 So that is the big picture of today energy equation. So energy equation whatever we have been applying whether it was momentum or mass whatever was applied to system we are applying it for control volume by using my E dot in minus E dot out plus E dot st is equal to sorry E dot generated is equal to E dot st. So similarly here also the same thing we are going to do we are taking a small control volume and I am taking the change in the internal energy is equal to the heat transfer to the system and the work done on the system. Now if I take that E change in the energy what is the change in the energy if I do the budget always we do this budget what is that here we are saying what is that we are saying there is net rate of energy increase in the control volume something E is getting in and something E is getting out. So how do I if I take again a cuboid of delta x delta y delta z we know this business quite well so I am not writing this on this. So what is that which is entering mass into energy. So rho u dy dz is mass flow rate into energy what is getting out same Taylor series expansion I do not think I need to write it again so we have been using left hand write this. So by now we are adapting this so rho e u dy dz plus del rho e u by del x into 1. Similarly I can write in the y direction I am not consciously writing this equation for z direction because it makes my equation quite huge so I want to keep it little minimal so I have put it only for x and y direction if I take in the y direction I have rho v dx dz which is the mass flow rate in the y direction and E similarly I can apply for exit. So now if I rate of energy increase within the control volume is del rho e by del t into dx dy dz getting in and getting out if I cancel out this will get cancelled out with this and this term I am left out with del plus plus is missing here plus is missing here plus del rho e u upon del x plus del rho e v by del y. So what did we take care d e we took care now we have so we will do some algebra here same term I am writing rho e plus rho e copy paste problem whatever problem was there here same problem because of copy paste. So now if I expand this if I expand this I get rho del e by del t plus e del rho by del t blah blah blah blah blah is that okay. So now if I rearrange that if I rearrange that what is this what is this continuity equation this is my yes some problem second line okay there is a problem first rho del e by del t plus e del rho by del t is that okay plus e del rho u by del x plus rho u del e by del x I am just taking two parameters keeping one as constant and differentiating the other second term is that okay. So now if I expand that so this is my continuity equation so this I can throw it out I am left out with rho del e by del t plus rho u del e by del x plus rho v del e by del y if I take out rho what is this this is the total derivative of e so I have rho d e by dt okay. So this energy is having the internal energy and also the finite. So e plus u squared plus v squared actually plus w squared also should be there but I am not taking w into consideration that is why I am stopping here as u squared plus v squared by 2 okay. So that is about the so let us keep let us keep this in the back of our mind this is my d e remember now heat transfer dq I have to get to heat transfer. So this heat transfer is straight forward this heat transfer is by conduction we have done this already this what we have done here is already familiar to us because this we have done in heat diffusion so what am I going to end up with k del squared t by del x squared plus k del squared t by del y squared I have not done that because to do that I have to make the assumption that k is independent of x and y is that okay. So what is the term left out now energy is over heat transfer is over work done on this system who is doing work on my system who is doing work on my system pressure forces normal stresses and the shear stress is that right. So now let us take the rate of the work done by pressure is that okay I know I am going fast but these are all now very familiar because we have done this several times in momentum equation all that I am doing is work done. So what is work done if I take the pressure what is the what is the work done by pressure force force into velocity force into velocity force is what in the x direction p into dy dz into velocity here it is gaining direction because of the velocity okay. So p dy dz into velocity what is getting out if I apply Taylor series expansion what will I get the same term plus del up by del x into dx dy dz similarly I can do it in the y direction. So what is left out if I do the balance this this this this term will get cancelled out and I am left out with minus del up by del x plus del vp by del y into only is that okay what is this this is the work done by pressure. So same thing I will have to do for so first let me take normal stresses normal stresses will be same as pressure only thing is that the direction is different. So if I I will do the same thing sigma xx sigma xx sigma yy sigma yy I am going to end up del u by del x sigma xx plus del v by del y sigma is that okay. Now shear stress same thing for shear stress shear stress here is sigma xy u dx dz okay. And here it is taking negative sign because why is it negative u is in the opposite axis okay. So now here you take apply Taylor series expansion you have sigma xx plus del sigma xy or instead of writing this it might be appropriate to write this as actually consistently that is what in the mail professor Arun has sent today. So instead of this we could we can write notation wise what is that I should write minus sigma xy u dx dz right okay in the top it will not now plus what it should be del by del by del del by del y del y of u sigma xy dx sorry dy dx dz expansion of this is what is written here you are getting me I have skipped one step and gone here because my feeling was we are 2 I have skipped this whatever I have written and expanded that here if you expand that you will get this I have skipped this because we are now familiar. So if I expand all this in x and y direction so I get del v sigma yx by del x plus del u sigma xy by del y dx dy dz is there always so now we budgeted work done by pressure work done by normal stress work done by shear stress. So we have to now add all of them if you do that if you do that so rate of increase in E rate of heat transfer rate of work done by surface and body forces of course body force also does work. So what is the work done by a body force effects into rho u mass into velocity into body force is that right. So now this is the whole lot of equation I get is that right. So let me take stock of each term what is this term I will randomly pick someone yeah Baluswamy quite young you look that is why I have caught you yeah. So what is this term this is energy this is internal and kinetic energy. So what are these two terms heat transfer by conduction what is this term work done by pressure force what is this term work done by normal stress x x y y work done by what work done by body force in the x direction work done by body force in the y direction z direction is there imagine it would become still in the air I have cut it short okay. So now we have to do some rearrangement why are we again I come back to my favorite question why are we doing what are we doing and why are we doing what are we doing we are trying to derive energy equation why are we deriving energy equation okay to answer this we derived momentum equation and conservation of mass why did we derive this to get the velocity distribution why do I need velocity distribution why do I need the velocity distribution I have not even come to heat transfer as on its own why do I need velocity profile what is the engineering interest of writing these equations okay what do I do with the boundary layer thickness what is my engineering yes shear stress why am I interested in shear stress I need to find the pressure drop then only I can decide my pumping capacity if I have to decide to how much pumping power that is whether I have to put a 0.5 HP pump or a 2 HP pump to pump from here to Puna let us say if I have to decide that who will decide that the pressure drop will decide that shear stress will decide how will I get that shear stress until and unless I have the velocity profiles I cannot get those gradient I cannot get the so engineering interest drives us to derive these equations on the same note okay to emphasize I want to emphasize I like this in fact this up this I have learned from Professor Bajans book he always reminds us what is the engineering interest of all these equations so the engineering interest of momentum equation and conservation of mass equation is what is the engineering interest of this equation to get the velocity profiles u v w and pressure once you get that you can from this you can compute the gradients right that will okay gradients let us write that gradients of velocity that will fetch me shear stress if you have to relate now itself what is shear stress v non-dimensionalized shear stress in terms of kin friction factor cf equal to tow all upon half rho u square u squared can be u infinity squared in external flow u average squared in case of internal flow to remind you f equal to 4cf f is Darcy friction factor cf is kin friction factor or fanning friction factor so why do I need this shear stress why do I need this shear stress for as ma'am told us it is pumping power the key for all of this is this we need to emphasize this to the students otherwise they will not understand why are we doing this equation what is going on they will get lost okay that is why I said for every derivation in between I have me I have tuned my mind you see I did not even plan I landed up asking question what am I doing because we tend to get lost in the derivation we should not get lost ourselves we should stop and ask ourselves it is a digression the same question we will ask yes how will I decide in fact the boundary layer concept itself came by Prandtl because he was asked to decide the blower capacity for blowing on one drying for a flat plate energy equation what am I doing what am I doing with the why am I deriving energy equation to get what temperature distribution what do I do with this temperature distributions what do I do with this temperature distributions what did you write for velocity after velocity what did you write velocity gradient here you are interested in temperature gradients why you want to get heat transfer coefficient let me remind you what is tau wall for fluid mechanics that is heat transfer coefficient that is heat transfer coefficient for heat transfer tau wall is the resistance offered by my solid surface for the flow to happen h is the resistance for the heat transfer to happen you can think analogically very straight forward manner no confusion on that score okay so h what is h you remember what was the h definition professor Arun had written very nicely minus k del t by del y at y equal to 0 upon t s minus t infinity I cannot get h until h definition is not q double dash equal to h into t wall minus t infinity h definition is this he told us yesterday so I will get from the temperature gradients I will get h which we usually write in terms of Nusselt number like I wrote tau wall in terms of cf or f okay now why do I need h why do I need h now you can answer heat flux or q t wall why do I say heat flux or t wall if you know heat flux you can compute what would be the wall temperature it is going to attain if the wall temperature attained is going to be more than the metallurgical limit in your heat exchanger your wall temperature turns out to be let us say 1000 degree Celsius you cannot so you have to either increase your Reynolds number so that your h goes up so that the wall temperature comes down so that is why so why we are doing we should not lose sight of this is pumping power or you can on the same note you can call it as maybe what can we call heat transfer rates maybe I think we will leave it like that okay so that is the that is the whole idea of doing this energy equation fine so that was good thing so this he this is done by Weijan before deriving he does this okay okay so now this was our energy equation now let us do some rearrangement why I reminded this equation do I see anywhere t no so I have to reduce this equation so that I need to t is there but still in e I have to write in terms of t so everything has to be energy has to be put in terms of t so momentum equation is given by this we have derived it yesterday Cauchy's equation it was called it is not I have not put for sigma xx I have put I have not put the velocity gradients this is the this is Cauchy's equation okay so now if I take this was my this is my rho du dt if I divide it by rho and multiply by u please listen to me carefully I am going to go fast I am not going to take you along with me until you make a conscious effort to be with me I am not going to take an extra strain because this is all algebra there is no concept involved so that is why I am going to go fast so if I multiply by u by rho throughout this is what I end up with so u du dt I can write as du squared by 2 and this is my right hand side so on the same note I can write v dv squared that is dv squared by 2 what is this y momentum equation so why am I doing this if I add this I will get u squared by 2 plus v squared by 2 and the right hand side is that okay now this is my equation b and this is my equation a so why am I trying to do that is I want to get rid of this u squared plus v squared by 2 if I subtract b from a b from a a b is nothing but u squared plus v squared by 2 on the left hand side so I get a minus b d by dt equal to blah blah blah is that okay huge equation but it is not it is not scaring us because we know what is each term so now d e by dt now I will go ahead and substitute for sigma xx sigma yy sigma xy I will substitute these equations okay if I substitute that and call this whole term as phi okay it is called as viscous dissipation I will discuss the importance of that little later but if I substitute for all this for sigma xx sigma yy sigma xy which are there here and call that as phi that phi turns out to be this what is this it is all what is this it is all gradients what is del u by del x stretching del v by del y stretching in y direction so what is del u by del x sorry no that is right this is right this is also stretching what is this del u by del y angular deformation angular deformation okay and shear stress of course agreed so now if I write this now I know that internal energy h equal to e plus we know h equal to e plus pv we write left and right so instead of writing v specific volume I have written it as p by rho why am I doing this because once I get in terms of enthalpy I can write enthalpy in terms of cp into t then I can get my left hand side also in terms of temperature all over the place there is temperature that is the exercises I am doing so nothing great I am doing so dh by dt equal to if I take total derivative dE by dt plus dt by rho by dt so let me expand d of p by rho by dt equal to 1 by rho dp dt minus p by rho square d rho by dt I have just differentiate kept key constant and rho constant so what is this d rho by dt what is d rho by dt what was my continuity equation d rho by dt plus rho del dot v equal to 0 so if I have to get d rho by dt that d rho by dt equal to minus rho del dot v so that is what I have reminded you and written for d rho by dt as rho del dot v which is here 2D I am writing so del u by del x plus del v by del y that is equal to 1 upon rho dp by dt plus p by rho blah blah so now let me substitute this here so what do I get what do I get what am I done can anyone explain me what I have done here in the next step because I tend to go fast I will catch someone can you tell me what did I do in the next step what am I trying to do in the first step I figured out what is dp by rho by dt why did I figure out dp by rho by dt what did I do I did what did I tell in the previous equation let us go slow then what did we write in the previous equation what did we say I am finding e in the left hand side I want to replace this e as h minus p by rho is that right but I have to take a total derivative of h minus p by rho first I have tried to find out the total derivative of p by rho that is what I get I am sure all of you have got this but now what am I doing in the next step I have done nothing I have written I have written the same equation which was there here rewritten the same equation I have done nothing now my next step would be for d e by dt I will write dh by dt minus d of p by rho by dt so that is dh by dt but minus dp by rho by dt becomes plus if I push it to right hand side and that is nothing but 1 by rho dp by dt plus p by rho del dot p. So this is the conduction term this is this term and this is the viscous dissipation term that is 5 by rho because instead of writing this big thing all the time I have written 5 by rho plus this 1 by rho dp by dt plus p by rho del u by del x plus del v by del y. So what do you see here this term and this term gets cancelled what am I left out with now it looks little small but the trick is I have written 5 that is why it is looking small but still 5 is a huge equation. So dh by dt is equal to k by rho del square t plus 1 by rho dp by dt plus 5 by rho we have not achieved what we are supposed to achieve what we are supposed to achieve now write for h cp t so if I take cp t and take cp is constant so I get rho cp dt dt rho cp dt dt plus k del square t plus dp by dt plus 5. Let us write this equation come on at least final equation we are supposed to write so what do I get rho cp rho cp dt dt is equal to k into del square t by del x square plus del square t by del y square plus dp by dt plus 5 okay plus 5 what will be let us expand the first term on the left hand side rho cp into can anyone expand this we have done this n number of times yesterday yeah del t by del small t plus u into del t by del x very good plus v into by del y actually plus w by w del t by del z but I am not taking w so that is why that is not there is equal to k into del square t by del x square plus del square t by del y square plus I would like to keep dp by dt as it is dp by dt plus 5 what is okay now let us take each one let us take each one what is the left hand side term what is the left hand side term convective term it is a material derivative of my temperature so this is the temperature variation for unsteady that is transient temperature distribution this is unsteady and this is the temperature variation by virtue of by virtue of velocity that is why it is called as convection term that is why it is called as convection term okay what is this diffusion or what we call as conduction okay so usually in all the equations in the undergraduate level for the energy equation we would not be writing these two terms in next couple of slides I will be saying why okay for now let us call this as this is called as pressure term people also call this as noise term noise term next is viscous dissipation term viscous dissipation term why is it called viscous dissipation if you go back what is it consisting of all velocity gradients followed by viscosity so it is the viscous dissipation these two terms are important only for high speed flows I will show you through non-dimensionalization why it is so you have these two terms we would take so I think we are now clear of each term now let us non-dimensionalize yes now we will use principle of similarity what is principle of similarity this is my energy equation this is my energy equation I am going to non-dimensionalize every length term it can be x or y by some characteristic length l okay and u star by some characteristic velocity if it is internal flow it will be average if it is external flow it will be u infinity okay and pressure term by rho u square rho v square and t by t minus ts upon t infinity minus t if I non-dimensionalize this each term each term okay so if I non-dimensionalize this each term I will end up I will not go through the algebra okay it is too lengthy it is just substituting that is to tell you what will I get for let us say now let us say if I take one of the terms rho cp u del t by del x let us do now all of you do now so that you know when you go back home how to do it so rho cp there is its rho cp only what should I replace u with u star equal to u by v so u will be getting replaced by u star into t so similarly t star was t minus ts upon t infinity minus ts so what will be del t by del x it will be del t star into t infinity minus ts upon del x star upon l is that okay so like that I have done for like that I have done for each and every term it is not okay some problem is the mungesh upon l why did I get x star I have replaced x I have replaced by x star into l l is constant that is why I have taken it out is that okay yeah today you are in comfort zone you are asking everything I am happy about it okay so now if I non-dimensional this is what the non-dimensional parameter will look like let us write this equation let us write this come on if we non-dimensionalize what is that I get u star del t star by del x star plus v star del t star by del y star equal to 1 upon r e p r into del squared t by del squared t by what do you expect del x star squared plus del squared t by t star you are right del squared t star by del del y star squared plus e c which is called as occurred number occurred number many of you have used the book occurred and Drake I saw in your syllabus that occurred only is this occurred okay so occurred into u star del p star by del x star plus v star del p star by del y star plus of plus of phi star into occurred by Reynolds number okay what is occurred number let us write each term what is Reynolds number rho v l by mu we know this what is parental number I am just reminding mu c p by k I would not like to remember like this rather I would like to remember as mu by alpha for reasons very obvious now after professor around spending so much time on importance of Prandtl number and occurred number is v squared upon c p into t infinity minus t s okay now now let us stop here what is the left hand side term we said in this equation what does it represent what does it represent sir what does the left hand side of this equation represent it is pressure term convection term what it is it is convection now what is the second term representing that is this term representing this is diffusion or the conduction but what are the non-dimensional numbers associated with the conduction term Reynolds number and now that means if I vary Reynolds number and parental number this term also becomes important okay these two terms now next is what is this occurred number what is this term by the way pressure term this is the pressure term and this is the viscous dissipation term this is also having occurred what is occurred number occurred number is v squared upon c p into t infinity minus t s now let us calculate occurred number for a velocity of 10 meters per second if the velocity is 10 meters per second and c p of air is 1050 okay and typically our temperature differences are of the order of when I am heating or cooling let us say where of the order of 10 degree Celsius t infinity and t s what is occurred number reducing to 10 squared upon 1050 into 10 it will be of the order of 0.01 or 10 to the power of minus 3 yeah 1000 upon 100 upon 1000 into 10 let me take like that 1050 let me make it as 1000 so what do I get not 10 to the power of minus 3 it is 10 to the power of minus 2 so what does it what does it mean this is just to show what does this mean because I said that I we usually do not take these two terms what one is the pressure term another one is the viscous dissipation these two terms in the energy energy equation we do not take these two terms why can now anyone answer this question very small for low speed flows subsonic flows incompressible flows it becomes only important in compressible flows that is high speed flow you take a Mach number of let us say 1 or 2 if you take a Mach number of 2 what will be the velocity Mach number 2 means velocity of sound in air is 330 meters per second roughly 330 into 2 is 660 okay let us take 1000 let us take 1000 velocity because Mach 4 is not at all a wrong Mach because when when when my special is landing back what is its velocity what will be Mach 7 Mach 8 it easily goes so 1000 is not an number thrown out of the world it is very much applicable to the real life world for high speed flows if I take 1000 square 1000 square divided by 1000 in fact T infinity minus T s also will be very high okay it will be very high as high as 500 to 600 okay you can compute how can you compute what will be the temperature you can compute no t0 equal to or h0 equal to h plus v squared by 2 so t0 equal to t plus v square divided by 2 cp okay so anyway I have just got digress the point is the well the temperature will be very high and this v square divided by cp into T infinity minus T s will be significantly low I cannot afford to neglect any more these two terms which is the pressure term why it is called noise term perhaps now you can understand why it is called noise term because it creates noise and noise is created at high speed okay flame you are taking a blowing torch sorry flame torch oxy acetylene welding torch let us say it is a very high speed velocity so their noise term is very important for all incompressible flows this term is not these two terms that is the pressure term and the viscous dissipation term are not important I hope I have answered your question v squared by cp delta t where does it come from kinetic energy okay what what is denominator heat supplied coming here actually I have given the I am showing the equation so that you can get the clue who is generating that temperature when it is coming that that high speed flows high speed flows the kinetic energy gets transformed into thermal energy thermal energy okay I take this example all the time Kalpana Chawla died because that one of the tiles flew off and the because of thermal energy or the friction the wall temperatures shot up and it caught fire okay so that is that is this so it is how much of this kinetic energy is getting transformed into thermal energy is being represented by that is the that is the physical significance of occurred yeah one we said in the numerator kinetic energy in the denominator that is thermal energy now you can know dissipation of heat to kinetic energy right smaller so v squared by 2 is smaller compared to cp delta t frictional heating maybe we can say frictional heating is so that is how we need to understand this it becomes very important if you are working in high speed force these two terms are they will create completely they will change things completely in fact h definition also will change let us not get into that so that is that is the importance of the energy equation I think I think we have understood this yes yes and we are adding the macro size dimensions but if suppose we are working with the micro and mini channels type of dimensions should we ignore these two terms their rarefaction comes into picture their rarefaction yes there we have to there we have to include this even if it is less velocity less velocity because why I have to include there when I say rarefaction what does it mean density variations come come into picture density variations last two terms are coming because of what let me let me clarify let me clarify I am clear now how to reach you this pressure term and the viscous dissipation term came into picture in high speed force high speed force it is very clear right in high speed force what is the character of high speed force high speed is gained because of lot of compressibility that is because of density variation again in rarefaction what is rarefaction I am only working at low pressures but they are also again my density is varying that itself suggests that these two terms are it depends on the fluid it depends on the fluid which fluid you are handling if it is water if it is water these two terms are important no if it is a rarefaction will come into picture in micro channel no sir when we are defining the nodes and number that is the value is less point correct so in the Rayleigh-Fakson effect will not come into picture no I understand what you are trying to say is then whether we are in slip flow or again in if the Knudsen number is very high of course we are not reaching everyone I am trying to reach only this guy but that is okay because he has asked me so Knudsen number is high only rarefaction comes into picture is that right so if Knudsen number if I am operating for high Knudsen number high means how high typically 0.15, 0.2, 0.3 for that case these two terms are indeed important I cannot really get them neglect them but as you said if it is air only Knudsen number comes into picture if I am operating water in a micro channel are density variations there are density variations there no then these two terms are not important so I cannot generalize in one shot and say that for micro channel these two are important it is specific to the oh yeah their velocities are quite less that is the way mass flow rates are of the order of centimeter cube per minutes Mangesh knows well because incidentally one of our student works on earlier he was working on rarefied gas so that is how I am little familiar with that okay so now let us come back to our we will try to non-dimensionalize we non-dimensionalize only the energy equation what are the non-dimensional numbers we arrived at we will forget here afterwards high speed flows we got Reynolds number and Prandtl number similar non-dimensionalization if I do for continuity and momentum equation how did I get this momentum equation how did I get this momentum equation quickly I have not I have not written all the terms if you carefully see there continuity this is steady let me remind you steady two-dimensional incompressible flow let us write those terms steady no I just want to get it registered in their minds steady incompressible two-dimensional flows steady incompressible two-dimensional flows what will be my continuity equation del u by del x plus del v by del y equal to 0 what is my x momentum equation here it is there for those guys who do not remember u steady del u by del t is not there u del u by del x plus v del u by del y equal to minus of 1 by rho or maybe I can okay minus 1 by rho del p by del x plus nu into del squared u by del x squared plus del squared u by del y squared let us stop for a minute here what is what is the scale of u to what scale you would be let me take flow over a flat plate let me take a flat plate and let me take flow over a flat plate which is having a velocity u infinity what will be the scale of u u is having a scale of x direction velocity scale means it is of the order of not exactly two meters it will be of the order of 1.5 1.6 so u is of the order of u infinity what is the order of x the length is l so u infinity upon l is the scale of del u by now these two terms should be there means what will be the scale of v I do not know the scale of v let the scale of v be v only what will be the scale of y what is the scale of y boundary layer thickness delta so what is the scale of v now u infinity delta by okay now let us see what is this let us put the scales of each term scale can anyone okay what am I trying to say we know that u is very high we will be higher or smaller than u v velocity in the way it will be small all of us register the question is how small is what we are asking so it is very easy to see that u is of the order of u infinity v I do not know how small it is how small it is definitely I can see the scale of x and y x is of the order of length of the plate and y is of the order of boundary layer thickness so now v is of the order of it is smaller by delta by l compared to u infinity each so now let us see the term of each scale of each term what will be the scale of left hand first term u infinity into del u what is the scale of that u infinity by del x scale will be l so what about that means I get u infinity squared by l now what is the scale of the next term what is the scale of v u infinity delta by l and what is the scale of u u infinity by delta so delta delta cancels out and I get what is that I get u infinity squared by l what does this mean both are coming u infinity squared by what does this mean both are of the same order I cannot afford to throw anyone both are important both are important now del p by del x del p by del x will let us keep it as it is let us not worry about that okay now what about del p by del x I do not know the scale so I am not bothered about what is the scale of del squared u by del x squared nu into what is the scale of del squared u by del x squared u infinity upon x squared is what about next term nu u infinity by delta squared so are these two how do you compare both of them second term can be second term is very large because delta is very small compared to so I can neglect this is that okay that is what precisely is being done in the this equation I am going to digress little bit and I will get to show you how to get to delta in terms of Reynolds number from this equation okay maybe I will do that when we do the external flows because for that I need to throw del p by del x I am not told fine what happens to the y momentum equation you have not written that I have not written that is right I have not written here y momentum equation x direction v is very small compared to u so it would not have that much weightage compared to x momentum equation so that is why y momentum equation is completely thrown off is that okay we will come to the pressure gradient when we handle external flows for now this is enough okay and consciously I have not put that you are right what you said this right okay so now these are the boundary conditions there are no sleep boundary conditions and constant wall temperature I have taken that is okay now if I non-dimensionalize them the same way what we did here it is easy to non-dimensionalize because the terms are compact so what do I see in the momentum equation Reynolds number what do I see in the energy equation Reynolds and Prandtl what does this mean I will ask you one I will ask you one thing I this time only I got the trick how to reach that Moody's chart what is that you plot in Moody's chart I am coming off but it is fine in Moody's chart what what versus what you plot friction factor versus Reynolds number why do not I plot and why does in Prandtl number coming there why does in Prandtl do I friction factor is plotted against Reynolds number why is it plotted against Reynolds number if I do measurements with Reynolds number with water and if I am doing measurements with water I can take the f for given Re and get the pressure drop let us say I have done that experiment and get the friction do I have to again do it with air or if I am doing again with oil should I be doing again with oil that is I am asking again with the Moody's chart f and Re is only there Pr is not there question is why trick is answer is here answer is here no one I have not reached anyone I am I will wait for some time I will wait for some time what is non-dimensionalized momentum equation comprising of Reynolds number so friction factor is coming from where no no no no no no no no no let us go slow in order to go in the morning in the morning we wrote what did we write in the morning in the morning what did we write I do not want to hurry up what is velocity profile no I have written in the morning I have written from velocity profiles I get gradients and from that I get shear stress and from that I get my pumping so how do I get this velocity gradients by solving my mass and momentum equation so what is the non-dimensional number sitting in my mass and momentum equation Reynolds number that means my friction factor has to be dependent only on Reynolds that is precisely the Moody's chart is plotting f versus Re but then you have to answer me the question should I again can I use that for oil can I use that for air can I use that for water yes it is independent of Prandtl number it is independent of Prandtl number that is what my non-dimensionalization is teaching me okay that is why we do not have Moody's chart for oil again water again air again we do not have that we do not have that even if you want to find out pressure drop pressure drop for any unknown configuration unknown configuration you do it with the simplest possible fluid or blower whatever facilities you have you do with that fluid that is valid for all that is coming from my non-dimensionalized parameter but is it true for is heat transfer coefficient independent so what is my energy equation telling temperature distribution is dependent on Reynolds and Francis that means my H is dependent on Reynolds and so what heat transfer coefficient I measure for air as a fluid is not valid for oil or water if my fluid is changing I have to sit down and do my measurements again that is what this non-dimensionalization is so is that okay did I reach you this is very very important for this work only professor Nusselt got his professorship which is what is embedded in oil textbooks and we read it I think we need to spend ample time with students on this concept on this concept because this is what is very important when they are planning their experiments even if you are running fluid it does not make sense again to calculate pressure drop if you generated pressure drop for air again to do it with water until I have this concept embedded in my mind so this dimensional similarity is going to tell me when to use which dimensionless is that okay any questions I can take I can stop and take because this is important okay fine yeah yeah I am not going to take any question on this filter I am going to spend ample time on internal flows I am no I am today very disciplined but that does not mean that I am not going to take questions but I am going to take questions but I am going to time it so that my effort to reach you is going to decrease okay that is my point okay so now now what am I doing is this is the non-dimensional same thing whatever I told I am telling that if I am doing for two different configurations whether water or air all that I need to do is Reynolds number I have to keep constant and Prandtl number I have to keep constant for temperature and for velocity I need to keep only the Reynolds number same that is all so now there is another thing what is called as CF so maybe maybe we can skip this okay here I think I cannot afford to skip because CF definition comes here and actually Nusselt number definition comes here okay so non-dimensionalization non-dimensional u star is a function of what see this equation u star is a function of x star y star and Reynolds number yes if I take dp by dx I am not taking dp by dx I am not taking because if it is a flow over a flat plate pressure gradient will be there only in internal flow and pressure gradient will not be there in external flows I will answer why when we reach external flows okay why dp by dx is 0 for external flows it is very easy to show we will show that but if dp by dx is not there then my velocity profile u star is a function of x star y star and REL what is shear stress mu del u by del y at y equal to 0 if I non-dimensionalize this for u and y u star and y star what do I get mu v by L so what is this del u star by del y star from this I can write that it is a function of x star and REL why y star is not there because I am taking it as y star equal to 0 only so you have mu v by L F2 of x star REL now if I non-dimensionalize this C of x equal to tau all upon rho v squared by 2 if I substitute this what do I get here 2 upon rho v L in the numerator mu so that is 2 by Reynolds that means what what is that I get 2 by Reynolds number which is a function of x star and Reynolds that means what C of x is a function of location and this is the inference we did intuitively already I told the Moody's chart example this is Moody's chart C of x that is for internal flow where there is nothing like x star but for flow over a flat plate from location to location my friction factor local is going to change is that understood similarly let us do for temperature T star equal to G x star y star REL PR H is what pressure is defined as K del T del y if I non-dimensionalize that I am going to get K by L del T star by del y star at y star equal to 0 so what is Nusselt number H L by K so if I substitute that here I am going to get del T star by del y star at y star equal to 0 what is this function of what is this function of T star is a function of go back go back to the equation do not have to hurry up T star is a function of Reynolds number Prandtl number and x star y y star is not there again because I am taking del T star by del y star at y star equal to 0 so what does this represent Nusselt number is a function of Reynolds number and location is that understood now so now I guess we know how to utilize non-dimensionalization so that is about the power of non-dimensionalization parameters that is how actually now after understanding this correlation flow how do I know that we have plethora of correlation hundreds of correlation all the correlations are of the form of constant into r e to the power of x p r to the power of some N why do I know that how do I know that I should be taking it that way because in the previous non-dimensionalization I have come to know that my Nusselt is a function of Reynolds and Prandtl we have to emphasize our students that this is what it is that is why we write correlations in this form Dittus-Bolter pressure set that is 0.023 r e to the power of 0.8 p r to the power of either 0.3 or 0.4 depending on heating or cooling how on earth Dittus-Bolter knew that he has to take that form he knew it because it came from Nusselt non-dimensionalization had Nusselt not told that Dittus and Bolter could not could not have come up with that correlation okay yes Mangesh. Sir, can we start the students about this correlation that it is coming from the Buckingham spy theorem. No I would say Buckingham spy theorem is little difficult Buckingham spy theorem I would say it is difficult because it is based on intuition how do you say that my T star is dependent on mu, Cp and K it is see that was hydrodynamics that was hydrodynamics when during time of hydrodynamics when Buckingham's and Reynolds did the analysis their friction factor and Reynolds number when they drew that Nicaragic did those experiments he had no idea of Navier-Stokes equation that was indeed done by the basis of your Buckingham spy theorem only whatever you told that was hydrodynamics that was not called as fluid dynamics after deriving Navier-Stokes equation and after deriving energy equation after Nusselt showed the dimensional similarity only our correlations have flowed in. So I would recommend that we should not show it through Buckingham spy theorem but we should show it through Nusselt's dimensional similarity because it is obvious no no I do not have to make any assumptions to take which parameter and which not parameter into my pitch it is quite difficult for energy equation why because historically it is like that historically it has come like this that is the reason. But sir the methodology is same in both the cases. Yeah but there there is into you know the answer that is why you are able to do it there but here I do not know the answer still I am going to arrive at the same answer that that is the difference between the two. See how to teach a student when you are taking Buckingham's theorem on the right hand side these parameters only I should take if you if he asks you let me ask you why I cannot take K for friction factor in my friction factor dimensional analysis if you ask you why K and Cp are not taken how will you answer it it is not possible with this you can answer that is the power of dimension okay. So I would historically it came we studied it is fine but that is not the approach we are supposed to follow to teach the students it is through writing differential equations and writing and non-dimensionalizing them you attempt this for any of your PhD problem it will work for you it will work for you that is what is called as formulation no any problem first write them in the differential form dimension non-dimensionalize them you get the dimension any problem you take any problem flow through a pipe or droplet spray you take two phase flow you take any problem first thing is write the differential equations non-dimensionalize them you get the dimensionless number then you can plan your experiments to arrive at this type of correlation you change the Reynolds numbers you change the parameter then it is mass production but the thought process is this so there is another interesting thing very very interesting thing but very simple very simple okay now I have what I have done is I have thrown the pressure term if you just see first two terms what is that I have done what is the first equation what is the first equation momentum equation and I have thrown del p by del x because this is valid for a flat place okay again I am postponing why del p by del x equal to 0 pressure is going to answer that next the energy equation in the energy equation I have made Prandtl number equal to 1 actually here R EL into PR was there so I will make PR equal to 1 do you see anything anything magical is there between these two equations those two look if you watch carefully even boundary conditions also will look so the solution of one should be the solution of the other that is what is called as Reynolds analogy so tau s equal to maybe so if you just see that the same thing I am not going to harp on that CFX is a function of extra and R EL Nusselt is a function of extra R EL now PR if it is made one these two functions are going to be same. So CFX into R EL by 2 is equal to so if I write NU by R E into PR NU by R E into PR PR is not there PR is 1 so that is NU by R E into PR is what stanton number so stanton number becomes equal to CFX by 2 what is the beauty of this equation what is the beauty of this equation if you measure CFX you can get the heat transfer distribution in fact I came across a wonderful application of this concept by one of my colleague Professor Menanges he uses reverse for compressible flow over a flat plate in shock tube he measures the heat transfer coefficient and he calculates the frictional drag by stanton from stanton number to CFX and it works for compressible flow also it works okay so it is a very very strong equation please do not relegate this as only mathematical or only bookish it works in real life it works in real life okay for flow over a flat plate whether it is compressible or incompressible it works because measuring pressures fast is difficult but measuring temperatures fast is easy for compressible flow so that is why he measures heat transfer coefficient the point why I took that example is that it is strong it is strong even in pipes even in sub cooled flow we have seen that by and large it works of course with some variations so with this I think I will stop maybe I will not handle this cold burn analogy I will leave it like that because for the positive of time I think I will stop here you can go through this yourself for cold burn analogy.