 So, good morning. Let's continue our lecture and this will be on boiling heat transfer as well. So, as a quick recap on what we looked into yesterday, first we started off on the basics of phase change. Then we looked into what is boiling and what is evaporation, what is the difference. Essentially boiling is the bulk phase change process. Then we followed it up with looking at what is required to ensure boiling. So, some amount of super heat is required essentially to try and maintain a bubble of radius r and that liquid scan exists in a metastable state and heteroflates fluctuations are required to disturb them and become vapor. Then we looked into what are the conditions required for bubble inception and how to calculate the nucleation size. We looked into an ideal case of bubble growth dynamics, inertia control growth, heat transfer control growth and then we looked at bubble growth at the wall. Essentially that a range of cavities can be activated and there's something called the bubble evolution cycle where the bubble grows, departs and a new bubble forms. So, in today's lecture we will just start with one or two simple problems on what we looked at yesterday. Then we look at a study that I did which is on DNS of bubble growth. It just gives you a perspective of what is being done today in bubble growth and follow it up with discussion on pool boiling curve and the correlations involved therein. Parametric dependence of this pool boiling curve to different operating conditions, some problems on pool boiling correlations. We'll then follow it up with two phase flows and look at pressure drop calculations and one or two problems on that and then we'll start on flow boiling. So, let's begin with the problems one or two from yesterday's study. So, this is the question. Consider a spherical bubble of pure vapor suspended in a superheated liquid form. On a PV diagram, can you tell me what kind of phenomenon can occur so that this bubble either grows or collapses? So, let's look at this from the PV diagram point of view. So, you have the saturation curve and if you remember an isotherm follows this trend where between here to here is an unphysical form. So, several such isotherms will exist. So, now we have a stable vapor in a state of superheated liquid say TL and PL. So, say your liquid is superheated. So, this is your metastable region you know. So, this is located say here and the corresponding pressure of the system is PL. So, this is TL, this is PL. And the saturation pressure corresponding to TL is this. So, your vapor pressure is going to be somewhere in between this and we know that your PV is somewhere closer to PSAT. So, this system exists. Now, what do you think should happen for the stable bubble to be able to either grow or collapse? So, I would like you to think in terms of the temperature at the interface, the temperature of the vapor right. So, a scenario what is your pressure and temperature? So, you look think of in terms of the hetero phase fluctuations. So, some form of fluctuation occurs so that your pressure and your temperature locally drops. So, essentially you fall on this curve a curve below the pressure drops. So, then what happens is the temperature locally drops. So, here is your T SAT for the previous case. Now, the temperature of the liquid locally has dropped. So, essentially the vapor interface sees as if it is at a higher temperature. So, it wants to lose heat. So, the minute it begins to lose heat it is going to collapse. So, essentially a drop in pressure is going to trigger a bubble collapse. Now, what do you have to do to make the bubble grow? Obviously, if there is going to be a jump in pressure, a violent shaking which causes an increase in pressure, then and it has to the temperature has to jump above your saturation temperature over here. So, the minute it jumps to a higher level say from here to this line over here, then the saturation temperature at this location is actually higher. So, what it is going to do? It is going to see that it has access to heat. It has access to extra heat. So, it is going to grow. So, essentially it is your pressure and temperature variations locally to the interface for it to grow or collapse. So, going on to the next problem for bubble growth in water at atmospheric pressure and 120 degree Celsius, estimate the bubble size corresponding approximately to your transition between your inertia and heat control growth. So, you remember that there are two forms of growth which is your inertia control growth which is a function of T and your heat transfer control growth which is a function of T power half. So, this is quite a simple problem. So, for a bubble growth in water 120 degrees, estimate the bubble size which lies in the transition zone. So, if you remember I gave you a correlation by Mckick which talks about modelling growth right from inertia control growth to heat transfer control growth. So, all you need to go is look at this graph that I have given you. I think in the previous lecture the graph is a little bigger and much clearer. So, what I want to point out here is that your inertia control growth is for a time T plus non-dimensionalized time T plus 0.01 below that. And for T plus greater than 100 it is heat transfer control growth. So, for transition control growth you can pick any T plus and corresponding R plus on this non-dimensionalized graph. So, in my case I have just picked R plus 1. You do not have to do that you can do something in this range. So, if you take R plus as 1 you plug it in on that R plus. So, your R transition size essentially is B square by A and then it is just a matter of plugging in the properties. So, essentially I want you to be able to use this graph and try to work out transition inertia control growth and heat transfer control growth. I think you will get after plugging in the values a transition size of 0.185 mm. So, you can just work this out and if you have some problem you can get back to me. Now, moving on to DNS of high pressure bubble growth. Why are we talking about this now? You are looking at trying to model a bubble growing. But an important assumption that you have made over here is the bubble is spherical all the time. Made be at the wall or in an infinite pool of liquid. But in reality your bubble is nowhere close to being a sphere. So, it is going to have random growth sizes. So, essentially when you are trying to model bubble growth it makes sense that you track the interface locally. So, nowadays computational techniques actually track your interface at every localized point. Now, why is this useful? A for the size, B I would like you to consider that you have only looked at atmospheric pressures up until this point. You have not looked at high pressure conditions and say for nuclear reactors or industrial systems. It is high pressures at say 100 bar, 150 bar and I have no idea how my bubble is growing. I cannot conduct experiments to look at it. So, I do not know how much heat is being transferred. And later on you will understand why this is such an important thing to be able to study what size your bubble is going to be. So, essentially I can only depend on a computational framework to be able to try and predict. So, that is the goal of trying to do a simulation. So, DNS stands for direct numerical simulation wherein you are trying to solve a form of the Navier stokes and you are also trying to track the interface. And you will be surprised to see that the equations are similar to what you have looked at at the core. So, that is what I want to highlight some of the similarities and the dissimilarities. So, when you are looking at first trying to predict growth for 150 bar pressures you want to be able to first validate the system. So, as a first exercise here I am just showing you the work that we did taking an experiment which was available for 45 bar a bubble growing they were able to visualize it and to be able to apply the DNS computations to observe the bubble growth and understand what the bubble growth is. Is it inertia control? Is it heat transfer control? Is there something else additionally involved in it? So, let us find out. So, first looking at the methodology like I told you we are looking at tracking the interface locally at the grid point level. So, how do we calculate the heat transfer? So, if you look at it you have the interface which is divided across the cells. So, essentially there is a vapor phase and a liquid phase. So, what we are trying to do is we are trying to calculate the gradient of temperature in the vapor phase and the gradient of temperature in the liquid phase. So, conduction to the interface. So, that is your QL and QV and your net transfer either it is going to be condensation or evaporation is going to be a sum of these two. So, essentially we determine m dot as a summation and then what do we do with that? We know say some amount of vapor is being produced. So, essentially we need to track the interface. What do I mean by tracking the interface? I am starting at a level 0 at time t and I have produced this much more. So, I am going to move my interface forward by a velocity. If you remember you push the bubble interface with u equal to you know you have an expression for u that you determine. So, something similar to that. So, you are pushing your void interface a little bit more with that m dot. So, that is what your second equation there does. And once you moved it your temperature has to restructure itself correct. So, you are solving a temperature equation after that. And if you remember this temperature equation is similar to the heat controlled differential equation that you saw yesterday. Where you derived r is a function of square root of t. So, there are some similarities in the method. So, we set up a simulation of a container with an initial bubble size and we try different grid sizes. Essentially because if you are trying to make this finer you are trying to calculate it more accurately. You are removing all the errors. So, essentially you try and go to higher cell sizes. So, you can try even up to 2 million we have tried it. But the catch here is the computational power. How long it is going to take? When we simulated it. So, this is the sample of the kind of contours that you get. And what do you see here? You see here that as the bubble grows it seems to take away heat from the nearby zone. So, there is some kind of heat removal from the wall. And as it grows and departs you will actually see the quenching take place. So, the evolution cycle the complete heat transfer can be characterized. More importantly our predictions were able to match the experimental values. And further investigation actually showed us. Now, this is the important part. For high pressure boiling the growth rate follows the trend that you saw for heat transfer control growth. So, the formulation that you learn for heat transfer control growth is applicable for high pressure conditions. So, when you are trying to design for nuclear reactors you are trying to go in for r in terms of square root of t. Now, the coefficient is the important part. So, is that all? No, actually there are some more intricacies involved in heat transfer. So, when you are trying to design for boiling at a surface till now we have just approximated it to be a sphere. And your heat transfer was how much evaporation takes place into the vapor. But the kind of heat transfer that you are saying is conduction across the thermal boundary layer and that is the important part. What we find out is that this is not all there is to it. There is actually a micro layer that is involved. What do I mean by the micro layer? So, by the micro layer I mean that your bubble forms at a cavity and is in contact with the wall. It is not able to move as fast of the rest of the interface. So, it forms a thin micro layer across which there is very high heat transfer. So, that is a heat transfer that we are not calculating in this expression. So, some people have gone ahead and developed a structure similar to this, but they given some empiricicity in the beginning. Why? Why can't you just do these DNS simulations to be able to predict? Because for example, to give you a timeline to make that bubble grow accurately, it took us at least 10 days to 14 days for that simulation to run. So, you are looking at a single bubble which is growing at a slow space. So, when you are trying to do atmospheric pressure with a bubble like this, you will have high resolution requirements with at least three weeks of runs going on for a single bubble. So, empiricism has its perks. I mean I know that people should complain that two phase flows is filled with empiricicity, but essentially this is what it helps you do. It actually puts in an empirical correlation C over here and you can calculate the rate at which the radius will change in within 15 minutes. So, essentially what this gives you is the rate, but you do get some input on the heat transfer magnitude, some amount of verification. Going on to scales of boiling. What do I mean by scales of boiling? Boiling we have looked at till now is just a single bubble, but we have several such bubbles forming. For example, let me just show you this video by Evelyn Wong in MIT. So, this is a pool boiling experiment. Look at how closely the bubbles are growing and how fast. So, it is hardly just that single bubble nicely growing at one corner. Eventually when you crank up your heat, you are going to have bubbles forming at very high frequency of different sizes. And when you come into flow boiling kind of approaches, can you see with increasing heat fluxes, this middle one has the highest heat flux. You have bubbles of different sizes and different moving velocities, different shapes going on. So, this is what you have to eventually try and capture. So, how do you extrapolate from a single bubble R of T to something that you are trying to model here? Series of bubbles growing at the wall. So, this they achieve by using the wall heat flux partitioning. So, you have an input heat flux and what they do is they divide it into the heat flux. So, they partition it into single phase convection to liquid, then there is an evaporation. After the bubble departs, there is a quenching phase. So, it is this that you will mostly be dealing with. So, when you try and calculate the heat flux on the surface, you will have to account for all of this. And in built into this is the RT term because your RT term will give you the amount of area, duration at which it overlaps the surface and the extent to which it grows. So, this still talks about heat flux locally on a surface. It does not talk about industrial scale problems where you are trying to model draw bundles. So, there we have to come to correlations again. Correlations which take some amount of this physics like a force balance or some amount of this growth rate relations, but essentially are fully correlations. So, that brings us to the pool boiling and the pool boiling curve. So, what is this pool boiling curve? Like you have looked at in that video, as they cranked up the heat, the bubbles that formed were closer, more rapid and more fast in their formation. So, this curve essentially talks about as you increase the wall super heat which is your x-axis, it is telling you how much your heat flux at the wall is going to be. So, let us just reproduce or deconstruct that graph more carefully. So, this is a curve that was generated by Nukiama in the 1930s. So, what he essentially did was he started with a low super heat and he cranked the surface temperature up and he visualized how boiling changed. So, essentially in the first portion as he heated up the surface as expected the natural convection curves were set up. So, till this point on there was just natural convection involved that is your first portion of the curve and then at one particular point boiling started. So, this boiling where just individual bubbles were formed in a very isolated manner. So, this was the isolated boiling region. With increasing heat more heat flux was ensured and higher number of bubbles were being formed. So, this continued on till a particular maximum heat flux was possible. Now, this region is the most favorable it is called the nucleate boiling region. Because you can achieve just for a small increase in wall super heat a very high increase in the wall heat flux. So, it reaches a top maximum point after which there is so much of bubbles around that they begin to coalesce. So, they start forming small vapor thin films. And when small thin films like these form what happens is that there is no further well there is lesser heat transfer that is possible across these vapor films. So, essentially your heat transfer or heat flux is going to drop and at one particular point over here and beyond. So, this portion is called the transition boiling. Then you go on to the final part where instead of sporadic vapor films you actually see a continuous vapor film formed and that is called the film boiling regime. So, essentially you see that your heat flux peaks at a particular point and is minimum at a particular location right. So, we know that we are only interested in maximum heat transfer possible. So, that maximum heat flux possible is an interesting location for us and we want to be able to predict that right. But it is actually more important than we actually thought. So, in the nuclear reactor case they have huge projects being funded to be able to just predict that maximum heat flux and that is known as the critical heat flux location. So, why is that of so much important? So, let us look at trying to instead of controlling the wall temperature let us look at trying to look at controlling the wall heat flux. So, I am increasing the wall heat flux and your surface temperature. So, this is for this graph here you see is for atmospheric water. So, I reach the maximum peak location point which needs a super heat of 30 degree Celsius right. Now, imagine I just increase my heat flux beyond that by just a little bit. You directly jump to the pool boiling regime where the surface temperature is almost 1000. So, what does that mean? You will appreciate this video I think. So, this can you see it because the temperature is increasing so high that at one particular point it is going to snap. So, essentially that is why critical heat flux is very important. In nuclear reactors they are scared that if that rupture happens because of such a high heat flux or heat surface temperature. Your nuclear reactor can have some break where your you know your radioactive fluid can slip out. So, what causes that? Essentially it is boiling. You have series of boiling happening with closed bubbles and just after your critical heat flux this immediately forms a film and there is a spike in the wall temperature and causes that rupture. So, you need to be able to predict from this bubble size growth rates to a film formation and that is why this is very important. So, people try so with that motivation in mind people try and develop correlations for different regions to be able to predict the heat flux that will put you in different regions and design systems accordingly. So, for the first region which is your natural convection region it is pretty much what you have learnt in your natural convection classes. Essentially all those correlations are only going to be used. So, you know that qv is equal to h delta t right. So, then you know this is given by your natural convection correlations as h t by k. So, all you need to do is take your natural convection correlation and get your h and plug it in here and that is all that is been done over there. So, all your nu correlations are what you see over there. So, essentially to calculate heat flux in this region you just need to use those correlations. So, going on to your nucleate boiling which is your most interesting most efficient region what you need to do is use you can use any correlation, but the most popularly used one is the Rosenhoff's correlation. So, Rosenhoff in his model comes up with at least he tries and incorporates some amount of physics. So, I like to bring your attention to the first term in the brackets there that is nothing but a balance between your buoyancy force and your surface tension force that is holding the liquid at the wall the bubble at the wall. So, that is a force balance takes care of the mechanical thing for the heat transfer you can see the next term that is c p delta t by h l v that is nothing but your Jacob number. So, if you remember that is a measure of how good your boiling is that is a number that you it will feature quite frequently in your correlations. So, he used these two let us call it physics in the equation and proposed some numbers in and around it going on to trying to predict the critical heat flux location. So, you have Zuber's correlation. So, I am just presenting one or two correlations you are free to use any correlation that you want for any specific reason. But I want you in your paper to just mention whichever correlation and why you have chosen that and I am I am flexible with what correlation you are going to use. So, in Zuber's correlation as far as calculating your CHF location is concerned he proposes this correlation, but if I like to draw your attention to what is in the brackets right. So, he tries to capture something over there and that is different from your buoyancy versus surface tension ratio. So, that brings us to how they conceptualized CHF there were many theories for CHF one of them is by Cole. So, he saw it that CHF essentially means a vapor fill in some cases right. So, there is essentially constant presence of vapor at the wall Cole saw it slightly differently. He said that it is because of a lack of refilling of cold liquid it is not that vapor is constantly at the wall. So, he was saying that your bubbles form depart in quick succession one after the other. So, they rise with the terminal velocity. So, essentially the force balance that he was interested in is the buoyancy versus the drag force. For him surface tension the bubble is already left the wall. So, he does that force balance and he comes up with that correlation there. I am not even going to try and pronounce that name, but that guy proposed a correlation which takes into account CHF for sub-cooling cases. So, in his case you can see that instead of proposing the standard Jacob number which is CP superheat by HLV. He takes into account that you are trying to raise the temperature from sub-cool liquid. So, essentially that is the kind of sensible heat that he is interested in. So, that is the formulation that he has tried to include in addition to your buoyancy versus drag force term. Coming to film boiling and this was I think another good video by captured by Professor Mudaga in Purdue and that is film boiling. So, that is just a wire and you have vapor forming and departing from the wire. So, essentially what you see is that there is a thick vapor film involved right. This is relatively thin, but essentially you will see a slightly thicker film. So, to model heat transfer there they say that film boiling has such huge heat that there is radiation component involved. So, you have a convection heat transfer component and a radiation heat transfer component. And your convection heat transfer component also has a phase change because your water is converting into steam there right to vapor. So, you have a HLV component and you have a radiation component and because it is bubbles leaving the surface, he is characterized it by the surface tension versus buoyancy force right. So, let us talk about how this curve will change for different conditions. Now this is a curve for saturated water at atmospheric pressure. So, when we try and look at how this is going to change for the first case which is with sub-cooling. Now imagine this is not a saturated curve, but for sub-cooling conditions what happens in different cases. We know that with sub-cooling natural convection will ensure more heat transfer. So, your curve is going to shift slightly a bit up. Then for nuclear boiling experimentally it is found that sub-cooling does not really help heat transfer significantly. Because essentially it in itself is quite efficient in forming bubbles and departing it is only bothered about heat transfer from the vapor. So, the nuclear boiling case experimentally they found does not help this pool boiling curve in the nuclear boiling region. But at the critical heat flux zone it seems to increase the ability. So, a system like for example, you are designing a nuclear reactor and instead of operating it at a saturated condition. Instead of sending water at saturated condition you send in water at sub-cool conditions. Then you are actually increasing the capacity of the system. You want the critical heat flux will happen at a much higher heat flux than at a lower heat flux. The target is essentially to increase the critical heat flux so that it never reaches that region. So, you find that most of the nuclear reactors will have a sub-cooled temperature inlet. So, your liquid is going to be sub-cooled and this is essentially the reason why. Because you have increased the critical heat flux. Then transition boiling again there is not much understanding because it is such a quick phase. It quickly changes to film boiling but people poichlate that it does increase the heat flux. And when you go towards film boiling the increase is there but it slowly decreases with sub-cooling. Because after a point yes there is sub-cooled liquid over here. It might cause some condensation locally but beyond a particular temperature it is the radiation which will take over. So, that is going to govern the case. So, your parametric dependence is that it essentially shifts it by a little bit. But you know not significantly in the nucleate boiling region and in the last portions of film boiling. First convection pretty much the same effects as that of sub-cooling. But only in the film boiling region it is able to ensure more transport of vapor. Because a forced convection is able to carry the vapor on forward. So, that helps the heat transfer coming to wettability of the surface. So, essentially when we were talking about hydrophilic and hydrophobic surfaces what we saw was that. So, when you have a hydrophilic surface this is vapor and this is liquid. Once the vapor forms and departs the liquid quenches the surface. But when you talk about a hydrophobic surface meaning liquid doesn't want to come into contact with the surface. Your vapor film will directly form it won't be a bubble anymore. So, this vapor film essentially is going to push you into the film boiling region directly. So, what people have seen is that this region disappears. And essentially you are looking at a film boiling region directly. Surface roughness essentially talks about you know trying to refill cavities. So, it is just an extension it is a minor point that I would like to point out. That once you have a lot of roughnesses it has a capability to trap air. So, that is going to promote nuclear boiling more heat transfer right. And in radiation people are not really sure because it helps heat transfer. But it can also affect because of its emissivity changes the radiation. But nobody has quantified it. So, with this background of pool boiling curve and pool boiling correlations. Let's just work out one or two problems in using these correlations. So, let's look at the first problem here. Water at one atmosphere is heated in a pan. The bottom of this pan is of a specified diameter and is maintained at 115 degree Celsius. Estimate the power required to boil the water in this pan. Determine the evaporation rate. What is the ratio of the surface heat flux to the critical heat flux. And what is the pan temperature that is required to get critical heat flux. So, let's just tackle this problem one at a time. So, looking at the first thing. You are given water at one atmosphere. So, your saturation temperature is 100 degree Celsius. Superheat is 15 and you have the properties of the liquid. So, I am not going to go into the properties of the liquid. And then you are asked to find out first the power required to boil the water in this pan. So, the first thing that you are trying to say is that I assume you want to initiate new create boiling. So, the first thing would be to come up with the Rosenov's correlation. So, you take the Rosenov's correlation plug in the values all of which which is it is here. And you have been given that it is a copper pan. So, in this table there is a better picture again in your textbooks I think. So, for water, copper say polished surface you have the value of CSF. Which is one of the coefficients that you need is 0.013 and n is 1. So, I think you know all the terms that are involved in that expression. So, once you calculate the surface heat flux you know that what is the surface area. So, it is the bottom side of a pan. So, you know the area. So, you have the power required and you will get something to the likes of 8.16 kilowatt. The evaporation rate. So, it says determine the evaporation rate. It is nothing but m dot h of g. So, you can just plug that in and you will get something to the amount of 13 kgs per hour if you are expressing it in terms of the hour. The next thing is what is the ratio of the surface heat flux to the critical heat flux. So, that means you take the correlation of Zuber's say which is the critical heat flux correlation. And you have all the values that is there plug it in and the ratio is 0.42. Now, it asks you to find out the surface temperature at which this critical heat flux is there. In this case it is water at one atmosphere which pretty much is just your pool boiling curve and you I have told you that the critical heat flux is at 30 degrees Celsius. So, in a later case you will be calculating the surface temperature in the next problem. But essentially it is 30 degrees Celsius here. So, going to the next question. Horizontal copper tubes 25 mm in diameter and 7.75 meter long are used to boil saturated water at one atmosphere. If the tubes are operated at 75% of the critical heat flux, how many tubes are needed to provide a vapor production rate of 750 kilograms per hour? What is the corresponding tube surface temperature? So, again it is just pretty simple. So, you have the critical heat flux which is Zuber's correlation and you plug that in solve that. So, you have 0.75. So, from the previous problem it is water at one atmosphere. So, in the previous problem this we get it to be 0.99 megawatt per meter square. So, your actual heat flux is 75% of this value. So, you have times this. Now, what do you ask to calculate the number of tubes? So, essentially this is into area which is n times pi TL. And that can be equated to m dot h of g and your m dot is given to be 750 kilograms per hour. So, it is just a matter of solving this relation and you get the number of tubes. I think I get something around 10 and you calculate the surface temperature. So, you have your heat flux known. So, go back to your Rosenhoff's correlation. You have the left hand side known. You can just calculate the super heat that is required. So, that is this problem here. The third problem, what is the critical heat flux for boiling water which is kept at a pressure of one atmosphere, but on the surface of the moon? Now, you know that the gravity of the surface of the moon is 1 sixth that of the earth. So, how does that lower gravity or reduce gravity condition what happens to your critical heat flux? So, when you take a look at, I wonder if that correlation is your, yes. So, it is your best correlation. If you take your, so essentially this becomes a comparison between your critical heat flux at on earth to that of moon. This relation pretty much holds because your QCHF is found to be proportional to one fourth. So, for the case of moon keeping everything else the same you have and what you will find that, so here we have already calculated it in the previous problem we were about 1.1 megawatt per meter square. So, you would actually find that the critical heat flux decreases in a reduced gravity condition. So, why does that happen? If you think about it, it is pretty much your buoyancy force which is pulling your liquid away from the surface and preventing and you know actually facilitating the quenching of the surface, but in reduced gravity conditions this will not happen so efficiently. So, your vapor is pretty much going to be lying around on the surface. So, essentially that means that your critical heat flux is lower. It is going to happen quicker because your vapor is going to be closer, your threshold is much lesser. So, essentially this is a topic of interest for ISRO because they are trying to do cooling of, cooling in their space stations. So, they are trying to you know try boiling is an efficient form of heat transfer for them, but in reduced gravity conditions they have to be very careful and they would like to investigate how boiling is occurring in reduced gravity conditions. So, coming to two phase flows right. So, two phase flows essentially is a more complicated form of pool boiling. You have looked at various regimes in pool boiling. So, you started with natural connection, nuclear boiling regime, transition boiling and film boiling. Essentially it is how these vapor phases are there in relation to your liquid. So, in the case of flows again the flow rate in itself complicates the flow pattern quite a lot. So, if you look at this video it basically shows you how the flow pattern changes with increasing glass flow rate and can you see that. So, you can see slugs actually going up slugs of at intermittent points. Then you actually have vapor coming down water falling on the side and then there is annular flow. So, people actually have seen these in experiments and they have tried to characterize them into different categories. So, they have actually when the first part where you see very little vapor bubbles forming, they call it the bubbly flow where you see those slugs forming, they call it the slug flow where you see those churning happening, they call it the churn flow and finally the annular flow. It is only that there is increasing amount of vapor and in different forms. Why does this matter? Because we are trying to calculate heat transfer in all these systems and the heat transfer in the first is going to be drastically different from the last. So, in any industrial case we need to know first what kind of flow pattern forms to be able to apply the requisite correlations. So, it is highly dependent on the flow pattern and the flow pattern is dependent on the input conditions of glass flow rate and the liquid flow rate. And why is this important? Because the first important calculation that you do is your pressure drop calculation. You need to know interfacial area because you need to know bubble breakups. Heat transfer involved there on and there. So, the first task is to be able to try and predict the flow pattern. So, what people have done is they have conducted many experiments and tried to develop maps like these called the flow regime maps. So, what are these flow regime maps? They have tried to chart out by you know varying your velocity of the liquid and velocity of the gas what different flow patterns can be observed. So, if you want to try and develop say flow through a tube you just have to go and see what is my mass flow rate for gas mass flow rate for liquid and find out the kind of flow pattern that is possible from one of these flow regime maps and then devise your equations accordingly according to the flow pattern that you will be observing. Going on to the mathematical modeling that is involved. So, once you have determined the kind of flow pattern that is there then you have the mathematical modeling of these faces. Essentially how people tackle these various flow patterns is that they try and average out the vapour. So, you will see complex vapour and liquid forms. So, essentially what they try and do is that they assume some amount of vapour and some amount of liquid in a segregated form. So, essentially these are such kind of numbers. So, first volume fraction or void fraction essentially is the volume of vapour occupied to the total volume. So, volume fraction of liquid or volume fraction of vapour synonymous with void fraction is an expression where which gives you amount of in a cross sectional area amount that is the volume that is occupied. The next three are quality. So, quality essentially can be of three types equilibrium quality, flow quality and static quality. So, let us the first is static quality is the simplest form in a cross section the mass of vapour to the total vapour. Flow quality talks about the mass flow rate of vapour to the total flow rate and equilibrium quality basically talks about the averaged enthalpy across a system to the total enthalpy. And slip obviously is the velocity ratio between the two phases. All these expressions that you will be seeing further will require some form of being able to transfer from say a void fraction to a quality and to a slip. So, this is just the interrelationship between all these terms which you will require in your calculations, right. So, to be able to model the flows that you saw essentially there are three popular models that are preferred one is your homogeneous equilibrium model the second is your separated flow model and the third is your two fluid model. So, when you are talking about your homogeneous equilibrium model essentially the assumption is that your liquid and vapour are moving at the same velocity they are just a mixture and you use properties that are mixture properties. So, that simplifies any kind of slip between the velocities. So, this probably would be useful in a place where you are trying to model highly dispersed flows in a sense very little vapour lot of liquid then you can actually assume it to be one medium and get a mixture property involved. The second is your separated flow model wherein essentially they try and model the liquid and the vapour phase separately the momentum equations and they try and capture this the slip between the phases. This is important if they are significantly equal amount of vapour and liquid the separated flow is preferred. And finally, the two fluid model in this case it solves two equations of mass two equations of momentum and two equations of energy and there will be interfacial interaction terms. So, your two fluid model is kind of the most up to date model which is being used by all your commercial cold softwares and is preferable but obviously it is computationally intensive. So, right now we will probably in this part of the talk we will just restrict ourselves to the first two models and for the subsequent lectures we will talk about the two fluid model. And the next part is where we go into the pressure drop calculations and I think I will pick it up from the next class and we will start with how to calculate the frictional acceleration and static pressure drops in a tube. Thank you.