 What is internal flow? I do not think I need to deal with this very much in internal flow, in internal flow I have the pipe the flow going through a pipe. The pipe can be circular pipes, square pipe, rectangular pipe, reposadal pipe or any shape which you can think of. Now the important concept which we need to understand in case of non-circular pipes is hydraulic diameter. Hydraulic diameter is 4 into wetted area upon perimeter that is what is stated here. Why do I multiply this 4 here? Because for circular pipe if I put area I get pi d squared by 4 upon perimeter wetted perimeter is pi d. So pi pi gets cancelled d d gets cancelled I get d by 4. To make this d by 4 d I multiply that with 4. Because for circular pipe at least I should be getting the hydraulic diameter as back as pipe diameter that is the reason we multiply by area by 4. And now you can go on doing this for square duct and rectangular duct. So thought for food for thought what will be the hydraulic diameter for flow between two parallel plates. I will take up this but I want you to keep thinking on that. Now coming let us come to mean velocity. There are two concepts which we need to understand that is mean velocity and bulk mean temperature. So in the mean velocity what is there in the mean velocity? We have what is mean velocity? Mean velocity is nothing but the average velocity. How do I get the average velocity? Most of the times you remember for getting the average velocity we measure the mass flow rate. That is if the flow is flowing in a pipe let us say in a tap water you just take it below the you take you collect that in a 1 liter box. You collect it for whatever time it takes for collecting 1 liter you note down the time and you have taken 1 liter you know the density. So you can calculate the mass flow rate. If you divide it by density and area you are going to get the velocity. What is this velocity? It is nothing but the average velocity. But is the velocity going to be same? Is the velocity going to be the velocity profile going to be same everywhere? No that is velocity profile is going to be parabolic if it is laminar and cap profile if it is going to be turbulent. Let me just draw it for the heck of it because I have not shown you so far internal flows. In internal flows of course we are going to derive this in today little later. We are going to get if the this is the flow we are going to get laminar if it is laminar we are going to get this the parabolic profile. This is for parabolic profile is for laminar flow parabolic if it is turbulent top hat profile I am going to get if it is turbulent I am going to get top hat profile. Let me see I think in the subsequent slides I have this let me first state those basics and then come back. So I need to give you some basics and then come back because we need to understand what are the basics of fully developed flow developing flow and things like that. So we know what is laminar, what is transitional, what is turbulent this is what we have studied in the last to last class laminar is goes in laminar and in transitional sometimes it is laminar and sometimes it is turbulent for turbulent velocity fluctuating with time and that is what is being shown here in turbulent flow. Now this was first found by Osborne Reynolds in 1883. So it is few hundred at least hundred years back. Now there is another important concept what is called as developing flow and fully developed flow. What is developing flow? In developing flow the boundary layer is growing the boundary layer is growing and once the boundary layer reaches the center of the pipe then subsequent to that the velocity profile does not change with stream wise direction or the direction of the flow till here you see it was uniform profile which is does entered here you see here you see at the center it is bulged and at the walls it is lesser why because this is uniform the velocity has decreased near the walls why as it decreased because the boundary layer now it has to compensate. So to compensate for that near the center it has to get bulged. So now that bulging goes on increasing as we move on because my velocity profile is becoming flatter and flatter and then it is going to become sorry it is going to not flatter and flatter it is going towards parabolic from the flat profile. So we get a parabolic profile once my boundary layer reaches the center line of the pipe. So from this the location at which it reaches the center line center location center of the pipe is that length is the called as the hydrodynamic entrance length or developing length. This is what I am called talking for velocity not for temperature this is hydrodynamic developing length. Similarly we will define what is called as thermally developing length this is hydrodynamically developing length. It has been found that L by D that is the hydrodynamic developing length is around 0.06 times the Reynolds number for laminar flow and for turbulent flow it is 4.4 or e to the power of 1 by 6 for turbulent flow because there is one question always my student ask always few students ask is ask us that is will I have developing flow for turbulent flow also or developing flow concept is only for laminar flow. Developing flow concept is there for both laminar and turbulent there is nothing specific for laminar or turbulent. So for laminar flow in fact it takes longer developing length compared to the turbulent flow. For turbulent flow general thumb rule is that you take 25 times the pipe diameter generally we get the fully developed pipe flow fully developed velocity profile that is the top hat profile for turbulent flow. But for laminar flow if the if the Reynolds number is around 2000 you can see that the developing length is quite large 120 times the diameter of the pipe. So it is very large as opposed to turbulent flow. So that is developing flow concept. Another important thing is that the pressure drop the pressure if I were to put the pressure tabs on the walls of this pipe and plot that pressure distribution that is the static pressure distribution. What is static pressure distribution static pressure is that pressure which is experienced by the fluid when I see is the pressure experienced by me when I sit on the fluid and I am travelling along with it. That means there is no dynamic component whatever molecular collision is happening on the fluid particle while it is moving that is static pressure. So that static pressure if I were to plot with length of the pipe you can see that there is a non-linear pressure drop in the entrance flow that is in the developing flow. Later on the pressure drop is linear. So this is what is we that is how we identify whether it is fully developed flow or developing flow very easily even not not even without measuring the velocity profile. So that is what is being shown here for turbulent flow naturally we have said that it picks up and then you have all sorts of initially laminar then transitional and then turbulent this is velocity with time or Reynolds numbers with time. So I think there is nothing new the point here I want to tell again here is that I want to define what is called as hydro dynamically smooth pipe. I know I am going to fluid mechanics back again because whatever fluid mechanics required for internal flow pipes I am trying to cover here. So what is hydraulically smooth pipe this is the term most of us would have come across what is hydraulically smooth pipe hydraulically smooth pipe is that pipe in which the surface roughness in which the surface roughness is lesser than the laminar sublayer thickness. If the surface roughness is less than the laminar sublayer thickness then we will be then it is called hydraulically smooth why because if it is hydro when I say hydraulically smooth the flow just smoothly goes over it there is no boundary layer getting disturbed because the surface roughness is lesser than the laminar sublayer thickness my boundary layer structure do not get disturbed because of this surface roughness only when my by my the surface roughness is greater than if the surface roughness is greater than the if the surface roughness is greater than laminar sublayer thickness then it is going to be what is it called what is it called hydraulically rough pipe. Remember this is not geometrically rough hydraulically smooth pipe is actually geometrically rough but it is not hydraulically smooth but here if the my geometrical roughness is greater than laminar sublayer thickness then it is called as hydraulically rough pipe that means it is breaking the laminar sublayer now I am going to have only the turbulent boundary layer or may be the buffer layer. So with this that is what again we come back and it helps us in understanding this hydraulically smooth pipe and rough pipe. So this was what was taught in the last class so now if I take I have not yet given the velocity profile so we will come to this velocity profile and the friction factor derivation little later with these fundamentals with these fundamentals let us go to the concept of mean velocity the concept of mean velocity. So here in the mean velocity what is that we are doing I am not going to write this so this is m dot equal to rho A v so if I take the integration area integral rho v r x d A c why r comma x I have taken because velocity profile can change with stream wise location and also with radius this definition is not specific to only for fully developed flow even in developing flow I can have the mean velocity. So average velocity equal to 1 upon rho A c in the numerator integral of rho v r comma x d A c so if I take a small elemental ring that ring diameter is 2 pi r and ring radius is d r so I have rho v r comma x 2 pi r d r integrated within the limits is 0 to r then what will I get so I get 2 pi pi gets cancelled out to rho rho if I take the rho outside that is rho is not varying with the location if I make that assumption if I make that assumption if I take the rho outside I get 2 by r square 0 to r v r comma x r d r that is for incompressible flow the very fact that I am taking rho outside means it is incompressible flow density is not varying with space that is the assumption I have made so this definition is for incompressible flow if it is for the compressible flow the density variation also I have to take into account within the integral. So I get 2 by r square 0 to r v r comma x r d r so that is what I get the average velocity so now let me go ahead now that I have defined mean velocity let me go for fundamentals of fluid dynamics and then come back to heat transfer. So if I take flow in a circular pipe this is a circular pipe I have v z r and theta direction r theta z direction this is polar coordinate or cylindrical coordinate it is varying with radius at a given radius it is varying with theta and this r and theta can be different at different z's that is the stream wise location instead of taking x I have taken it as z now what are the continuity equation del v r by del r plus 1 by r del v theta by del theta plus del v z by del z plus v r by r equal to 0. Now v theta is not there it is viscous incompressible steady flow I have for steady flow this is the equation what I get for v r and v theta for a fully developed flow if I take fully developed flow v theta is 0 v theta and v r are 0 only v z is varying with radius that is the stream wise velocity is varying with radius v r and v theta is 0. So v r and v theta is 0 v z is only a function of r so continuity equation in the continuity equation what will happen del v z by del z is equal to 0. So I get only del v r and del v theta by del theta is also not there so then because it is symmetric and then in this equation what will happen for this equation this equation is not there because v r is not there and this equation is not there because v theta is not there that is momentum equation for v r and momentum equation for v theta is not there then I have to take the momentum equation for only v z and pressure is only a function of z not r and theta because there is no theta momentum equation and r momentum equation. So if this is my equation so I get v r del v z by del r plus v theta by r v theta is not there so this term is vanishing and what about del v z by del r it is symmetric so this is 0 so this is not there and del v z by del z if I take fully developed flow velocity is not varying with stream wise direction so this term is not there and v z is not a function of theta so if I with this thing so full left hand side is 0 I get minus 1 by rho del p by del z plus nu of all these terms in this again v z is not a function of theta and v z is not a function of z v z only is a function of r that is the definition of fully developed flow. So if I integrate this is the equation I get if I integrate this and apply boundary conditions that is at r equal to 0 d v z by d r equal to 0 that means it is symmetric and at r equal to r v z equal to 0 this is no slip boundary condition if I substitute these no slip boundary conditions I am going to get the velocity profile like this which is nothing but paraboloid that is minus r square by 4 mu d p by d z still d p by d z is not known let us keep that if I take in this as z equal to that is r equal to 0 v z maximum that is at the center line velocity I am going to get this as minus r square by 4 mu d p by d z that is what is v z maximum if I substitute this here in this equation I get v z equal to v z maximum into 1 minus r by r whole square this is the parabolic velocity profile why did I come here because I wanted to show you how to compute the average velocity which is what I just defined 2 minutes back the average velocity is 1 by pi r square 0 to 2 pi 0 to r v z r d r d theta there is a small elemental ring that is if I integrate this d theta I get 2 pi r I had taken in one shot there 2 pi r I am showing here it is 2 pi r that is r d r you get 2 pi upon pi r square if you integrate this all through which you can do it yourself if you integrate this you are going to get that as v z average equal to v z maximum by 2 so this is the basic thing which is what we check when we do the velocity profile so then you can calculate the shear stress also tau equal to mu d v z by d r you are going to get that shear stress in terms of average velocity and if you reduce that in terms of Reynolds number you will be getting 16 by r e this is fluid mechanics this is not heat transfer that is why I have gone quite fast I think it is reasonable to go fast here I would expect you to work out all these steps we have not skipped even a single step plus and minus multiplication everything is there in transparencies 22 to 29 please sit down and calculate this you will realize every step is self explanatory if you get any doubt please put them across in moodle we will answer that that is C f equal to 16 by r e only one thing I need to tell here there is another thing which is defined what is called as friction factor skin one is skin friction factor another one is Darcy's friction factor Darcy's friction factor is four times the skin friction factor in most of the text books both are used as f only but we need to realize whenever you see 16 by r e it is 16 it is skin friction factor whenever you see 64 by r e it is Darcy's friction factor so this is what always confuses us we should not be getting confused yes so this is non dimensional C f is the non dimensional shear stress and f is the non dimensional pressure gradient so I think we have studied this from fundamentals of fluid mechanics now with this what I will do is so what I have covered so far is if I have to take a recap of what I have covered what did I do I did I defined what is internal flow and I said what is hydraulic diameter I defined hydraulic diameter and I said I defined what is the average mean velocity and then I went to the fundamentals of fluid mechanics that is I went to what is laminar turbulent transitional on a quick note and then defined what is called as hydro dynamically fully developed flow that is developing flow and fully developed flow and how do we differentiate or how do I identify them and what is the developing length given by we said and with that and I said what is I defined what is hydraulically smooth pipe and hydraulically rough pipe I defined and then came to the velocity profile in a circular pipe and took the navier stokes equations and reduced these navier stokes equations for fully developed flow and solute that velocity distribution a solute the navier stokes equation along with continuity equation substituted appropriate boundary conditions got the velocity profile and realized that it is parabolic in nature for laminar remember this is laminar flow and we realized that shear stress is given by our skin friction coefficient is given by 16 by R e and Darcy's friction factor is given by 64 by R e. So, with this we will move on to what is called as bulk fluid temperature or bulk mean temperature which is what professor Arun is going to teach us. So, good morning we will study now internal flows from a heat transfer perspective. What we mean by that is so far professor had introduced fluid mechanics aspects solved and obtained the velocity distribution average velocity friction factor coefficient of friction so on and so forth. And then also there was a definition which is very important from a point of view of average velocity which is nothing, but the number that we come across in most problems you know. So, actual velocity we know is going to vary with R with radius and z, but we are saying this so called average velocity is a constant which is defined like this. Now, if average velocity is defined like this what about so called average temperature. So, when we say water flows through a pipe with a velocity of 2 meter per second or mass flow rate of 0.2 kg per second and at temperature of 25 degree centigrade or let us say let us take a higher temperature of a temperature of say 80 degree centigrade. So, this value that you get is nothing, but what is called as an average bulk mean or text books will use this concept called as mixing cup temperature average bulk mean or mixing cup temperature. What does this mean see this thing is a little bit different from a fluid mechanics definition. So, how is it different let us see normally what do you do when you have a pipe and you are supplying heat all of us have done high school physics water enters at m dot flow rate and say 25 degree centigrade and say leaves at 60 degree centigrade this is a regular high school problem in physics. So, we will write m dot C p t out minus t in is the amount of energy that has been supplied to the water and this heat is coming in by means of say an electrical current. So, some heating your household water heater geyser that you have in the bathroom is essentially this kind of a device. So, you increase the flow rate the water becomes because the wattage associated with the geyser is fixed. So, this is the power rating that you get from the manufacturer. So, if you are talking of a bathroom for a house which has 4 people the rating will be very different from what you have for a say a hotel where there are about 100 rooms. So, we are talking of the rating which is fixed now in this case and as I vary the m dot my exit temperature is going to change according to the inlet temperature. So, that is why when you want hot water you reduce the flow rate because you cannot change the power. So, you reduce the flow rate and as you reduce the flow rate the water is going to get hotter and hotter. Now, conversely if you want cooler water you increase the flow keep the tap at as close to maximum operating condition as possible. Of course, in summer if you see the inlet temperature of water in your overhead tanks in the building water is already gotten hot by 10 o clock in the morning. So, if you are putting on the geyser and trying to get hot water even the regular tap water sometimes is hot enough. Therefore, if you turn on the geyser even at highest flow rate this T out will be slightly more than what you can bear. So, this T in T out what we give to students what we get in engineering application is what we call as average or bulk mean or mixing cup temperature. Now, let us understand this from fundamentals point of view see we all we always talked about this concept of boundary layer what does that mean before we go to boundary layer. Let us say if this is the pipe water is getting this surface is let us say a temperature 80 degree centigrade I am doing some by some mean I am keeping the surface temperature constant. So, I know that the fluid closest we are talking of a circular pipe. So, the fluid closest to the wall is going to be at about 80 let us say this is 79 next layer would be at 75 and then 60 and then 40 and say the center most is at 35 this is your room temperature water. Now, the same fluid the same fluid I am taking the same fluid and moving. So, this is called as Lagrangian frame of reference all of us have studied this in fluid mechanics Lagrangian frame of reference basically is you are standing and moving along with the fluid particle a passenger on the roof of a train or a passenger in the train is basically travelling at the speed of the train. So, that is Lagrangian frame of reference. So, everything you see is with respect to that particular speed. So, you are a particle in the flow field the person in the train is a particle the train compartment is the the moving train is the flow field. So, Lagrangian frame you travel along with the fluid particle. So, you are sitting Eulerian frame is you are standing on a platform and seeing the train pass by that is Eulerian frame because the control volume the region of interest is limited by your field of vision and you are seeing what is coming in and what is going out. So, now I am going to sit as a fluid particle and go from say Z 1 location to Z 2 location. Obviously, common sense tells me that this fluid is going to get heated because it has gotten heated from 25 to 60 somewhere. So, it is going to get heated as it moves. Now, if it gets heated definitely we know that no slip condition or no temperature gradient condition will impose that the first layer is at 80 that does not change, but if heat is coming in what is happening next layer of fluid which was at 79 will be at 79 these in inside layers which were at 75 and all will now start to increase in temperature. So, what I get is the layers which were which were cooler here in the center also will start to become hot. So, essentially what will happen is with time this fluid for this I have to explain thermal boundary layer let us just explain the sorry let us just explain that first. So, when a flow is coming into the flow field into the pipe with T with velocity u infinity and inlet temperature T infinity let us say this is T s is constant what happens we are talking of internal flow. So, the first time the fluid enters here this is z equal to 0 or books will call x equal to 0 I do not care whatever this is this is the flow direction this is my radial direction r and sometimes book will refer to as y where y is equal to r minus r and capital R is the radius of the pipe. Once the fluid enters just as you have a hydrodynamic boundary layer form the thermal boundary layer will also be formed what does that mean the layer of the fluid which is in contact with the heated surface will have the same temperature. The next layer will see a different temperature following and further and further you will see temperature which is close to the inlet temperature. Now, in fluid mechanics we had only two quantities u mean and u infinity correct the variables were like that. And in fact, let us use v or u let me just go back I do not want to change notations we have used v here. So, we will stick with v in this also. So, v mean or average velocity and u infinity. So, I have only two things to deal with and when we said fully developed hydrodynamic flow we said d v by d z is equal to 0. So, the velocity profile does not change with respect to the flow direction after the hydrodynamic entrance length is reached. I hope you are with me on this let us just go back here see this is the boundary layer velocity boundary layer which is formed. So, we all know that this is uniform velocity inside the boundary layer there is a gradient outside of the boundary layer in this part the flow remains at velocity u infinity is it u infinity or greater than u infinity it is going to be greater than u infinity in this portion central core because the fluid will get accelerated because the fluid has slowed down inside the boundary layer m dot has to remain constant whether you do m dot integral or m dot by this average value the mass flow rate is constant. So, because the flow has gotten slow in this part imagine a circular pipe. So, in closer to the walls the fluid has gotten slow. So, the fluid will accelerate in this part and after that slowly the central core will get slower and slower and finally, the boundary layers will meet at the center and beyond that point we call as a fully developed flow very nice very easy mathematical condition is d v by d x or d v by d z equal to 0. Now, what I am saying is in fluid in fluid mechanics things were good in heat transfer what is going to happen as I said a boundary layer would be formed this boundary layer is going to be formed because of a difference in the fluid and the surface temperature. So, in this part let us say T s is greater than T infinity in this part let us say the temperature was like this T s fixed. So, the temperature was T s in this part it is going to be lower than T s lower than T s lower than T s. What happens here based on the amount of heat that is put this temperature would basically the boundary layer will be formed or the fluid will feel the effect of this heat transfer based on its properties whether the boundary layer is this thick or this thick will be based on the fluid property which is over alpha essentially. So, what this tells me is in this part the temperature remains like this. Now, if I go further obviously, because the fluid is being heated what is going to happen we have a constraint of this surface temperature being fixed. So, the surface temperature is fixed let me go I will have to draw it again I will take a long pipe let me draw the boundary layer like this and let us say in this part this is the temperature distribution. Because this has been fixed at T subscript s at a next location you have transferred heat from the outside to the fluid this also has to be at T surface s and the temperature distribution will become a little bit more fat. And this fatness or this size of this profile is essentially a measure of the heat content or heat that has gone into the fluid now what happens is because the flow is an internal flow the boundary layers will tend to meet the point of the location where the boundary layers meet is going to be called as the thermal entry length. This thermal entrance length is essentially similar to your hydrodynamic entrance length entry region where the boundary layers have come together after that what is happening the entire region entire flow field has become viscous effects are important here in this portion this was the inviscid portion of the flow this was the viscous portion of the flow after the merger of the boundary layers this is in the hydrodynamically fully developed region the velocity profile is parabolic this is completely viscous flow. Now here what is happening this is being shown for a case where T s is smaller than T infinity that means the fluid is losing heat nevertheless the location of the thermal boundary layer is shown this is the thermal entrance length. And after that we have this region called as a so called thermally fully developed region I say this so called thermally fully developed because there is a problem associated with it. Now why all the story the story is being told because of 2 or 3 reasons first thing fluid mechanics we had u m and u infinity here I have T surface I have T surface I have T infinity which is the inlet temperature which I generally do not care too much about T local that means what at every location here here a b c the same a would have had a new temperature now a prime b prime will have a new temperature. So there is a local temperature which is a function of r comma x and unlike my a velocity where I could define a u m here I have to define the so called bulk mean temperature which is a representative temperature that you as an engineer has to give for application. So in heat exchangers all of us have studied oil enters at so and so temperature it is flowing through a pipe we know for a fact that the temperature closes to the wall at this location is going to be different and it is going to be different at every point. So this bulk mean temperature we are going to say what it is. So this bulk mean temperature is essentially obtained by energy balance what are we doing we take energy essentially is m dot c p T mean or bulk mean temperature this one that is equal to. So what is what is this mixing cup temperature is you take this fluid out of the pipe it has a temperature profile take it into a cup and stir it mix it and put in a thermometer whatever the thermometer measures is called as a mixing cup temperature that is the name mixing cup the fluid is well mixed and it has a uniform homogeneous temperature in from thermodynamic point of view we say that thermal equilibrium is there otherwise what is happening every molecule of the fluid which is at a higher temperature will give energy to the molecule at a lower temperature and therefore there will be some heat transfer between the fluid molecules. So this is the mixing cup temperature physically that is the mixing cup temperature now mathematically what do I do. So as professor has pointed out if you take a ring element if you take a ring element that ring element there is an associated mass flow rate. So this is d a there is an associated mass flow rate there is an associated temperature this is what is a local temperature now imagine the pipe if this ring is closer to the wall the local temperature is going to be higher in case of heating if this ring is closer to the center the local temperature is going to be lower. So this energy content of this fluid element remember we are talking of a fluid element. So this d m C p m C p t so I am going to write d m C p and temperature this when I integrate over the entire area that is going to be the same as the energy content of this fluid element and that is what is shown here e fluid e dot fluid energy content of the fluid element is m dot C p t m is nothing but integrated over the mass entire mass which essentially translates to integration over the area d m dot C p local temperature t and this when I write mathematically it is just going to be m dot C p t m is equal to integral rho d a is in u is going to be a local velocity d a is 2 pi r d r for the pipe C p t of r comma x. So I have a local velocity and I have a local temperature. So we are talking of product of two distributions which has to be integrated from r is equal to 0 to r is equal to capital R this m dot is nothing but rho pi r squared if I am talking of a circular pipe rho pi r squared u mean or v mean bulk mean velocity rho a v. So I can write therefore, next page therefore, from this equation I can write t mean or bulk mean temperature is essentially what integral 0 to capital R rho C p u of r comma x t of r comma x 2 pi r d r this divided by m dot C p coming from the left hand side which is a rho pi r squared u m and C p and if I am saying it is constant properties which is what we are going to assume this gets cancelled pi gets cancelled here cancelling pi here. So I am left with 2 by u m r squared integral 0 to r r u of r comma x t of r comma x d r this is your bulk mean temperature or mixing cup temperature or average temperature which you are going to give in your problems and see it looks so complicated because it involves a velocity distribution and let us think of it logically if the flow is going to be very very fast very high speed flows or let us take the example of your geyser at home when the flow is very slow velocity is small the fluid is going to take there is going to be time for the fluid to take the heat therefore the temperature is going to be higher. So this is where the velocity distribution comes in this is where the temperature distribution comes in. So fluid mechanics is completely involved in this part. So unless I know the velocity field I cannot get this so called bulk mean temperature and this bulk mean temperature is essentially this definition. Now do we have to mug up this definition absolutely no it can come from fundamentals which we have shown. So please understand that it is a coupling of the velocity and the temperature distribution that is what is shown here. So with this definition of bulk mean temperature let us move ahead. So this bulk mean temperature where is this going to be used I said in heat transfer we have lot of headache because I have T surface which can be a constant or varies with x if I have a constant heat flux condition heat flux is equal to constant condition this would imply that wall surface temperature is not a constant it is going to vary with x we have a local temperature which is r comma x and now we have this new concept which we like because it is more convenient to use that is bulk mean temperature and this we know is a function of x alone how do I know it is a function of x alone because it is obtained by a integration in the integration in the r direction. So because of this integration this r dependency of T is completely lost so I am going to get T m of x is going to be a function of x alone. Now mathematically we are telling physically realistically what we are saying if water comes in at 25 degree centigrade and the surface is being heated by electrical heating whatever be it we know that the there is going to be a local temperature difference the profile is going to be like this. But bulk mean temperature if I put a thermometer at this point versus a thermometer at this point they are going to measure this T 1 is going to be lower than T 2 bulk mean temperature is lower here as compared to this point so fluid is getting hotter obviously. So this is going to be only a function of the x direction because by definition the radial dependency is absorbed in the definition of the bulk mean temperature. But local temperature of course is going to be a function of r comma x and this is something which we have to emphasize to our students maybe students do not understand it or teachers do not understand but we have to emphasize this point that bulk mean temperature varies only with respect to the axial direction or the flow direction. Now what is this business of thermally fully developed so there is a little difficulty in understanding this from because what is fully developed hydrodynamically fully developed let us just go back to our drawing hydrodynamically fully developed we have understood that d v by I do not have it here so anyway I will write it again hydrodynamically fully developed we wrote this condition d v by d x is equal to 0 velocity does not change in the x direction. So if I take a picture or velocity distribution at this point when the flow is fully developed at some other location the profile is identical. Now let me argue from a thermal point of view I have a fluid the boundary layers have met my temperature distribution has come like this at this point x 1. Now let me I am supplying heat into this pipe this pipe length is not of concern to us let us say the pipe length is very long I do not care about the length so I am taking two locations x 1 and x 2 and I am going to argue something what we are going to argue let us see. So I am supplying heat the boundary layers have met velocity has become fully developed. So let us say Prandtl number is of the order 1 so both hydrodynamic and thermal boundary layers are of the same thickness which means they are going to meet roughly at the same point. So let us say this is your hydrodynamic entry length and your thermal entry length. Now from a point of view of heat transfer let us ask this question I am supplying heat this is my temperature distribution let us say the surface temperature was 70 degree centigrade and the center line temperature of the fluid was say 35 degree centigrade. Now if I add more heat logically I am since I am going to heat the fluid if heat fluxes constant the wall temperature will have to rise. So this temperature obviously at a different location is going to be let us say 75 degree centigrade and because the fluid has gotten heat hot the center line the center portion of the flow also would have got a hot. So this let us say has gone up to 40 degree centigrade. So what is happening to this profile this profile is essentially swelling it is swelling in the direction of flow and finally if the length is very very long you will come up with the temperature distribution which looks very very fat like this. What it tells me that because of heat transfer if I have q is equal to 0 all this business does not happen at all where hydrodynamics was very good because once the boundary layers have met fluid does not care what happened to the inlet or what is the boundary condition. Here I have a problem because I am supplying heat there at every different axial location for every different axial location there is a heat addition. So the temperature profile T of r comma x at this location is going to be different at this location. So now fully developed d v by d x is equal to 0 I could write now can I write d T by d x equal to 0 we have argued now that velocity profile becomes thicker and thicker. So d T by d x I cannot write that equal to 0 because local temperature changes along the flow direction. So let us not even worry about thermally fully developed well that is not the case thermally fully developed flow can be understood can be established by understanding this concept what it says is this as the flow progresses from here to here. If I define what are the variables what are the variables associated with the flow I have written this several times let us write it again here T s T local T bulk mean please remember that the free stream or T infinity is long forgotten that is no longer going to come into the picture. So I have three different temperatures that I have to deal with and T s of x T mean of x and T r comma x. So from this it looks that flow will never become thermally fully developed but we can say let us do this carefully if I write a temperature non-dimensional temperature which looks like this T T s of x minus T r comma x I will tell you what this is T s of x minus T m comma x if I write this temperature what is this this let us take the pipe again T s is the surface temperature T is the local temperature and T m is this bulk mean temperature which has the integrated effect of u and T in fully developed we are we are trying to define thermally fully developed flow situation. So what we are saying when can a flow be called as thermally fully developed that is after the boundary layers have met when the thermal boundary layers have met if I define this T s minus T local divided by T s minus T m this ratio has been found to be invariant with respect to x. So d by dx of let me just go back and show this to you again d by dx of T s minus T local divided by T s minus T m is equal to 0. What it means is that this temperature profile which we kept drawing repeatedly this temperature profile if I draw once more time once more here this is a profile which is like this this is another location where I am going to make it little bigger purposely and make it a little fatter. So T s 1 T s 2 this is a local temperature T s I mean sorry T of r comma x this is another T of r comma x there is a T bulk mean 1 T bulk mean 2 what it tells me at x 1 and x 2 if the flow is to be called as thermally fully developed this derivative of this non dimensional temperature difference. So T s 1 or T s 1 minus T local divided by T s 1 minus T m 1 is equal to T s 2 minus T r comma x divided by T s 2 minus T m 2 that means what this ratio does not change with respect to x. In other words this is mathematical physically what it tells me is that this thing is becoming fatter as it moves I am talking of heat going into the fluid into the fluid fluid is getting heated as this thing is getting expanded as it is getting bloated up it gets bloated up in such a way that this condition mathematical condition is satisfied what it means is that this surface temperature has increased this fluid temperature locally also would have increased and correspondingly the bulk mean temperature which is obtained by integration of this local temperature distribution that has also changed and the change is in such a way that this ratio does not this ratio derivative of this is equal to 0 or this ratio does not change with respect to the flow direction this is what I will call as a thermally fully developed condition. Now what it means is that the shape of this profile the shape of this profile is going to be such that the heat transfer characteristics will be such that this condition has to be satisfied. So, the non-dimensional temperature gradient this is not come from magic this is come because we in fluid mechanics we could write d by dx of V m is equal to 0 I had to deal only with V m I did not have to deal with any other quantity this is the average velocity, but here because this T m is dependent on local as well as the surface boundary condition which is the T surface which is not going to be a constant which in constant heat flux condition it will vary in constant wall temperature condition even though this is constant because heat is being transferred the fluid is going to get hotter or cold depending on the direction of flow. So, this represents mathematically the thermally fully developed condition there will be lot of questions on this we will take it later on. So, beyond this point I can write mathematically that this non-dimensional temperature profile would remain the same this non-dimensional temperature distribution let me call this is going to be invariant with respect to the axial flow direction. So, much for that. So, if now comes the mathematical aspects of in a thermally fully developed flow this derivative T s minus T divided by T s minus T m with respect to x is 0 we have written that mathematically this derivative is equal to 0 what does it mean if a derivative is 0 that means this quantity is not a function of x is independent of x obviously it is going to expand in such a way that this non-dimensional ratio is going to have the same value at different x locations. If this is this whole thing is not a function of x then obviously its derivative with respect to the other flow direction r must not be a function of x what am I saying if D by D x of T s of x minus T of r comma x divided by T s of x minus T mean comma x is equal to 0 means this bracket T s of x minus T of r comma x divided by T s of x minus T m of x is not a function of x. If this is not a function of x derivative of this quantity with respect to r D by D r of this is obviously not a function of x mathematics you can take any function and see it is going to be independent of the other variable. So, because this is not a function of x why am I doing this let us just understand why am I doing this I am doing this for a very simple reason because my heat transfer coefficient is defined how minus k D T by D y at y equal to 0 divided by T s minus T infinity. Remember I had drawn the pipe and I had changed the coordinate axis also this is I called as y and this is r and I said y is equal to r minus r. So, D y is equal to minus D r. So, my heat transfer coefficient therefore I can write as minus k D T by D y at y equal to 0 would become k D T by D r at r is equal to capital R divided by T s minus T infinity this is also this is exactly the same thing written in terms of the r variable not the y variable just by using this coordinate transform when y equal to 0 r is equal to capital R that is what I have used. So, I want this D T by D r when do I want this D T by D r I want this D T by D r for thermally fully developed condition if flow is not thermally fully developed I cannot do anything this is we are trying to get some kind of a logical solution for the heat transfer coefficient when the flow is thermally fully developed and thermally fully developed condition comes from this huge expression which says that this is not a function of x. So, this I want to take this derivative with respect to r of this quantity I cannot take directly of this quantity. So, I have to take for this quantity which represents the fully developed thermally fully developed condition. So, when I do the maths here I am going to get the denominator is essentially a function of x. So, let me write here denominator is only a function of x. So, it comes out it is not going to be involved in this differentiation this is going to be a function of x. So, derivative of this is equal to 0. So, I essentially have this quantity and what am I saying I am saying that 1 over T s of x minus T m of x times d by d r of the second term in the numerator. So, it is going to be minus d by d r of T of r comma x is not a function of correct how did we come up with this we argued in the last set of slides we said that because this is not a function of x the derivative of this quantity with respect to r is also not a function of x. So, I will write which means minus d by d r of T of r comma x I am going to stop writing this r comma x evaluated at r equal to capital R is not a function of x. And now in couple of steps I am going to come up with this definition of heat in Nusselt number minus k d T by T y at y equal to 0 this is nothing but k d T by d r at r equal to capital R this quantity I am going to recast here in the form of Nusselt number q is h times T s minus T m I will come up come to this derivative this thing q is therefore nothing but heat flux heat flux is minus k d T by d y at y equal to 0 it is nothing but k d T by d r at r equal to r. And this k d T by d r is nothing but this quantity I will substitute from here. So, I can write let me just go back here heat transfer coefficient is minus k d T by plus k d T by d r at r equal to r divided by T s minus T m why T m because we no longer need to use T infinity because in internal flow T infinity has no meaning T m takes over the same role as T infinity in external flow T infinity was a constant the driving heat transfer the heat transfer is driven by this temperature difference T s minus T infinity in internal flow because T m is going to change at every location heat transfer is going to be governed by this temperature difference. So, the representative temperature difference is what I am going to deal with. So, h times T s minus T m is equal to k d T by d r what does this tell me d T by d r I have shown is not a function of x. So, this quantity therefore, heat transfer coefficient k d T by d r by T s minus T m this quantity is not a function of x. So, in thermally fully developed region this local convective heat transfer coefficient is a constant is a constant means it is independent of the x direction. So, mathematically we have shown this h by definition is let me go back h by definition is k d T by d r at r equal to r divided by T s minus T m correct. So, this quantity we said is not a function of x. So, it is whatever location in the pipe that you are taking after the flow has become thermally fully developed this quantity is going to be a constant. Therefore, I will say this if this is a constant this also has to remain constant. So, h is independent of the axial direction in case of thermally fully developed situation. So, this is a major this has lot of implications actually in concepts I mean in practical applications heat transfer coefficient remains constant. So, if you have understood this we try to plot this quantity variation of friction factor and heat transfer coefficient in the flow direction. This is the pipe and you have thermal boundary layer we have drawn this for p r greater than 1 hydrodynamic boundary layer is slightly thicker than thermal boundary layer. This is the hydrodynamic entry length where the flow has become hydrodynamically fully developed this is the thermal fully developed entrance length after which the flow is thermally fully developed. We are showing the friction factor is constant in this part for laminar flow heat transfer coefficient is constant beyond this point for the thermally fully developed. So, what and the of course, these are correlations hydrodynamic entry length is 0.05 re times d thermal entry length is 0.05 re p r times d these are just correlations. So, variation of Nusselt number along the tube in turbulent flow for both a uniform surface temperature and heat flux this we will study a little later why this want to tell you couple of things in real life situations internal flow real life problems. We deal with two boundary condition q s or q wall is constant and T s is equal to constant T s or T wall surface temperature why these two because these two situations lend themselves to some kind of a logical analytical solution. Most real life problems are neither this nor this, but we can model them to be either one of these conditions. So, that things are easier to control from experimental point of view you can at least try to get some kind of a comparison. So, that is why in our study we are going to deal with one of these two boundary conditions that is either wall heat flux is constant or wall surface temperature is maintained constant. This is what we will first talk about we will then go to the surface temperature equal to constant case.