 Good morning again at the point where we left off last week we were talking about the difference between probability and probability density. How probability can be the theory of probability or actually the whole concept of probability can only be applied to where the number of outcomes is finite and countable. Whereas you need to go to a concept like probability density when the outcome is on the real axis. So, any real number between some limits is a possible is a possible choice for an outcome. So, without going much into the theory and this is actually a an unresolved question in space we will for now assume that drops can be any size between some minimum size and a maximum size. The minimum size could be 0 maximum size could be infinity for all we care. But we will just assume that they do have some finite limits and even otherwise nothing in what we discussed the other day is going to be different as long as there are some limits on which these two on which all the drops exist. So, we will continue that discussion and start looking at sprays in some detail. So, we know that we define a pdf. So, f of x is a function and f of x times dx is the actual probability of finding the value in the range x to x plus dx. So, that is if this f of x is defined in this fashion then f of x is called the probability density function. So, this we looked at the case the other day when we looked at a circular hoop and we said what is the chance of one of the angular positions showing up on top and really speaking the probability density could be any value. In fact the probability density does not have to be between 0 and 1 because this is the actual probability of any set of outcomes has to be between 0 and 1. But the probability density itself can be any number we will see in a short while. In other words the function has no upper limit on the values and it can take as opposed to probability fine. So, applying this principle to sprays what sort of intuition can we bring to the forms of the function that this f of x should have in a general spray. Let us start thinking about that for a little while. So, if I will now use a slightly different notation I will say p of d p of d is the probability density function of say a drop diameter in a spray and I will be even more specific we will just say at a point for now at a spatial point. So, this is sort of the easiest to understand we will start with this. So, if I did my temporal sampling. So, I was sitting at one point in the spray and I sampled every drop that went by me and I accrued statistics of a large number of drops and we discussed how to construct a probability density function from a large set of such drops. So, I did that and I got this function p of d p which is a function of d what sort of functional forms what should the graph of p of d look like. So, if d is generally speaking what would you expect the graph to look like let us draw some simple features the first sort of physical limitation is that the drop size cannot be less than 0 correct. So, on the right side on the left hand side of the graph the probability density clearly has to be 0. Now, can I find a drop of size 0 there is no such thing as a drop of size 0 the drop is of size say some minimum possible size in the spray. Now, we will see that in we will see may be in a couple of classes from now that that number is actually a number that is much greater than 0 much in the sense of atomic scales. So, if one molecule of water is a few tanks from the drop of a drop of water the smallest drop of water that you can create using a spray process will be much bigger than this size and there are reasons for that we will talk about that, but let us for now say there is a d min below which I do not expect to find any drops what about a d max let us talk of the limits. If I take a typical spray we looked at the perfume spray in some detail there is a hole through which a hole on that perfume can through which drops are delivered into the spray right. Now, is it reasonable to say that I there can be no drop bigger than that hole diameter. So, in other words if I have an injected diameter I cannot produce drops greater than that injected diameter strictly speaking I am wrong I can produce drops greater than the diameter if I have other processes going on later on. So, if two of these drops want to come together and coalesce I have no reason to prevent that from happening. So, technically speaking there is I cannot say that I cannot find drops greater than that size all I can say is the probability has to continuously decrease as the size decreases as the size increases. So, as the size say for example, if this is my orifice all I can say is as I go further away from this d o this probability has to somehow decrease. So, that is going to have to be the form of the graph going away it has to asymptotically reach 0. Now, this is as far as our understanding without really having a spray in front of us no data nothing say now can I actually allow this graph to come intersect. So, if I is this is this a reasonable possibility that the graph will intersect this axis beyond that the answer is that there are no drops greater than that particular size all I can say is I cannot make this generalization for every spray. So, at the moment whatever we are saying we want to leave it sufficiently flexible to be able to apply to every spray that is sort of our approach for now. So, at the moment all I can say is that this has to asymptotically approach the diameter axis. So, the probability density asymptotically becomes 0. What about the shape in between this and this? Now, first of all I know the shape of the graph near d min it has to increase that is the only way it can go probability density also by the way cannot be a negative number right just like probability. So, probability density is a positive number with no upper limit lower limit is of course 0. So, with these as a thing now I can there is only one possible general shape for this graph if I say this is the starting and that is the ending there is only one way the graph can go which is if the probability density function is a continuous function and differentiable everywhere this is one possibility which means that there is one point which is the at which point the probability of finding a drop around that point is the maximum this is not the average drop size. So, we will look at that in just a moment this is the what we will call the most probable drop size. So, you can sort of just although this is not a widely used way of referring to it we will very soon make the distinction between average and this most probable drop size. But so essentially I have this function p of d which could be which now has these general features if I want to write a mathematical equation that fits this there are many different mathematical forms that are possible that initially increase and then decrease with or without with d min being 0 with whatever it is there are many mathematical forms that are possible as many mathematical forms. So, many proponents in the literature for those mathematical forms meaning this is there is at least like 5 or 6 very widely used forms of distributions that people claim are fundamental to sprays and each of them has their own valid reasons for those claims. So, this is but this is the shape that is generic to all of those mathematical forms we will come to that in the next class. But before that I want to define a few parameters that are not dependent on the specific mathematical form. The whole point is that ultimately if I say what is it that I have achieved by defining this I have defined the spectrum of drop sizes that I can encounter at one point in the spray over some period of time how useful is this very useful because I now know what the spray looks like from the inside like we said before right. But in many instances this is too much information I want to condense it if I have to now move to another point I have to find this all over again for that other point. So, I want something that is more manageable and therefore, going from this full this is the full probability density function we want to define we want to define what are called moments of the distribution. The simplest moment to define before we go to defining moments I want to make one obvious point one more time that if I take any of these the area under this the area going 0 to infinity of that probability density function in d d it is a little tricky, but I cannot help it I can probably make the point over here as well just to sort of because integral 0 to infinity f of x d x. Because f of x d x is a probability of finding the draw finding the point in the set of outcomes x to x plus d x if I do this addition over all the say all the x that amounts to integration and that is what I get this is the only requirement of any probability density function all right. So, I now want to define a set of moments of this the simplest is what we call the average diameter you know I will move away from this d d business and I will say moments of a p d f f of x and I will define x is drop size in this case first constraint is that f of x d x equal to 1 and then if I define a mean drop size let me write down what this is. So, the probability of finding a drop in the limits x to x plus d x is that so that probability multiplied by the drop size itself added over all of the various sizes possible is what gives me the mean drop size. So, just to sort of illustrate the point if I have let us say different sizes x i x 1 x 2 dot dot dot some x i x n I have n drops of sizes then the mean drop size for this sample we all know would be x 1 x 2 that all you are saying is that each drop has a probability 1 by n. So, you are essentially multiplying 1 by n times x 1 plus 1 by n times x 2 now if some other if x 1 and x 2 are the same for example, in this set then that would become 2 by n x 1 2 by n x 1 because your 2 drops of size x 1. So, essentially what we do by average is the same as what we are doing in the sense of probability density. Now, we are going to get used to this notation of using the angular brackets to denote mean. So, this is often called the first moment I can define any moment of this distribution by doing exactly that. So, if I take x power p, p could be some number then x power p just like if I have drops of size x 1 x 2 x 3 etcetera then I can define if I say what is the average surface area of these drops. If I choose p equal to 2 in this case that gives me the mean surface area barring some factors like pi or pi by 4 etcetera you know that does not I mean we are not really worry at that level of detail and we are not concerned about that also usually. If I choose p equal to 3 what do I get I get the mean droplet volume. So, I can define any of these moments you know I can define a p equal to 4 now what would that physically mean I do not know, but I am sure there is a reason to define it and people actually use it will see in a moment one such use. So, what we want and by the way p equal to 1 gives me the number based average. So, if I said p equal to 1 that is simply the mean diameter I can define an average called x p q from this same f of x as the following. So, if I say x power p f of x d x integral 0 to infinity divided by x power q f of x divided 0 to infinity if I raise this whole thing to the power 1 over p minus q. Let us think about what this is x power p this is the p th moment and the numerator is the p th moment the denominator is the q th moment of this same distribution. If I take the ratio of the two that gives me sort of the value of the p th moment in comparison to the value of the q th moment that is essentially what ratios are, but the funny thing is if x has units of say micrometers or millimeters x power p has units of millimeter power p x power q has units of millimeter power q. So, this ratio will actually be a dimensional quantity of units millimeter power p minus q which is like a funny unit I do not want to deal with that. So, if I raise this whole thing to the power 1 over p minus q that gives me something in the units of diameter. So, this is also some moment of the droplet size distribution and is a diameter and it has the same units as diameter. So, I can now relate to it physically that is the only reason to do the power 1 over p minus q. Let us look at some simple possibilities. If I said q equal to 0 and p equal to 1 we get our first moment p equal to 2 q equal to 0 gives me the second moment just the way we have defined, but if let us say p equal to 2 and q equal to 1 what do we get raise to the power 1 over 2 minus 1. So, this is essentially the mean surface area divided by the mean diameter. So, going back to our original this thing I want to now write everything in terms in this p q notation x 1 0 is the mean diameter you can see that x 2 0 actually let us be clear about this I am sorry we will say this is the mean. So, just simply a number based diameter because I am just doing an arithmetic average of all the drops that is essentially what x 1 0 is if you go back look at this notation here of x p q. If you look at this notation of x p q the numerator is x f of x d x which is just like saying I am doing an arithmetic average over all the population of drops that I have x 2 0 is the mean surface area based diameter and just for the sake of equivalence I am going to do a 1 over 2 minus 0 to make it look like a diameter. So, I have taken the mean surface area in the spray at that point and divided and taken the square root essentially to give me back a number that has units of length of size likewise x 3 0. So, these are all diameters that is the point to note here they are all they have the same units as diameter, but they are all different physical quantities because now the x 3 0 tells me sort of their average volume. So, it tells me where the volume is concentrated another x 3 2 I want to write this out. Now, this has a special name this is often called the Souter mean diameter. Now, when you see Souter mean diameter in the literature they do not use x they use d for diameter I am just using x because that is easier to do d x in this kind of notation, but imagine d 3 2 you will see this written very often d 3 2 is this Souter mean diameter at that point in the spray. Now, this is very widely used as a measure of a spray now remember it is just another moment of the p d f that is how you get the Souter mean diameter, but what makes this number so special it is essentially the fact that the numerator is the mean droplet volume and the denominator is the mean surface area. So, both those quantities being physical if I want to imagine a situation where I want to release the volume contained in a liquid drop into a gaseous medium it has to go across this interfacial area. So, any sort of a mass transfer problem that is the simplest to imagine, but it also applies to heat transfer any sort of a interfacial transfer problem involves volume because that is your source of either mass or thermal storage in the form of rho c p rho v c p or m c p that is the total thermal inertia or thermal storage capacity of a given drop. So, that m or rho m or v is an indication of the volume in the which is in the numerator and the denominator is what the mass has to go through the total interfacial area that is available to a certain given drop for that mass to become let us say evaporated or heated or whatever interfacial process the has to happen. So, this is essentially also referred to as the mean this is essentially like a mean volume per surface area for the drop or 1 over this is the mean surface area per unit volume you know in our very first lecture we said the whole objective of a spray is to increase surface area here is a direct measure surface area of what surface area of a given volume of fluid. So, here is a direct measure of that increase in the surface area per unit volume. So, if you in some very crude way take 1 over this and multiplied by your volume that you had in that 1 with that automatically gives you the total surface area that you have generated. So, that is this is the this is one of the more commonly used measures another sort of commonly used measure is called the debrock your diameter notice that if p minus q is equal to 1 and automatically you have units of diameter. So, this is used in cases where in like spray drying where evaporation is important. So, it is like x square gives you the surface area and x power 4 is a moment that is related to it. Although it is really speaking this outer mean diameter is what you will see reported in many many different applications. In fact, some of this theory of you know handling distributions came about even before they had the idea of characterizing sprays using distributions. The first origin of idea of distributions came from pulverized coal when power plants when coal fired power plants were being built the most you know some it was figured out that the most efficient way of burning coal was to pulverize it to tiny particles and then burn it. So, I now had to have some measures of the of the you know what the result of my pulverization process. So, all of these distributions all of the moments that you see here d 2 0 x 3 2 or d 3 2 were all developed for particle sizing in coal in pulverized coal applications and they are just as applicable for this case as well. So, here we have I have what I am claiming now is that using let us say I will start to number this just to give you a quick count of where we are let us say there is 1 there is 2 there is 3 I will also add x 2 1 here which is the mean surface area per average drop. So, how much so that is like a another measure of surface area per average diameter there is 5 and there is 6. So, we are now somehow claiming that instead of giving you a whole pdf if I give you either these 6 numbers or even a few of these 6 numbers you can have you can get a reasonable estimate of what that distribution will look like without really having a full functional form at your disposal. Knowing the limitations that we started to talk of if I just tell you an x 2 0 and did not tell you it was from a spray really could not make much of it. If I tell you it is a if I tell you it is a drop size distribution then you already have all this intuition that we started with that the drop size distribution has to tend towards smaller and smaller values as you go forward as you go towards infinity it has to have an almost finite cut off although d min can be 0 there is no reason to not be 0, but we usually find it to be a finite value and it has to reach a maximum value at some point and then drop off. This intuition taken alongside that number now has now tells you some shape of this distribution like for example, I can clearly say I mean the simplest intuition to start with is to this would have a greater x bar you can already see that. So, if I just tell you x bars of two of the spray at two different points you can imagine what these distributions would look like they have to be something like this. Now spray themselves spray itself is not a one dimensional entity I am not talking spatial dimensions or time the set of drops at one point in the spray are characterized by more than one variable we looked at a whole list of variables in the very first class. So, I could have diameter be a characteristic temperature is another scalar characteristic concentration is another scalar characteristic velocity is a vector characteristic I can treat the velocity as being let us say at most three most three more scalar characteristics scalar properties. So, I could have all these scalar properties of a drop apart from the size itself. So, if I want to understand what the set of drops look like in that space I need to go to multivariate distribution. Say for example, I will simply I will take a simple case of drops in a pipe let us say they are all moving only in the x direction there they could be of different sizes, but they are all only moving in the x direction. So, I have a velocity u. So, if I do what I told you before if I sit at this location y y or b b I will just call this b b if I sit at b b and sample all the drops going by basically measure its velocity and diameter I have n drops I can sample the diameter and velocity of every drop I can now construct a pdf in two dimensions in d and u. I will put this in quotes using the word dimension only to show the idea that I have two orthogonal directions in which pdf could vary the function itself is a function of two variables. So, essentially it is a two you have two independent variables in the argument list of the function which is d and u. So, if I will try to draw a quick cartoon sketch of this. So, essentially this f of d comma u now becomes a surface. So, I could have in this direction it could be something like this in the other direction it could be something like that I am just drawing sort of images sort of shadows of this distribution in two different directions these are all the I could have this surface in the in three d in f of d comma u is a function of d and u and this surface gives me the total it gives me the probability density in this two variables. Now, again all by doing this I only made the problem more unwieldy. So, I need to I am better of going back to my moments. So, I want to see if I can define some moments of this 2 d distribution in the context of diameter d being diameter and u being velocity. So, the simplest thing is to say that if I want the average diameter just like. So, let us before we go this far I want to rigorously define what this is can let me just quickly rewrite this in terms of x I think that makes more sense x is size u is velocity. Now, the mean size in this at that point is simply the integral of this. So, if I take this x f of x comma u d x d u this gives me the probability f x f of x comma u times d x d u is the probability of finding a drop in this in this range that in the set of values that that I have defined. If I do this two variable if I do this integration that gives me the mean value over all the sample of drops this is like just like an average thing except since I have a second variable of integration u I have to do it I have to do a double integral. Likewise, I can now define an average velocity and I can define all the moments that I described back then. So, I can define like a x 3 2 x 4 3 you know all of those that we defined earlier are also applicable except each one each term the numerator and the denominator would now be a double integral. Now, there is only one more additional physics that is introduced in going to two variables in the probability density function that is something we talked about in the end of I think two classes ago which is this idea of size velocity correlation. How do I understand this size velocity correlation that is are the larger drops moving faster than the smaller drops or vice versa or there there is really no relation between the two. How do we go about that the way to do it is to define an average velocity that is conditioned on the size. So, I will put this that is if I do this integral f of x comma u d x sorry f of x comma u d u f of x comma u d u is the total is the probability density it is still a probability density because I have not eliminated the other variable x it is a probability density in the space in the x space. So, this gives me and that times u gives me a mean value that is conditioned on the size. So, this tells me the velocity while x is still a function of the. So, if I take this and divide it by the average velocity this gives me a function of x because the numerator is not is a function of x the numerator is a function of x the denominator is just the mean velocity like we have defined in the past. The ratio of these two would remain a function of x and this tells me the average velocity of each x of each set of drops in the size x to x plus d x. So, it is like an average velocity of that population of drops and as it turns out if this is a straight line the if this average velocity is not really a function of x, but is relatively the same for all x that tells me there is no size velocity correlation. So, one way to another way to understand this is that I have this two dimensional function I want to look at two I want to see if I can split it up into two independent if I can separate the variables out. So, we will talk about this in the next class if I start off by saying I can do this that means the velocity distribution in the u coordinate is independent of the distribution in the size coordinate. So, this tells me that there is no size velocity correlation. So, if I can so both are equivalent if I through the first measure find that there is no size velocity correlation that means I am able to separate the function f of x comma u into this form. Let us quickly recap what we learnt today and then we will stop here. We started with moments of a pdf and then looked at multivariate multivariate distributions and along the way we built some intuition into pdf forms in sprays. So, pretty much every pdf you see will have that up and down and asymptotically going to 0 kind of a trend that we will stop here.