Welcome back, we will continue our discussion of ah multi phase modeling of sprays ah towards the end of last class we had sort of arrived at a taxonomy of multi phase models to see what a possible way by which you could based on the physics of your given problem choose an appropriate model whether you should be using a Lagrangian model or whether you should be using an Eulerian multi phase or a mixture model or the exact model that we discussed as our choice number 1 ok. So, we will start from there and look at the mathematics of some of these models ah and see what underlies the mathematics ok. So, we said based on the particle relaxation time and the mean free path these are the two parameters we defined based on which you could choose between the different models and in the gray area of course, you have to there is some room for engineering judgment apart from that of course, we said the exact model would be applicable ah in all these situations. So, let us look at the the governing equations for the mixture model ah if you recall the mixture model is essentially a single phased version of a multi phase model that is in reality we have drops laden in air that is our spray. But we believe due to certain physical ah laws that the say that the spray seems to obey that we are able to make a simplification and look at a third equivalent fluid which is a mixture and we will only write our balance laws for the mixture. We are not at this level going to be concerned about individual droplets or air as being separate we are only going to write ah balance law for the mixture. So, the first balance law is the conservation of mass that you will see here, but before I go to the conservation of mass we need to be clear about our definition of alpha. Alpha is called the volume fraction and this is the fraction of a given space ah given infinitesimal spatial region that is occupied by either phase 1 or by alpha i is the volume fraction of the ith phase in the mixture where ah a certain volume of an given infinitesimal volume is occupied by the ith phase and that fraction is given ah as alpha i. So, remember this is a field property that is it is a function of x, y, z and time in general which means that at every point I am looking at a tiny infinitesimal volume surrounding that point and I have I envision alpha i as being the fraction of that infinitesimal volume that is occupied by the ith phase. So, we already we we sort of touched upon this that the idea of our infinitesimal volume now is not our old calculus mathematical infinitesimal volume, but it is a volume small enough in comparison to the rate of variation of alpha i, but big enough in comparison to individual drop sizes. So, we cannot be talking about an infinitesimal volume that is on the order of the drop size we are looking at an infinitesimal volume that is bigger than the drop size. So, I can talk of a smudged out volume fraction. So, the first constraint that that we have from that is this fact that the sum of all volume fractions has to equal 1. So, this is essentially what you might call occupancy of space I cannot have empty space in a continuum formulation there is no such thing as emptiness. So, I have essentially all the space occupied by either air or my drop phase we are not talking drops we are talking drop phase that is our. So, if I was to only talk of a two phase problem in the two phase case m would be equal to 2 and then I can define alpha 1 and alpha 2 and alpha 1 plus alpha 2 has to equal 1. Now, based on the local alpha i I can define a mixture density and alpha and rho i is what we will call the material density. So, that is the actual physical density of the liquid I am spraying and actual physical density of air the material those are the our those are my rho i s rho m is the local mixture density that is the density of the constituent the mixture phase that is made up of these two materials. So, now I have my mixture density and so we will come back and define our conservation of mass which is which looks very similar to our single phase conservation of mass equation that we usually have accompanying our Navier stokes equations that the partial derivative of alpha i rho m with respect to time plus the gradient of alpha i rho m v m is equal to 0 v m is the mixture velocity field. This mixture velocity field is basically based upon the idea that locally at that point both the air and the drop phase are moving with the same velocity v m this is a velocity field of course. So, it could be a function in general of x y z and time and alpha i rho m v m gradient of that quantity plus the rate of change with respect to time of alpha i rho m is equal to 0. Now, notice how I still have the subscript i which means that this is an equation that we write for each of the two phases. So, alpha i rho m for each i you have a conservation of mass equation. So, just to be clear about that fact I will say this is a conservation of mass equation for phase i. Now, if I take the simple two phase formulation and I take an equation just like this with i being equal to 0 and i being i being equal to 1 and i being equal to 2 and add them up. Let us see what we find if I take summation over all i what I find is that I have rho m summation alpha i plus gradient of rho m v m summation alpha i is equal to 0. We already know summation alpha i is equal to 1. So, I can write this as gradient of the mixture density plus sorry derivative of mixture density with respect to time plus the divergence of rho m v m is equal to 0. In fact, this is exactly the mass conservation equation for a compressible fluid which means all we are saying in the mixture model is that this effective fluid is a compressible fluid and the constituents of the compressible fluid are these alphas and this rho m being equal to alpha i rho i summation is like my equation of state. Remember, we need an equation of state for a compressible fluid to close the set of equations. So, essentially this rho m being equal to summation alpha i rho i is like my equation of state it is not yet complete completely formulated as the equation of state because I need another property that says rho i itself is a constant. So, if I know that rho i itself is constant. So, if I treat my air and my liquid as individually being incompressible in the material sense rho m being equal to alpha 1 rho 1 plus alpha 2 rho 2 is a is a is an equation of state that relates the density of this mixture to a property a field variable called alpha 1. Remember, I do really only alpha 1 is an independent variable alpha 2 has to be 1 minus alpha 1 in the 2 phase in the 2 phase description. So, rho m is equal to alpha 1 let us write this out and just be clear about it. So, rho m is equal to alpha 1 rho 1 plus 1 minus alpha 1 rho 2 if I let this be accompanied by rho 1 equal to constant and rho 2. So, if I say rho 1 and rho 2 are individually constants this is saying both the individual phases are in themselves incompressible then rho m equals alpha 1 rho 1 plus 1 minus alpha 1 rho 2 is the mixture density and this is essentially sufficient to be called an equation of state because I am relating the mixture density to a field property called alpha ok and I have an evolution. So, in as far as this mixture conservation equation is concerned this mixture conservation equation is my is the compressible is a the mixture is a compressible fluid and this is a conservation equation for that compressible fluid the property of that mixture is related to another field variable called alpha 1 for which this is my transport equation this is my conservation equation for that alpha 1 ok. So, this is now as far as conservation of mass is concerned the we have defined what the fluid is looking like. Now, in the momentum sense this is essentially my conservation of momentum equation partial derivative with so this is again familiar there is two parts to this I have a now a mixture pressure and a mixture stress field shear stress field these have to be related to my field properties. So, the pressure let us just quickly think of what this pressure is this is the pressure in the mixture that is arising out of the stress field in the air phase plus some super post stress field in the drop phase that is if I now separate the two phases the stress field in the air phase is easy to understand it is what we have traditionally understood as the hydrostatic pressure field in a single phase. The stress field in the drop phase is what we talked about earlier this is a result of the collisional processes that occur between the individual drops. So, it is like it is a stress that arises it could be a normal stress that arises out of drops being in some sense crowded in one place or drops being agitated locally in some point. So, I have to have a model to relate the this drop phase stress field to some other drop phase properties such as for example, alpha gradients of gradient of the velocity field I could even define sort of a temperature which is a measure of this agitation level in the drop phase. So, I could locally have a very high even though the mean velocity may be 0 if I have a locally very high movement within the drops this is like a granular temperature or a dispersed phase temperature that you could define that is very often defined in granular phases granular flows as a matter of fact. So, I could define all these field variables and relate and come up with relationship between the mixture pressure and these other properties mind you the hydrostatic pressure. So, if I was to do the exact model the actual hydrostatic pressure in the air is only one component of this mixture pressure and then there is also a stress field that has to be related to some argument list and these are called constitutive relationships. So, these relate you know you will recognize the mixture shear stress being related to the gradient of the velocity field is like our Newtonian viscosity model. So, if I make that viscosity be a function of the local volume fraction of alpha 1 let us say in the simple two phase formulation that is an extension of our Newtonian viscosity model to a mixture level ok. Now these constitutive relationships are required before I can close the set of equations because if you look at just these I have two equations in two unknowns I have my unknowns are alpha 1, alpha 2, V m, P m and tau m and I only have three equations in the form in the conservation form these are my as conservation equations these are my conservation of mass for a phase alpha phase 1 phase 2 and the conservation of momentum for the whole mixture. If you look at the mixture conservation of momentum equation that does not have any subscript I. So, we are only writing one momentum equation one balance of momentum for the whole mixture. So, we have to have two more relationships which are my constitutive relationships to complete the to close the set of equations and have P m related to some other fields alpha I gradient of V m etcetera ok. So, this part is an area of active research even today if I want to use mixture model you if you look if you look at the literature there is lots of papers that use mixture models to study sprays or bubbly flows there is lots of different multi phase flows that use mixture model. But you will see the level of simplicity that goes into these constitutive relationships and if you begin to question physically the physical reasons for such simplistic arguments you will you will find yourself in an area where in you will find yourself in unchartered territory because the reason such simple forms are employed is because that is the best we can do as far as trying to employ our intuition from single phase flows we are basically in the process of extending our intuition from single phase flows to multi phase flows ok. Now how do I get these constitutive relationships really one way usually is experiments, but in this case experimentally to get these kinds of relationships experimentally is very difficult. So, the other way is to go to the exact model and do like fully resolved droplet level simulations and extract the local pressure field as a function of some other arguments. And that is a tractable approach and is quite widely being used today to extract realistic constitutive relationships that relate the stress field to some other field variables ok. This is the simplest of multi phase models. So, if I take the constitutive relationships and say my P m is independent of the simplest of constitutive relationships could be that P m is independent of all alpha and gradients. So, it is just like my hydrostatic pressure field in the air phase ok or I could relate tau m to the simplest forms would be relating P m to only alphas and relating tau m to only gradient of B m just like a Newtonian fluid would be would behave ok. So, let us move on and we will look at the next level of complexity which is my Eulerian multi phase models. The Eulerian multi phase model again starts by writing a conservation of mass equation for each of the. So, this is for the i equal to 0 phase. There is a particular reason I have written this in this form, but really it does not matter I could have let this i go from 0 to m and it would not make any difference ok. So, now I have this conservation of mass for each of the phases constituted constituting the mixture, but the difference now is that we are going to write a separate conservation of momentum equation for each of the phases. What have we done in this process? Let us look at our list of dependent variables we have alpha if I simply take m equal to 1 here m equal to 1 would mean a 2 phase flow because i equal to 0 is 1 phase and i equal to 1 is another phase. So, I write an equation like the top 2 here for my i equal to 0 and an equation like this for i equal to 1 and I have alpha 0 is my is one of my unknowns rho 0 is just the material density of the fluid v 0 is a velocity field that is unknown for the phase 0. Likewise I have alpha 1 v 1 I have p 0 p 1 tau 0 tau 1 these are my unknowns. I can come up with a relationship or we will have to come up with a relationship between p 0 tau 0 and alpha 1 alpha 0 the gradients of the velocities that is what we called our constitutive relationship. We also know that alpha 0 plus alpha 1 has to equal 1 that is our occupancy of space relationship. Now in addition so, we have essentially created as opposed to the mixture model we now have one more new velocity field here which is my v 1 which is new. So, we have in 3 dimensions we have 3 velocity components and alpha for each of the 2 phases. So, this is I have 8 unknowns at every point in space apart from the 2 pressures apart from the 2 pressures. So, once I relate the pressures to some other field variables I have 2 pressures 6 velocity components in all and 2 volume fractions that is my complete list of unknown variables as opposed to the mixture where mixture formulation where we had 2 volume fractions 3 velocity components and one pressure. So, we have essentially added 4 more unknowns to the list. The problem is not just the addition of 4 more unknowns it is the numerical complexity and the solution complexity that comes with it that poses some convergence challenges for this kind of a model formulation. The simplest assumption made so, now look at the pressures for a moment we are able to postulate an individual pressure field in each of the 2 phases in this kind of a formulation, but they are not really completely independent of each other. And so, you the list of unknowns becomes smaller through algebraic relationships that exist between these kinds of other variables like for example, alpha 0 plus alpha 1 has to equal 1. So, that through relationships between these individual field variables the total number of independent unknowns comes down slightly, but we have not yet touched upon this term here that did not exist in the previous formulation. So, now that I have 2 phases let us just take i equal to 0 and i equal to 1 and the 2 phases are moving with different velocities they become sources and syncs of momentum to each other. So, I can have interface momentum transfer that now becomes important notice the form of this term k 0 i is like a drag coefficient and that coefficient multiplied by the differential velocity the relative velocity between the 2 phases is a simple constitutive model for a force a drag force that exists between the 2. So, drag is the simplest of these interface momentum transfer terms now again if I assume i equal to 1 there is a k 0 1 which is dependent on the relative velocity between the 2 phases that takes on a positive sign in one equation and a negative sign in the other. This is basically saying the force acting on one phase is equal and opposite to the force acting on the other and if I take all the if I take all the equations all the balance laws for each of the different phases and add them up that gives me the even in this formulation I can derive a mixture level balance law, but that would have the individual velocity components remnant in it. So, I could still define sort of a velocity of the center of mass of the mixture I know now that the 0 phase i equal to 0 is moving with 1 velocity v 0 i equal to 1 is moving with a different velocity v 1, but I have based on the relative volume fractions and mixture densities we can define what is called a barocentric velocity of the mixture and I can we can write a formulation for the barocentric velocity of the mixture and in that formulation these interface drag terms vanish. The interface drag cannot affect the barocentric velocity of the mixture because it is a force internal to the whole mixture, but it does affect the individual phasic movement velocity fields in the individual phases. So, let us let us for a moment look at this shear stress terms tau 0 and tau i tau i specifically what is tau i tau i is basically the shear stress field that exists in the drop phase. So, if I take a collection of drops with no air in between if I try to put a hook on one of the drops and drag it does this drag other drops with it that is the basic definition of shearing right. So, if I pull one drop with me or a small group of drops what does it do to the other drops if there is no effect on the other drops that means the effective viscosity of this collection of drops is 0 there is no shear stress that can be sustained in this mix in that collection of drops. So, if I now take this collection of how do I now find the viscosity of this collection of drops one would have to do or the effective viscosity of this collection of drops one would have to do fully resolved simulations to look at the effect of collisions between drops which is basically responsible for this effective viscosity and derive from sort of an inverse from a regression these kinds of constitutive relationships. And this is like I said an active area of research in itself that would be worth pursuing to complete to bring these model formulations up to some level of applicability where they are more realistic for dense sprays. So, why do I say dense sprays if I look at rare if I sprays and do the same thought experiment of putting a hook on a small group of drops and dragging them because there are there is a because they are a very high Knudsen number collection of drops the effective viscosity of something like that would be basically 0. Whereas if you are in the dense spray region where collisions are more frequent then if I try to do this thought experiment in the dense spray region you can imagine that the effective viscosity of this mixture would be non-zero and so this shear stress term sustains itself in the dense spray region while it goes to nearly 0 in the rarefied region okay. I want to look at I want to pause here for a moment and show you one possible way by which you could look at modeling what use are these kinds of models in a real spray what use are these Eulerian models in a real spray. So, one of the areas of applicability of these Eulerian multi phase models in sprays especially when you have break up and coalescence is to look at population balance modeling. So, we look at what population balance modeling is for a moment let us say I have a little box with lots of different drops okay and just for the sake of convenience I am going to derive divide these drops into different bins. So, I have let us say drop spanning from some D min to some D max I am going to divide this into different bins.  So, if I want to look at how to model break up let us say. So, initially all the drops are of this large size and there is some process that is causing these drops to break up. So, if I take the number of drops in the ith bin and write a and that is some n I why would n I change n I changes because of two reasons because drops in the ith bin are breaking up. So, if gamma I which is often called the breakage frequency okay. So, if the ith drops are breaking up at if the drops in the ith bin are breaking up at some rate. So, gamma I is a rate of break up of drops then gamma I times n I with a negative sign is the rate of break up of drops and that is essentially with a negative sign taking drops out of the ith bin. The other possibility is that I have some drop in the I plus pth bin okay breaking up and putting some mass putting some number of drops in the ith bin. So, when a drop that is larger than the ith size class breaks up it puts drops in the ith bin. So, I have some other jth size class breaking up at the same rate gamma j n j and a fraction of that gamma j n j which is given by this n I j coming into the ith bin and I could have this go over all the drops from I to m really speaking it should be I plus 1. I cannot have the ith bin breaking up to give drops in the ith bin. So, at best I could have the I plus 1th bin breaking up to put drops in the ith bin or I could have the I plus 2th bin breaking up putting drops in the ith bin etc. So, this n I j is a fraction of drops that result that end up in the ith bin from a breakage event in the jth bin. So, this n I j is a fraction of drops that end up in the ith bin originating from a breakage event in the jth bin. Some drop in the jth bin, jth bin broke up and it breaks up into lots of fragments it could break up into two equal halves it could break up into one big drop and one tiny fragment flying off it could do lots of different things and this break up event now puts some number of some fraction of those drops originating from the jth bin into the ith bin. So, this is a balance of the number of drops in the ith bin which due to break up. So, when I have some break up process causing these drops to break up some physical process we do not know that physical process has to obey something like this. Now, this is called a phenomenological model notice how I have not yet brought in any physics of what is causing this break up. The physics is essentially hidden in these constitutive coefficients gamma i and n i j this is my opportunity to bring in the physics. Say for example, I might believe through a physical process that a larger drop is likely to break up more frequently than a smaller drop. So, for example, if I believe that there is a smallest size drop that just does not break up then I might say that gamma 1 which is the break up frequency of the smallest size class is 0 and I might write a relationship between gamma i and d i the size of the drop or I might write a relationship between gamma i and some other property that is imparting this energy for break up to the drops. And n i j is our phenomenal phenomenology of how a given size of drops break up breaks up. So, if a drop breaks up is it likely to break up into 2 equal halves in which case my n i j. So, if it is exactly breaking up into 2 equal halves then say for example, n 2 4 would be 1 and all other n i 4 where i is not equal to 2 is 0. So, if for the case of j equal to 4. So, I am just looking at drops originating from break up happening in the fourth bin and if let us say for a moment that all the diameters are spaced equally in an arithmetic progression then a drop breaking up in the fourth bin only puts drops in the second bin because it is breaking up into 2 equal halves and it cannot put drops in the third bin or the first bin. A drop breaking up in the hundredth bin puts drops only in the fiftieth bin and in no other bins. So, this is a phenomenology that is based on some physical understanding of that particular break up process. So, the population balance model that we wrote this part is only a framework it is a phenomenological framework. The physics comes in the form of these gamma i s and n i j s and over the course of the last let us say decade or so these various models have been developed for both gamma i and n i j to accurately capture the physics of a given physical problem. Say for example, we have a multi phase reactor bubble column where you have bubbles of some size coming in at the bottom of the column and they may break up and leave and we want to look at the phenomenology of the break up of these bubbles because that is important to creation of surface area the creation of surface area is important to mass transfer. So, if I look at the mass transfer process and if I want to model that mass transfer process I need to model the evolution of the size distribution of let us say drops or bubbles or whatever the case may be. So, what we are going to do now is look at how we can take our regular Eulerian multi phase model governing equations and include these population balance terms. So, essentially we looked at drag as being the only coupling between these two in fact if I take a two phase formulation of this there is only two coupling mechanisms between the individual phases as it is apparent in this case one is through drag and the other is through occupancy of volume. So, alpha 0 plus alpha 1 has to equal 1 if I now look at what happens if I include population balance I have these drag terms, but I also have additional coupling in the form of these population balance source terms. So, think of what this means if I have the drops in a box the left hand side of the drops in a box population balance was just d n i d t which is the rate of change of the number of drops in the ith bin as a function of time. When I now have a flow field super imposed over these drops I have to go I have to write the material derivative of that of that flowing field. So, in the in the Eulerian formulation where I have these drops coming into a control volume and leaving while these break up processes are happening the first thing that you see is that the left hand side is now of the form of the material derivative where alpha i is the volume fraction of the ith phase and if you look at what you see on the left on the right hand side of these equations you see this alpha i by x i x i is essentially the size of the size of drops in the ith bin it is actually let us be even more clear it is not just any size it is the actual volume of a drop in the ith bin. So, volume of a drop in the ith bin so alpha i is the total volume fraction alpha i by x i gives me the number fraction of drops in a given at a given point that reside in the ith bin. So, you see that the rest of this equation looks very similar to the population balance form except I have this I have this gamma i n i what looks like n i and gamma j n j times some n i j which is a fraction of the drops that come over into the ith bin and because I am looking at a multi phase formulation where I say each of the size classes is also allowed to move with its own velocity field if we go to that level of generalization of the flow field then drops breaking up in one phase resulting in showing up in a smaller size class are also bringing in momentum with them. Because total momentum over the entire mixture has to be conserved the break up of drops from a larger size class putting drops into a smaller size class is completely internal to the whole air plus drop mixture. So, whatever momentum exchange processes that happen are supposed to be equivalent opposite. So, that is the reason for these two terms that you see in the red ellipses where any momentum leaving the jth phase accompanying the mass transfer from the jth phase to the ith phase ends up in the ith phase. Likewise any momentum leaving the ith phase due to break up of drops in the ith phase leaves the ith phase ok. Again these are only phenomenological. So, I just this is some break up process this is some process it is like a balance law that tells me mass coming into the ith phase and mass leaving the ith phase mass and number of drops are sort of related. So, how do I let this mass come in and go is where the physics of the problem comes in. So for example, in a general flow situation like this the actual break up of drops is due to a relative velocity between the air and the drop phase. So, we could relate the gammas to some kind of a relative velocity in the drop phase and when you do that essentially this is a model of secondary atomization that follows your primary atomization process. So you have some primary atomization that produces a distribution of drops and from there on you could look at studying the secondary processes using a phenomenological model like this. This is where you can take multiphase models and apply it to atomizing sprays without this atomizing part this red is essentially the atomization part of the multiphase formulation without the atomization part these are my governing equations these are my governing equations where if I have the balance laws and the appropriate constitutive relationships and I know the size of the particulate phase because this drag coefficient is going to require some information of the size of my dispersed phase drops. So, if I know the size of the dispersed phase drops I have this equations complete that I can look at transport of drops in the spray. I cannot include how the process of these drops breaking up without the help of something like a population balance model. This completes our discussion of multiphase models and you know mixture model Eulerian model, but I just want to leave you as a leave you with a sort of a thought we went through this process of asking ourselves some physics based questions and finding and arriving at a model choice that is appropriate. In reality though most of our models are neither here nor there or they involve both. So, I just take a simple hourglass that is flowing sand. If you look at the Knudsen number which is the distance traveled by a given particle between two collisions. If I look at the Knudsen number in this region versus the Knudsen number in this region they are vastly different. This is a case where if you imagine the sand particles are all jiggling and this jiggling sand particles are going to collide very frequently. So, this part is right for our kind of a continuum model. This part after you have gone past the constriction is essentially where individual particles are moving with no knowledge of their neighbouring particles. They are all falling in the same gravitational field with about the same terminal velocity if they have reached a terminal velocity. But the mean free part which is the distance between collisions has gone up significantly during this process. So, if I ask the question what model is appropriate for this? This is actually a very very close representation of our real spray. We have a very dense spray region where we require a continuum model where we I mean because of the frequency of collisions you either require a model to handle the frequencies of collisions in the individual particle like in a discrete particle model or you go to a continuum. And on the other count when you have reached into the rarefied region you require something that defaults to where individual particles are moving with no knowledge of other particles around. This is an area that is an active this is an area that is research wise very active. To patch rarefied spray models like Lagrangian particle tracking is a very good choice for rarefied spray models. But you require some kind of an Eulerian model in the discrete in the dense spray region. Patching these two together is what real life research these days looks like. We will continue our discussion in the next class.


Where do we get started? Meatless Mondays, vegan Fridays, what's next? Tofu Tuesdays, plant based Wednesdays, sweat boy Thursdays, how about puss boy weekends? You know, the craziest thing about all of this to me is how gullible people are. Most will admit that meat is bad for you. Fruits and vegetables are healthy. And this is a huge problem because the masses need to understand that meat consumption, animal protein, is vital for your health, especially during developmental stages. Beacons love to argue that the only reason to eat meat is for taste. Oh, meat tastes good. No, it's so your brain will work, function properly, something vegans aren't used to. Honestly, has anyone seen a healthy vegetarian or vegan? Women are notorious for becoming anemic zombies when removing meat from their diet, but are so gaslighted that beef is the devil, they would rather wither away.  Girls stand up or get up out of bed and they feel like they're gonna pass out. It's crazy, young women in their teens that are supposed to be healthy. With a new mayor comes a new menu for New York City public schools. Tomorrow, the city will roll out vegan Fridays. CBS News lay at Michigan with reaction from parents and students to the new veggie forward menu. Put down the pizza and that burger. Students grades K through 12 are going vegan on Fridays. I didn't know that, actually. The new initiative will kick off tomorrow with vegan veggie tacos for high school lunch. Students outside LaGuardia wondering why no meat. They already have a vegan option, so why should they force everybody to go vegan on Fridays if there is a vegan option? They have to have chicken. Come on, you gotta have chicken beef. New York City's Department of Education says you'll be able to request non-vegan items if you choose. I think it's great. This mom of two says her family is vegetarian vegan half the time. Try to give them a variety of diversity of food, flavors, vegetarian vegan. There's some meat nights. She's happy schools are offering different options. Her pre-K sons favorite food? Apples. Vegan Mayor Eric Adams says this initiative is a way to expose kids to healthier food. This month on Fridays, they'll experience Mediterranean chickpeas served with rice or pasta and black bean and plantain rice bowls. In one voice, we talk about fighting childhood obesity, diabetes, yet you go into a school building every day and you see the food that feeds our healthcare crisis. I'm going to do the best I can to give them the options of a more healthier diet. I think that's fantastic. They throw in cool things. The DOE says meatless Mondays and Fridays have been a success. This builds on it, they say, with a nutritious menu that groups of students have sampled and approved. On the Upper West Side, Leia Mishkin, CBS 2 News. And the Mayor, of course, has spoken many times about how being a vegan has changed his life. He credits this plant-based diet with reversing his diabetes diagnosis. I like the idea. I have a lot of feeling, though, that kids are gonna come home hungry.  Exactly. That's why you gotta eat it. May take some time to get used to. That's right. Hey, there's the truth. They lie to you for two minutes straight.  This is comical. The clowns that caused the healthcare crisis with their toxic foods are replacing it with the same toxic foods just without the meat.   And I didn't realize that these kids were being forced to go vegan. I thought it was just gonna be an option, but they flipped the script. Everything is vegan, and then they have to ask for the meat option. And I'm sure they'll be guilted, oh, you're destroying the planet by asking for meat as a 12-year-old kid. It's despicable. Despicable. I mean, we've seen this over and over again. Typical media blitz BS. Everyone needs to know about how healthy the New York City Public School vegan kids are gonna be. Now, this public school situation was horrible to begin with. And we all know with the, you know what? Ahem, ahem. For the past two years has just made this worse.  I won't talk about meatless Mondays because we already did a video a few years ago if you want to check that out. And I think they're literally gonna turn everything vegan at some point. This kind of reminds me about how I really wanted to open a soup kitchen after getting that slaughterhouse last year. There's a ton of extra meat from meat processing that is not expensive enough to sell or ship, especially the bones. You know, so it would have been nice to offer people nutritional beef that they couldn't normally afford. Too bad it looks like I'll be in Columbia soon enough with how things have been going on the business end. But thank you guys for joining me. I saw this a couple hours ago figured. Do a quick video if you guys are not in the loop on all the crazy nonsense going on in New York City. So if you could please drop a like, leave a comment down below, subscribe so that YouTube can unsubscribe you next week, and be sure to check that notification bell so they don't notify you of my videos. Therefore, you can check out frank-sefalna.com to support me through all of my businesses before I completely lose my mind. And this is what I do on a Friday night guys. I couldn't find a Mexican girl to lock me up in her basement, so I'm dressed up in silver clothing in my parent's house hiding from radiation.


Hello and welcome to the series of video lectures on the subject digital techniques for secondary IT students. I am Dr. Sri Shalasarad Kajbar and in this video lecture we are going to study identifiers, keywords and data types that are used in Veriloc SDL language. At the end of this session you will be able to identify identifiers, keywords that are used in Veriloc programming. You will be also able to describe data types that are used in Veriloc programming. Identifiers in Veriloc are nothing but the names given to the objects. The objects in this case are nothing but the, it may be modules, it may be instances, it may be nets, registers, parameters, system tasks and functions, blocks, etc. Here to the C programming language they may be made up of any number of letters, digits, underscore and dollar signs are also allowed. However, there are certain rules and these rules are the first character of an identifier must be alphabetic or you can also use the underscore character and the rest may be the combination of alphabetic, numeric, underscore or dollar. However, an identifier cannot start with the digit and identifier may be as long as one million characters and one thing you have to remember all the identifiers are case sensitive. The examples are given here, here the identifier name is n dollar 6.7. As mentioned earlier, an identifier may start with the underscore, here the name is underscore bus 2 3, the another name is shift rate underscore 8. Now many a times a programmer wants to give the name of his choice and in that case he can use the escape identifiers. The escape identifiers begin with the backslash and the name is considered until the first white space. Using escape identifier, one can use any printable ASCII character as a name. For example, here the escape identifier name is a plus b minus c. One thing you have to remember that it starts with the backslash. So anything that start with the backslash is considered as a escape identifier. Similar to the C programming languages, there are variety of keywords in verilog hdl language. Keywords in verilog are nothing but predefined non-escape identifiers and escape identifier is not treated as a keyword. For example, although begin is a keyword, backslash begin is not a keyword. Following is the list of various keywords. In the last two video lectures, we have seen some keywords such as assign, we have seen module keyword, we have also seen end module keyword, we have also seen the end module keyword and these are the remaining keywords. Here is another list of the keywords. We have also seen this wire keyword to define the wires. Now pause the video for two minutes and write down the answer of the following question. I hope you have written the answer. The answer is the first entity given was underscore wire and in this case it is an identifier because although wire is a keyword it starts with the underscore so it is an identifier. The second backslash module is an escape identifier although module is a keyword since it starts with the backslash it is an escape identifier. And the third example was end module and in this case it is nothing but a keyword. Now let us see the data types that are available in verilog hdl language. In verilog every signal, constant, variable and function must be assigned a data type. Verilog provides a variety of predefined data types. Now before considering the data types in detail let us see the values that a data type can type and we refer to them as a value set. Every data type in verilog can have either of the four values that is verilog supports four basic values that a signal can take and those are 0, 1, x and z. The zero in this case describes a logic zero or false condition. One indicates a logic one or true condition small x or capital X indicates it is an unknown or uninitialized value z or capital Z indicates high impedance it may be a tri-stated or floating value. The next data type is net data type and this data type models an interconnection between components. A signal with a net data type must be driven at all times the meaning of this is that consider this diagram here there are two AND gates one NOT gate and one OR gate in this case the interconnection between this first AND gate and OR gate is through this T1 and this is nothing but a wire because it models an interconnection between components. The meaning of this is that whatever the value this T1 wire takes it must be present at all times and this value may be 0, 1, x or it can be a z value. The most common synthesizable net data type is a wire data type for example consider this circuit suppose you want to declare the net data type in your module definition then you have to write as wire which is the keyword followed by the names of the wires that is T1, T2, K. Verilog also contains data types that model storage called variable data types and they can take value 0, 1, x and z they hold the assigned value to them until next assignment and the different variable data types are given in this table a reg data type indicates a variable that models logic storage there is also data type called as integer it is a 32 bit 2s complementary variable representing whole numbers between this big range there is a real data type also which represents 64 bit floating point variable there is also a time data type which represents an unsigned 64 bit variable between this range now let us see the vectors data type it is nothing but an array of only elements all the values of the net data types in addition to the variable data type reg can be used to form vectors a syntax for defining a vector is as follows so there will be first you will define a type followed by in the square bracket MSB index colon LSB index and finally you will write the vector name for example here the first example is wire the wire in this case indicates a data type followed by in square bracket there is 7 colon 0 followed by the vector name as sum here the MSB is given as the index 7 whereas the LSB is given the index 0 ok this another example this defines a 16 bit vector called Q of type reg this another example reg data type is defined however in this case the 0th bit indicates MSB whereas the 15 bit indicates the LSB of this data type R now let us see the arrays data type and it should not be confused with the vector data type a vector is a n bit void quantity whereas array may have multiple elements that are 1 bit or n bit void array data type declaration is allowed for reg integer time and vector register however it is not allowed for real data type also virilog does not permit declaration of multi dimensional arrays now let us see the examples here in the first example the data type is integer the identifier name used is count in square bracket 0 colon 7 is given the meaning of this is that 0th bit will be used as an MSB bit whereas 7 bit indicates an LSB so this defines an array of 8 count variables each of which is 32 bit void because the data type is integer in the second example it indicates an array of 8 port IDs each of which is 5 bit void as you can see here port ID in square bracket 0 colon 7 means what it indicates 8 port IDs and each of which is 5 bit void so this is an array data type these are the references thank you very


Yeah, we gonna talk. Say you have another question. I gotta get this here Okay, go ahead.  I got calls from some individuals Internet savvy say man, our checks never did stop man.  Yeah, they the writer strike is here. What do you think about the internet people still was making money? Wait, what are you saying the check checks at the sex check didn't stop. No, okay, but You guys couldn't work Oh, yeah, yeah, these guys continue to work.   Explain. What what did he say door?  I'm like stop listening to these goofies $50,000 is no money in this business. That's real I get it. I get it So And I was like I'm telling um scruncho Because he said a minority said there's a door you go through. I was like, let me tell you something, man What door is a group? This is something on the internet.  You niggas ain't got no money Why the people that owned the building here didn't go to that door? It's real So they only want low level motherfuckers that make $50,000 Don't make sense. You see what I'm saying?   You see I'm saying what what made nor even because he's a goofy What made him say it? I don't People want to believe there's this mythological door of Evil To get to where you want to go to people are just evil.   No, you can be evil without going through the door And you can make money without going through a door You don't have to go you don't have to do that shit I felt you when you said that there was another level to it when you talked about the fluffy I didn't feel that because like I didn't think of it that way the Dave Chappelle's and all those guys And and then thinking about what they're setting up for the future versus what you see when you're just doing it from a We just starting out on the internet standard I mean, but those guys that's a springboard.  Could he go to Dodger Stadium and sell it out twice? But you gotta understand I think the skits and all the stuff that country Wayne done got him to the netflix special Okay, I'm glad you said that's what that helped them get there It's not hard to get an effort special But it's a thing that they feel I mean you got to realize people get on there Remember how monique how monique wanted to do her thing. Yes, but she's a part of That level What level that I need to be on netflix Have you heard of russell peters? Haven't heard of russell. Exactly. I'm gonna look him up though. Exactly. Who is russell peters? He's one of the top comedians that tours the world He's on fours He was one of the first on netflix.  He's other places Netflix is just a familiar name. What does it do for you when you go on netflix in your Country Kevin country Wayne Country Kevin country Wayne whatever the name is. I think you've got to have the country's birthday names But I'm just I mean do you does it does it push you like in the limelight to where it pushes you to the next level What do you do when you why did monique want to go on there and once she got on there She thought that was gonna solidify her But she didn't know she was already there. Yeah It really didn't expose her expose country kevin Now he's sitting on the stage. He's looking like a christmas tree With no jokes Just think about it Has anybody talked about country kevin's? It was mixed reviews on it Jamaica, what did you hear about it? That it wasn't good. Yeah When I call him I heard him He's gonna he known for it See y'all y'all going to bed. I'm trying to be there. I gotta look at the I gotta look at both sides Yeah, because people that follow him still watching that was they love to see it. There's not that many following him So you don't think when we went in but there's a lot of people 3600 seats. Listen to me. What he's filled 3600 seats. Okay Okay We was in use and I seen it 3600. He was there by himself Not yeah, he had three under under card Under cards, it wasn't nobody else big with him. Oh, he was he was the headliner. So he he filled the 3600 So he did that. Yes, you think he could go to chicago do that I think he can Has he I don't know. No, he cannot Why do you say chicago? You say it's always something Some people have fans in certain cities and some spaces they don't know that's what I'm trying to tell you when you're A worldwide comedian. You can go any city. Yeah You didn't go to seattle and do that, but he had the church people. It don't matter. You gotta be funny polyperi What's that? What is that? Network Was that russia russia peters? Yeah makes 20 million a year in comedy and most urban people never heard Wow, and I had I'm contrawaying him You don't do no skits. No, he just he don't do skits. He lives in Beverly Hills He lives in Malibu.   But you do you think you was discouraging them or encouraging them when you say what you say have you ever have you ever played football? Yes, hell yeah, your coach say you did a great job. No, sorry motherfucker.  Come on guys.  Who won four to run this ball? That's the the where you guys are from playing soccer and everybody gets a Everybody wins.


Yeah, thank you all, everybody, for being here. There is going to be, I believe, a post-reception. Yes, there's an evening reception, 7.1, yeah. Yeah, down at 7.1, correct. So if you all go downstairs, there will be the reception there. Look out for some of the SEO folks and come and chat. We're all going to be there around. And we're looking forward to it, everybody. Thank you for attending. Yeah, thank you so much.  If you search SEO on the schedule, you'll see a bunch of talks pop-up. We also have a project booth. So definitely to drop by at the project booth when the exhibit opens up tomorrow. Yeah, and thank you so much for sticking on the last talk of the day. Thank you. Yeah, thank you all. And the project booth is booth 0B in the Project Pavilion. Well, great job. Remember that. Thank you all.


So about a week ago, I put out a poll on my YouTube channel asking people what Facebook ads targeting strategy they've been getting the best results with. Ever since the iOS 14 rollout, there's been a massive shift and change in the way that people have been advertising and the way they've been targeting people. And what that's done is it's created a big debate over what is better for dropshipping or what is better for e-commerce. If you're selling a physical product, what is the best type of targeting to do? Is it detail targeting or is it broad targeting? The other option I added to the poll as well is look-alike targeting because that seems to have been forgotten because it's becoming increasingly more difficult to track people once they leave Facebook, then look-alike audiences have kind of been left behind. So what I thought would make an interesting poll would be to put it out to everybody who watches my channel because everybody who watches this has some form of interest in e-commerce. They're going to be the types of people who are spending money on Facebook selling physical products. So it'll be a good study to find out essentially what's working best. So in this video, I'm going to cover exactly what those three types of targeting are. I'm going to give you, in my opinion, the best times to use them based on my own experience. And I'm also going to explain at the end of the video why you should be using them like that as well. Probably the boring part of the video, but in my opinion, the most important. It's okay to watch step-by-step YouTube videos and follow them step-by-step. And sometimes it'll work out for you. Sometimes it won't. But in the times that it doesn't work out for you, unless you understand that why and how things work, then when they don't work out, then you'll become stuck. So if you do want to be successful at this and you have a commitment and you want to make it work, make sure you stay tuned to the end of the video because the explanation of why these targeting strategies work is what's going to help you next time you watch anybody's step-by-step video, not just mine. And if it doesn't work out, then you'll know exactly what to do next. Okay, so the first type of Facebook ads targeting strategy is detail targeting. If we go back all the way to 2016, when I first started running Facebook ads, detail targeting was king. That's what I first started out with. I started selling what I called passion products. And these were the clues in the name. They are products that people are passionate about. So these are pet products. These are pieces of jewelry with certain animals on. And the reason why those products worked so well back then, and they still do today, by the way, is because you would target people based on their interests. When you go into your Facebook ads manager and you create a campaign and you get to the ads at level, and you get to that detail targeting area, you can put things like behaviors, what people do for a living, what Facebook thinks their average income is. But what you can also do is put what interests they are linked to. So Facebook will track people as they move across their platform and interact with different posts and different pages, their likes, that sort of thing. And then they'll categorize these people into different interests. So if somebody has been interacting with dog related pages and been engaging with dog related videos, when you target dog as an interest, they will then be included as that audience. So back in the day when I first got started, it was brilliant because if you wanted to sell a piece of jewelry that had an elephant charm on it, you could target people who were interested in jewelry and interested in elephants and it was just a match made in heaven. And that worked absolutely beautifully until the iOS 14 rollout came along. What the iOS 14 rollout did, kind of like in a nutshell, I could probably record a whole video on this but I'll be behind the times by now, is that once somebody had left Facebook, there was an opportunity now for them to opt out of tracking. So once they had left Facebook, it made it increasingly more difficult. There are workarounds, there are certain apps like Cometly, which will do it for you. However, the amount of information basically that is tracked once they leave Facebook was severely reduced. So what this meant basically is when you would then use detailed targeting, is it somewhat restricted your results. And from then, this is where broad targeting was born. Up until then, it had never really been heard of, at least I'd never heard of it. And the reason being is because if you target the dog's audience, and there's 2 million people in that audience, and Facebook can only track 10% of those people, if you allow Facebook to target the whole entire UK, which is about 60 million people, and it can track 10% of 60 million, there's obviously a lot more people that it can track. And therefore, it can optimize your results a lot faster, because there's more information that the pixel is tracking essentially. That was the argument anyway. That is why people would say broad targeting work best. But we'll take a look at the poll results in a second, which may say differently. So we've got detailed targeting, which is where you select interests. Broad targeting is basically the way you say to Facebook, go out and show it to anyone. And then you have look like ordinances. So look like ordinances is an audience that Facebook creates based off a pre-selected audience. So if you create what's called a custom audience, and in that audience, you put everybody who has visited your Shopify store, you can then say to Facebook, create a lookalike audience, which is essentially an audience of people that are most similar to that source audience, so the people who have visited your Shopify store. And typically lookalike ordinances are always used as a way to kind of scale up your store because everybody starts from scratch, going out to cold. So ordinances, people have never interacted or seen your business before. And once you built up that bit of traction, and you had those people who had interacted with your business one way or another, then Facebook would soon start to learn who your ideal customer was or the types of people who were interested in your business. And once you had enough people on your list, you could then ask Facebook to go out there and find more people essentially that were most similar to those. So the poll, I put it out recently. There's only 30 votes, so it's not a massive kind of set of data that we derive in a 100% set in stone result from. So if you feel differently to this, make sure you comment down below and let me know what's been working best for you. But we can see from the 30 votes that detailed targeting 43%, broad targeting 40%, and then like I mentioned earlier, lookalike ordinances have kind of been left in the dust with 70%. Okay, so now that we've established exactly what these different tags and options are, and we've seen what's working for people right now, then when would you use them? And please keep in mind this is based on my own experience. The ultimate takeaway, I suppose, from this video is to test a little bit of everything and see what works for you. Okay, so when should we use detailed tagging?  Once you've scaled above that and you've got 100, 500, 1000 purchases, then you could afford to start targeting a bit more of a broader audiences and kind of rely a little bit more on the Facebook algorithms to do its job. And the reason being for this then is I've come up with a metaphor that I'm going to try and explain to kind of put it across in simple terms so you have an understanding of how the Facebook algorithm and Facebook pixel matures. Okay, so bear with me, it will be worth it, I promise.   It's obviously going to fall on deaf ears. You're not going to have much success. And this is what broad targeting is. If, however, you compare this against detailed tagging where you can pre-select what people's interests are, instead of trying to sell a dog product to the public in your local city centre, the equivalent would be going to an expo that is purely for dog owners about dog training and then walking around inside that expo trying to sell your product. You know people are there because they have an interest in dogs and therefore as you're walking around talking to people, the chances are those people own dogs and therefore you're going to have a higher success rate. So in its essence, that's what detailed tagging is. And then the previous example is what's broad tagging. So the reason why I recommend anybody with a new or fresh Facebook ad account start with detailed tagging is because Facebook becomes better with the more information it has. As a new ad account starting from fresh, it has zero information about who the types of people are that want to buy your product or interact with your business. And therefore if you send it out to a broad audience, it's going to go up to random people or it's going to show your ad to random people. It's going to put your ad in front of random people on their newsfeed, see what kind of reaction they have and it's going to take a lot more money and you're going to have to burn through a lot more people or show your ad to a lot more people if you're tagging a broad audience before Facebook learns who the right sort of people are. If you target a detailed audience, then you're going to have a higher hit rate. I suppose the best way of explaining it is is that every time your ad goes out, it's going to be put in front of people who have an interest in dogs and therefore we have a higher chance of getting a response from that person and therefore Facebook gets more data to learn who your ideal audience is. And in a nutshell, hopefully that all made sense. Any questions do please leave them down below. That is why I would use them that way and that is why I continue to use them that way. Once you have an established Facebook ad account and Facebook's saying themselves they put a benchmark of 50 conversions a week, once you're consistently hitting that, then you can start to kind of remove the interests and then give Facebook a bit more freedom, I guess so to speak. So they have a larger pool of people to choose from. To wrap the video up then quickly, look like audiences. When would I use these from day one? The second you've got enough people on your list to start using them, start testing them. Small budgets, see what kind of results you get. If it works, keep it going. If it doesn't work, kill it. Wait a week until you've got more information on those source audiences. Run them again, see what kind of results you get. And again, just be super strict and super harsh with them. Cut any emotion. If it's profitable, keep it running. If it's not, shut it down. And so with that being said then guys, I'm gonna wrap the video up. I didn't expect this video to be this long to be honest. Hopefully I've explained things in a way using terms that make sense regardless of kind of what stage you're at in your business. Hopefully I've offered up some clarity or made some of these subjects a bit more clear for you. Like I said, any questions at all, I wanna try and help as many people out on this because this is a super important part to your business. Leave them down below, I will get back to you. If you need any extra help, check out my free training. 100% free won't cost you any money. It's a top link in the video description down below. It'll show you all the necessary steps you need to take to launch your business in the next 21 days. And as a thank you for signing up and watching as well, I'll also give you a 100% free copy of my PDF of 194 profitable product ideas for 2023. Thanks for watching guys. Don't forget to comment, like and subscribe and I'll see you in the next one. Cheers.


If we start MPV with the command dash dash input dash IPC dash server and give it a file name, we can then play all the files in a directory and control them by sending commands to this file using SoCat. For example, we can play the next song in the playlist. We can then also cycle play pause, pause it, unpause it. We can go back a song in the list by saying playlist previous or preve. We can also seek forward in the song by seconds. We can go forward five seconds forward five seconds or the same command with a minus we can go back five seconds. In an upcoming video, I'll show you how to use this technique to control your media using the media keys on your keyboard or remotely through a network on a mobile device.


Whatever problem we have maybe it's a marital problem. Maybe it's a problem with one of our children. Maybe it's work Bring some vicar to that Sprinkle some vicar to that If you're if we're having marital problems, let's get together as a marriage and let's do vicar together me and my wife Me and my husband and if we're thinking right now. Oh, that's really gonna help Say in Israel say in his whispering right now Think it is real. They get as powerful whether we comprehend the benefits whether we see them They're metaphysical. We don't understand them, but these are divinely promised as Staghfirullah brings Rizq brings brings Rizq brings providence providential care la ilaha illa Allah removes rust from the heart Allahumma salli ala Sayyidina Muhammad wa ala alihi wa sahbihi wa sallim salawat and the prophet Muhammad say salam relieve you from your your problems and Removes you from darkness to light and if we're thinking. Oh, yeah, really. How is that happening? Remember We don't just believe what we see. We don't just believe what we see We see what we believe we're believers. We're different. We're Muslim. We're different the first description of believers in the Qur'an You mean on a bill guy, they believe in the unseen realm and so if we have been fed for these many years and decades in 2021 if we have been fed the the the illusion That vikr is not something that can help me that is powerful. It's not just like I'm getting points I'm getting points in my book to get on the day of judgment vikr is more than that When you mention Allah Allah mentions you and so we ask ourselves today How am I doing enough vikr? Am I doing enough vikr? and So we're gonna do that in two ways number one. We are going to break these ideas These notions that may have been planted in our minds That vikr is this lofty mysterious thing That's in some other realm. We're gonna break that notion and we're gonna accept it as a tangible tool We're gonna accept it intellectually as a tangible tool Like God said with the two ingredients feth butu wa thgurullaha kathira and then you'll find success The only thing the only thing in the entire Qur'an that Allah says to do a lot of Is vikr? Is vikr? The only thing and so we ask ourselves am I doing enough vikr that I teach my kids to value vikr and It's easy.   Don't be heavy on your children You can do it more than them, but when they grow up and become 815 and your influence decreases on your children When they become 18 or 20 and 21 and they leave the house, they will remember my father was a man of vikr Every Friday in my life my father did vikr and Every morning when we wake up What vikr are we doing find a qualified teacher and ask them what the kid can I do every morning? For your own sake you're not get you're not you're not helping God through vikr. You're helping yourselves You're helping your children You're helping your grandchildren because the but I got flows This is real And so this is an invitation for us today to bring make our house a house of vikr Our marriage a marriage of vikr our home a home of vikr and my my body a body of vikr And if we're too busy out the the vikr beads The vikr beads are a tool To remind the tongue and the tongue is a tool to remind the heart And even if we don't satan's gonna come the devil's gonna come whisper and he's gonna say What's the point of this you're not feeling anything? La ilaha illallah la ilaha what you what's the point of this you're not feeling anything You're not doing it. What's the point of this that satan taking you away from doing good And perhaps if you continue doing it Vikr, Allah will elevate you from a state of doing vikr without feeling To a state of vikr with feelings Don't give up on it start with the minimum every morning Of some staghfirullah of some la ilaha illallah Of some salawat on the prophet sallallahu alayhi wasallam and find a qualified teacher and incorporate this in our lives