 Hello at the end of the last class we made a list of a few applications of sprays and what we are going to do today is sort of zoom out a little bit and look at spray as a entity spray is a collection of about a million billion may be even a trillion drops okay and we want to see what sort of measures can I bring to a spray to gain a quantitative understanding of a spray. So we will start to make a list of a few different spray characteristics. Now before we turn to the audience and try to ask this question of them I want to suggest that there are basically two kinds of spray characteristics one that are macroscopic in nature and a second that is microscopic okay so we will make a list of a few different macroscopic qualities first and then move on to the microscopic qualities we will just take a steady spray so as opposed to a little perfume spray perfume spray is where you push the plunger down a whiff of perfume comes out there is a start and an end to it okay so we are going to talk about a spray that has no start and an end but it is just like a continuous perfume spray it is just easier to understand these qualitative aspects in that context then we will talk about an intermittent spray also and talk of some qualities of an intermittent spray as well okay so let us start listing a few different macroscopic qualities. So we will just say spread or span of the spray spray angle very nice okay I will also distinguish this between this and cone angle okay we will talk about that also penetration length droplet size okay we are going to list this in the I will distinguish this because this is going to occur in both the microscopic and the microscopic list okay so we will talk about talk of this in just a moment microscopic measure okay any others that we want to include velocity so we will just say velocity as a again a microscopic measure so we have already made a list of a few different things here the so and I want to add one to it which is one that is relevant to many different processes actual mass flow rate of the liquid we can add a few more but this is a good enough list okay so if I just take a typical spray let us say I have a spray nozzle from which I have the spray emanating now first of all I have these all these drops in the spray we have seen this many times the actual flow rate coming out in a steady sense is the liquid mass of the liquid flowing out per second this is relevant to surely many applications okay now talk of velocity there are two kinds of velocities okay the reason I wrote down microscopic measure is because I want to know sort of in a bulk so if I take a whole bulk of this spray what is like a velocity vector of that bulk okay but really speaking it is not of use to know what this whole bulk is doing I want to know what a what different points in the spray are doing okay so therefore there is a need to understand sprays using microscopic measures back to the mic microscopic I just want to complete the list here spread or span like for example that width at a target surface say for example if you are spray painting a wall I have the paint can about let us say 10 inches away from the wall when I push the plunger and this is a continuous spray because I can just walk along the wall and paint the wall when I am walking along the wall there is a certain width that the paint impact will cover that is important because I want to know how many passes I have to make of this can so if I look at a measure like that that gives me a width and clearly that is also tied to the cone angle or the spray angle so there are two kinds of angles and we will talk about them maybe about a week from now but essentially if I call that angle to theta to theta is what we will call the spray angle the cone angle is slightly different we will look at that in some detail in the sense that the angle very close to the nozzle here would be slightly different for different fluid mechanic reasons and that is often referred to as the cone angle so you have a cone angle that is usually larger than your spray angle penetration length now again some of these measures are all tied to each other because if I take a spray of a certain mass flow rate and somehow increase the width which also means increase the spray angle I am naturally going to decrease the penetration length because the drops on sort of the mass of drops are going to slow down so the penetration length is a measure of that all right so let us put this intuition to work we will look at a little video we made in this in our lab here this is a simple perfume spray that is going to go from start to end okay so let us play this video and see what we learn from it I want to pause it there to point to a few different things you look at you look at that angle there that is what that is the spray angle now as you go further downstream you can see that the spray angle is becomes less and less defined so as I go further and further downstream the spray angle becomes less and less defined because there is no spray edge the idea of a spray edge is an illusion it is not like the edge of this pen it is where the edge of a pen is well defined edge of a spray has to be defined by us and our definition of the spray edge is going to also influence our definition of the spray angle so if I say I want to go as far as I find no drop at all that is going to give me an almost 180 degree cone angle in many different sprays it is going to be a very large number in comparison to where I said I am going to go to a point where the mass flow rate drops below some critical value so mass flux okay that is the rate of mass flow per unit area drops below a certain value so I will say okay when there is very very few drops almost a mist I am not going to count that in my spray that is a simple definition of a spray edge we can find more quantitative definitions as well but our idea of a spray angle is clearly tied to that okay all right so here let us go on and continue to play this video I want you to understand two or three different aspects here you can see that there are some big white flashes right where my mouse is pointing okay those are in a sense big drops so I can go back to the start and show you where they started you see that that drop there that started is going to continue forward I want you to notice two differences here one of course this is not a steady spray but this idea also applies to a steady spray so the concept of having big drops alongside tiny tiny drops is not something that is out of the ordinary every spray of any commercial interest will have a range of drops that the spray has so typically the range is you is on the order of at least two or it is at least two orders of magnitude in span more often it is three orders of magnitude in span so the idea that you have drops that are ranging from nearly let us say a fraction of a micron of micrometer in diameter to hundreds of micrometers in diameter is not at all out of the ordinary in fact you can see this in this image okay now as you keep going you can see that the perfume has practically stagnated around and there is not much movement of the drops past that point so you spray and the spray goes into air and sort of the drops reach due to the drag from the surrounding air they sort of equilibrate they may continue to move forward by momentum conservation but it is diffused quite a bit so your spray width starts to increase as you go far downstream okay so now when I look when I go back and look at that like I said there are many different microscopic properties that are actually more important and relevant to a spray like for example at every point we said you are going to have a drop size distribution so I said point but we will qualify that in just a moment the idea that a spray is composed of several sizes of drops this is sort of obvious to us by now not just that that there is also a velocity distribution again I will qualify this to clearly distinguish it from the air around okay we are looking at droplet velocity distribution so you have drop size distribution and droplet velocity distribution that are all microscopic properties so these are now like I am looking at the individual droplet level I am almost looking at somehow understanding the spray from within as opposed to from without from outside okay what other microscopic properties can I list here I can look at in a typical spray like a perfume spray we looked at this yesterday as well in the last class we are interested in like a rate of evaporation or temperature dependence which is clearly temperature dependent okay so from there I can say somehow I am also there is also this idea that there is a droplet temperature distribution okay so these are all in some sense individual variables like a drop a single drop has a size it has a velocity and it has a temperature these are all at the droplet level they are properties that you can assign to a single drop now at the spray level what do I how do I convert this droplet level information into a spray level information okay there are two ways of doing it one I do this I do just this so I sit here play this video or I have a spray like this going I pause okay I pause the picture and then look at all the drops in this frame now this particular video is not particularly good to zoom in and zoom out although you could to some extent it is not quantitative but clearly that is just a question of camera resolution so if I have a sufficiently high resolution camera and these are not out of the ordinary also I can pretty much image every drop in this frame right now okay and from the image of every drop I can reconstruct the size of every drop so let us say we did this calculation last time there are approximately a million drops in this picture right now probably slightly less than a million I can image every drop get a size and get million numbers okay from that I can construct a histogram of what to do with this so once I go to the million drops in the picture so sample so let us first pose a question and then we will answer what is drop size distribution so I want to understand what is what do I really quantitatively mean by drop size distribution and how do I measure it that is what I want to get to we will spend quite a bit of time on this how do I measure it thing but I want to first start with what is drop size distribution and then that will naturally lead us into how do I measure it if I so I can take a freeze frame count every drop in the frame and measure its size okay this will give us essentially like a million numbers or slightly less than a million numbers and from there I can construct a histogram we are all familiar with the concept of a histogram okay I want to spend some time on that because it is there is some very important mathematical tools that we need to acquire to completely understand what the real histogram and what we can do with a histogram okay so let us say I do this histogram and it gives me something like this so this is diameter and this is count I can get the count versus diameter of every drop in this frame and that will give me a distribution clearly this is a drop size distribution in the spray this kind of an information is what we will call spatial drop size distribution because my independent coordinate of acquiring this information was x and y space another way of thinking at it is thinking about this is for every drop that I that I capture or count in the freeze frame I can also get its x and y coordinates okay so that all that automatically means I am in the process of acquiring the spatial drop size distribution because those x and y coordinates would be different for different drops okay another way of getting the same drop size distribution is to sit at a point in the spray so look at where my cursor is I will sit at that point in the spray and I have a way of just you know sitting in a launcher and counting every drop that is going by me okay so I am not moving in spatial coordinate I am just sitting at that one point and acquiring information of the droplet size droplet velocity droplet temperature if I want to really get specific about the spray of every every single drop going by me okay so let us let us look at that another way of doing the same thing so I can look at every drop in the spray that is going by me and get its drop size velocity in this case it could be a vector it could be both the x and y coordinate and temperature if I really want to get fancy of every drop now the independent coordinate that distinguishes different drops here is the time of arrival there are succession of drops coming through so every drop is going to have a different time stamp of when it arrived at my location okay as opposed to the previous way of sampling drop size distributions where I had a the distinguishing feature it was as x and y coordinate of the drop okay so I can now take the same let us say I sit there for 10 seconds and in 10 seconds like you know typical perfume with is like let us say 2 seconds and in 2 seconds I was able to sample let us say again a fraction of a million drops and I have now statistics of size velocity of every drop coming through there so I will just for now ignore the other quantities and look at only the size to sort of illustrate the point so I can now do the same histogram okay I will again the independent coordinate of this histogram is the same draw I am counting the number of drops in a certain bin okay now this distribution is what we will call the temporal drop size distribution okay so we started to talk of microscopic measures of which drop size distribution is one measure and when we when we listed this here it was actually a fairly simple thing to list you know there is different size drops we want to understand some information about the size drops we quickly found out that there is not one but there is two different drop size distributions now we are left in a limbo okay are they the same or under what conditions can we expect them to be the same under what conditions will they be totally different because they contain different pieces of information okay so before we completely understand under what conditions are they the same and under what conditions are they different okay I want to sort of make the equivalence between these two ways of sampling the same problem sampling the same physical system you are essentially measuring something about a physical system there are two ways of understanding any physical system okay one is what is called a random field approach especially a statistical system like a spray that is like some sort of an uncertainty some sort of a you are you have to resort to probabilistic measures at some point and under those situations there are two ways of doing it either a random field approach or what is called a point process approach okay we will talk about this in some detail but I want to sort of draw the equivalence between a field approach that is where I am looking at a spatial distribution of some quantity and a temporal distribution of the of another quantity as being two completely different ways of looking at the same system so in general we do not expect them to be the same they are completely different okay under what they will present one simple situation under which they will exactly be the same so if I say for example when are the the same is a very restrictive criteria okay I want to start with that because it is the easiest to understand they are exactly the same when you do two things one if I take a spray and in this case I cannot do a spray like a usual spray I take drops in a box so this is a box that has all these drops in it let me not let me not clutter the box with it okay there are different size drops if every drop in this box was moving to my right with exactly the same velocity okay so all the drops are moving to my right with exactly the same velocity then if I had a way of staying at this location I will call this XX sample drops so I am not sampling drops at a point but I am sampling drops passing through a certain cross sectional area and this is also possible measurable and quite routinely done so if I am able to sample drops passing through a certain cross section and I and I construct a drop size distribution from the sampling at a certain cross section this is like my temporal drop size distribution and if all the drops were moving with exactly the same velocity okay then essentially in the time let us say the width of this box was some delta divided by you is the time is the time taken for all the drops to pass through my cross section so whether I take a snapshot at some time t equal to 0 of this box or and then construct a spatial drop size distribution from all the drops in that box or whether I sit at location XX and sample all the drops come through that cross section and then construct the same distribution I will get exactly the same answers okay so really speaking there are two assumptions I made not one one assumption is that all the drops are moving with the same velocity okay I want to I showed you where I am going to show you where I cheated you into another assumption okay so one assumption the second assumption is this let us say this box is my frame correct this is the frame that I used to construct my image frame that I used to construct my spatial drop size distribution and then I am sampling for a certain time t capital T that is going to tell me the time over which I can sample to get the temporal drop size distribution okay so if delta is the width of this frame and capital T is the time of sampling it is only when capital T is exactly equal to delta divided by U that I get this if I choose capital T to be some other number other than delta divided by U let us take the simplest situation where capital T is less than delta divided by U okay what happens if capital T is less than delta divided by U I would have only sampled a part of this frame I would not have sampled the other half of the frame or other part of the frame which means now is there a guarantee that the drop size distribution I construct from half the frame is the same as the drop size distribution in the full frame there is no guarantee I could have had a completely different mix of drops in the back half of the frame I do not know that right so this criteria that capital T has to exactly be equal to delta divided by U so in other words if you made a freeze frame of a spray and you reconstructed a drop size distribution even if all the drops were moving with the same velocity I am constrained to sample at this capital T only to get the exact same distribution which is again very restrictive so T has to be exactly equal to delta divided by U this is not so obvious okay now clearly the first assumption and the second assumption are different in the basic philosophy one assumption number one I have no control over what actually happens in the spray all drops have to move at the same velocity I do not get to control that but the second is second looks more like a measurement thing okay the second looks more like so even if all drops are moving with approximately the same velocity can I make this equivalence the only way to make that equivalence is if somehow even if you make spatial drop size distribution measurements with some delta I make with another T okay I do not know what delta you used but I make it with some other T what sort of delta and T will give me approximately the same information okay or at least on what sort of delta and T relationship will give me information will allow the two kinds of distributions to tend towards each other the answer to that is as I keep making my delta larger and larger so your initial spatial size distribution is composed of a very large frame and you had enough resolution to actually size every drop in the spring and then I can take a time sampling that is also very large so I can sit there forever and ever and ever and then sample all the drops so if you if delta became infinitely large and T was also infinitely large all I need to know is then they are going to be approximately the same given that the drops are all moving with the same velocity okay so essentially the second criteria is an artifact of the idea that we are constrained with finite domain sampling and finite time sampling okay the fact that I cannot sit there and measure forever and ever I have to do my sampling in a finite time likewise my camera cannot zoom out to the entire region of the spray I am only constrained to a small part of the spray so these are the sort of the assumptions that underlie the equivalence between the spatial and the temporal drop size distributions now what about velocity let us talk a little bit about velocity as well and then we will move on to the third part so we will look at what velocity distribution is again if I go back to my image I can pause the image I can this is a paused image I can take another freeze frame just like this a short time thereafter and use some fairly simple algorithms simple or sophisticated to find the displacement of a drop so it is like we are going to get into this in some detail towards the end when we talk of measurement techniques but essentially I can image a drop image a set of drops now and a short time later and from those two images I can reconstruct a velocity field okay so one way is take images a small time apart another way of doing the same thing see how this is equivalent to the spatial information what distinguishes every particle here is its original x and y coordinate in the image so if the original x and y coordinates are of every particle in this in the original frame are different like we started to do with let me discuss with the drop size distribution the second way of doing it is exactly like the previous temporal size distribution which is I sit at one point and measure the velocity of every point coming through every drop coming by me okay so I can sample the second way so these are two different ways again like we discussed with drop size distribution it is not obvious that they have to be the same in fact in the case of velocity there is no reason to even believe that they will be the same because drops at different points are different in other words I cannot even start to ask the question under what conditions will these two be the same because they are two completely different pieces of information temperature and other like we discussed towards the end of the class last time we talked of concentration as being a parameter that qualifies each drop so if I have a fuel that I am spraying that has multiple components concentration of a certain component in a given single drop is a measure is a scalar measure that is associated with that drop it is like a scalar property not a measure scalar property associated with that drop so I can have this temperature is also another scalar property associated with the drop vector is a velocity is a vector property associated with the drop size is a scalar property I can get into lists of vector and scalar properties associated with these drops but the point is all of those properties can be measured in a spatial sense or in a temporal sense okay. Now the basic requirement for equivalence between these two is that we will only focus on the first one that all drops have to be moving with approximately the same velocity if not exactly the same velocity that principle is very often violated in a real spray and therefore these two measures are completely different and that violation is quantified in terms of what is called a size velocity correlation correlation the word has a meaning that there is somehow there is a if not a causality meaning somehow there is there is a sense that a certain size drops can be expected to move with a certain velocity that is known a priori okay that is the idea of a correlation. So I can now take this spray for example and when I played you can see in here that the larger drops for example right in the middle here are moving with a slightly higher velocity compared to the smaller drop at exactly the same spatial location or very nearly the same spatial location. So at least as far as this spray is concerned larger drops are moving with a higher velocity in comparison to the smaller drops and that is basically information that I can take to if I measure that in this perfume spray I can transport that information to another perfume spray that is somewhat similar in construction and I can apply this information to other processes as well where even if the nozzle design was not the same fluid mechanically if they were similar then larger drops in that other spray can also be expected to move with a higher velocity in comparison to the smaller drops and in general in many many different sprays spanning many different nozzle designs at any given point the large drops will tend to be moving slightly faster than the smaller drops on average okay. So this is the idea that this is a very important word because I cannot draw any conclusion on any two given pairs of drops if I give you two drops at a particular point and ask you the question is the larger drop moving faster can you for sure predict if the larger drop is going to move faster you cannot make that prediction all you can say is on average in a fairly large sample of similar drops similar large drops similar small drops the similar set of large drops is expected to move slightly faster than the similar set of small drops that is the only prediction you can make and that is the only idea that you can take away from a size velocity correlation all of the measures that we talked about all of the spray characteristics both macroscopic and microscopic that we listed thus far are only sort of steady spray characteristics. So these are we have discussed them thus far in the context of a spray that has no beginning and end in time and that is relatively unchanged in time okay. So let us sort of make sure we completely understand what we mean by steady because it is a very important you know we say we are making the steady flow assumption all the time let us understand what we mean exactly by that word steady spray in the next few minutes let us take the spatial distribution as our as our means of understanding steadiness. So I take a picture and I have reconstructed this drop size distribution I take a second image and a short time later let us say one second later and these two images are exactly alike now clearly they would not be exactly alike all I have to say is they are alike. So without going into the microscopic detail of which drop is at which x and y location in the image if I can take two frames okay that I took some time later and I am unable to tell which one was taken first so the timestamp the timestamp is indistinguishable the timestamp is not encoded into the picture itself that is the case of a steady spray okay. So if I go back to my old high speed video if I took this picture and then this picture the first image had the spray only about that far the second image has a spray almost further downstream that is the nature of our perfume spray works this is the clear a clear case of violation of my steady definition but if I drag this further down into my high speed video and now play it I took this image or I will go back okay I took this image and then another image a short time later now this is still a transient spray as far as my eye is concerned when there is a start and end but there was a time in the middle say about one second when I could not tell two images that came a hundred milliseconds apart okay so these in that time span there was a time span in the case of a perfume spray when I can treat the spray to be relatively steady okay so basic definition of steady okay for the case of this class okay is that I if I take two pictures of a spray and I cannot from all macroscopic measurements okay I am starting off with a less constrictive criteria first and then we will talk of the microscopic case I cannot tell which one came first okay then I am beginning to make the case that they are steady then if I make microscopic measurements of the drop size distribution so I am now recovering the statistics of the spray be it temporal or spatial if I can still claim that I cannot tell the difference as to which one came first or which one came next from the microscopic measurements of distributions I am still okay I can still call it a steady spray when I can begin to tell the difference like the first example we saw with a startup I can from a macroscopic measurement of a penetration length tell you that they are not the same I can tell which one came first so that is not the that is a clearly unsteady spray so if you take a clear case like a diesel engine where you have your injector operating a few hundred times a second can I treat can I use any information from all this literature concerning steady sprays for diesel sprays the answer is yes the answer is pick up that time window where you cannot tell the difference and there is a time window as long as you restrict the application of these models to describing the spray in that time window you are doing quite okay so just as a quick recap we talked of different measures or different characteristics of spray and then we talked of spatial versus temporal measurements and the equivalence thereof okay and then finally we made the case for a steady what do we mean by steady spray okay we will continue this discussion in the next class.