 Good morning. We will continue our discussion of sprays as multi phase flows. Towards the end of last class, we had derived a co-moving derivative in a slightly general formulation involving not just advection through a spatial coordinate x 1, x 2, spatial coordinates x 1, x 2, x 3 but also through a velocity coordinate v 1, v 2, v 3 and we said f 1, f 2, f 3 would have to be the force vector per unit mass for this advection through the velocity coordinate. The simplest of such force vector that you can think of let us say is gravitational force. So I have an ensemble of particles that are sitting with no interaction and let us say they are just remaining at rest and f 1 and f 3 are 0 and f 3 is some minus g. So that would cause particles that were at rest at some time t equal to 0 to begin to accelerate and acquire a velocity and if I was to study this in the context of probability density function of these particles, it would be where the initial velocity v 0 which was which in this particular example is 0 would have become some v 1 at some later point in time following the rule that v 1 is v 0 plus a dt a in this case being minus g. So the advection through the velocity coordinate happens due to an external force not due to forces that may originate due to interaction between these particles. So let us be clear about what f 1, f 2, f 3 are I will mark this in red is this is the external force per unit mass or acting on the ensemble of particles on this population of particles. If I look at what happens due to the interaction between the particles, so we started to look at another example between a pair of particles. So let us say I take a particle of some mass m and another particle of mass m and my system now is this two particle system and if I allow these two particles to come collide inelastically the momentum is destroyed due to these forces that may originate at the point of collision and all the kinetic energy in these two particles is converted to heat that is our classical mechanics understanding of this kind of a collision. Now when I look at this in the probability density space I have one direct delta function at plus v this is actually half a direct delta function because the sum total f dv going minus infinity to infinity if I am only looking at one velocity coordinate then f dv would have to equal one. So this would be half a direct delta function at v I will call this v0 just to distinguish it from the coordinate v and this would be half v minus minus v0. If I was to define f to be the summation of these two that would ensure that the probability density function integrates out over its appropriate limits to 1. So when this collision happens the this probability density function which I have indicated in black becomes one probability density function given by delta v minus 0. The sum total area under the distribution of course remains the same but it has advected from this two peak probability density function towards this one peak at 0. What is the cause of this advection? If I go back to the formulation I had for the total derivative none of these terms can be responsible for this because it was entirely an internal force. So essentially this total derivative of f is has an effect due to collisions and any other systemic effects. So if I rewrite the system equivalent of this it has to equal some contributions due to collisions plus any contributions due to changes at the system level that is if I had an ensemble of droplets that were let us say you know of total mass m. If I was to draw a system boundary and by definition a system boundary is where you do not allow mass transfer across it. What sort of effects can cause the system mass to go up if I do not allow mass transfer across the system boundary? One such effect is if I have a nucleation of droplets inside due to some condensation process. So I had some vapor inside and the vapor condense to give me a droplet. So now as far as the droplet phase is concerned there is an increase in the mass due to this nucleation and condensation process. So I could have some droplet generation. I say generation but it could also mean destruction due to evaporation. So these are all the effects that can happen at the system level. These are all the reasons by which the probability density function of this droplets in the size velocity space can move can change. So if I combine the system level equation with the version we had before from the advection form we find and I will write it in a slightly compact fashion. This is in the spray literature this is called the Williams spray equation due to a man named Foreman Williams. And we must also mention that this is very similar in fact has a very strong connection to the Boltzmann equation that physicists use to study gases. So you can think of gases which is made of molecules which are made of molecules and we can write a probability density function for the ensemble of particles. For an ensemble of particles in the size velocity space and that advex through the velocity coordinate size coordinate due to respective forces and is altered due to the collision term. So this also has connections to the Boltzmann equation which is used in kinetic theory. So now we started to define this F as being our probability density function and we now have an equation that tells us how F changes in the spray. If I go back to what F is for a moment we said F is a probability density function we did not quite precisely define if it was number probability density or mass probability density. We can define F to be either of the two and it would work just as well nothing in the equation would be different if you make this a mass probability density function as opposed to a number probability density function. Now if you look at this as a mass probability density function so I am now going to define F of R of course I forgot to write my R term so let me include that. So I will describe what each is this is basically due to growth of droplets. This is due to collisions this is due to birth of droplets or death this is the advection in the velocity coordinate the explicit time rate of change. So if I now continue on and look at what the forms if I define F as a mass probability density function and I will use a vector notation just to so this has just to be clear it has units of kilograms per micrometer of the drop size per meter of physical space per meter per second of velocity. So this is the units of this probability density function if I integrate this over the appropriate limits of R let us say so I will take this equation. So F is a probability density and I will just quickly write out the integral condition that we have in an explicit form that is the probability of finding a drop in the size 0 to infinity with velocity components. So really this first integral is a triple integral at a given point the probability of finding a drop from 0 to infinity and components minus infinity to infinity is 1. Now this is a little different from the form that I developed earlier that I have written down here where the normalization factor for this probability is the total ensemble of drops in the volume x to x plus dx. So in other words the problem if I take the normalization factor as being the total set of drops in the entire spray then I have a probability of finding a drop in my volume x to x plus dx. So probability of finding any drop in my volume x to x plus dx itself being less than 1. So if my normalization condition is the total set of drops in the entire spray then the probability of finding a set of finding any drop in the volume x to x plus dx is some number less than 1. Of that count this integral minus infinity to plus infinity and 0 to infinity of f dr dv is a smaller is this is the total number of drops found at that point in the volume x to x plus dx. So really speaking this is normalized to the mass of drops to the mass to the volume x to x plus dx. So if I take the normalization the denominator in my normalization condition as being the total mass of drops in the volume x to x plus dx then you know that is how you get this integral to be 1. In fact this is the total mass of drops. So the units on this f are not the same as the units that I have written here you would not have this per meter. That per meter is a probability density function in the spatial coordinate. So if I want to ask the question I have a spray I want to ask the question what is the distribution of droplets in this spray. I do not care what their size is I do not care what their velocity is how is the mass distributed in this spray okay which is f is a mass pdf. If I ask the question how is the mass distributed I may find that there is more mass here in the edges and that there is less mass somewhere here in the middle let us say. So in a contour sense I might say there is a contour inside which there is low mass density outside which there is high mass density okay. So this is a spatial mass pdf. If I take this spatial mass pdf and if I take a small point and I look at what the spatial mass what the distribution is within the mass that is concentrated at this point. So I am now not concerned that there is less mass here but if I said if I took the mass concentrated at this point and I want to see how is this mass distributed in the size and velocity coordinates. I can create a pdf and that pdf is given by this integral okay. So if I now say so this mass pdf is defined point wise. So the difference is if I treat x as a pdf variable that is as a variable in which I am also accounting for the probability density then I get this per meter actually is if I am only dealing with one spatial dimension but per meter cubed if I am dealing with three spatial dimensions just like this would be per meter per second if I am only dealing with one dimensions. So this is for a 1D case this is the units are this. If I am not dealing with space with the spatial coordinate explicitly as being part of my pdf that is I am not interested in how this pdf how the mass is distributed a probability density of mass distribution spatially but I will treat space just like time and take a snapshot. So this is how we did our earlier discussion. So if f is a function of r and v with x and t as parameters okay. So if this is the case then only r and v are my parameters whereas if this is the case then I have another integral another triple integral minus infinity to infinity of dx. If I have to normalize if x is also part of my pdf independent variable list then this is the condition that I have to satisfy and this will also be a triple integral. On the other hand if I want to treat x as only a parameter so at every point I create a pdf of only size and velocity. This is how let us say we do our pdpa data set we are not doing a pdpa data set treating x as a as a as a depend as one of the independent variable list in the pdf because pdpa by definition is a point measurement instrument whereas if I am looking at something that measures spatially I may be able to do the x I may be able to find x as part of my independent variable list okay. So if I now write a revolution equation for this f or not evolution equation as a normalization condition for this f then and this is true at every point again this is a triple integral if I now continue if I take my Williams spray equation I am going to take the case of a dilute spray with no evaporation condensation. So the sc and st are 0 and this is what we call a dilute spray dilute non evaporating spray dilute essentially means no chance of collisions happening. So the collisions are so rare in occurrence that I can ignore that effect okay. So if I set this to be equal to 0 on the right hand side and perform this integration over the size coordinate. So I want to now see if I only integrate this equation in the r coordinate we saw from our discussion of multivariate probability densities that this will also be a probability density in only the velocity coordinate because whatever this function is the function of only the velocity v and that will also be a probability density because integral of that over the appropriate limits of velocity would become 1 we saw that in an example earlier notice how in some instances I brought the integral inside the integration and in other instances I did not the reason for this is just to understand that in this case the integral integration over r and time derivation are commuted. So I can exchange the two operations in order. So if I now define this as being some sort of a probability density function in velocity correct. So since we said this will have all the status of our probability density function we are able to do this. So if I rewrite this what do I get notice how I have still left with some f terms I have not I do not have an equation only in g terms because I know integral 0 to 0 to infinity let us say I have replaced integral 0 to infinity f dr with some g of v it is like a probability density function in the velocity coordinate and I am able to take care of the first two terms but the last two still have this f part remaining in it ok. If you look at especially the third term that I still have f because if fj the force acting on a set of particles external force is a function of r then I if it is not a function of r I can take that out and write this as fj times g. So if fj is not a function of r I can write this as fj times g if it is a function of r then I am in trouble because I have fj from my earlier formulation as the external force per unit mass acting on these particles and if there is an size dependence on it then this becomes a little tricky to write. If we think of an external force per unit mass on a set of droplets where there is size dependence the simplest of such forces is drag when you write the drag on a sphere for example the drag on a sphere scales as the size of the sphere. So you essentially have to have I mean so this is not something that is out of the ordinary it is very commonly found in space that this fj is a function of r and as a result I am not able to do this integration right away this is the start of what is often called the closure problem. So this part say for example I will set r dot to 0 we will come back look at that later but this part here where I am unable to write fj this integral fj fdr in terms of g is what is the reason for what we call our closure problem okay. So what we will do in the next class is we will start from here and the governing equations that we are used to seeing are our Navier-Stokes equations. So we want to see how to go from something that looks like the Boltzmann equation or in this case our Williams-Prey equation and arrive at a multi phase or even a two phase set of equations two sets of equations describing the gas phase and the droplet phase. So we are going to do that mathematics in the next class but before that we want to understand what is a phase. So if I take a spray we have done this many times you know we have air in the interstitial space between drops but if I take all the drops with vacuum in between and the air alone in some sense they look alike in the sense that drops are macroscopic distinct entities that are separated by emptiness in the case that I talked about where the drops are taken out with vacuum in the middle. Now if I really really zoom into the air part the air is also discrete molecules with emptiness in the middle. So really the only difference between air and the drop phase is the size of the particles where molecules of oxygen or nitrogen may be a few tens of angstroms drops of liquid may be fractions of a micron to several hundreds of micrometers and the interstitial space also scales but other than these scaling effects the nature of nature or morphology is somewhat similar. We will use this argument as our building block to replace the drops with a continuum just as we replaced as discrete a set of discrete oxygen molecules and nitrogen molecules with this continuum idea called air. We will use we will replace a set of drops with a continuum called will for today call it a spray or call it a drop fluid. So this drop fluid plus air together put together makes our spray. We will see how to get this far from the Williams spray equation in the next class.