 So, let us come back we were working through a design procedure for the pressure swirl or simplex atomizer and we arrived at the following. We had an expression for CD which is our expression coming from the basic definition itself then we found that CD is equal to this quantity where x is a0a divided by a0 and from there we have k which is equal to ap over ds d0 being equal to pi squared over 32 1-x cubed divided by x squared r i divided by rs squared sin squared beta. So, for a given exit orifice diameter r0 if I assume r0 I can calculate CD and from the next expression I can calculate x and calculate k. So now I have some estimate of ap relation between ap ds and d0 I do not yet have unique values for those. So, let us see if we can go through an example to actually get to some unique values what do I have. Let us say I want to design a pressure swirl atomizer to operate from a supply of about 7 bar. So, just to give you some flavor for English units as well that is approximately 100 psi and I want a flow rate of 1 gallon per hour which is approximately 3.875 liters per hour of some very light weight fuel. So, that is almost like jet a essentially this is what is given to me. So, I say can you design a nozzle that will do this. So, what do I start with I start by assuming an r0 which I am going to say is about 0.5 mm for a flow like this and r0 of 0.5 mm implies a0 is pi. So, I am sorry if I choose the diameter to be 0.5 mm I am going to work this example out in metric units. So, we can all follow through q in this same units is 3.875 into 10 power minus 3 divided by 3600 meter cube per second. So, my first step was to start by assuming an r0 I have done that I have chosen a d0 of 0.5 mm I can now calculate a CD q is 3.875 into 10 power minus 3 divided by 3600 equal to CD times the supply pressure delta P is 7 bar 7 bar when converted to Pascal's is 7 times 101,325 Pascal's the density happens to be 875 and from here I can get a value of CD which will come out to 0.14. Now, if I use the expression that relates CD to x if I simplified I am trying to solve for x if I make some rearrangements I get an expression involving x alone from here I can find there is only one real root of x it happens to be 0.69 abbreviate that it is not really needed that I keep all those decimal places. So, x happens to be 0.69 now this is typical that like in this case I am finding that 70 percent of the cross sectional area is taken up by the air core and only 30 percent of the cross sectional area is available to the liquid film. So, let us make some simplifications x by definition is the cross sectional area of the air core divided by the cross sectional exact cross sectional area for the flow. So, this would be r 0 a square divided by r 0. So, r 0 a divided by r 0 is square root of x which in this case is 0.83. So, if I want to calculate the film thickness T is r 0 minus r 0 a I will just divide and multiply by r 0 and this is what I have r 0 is 0.5 mm divided by 2 I have chosen the diameter to be 0.5 mm times 1 minus 0.83. So, I end up getting film liquid film that is only 40 microns in thickness. So, the resulting drop size is also going to be much lower is going to be on the order of 40 microns at the very highest most of the drops are going to be smaller than this or on the order of 40 microns. So, here is a nice way to control the drop size where you had an orifice that was 500 microns the drop size is only on the order of 40 microns. So, that is the advantage of the basic advantage of using a simplex design. Let us continue our thought this is not you know the end of it. We have a value for x we have to find the rest of the design variables. So, if I go back to the expression I had for k which is given by this it is actually k squared is equal to this right hand side. If I substitute that I have k squared equals. So, I can take the square root of this in fact k squared if I do this computation it comes to be 0.0205 times r i over r s squared times sign squared beta. So, all I now have is I can relate this a p over d s d 0 r i over r s and sign beta all I have done is taken the square root on both sides. So, I have all of these design handles at my disposal beta r i r s a p d s d 0 etcetera. And I have one more relation between these variables you can notice that in this particular instance r s and d s cancel out to leave a factor of 2 d 0 is something we identified up front that is 0.5 mm. So, essentially I have an equation I have a way of relating r i to a p that gives me this flow rate. Then lastly if I take sign theta I can make the substitutions I know c d from the earlier calculation beta is a design variable that is still at our disposal time k that we have calculated from the previous step k is this 0.14 r i over r s sign beta. So, if I make that substitution this particular this cancelling does not happen automatically it just numerically happens to happen in this particular example. So, you will see that this gives us a value for theta that is almost 90 degrees with this particular. So, sign theta comes out to be very close to 1. I prefer using the expression for tan theta because that allows your you can without loss of ambiguity find theta that goes beyond 90 degrees. So, these are so this is how we started with the certain set of inputs in the form of the pressure the source pressure q the volume flow rate density and by assuming a certain exit or f s diameter we were able to go through the calculation to get all the way to the spray angle. So, we were able to find the sign theta or the cone angle for a given set of input conditions. Now, the we said we were also able to relate the SMD to the film thickness this can be done either through linear instability analysis we looked at ways of doing this or through empirical data. Now, the empirical data path has shown that you can find the SMD to approximately scale as t power 0.4. So, you have essentially two ways of going from the film thickness that you get from x all the way to the souter main diameter in the spray as well as the cone angle. Now, let us go back to some of these expressions that we derived actually that fell out of the analysis. If you remember this is what we had for CD and this is what we obtained from the so called principle of maximal flow maximum flow. In reality if you take a if you take experimental data measured x and CD. So, if I take a range of spray nozzles and I am able to measure CD on those spray nozzles by simply measuring delta p and q and measuring x by getting the air core diameter. And what we find is that this in fact does show a linear CD in fact does show a linear scaling with this grouping of x except it is off by about 15 percent in absolute value. So, this is the empirically corrected version from the theoretical expression. So, it is not off considering that all of what we did was starting with Bernoulli's equation and some simple physical arguments to have a design that is within 15 percent is quite remarkable. Now, another correction we need to make has to do with the fact that you have the we made the inviscid flow assumption. So, if we ask ourselves the question we will bring in the effect of viscosity, but in the form of Reynolds number mu is the liquid viscosity u is of course the speed it is not the axial velocity. There is an expression due to Jones which includes this effect of viscosity where CD is given to be 0.45. This is obtained on a wide range of atomizers mainly from the power generation sector and this is one of those correlations that has withstood the test of time. Now, I want you to draw your attention I want to draw your attention to some of the exponents in this correlation CD is essentially what we had for Reynolds number raised to the power minus 0.02. It is in fact the smallest in magnitude of all of the different exponents in the correlation and that clearly shows that liquid viscosity has no effect on the flow rate it is quite surprising and we talked a little bit about this as to why it shows this insensitivity. One reason is that as the liquid viscosity increases you have essentially an increased fluidic resistance because of which the axial velocity decreases and your swirl velocity is effected even more than axial velocity because of which the film at the exit is now thicker. So, you have two contrast two sort of effects that are acting to oppose each other fluid viscosity like always is trying to decrease the flow rate and that same fluid viscosity is trying to increase the film thickness which effectively increases the flow rate. So, these two effects counteract to give you a very weak exponent for the viscosity. So, in fact if I write this correctly the way CD is mu power 0.2 0.02. So, it is as the viscosity increases CD increases to the power 0.2. So, if any as the viscosity increases the flow rate through the nozzle increases albeit only by about only by that power 0.02 this is observed in experiments quite a bit in fact this correlation is based on experiments. Now, there are other terms in this other dimensionless terms in this correlation let us talk a little bit about those different terms L 0 over D 0 also does not have much of an effect on the flow rate. So, what role does it play in the design itself you essentially want to have a sufficiently long orifice for a given D 0 such that memory of the slots is wiped out. So, you have the fuel coming in the form of discrete slots. So, let us say you may have 3 or 4 or 6 or like I showed in the schematic two slots that are 180 degrees apart on the page. You do not want your spray to still have remnants of the number of slots that brought the fluid into the swell chamber. If you did that you would see essentially if I had 6 slots coming in I would if I took a circumferential measurement of the fuel flux the liquid flux I would see 6 angular positions where the flow rate was higher than at the other places which is not good for uniformity of the spray. So, I want the length of the orifice to be sufficiently large that memory of these slots is wiped out other than that it really does not play much of a role and. So, just as a rule of thumb anything that is greater than about 0.5 is sufficient I will put slots in quotes. Let us look at the other one which is L S over D S. If you go back this was our length of this swell chamber divided by the diameter of that swell chamber as you can see from this correlation it really does not have much of an effect on the flow rate again, but it does have an effect again on the circumferential uniformity of the spray that you get. Now some of these exponents should not come as a surprise to us when we go back to our inviscid theory. If you just think of the nozzle as an inviscid as if you go back to your nozzle as though you are only spraying an inviscid fluid the only time pressure or velocity change is if there is a change in the area of cross section other than that length of the orifice diameter of the length of the orifice certainly has no bearing there because you do not have any pressure drop associated with a long pipe in a flow of an inviscid fluid. So, as a result what you find is that all these length involve terms involving the length really have very weak exponents and this argument is based on inviscid theory, but if you go back to the Jones correlation it shows that the effect of viscosity itself is small which means even further that an inviscid fluid approximation is sufficient for a real fluid. So, all of our logical arguments which will work for inviscid fluids will also work for real fluids. So, therefore, the length of the orifice really has no bearing as far as the flow rate is concerned it has a very weak bearing, but it does have an effect on the circumferential uniformity of the spray. So, as far as L S over D S is concerned again the idea is only sufficient to wipe out. So, just the fact that I had these discrete slots coming bringing in the fuel I do not want those discrete slots to still have an effect as the fluid goes through the nozzle and comes out in the form of a spray. So, essentially this L S over D S being greater than about 0.2 is sufficient to ensure that that effect is wiped out. Likewise again in the same distinct alpha is usually on the order of about 90 degrees to about 120 degrees alpha happens to be the convergence angle going from the swirl chamber to the exit orifice. Now, again without any surprise C D is most influenced by this ratio A P over D S D 0. So, if I want a particular C D that is the parameter that tells me what it is that that is my design group. So, I can find a value for this A P over D S D 0, but I do not still have a way of identifying individual dimensions how do I do that say A P is N times W times H we wrote this down N is the number of slots W is the width of the slot and H is the height or depth of the slot I think we said D back there. Now, you can again see the only parameter that makes a difference to the flow rate is A P which is again N times W times D fluid mechanically it makes more sense to have W approximately equal to D. So, you are bringing in the fluid without any choice of choice of an aspect ratio choice of a preferred asymmetry to the fluid flow inside the swirl chamber. So, essentially square slots are preferred N is chosen usually in the square slots based on manufacturing issues say for example, I go through this design process and I am able to identify a value for A P. So, I can now choose N let us say as a number like 4 in which case W times D becomes I can find a value for the cross sectional area and from the argument that I wanted to be square I can find individual dimensions if N instead of being 4 became 6 then W times D is now smaller which means I have to now machine smaller dimensions. So, you choose N such that those machining that your current machining practice is able to machine those dimensions not just that you also do not want very very small fuel passages inside the nozzle because most of these nozzles especially the ones that go into combustion applications could be in a high temperature environment. So, you do not want any sort of a reaction to be initiated inside the spray nozzle although that is unlikely because it is being cooled by cold fluid all the time. But what you certainly do not want is some of those slots becoming clogged with dirt coming from the coming past your fuel filter most of these spray systems have a filter on the line before you come to the spray nozzle. Now, the filter is only able to tolerate certain classes of particles and the smaller ones do get through and these smaller particles are still likely to clog up slots in your spray nozzle even though the slot is wider than any one individual particle. So, let us say I have a fuel filter that is able to keep out particles on the order of tens of microns and if the slots are now let us say hundreds of microns in size I could have these ten micron particles actually agglomerate and clog up particles even though the slot is on the order of hundreds of microns. So, essentially N is chosen from manufacturing issues having determined N I can find W and D from the fact that I need a square slot I can find R I and R S. Now, if you go back to most of these the correlations that we had for CD and K you have this ratio R I over R S that occurs many times over. So, the closer you make that equal to one the more efficient the usage of energy because if I take a slot that is coming in at a certain radius R I with respect to the center, but I have a swirl chamber that is let us say much larger than this R I then R I the ratio R I to R S is essentially going to create a fluid motion inside the swirl chamber that is not useful to me. I have a swirl chamber that is much bigger, but I am bringing in the fluid at a at an intermediate radial location. If my thumb is the center line and if my index finger is the wall of the of the swirl chamber I want to be as close to the wall as possible. The only manufacturing problem with that is if I go to this very first figure that I had if this is R S if this diameter is two times R S it would be very small. So, it would be very difficult to draw a square slot where R I becomes very close to R S. So, some practical issues related to manufacturability overtake efficiency considerations and therefore, you have to live with R I over R S typically on the order of about 0.6 to 0.7 you cannot really go beyond that. Now, sin beta also has a bearing as you can see now this is the angle at which the fluid is brought into the swirl chamber I will go back to my very first sketch. If the fluid is brought in at some angle beta then the u times sin beta is equal to sin beta is the tangential velocity and u times cosine beta is the axial velocity that is coming in. So, beta is a nice direct way of independently controlling the spray angle because I can now control the swirl momentum flux to the axial momentum flux I can control the ratio of the swirl momentum flux to the axial momentum flux. So, beta gives me a direct handle on the swirl on the spray angle the cone angle much more than any of the other variables. So, that is the handle that is often used, but in reality again coming from manufacturing issues we find that there is a nice ingenious way of achieving a spray angle we will talk about that for just a moment. So, if this is my diameter d 0 let us say this is alpha. So, I have the swirl chamber upstream of the figure of the sketch that I have drawn here and the liquid is coming out let me draw this. The liquid is coming out in the form of a film and this may be the position where the air core ends. So, all of this is the air core. Now, what is usually done here at the exit is that a feature that looks like a trumpet is added on. So, it is essentially of some very large radius and axisymmetric about the center line when you add this feature smoothly to the exit orifice what you find is that this film sticks to the exit orifice. And by mass conservation constraints if you it actually becomes thinner because the cross sectional area available to the flow is now increasing the film by mass conservation becomes slightly thinner although that is not the intended effect it is a desired desirable side effect. Now, if I take these if I create a sharp edge at this corner this film is no longer able to make this sharp turn it is essentially what is called coanda effect in surface tension literature. Just as if I am trying to pour water out of a glass jug the water tries to dribble around the radius edge of the glass jug for have a sharp edge on the glass jug it is able to perfectly flow out it is entirely using surface tension. So, I can now have this liquid film depart from there and what I have what I have happening let me be more quantitatively accurate what I have happening down here is that I have I start to get a spray. So, by controlling this radius r and this depth delta you can see that if I take a certain radius the deeper I go the more radius the more of an angle I create which widens the cone angle automatically. So, by controlling delta and r there is a very nice elegant way of controlling the cone angle I can do whatever I want upstream and I can dial in a cone angle at the by controlling the exit geometry which is always good from a manufacturing perspective. Again one more little trick that is often employed which is kind of neat is that in order to further facilitate this coanda effect I can create a very sharp exit orifice. So, this is almost a sharp corner or knife edge. So, I can get a very precise spray angle for any flow rate that is upstream all I need is that I have sufficient centrifugal force and or surface tension coanda force one of the two is enough I can create if I have if I do not have enough surface enough centrifugal force upstream in this in the nozzle surface tension may be sufficient to cause this liquid film to stick to the wall as it exits the nozzle just like water being poured out of a jug. So, I can control the this is how typically the spray angle is controlled in a commercial pressure swirl atomizer. So, you design the spray nozzle for a given flow rate and film thickness. So, the one parameter that I still need control over from the geometry upstream is the film thickness which then has a role to play in the south remain diameter. So, I am able to dial in a certain flow rate south remain diameter cone angle for the spray for and then satisfy the given set of constraints. So, this is although this is how the design process for a typical pressure swirl atomizer has evolved and most of this literature goes back to at least 40 years the chew et al paper that I discussed is more recent, but it is more of a validation of the literature from back then in terms of the insensitivities of the flow rates to the parameters that we discussed, but if you were a spray applications engineer today one would suspect that you would not have to go back to the drawing board to do a design of this go through a design process such as discussed today. It would be sufficient if you go to a selection table to choose a nozzle. So, we did discuss that as well in one of the earlier classes that most of most applications today is essentially a selection not a design exercise.