 Hello, I am just taking questions here and there. Nirma University, Amdabad, you have any questions? What is different between conservation equation and non-conservative equation? Okay, the question is what is the difference between conservation equation and non-conservative equation? Okay, actually we will get back to you. All that I will tell is whatever the form, there is nothing like non-conservative equation. There is nothing like that, that is first thing. Second thing is that we have conservative form and non-conservative form. We will write the equations in the non-conservative and conservative form and get back to you. This is a good question. I would request you to put it in moodle. Sudeep, please note down. There is nothing like non-conservative equation. The only the equation, it is conservative or non-conservative form. That is to put it in a very simple term. You can see these terms getting conserved in one form and in another form you cannot directly see them conserved. Nevertheless, the equations are conservation of mass, momentum and energy. That they cannot violate. None of these equations can get violated. Is that okay? Any other questions? Shown on page number 64, can you explain that significance of the arrow you have shown in with written that pipe central line? 64 of which convection 1, convection 2, convection 3? Convection 2. What is the significance of this figure? That is the question. See, as I said in the previous transparency, we have u plus, we have y plus. Of course, they are not u and y directly the dimensional form. u plus is u by u tau and u tau is square root of tau all by rho. Incidentally, u tau is also having the unit of velocity and u plus is indeed non-dimensional and y plus is also non-dimensional y u tau by nu. y u tau is meter squared per second and nu is meter squared per second. All that what people found is of course, this can be although it has been stated and said that as if it has come from empirical, it can be derived from fundamentals. You can see this derivation in Professor Bajan's book, Convective Heat Transfer Book. So, or even in Sebastien-Brodshaw boundary layers by Sebastien-Brodshaw, the point here is if I take any measurement of shear stress along the boundary layer at different y locations, if I take those measurements and non-dimensionalize my velocity and shear stress in these lines, the profile is essentially going to follow initially linear in viscous sub-layer and then logarithmic in turbulent boundary layer and also in buffer layer. That is what we are trying to say. Over to you, if you have any specific question in this, I would like to take up that question. The pipe centerline and the arrow you have shown there, can I know the significance of that? The question asked by one of the participant is that, what is the significance of arrow mark in the pipe centerline? That means the significance of this arrow mark, this only means that I am going towards pipe centerline. This is the pipe wall, this is the pipe wall for internal flows or for a floor flat plate, this is y equal to 0 just above the flat plate or just on the pipe wall. As I move away from here, I am moving into the boundary layer, first viscous sub-layer, then buffer layer, then turbulent boundary layer, this is towards the pipe centerline if it is pipe case. Is that okay? VIT Pune, any questions? My question is, while non-dimensionalizing the momentum equation, may it be steady or unsteady, we end up with one non-dimensional parameter that is one by RE on the other side of the equation. While non-dimensionalizing this energy equation, we have assumed that flow is a steady and we have ended up with the non-dimensional parameters like RE, PR and ICART number. Now if we consider the unsteady energy equation, do we lead to some other parameter? Yes. See, one of the participants question is that, if we take steady flow, I get RE as a non-dimensional number in momentum equation. If I take steady energy equation, I get the non-dimensional number as Reynolds number and Prandtl number in energy equation. See there is, I do not have to assume unsteady. Yes, if I do not assume the flow as steady, I get another non-dimensional number what is called as Fourier number. Whether it is momentum equation or energy equation, most of the times as you and I know, we handle steady flows. So that is the reason, we do not consider that Fourier number. Sorry, not, what is it? Froude number. Sorry, the non-dimensional number is Froude number. If you take unsteady flow irrespective whether it is momentum equation or energy equation, you get what is called as Froude number. But I would think this is a very good question. So for unsteady flows, we get non-dimensional number as Froude number in momentum equation. But in energy equation, we end up getting Reynolds and Prandtl. The point is as long as the flow is steady, you get Reynolds number in momentum equation and Reynolds and Prandtl in energy equation. If you take unsteady, you take, you get Froude number as the non-dimensional number for the unsteady term. I think that is a good question. Then you may ask me, what is Froude number? I do not have right now the Froude number definition. It goes something like this, u squared upon square root of lg. I will elaborate this question. I will elaborate the Froude number definition in the next session. Is that okay? Professor Sevatkar. Thank you, sir. Thank you. NIT 3G, any questions? So in boundary layer formation, we have discussed flow over a smooth flat plate. If our surface has any, if our surface is, we have any surface roughness factor or it is not a smooth surface, such a case is what would be? Okay, one of the questions is if the surface is roughened by rougheners, what it would be? Okay, we have answered this several times and we will be re-dealing with this in internal flows again. Nevertheless, I have been telling time and again, if the surface is roughened, if the surface roughness height is such that it breaks the laminates of layer and which is what we generally want to do, if that is the case, my surface would have the boundary layer completely turbulent. That means the heat transfer coefficient will be very high because my friction factor has gone up. Because my shear stress at the wall is essentially minus rho u prime v prime bar. So, laminar sub layer is not going to exist. It is going to be completely turbulent. Is that okay, Professor? Sir, one more question. Yeah. So, in cricket, during fast bowling, compared to England and Australia, our cases, the swing is very less in the Indian conditions. Is there any role of a boundary layer formation in that? The question asked is, when we play cricket, the swing in India is much lower than the swing in England. That I do not know the answer, why in England it is more, maybe because of the wet condition. But all that I can definitely say is, what is essentially happening in the ball is that, while there is a linear motion, there is also a rotational motion. Because of this rotational motion, there is a slight lift generated, which is what is called as Magnus effect. This is what we study in fundamentals of fluid mechanics. So, in fact, few people in early days, they wanted to use this Magnus effect and move the ships, but they failed. But this Magnus effect is the one which is actually generating the lift for swinging of our tennis ball. Now, if you ask me the question, why in England swings more than in India, I think it is something to do with humidity or the wetness. So that is essentially, what is different is the density. Density, I think, but it is a very specific question. Why? I have not answered one part of your question. That is, why in England it swings more than in India? I am hand waving, but I will come back to you. This is a very good question, but there is a lift. You have an answer? Yeah, professor Atul wants to answer this. Why do not you pitch in? Basically, your question is the effect of seam conditions. If you remember, if you play any of, I mean, if you watch any of these cricket games, if you see the bowler, he tries to hide one side of the ball. So, that one side is hidden just because they do not want to show the condition of the ball on that. So, what they do? They shine the one side and they roughen the other side. That is why you have these incidences of ball tempering and other things using your nails and the caps of water bottles. So, they try to roughen the other surface. So, when you have seaming conditions, the role of boundary layer or the effect of seaming conditions on the boundary, on the ball, on the two sides of the ball would be different. So, the boundary layer formation on the roughened side would act in a different manner as compared to the smoothened side. So, that is why you have the swing. The ball tends to move towards the roughened side. So, I mean, seaming conditions are as Professor Prabhu has told. The wet conditions compared, I mean, in England compared to India, they definitely play a role. But if you have the condition of the ball, which is equally good on both sides, the seaming effect would be, would be subsided. So, in order to have the swing of the ball, you need to temper or you need to have different conditions on either side of the ball. Sir, this is, thanks Professor, thanks Professor. Kolhapur Institute of Technology, any questions? Sir, in case of oil bearings, the viscosity is going to change with respect to temperature. But do we take into consideration the mu value is constant or is it changing with temperature as a function of temperature in case of these equations? So, the question asked is by one of the participants in bearings, whether it is ball or roller or whatever, whether the viscosity varies, because the temperature in the bearing changes as it is being used because of the friction, the temperature of the oil inside the bearings will keep changing. So, whether the viscosity we have to, whether should we be taking the variation of the viscosity with temperature or not in the equations? Yes, one has to take the variation of the properties with temperature. In fact, as we go along in convection, we are going to see that there is significant influence of the property variations with the temperature when we are calculating the heat transfer coefficient. So, what you asked in bearings? Yes, viscosity variation with temperature has to be taken into account in my set of equations. Okay, next question please. Thank you sir. Sir, one more question sir. Sir, we discussed the universal profile for flow over the plate and flow in the pipe. Can we apply that universal profile in agitated vessel? No. So, the question asked is one of the participants question is that can I apply this universal profile whatever has been taught this u plus y plus the viscous sublayer buffer layer and the turbulent boundary layer it is, we said that it is applicable for flow over a flat plate and for flow in internal pipe. Is it applicable for agitated flows? No. See that is why I said this is only applicable for these two class of flows, because that is what it has been tested for. Generally for any other special or specific problem this is not applicable as I said even for jet impinging flows for example this is not applicable. People are struggling or trying to come up with some other profile but that is not working. So, they have tried at least to the extent possible I know to the best of my knowledge velocity profile they have succeeded but not for temperature. In fact we have similar temperature profile also which we will be studying little later but universal velocity profile they have slightly succeeded for impinging flows but that profile looks different from this. So, that is why then it is not going to be universal at all. So, there is nothing like I mean we should not be calling this profile as universal. So, the point is for each class of a problem the stress distribution with velocity is going to be different. So, to answer your question for agitated flows this is not applicable. So, now we will stop this question and answer session and professor Arun is going to teach us from where he had stopped. We had introduced or come up with the non-dimensional form of the equation. A quick two minute recap why was this done? See we will go back to the dimensional form of the equation this is the dimensional form of the energy equation. Please do not see this in isolation see this in conjunction with the x, y momentum equation and the continuity equation and we said that there are four variables of interest that is u, v pressure and temperature. So, we have four equations one continuity equation one x momentum equation y momentum equation and the energy equation. So, four equation four unknowns this problem is a nice closed form problem because whether we can solve it directly for all situations that is a different question. Then we said this energy equation is nothing but first law of conservation of energy and then this first law of thermodynamics and from here we can derive the diffusion or conduction equation when this term u and v is go to 0 pressure work goes to 0 viscous dissipation goes to 0. That means when there is no flow associated with the problem you can transform this equation to the transient 2D conduction equation with constant properties. We saw that also and then we said let us non-dimensionalize these using a certain set of non-dimensional variables. These are chosen with an idea that the non-dimensional parameter that is obtained x star y star u star v star p star and non-dimensional temperature is all varying between 0 and 1 a general rule of thumb. What this l is what this v is all those things will be problem specific situation specific to the problem whether it is average velocity free stream velocity or any other velocity whether it is the length of the pipe diameter of the pipe length of the flat plate so on and so forth that depends. We said in fluid mechanics we have u local and u infinity as the 2 variables I mean 2 quantities related to velocity. We do not have anything like u surface whereas in heat transfer you are going to have T s in addition to a local temperature and T infinity. Therefore, we have to encompass or take into account all these 3 quantities to define a non-dimensional temperature. After all this algebra which we hope you do it later today you come up with the nice non-dimensional form of the equation which looks like this and we said that this occurred number which is representing the inter conversion from kinetic energy to heat becomes dominant in case of high speed flows typically because you are talking of this ratio. When T s minus T infinity is large or C p is large also these things go small and therefore, we tend to drop this term and this term viscous dissipation term and pressure work terms are neglected they are not equal to 0 they are neglected in our normal heat transfer problems that we are going to do in this course that does not mean these are unimportant these become important in high speed flows. Somebody I had a question on if I have a unsteady term what is a non-dimensional number that is going to affect this part also. In fact, if you go to aerodynamics the same energy equation this term will become far more important or this term will be of comparable order of magnitude and somebody had asked in the Moodle also can this I do not remember this thing about high speed or something like that we had mentioned about this occurred number which is valid for high. One question in the Moodle that when can I take the flow as compressible and when can I take the flow as incompressible and one of you had answered also very rightly that is for all Mach numbers less than 0.3 the flow can be considered as incompressible and for all Mach numbers greater than 0.3 the flow can be considered as compressible and of course why Mach number less than 0.3 can be considered as incompressible that is because essentially the density variations that is delta rho by rho the density variations is less than 10% for all Mach numbers less than 0.3 and for Mach numbers greater than 0.3 the density variations become significant density variation is what we have been telling for compressibility into account. So for compressibility and incompressibility other criterion which can be taken into account which is what usually is done is Mach number. Mach number less than 0.3 is incompressible and greater than 0.3 is compressible because density variations become significant for all Mach numbers greater than 0.3. So now what we are saying is you have a non-dimensional energy equation what is the use of all this what are we going to get at the end of this. So before we understand what we are going to get let us just look at these three equations that we have. This is the continuity x momentum and the energy equation and y momentum equation if you see what is the y momentum equation for those of you can write down y momentum equation rho v du by dx plus sorry rho u dv by dx dv by dy equal to mu d squared mu d squared v by dx squared plus d squared v by dy squared minus dp by dy right now minus dp by dy ok. So what happens to this for flow when we have this kind of a situation we are talking of steady incompressible laminar flow of a fluid with constant properties this is a set of equations and the boundary conditions. So at x is equal to 0 what is x is equal to 0 x is equal to 0 represents let us say I have flow over a flat plate this is my coordinate axis x is equal to 0 represents this point. So when the flow is not or just come in to the region of interest velocity u at all wise is u infinity free stream velocity temperature at x is equal to 0 for all wise is equal to t infinity. So this fluid has not started to flow over the flat plate it is just coming into that region of interest ok. Second thing y equal to 0 y equal to 0 represents the flat plate surface x axis this is the no slip condition u is 0 v is 0 and temperature at all wise for temperature at y equal to 0 for all x is equal to the surface temperature. So for this surface if t s is greater than t infinity I can always say that velocity u v both are equal to 0 at wall and t is equal to t s at so this is what is given by the second boundary condition third boundary condition is y tends to infinity what is this y tends to infinity it represents the region far away from the influence of the solid surface ok region far away what is this so called far away this so called far away refers to a region much beyond the boundary layer. So if I have a boundary layer that is firm the region beyond that y tending to infinity I will have the governing boundary conditions as u at all x locations at y tending to infinity is u infinity. So the fluid does not know does not remember anything about seeing a flat plate somewhere far below 8 temperature is also unaffected t at y tending to infinity equal to t infinity are these enough are these enough all that is what we have to see we have u in terms of x we have u in terms of y so d squared u by dy squared is there so two boundary conditions in y I have two boundary conditions in y one boundary condition for u in x I have that one boundary condition for v in terms of y I have that then the remaining are all related to temperature one two three conditions for the temperature so this is a well posed situation what happens to the y momentum equation why was that not considered it I want to leave this as an exercise for all of you what will happen to the y momentum equation is should I leave it yeah you will see that terms will get cancelled off will get dp by dy equal to 0 if I urge all of you to try this if you do not get this and you are unable to proceed please ask us or please put this on model we will reply to all of you together okay but y momentum equation simply is going to vanish okay so it is not going to give you any information which is going to be relevant except the fact that dp by dy equal to 0 and what does dp by dy equal to 0 mean that also you should interpret partial derivative of something with respect to a variable is 0 means pressure is not a function of y so that is something which you will have to interpret and then you will be able to appreciate that we have only three equations we urge all of you to write down the y momentum equation and please do this mathematics okay so I think it is there in the tutorial today okay so we will cover that we are not going to do it here explicitly so with the same non-dimensional parameters that was used for the energy equation let us non-dimensionalize the continuity and momentum continuity as we quickly saw it would have been v u star is equal to u so I will get v I will get l here so v by l du star by dx star plus v by l dv star by dv star is equal to 0 that l by v goes off momentum equation u star u du dx I will just spend half a minute writing this because people get overwhelmed very easily when I non-dimensionalize this is v u star d by dx will have a l dx star du star will have another v du star okay so this becomes v squared by l u star du star by dx star I like to keep the functional form of the term the same u du by dx u star du star by dx just a change in the nomenclature on paper but what is coming in front of it is this v squared by l when I do that for all the terms what do I get the similar term for the similar form for the second term is v du by dy will essentially become v du star by dy star that we just put it here v du by dy becomes l yeah v is equal to p star v times v star du star v l dy star so this will be v squared by l v star du star by dy star essentially the same functional form with this v squared by l sitting in front then the right hand side I have mu d squared u by dy squared so that is going to be taken care like this mu d squared u by dy squared is going to be mu d by dy second time I am going to do this so I would have d by dy of du by dy this is mu y is what let us recall y star is y by l so this is what I am going to use this is v du star by l dy star and again I would have your l dy star I hope all of you are with me here so I will get this as keeping this in mind I will get mu v l squared d squared u star by dy star squared something in front the mu is also there the functional form is essentially the same mu d squared u by dy has become mu v by l squared so on and so forth and let me go back to the last thing dp by dx will be non-dimensionalized in this form dp by dx is nothing but pressure was non-dimensionalized as rho v squared dp star this is l dx star so if I keep all these terms together in the non-dimensional form I would have write it one by one v squared by l u star du star by dx star plus v squared by l v star du star by dy star equal to mu v by l squared d squared u star by dy star squared minus dp by dx which will give me minus rho v squared by l dp star by dx star so if I that this is divided by v squared by l I hope all of you can see that so when I do the jugglery cancel of v squared by l on this side I would be left with u star du star by dx star plus v star du star by dy star equal to l l cancels v cancels here I would get a mu l v rho I think I have missed somewhere let me just go back yeah there is a rho here sorry about that there is a rho which is there so this rho keeps coming all the time so there is a rho here there is a rho here there is a rho here so that will be there and I would have here minus v squared by l cancels off so I will be left with a rho dp star by dx star divide through by rho I would have rho here this will cancel off and what am I left with I am left with this form rho v l by mu is my reciprocal of Reynolds number so what do I see u star du star by dx star plus v star du star by dy star is equal to 1 over Reynolds number based on some l some characteristic dimension which is not volume by surface area d squared u star by dy star squared minus dp star by dx star okay so what was my original equation my original equation which was not before this was rho u bring this rho down you get u du by dx plus v du by dy is equal to nu d squared u by dy squared minus 1 by rho dp by dx and that is what I have here in the non-dimensional form as the reference side is same except for the stars I have 1 by Reynolds number which is something which is very important and dp by dx is just left as it is and we have done the same thing for the for the energy equation and we showed this just before we went for t that it is of this form if I neglect the viscous dissipation and the pressure work term my energy equation reduces to u star dt star by dx star plus v star dt star by dy star equal to 1 by re l pr d squared t star by dy star squared please keep these two in mind this is probably the heart of convection I cannot over emphasize this because now if you look at these two equations forget whether the independent variable is u I mean sorry dependent variable is u or t look at the left hand side if I cover the right hand side look at the left hand side identical instead of u here I have u star I have t star point sorry instead of u star I have t star otherwise it is exactly the same thing left hand side same very good if dp by dx went off to 0 that means I throw this off here I have d squared u star by dy star squared d squared t star by dy star squared this is just the constant coefficient in front of it Prandtl number let us say it is equal to 1 for just for the sake of simplicity it is identical right so let me for the sake of being repetitive I will just bring this up here again the functional form of the governing equation non-dimensional governing equation looks identical when dp by dx is made to 0 and pr is made to be equal to 1 essentially identical form I have non-dimensionalized the governing equation so I have to non-dimensionalize the boundary condition which is a matter of which is very straight forward all of you can do it x equal to 0 x star equal to 0 y tending to infinity y star tending to infinity t you just put it appropriately you will get 1s and 0s so that is also non-dimensionalized here okay what am I saying because of all this is something which is so so so important is the following let us just go back here the same equation I have recast when do not read all this Chilton Colburn do not read all that for now dp by dx when it is equal to 0 and pr is approximately equal to 1 the non-dimensional momentum equation and the non-dimensional energy equation are identical in its functional form if I have an identical equation differential equation and the boundary conditions are identical the solution is also identical what it means now if I just go back step by step to the boundary condition here u star at 0 comma y star is 1 where is this coming from this free stream velocity I have another thing here free stream temperature this essentially is u star becomes 1 because I have non-dimensionalized using u infinity which is the representative capital V that we chose okay that instead of capital V you can use u infinity which is the maximum possible velocity in the flow this t star by definition is going to become equal to 1 am I right t minus t s divided by t infinity minus t s go back here when in doubt go back to this t infinity minus t s divided by t infinity minus t s gives me 1 so non-dimensional temperature will be 1 that is precisely what is given here these two okay then what else is there u star at x star 0 t star at x star 0 is 0 x becomes x star here also 0 remain 0 t s minus t s you will get in the numerator so that ratio is going to become equal to 0 non-dimensional temperature is equal to 0 for this boundary condition this one then remaining 3 are here y tends to infinity this one all x so x star infinity 1 because this is again u infinity x star infinity this is going to be 0 sorry x star infinity this is also going to be 1 t the ratio is this is t infinity and the ratio is t infinity minus t s divided by t infinity minus t s so that is going to be 1 v was non-dimensionalized by again by the same scale so this will be x star 0 equal to 0 so what am I getting here I have three conditions for you u star u star u star t star t star t star forget the v part for a minute these are identical am I right so when Prandtl number is of the order 1 of the order means roughly equal to when dp by dx is negligibly small can be neglected or made equal to 0 then the non-dimensional I am going I am repeating again then equations are not the same the non-dimensional x momentum equation and the non-dimensional energy equation have the same functional form and the non-dimensional boundary conditions are also identical okay so this is probably the heart of what we have studied we are going to study the non-dimensional equations are of the same form the non-dimensional boundary conditions are identical that means logically I know that the solution nature of the functional form of the solution is going to be the same I think this cannot be any better explained because you take any differential equation d square y by dx square equal to constant or d square t by dx square equal to constant you will get the same functional form same thing here with the same boundary conditions this equation and this equation when in the non-dimensional form are identical therefore my non-dimensional velocity distribution will be identical to the non-dimensional temperature distribution okay I will restate this non-dimensional velocity distribution would be equal to the non-dimensional temperature distributions not written here but please write this down and take this away with you I think this is something which our students at an undergraduate level are not taught it is not appreciated and what we say some Reynolds analogy is given write short notes on Reynolds analogy this sort of a question is asked in an exam write short notes on Reynolds analogy what is Reynolds analogy it is essentially this part it is telling you this when the non-dimensional energy and momentum equations with identical non-dimensional boundary conditions are there that means it is of this form we start something in front this is the nature let me go back here this is the nature I have written on the white board just brackets the square bracket can be u or t let me call it a a a this a can be u star or t star okay when Prandtl number is approximately equal to 1 and dp by dx star equal to 0 the form of the equations non-dimensional equations are the same boundary conditions are identical it means non-dimensional temperature and velocity distribution u star and t star this is something which we have to carry with us at the end of heat transfer convection introduction this is what the student will have to carry what does this mean it has far reaching implications what does that mean let us see here when Reynolds number what is this use why is this done this is essentially done because I want non-dimensionalization gives me this freedom of collapsing multiple variables L V E T infinity nu alpha into two simple non-dimensional number Reynolds and Prandtl number that is why we are interested in non-dimensionalization okay that apart what we are saying is this when I have the non-dimensional momentum equation which is given by this okay solution of this two equations is going to give me u v okay and p also because p we have absorbed in the y momentum equation already so u star which is the non-dimensional x component of the velocity is obviously going to be a function of what it is going to be a function of Reynolds number local r e l it is going to be a function of x star it is going to be a function of y star correct because I want u d squared t by d squared y by dx squared plus d squared y sorry d squared f by dx squared plus d squared f by d y squared f is going to be a function of x and y same thing here u is going to be a function of x star y star and this Reynolds number that is what I am writing here so in general this is going to be the relationship in general t star non-dimensional temperature is going to be function of same thing x star y star r e l and there is one p r sticking there some constant which is there that also is going to influence the temperature distribution so just look at these two I am going to flip back and forth between these two identical solution form what does it tell me if I fix the location if I fix the wall as the location y equal to 0 y wall why not y equal to 2 centimeter from the wall it is because y equal to 0 represents a very important condition for us shear stress is evaluated at the wall that shear stress is given by what mu du by dy so if I fix y that means y star is fixed this is no longer a variable of concern so I can write this du star by dy star is essentially mu v by l right because I am non-dimensionalizing u so du star will be v du star y will be l dy star at y star equal to 0 that means this is going to be some functional form of x star and r e l because y has gone in it has been eaten up so mu v by l is what again mu v by l I can recast this multiply divide and do something and get it in terms of Reynolds number we will see that c f x coefficient of friction is therefore our definition one half rho v squared that is in the denominator tau so tau is mu v by l divided by rho v squared by 2 f of x star r e l and this is nothing but 2 divided by Reynolds number what does this tell me so important so important so elegant coefficient of friction for a case with dp by dx equal to 0 is nothing but a function of the local position and the local Reynolds number thus coefficient of friction can be for a given geometry can be expressed in terms of the Reynolds number and the dimensionless space variable local position x star x star is nothing but x right so x star alone instead of being expressed in terms of v rho etcetera this is so important I do not know how much more to emphasize this coefficient of friction is a function of this now people had asked questions you know what happens to the boundary layer thickness as you move along the length of the plate why should it increase this is the answer this is coefficient of friction we can do a similar thing for boundary layer thickness and you will see it is a strong function of Reynolds number Reynolds number is what defined in terms of the local location that is going to increase now let us not forget that we have to look this always with the temperature distribution non-dimensional so non- dimensional temperature distribution we got from this how did we get this just by looking at this equation t star is the dependent variable it is dependent on x star y star r e l and p r that is what I have written here heat transfer coefficient was defined first lecture minus k of fluid d t by d y at the wall divided by t s minus t infinity this on non-dimensionalization is going to give me d t by d y would be minus k I will just do this because this is important so I will just do this very quickly h is equal to minus k d t by d y at y equal to 0 divided by t s minus t infinity this is going to be minus k t infinity minus t s times d t star divided by l d y star I hope I am doing this right y equal to 0 translates to y star equal to 0 this divided by t s minus t infinity cancels off there is a minus sign already sitting there so it will become plus minus sign is already sitting there this is t infinity minus t s this is t s minus t infinity so there is this is going to become plus so h is equal to one of the most important findings k d t star by d y star at y star equal to 0 what is this k k of f now one more step and we are away away from something which has been bothering us so much take this l and k on the other side h l by k fluid what is this just go back here this is what we had this I recast in this form nusselt number therefore is d t star by d y star at y star equal to 0 far reaching implications just go back one step c f f is nothing but non-dimensional War-shear stress that is equal to 2 by r e f of x r e l basically what am I doing here here also I have taken the derivative of temperature with respect to y and evaluated it at y equal to 0 or y star equal to 0 so this derivative no longer is a function of y star is this derivative is going to be some g 2 times x g 2 function of x star r e p r now all the questions all the correlations in the world that have come for force convection this has to be the functional form nusselt number is going to be a function of the local Reynolds number which involves the position x star and the Prandtl number associated with the fluid now our u g curriculum where you are given a bunch of formulae and student have to memorize each and every formulae associated with a geometry there is no need to memorize anything ok in fact that is a stupid way of testing you should give all the formulae the student should be asked to choose you give force free convection mixed everything does not matter the student if he is smart enough will choose that for a force convection problem the the functional form of nusselt number has to be r e and p r alone it will not have anything else that is testing ok so nusselt number which is equivalent to a non-dimensional temperature gradient at the wall this is the definition of nusselt number what is c f x non-dimensional wall shear stress non-dimensional wall shear stress this is non-dimensional temperature gradient at the wall both are quantities evaluated at the wall both are derivatives with respect to y but both are non-dimension we are not saying d t by d y or d u by d y we are saying non-dimensional force free gradient non-dimensional temperature gradient now let us say why is this important why what a why am I marrying these two again and again it is because one step before this we said if the governing non-dimensional governing equations are same identical in form and the non-dimensional boundary conditions are identical the functional form the distribution of the u star and t star are identical if the distribution is identical then the derivatives are also going to be identical the derivative evaluated the wall is going to be identical except for the constants which are associated I am going to say for sure that the non-dimensional temperature gradient at the wall and the non-dimensional wall shear stress distribution or c f x coefficient of friction is going to have a similar form okay and that is what is explained by right short notes on Reynolds analogy this is Reynolds analogy that is this Nusselt number how how did this come this came from just these two Nusselt number is g 2 of x r e l p l p r this is f 3 of x r e this one these two are equivalent are equal because the non-dimensional temperature I am not saying where in mind students come back and ask is velocity distribution temperature distribution the same no the non-dimensional velocity distribution the non-dimensional temperature profile are the same are identical because boundary conditions are identical governing equation is identical assuming p r equal to 1 and d p by dx equal to 0 this thing is called as Reynolds analogy what does this tell me if I know the coefficient of friction fluid mechanics experiments are relatively easy to do compared to heat transfer if I know this and if all the conditions that were used in this derivation are satisfied then I can get local heat transfer distribution if I know the local coefficient of friction if I know the velocity gradients at the wall at different location then hopefully I can translate that to local heat transfer coefficients by this so called Reynolds analogy so this is something which we have to see again and again Nusselt number is equivalent to the dimensionless temperature gradient at the surface and therefore we say all correlations typically will have this functional form now let us just quickly go back to that transparency on Nusselt number that we had convection 1 here you will see this Nusselt number we wrote it we didn't we quickly didn't do this transparency nu is hlc by k we wrote it just like that now we have shown nu is hlc by k just go back here hl by k we have shown that non-dimensional analysis and what is it it is not something which is a non-dimension number which just comes out of air in fact you cannot calculate this because the actual aspect actual thing that you do is calculate Nusselt number first by using Reynolds and Prandtl number from getting Reynolds and Prandtl number you use the appropriate correlation get Nusselt number then use the definition of Nusselt number from here and calculate h it is not the other way given h calculate Nusselt number no that is not the nature of problem the problem is the other way calculate Nusselt number from Reynolds Prandtl and then calculate h from knowing the Nusselt number so this is essentially the definition heat transfer through the fluid layer will be by convection when the fluid involves some motion and conduction when it is motionless essentially this is the definition so Nusselt number represents enhancement of heat transfer through a fluid layer as a result of convection relative to the conduction larger the Nusselt number more effective is convection nu equal to 1 for a fluid layer then the heat transfer is by pure conduction so this I think will now correlate marry everything that you had all questions hopefully which you had relative to this Nusselt number fully are understood answer because we are definitely going to take other questions it is not that we are not going to take let us quickly one last slide we just complete that and I will hand it over to Professor Prabhu so this is Reynolds analogy similarly there is what is called as Stanton number which is essentially relationship between CFX and see this is CFX REL by 2 Nusselt number and then the Stanton number is defined this so I put these two here because Nusselt number is a function of REPR you get this Stanton number and this forms the crux of what we call as Reynolds analogy u star is equal to t star du star by dy star at y equal to 0 or y star equal to 0 is dt star by dy star at y star equal to 0 and this one ok so for laminar flow over a these are some kind of correlations that you have seen so anyway we are not going to detail going to details over this one word of caution dp by dx star is equal to 0 was a necessary condition for the non-dimensional equations to be identical in form for a pipe flow there has to be a pressure gradient along the flow direction if any flow has to occur so in that case we cannot apply this Reynolds analogy business to this kind of pipe problems that is the word of caution root force do not try to apply it everywhere please understand under what limiting laminar flow condition this was there dp star by dx star was equal to 0 that was the limiting condition ok