 Earlier we used to have a full fridge to convection course long back like this 40 hours, then we compressed it into advanced heat and mass transfer course. Some of you have taken that course and now this has been made into an elective, so that more of convective topics can be covered basically. In that what Dr. Arvind has done so far is given you the for almost 30 hours laminar heat transfer, convective heat transfer study is incomplete without turbulent without the study of turbulent heat transfer, but turbulent heat transfer cannot be done without the knowledge of turbulent boundary layers which cannot be done without the knowledge of turbulence per say. So, this is a very unique and complex kind of a convective heat transfer compared to the laminar convective heat transfer which you have done so far. In your course on incompressible compressible flow you have done incompressible, so boundary layer theory must have been done there, turbulence was done turbulent boundary layers was it one classes we will try to do something about it although the time is a little limited, but we will try to do this. But before I go into turbulent heat transfer and free convection Dr. Arvind I can do the quiet flow you said quiet flow can be done. I would like to tackle one other particular problem in convective heat transfer which is called the quiet flow. It is a very simple geometry it is not a boundary layer flow, but it gives us tremendous amount of experience actually exercise experience in simplifying the total overall complete Navier-Stokes equations to a very very simple set of equations. And then you are able to solve it by hand in the class without necessity of any other complex solution methodologies like series solutions or transformations numerical methods especially you can get with certain assumptions which are very reasonable you can get solutions for heat transfer in a very direct complete fashion without resort to computers or any other complex mathematical techniques. Number one number two what is so great about it why should we do that we have we have very nice codes available today you know you put in all these things into your code whether a finite difference technique finite volume or finite elements you will get the answer that is true. But the beauty of this particular problem as I see it and it is really exciting that very important concepts in convective heat transfer at least couple of them three of them come out of this very simple solution. So basically there are three reasons why this particular flow situation that I am going to do is important one it is considered in literature as one exact solution of the complete Navier-Stokes equations if you please make a difference between the complete set of Navier-Stokes equations and the Prandtl's boundary layer equations what is the difference between these two by the way I will be asking some questions please respond I do not write too much on the board I want you to write down what is the difference between the Navier-Stokes equations and the Prandtl's as the name suggests boundary layer equations what is the boundary layer yeah what is the boundary layer what is defined a boundary layer do not say all that you be a little bit more firm yeah you are right so where the viscosity effects therefore what particular force does viscosity affect no you have to answer me what force does this property of viscosity of fluid affect control share no you should not take so much time to answer this question you know so viscosity property of the fluid there is a force share force these together along rather along with the velocity actually give you the shear stress fine so there is a layer as you said very close to the surface where when you say viscosity is important it is not viscosity that is important actually it has an effect it is a shear force shear stress that is important and these shear forces are of a magnitude much higher than what it could be outside the boundary therefore they are important the property of the fluid which comes into picture is the viscosity for you to have a shear shear force shear stress it is not simply viscosity that is enough there should be a velocity there should be a relative velocity if you look at what is the fundamental expression for shear for viscosity defining viscosity or the shear stress equation give me the complete thing say it loudly if we make mistakes it is ok no problem so you must have the viscosity there should be a gradient it is not simply the velocity the gradient so tau equal to mu du by dy du by del y is the what is this law called Newton's law of viscosity so it is actually a definition for viscosity in one way so what does the Prandtl's boundary layer equation how does it now differ from the Navier-Stokes equations if you want I will go back a little bit before Navier-Stokes equations what kind of equations were present were used for fluid flow what is that Euler's equation so Euler's equations now can you tell me what is the difference between Euler's equations and the Navier-Stokes equation that they look very very similar almost identical except for something what is the difference between Euler's or Euler's equations and the Navier-Stokes equations viscous term is coming into picture in the name that is a very big achievement find out when the Navier why is it called Navier-Stokes equation please find out if you have not when was it actually proposed please find out I want all of you I guess I am interested all of you to be interested in the history of these things history of science per se history tells you a lot today when I am talking about this Dr. Irwin this talking about it is as if we know about it forever it has been as a given thing for us to come to this stage lot of work lot of thinking lot of research has been expended in the last 200 years and what we are giving today is as if it has been there forever you should really look at how the history of fluid mechanics has developed history of heat transfer has developed who have contributed to all this and what were they doing today the moment I talk about heat transfer you will get an expression what were they doing when they did not have the equation who developed this equation for example you know I want you to please think look at history today everything is available at your fingertips so Navier-Stokes equation fundamentally was a great jump in knowledge as well as utility compared to the Euler equation this was in the 19th century 19th century was the time when tremendous advances were made in both in fluid mechanics to an extent in heat transfer but the real advances in heat transfer have been in the 20th century real 1900 to 2000 has been a great period for advance advances in heat transfer per se but as per at least we are concerned convective heat transfer the real foundation for that was in the 19th century starting from the Euler's equations in the Navier-Stokes equations what was what is the fundamental heat transfer equation which was developed in the 19th century which you use it all the time in convection also use it four years of heat conduction find out go to the original paper if you have a chance go to the paper Fourier James Fourier the French mathematician he gave this heat conduction law Navier's Euler's equations were there without discuss terms in Navier-Stokes Navier and so these two are different people by the way find out why there are two names there Navier is different Stokes is different you might have heard of the Stokes hypothesis you have not heard of the Navier's hypothesis so find out why this was called so so Navier Stokes equations finally in the late 19th century really gave a push to the science of fluid mechanics from one particular point of view that is consideration of the shear stresses and only then really the aeronautics industry developed aeronautics industry loved once this point was understood that even for air which has a very low viscosity shear stresses could be important under certain conditions tau equal to mu du by dy so you can neglect the shear stress in a flow either when du by dy is 0 or vanishing or nu is vanishing air has a very low viscosity but even though air has low viscosity and many engineering several engineering fluids including water low viscosity in their flow shear stresses become important when velocity gradients become important that's the point you should look at otherwise viscosity was known they did not know how to bring this into mathematical form what is the D'ambert's principle or D'ambert's paradox if everything is considered as ideal you know in machines ideal fluids so a solid body can move through it forever without being retarded but experimentally everybody would see everybody could see it was retarded and that was what was given tremendous retardation tremendous slowing down of any vehicle which moves in the air including the automobiles for example so there was tremendous research that was going on at that time and the credit for bringing in shear stress into our engineering consideration and mathematical possibilities is given to Navier-Stokes and the Fourier's law of heat conduction from the heat transfer point of view but in 1904 something extremely important also happened 1904 so Navier-Stokes equations were available 1904 something else happened what is that Prandtl let with Prandtl please find out about his life you have a book by him in the library green bound book please go on study that let with Prandtl proposed after studying Navier-Stokes equation after lot of experimentation he was the first one to propose the shear stresses are very important it's all very fine but these are important only adjacent in a very thin layer adjacent clear solid surface on which the fluid is flowing this was a it would now appear to be as a as a what is it as given but for him to come out with that physical idea concept and then take care of the complex normally a Navier-Stokes equation that was a huge achievement of those times now you are coming to that so from Euler you graduated to Navier-Stokes from Navier- Stokes I would say you graduated upwards to Prandtl's boundary layer equations because he simplified them tremendously he said all these complete Navier-Stokes equations need not be considered in their completeness in the layer very close to the surface it is enough if I take certain terms certain assumptions I can make and then a mathematical equation I can derive a mathematical model the solution of which will give me the shear stress what is the major major mathematical impediment I was in difficulty with Navier-Stokes equations the complete Navier-Stokes equations look at the left hand side u del u by del x v del u by del x nonlinearity and nonlinearity and this is the problem with Navier-Stokes equation mathematical point of view for engineers while pure science is important pure mathematics is important but for engineers certain assumptions are okay certain overall picture is okay as long as we get engineering wise applicable solutions without bothering too much about how they originated a scientist need not know engineering but an engineer should know the science behind boundary layers science will fluid mechanics then you will be able to appreciate it better but more than that you will be able to correct yours and later also so let me Prandtl simplified the Navier-Stokes equations into what now we call as Prandtl's bound they are referred to as Prandtl's boundary layer equations meaning they are fully applicable in that boundary layer which is a very thin layer next to the surface this he made it only for the shear stress not for the heat transfer actually although we then expanded up extended upon it is one part I'll come back to this what now we'll come to the heat transfer side what is the fundamental objective of convective heat transfer h why do we want it why give me a complete answer yeah that's the result number is then you know simply a non-dimensional what is that equation give me give me the equation I know you know all of it not that you don't know I'm trying to I'm the my my first my first class I'm the developing region I want to develop by tomorrow day after into my fully developed flow conditions q is equal to and what is it called is it really a law that's the point now I wanted to ask can you okay now just go back and what are the modes fundamental modes are mechanisms of heat transfer so there is these are the leftists these are the rightists here they don't want they say only conduction radiation they say conduction conversion radiation yeah actually convection if you see is a modified form of conduction basically the ultimate mode of heat transfer right at the interface at that first micro layer is conduction how do you know this mathematically what do you do how do you know this no sleep okay no I'm looking at that so how do you connect conduction to convection with this concept let me put it the other way how do you conduct connect conduction to your heat transfer coefficient conduction at the wall at the interface to the heat transfer coefficient what is it so h delta t is equal to minus k dt by dy equal to 0 now on both sides what is the difference in the in the coefficients in terms of the future the characteristics of the coefficients h delta t you have k dt by dy no unit twice okay but anything more than that okay one on the right hand side dt by dy is a gradient and on the left side it's not a gradient okay number that's very clearly seen but I want you to tell me the difference between the other two factors the other factor on the right hand side on the left hand side what is property now you come k is a property of the fluid h is not h is a function of what then you give me the variables that's the flow condition yeah geometry and the properties therefore it is not a property it is a characteristic now therefore now I'm coming to this point q is equal to h delta t or h a delta t we should not be strictly calling it a law now come to conduction write down the expression for conduction for your slug you know that if you want to q is equal to minus k dt by dy write down for radiation q is equal to sigma epsilon t14 minus 324 write down for convection h delta t now look at these three expressions for conduction you have k is a thermal conductivity which is a property on the right hand side for the radiation you have epsilon property coming into picture and in both of these cases of course there is no flow in the convection case you have q is equal to h delta t where there is no property alone properties incorporated in h while you call correctly four years law of heat conduction six siphon words one's law of radiation it is not strictly right to call these newton's expression as newton's law of cooling because it depends upon geometry flow and the property you might say what is so great about it I want you to understand the difference you don't have to accept everything as it is said you know you have to understand that convection is not a in a way not a fundamental mode of heat transfer it is a modified form of conduction it doesn't matter in your calculations but you should understand as scientists engineers similarly newton's law of cooling so then if it is not a law what is it it is simply a relationship it is simply a definition of h no like tau is equal to mu du by dy newton's law of cooling it is a law of cooling it is a law because mu comes into picture it is a property q is equal to h a delta t or h delta t flux h is a characteristic it is a definition of h this was a very fundamental thing find out how this came out of you called it newton's law anyway already so although heat transfer developed tremendously in the 18th and 19th 19th and 20th centuries newton himself has contributed certain things especially newton's second law of motion you are going to use newton's law of viscosity you are going to use and the so-called newton's law of cooling so called i am saying so some of these concepts came from the time of newton that is at the in the 18th century 17th 18th century there there is not much of work then four years law of heat conduction came into picture Euler's equations came into picture graduated to navier stokes equation then came this seminar year of 1904 when prancer said i do not have to solve the entire navier stokes equations because these the engineering at that time the engineering objective of the study was sheer stress as far as the aeronautics is concerned that was the the drag coefficient that they wanted that was the purpose drag coefficient cannot come from your Euler's equation is just does not exist there the complete navier stokes equation has discussed some so it has to be solved then prantle said no you do not have to solve this let me try to simplify order of magnitude analysis scale analysis finally he gave those even though they are non-linear still they are much less formidable than the navier stokes equations u del u by the x plus v del u by del y equal to new del square by del y square minus 1 by rho dp by dx so many terms mathematical terms he cancelled out with certain argument assumptions some very right some approximately right but his intent was can i handle the navier stokes equation in much simpler simpler way for me to get my skin friction coefficient that was only from the fluid mechanics part of you then came of course much later the energy equation the same concept of boundary layer for viscosity and shear stress was extended to the energy equation to then term a thermal boundary layer they simply surmise as viscosity is important in a very thin layer maybe k is important only in a very thin layer although these two may not be equal but they just like a hydrodynamic hydrodynamic is not a nice but hydro is water actually so it is a aerodynamic boundary layer fluid better would be a fluid mechanical boundary fluid dynamic boundary layer like that there is a thermal boundary layer these two concepts which came only in the 20th century 1904 prantles he he presented it in an international conference nobody bothered about nobody bothered till 1934 in one of the international conferences there were six papers on boundary layers 1945 there were 280 papers and by 2000 there is no year in which there is less than 1000 papers on boundary theory that is the import the significance the importance i would even say greatness of this particular concept which we today talk as if i i i came up with this concept i tell you this exists i have never seen it we have done experiments that is a different matter so 1904 it is already 110 years old along with him at the his contemporary was nusselt nusselt now you know he he is remembered with his he died only in 1950s a nusselt number i am now going one step ahead that he transfer coefficient which is a whose definition is given by the newton's law of cooling was given a non-dimensional form by nusselt in 1925 he said he came through non-dimensional he didn't know about the biking ampy theorem at that time he simply made a scale analysis he found this group hx by k was non-dimensional and then he said i i should be able to relate it by that time Reynolds number was already when did the Reynolds number when was it proposed Reynolds number i know the Osborne Reynolds 1883 at another major experiment by Reynolds experiment please look into Reynolds experiment where he injected a colored die into a pipe flow under extremely low velocity conditions and he could see a very nice as now we call a laminar filament of a layer and then when he increased the flow he started saying fluctuations he said what is happening the velocity at every point is changing with time and space that's how the idea of turbulence was actually visualized and then that particular parameter or a period of time it was given the name Reynolds number what does Reynolds number signify what does it physically signify inertia by viscous forces so as you increase inertia is your u del u by del x actually you please make it out Reynolds number is inertia by viscous so rho u du by dx divided by mu d square du by dx square if you do that you will just get rho vd by mu that is vd by nu v being a characteristic velocity Frankel did another thing I mean he he he didn't do it per se but when he non-dimensionalized I mean he made a scale analysis he got this term nu by alpha actually mu Cp by k that is called the Frankel number today now you see over a period of time from the fluid mechanics point of view 1883 Reynolds turbulent laminar turbulent transition came into picture and a number Reynolds number came into existence then Frankel gave the Baudelaire theory in 1904 Frankel talked about Musselt were around 1925 Frankel by the way Frankel did another thing in 1925 we have not had gone to turbulent Baudelaire but I'll tell you now he developed the first what we call the zero order model for turbulent flow called the Prandtl's mixing length theory please note down we will talk about it later so Musselt then gave us the non-dimensional form of x there is nothing more to it we can talk about convection by conduction we can then improve give physical significance they have physical significance so over a period of time you finally got convective heat transfer to be represented by Musselt as a function of Reynolds and Prandtl Musselt is a non-dimensional form of the heat transfer coefficient which came from Newton's so called of Coulomb Reynolds number from Reynolds because of the laminar that that number differentiates between laminar and turbulent flows and then Prandtl number what does Prandtl number signify when I give you a Prandtl number what would you what information will you get fluid properties yeah take off yeah it identifies the fluid that is the thing yeah it's fluid properties therefore it identifies the fluid there is nothing you can't miss anything if you say Prandtl number is equal to 0.72 it is air and most of the gases somebody need not tell you find out whether it is a oil or a liquid metal or water no it is gas end of the matter on the other hand if I said this Prandtl number of this fluid is 0.01 not 0.01 you know it is a liquid metal mu C p by k k is extremely high on the other hand if I say the Prandtl number is some thousand to fifty thousand mu C p by k where k and mu are varying with temperature and that has to be oils so you have oils light oils heavy oils so the moment that is the beauty of this particular number Reynolds number gives you the type of flow non-dimension missile number gives you the non-dimensional heat transfer coefficient which is hd hd by k no you can write it as h over k by d so it is a convective heat transfer coefficient divided by the conduction part it is a relation between these two and then Prandtl number signifies the fluid this is extremely important so if you look at this general expression Nusselt as a function of Reynolds Prandtl you have everything coming into picture here left hand side is what we want that is the objective of convective heat transfer study getting you said Nusselt number in the beginning getting the heat transfer coefficient in the form of a non-dimensional number but it this is a function of the flow the geometry and the properties flow on the geometry property come through the Reynolds number it will tell you a lambda or a turbulent and the Prandtl number says this is the fluid you cannot therefore use this is what I tell a lot of students so do not think the equations are available in the some tables so why should I attend my class I can simply take any equation you will get all equations in force convection and Nusselt as a function of Reynolds and Prandtl but you should know what is the background of this what is the significance of all this how do how does H which is our major goal in convection how does it vary with flow that is the velocity the geometry and the properties of the fluid so the engineering objectives are two in convection heat transfer actually it is not one we should say it is two because you have to get the skin friction also that is a purely fluid mechanical thing but when you are talking about convective heat transfer skin friction coefficient determination is important and heat transfer coefficient determination is important when do you think viscosity or do you think viscosity will affect the heat transfer or not viscosity is a purely fluid property heat transfer coefficient is a function of the property these two are the objectives do they inter relate do they affect each other that affects the whole process do they affect each other how does heat transfer affect fluid flow now if mu is a function of temperature not so much of pressure rho is a function of pressure and temperature so heat transfer is very important therefore we talk about low temperature heat transfer high temperature heat transfer and the effect of properties which are a function of president properties are a function of pressure and temperature so we should look at the president temperature how do they affect the properties whether they will affect my skin friction on the other hand how about friction viscosity affecting heat transfer coefficient what is it resisting that is a new term maybe we mean the same but I am not able to get the yeah viscosity affects the flow okay yeah this is like in all our the same thing we look at different angle so you should give me different angles to this viscosity causes friction I want I was looking at from the friction generates heat sometimes it is negligible most of the time we want it to be negligible but not all the time when frictional heat is not negligible it affects your temperature profiles this effect is called the effect of viscous dissipation so actually you should it is mathematically already there in your energy equation do you remember u del u by del x plus v del u rather u del t by del x plus v del t by del y is equal to alpha del square t by del y square simplest of the boundary equations plus q triple prime by rho cp if you have the heat generation and the last term is what mu mu phi phi the dissipation most of the time we neglected we say you are causing a lot of problems so we will let me not bother you but imagine if that is important your whole energy equation is affected by that again it is not simply the viscosity that viscosity along with the velocity gradients that phi is nothing but all velocity gradients as you will see so they are all mutually affecting each other so it is possible in certain conditions we consider sometimes we do not then as scientists we should say when do you consider viscous dissipation when we do not that is from viscous dissipation point of view now if you come to the flow itself you know laminar turbulent we talk about this but there is a in between zone the transitions actually laminar what is the turbulent what is the critical Reynolds number for flat plate for flat plate 5 into for pipe when you say 5 into 10 where you got it from Bhagavad Gita is it where did you get it from one beautiful value actually if you go a little bit detail there is a transition zone so there is a initial Reynolds critical Reynolds number and a complete critical number it depends upon so many things including the surface type of surface obviously so it is not that suddenly the flow which has been very nice well behaved it jumps to turbulent flow it happens over a Reynolds number range so initial Reynolds number in our general handling of convection we may not bother much about it but you should know that there is a range and it can be controlled in fact a whole lot of things can be controlled by controlling the flow that is possibly in some ways it is under in your control and especially also controlling turbulence itself and the transition to turbulence so if you know some of these fundamentals you would like instead of simply accepting whatever there is shear stress or heat transfer coefficient you may say can I control this can I control the flow for whatever reason can I control the heat exchanging process within certain limits you can do it that is why we should you should try to go as deep into the fundamentals as possible as an engineer finally an actual technical person may not bother about these things but he knows if you go to a person who is handling heat exchanger and he knows how to increase the heat transfer rate how to decrease he may not know the basis of that sometimes he increases the flow he is actually increasing the Reynolds number basically there are many ways of controlling heat transfer and if you know that Reynolds number of the flow from the flow point of the controls heat transfer in a unit exchanger you can do something with the velocity you can some do something with the type of the surface that is the reason why I would like you to Dr. Arvind has done lot of thing on a flat plate and pipe I am just adding to that if he has not done that please read about the history and try to get a picture of where we are today why we want to study convective what has been the background that will give a little bit more I would think excitement understanding of the whole process otherwise it will become a little dry I will again talk about similar things I want you to draw start draw writing down drawing a convection chart a convection chart you must be able to do it on your own I will just guide you on that I like this particular thing I always emphasize for us to get a full picture especially for beginners I know you have already done quite a bit of convection but still I want to do this on the right is a chart chart or a sequence of events let me say on the right hand extreme you write the purpose of convective study determine and I want you to write what is it determination of I want you to write this no I want to write purpose of convection study is what determine h write down that and go one step ahead put an arrow there how do you get this edge in our in our scientific approach what do you need to get that edge model wise or experiment wise temperature difference no I have done all this I am actually trying asking you to put whatever you are done in a this chart fashion or a table fashion how do you get h actually from your analysis what is it that you use what expression for you to get h from all the analysis laminar itself what all you have done yeah what tell me what 4 years law at the surface no this is what I wanted at the surface you apply the 4 years law write down so 4 years law comes into picture minus k dT by dy at y equal to 0 equal to isn't it that is how you are going to get it that is the second what box in my chart where do you get the dT by dy at y equal to 0 you should always say dT by dy at y equal to 0 it is at the wall we are looking at convection heat transfer between a solid surface and a flowing fluid for me 3 conditions are to be satisfied in convection can you tell me what we are talking about a solid surface and a fluid okay I want 3 conditions for me to say this is convective heat transfer conditions is a very big word situation but I would like to use the word condition there is a solid surface there is a fluid when do we talk about convective heat transfer maybe I have not motion relative motion that is very important but before that something else correct relative motion is one of the conditions for me temperature gradient a temperature difference to the third one I am saying purposely convection is very simple you will say oh sir we know about it but that doesn't matter can convection occur if the surface is here and the fluid is flowing here they should be a contact this is what we are talking about so convective heat transfer is that type of heat transfer which occurs when a fluid and a solid surface are in contact with each of the number one there is a relative velocity and there is a temperature difference all the three must be why I am saying this if a solid surface and a flow of different temperatures are not in contact with each other also there is heat transfer occurring but what is that heat transfer will be radiation and if the fluid is in contact with the surface there is a temperature difference but there is no fluid motion even then heat transfer occurs what is that heat transfer huh so conduction can occur radiation can occur between a solid surface and a fluid depending upon whether in contact no motion no contact no motion but convection is contact and motion motion is created by the temperature difference motion is created by nature nature where we are talking about the G effect actually but it's better sometimes to call it free convection but you are perfectly correct what you can talk about is in forced convection there is a imposed flow a flow is forced by some means in free convection that flow is created inherently within the fluid itself because of the existing temperature difference so it's a good point that means I mean we know that for heat transfer first of all temperature difference must be there of course there is if there if these two are in thermal equilibrium there is no heat transfer so that is a but I am your point is well taken I am will emphasize that so convection is that mode of heat transfer between a solid surface and a fluid when they are in contact when there there is a relative motion and when there is a why I emphasize this is a following let us say there is a contact between the solid surface and the and the fluid there is a temperature difference but if they are moving together at the same velocity there is heat transfer but that's my conduction only there's no relative motion there's no boundary layer there so conduction can occur in a fluid I mean between a solid surface and a fluid when they are static both of them or both of them are moving at the same velocity that's why I have to bring in the term relative motion must be there even if there is a point one centimeters per second difference we say there is a relative motion but then you can also see the convert the heat transfer due to the fluid motion is very mild it's not that much in terms of Reynolds so that is my as you students at take on convection is it convection only between solid surface and a fluid under these conditions do you have to stress on solid surface it could be heat transfer it could be my sense and the point here is that it's not necessary that we always talk about a solid surface you know now there is a big water shortage in the city and we are going to feel it you know why one of the water short reasons for water shortage of course there is no rain but if you go to any of our reservoirs they are all open surfaces of water you think there is nothing happening there at the surface there is a wind blowing in fact at all o'clock the wind sets in the the sea breeze sets in and there is a wind flow over these surfaces what is happening at that time vaporation there is a mass transfer so when there is a concentration different there is a mass transfer that we call it as a convective mass transfer so there is a concentration boundary layer and similarly if there is a temperature difference between these two heat transfer will occur and sometimes you can have isolated cases of I am sorry isolated heat transfer mass transfer but there are cases where both of these are possible heat transfer mass transfer simultaneously but in convection fundamental to heat transfer and mass transfer is the fluid motion fluid motion plus conduction Fourier's heat law of conduction is your convective heat transfer fluid motion plus fixed law that is your mass transfer convective mass transfer so I would also therefore like to say convection is not necessarily between a solid surface although we focus on that between two fluids as long as there is contact temperature difference fluid motion relative velocity is another point about it now if this flow is from an external engine that is force convection free convection there are mixed convection cases also to get a handle on quantitative handle handle on heat transfer coefficients and skin friction we do two things either we follow a science model or a engineering model heat transfer as a as a subject has two aspects science and engineering engineering means you do actually experiments you get the heat transfer coefficient or whatever you want measure temperatures everything but even basis for those experiments is your actually theory part of it so when you talk of getting the heat transfer coefficient theoretically you develop a theoretical model I want you to from this come to that point that chart so heat transfer coefficient you get by equating minus k dt by dy at y equal to 0 y equal to 0 is very important y equal to 0 is the interface is equal to h delta t tw minus t infinity now go to the next verse next box what do you have to get in this equation what is it that you have to get temperature no first you should say I want you to say you want the temperature gradient at the wall you do not really care about the rest of the thing I want dt by dy at y equal to 0 this is very important for me dt by dy is there from y equal to 0 to y equal to delta but I do not bother dt by dy y equal to 0 wall gradient now you go to the next how do you get the wall gradient at y equal to 0 only this is what is making all your convection a big issue to use a very mild wall all you need it dt by dy y equal to 0 therefore when you do experiments ladies and gentlemen you do not have to measure the entire temperature profile for you to get the heat transfer coefficient just close to the surface is enough actually you do not have to do go for the entire boundary thickness so for but mathematical that is from a theoretical point of view for you to get dt by dy equal to 0 you have to get t as a function of y that means now you have written the temperature profile now how do you get the temperature profile now we are talking about the boundary how do you get the temperature profile go to the next box what is the next point how do you get the solving you cannot simply say energy equation energy equation is not enough by itself for you to solve the energy equation you also have to solve the y what is there in the energy equation which makes you so yeah you have you that is the difference between your conduction equation which is the energy equation and this equation fluid flow affects your energy equation so u del t by del x v del t by del y for you to get this you have to solve the momentum equations and momentum equation you cannot solve it on their own you have to solve the continuity equation so when you solve the continuity and the momentum equations you will get the velocity profiles when you take those velocity profiles and put it into the energy equation you get the temperature profile when you get the entire temperature profile you just take the dt by dy at y equal to 0 multiplied by k equity to h del t then you will get get h now from where do you get that profile which equations now you have to get the profile so from where do you get the equation you have to solve the plant I want I want answers that I want plant it's boundary layer equations you have to solve not the not in a never stokes equation that's what I am coming to at this point you have to solve the plant it's boundary layer equations at the entire thing momentum equation energy equation of continuity it's it is understood that you have to do that so these are already simplified where do you get how do you get the plant boundary layer equations from where from so write down never stokes so simplification of never stokes use is it all this you have done I am just putting it in a certain form which I like where do you get the never stokes equations what is the basis of the never stokes equation Newton's second law I want I want start from the beginning I say first equation first continuity equation law of conservation of mass then next three equations yeah okay I will leave it at that three equations and then the last one law of from where does love conservation energy come which law first law of course if you are looking at mass time so you also at this space is a question will present on what physics do these equations depend on I am trying to get you related this edge that you want to engineering wise require to nature basically now look at this loss you are talking about first law of thermodynamics second law also issues a second law is important minus k dT by device for second law of thermodynamics expression actually and then all the velocities come from the momentum so law of conservation of mass now look at the how who developed this when were these developed what is the proof of this law of conservation of mass you write down you know immediately a mass cannot be destroyed created you say Newton Newton has done tremendous Newton was a really great scientist he has done in work in optics heat transfer fluid mechanics actually he was a philosopher who was looking at the celestial bodies he started developing equations for the motion of celestial bodies although he was just a banker he was in charge of the treasury of England actually at that time so he came out with these equations which happened for us to be applicable to fluids also on the earth and then Prandtl made it to the border layer and thermodynamics now where is the proof of the first law of thermodynamics I am talking about a theoretical proof where is the theoretical proof the second law of thermodynamics now you see in science we always say any model any law must have experimental validation now these are all called natural laws because nature itself is a proof for this on the other way we say in nature we have not found anything against this people have tried to prove the second law of thermodynamics actually theoretically but they got stuck what is the purpose of this for me this collects I want to connect the engineering aspect and nature finally we are dependent upon these nature in terms of the loss of nature it is not that one man came up with this loss whereas you have a four years law of heat conduction you have so-called Newton's law of cooling Newton's law of viscosity stiffen Boltzmann's but here you don't get why any one name so-and-so first law of thermodynamics he evolved over years from the 17th century and really find out when the first law of thermodynamics was given a mathematical representation try to find out on your own now in most of the classes what we do without you know telling you this we start let us look at the law of conservation mass and then derive I am trying to tell you the reverse of it you want h and you want also tau w by the way shear stress so the same thing you can relate tau equal to mu del u by del y but you want the shear stress at the wall tau w equal to mu del u by del y at y equal to 0 never forget that at y equal to 0 point for convection that du by dy I get from the velocity profile now you see velocity profile you have to get from your momentum and the continuity equation for shear stress that is the end of the matter as per speed mechanics is concerned but if you think heat transfer can be obtained without records to this it is not possible because your energy equation contains u and v so a natural law by the way this first law second law Newton's law these are called general laws physical laws general laws universal laws they will never be disputed anyway but your four years law Stephen Boltzmann's and Newton's law of viscosity these three are not general laws why tell me just look at the equation and you can make out tau w equal to mu del u by del y q is equal to minus k d by dy a q is equal to sigma epsilon 1 to 4 what is common in all these three equations common in terms of the nature of the equation there is a property the moment you say property there is a medium coming into picture so these are medium oriented medium based medium controlled equations though they are called particular laws so we have particular laws general universal natural or physical laws from a combination of these laws now combination of these laws lead you to the equations which we would refer to with all along with the boundary conditions the mathematical model so from the physical model where you take a control volume that is exactly what you are doing it control volume or a system apply these general laws you come down to very broad equations they are not enough for you to get heat transfer they will you give you only the profiles for you to get the heat transfer coefficient you have to come out with a particular law which is minus k d by dy for you to get the mass transfer you should have a fixed law for you to have the shear stress you must have Newton's law of viscosity so a combination of general laws based on nature and particular laws based on peoples they have developed it please read Fourier's law of heat conduction very nice paper it is in French some English translations are available how what now you and I take it to be so simple q is equal to minus k d by dy what is there please look at that particular 18 1829 or so it gives you tremendous information on what was his thinking at that time when you put all of these things together ladies and gentlemen we can then say we are ready to find out rather determine engineering wise heat transfer coefficient and the shear stress I will end now