 Okay, we will continue our discussion of drops and drop size distributions in sprays and we will begin with a few questions and try to answer them. Drop size distributions we came across spatial drop size distribution and a temporal drop size distribution. What is the primary difference between the two drop size distributions? Okay, the question relates to different measurement techniques of measuring drop size distributions in sprays. We discussed spatial sampling and temporal sampling, we want to see if we can maybe use an illustrative example to differentiate between the two in some in a but it will be a simplistic illustrative example but sufficient to understand the complexities involved. Okay, let us take a couple of mono disperse droplet generators. So these are mono disperse generators, I will number them 1 and 2. The whole idea of mono disperse means that they are producing drops of one size. Let us say these drop the generator 1 and let us we will just call this example problem number 1 for now. Let us say droplet from generator 1 is 100 micrometers. So essentially it is producing a stream of droplets that are all diameter D1 moving with some velocity V1 and they are produced at some let us say we will give these some numbers let us say 1 meter per second and the number rate of production is let us say 10 per second. So we are producing 10 drops of 100 micron diameter per second and each one is being ejected out from the needle at a speed of 1 meter per second in the axial direction. I want you to note that the number frequency is nothing to do with velocity, they are completely different measures of how fast the drop may be moving. Likewise we will say D2 is much smaller let us say it is only 10 micrometers. We will start by looking at the situation where V2 and N2 are the same. We will see what this generator would give us in the for a spray. So essentially my atomizer is composed of two little needles one producing 100 micron drops and another producing 10 micron drops. If I take a region of interest that includes the tip of this nozzle let us say something that is 1 meter in diameter and the width really does not matter this is the this is my imaging is the imaging interrogation area or imaging area. So that is the area that my camera is capturing. So a simple way could be that I have a backlight and the backlight is illuminating the drops and when I view from the front I see the shadow of the drops. So the individual shadows form circles in a snapshot in a frozen picture and from those individual circles I can get the size of each droplet. Once I am producing so let us see if each droplet is moving at 1 meter per second and the spacing between droplets is one-tenth of a second. So the production spacing between droplets is one-tenth of a second and so if I take one snapshot that is one meter in height let us say the time spacing between drops is one by ten seconds. This is coming from the fact that N1 I will call this N1 dot that is number production rate okay. So if I take N1 and N2 both are 10 per second which means I am producing drops one drop every one-tenth of a second which is the same as saying the time spacing between the drops is one-tenth of a second. Since the velocity of the drop is one meter per second if the interrogation area has a height of one meter the drop is going to be resident in there for one second okay. So the second thing is to do with the residence time is one second which means I am going if I take a snapshot I am going to have ten of diameter D1 and ten of diameter D2 okay. So if I now look at in any given snapshot I want to calculate let us say the simple arithmetic mean diameter which is sum over all the drops Di divided by N okay. If I do this I have ten times 100 micrometers plus ten drops of ten micrometers divided by 20 which happens to give me a number like 55 micrometers. This is the diameter I get I will put this DS for spatial sampling for the same situation if I do temporal sampling. So this is same spatial sampling mean diameter equal to 55 micrometers. For the same spray if I do a temporal sampling what do I get so the whole idea of temporal sampling is that I am going to remain at some cross section here let us say okay. This is my sampling position and I am going to accrue statistics of drops passing through this point I will call this cross section AA okay I am going to measure. So the next thing we will talk about is temporal sampling the steps involved are measure the diameter passing through AA over a fixed this is very important period of time okay. So what happens if I do this now I know that I am producing ten drops per second which means if I sample for one second I will do it just to move away from one if I sample for two seconds I would have sampled twenty hundred meter hundred micron drops and twenty ten micron drops. So again I can calculate that the temporal mean diameter so as it turns out the temporal mean diameter is also 55 microns. Now we have two parameters characterizing two separate measures of a time associated with the drop one is the velocity another is the frequency of production. So we will keep the same frequency of production and see what happens if we change the velocity that is our first test. So I will now take the exact same situation except I am going to make this V2 so I will call this my example two okay example one was fairly straight forward we got the same mean diameters whether you sample in the time in time domain or whether you sample in the spatial domain. So now if the drop is moving fast and I will go back to the same exact spatial now the drops are moving fast these drops are moving at ten meters per second in relation to these drops that are only moving at one meter per second but the time spacing in the production is exactly the same. So if I did a spatial sampling just like I did before consider spatial sampling first if I if the drop is moving at ten meters per second and the length of this whole thing is one meter it takes one tenth of a second to traverse the whole meter okay whereas I am producing one drop of ten microns every one tenth of a second. So what are all the possibilities for how my right side of the image will look like okay so I want to distinguish between the right and left side because it is like a very simple spray it is like a by this two mono dispersed sprays so super post and if we understand how this works we can get a feel for how the real spray works. So the drops that are ten microns in diameter are moving at ten meters per second that is a given drop is in this frame for only one tenth of a second. So I have three possibilities for the right half depending on how my one depending on whether I am when I take a snapshot I could have one drop somewhere in the middle of the frame like that okay and there is the time before the next drop is produced is like further in the future compared to my frame and the previous drop has already exited the domain. So I could have one drop in the frame or there is also a very remote instance say for example if this was not one meter but point nine meters okay or slightly less than one meter I could also have the instance where the drop has just exited and I have not yet produced the next drop on the right half so I will have no ten micron drop I could also have again if it is slightly greater than one meter I could have the instance where I have one drop just about to exit and another drop just about to be produced. So really speaking I have this one drop in the frame somewhere in the frame or I could have no drop or two drops these are all remote possibilities and only when l is not equal to one meter exactly okay if l equal to one meter then if this is exactly mathematically correct then it has to I will only see one drop the drop has to exit before I see the other drop okay. So really speaking the only possibility we want to look at seriously is this one drop of ten microns that is the only possibility on the left hand side nothing has changed I will still see ten drops right because the time production of each drop is the time separation between each pair of drops is one tenth of a second and they are all queued up one behind the other so in one meter physical space I will have ten drops one behind the other if they are all moving at one meter per second okay. So essentially I could have if I do the spatial sampling correctly ten hundred micron drops and one ten micron drop so if I do my spatial mean average approximately 92 microns if I do the time sampling let us see what we find so temporal sampling on the same system is based on the fact that I have one drop coming through here every one tenth of a second and so if I again sample for two seconds I will get ten or ten times to twenty hundred micron drops and twenty ten micron drops for a sample for two seconds which means again if I do the so if I look at what happens what is it that I am measuring when I do spatial sampling what is it that I am measuring when I do time sampling if I time sample at for two seconds that means I am getting twenty drops that are moving that are at hundred microns now you have the sampling and it is like a sampling cross sectional area in which I am allowing these drops to go through so every time a drop goes through I have a way of getting its diameter measuring its diameter so the rate at which drops can go through here in a steady spray has to exactly equal the production frequencies whatever be the production frequencies that is what I end up measuring here okay. So what you get in a time sampled in the time sampling in the time domain is it is basically related to your n1 dot and n2 dot we can look at an example three where we go back to v2 being one meter per second and n1 dot or n2 dot being let us say hundred per second or one per second okay you will see that so let us actually complete that example just so we gain our complete understanding of the whole problem I want to go back to example three where v2 is back to being one meter per second but we end n2 is now down to one per second when you do that you end up seeing now I am going to erase this part here because it corresponds to the previous example but what do I what could I possibly look for when I do spatial sampling in this the fact that I am only producing that all the drops are moving at one meter per second okay so in any the drop is resident in this frame for one second but I am producing only one drop per second okay. So if I look at what I will end up observing so if you look at the rate of production of the drops of size two in this example three it is ten times slower than the rate of production of drops of size one so if I do spatial sample sampling I am producing one drop per second and that one drop is going to move at one meter per second that is the drop that is produced will remain I only have one possibility that there will be one drop in the frame the other three where it is just about to exit or just about to be produced are sort of rare events so two drops and no drops are both rare events so in the in the one meter frame that I have the only possibility is that I will have one drop resident in there it is moving at one meter per second and it is going to take one second to traverse through the frame and it is not it is going to be one second at after this drop is produced that the next drop will be produced so whereas on that is as for that is the story for the left half for the right half for the left half the story is not changed so if I do the spatial sampling what I have is ten drops now does the size of the domain make a difference so if I have instead of one meter if I sampled over two meters will that make a difference chances are again the drops are spaced so the for any given drop di okay so let us say in this case I take a situation where vi divided by ni dot is and let us say l is the sampling domain height I will take a situation where I have l is much much greater than any of vi divided by ni dot okay so let us say I will recreate this situation where vi is not one meter per second but let us say ten centimeters per second or one centimeter per second and ni dot is like one per second or ten per second and we end up packing the drops much closer now okay will this if l is very large in comparison to the max of vi dot vi divided by ni dot what you end up seeing is that your you will have these kinds of possibilities of whether I have 99 drops or 100 drops on the right hand side as opposed to 1 or 2 so the percentage error associated with the number of drops being slightly different on one side versus the other keeps going down so the spatial sampling gives you a result that is closer to the production rate sampling to the time rate sampling think of an extreme situation where I could sample an infinitely long domain in which case every drop produced is all is at the moment present in the domain so if I took a snapshot of this domain it cannot be different from having sampled every drop produced in time so essentially when we say spatial sampling and temporal sampling they are different as far as finite times of sampling are concerned it is only when I say I am sampling over one meter sampling over two seconds that you will get two separate answers if you go to the extreme case of sampling over an infinitely long domain or and an infinitely long time those two answers are exactly the same those are true representations of the production rate that is the SMD or that is a mean diameter based on the production rates is what I can completely tied to the atomizer performance everything else could be an artifact of the way I am sampling in the spray right so essentially if I take these two sprays and if I have L be much greater than V I divided by N I dot then the spatial sampling and the temporal sampling become very close to each other and I will say that both of these would end up becoming equal to the production sampling actually production sampling is something that is already exactly equal to the temporal sampling but just to sort of have a physical feel for sampling in time being different from sampling at the nozzle exit if I now look at I can now come back and quantify this effect of the size of the domain so in going let us say instead of one meter if I went up to ten meters same exact system what are the possibilities that are the possibilities that I have here instead of three possibilities I actually have many more but we will rewrite this if I am sampling over a ten meter long domain same rest of the performance characteristics are the same I am producing one drop every second and I have ten drops of N I N 1 dot being produced so essentially I have ten drops that are ten microns in size and ten drops that are hundred microns the spacing between sorry one ten so I will have hundred drops that are hundred microns in size the spacing between a pair of drops being the same for the L 1 is 0.1 meters I will call this L 1 so if the size was ten meters I will still end up getting the proportion of the drops D 1 to D 2 would still remain the same because the physical spacing between them is still the same. So the mean drop sizes tends towards the temporal sampling mean drop size the spatial mean drop size tends towards the temporal drop size as long as the sample remains distinct as long as the sampling domain is of a finite size as you go to a larger and larger domain size the two values converge to a particular unit.