 What is the boiling curve? What is the slope of this line everywhere? y axis is Q, x axis is delta T. So, essentially the nature of this curve gives you the slope which is the heat transfer coefficient directly. So, that is why this part which we call as the natural convection boiling is essentially what you have seen in natural convection nothing new. In this part of the curve T wall minus T sat this is a thought experiment which is done by controlling the all of you are teaching this that is why I am not answering. I can do a boiling experiment to generate this curve by two ways. One is heat flux controlled one is surface temperature controlled correct or wrong? Is there any difference? Is there any difference heat flux controlled versus surface temperature controlled in terms of what you see in the boiling curve? Automatic automatic means what correct or automatic? What is automatic? Ok let us for you got your idea you know what you are talking about what happens is if I do this generation of boiling curve let us say with heat flux controlled experiment means I am changing the heat flux trying to keep the wall temperature and anchor. So, what I will get with progressively increasing heat flux that means my gas is slowly being turned higher and higher making T we are still making T only. What I get is essentially if I measure this I will get a nature of the curve which shows A to C. This A, B, C essentially represents what we call as a nucleate boiling region. What it means is they are segregated into bubbles collapsing etcetera we will call it subcooled boiling and saturated nucleate boiling etcetera they are not concerned with those definitions at this point. This is a region of good heat transfer coefficient why because essentially latent heat conversion is taking place and the process is occurring roughly at constant temperature T wall minus T sat may change a little bit, but by enlarge this delta T is constant lot of heat is being removed what is the Q double prime balance by m dot h f g m dot is the rate of evaporation or rate of change of liquid to vapor phase. Now, when I come to this point C I do not come to this point directly when I come somewhere here you know near about we do not know when we are doing the experiment what is this maximum. What is the typical heat transfer coefficient you can expect in nucleate boiling? We can just see 10 to the power of 6 to 10 to the power of 4 I can forget 10 to the power of 4 10 to the power of 6. If I assume linear although it is not linear logarithmic its logarithmic if I take 10 to the power of 6 divided by temperature difference of 20. So, it is how much 50,000 around 50,000 you will get. So, heat transfer coefficients are very very high. So, if I keep on increasing the heat flux I reach a point where with the increase in heat flux what happens? I go to a wall temperature which is suddenly become very very high. So, from C I directly jump to E. What it means? What does it mean? How will you tell this to your students? Very nice. Can you give me having told all this I do not know where this film is coming from. Can you give me one very very very common example? All of us have seen ladies have done this at home this experiment exactly. Very useful easy to understand when your tawa becomes very hot you are not able to make the dosa proper. Why? It is just that. So, you splash water and if the tawa is so hot you see this dancing of problems. Why is it dancing? Because there is a film and then it evaporates. When the surface is not very hot the water which is sprinkled on it will stay at that point and evaporate. That is film. You give this example please it is such a useful easy to follow example. All of us have eaten dosa all of us have seen dosa being made either at home or at the roadside. So, we know what it is what how to explain this. This is film boy everything else experiment though that is need and first you will tell later. But the classic example of your tawa becoming very hot when you are making dosa you will see that you will be unable to move the batter freely. So, what in the on the roadside stalls if you have seen you will have water and you will mercilessly sprinkle water on it. Why he is doing it is to reduce the temperature of the tawa to a lower value. If the flame is very high and it is gotten very hot you get a phone call you are talking of forgotten about the gas it has become very hot. So, when you sprinkle water you will see these droplets dancing all around and then they evaporate eventually. This dancing is because of the layer of vapor or steam which is formed between the liquid and the tawa you are not able to have contact of the liquid with the heated surface. Therefore, this nature of such heat transfer is very very bad because you are having an insulating or a vapor film which has a very very poor thermal conductivity which is going to have a very very low value of H. What feeds what we are not concerned at this point. So, what happens is the wall temperature has become very high. Therefore, your heat transfer coefficient for the same heat flux has plummeted down because delta t has increased tremendously. When the tawa is not that hot when you sprinkle the water the water will remain as droplets locally and will eventually evaporate. So, this example I like to give and it is well understood by everybody even who is not an engineer will understand this. So, this point therefore, will see a sudden increase in the wall temperature for the same value of heat flux. And then if I reduce the heat flux I will not follow this path I will come down along this and trace this so called I will not trace this transition boiling path CD. I will come back to a point here from D. So, for a heat flux controlled experiment I will trace the curve A, B, C, E, B back to some F which is between A and B. So, this heat this regime or the boiling curve therefore gets divided into roughly 4 regimes natural convection boiling, nucleate boiling, film boiling and transition boiling. This film boiling and transition boiling are poor boiling regimes or poor heat transfer regime. How is this transition boiling demarcated? It is demarcated by the point at which the so called film boiling is gone. That means liquid no longer is sorry vapour no longer completely occupies the heated surface. There can be intermittent contact of liquid with the heated surface. Wetting of the heated surface becomes possible at points between C and D till here from E to D it was only vapour, vapour, vapour, vapour. That is why this is called this is that is why called as minimum film boiling temperature or in order of laden frost called laden frost boil. Minimum T min it is called minimum film boiling temperature afterwards it comes like this. Capturing transition boiling is very difficult. So, in transition boiling you would have seen pictures and text books also alternate vapour and liquid will be there reaching the heated surface. So, you will have an oscillatory behaviour for the heat transfer coefficient nothing stable you will get. So, it will look like noise if you are taking data it will look like noise suddenly there will be a very high temperature suddenly it will be a low temperature. Now, this is all for boiling curve. What is this? Why is this happening? Why is this excursion from C to E happening? What is this critical? How do you explain this critical heat flux? Everybody uses this word so loosely. Critical heat flux I will give you other words which people use dry out critical heat flux burn out if you little bit advance departure from nucleate boiling all these will be used interchangeably wrongly. What is the difference? What is happening in the so called critical heat flux point and then rise in the temperature is the cause or the effect? It is the effect what is the cause? Surface is not melted anywhere. So, when the surface gets destroyed because of it being unable to get cooled we will call it burn out that is burn out everybody understand the surface is destroyed irreparably damaged burn out let us leave it. It is just the term which has I do not think any scientific meaning other than this, but from a point of view of fundamentals what is happening there? So, if I have a sudden increase in the heat flux I have a pool I have a pool boiling situation the same team making experiment. You increase your you have started doing very nicely patiently you have observed all these things and suddenly you get frustrated you say what is this here? How long I have to keep looking increase the gas to what happens then? Suddenly there is a burst of activity you would have seen this also all of you have seen suddenly there will be explosions of bubbles. So, what it is telling is when the rate of bubble when this bubble formation is accelerated so much rapidly you know one bubble is formed left another bubble is formed left etcetera etcetera it is continuous such that what happens after one bubble has formed and before the next bubble comes at that point there is cold liquid. So, the boundary layer is completely destroyed because this bubble has left the place cold liquid has replaced that location it will take a certain amount of time for the cold liquid to come back through this cycle. In fact bubble growth and nucleation cycle will have a waiting period that that time where there is low activity which you have seen when you make tea you will see that there is a distinct time lag between true successive bubbles at the same location and the time lag progressively decreases and then you start to see rapid bubble formation. Now instead of allowing for this slow decrease in this time I suddenly increase the heat flux I am going to have sudden burst of bubbles formed when the sudden burst of bubbles is formed there is no enough time for the cold liquid to come and replace this what is happening is this bubbles will be formed, but it will be formed so rapidly that it will be like a cushion and cold liquid will be unable to come and enter that place that is the cause. So, for a high heat flux situations dry out why I am saying is used wrongly is because of this dry out means what it is dry is this dry no liquid is still there very much there if you measure quality quality will be 20-30 percent here that is why CHF can occur even in low quality situations. CHF can occur in high quality situations also so if I have a flow for example I have shown this to you film becomes annular evaporation is the source of heat transfer quality if I take the cross sectional view there is liquid here, liquid here progressively liquid film is getting depleted and there exists a point where there will be only liquid droplets in the flow no liquid film attached to the wall correct this I can call as dry out why because there is no physically no liquid present what is the thermodynamic quality in this situation almost one very high liquid droplets are there that is all so what I am trying to say is we may not teach all this to students at undergraduate level, but let us have in the back of the mind critical heat flux represents that value of heat flux at which such a thing can happen beyond which such a thing can happen let us take it like that it is not that the surface is going to get burnt it is not that it is going to get deteriorated it is just the situation where the heat transfer coefficient just drops completely two possible reasons are there one is critical heat flux when it occurs in high quality regions we will call it as dry out critical heat flux when it occurs in low quality regions of the order of 0.2, 0.3, 0.4 we will call it not as dry out we will call it as a departure from nucleate boiling because it has just moved away from auto it was doing very nicely nucleate boiling this is called as BNB departure from nucleate boiling this is called as dry out the whole phenomena or the whole thing which we are classifying as critical heat flux is actually these two mechanism of reaching critical heat flux is dry out mechanism of reaching critical heat flux is departure from nucleate boiling am I clear I think this is a very very important concept which we have to take home. This is BNB in pool boiling. In pool boiling what do you think the critical heat flux will be BNB or dry out. So, if I were to draw the boiling curve for this situation will the curve be any different will the curve be any different I cannot classify it. So, this pool boiling curve which is drawn is essentially a representation of q versus delta t it will not tell us the mechanism when I have the boiling curve drawn for flow boiling it will almost be identical except that the convective component of the heat transfer coefficient will add to the existing heat transfer coefficient that curve will be just shifted a little bit higher that is all. So, from this I will not be able to tell a mechanism associated with the critical heat flux another concept which is I have written down made a note of some 8 to 10 things which I should tell, but I do not think I will be able to tell everything. This whatever we can I get this C to D then I will never get C to D the way professor has told us. Now, we have traversed through the path A, B, C, E, D and a point between. So, then how did I get this C to D? If I control what? What was I control? How do I get C to D? If I do an experiment with constant wall temperature then only I can traverse through the path A, B, C, D, E. This has to be told to the students. This has to be told to the students. If a single will have a picture of both the boiling curves I think it will I do not know it has I do not know that it is here, but it has a picture of both the boiling curves. One other thing just in view of time because by really running short of time, but some things which fundamentally we have to know I am making a note and I will try to tell that to you. One I told quality and I held back the definition for all of you. You do not have to tell this to students, but this is something we need to know. Thermodynamics taught us only one quality. We said x is equal to m dot g by m dot. In fact, we did not use m dot also. We m, mg by m dot. Two phase flow there are three quality definitions. One is flow quality which is m dot g by m dot. Static quality which is this one. For a fixed control volume how much is the amount of gas phase present and over how much is the total mass. And the third one is called the equilibrium quality. This assumes thermodynamic equilibrium. Yes, I know I can see that question in your eyes. This is called as flow quality. This is called as equilibrium quality and this is called as static quality. This is what we defined in thermodynamics. Mg by mg plus mf. In fact, at that time none of us ever questioned. You did it for closed systems and that you are using the same thing for flow. Turbine, etcetera, etcetera, flow is happening. We still use the same quality is 0.92. That was and then we are using this also. We were using this also. Am I right? All we were taking it equal. We were assuming everything was in equilibrium. So, all definitions are interchangeable we were using it. But in two phase flow we have to be careful. We rarely use, we do not use this that much because we are almost always dealing with flow. Equilibrium quality will essentially be used in energy balance. Flow quality also we will be using. These two will be equal when you have thermodynamic equilibrium. So, this you keep at the back of your mind. Another thing which is very important which is not there in single phase flow is what we call as void fraction. How many of you have heard this word void fraction? Void fraction what is the definition? Sunderam only sir. Not remember. It is okay. Anybody else? Void fraction. Volume occupied by the vapor phase divided by the total volume. If I take a control volume of the finite thickness, I can approximate this to be Ag by Ag plus Af. Void cold drink when you shake the bottle of thumbs up. The liquid amount is the same. But then that fees which comes out, it is all gas that is the void. That is the space which was originally empty occupied by air. Now when you shake it, the whole bottle seems to be full and the top portion is also occupied by the fees. That is the void. It is like sponge porous. So, this definition was not there ever in single phase. What is the difference? What is the relationship between quality and void fraction? Let us just two steps. Just for completeness, I am going to put this in perspective. This is correct, which is nothing but rho G Ag Vg divided by rho G Ag Vg plus rho F Af Vf which is nothing but 1 over 1 plus rho F by rho G Af by Ag Vf by Vg. And then I will just use fundamentals. Therefore, x is equal to 1 over 1 plus rho F by rho G Vf by Vg and I will multiply and divide by total area Af by A divided by Ag by A. Af by A will be 1 minus void fraction. Ag by A would be void fraction. Am I clear here? So, what is Vf by Vg? Is this clear? Have I reached all of you with this definition and relation? There was Af by Ag. I am multiplying and dividing by total area of flow. So, Af by A is 1 minus alpha because Ag by A we have defined as alpha. Here, if you see, this is Ag by total area. This is the definition of alpha. 1 minus alpha would be Af by A and that is what I am using here. Now, tell me if I know the, if I mean thermodynamic equilibrium, x let us say is known, can I get void fraction? Vf by Vg, is it 1 ever? Physically, does it travel at same speed ever? Never. You can make this assumption of so-called homogeneous flow under what situation? Can you give me an example where you will approximate the velocities to be the same? Liquid droplets are dispersed. What do you mean by dispersed? When liquid droplets are dispersed by bubbles in vapor. What do you mean by dispersed? Atomized and spayed. So, when the physical size is very small, it can move with the velocity of the gaseous. Other example, other extreme bubbles, bubbly flow. You are sucking cold ring. That bubble is also flowing with almost the same velocity as your cold ring. There, two extreme conditions I can approximate these to be equal to 1. This ratio to be the same. Such an approximation is very, very, very, very powerful. It makes life so much easy because only for this case, will I get a relationship between x and alpha. In such a model, it is called as a homogeneous flow. I am not going to do anything beyond the definition. This is what it is. Thermodynamic equilibrium is there. Temperature of the liquid phase and the gas phase are the same. V f is equal to V g is what is that. Makes life beautifully simple. Why does it make life simple? Let us, if you know this, I think you have understood a little bit of two phase flow. Because of this, what can I, how does life become simple? Velocity is taken care of. Reynolds number can be defined. If I, properties, I will weight them somehow. Specific volume, I know the weightage with quality. So, density is known. Viscosity, either I use the reciprocal one or the similar one to density. It does not matter. I will have a form. Rho v d nu. I can get some form of a Reynolds number, some form of a friction factor, some form of a delta v, particular, particular phase. But at least I will be able to do a static. And if I, if I do this pressure drop calculations, this, if I can get the friction factor and the delta p associated to that particular phase, then I will use a Fudge factor or a multiplier or something to say delta p for two phase is equal to this number times delta p for single phase that is what is flowing through the entire pipe. Those factors, etc., we are not going to one other concept. One other thing, this is called as slip vg by vf is called as slip velocity or slip. This is something which will come across in literature. Another point again just, we are not deriving. I am telling, please take it. In single phase flow, delta p, what was it? Frictional pressure drop plus rho gh. If it was a vertical pipe, you had had Bernoulli equation. All of us have applied p1 plus v1 square by p1 by rho g plus v1 square by 2g plus z1 is equal to p2 by rho g plus v2 square by 2g plus z2 plus hf. And z1 and z2 are different. So, delta p essentially was balanced by frictional head loss and the gravitational head loss. So, only two components were there. Now, because of the presence of the second phase, I am just going to state this. Please accept it. Do not question me. You can question me, but we do not have the time for discussion. Because of the change in quality as the flow progresses along the pipe, quality is changing because of the presence of the second phase and or because of the change in the cross-sectional area, this we will be able to appreciate. Even in single phase flow, what is happening in a nozzle? Flow will accelerate, diffuser flow will decelerate. Now, do a two phase flow through a nozzle. Both phases are there. It is going to be obviously there. Compressibility effect, if pressure change is appreciable along the length, then also density changes are there. If because of either or or a combination of these, there is an additional contribution to the pressure drop, which we call as momentum or acceleration pressure drop. Therefore, dp by dz total two phase is equal to dp by dz friction plus dp by dz acceleration plus dp by dz gravity. So, already it was complicated. Now, we have introduced one additional thing in something so simple as pressure drop. This is a nightmare. So, I will leave with that one last statement. Acceleration pressure drop in case of boiling is going to contribute to increasing the pressure drop. In case of condensation, it will reduce the pressure drop because of conversion of vapor to liquid flow will decelerate. Think of it in that way. Deceleration pressure drop it is going to be, the sign is going to be opposite. Therefore, life is a little bit easier. One last word of thought, think of this also. dp by dz is a function of the nature of flow, orientation, vertical or horizontal. What is touching the wall matters greatly in determining the frictional pressure drop. How much area is wetted by the liquid and how much is wetted by the gas phase influences your wall shear stress, which will influence your pressure gradient at the wall. So, actually just to touch upon this Karoo, actually many people call this as New Kiyama Karoo. Why because it was done by Professor New Kiyama. We will upload that paper which was published by Professor New Kiyama. In fact, he had written somewhere in 1910s or 1905s, but IJHMT journal specially requested him to republish that paper again. So, it has come back again in 1972 or so. That is why we have that paper. Maybe there will be some copyright issues, but still I will upload that in the moodle. Within the moodle, do not mail it to anyone to keep it in the moodle, so that you can hard copy whatever you want, but do not mail it anywhere. So, New Kiyama was the first one who did these experiments and Americans were after him for the design of boilers. It is told that he was supposed to come, what was the capital of Japan? He was from Japan, Tokyo. He was asked to, he was not in Tokyo, he was asked to come to American embassy of Tokyo. They then the issue that he was very much bitter with Americans because of world war and all. He said that no, I want to help only my countrymen, but later on he repents for those words after long, long time. The point is he was considered the expert of two phase flow at that point of time. So, this is, this can be or this should be appropriately called as New Kiyama curve. In fact, this curve is generated if we read that paper, we will realize that he has done heater wire, he has done so many experiments and that is the paper which I have seen where in which repeatability has been checked so many times. The wire has been changed from nichrome to stainless steel to some and wire diameter has been changed. He will do all that to check and make sure that no matter what material uses and what diameter uses, he is going to end up with the same. So, that is very, very meticulously done and reported experimental data. So, I would think that all of us should read New Kiyama paper, I will upload that. Coming back, so professor has made us visualize, so I do not think we need to spend time, what is nucleate boiling, what is transition boiling, what is film because we can visualize all that. But for the students, I guess these pictures are useful, I do not know where I have taken this from, but I think one of the books only, this is I have taken from, in fact two phase flow, now only I realize when I see these figures, it has been very well dealt in a simple manner in textbook by Mills, heat transfer by Mills, M-I-L-L-S. When I saw this blue colored picture, I recollect. So, heat transfer by Mills, maybe now new co-author got added up, but Mills, good thing is even if you do not have that book, you do Google book search, you will at least see these pages, you can read them, is that okay. So, these figures are taken from there. So, of course bubbling, bubble nucleation, boiling inception professor has taught us so much, so I do not need to spend time. So, all that now, Rosenhow did is that, now again we want to get back to our Nusselt number, Reynolds number and Prandtl number. So, that is what we put today. So, Nusselt number is again hl by kf, f here subscript represents for fluid that is liquid and Reynolds number is rho f u l by mu f, problem is do I know u here, do I know the velocity, no, the liquid velocity I do not know, the velocity is taken as the liquid velocity in towards the surface which is to supply vapor which is being produced, that is what explains us, what does that tell, what is the supply heat flux, what is the supply power for generating this liquid velocity or the vapor velocity sorry, see I am applying the heat flux, the vapor velocity is because of this heat flux, had I not given this heat flux, will the vapor velocity would have been there, no, higher the heat flux, higher is the vapor velocity, that is all I am doing, through q double dash upon rho f hfg because this velocity is dependent on the latent heat, so that is how you took, this is all intuitive, so you took the velocity as that and length scale, length scale here you see, length scale he is taking, we were mentioning in the beginning surface tension, so comes back here surface tension, surface tension upon g into rho f minus rho g, it has to be balanced, actually I cannot teach this, all this because this has to come from Rayleigh Taylor instability, that is I have to take, I have to take liquid vapor interface as a wave and then figure out when that wave will break because of instability, that is what, that instability criterion is rho f minus rho g into g upon sigma to the power of, so I, we can only tell the students like this, this is what I also do at UG level, if at all, if I used to teach in IIT, here I am not teaching, whenever I used to teach, all that I used to say is that liquid and vapor is going to have, I can imagine like a wave, so that wave, if it is getting disturbed too much, it becomes unstable, then only my bubbles can move up, so that instability criterion is rho f minus rho g into g upon sigma to the power of half, this is what is called as critical wave number, so and another book what professor all taught for two phase flow, very simple book, very small book, not scaring book is PB valley, Oxford University Press, it is very expensive book, very thin book but very beautiful book, two phase flow by PB valley, W H A L L E Y, very beautiful book, it is not available on Google book search, it is very expensive, very thin book, do not get scared, you add it to your library, you add it to your library because you can read all these things, whichever we have told, notes, it is like a small notes, very nicely pages, that is it, boiling curve, you read in that textbook, you will understand much better, so title is two phase flow, two phase flow and heat transfer, if I am, PB valley you put in Google, you will get it, fine, so see when you see the material only, you will remember the textbook where from you took, so now another thing we need to tell students is, whatever we are teaching, we should tell them the source where from we are teaching it, to that extent we need to be very honest because I may not be telling exactly the same way Professor Bejan is telling, if I tell that okay I am doing it from Bejan, he can always go back and read that if he is interested, if he is interested, so let us share where from the source, definitely I cannot tell out of the blue, it cannot come from sky, it has to be referred from somewhere, right, how are textbooks written, they are referring to journal papers, they are collated, understood, properly presented, that is what we are also trying to do, so please share your source, that is why I am telling wherever I come across any new textbook where from I have taken, I am telling you that okay, so now putting that HL by K as Nusselt number defined that way and Reynolds number defined that way, Prandtl of course is Mu F C P F by K, what Rosenhow did is that, Rosenhow from MIT did is that okay, Nusselt number equal to R e to the power of something, P r to the power of something, he did the Karoo fitting, how did that Karoo fitting come from? Based on the experiment with that, he gets N equal to 0.33 and 1 plus M equal to 1 for water and 1.74 other fluids and this constant CSF, CSF depends on the nucleation sites, how the nucleation sites are formed, see if you see that, let me handle that CSF little later, what am I doing? How did you get this N and M? I am plotting, if you see here, please see, C P T wall minus T sat upon H F G and that is on the x axis, what is there on the y axis? All of this term, all of this term is on the y axis. So taking the slope of this, I am going to get the, is that okay? So this he will do for various fluids and for different fluids he is going to get different M, that is how he is going to substitute for M, okay. So this is done for different operating pressures, so it is, so that is how Rosenhow or how I do not know how should I pronounce it, so he gave us this equation and he found that for different surface conditions, you see for water itself on copper, scoured means if you have just put with a hacksaw blade, MRE polished paper, you get a different CSF, why? Why am I getting different CSFs for the same fluid but different surfaces? Why? The way the nucleation sites are created are different for different, so that is why CSF comes into picture. Actually you can see in the literature if you are interested, you will see on CSF itself hundreds of papers, even now in micro channels work is going only on CSF, okay. So anyway, so that is about, I think that is where I would stop, that is where I would stop. I think this is all it is enough for undergraduate teaching but whatever professor Arun has taught is, he has tried to give us insights for two phase flow. If you do not, if he had not taught all of that, if I had just shown you this figure, you would not have understood this. Because he made us understand so well that air is required, if there is no air there cannot be boiling, boiling cannot get initiated at all period, that has to go into the minds of the students, that has to go into the minds of the students, that is very, very important. Otherwise how will boiling occur? Same problem in casting also, nucleation, there has to be a nucleation. If there is no nucleation, how will it get initiated subsequently? That is not possible. So that is why, in fact in two phase flow he goes in length and goes on deriving the conditions, what should be the T wall minus T sat, all that let us not get into, all that let us not get into. There is another relation for critical heat flux, people have attempted almost same way, it is directly giving q double dash equal to, this you can see, this we have seen, this is sigma rho f minus rho g into g, h f g rho g, people have attempted little, this is critical heat flux, sorry, okay, okay, okay. There are lots of things but I do not think, I do not think all this is required. See because this we teach in heat exchangers, in fact one can compute this, we had computed various conditions, I do not think all this is required for u g guys. So I think we can just stop, still further, flow boiling, this is you have already covered, that is it, that is it, I have truncated, it was there. Because you see we used to teach all this data was generated for our heat exchangers course, thanks to that course, we taught it again together incidentally. So that is where we had to teach all these fundamentals to them. But I think this is enough for us, this q minimum, that lead and frost, q minimum that is also given, but again everywhere in all these correlations, one common thing you are going to see, what is that? Density difference and surface tension, surface tension. If there is no surface tension, there is no boiling, surface tension is because between air, air has to be there. Whenever you coat surface tension, you have to have with reference to medium. So how do you explain surface tension? How do you explain surface tension? All of us coat this example, I take a blade, put a steel water, I carefully put the blade on the steel water, it stays, why I ask, then students will answer me, surface tension. It is like having a surface which is having a tension, which is tense. So that is what any liquid surface interface, it is between the liquid and the air. It is between the liquid and the air, there is a surface tension. If this surface tension is not there, then there is no boiling period. That is why everywhere in every relation, you invariably end up with surface tension. We have to impress upon surface tension. This is the only place where sigma comes into picture. In every other place, you have only mu, rho, Cp, k thermo physical properties. But this is where sigma comes in two phase flow, you have to bear with sigma. Micro channel, I have, okay, what is that? Non boiling type of fluid. Air and water, you have to have interfacial. I do not know, off the hook, I cannot answer. But my understanding is that adiabatic flow. You are talking about adiabatic, it is understandable. Now I am coming back to the question which we had left yesterday, natural convection. It is delta by delta T to the power of Prandtl to the power of N is valid even for natural convection. So I am showing you Bajan. Again, please do not get bored with Professor Bajan. So I like Professor Bajan very much. Even if you send a mail, even if you send a mail, you will get response within 5 minutes, okay. So prompt ease. See, coming back, what was the question we posed ourselves yesterday? We had a doubt that the velocity boundary layer cannot exceed thermal boundary layer. That was the perception. It is not like that. Delta by delta T is of the order of Prandtl to the power of N where N is a positive number is still valid for natural convection. That was the gut feeling I had because I was biased because I had taught this in convective heat transfer. So I am just flashing the book itself. Why? Because I could not generate the, let us just focus on this figure, okay. So what is this? I am having, this is a case in which Prandtl number, which case it would be? Prandtl number is greater. So that means delta is greater than delta T. So here what is happening is although delta T is there, velocity has picked up. It takes a while again for my velocity to get back to 0. See in my forced convection, what is the boundary layer definition? My velocity ultimately has to get back to u infinity. Here it is not like that. It has to come back to 0 because it has to talk to still layer Prandtl number less than 1. It is very easy to think of. That also I am going to show you. So at least I am happy that one of the questions, okay. So this is for Prandtl number less than 1. So delta T is larger than delta, the inner one. This is my delta and this is my delta. So here there is no confusion. As you can see temperature profile is there, but my velocity profile is, now it looks obvious. Yesterday of course we were all worn out. We could not fight it out, but now no confusion. Now I am going to discuss with you on a personal decision. So this is clear. No confusion on that. Point is, summary is delta by delta T is of the order of Prandtl to the power of n is valid both for forced and natural convection. I think we will sign up. We are coming back to two-phase law. Okay. You said pure substance? Pure substance. Liquid you mean with impurities. No, when you say pure substance. Correct. That is what we are talking. Okay, go ahead. Like what? We are handling two phases only. When I say two phase, what do I mean here? I mean liquid and another important thing. Because of impurities. Yes, people are using. He is working in an area. One of his PhD student is working with mixture of refrigerants where condenser he is designing. A operator design. So now how will you design that? So now again boiling temperature and condensation of the high boiling component in the liquid phase. The boiling point is 10 towards the boiling point or that is the point we are talking about. But the physics, nucleation, etc., we will give a unique horizontal line which we bought for 3B, TV diagram for that in general. That is one. You have a space. You have a space over which boiling is not a line in the space. I just want to add another point. This liquid metal convection. See that is again natural convection. Liquid metal. See because I was wondering yesterday evening whether liquid metal will it have any natural convection but later on only it occurred to me. If I am taking melting of lead which is what we are doing in our lab. If I take melting of lead in a container with one wall being heated. With one wall being heated. So natural convection dominates. How does this solid? What am I saying is let us say this container is filled with thoroughly lead. Now I start heating from only on one wall. So actually if it is pure conduction problem how will the interface move? It will move like this. That is actually called again as Steffen problem. Because Steffen was the first person as I told yesterday moving boundary problem he had solved. So it will move like this. But it cannot move like this. Because of why will it not move like this? It is not pure conduction problem. What is the other mode which takes over? Natural convection. Actually my interface will start like this. Will go like this. Will go like this. Will go like this. With the type. So it will. We have seen this through experiment. We have seen this through experiment. Okay. Point is natural convection can dominate. The melting time will be almost less than half or even less than that compared to pure conduction. Natural convection is heating. Why I am quoting this example? Because you can quote that liquid. Because usually we think that liquid metal is not useful. We rarely use that. It is not out of the world. Liquid metal melting is having lot of significance. Tin melting. Aluminum melting. All molten metals. All metals more or less are going to have tranquill numbers. Okay.