most likely going to break off. So here is a way I can relate the nozzle performance or the parameters of the associated with the flow inside the nozzle to a structure that could likely break off from the liquid sheet which could further coalesce to form a drop further downstream. So to start with I have this linear instability modeling allows me to go from internal flow inside the nozzle to external flow we looked at the models due to Giffen and Murazov and Shu et al where you are able to take internal flow the swirling flow inside a swirl chamber and predict a film thickness a film swirl velocity a film axial velocity and those will form inputs to your linear instability model which could give rise to a dispersion relation that in turn gives rise to drop size. So this is following a track of almost completely analytically modeling the system. So we are only sticking to looking at the system as being enveloped by an analytical model the other approach the complementary approach has been to rely purely on experimental data okay. So while this approach takes us from internal flow in the nozzle to an external spray characteristics such as drop size okay using pure and using only analytical tools for the most part other than that 1.17 factor that we put in front of the CD equation if you go back to the last lecture everything else was purely analytical and the model worked even without the 1.17 fact correction factor okay you are just off by about 15% which is not too bad okay. The other complementary approach is using empirical evidence. So what do I do in this approach I have let us say a fluid line in which I am able to monitor the pressure I will call this P1 and this fluid is coming into some sort of a spray nozzle that is creating the spray and let us say the pressure here is P2. So delta P P1- P2 is the pressure differential driving this flow. So in return for the pressure differential I could get some Q which is a volume flow rate I could get a spray angle theta so I will call this theta over 2 and I could get some SMD which is this outer diameter if you go back to our old notation from PDF's SMD is essentially D32 which is defined as d cube times f of d dd divided by 0 to infinity d squared f of d is my drop size PDF and D32 is defined as the third moment which is integral 0 to infinity d cubed f of d dd divided by 0 to the second moment which is 0 to infinity d squared f of d dd okay now clearly you can see this has units of diameter and the this is this usually has a special relevance when it comes to sprays because this is the total volume per unit surface area we looked at this early on and the total volume per unit surface area has a relevance to a lot of spray applications. So I can now let us say we have looked at Q the volume flow rate in terms of the pressure and the Q is not directly related to the pressure but the way we found Jones correlation for example gives us a way to relate CD the discharge coefficient to everything else that is related to the spray geometry okay. So I can similarly find a correlation for the spray angle in relation to the other geometry features we in fact did that although we found an analytical expression based on inviscid flow theory that seems to work reasonably well and we also found that really speaking in a commercial spray nozzle they do not rely on the fluid mechanics inside the nozzle to give us a certain spray angle they use co and effect which is much more effective and provides much finer control over the spray angle at the exit okay. What we are going to do today is look at Sauter mean diameter and how Sauter mean diameter is in turn related to the properties inside the nozzle okay. So again like I said there are two approaches the first approach is using purely analytical tools such as inviscid flow theory and linear instability theory that does go fairly close to predictions fairly close to measured values but the best bet for most design level tools would be to use empirical correlations and so we are going to look at a few such empirical correlations today. So what sort of empirical correlations are available for SMD in simplex nozzles in precious whirl nozzles the one of the first in early works is due to this man Radcliffe who gave us this formula okay. So just to get the nomenclature right sigma is surface tension new L is liquid kinematic viscosity m dot L is the liquid flow rate mass flow rate and delta P is the pressure difference now we look we will study this a little carefully and see what we can learn the first thing that Radcliffe is trying to tell us is that surface tension seems to have a significant effect as far as the sheet breakup is concerned because the exponent associated with that has the highest magnitude 0.6 liquid viscosity on the other hand has a small positive exponent 0.2 which says that as the viscosity of the liquid goes up the drop size is expected to go up although not very strongly okay. As the mass flow rate goes up on the same atomizer now how does the mass flow rate go up on the same atomizer it goes up because delta P goes up there is no other way because this is only a single fluid atomizer. So as delta P goes up mass flow rate goes up and as a result the increased mass flow rate gives us a south remain diameter that is slightly higher but the delta P is raised to the power minus 0.4. So really speaking this is supposed to be valid over a large range of atomizers that go beyond just a single atomizer but if I were to apply this over one single atomizer I have to figure out how delta P is related to m dot L and that is usually using the Jones correlation using CD where you can relate the volume flow rate to delta P or the square root of delta P more precisely and that gives us some way to relate all these parameters. There is one serious problem and shortcoming with writing an equation like this that is the fact that an equation like this is dimensionally incorrect that is all of your quantities sigma nu L m dot L and delta P are all dimensional quantities. So this number 7.3 is expected to have some weird units to it such that the right hand side all of the quantities of the right hand side raised to those powers give you a length scale give you units of micrometers or some drop size unit which is what is on the right on the left hand side of this equation. So one of the biggest problems with an equation like this is that it is dimensionally incorrect while it is of engineering use meaning I can convey information to a fellow engineer through an equation like this but I will have to give the fellow engineer a lot of other information about what unit system should be employed for sigma for nu L for m dot L etc. Otherwise the number 7.3 would be different it would depend on the choice of the unit system. So there were right around the 40s 1940s and 50s there were a lot of these kinds of correlations that were developed that gave us dimensionally incorrect correlations that in order to reconcile we have to go to some sort of a non-dimensionalization. So that was the need for non-dimensionalization comes from needing to convey souter mean diameter like information without needing to convey a system of units that without being constrained to a single system of units. So let us see how to do that. If I go back to my simple model of a liquid film exiting a simplex nozzle and I want to make a list of all the quantities that are my design variables. So in this simple problem let us say list of dimensional quantities that are important you start to see rho the density of the liquid sigma L which is surface tension mu L which is liquid viscosity m dot L which is liquid mass flow rate UL which is like a liquid characteristic velocity and let us say d0 which is that distance exit or if is diameter these are all quantities that one would expect will play a role. So I am going to transcribe these back on the other end here so deriving dimensionally correct SMD correlations involves taking this list of quantities and finding a set of dimensionless groups that are important. So if I go through this I have 1, 2, 3, 4, 5, 6 if I go through the dimensionless grouping I am expecting to find 3 dimensionless groups which are important. How did I come up with this 3? I am expecting to find n-3 groups and for this case n equal to 6 and I have 3 primary dimensions. Now I can go through the so called Buckingham pi theorem analysis or I will just write down the result because this is standard UG classroom exercise I have the Reynolds number Weber number which is given by rho L and the last one in fact is like our CD okay. I will call this alpha which is essentially m dot L divided by rho L UL D0 squared coming to think of this the denominator here has units of mass flow rate it has units of kilograms per second the numerator is the actual mass flow rate m dot L okay. So if you go back this essentially represents is this is like our X which is the area of the air core divided by although not exactly the same it is like our X essentially it gives us the effect of the air core at the exit. Now if you go back I need we can now figure out a way to correlate all of these variables to their respective dimensionless groups and the one due to Wang and LeFevre is the most general in this regard and what that gives us T here is a film thickness that can be obtained from inviscid theory. Now these quantities that you see in this parentheses actually I need to place that so that gives me the dimensionless film thickness these quantities inside the square parentheses this part and this part are essentially dimensionless groups that are functions of the Reynolds number and Weber number. So some combination of Reynolds number and Weber number give us and alpha give us these groups. So there is two different approaches to predicting spray performance one or drop size in a spray actually one the use of inviscid flow theory inside the nozzle to give us the boundary conditions to go into a linear instability analysis and the length scales that are predicted from the linear instability analysis giving us initial estimates of drop sizes. The second approach is to purely go to an empirical correlation based approach where we relate the input parameters to the nozzle to the parameters of the performance of the nozzle meaning SMD. So SMD I can write down a correlation of the form SMD over D0 or SMD over T film thickness which is so I can either choose this equals C1 times RE power A times Weber number power B alpha power C. One of the biggest advantages of writing this down is that this would be dimensionally correct not just dimensionally correct but unit system independent and unit system independent because I can choose whatever system of units for the dimensional quantities the Reynolds number comes out to be the same. Now there are different studies out there where you can take a wide range of sprayed measurements and correlate this quantity SMD over D0 or SMD over T plotted versus RE and Weber number and alpha and these coefficients C1 AB and C can be obtained through regression. So now let us move on this is as far as understanding spray performance for simplex nozzles are concerned. So let us talk about air assist atomizers now air assist atomizers are much involve a wider class that includes the so called pre-filming air blast atomizer we looked at this design in one of our earlier lectures. So essentially what we have here is air coming down the middle and liquid that is spilling over from this passage. So you have a liquid film that is spilling over that you have that is impacted by by the air on the inside and outside. This is our simplest air assist atomizer let us take a case where there is no air on the outside. So we are going just for the sake of our argument we will keep it simple we will look at air only on the inside that is swelling and we have a liquid film coming out. So right away I know that one of the indicators of drop size is going to be this thickness T that is called the pre-filmer liquid thickness. The pre-filmer liquid thickness is going to determine the thickness of the liquid film that is spilling out of that annular passage and of course we have the liquid air velocities and liquid velocities. So if I go through the same argument as to making a list of parameters that are important. I have liquid density, liquid velocity, liquid pre-filmer thickness, liquid viscosity, surface tension potentially. But in addition I also have the air side properties the air density air velocity the air side length scale like a diameter of that inner air passage actually I could even include air viscosity but we are going to experiments have shown that that effect is small but let us just say we will keep it simple and this is going to give us 1, 2, 3, 4, 5, 6, 7, 8 dimensional parameters which means I am expecting 5 dimensionless pi groups. So I am going to write down the ones that people have shown to have a relevance. The first one is of course we will make it simple and define this based on D0. Now I also want to point out that Ohne's Orga number is very often defined as the square root of the Weber number divided by the Reynolds number. But if I have Reynolds number and Weber number as two independent pi groups then Ohne's Orga number is only formed by a combination of these two pi groups and therefore cannot be regarded as a third independent pi group. Now I am expecting 5 I have written 2 there are clearly others first one that is important here is what is called the air to liquid ratio which is the mass flow rate of air to the mass flow rate of the liquid which if I write in terms of which in terms of the liquid densities and liquid velocities can be written in that form although it is easiest to treat this group in terms of the mass flow rates itself because this is directly measurable. Anytime you do a test you have mass flow rate measurements and therefore it is much easier to relate them directly through the air to liquid ratio. Now there are also other dimensionless groups called the density ratio or the momentum flux ratio. So you have mass ratio momentum flux ratio density ratio these are all variables that are these are all pi groups that are independent of the others that have that have already been presented to you. Now I can form a capillary number just like that but that would be a pi group that is already that can be formed from the Reynolds number and Weber number. So these are not dimensionally these are not independent pi groups but the ones that I have listed here the Reynolds number Weber number and the rest of the groups here are all independent of each other I cannot form one from simple combinations of the others. But if I define ua to ul which is simple velocity ratio I can form this using beta and gamma. So if I take gamma divided by beta square root gives me this delta. So this velocity ratio would not be an independent pi group but beta gamma and the air to liquid ratio are all independent pi groups. So how do I relate this to performance we will take a couple of examples of correlations here is one correlation due to L chord et al that is dimensionally correct as you can see they only involve dimensionless groups and this number 51 does not depend on the unit system used that is the sort of acid test of what is the correct correlation. Now as the Reynolds number increases you expect the drop size to decrease as the Weber number increases the drop size decreases as the air to liquid ratio increases drop size decreases again sort of intuitively correct trends appear in this correlation when I include the effect of the so this is for a classic external mix air assist atomizer. If I want to look at the prefilmer effect of the prefilmer in the air blast system you can look at the favours textbook for a correlation based on his work DH is the hydraulic diameter of the prefilmer passage. So if I take the prefilmer this is the area of the prefilmer the hydraulic diameter is approximately equal to 2 times T. So this is essentially 4 times area divided by perimeter that gives us for a circle DH would exactly be equal to the diameter of the circle. The numerator here is the cross sectional area available to the fluid flow which is pi D0 times T the thickness of the prefilmer. The denominator is the total wetted perimeter which is actually equal to pi D0 on the inner side plus pi D0 plus the tiny thickness pi D0 times pi plus pi 2T the wetted perimeter in this case is that and so if I ignore the thickness in favour of the diameter I get 2 times pi D0. So the hydraulic diameter being equal to twice the thickness is a reasonably good estimate where the where the prefilmer thickness is small in comparison to the to the inner core diameter D0. Now there are two aspects of this that you have to take into account. This first of all is a very comprehensive correlation obtained from experimental data. That includes the effect of auras organ number, beta the density ratio and Weber number as well as the air to liquid ratio. So we made a list of five parameters the only one that is not included here is the momentum flux ratio and for this set of experiments they have not reported the effect of momentum flux ratio. Essentially if you look at this correlation as the air to liquid ratio increases this 1 over air to liquid ratio decreases which means the south remain diameter is going to decrease and even for an infinite air to liquid ratio even if you have a very high air to liquid ratio this pre multiplier 1 plus 1 over ALR only tends towards 1. So what we know there is that there is a limiting performance associated with increasing the air ratio that does not yield benefits beyond a certain beyond a certain value of ALR. So typically for a classical air blast or a prefilming air blast that is at about 10 percent. So beyond about 10 to 15 percent you really more air does not give you the benefit of that increased energy input in terms of reduced drop size. The second part that we want to observe is this Weber number and the power associated with that which is minus 0.6 that as the Weber number increases the drop size is going to decrease for the same air to liquid ratio. And as beta the density ratio increases again just for our own understanding density ratio is the density of the air divided by the density of the liquid. As beta increases the south remain diameter is going to decrease as the power 0.1 minus 0.1. Now a density ratio is important in an aircraft or a power generation application where this air blast atomizer is spraying into a high pressure chamber. Essentially beta is a reflection of the chamber pressure because the liquid density does not change much between atmospheric condition to the high pressure condition rho L remaining the same rho A goes up proportional to the pressure in the chamber which means that the same atomizer is going to give you a finer drop size inside the combustion chamber than measured outside in an atmospheric pressure test rig. And lastly there is this almost linear dependence on the owner's organ number that as the owner's organ number increases the south remain diameter is going to increase but the pre multiplication factor is very small and notice that this is an additive effect. So, in comparison to that pre multiplication factor the effect of viscosity is very small. So, liquid viscosity in general does not play much of a role in the spray process which is why it is which is why inviscid theories give us reasonably good answers when we use it for real spray applications. We will continue this discussion in the next class.