 So, good morning. My name is Janani Sree Murli Dharan and I am an assistant professor here at the mechanical department. So, as part of this heat transfer course, I am going to take four lectures on phase change heat transfer and the first two lectures is going to be on boiling and Thursday and the subsequent Monday is going to be on condensation heat transfer. So, when you are looking at phase change heat transfer, I would like to highlight certain things. So, you would be used to seeing a lot of correlations, empiricicity and pretty much nothing much beyond you know trying to see vague numbers and plugging them in and getting out your answers. So, the reason behind that is phase change is a highly coupled phenomenon and we really do not understand how to model them in a more physical way. So, we end up doing experiments and trying to you know characterize how the pressure or the temperature varies with the bubble size and we try to kind of use numbers that we have got out of specific experiments and kind of extrapolated to other conditions. So, as part of this lecture what I have tried to do is give you a little bit of the background of the physics maybe not go through the entire derivation because it can get quite complicated, but just to give you an appreciation of where the reasoning starts from. So, you probably will be looking at setting up the problem and going through some of the physics, but not all the way of working out the answers. So, when we talk about phase change in today's lecture we will be addressing these topics. So, I am just going to talk about some fundamentals of phase change little bit of boiling basics followed by what are the conditions required for nucleation to occur. Then after it is nucleated what are the bubble growth dynamics in an ideal scenario. Then in a case where we are really interested in which is bubble growth at the wall because we are looking at heat exchangers. So, bubble growth at the wall and probably two simple problems to finish with. So, all of you must be familiar with this phase change diagram and it essentially shows that your three phases solid, liquid and gas are governed by you know pretty much specified by a specific pressure and a specific temperature. And the boundaries that you see are essentially your equilibrium curves across which the transition of phase the phase change actually occurs. So, along these lines when you are at any of the pressure and temperature what we will see is that at that point your liquid and say the vapor phase is going to coexist. So, essentially at that phase change instant there is going to be an overlap of the phases right. And as I have told you we pretty much will be concentrating on the right hand most of the curve which is your vaporization and condensation because most of your heat transfer applications and industries are focused on boiling and condensation. So, before we go on to phase change details I just would like you to think about certain everyday applications where you see phase change. So, for example you are down with fever and four five days down the line your fever has to reduce. So, because your infection is gone and your doctor says it is a good thing that you sweat right. So, what happens why does your body trigger the sweat glands to have water on your skin surface. It is essentially because it is trying to lose heat it wants to bring down its fever. So, when it produces these water droplets on your skin what essentially happens is hoping that phase change happens and heat is removed away from your body and your fever comes down. An extension of the same problem is on a highly humid day when you are sweating profusely and the atmosphere just refuses to dry you up because it is saturated with moisture and it cannot take in any more liquid. So, phase change heat transfer does not occur and you are feeling miserable you cannot cool yourself down. So, that is another application of phase change. When you are trying to do some cooking in the kitchen and you are boiling water and you spill water on your finger you are going to end up burning yourself right. But when you open a pressure cooker and the steam comes in contact with your finger do you think the nature of the burns are going to be different or are they going to be the same they are going to be different. Why? Yes, exactly. So, essentially it is going to transfer more heat to your finger even though it is at 100 degree Celsius. So, it is going to lose that bit extra bit of latent heat to your body. So, the final scenario is you are trekking in the Himalayas and you are stuck in a stone storm right. You have your electric stove you can do something with it you have battery powered stuff but you do not have water with you you have run out of water. It is a snow storm I mean how bad can it get you probably could just stuff the snow down your throat and it should become water right. But strangely people do not advise this it is better to waste some amount of you know your stuff your gas and heat up the stuff. Why? So, this is a little bit of the numbers involved. So, if you are trying to just take water heat it up in a stove you are probably looking at just spending 61.9 kilo joules to heat it up to a temperature of 37 degree Celsius. But say you have snow at minus 5 again to heat it up to zero degrees is just 4.1 but look at the magnitude of the latent heat it is 133.6 kilo joules. So, if you are going to just put snow in your mouth your body would be expending that energy and essentially what will happen is you will try you will actually if you consume quite a bit you are going to freeze to death. So, that is not advised. So, what I want to actually bring to your attention is that it is not that phase change gives you a mode of heat transfer it is the magnitude of heat transfer. So, essentially this magnitude it is what is sought after by industrialists when they look for an efficient heat transfer procedure. What requires this energy for latent heat transfer? So, everybody knows that you have liquid bonds and vapor bonds and the nature between them are quite different. So, your liquid is held together more closely more tightly in a more regular fashion. So, essentially to break these intermolecular forces you need to provide that kind of energy to be able to transfer the liquid into a vapor phase. So, when this happens for example for water you are going to pump in energy and provide it just enough energy to be able to you know move break those bonds on the surface the surface tension and jump into the vapor phase. But you will have to ensure that this happens at an equilibrium case. So, essentially there must be a saturation pressure saturation temperature involved and your chemical potential has to be same right. So, when I talk about a saturation temperature required for phase change then what about evaporation? So, you have a puddle of liquid on this on the street and it evaporates by itself you are not heating it up to saturation temperature. So, the key distinction between boiling and evaporation is that boiling is a bulk phenomenon. So, I told you that molecules are on the surface of the puddle and when the sun is heating it up one or two discrete molecules manage to capture that energy required to jump across the phase and become vapor. But it is not a bulk phenomenon it is not an entire mass of you know coherent liquid getting transferred to vapor and that is what we want to achieve and to be able to do that we need to boil the liquid. So, when we talk about boiling there are certain factors which are essential. So, we know as of now that you need to retain saturation temperature pressure etc. But that is not all that boiling is. So, when you put a pan of liquid to boil on a stove what do you see happening? Where does boiling start? It starts obviously at the bottom surface which is getting heated right. So, what if you plunk you know a ceramic mug in a microwave and heat it up just to the same amount of temperature. More often than not you would not be able to find it to boil it will just be a quiescent liquid and you take it out and you are like it is not boiling and then you decide that okay let me open a coffee powder and you drop it in you would see that it instantly boils over at that point or even if you just stick a stirrer in it is going to boil over. So, essentially I am heating the same liquid in a different fashion. This was a localized heating process and what happens does natural convection currents that are being set up which is going to kind of disturb the fluid push it and disturb cause fluctuations whereas in a microwave it is a bulk heating that is going on. So, we understand that some kind of disturbance is required for boiling to happen for the phase change to happen right. What about different surfaces? Now, we have found in experiments that if you have different surface textures like hydrophilic or a hydrophobic surface different temperatures are required. More often than not it is a superheated temperature it has to be a temperature more than the saturation temperature. So, why is that the nature of surface play a role in boiling and how does it control boiling? Does it make it more efficient or inefficient? Right. So, I mentioned here that superheating is something that is required that is more than your saturation temperature. So, if you are looking at something that is more than saturation that sounds quite improbable. I just have a video here which is of a super cooled liquid. Now, this is occurring in all phase transfer procedures. So, you have something some water which is less than zero degrees. So, it exists as water and it is still not become ice. So, we do not know what will trigger this. What will trigger this phase change? It is super cooled. So, why hasn't it yet not changed into ice? So, these people try two methods. In the first case they just shake it violently. So, if you see that super cooled water and it's still water it's not become ice yet and you can see that that shaking has helped transform. Now, in the second case all that shaking is not required. We will see what they do. So, what they do is they just drop in a single crystal of ice and that was sufficient for the phase change. So, you need some kind of super heating. You need some kind of fluctuation and you need some kind of external presence to ensure boiling. So, what is this nucleation process? One has to understand. So, essentially nucleation talks about a self assembly of similar molecules in a particular phase. It could be homogeneous or heterogeneous. So, what do these terms mean? When I talk about homogeneous nucleation what I am trying to say is that this phase change does not need any external surface for assistance. It doesn't need a stirrer. It doesn't need an ice crystal. It will just change phase. But in experiments it has been found that extremely high amount of power is required to affect this process. So, essentially all normal earth events what you see is heterogeneous nucleation. It is nucleation at a particular surface aided by something. Either it is just your pan which is providing that surface for boiling or it is a nice crystal right. So, what is the energy that we are talking about? So, this is some kind of energy that is required for formation. So, essentially the formation energy for nucleation comprises of three things. One is your chemical potential energy. Second is your surface energy. So, some kind of favorable surface energy balance has to be reached. And the third one is your pressure volume work. So, when I talk about chemical potential what I mean is that you might have different species and they might be reacting in between. So, in essence some kind of energy must be provided for say diffusion of events for proper redistribution of you know the substance if there is different species. So, the chemical potential has to be the required requisite chemical potential has to be provided. Surface energy has to be provided and pressure volume work. So, in that context when you talk about hydrophilic or hydrophobic surfaces when you have different surfaces essentially it is a surface energy that is changing. So, one surface might be more favorable for nucleation and hence requires lesser super heat whereas an other surface might be requiring that you pump in way more heat for nucleation to occur. So, when you talk about nucleation to occur in terms of energy I mentioned something of super heating and we saw that super heating alone or super killing alone is not sufficient. So, how does a system exist beyond the saturation temperature in a super heated or super cold state. So, that is called in literature as a metastable state. So, what is this metastable state essentially what the liquid sees for example is that it is on that equilibrium curve that you see and it has become 100 degree Celsius but it is happy in itself it is an equilibrium with itself. It does not know that there is a vapor phase which it has to change into. So, essentially it is a locally minima which you see which is like an equilibrium process and it has to be nudged from that by giving some amount of energy may be violently shaking it or adding a nice crystal saying listen there is some other surface here and then it wakes up to that fact that it has to actually change its face. So, this metastable state can be disturbed by giving fluctuations and those fluctuations are known as hetero phase fluctuations. So, those random shakings that you see are called hetero phase fluctuations. Look more closely at what this metastable state is. So, let us draw pressure curve saturation curve or pressure volume diagram and we know that the vapor is on the right hand side. So, you have liquid here at the saturation pressure and then it transitions into vapor at this point and this happens at say the constant temperature. So, this is your constant temperature line. So, this is your isotherm. In literature we are trying to find out what is its existence beyond this saturation curve. So, how does the medium exist inside? So, there is some equations which you saw to get the distribution to be something of this. So, what happens here is that this region AB this path that it travels is unphysical. So, the only physical parts of this curve is your say C and D. So, you are talking about AC which is physical and BD which is physical. So, I am not going to the derivations of how these equations, but it is just to make you understand that. So, you can see over here that. So, I have drawn AC, B and D. So, in this BD is considered unphysical. So, from to this point here is considered unphysical and somewhere between A and B you can prove that there is mechanical stability. So, mechanical stability in this portion. So, let us forget about this. So, in this portion there is mechanical stability meaning there is a system can exist stably, but this portion here is not physically possible. So, any kind of fluctuation that will disturb the system will push it to the size side quite quickly and the next nature of the liquid is to jump to the next vapor phase. So, essentially your metastable state in your pressure volume curve is the state over here. So, that is your metastable state and when you disturb the liquid your fluctuations essentially push it into a region where it cannot exist and it jumps to the vapor phase. When you talk about formation of vapor what is the kind of yes. No, this is not in really effect of a correlation. So, this is properly you derive from the Gibbs free energy and then you show this path and of this path. So, you do a theoretical analysis and then you say you are able to prove that that mechanical stability here does not exist, but this is mechanically stable and a metastable state exists. So, that is why you are super heated or super cooled liquid can you know exist physical meaning mechanical stability is not there. So, for example, the condition for mechanical stability here I have shown is change of pressure with respect to specific volume should be less than 0. I am not going into the details of the derivation. So, if you see here this violates that it is greater than 0. So, it is a violation of the requirement of stability. So, you will not have a system it will just bypass that that condition it will not go through it. So, though you are theoretically deriving it through equations it is a non physical state right. No, here you will not it won't go through the state at all. So, when you are talking about metastable state then we want to know about super heating. So, I am I have talked about having some kind of external system having a surface energy being balanced and all of this sort. But, I if what if you can give all this energy when the system is super heat when the system is just saturated why do I need to super heat the system I do not have to super heat the system. Yes, one thing is it is easier to disturb it to push it. So, lesser energy might be required, but that all that all cannot be the driving factor. So, the important fact is we are talking about nucleation a bulk transformation. So, your super heat is required for this case. So, let me do just a small derivation here to show you how that is the case. So, for phase change to happen there are three things that you would be repeatedly using. So, you have a system and you are trying to nucleate a bubble say pressure p v and the system is super heated at t l and at pressure p l. So, you would repeatedly be dealing with three equations which is p v minus p l is 2 sigma by r r being the radius here. So, this is essentially your mechanical balance to maintain that pressure difference. The second thing is your kind of thermal equilibrium formula which you will be using which is your Clausius-Clapeyron equation which talks about the change in pressure with respect to temperature and this is specific volume change. So, this is your second equation and third equation is talks about the chemical equilibrium with the Gibbs-Duhem equation which talks about the change in the chemical potential with respect to the system properties. So, essentially it is three equations that you will be briefly revisiting at different points. Okay. So, as far as this case is concerned when you are trying to do trying to understand why super heat is required we will go and access this particular equation here. So, we do know that for such a system to exist yes you need your mechanical equilibrium to sigma by r but when you come to your chemical equilibrium now let us look at the liquid phase. The liquid phase exists at a superheated temperature T L and at a pressure P L. The vapor exists at P V pressure and the temperature is at the constant temperature case. When we try and apply this over here it is a constant temperature case. So, this goes to 0. So, essentially you have D mu say from any saturation case to the liquid T L situation that is your this state over here and V D P integrated along that same way. So, you will be actually doing this integration first for the liquid which will give you the equation mu L is equal to mu sat at L plus V P just get confused with the differences. So, this is when you integrate this for the liquid case and for your vapor case you will have a similar equation but you are assuming ideal gas. So, you will be looking at RT L ln P V minus P sat. So, you have equation 1 equation 2 and 3. Now when in a real case you plot this on a graph what you get is something like this. So, essentially when you come to this graph over here. So, let me write equation 1 also over here. So, when you look at that graph over there that is the graph you will get on plotting the mu equation. So, it goes something like So, what does this graph represent? Essentially AC corresponds to the change in potential for liquid. So, AC is the liquid line whereas D B D B E is the vapor line of these lines the stable portions are A and B A to B for liquid and B to E for vapor. The portions that you see that are left over are the metastable state portions and this common point B is nothing but your saturation point P sat. So, when you go through all those three equations you end up with the relation that you see over there and the form of that relation is such that it essentially indicates let us not go and go into two details essentially indicates that your vapor pressure that is your vapor pressure is close to your saturation pressure. So, essentially this is your saturation point and your vapor pressure tends to be on the vapor line on the stable vapor line at this particular location over here. Let us look at our system again there are two conditions that has to be met I said for phase change to occur you need your chemical potential to be the same and for nucleating a bulk phase you need PV minus PL equal to 2 sigma by R. Now you know the vapor phase PV over here and that has a mu L where does it touch the mu L part so it has a mu V but that has to be equal to mu L. So, it contacts the liquid line only in the metastable state secondly so you can say can I extend it and is there going to be a second intersection or something of that sort but another second point is going to be that PV minus PL must be equal to 2 sigma by R. So, this distance is fixed so all this is only to show you that you need super heat to be able to nucleate a bubble of radius R that is why you need the super heat. So, I did not go into too much detail of the math but I think this kind of puts in perspective why you need that super heat. So, now you have talked about nucleating a bubble so once you have nucleated a bubble with a super heat T super heat what you need to do is look at how it grows because that is an important part of heat transfer. So, when you are looking at bubble growth dynamics then there is something that we have to understand the bubble growth pattern can be of two types one is your inertia control growth and the other is your heat transfer control growth. So, we have a liquid which is super heated and you have a bubble that is just nucleated with that super heat. So, at this stage what happens is that your liquid right next to the interface is still going to be super heated. So, there is still lot of heat that is involved right at the interface and if it wants to grow all it needs to do is access that super heat right next to the interface. So, what is controlling the growth? Essentially it is ability to push the liquid around it. So, it is a momentum controlled or inertia controlled growth initially. So, that is the beginning process. Now as it consumes heat at this location the temperature here is going to drop and what happens is that a thermal boundary layer slowly develops and the temperature across this thermal boundary layer is going to vary from T sat to the super heated liquid temperature. So, once this medium is reached what happens is that your growth is no longer controlled by the infinite axis of heat or momentum you are controlled only by the amount of heat transfer that can be achieved across this thermal boundary layer. So, that phase is called the heat controlled growth. So, there are two extreme factors one is your inertia controlled growth and the other is your heat controlled growth and somewhere in between you have the transition period which is modeled differently. To look at how this bubble growth in an inertia controlled case happens. So, you have a bulk vapor that is formed P L T L and there is P V. So, if you look at how this interface is growing so, radius r and there is liquid surrounding it and this interface at this location is moving with a capital velocity u. So, mass conservation meaning the interface is pushing the liquid and the entire liquid is moving forward. So, what that requires is 4 pi r square capital u this has the unit of meter cube per second right and this pushes the liquid and say the liquid has a velocity small u at a distance r can be equated to 4 pi small r square right. So, you can get the relationship. So, you can essentially get a formulation for your velocity at. So, this becomes capital u capital r square by small r square this is nothing but at this location r capital r by T. So, this is one expression here that we have. Now what happens when this liquid pushes when this vapor interface pushes the liquid you are imparting kinetic energy right. So, the kinetic energy equation can be written as integral from r that is your interface to infinity half rho L u square dv right. So, you substitute for u square into that and your v is nothing but 4 by 3 pi r cube because it is going to be traversing through that. So, when you integrate this what you get is. So, you will get an expression something like this not going through the integration it is pretty simple d r by dt the whole square r cube. So, essentially your interface as imparted this kinetic energy into that liquid ok. So, for this interface to impart it what does it have to do it has to push against the liquid. So, what is the work done by that interface this work done is a work done for it to form, but all of it does not go into adding kinetic energy right. There is some resistance from the outside fluid. So, whatever kinetic energy is left over in the system is after it has overcome that surrounding resistance. So, minus that resistance work. So, this network is what comes as the kinetic energy of the fluid. So, this work done can be written as integral 0 to r the pressure right near the interface is PLI because that is the kind of work that it has to do PLI 4 pi r square or capital R square because we are talking about the bubble d r minus the entire resistance is nothing but 4 by 3 pi r cube into p infinity p in or p l here in this case. So, essentially if you equate this to this meaning whatever work your bubble has done has gone in as a kinetic energy and this when you solve you will get a complex well not a complex well it is quite involved. So, you will land up with and then you differentiate it so on and so forth you will land up with that equation. So, the beginning parts of it is this it is essentially that you are trying to do a work and energy balance and this is known as the Rayleigh's equation of bubble growth and this is inertia control growth you have looked only about the kinetic energy and the work done. And you make a simplification over here because it is inertia control growth meaning it is something that happens right at the beginning what happens is that your bubble is very small meaning your pressure difference is pretty high. So, we make the approximation that the p v minus p infinity p infinity here I mean is nothing but your p l over here 2 sigma by r. If you do the 2 sigma by r and then solve the differential equation eventually you will arrive at this here what is this formula it is the rate of change of your bubble with time how does your radius of the bubble grow with time and this is the inertia controlled growth formulation. So, essentially it is this formula that you need to know and if you notice the relation it is a linear relation. So, with time it linearly increases as in for heat transfer control growth you will see that it varies as a square root of time meaning the growth rate is slower at a heat transfer control growth because it is waiting for that heat to be provided across the thermal boundary layer. Now, for the heat transfer control growth gets a little bit more complicated. So, what happens is that as I told you there is a thermal boundary layer that is formed and it varies between the saturation temperature to the superheated temperature. So, you will have to solve the energy the temperature equation across this thermal boundary layer which is there. So, you can see that we have taken in care of some amount of connection and some amount of conduction the mass balance or the mass conservation is still there you have the expression over there and let us look at the inertial and boundary conditions here. So, the initial condition is that everywhere it is going to be superheated temperature it is shown as T infinity which is essentially T L over here and after some time at your interface it is going to become T saturated whereas in the far off liquid it continues to be superheated. An important condition is that at this interface your heat transfer is controlled by the conduction across that thermal boundary layer. So, this is the key thing that you need to know that the conduction heat transfer across the thermal boundary layer will govern the phase change and this again gives up comes up into a highly complicated thing and people do not actually solve it entirely and they have opted for some kind of approximations simplifications and they come up with that kind of an expression over there which is purely based on a lot of approximations and some experiments. So, you have as you can see the radius as a square root of time an important feature I would like to point out here is that your JA is known as the Yaqab number and it is a non-dimensionalized number that you will find in a lot of places as far as boiling is concerned. So, what is this ratio? It is the ratio of sensible heat that is rho Cp delta T. So, sensible heat required to keep a medium at its superheated level to the amount of heat at that point taken for phase change. So, it kind of gives you effectiveness of the boiling. So, now when you are talking about two limiting factors inertia controlled and heat transfer controlled. So, Yaqab number essentially is the ratio of the sensible heat that is required to keep a system in superheat. So, when you have phase change it is going to slightly drop in temperature right. So, there needs to be some amount of superheat that still needs to be retained. That to how much heat transfer how much latent heat is required for a phase change at that saturation temperature and pressure. So, when you are talking about a transition now. So, you have looked about inertia and heat transfer. So, in this graph you can see that inertia belongs to one limiting end and diffusion or heat transfer controlled is to the other limiting end. What we need to do is try and build a model which does not stick at two different ends, but which merges all this and transitions. So, that is what Mickyq did and he came up with this model over here wherein it can model both inertia, intermediate that is your transition and your heat transfer control growth. So, he has one thing I like to point out here is that he has a dimensionless time over here and if you note carefully the dimensionless time of approximately 1 falls in the intermediate case 0.01 below onwards it is inertia controlled and above hundreds it is diffusion controlled. So, in case you are asked to find out what the radius is like in an intermediate control growth phase then this is the kind of non-dimensionalised timelines you are looking at. It is just to be good to be aware of the timelines for bubble growth right. So, we have talked about highly idealised conditions of growth which is you know in an infinite pool of liquid growing at you know inertia control growth, heat transfer control growth, but what we are really interested is this bubble growth at the wall because we need the heat transfer to be you know useful in industrial situations. So, heat growth or bubble growth at the wall is a cyclic process. So, we know that bubble growing on a surface the surface has a lot of roughnesses and the way boiling people look at it they look at it as crevices on the wall or they call it as cavities on the wall. So, essentially your surface is going to be something like this but they view it as cavities on the wall and the theory is that you have trapped air inside these cavities inside your pan and when you heat up what happens is that your liquid slowly gets heated up that is your natural connection on the right hand side. That kind of provides the super heat and in combination with this external surface and this trapped air similar to that ice crystal it is able to start nucleating. So, vapor phase happens over here and the bubble slowly grows and afterwards as it grows there is going to be a force balance that is required that is your buoyancy force versus your surface tension force which is holding the liquid to the wall once the buoyancy force is significant enough it is going to depart. So, when it departs there is going to be cool liquid quenching the wall and then the process continues. So, this is a cyclic process and this is called a bubble evolution cycle which essentially talks about how the cavity gets replenished with air how it continues to grow. So, when you talk about this what we need to know really is there is a cyclic process but what kind of bubble sizes can I form at a particular super heat essentially you would like to know that. So, let's look at this very simple question over here for water at one atmosphere estimate the critical bubble radius for liquid super heat. So, for super heat levels of 2, 10 and 40 degrees. So, you have liquids at those three super heat what is the kind of bubble sizes you are going to form. So, again bringing your attention back to those three equations I gave you which is the Young's Laplace equation pv minus pl equal to 2 sigma by r right this is your mechanical stability and you have your Clausius-Clapeyron equation. So, to have a super heat of dt to have a pv minus pl. So, this is t sat t infinity because so all you need to do is take this equation club it with this equation and you've got a radius and that is that equation. So, that's the formula to get the radius of a bubble to form for different super heats. I have done the calculation for the three super heats and I'd like to point out an interesting thing you can see that with increasing super heat smaller bubble sizes can be formed smaller nucleation is possible. What do we get out of it from a normal surface? In boiling what people think of is that when you have a cavity they assume and it's a very big assumption that they make they assume that your initial nucleation size the radius is same as the cavity size. So, this is a strong assumption that they make and I told you that the surface is filled with different cavity sizes. So, it is logical to think that different cavities are going to get activated and you have bubbles of different sizes forming. So, if you would look at say this is the surface and there is a t super heat that is maintained and the far off liquid is at a t saturation temperature and this is the distance y. So, initially think about the bubble evolution tire cycle which I talked about initially when you're just having the liquid quench and starting to heat up again your temperature profile is going to be like this. Slowly the liquid is going to get heated up right? So, it's a transient process where your temperature lines are going to follow this pattern. Now, going back to this assumption that a bubble nucleates at the size of radius r it follows that I need superheated temperature till this point till this distance r. So, the important thing that I would like to tell you here is that. So, initially you might have you know cavities of a smaller size which have temperature up to this but as the temperature extends and becomes higher bigger bubbles can form. So, essentially there is a range of cavities that can become active on the surface and it is given by that formula there and I think we are up with time and I like to finish at this point and we will pick it up from here in the next class. Thank you.