 So good morning all of you so now that quiz one is over I am sure that many of you are feeling a little bit familiar to whatever we have covered until now so that was on external boundary layer flows external laminar boundary layers so from today onwards we will start looking at internal flows okay internal laminar flows internal flows are very important as far as you know engineering heat transfer is concerned because most of the heat exchangers that you are looking at tube in tube heat exchanger shelled in tube heat exchangers so they are all having internal flows and how do you design these heat exchanges once you know a corresponding relationship between the Nusselt number and the flow conditions such as given by the Reynolds number as well as the properties given by a Prandtl number so you have to develop suitable correlations between linking the heat transfer coefficient and the flow and the properties in some manner so we will look at developing such kind of correlations for internal flows so far we had developed these correlations for external flows the technique for internal flows are a little bit different not exactly like the way we are going to do external flows where we had a clear boundary layer kind of a flow pattern so we were giving boundary conditions like at y equal to 0 you have whatever temperature and away from the wall you are giving free stream condition that is y going to infinity far away you are giving the free stream condition but in the internal flows there is no classical boundary layer kind of a pattern that you can follow especially when the flow is developing fully developed so there in those regions you cannot look at regions which are far away from the boundary layer because in the case of fully double flowed flow everything is boundary layer itself okay the boundary layers from the bottom wall the top wall of a channel can merge and they can everything will be a viscous region and the same way in a duct you can have boundary layer growing simultaneously from all the walls of the duct and you cannot really go and identify a point where you can look at potential application of potential flow and things like that so therefore the solution procedure for the laminar internal flows are slightly different from the external flows so therefore we will start looking at some basic aspects of internal flows before going into the mathematical theory of how to solve the laminar internal flow heat transfer I think most of you are familiar with the basics of internal flows from your fluid mechanics classes and your earlier heat transfer class so I am going to give an overview very brief overview and we will look at basic fully developed case first okay where we can get the solution for velocity profiles very easily okay and then we will slowly gradually go to the case where we have heat transfer so we will look at different regions again so first we will start with the simplest case that is both hydro dynamically and thermally fully developed case okay so before doing that let us let me give you some kind of an overview to laminar internal force convection okay so classically I would like to illustrate internal flow by means of flow between two parallel plates for flow you can take a sectional view of a duct and then look at how the flow pattern is so here when the flow is entering the inlet of the duct so you have a uniform velocity across the cross section and then you see a boundary layer growth from say the two walls of the duct like this and this growth keeps going on till a particular location which is at the center of the duct diameter okay so exactly at the center the two boundary layers will merge and after that the look the region whatever you are looking here it is completely dominated by viscous effects so whereas at the inlet you have a very small region close to the plates where your viscous effects are important outside still the potential inviscid flow theory is valid whereas once the two boundary layers merge in this particular region everything is viscous dominated okay now if you look at how the profiles are drawn somewhere at the inlet you can have a profile something like this now as you keep going downstream the gradient becomes better and better so as you keep going downstream the gradient becomes lesser and lesser here you have very high velocity gradients at the wall and slowly it gets smoother and smoother now once you reach this particular zone here both the boundary layers merge and the velocity profile becomes what is called a parabolic velocity profile okay so therefore looking at the different regions here you can classify the flow based on whether you are looking from the entrance till the region where both the layers meet okay from the entrance of the duct so this look this region this length is called the hydrodynamic entrance link okay so you can use the notation L subscript H to indicate that this is a hydrodynamic entrance length we will also have a similar entrance length for temperature profiles so after the two boundary layers meet now the velocity profile looks similar wherever you go downstream it all becomes parabolic velocity profile and this region is called fully developed region therefore you can classify the different regions of laminar internal flow so now you can look at this region as entrance length region or entry length region or hydrodynamically developing region okay so therefore here you can see the boundary layers thickness is a function of position now once it merges then everywhere it is viscous dominated and the velocity profile is invariant of the axial location all right and this becomes completely fully developed so this is the characteristic of internal flows and in order to really understand how to calculate the length the hydrodynamic entrance length there are some several hand waving correlations there is no exact rigorous theory to determine the entrance length okay strictly speaking you have to solve the equation and understand where the velocity profile becomes parabolic and becomes invariant okay so just to give a empirical thumb rule for laminar flows the non-dimensional entry length so this is non-dimensionalized by the diameter if you have a duct or if you have a channel so this is the separation between the two plates of the channel okay so LH by D is roughly 0.05 times RE based on the diameter of the duct okay so here the Reynolds number for internal flows is defined based on so how do we what kind of characteristic velocity that we have to use that is a question so they kind of an average because here you see even if you look at a fully developed case you have a profile which is a function of Y okay so you do not have a fixed velocity value like in a case of external flow like a free stream velocity therefore so you can use what is called a mean velocity which we will define shortly okay so this is your mean velocity so if you replace this parabolic velocity profile with the constant profile everywhere across the cross section so this is referred to as the mean velocity so naturally how you have to define the mean velocity volume flow rate or the mass flow rate has to be constant okay so therefore the mass flow rate defined by this let us say rho into U M into the area for a circular duct the area is pi into R0 square okay where R0 is the damage radius of the duct so this should be equal to the flow rate obtained with this particular profile parabolic profile so that is rho into U into now if you if you say area is equal to pi R0 square my DA will be equal to 2 pi R0 DR0 okay so I can average this integrate this over the entire cross sectional area okay so this will be 2 pi R0 DR0 correct so I have to integrate from so if I take the coordinate from the center my radius right from the center this is 0 and this is R0 okay so I integrate my profile across the entire cross sectional area so that will be rho U into R0 into DR0 so this this will give me the mass flow rate and this mass flow rate given by this profile should be identical to a mass flow rate if I replace this parabolic profile with a uniform velocity okay so from this I can define my mean velocity or the bulk velocity okay my pi cancels so this is if you look at incompressible flows my density is also constant so that therefore this will be 2 by R0 square so here my R0 so this is basically my R this is pi R2 okay so this should be 2 by R0 square integral 0 to R0 U into RDR so this will be the definition of my mean velocity or what I call as bulk velocity in internal flows okay so this will be used as a characteristic velocity to define my Reynolds number and it is also based on the diameter of the duct divided by the kinematics class okay is it clear okay so this this is the area of at any location R okay so if you if you look at any location R so this will be the cross sectional area pi R square okay so this is what I am using it so if I look at any differential area that will be 2 pi R DR so I have to integrate it over the entire cross sectional area so that is why that is why I am doing this integral right here okay now based on the Reynolds number you can again classify now when I say that for laminar flows this is the relation to calculate my entry length so how do I first determine whether the flow is laminar or turbulent so once again I use the definition of Reynolds number to check if it satisfies a certain cutoff or critical Reynolds number okay so for typically duct flows or flow between two plates the critical Reynolds number for transition to turbulence will be approximately 2300 okay this is a once again an approximate value you know it need not exactly become turbulent at 2300 in fact if you maintain the walls of this channel to be extremely smooth it can remain laminar even as high as 7000 or 8000 okay so if the turbulent intensity at the inlet is so small that it cannot trigger early transition to turbulence and if the walls of the channels are very smooth it can go as high as the critical value can be as high as 8000 or 9000 okay so if you check your RED is less than RE critical so this is an approximate thumb rule for classifying the flow as laminar okay and otherwise you can look at either in a transition regime or turbulent regime okay as far as laminar flows are concerned this is the thumb rule for calculating the entry length alright and for turbulent flows interestingly what happens the entry length ceases to be a function of Reynolds number okay so the entry length non-dimensionalize comes out to be approximately some 10 okay so imagine that you have a Reynolds number of say 3000 okay if you classify that as a laminar flow thinking that the flow is still streamlined enough to be classified as laminar so there what should be the value of LH by D 15 okay now in the case of turbulent flows if suppose the 3000 was now turbulent the LH by D would be 10 so now you can infer that the entry length entrance length for turbulent flows is actually smaller than the entrance length of laminar flows okay so the reason is the turbulence promotes lot of intense mixing or diffusion of flow so the mixing takes place due to the gradients along the y direction so there will be intense mixing and therefore the profile can reach a fully developed state much earlier in the turbulent case than in the laminar case okay so therefore the entry lengths are much smaller in the case of turbulent flow than the laminar flow so now the condition that you have to apply to determine that the flow is fully developed okay now you can see from the shape of the velocity profile that if you plot this velocity profile somewhere downstream they are going to all look very similar so we have to introduce a mathematical criteria okay so what should be the mathematical criteria how do I say that I am in a fully developed region like that yeah so you look at the velocity profile if the gradient with respect to the axial direction this is my axial direction x this is my radial direction if this is 0 so this clearly tells me that I am in the fully developed region okay so this is the criteria as well as the flow is concerned okay now associated to the flow we can also have heat transfer now in the case of heat transfer suppose I take a case where I have Prandtl number greater than 1 okay so I have a growth suppose I apply either a uniform heat flux or a uniform temperature to the duct walls so either I say q walls in either of this case now associated to the boundary layer growth of the velocity profile you can also have a thermal boundary layer growth now if my del Prandtl number is greater than 1 my delta T will be less than delta and therefore if you see the growth of the thermal boundary layer it will be smaller than the velocity boundary layer growth and therefore the point where they merge also will be slightly downstream the then the hydrodynamic entry link okay so somewhere they will be merging here so this will be my delta Tx now so similar to the hydrodynamic entrance link I can also define what is called as a thermal entry link the point where the two thermal boundary layers merge okay I can use the notation L subscript T to say that the length corresponds to the thermal boundary layer and if I draw the profiles okay so typically if I draw a profile somewhere here so now the wall temperature is higher than your bulk temperature or the temperature of the fluid inside so it should it should look something like this right so the magnitude of your wall temperature will be higher than the magnitude of the temperature of the fluid at the center correct so this will be your temperature profile T as a function of why so now when you say that you have a hydrodynamically fully developed region and you have a hydrodynamic entry length similar to that you have a thermal entry length okay and you should also have a thermally fully developed region okay just analogous to your velocity profiles where the velocity profiles are invariant of the axial location however you can very clearly see when you apply heat transfer okay the for example if you apply a uniform heat flux what happens the wall temperature keeps on changing with respect to x the wall temperature will not be the same for example at the entrance and somewhere downstream because you are continuously adding heat so therefore due to conduction so the wall temperature has to keep increasing downstream and also consequently the fluid also keeps getting heat continuously so the fluid temperature also has to vary along the x so therefore unlike the velocity profile which shows a constant behavior the temperature will never be a constant value correct so you will find the temperature of the fluid anywhere that you plot will keep on changing with respect to x at any y location or even at the wall that also keeps changing so how do we now look at a region and tell that that region corresponds to thermally fully developed okay so the criteria is we have to now develop a definition to call the thermally fully developed region okay so how do we now use that criteria okay so thermally fully developed so I know that due to heat transfer my wall temperature is a function of x keep changing my fluid temperature anywhere is also a function of both x and R okay now I have to same way similar to the way that I have defined what is called a mean velocity now I have to define another velocity which is something similar to a free stream velocity in external flow for internal flows okay so I will define what is called as a mean temperature so I need all of this to define a non-dimensional temperature correct so I need to define another mean temperature now this mean temperature should be function of only x so therefore it has to be constant again something like this right so this should be my mean temperature the same way I define my mean velocity based on continuity principle I have to define a mean temperature based on what energy conservation so what I should say is that the enthalpy of this uniform temperature profile should be the same as the enthalpy of the actual profile that you are getting okay so how do you now define enthalpy based on the temperature CP T now this is the specific heat capacity so the total enthalpy will be you have to multiply by the mass flow rate now this is the average enthalpy if you replace the temperature profile with the uniform profile correct now that should be equal to the enthalpy of this varying profile how do you calculate the enthalpy of that integral 0 to R0 so CP my m dot now will be now there is a velocity profile also here please remember corresponding to this temperature profile there is a parabolic velocity profile so my m dot there will be rho u into you have already dA which is your 2 pi into R dr okay and of course you have your temperature in the into the enthalpy so this will be T into R dr so this is the balance of enthalpy no so this is satisfying enthalpy conservation so therefore for an incompressible flow with constant property you can so this is nothing but your okay so let me just write it right now so I can knock off this and my Tm will be 2 pi integral 0 to R0 I can take my row out so T into u into R dr divided by so my flow rate there again will be a parabolic velocity profile so I can write this as integral 0 to R0 into u into R dr right so I will have a 2 pi into R0 again so these will get cancelled off so therefore this will be 0 to R0 you have T into u into R dr divided by integral 0 to R0 u into R dr so this is my definition of my mean temperature at any axial location mean temperature or sometimes people refer to this as bulk temperature sometimes they also refer to this as mixing up temperature so there are different names to the same mean temperature so so this mean temperature is defined based on the enthalpy conservation the enthalpy if you replace your varying temperature profile with the uniform temperature profile okay so therefore now we can see that we have defined all our necessary temperatures here so you have T wall which is a function of X Tm which is a function of X and you have T and therefore we can define a non-dimensional temperature ? just similar to the external flows where we defined as T minus T wall by T infinity minus T wall we can define T minus T wall so T is a function of both X and R divided by T mean minus T wall okay so once you define a non-dimensional temperature profile like this now we can look at a region where this non-dimensional temperature profile is invariant of X because in that region your T minus T wall will be a particular function of X Tm minus T wall will be a particular function of X such that the numerator and denominator are very in the same way so therefore there will not be any variation of ? with respect to X so the condition for defining thermally fully developed flows will be d ? by dx should be equal to 0 okay so once I identify that region where my d ? by dx is equal to 0 so then I can call that this is a thermally fully developed region so I have to monitor my temperature profile ? non-dimensional so if I plot the non-dimensional profile so although the dimensional profile will be something like this okay the non-dimensional profile will be something similar to the velocity profile okay so at wall it will become 0 right at the center it will be T mean okay so therefore it becomes so so it will be something like this okay so you have this is your dimensional profile and this is your non-dimensional profile this is your ? and this is your now once you plot ? okay so this ? should be only a function of R now in the thermally fully developed region because it does not the non-dimensional temperature profile now should not vary with respect to X that is a condition which satisfies the thermally fully developed region okay so this should be a function of only R now based on this we can now define a criteria for the heat transfer coefficient in the fully developed region okay so therefore in the thermally fully developed region my ? is a function of R only whereas in the developing thermally developing region here thermal entrance length region my ? is a function of both X and R okay but once it becomes thermally fully developed it does not become a function of R X it is a function of only R provided your velocity hydrodynamically is fully developed okay so this is a condition provided your hydrodynamically also fully developed okay so it should be hydrodynamically fully developed first and that is where we have taken a variation like this you have a parabolic velocity profile and on top of it if you if you put a condition that d ? by dx is equal to 0 so then you are looking at a location where your ? is a function of only R okay so for this particular region you can calculate your heat transfer coefficient as minus k dt by so now it is not Y anymore it is R and this will be at R equal to where the heat flux should ball heat flux should be at R equal to R not okay now you have to be careful with the sign because the coordinate is going from the center it is not coming from the one okay so therefore you see the temperature profile the temperature profile increases the temperature increases with increasing R so you do not have to put this negative sign here the gradient will be anyway positive okay so whereas if you have a coordinate from like this then the gradient will be negative so you have to put a negative sign okay so now your coordinate system is from the center of the duct so therefore you can just say this is k dt by dr at R equal to R not divided by P wall minus T mean this is how you define your heat transfer coefficient for internal flows okay so now we can write this in terms of the non-dimensional temperature so I can write this as d by dr since this is a derivative function with respect to R and T wall and T mean they are functions of only X okay so they can be directly taken inside this and you can also introduce T minus yeah so T minus T wall by Tm minus T1 and you can put a negative sign I am just flipping this as T minus T1 okay so since T wall Tm they are only functions of X I can just introduce this as T minus T wall by Tm minus T wall which is the same as this correct so this is nothing but ? so this will be minus k d ? by dr now this is at R equal to R not now if you look carefully my ? is a function of only R okay so therefore the slope of the profile at R equal to R not that is a slope at the wall so that also has to be a function of only that is a fixed value the slope at any point will be a function of R at the at the wall that will be a fixed value it now since ? is not a function of X the slope also cannot be a function of X okay so whatever slope I calculate with respect to the profile whether it is here or if I use the non-dimensional ? so the slope I calculate will be the same because the profile is going to be the same the slope here and here they have to be identical so therefore so this d ? by dr at R equal to R not has to be a constant value so which tells me that my H is a constant so therefore it is not a function of X so this is a very very important conclusion so as far as laminar internal flows both hydrodynamically and thermally fully developed so your heat transfer coefficient will become a constant it is not a function of X so that is going to simplify your correlations very drastically okay so this is a very important principle and this does not depend on the boundary condition that you employ whether it is a constant wall temperature or constant heat flux irrespective of that we have not used any boundary conditions here irrespective of the boundary condition this is the fact that your heat transfer coefficient is a constant for laminar internally fully developed flows okay so with that this kind of gives you a brief overview what we are planning to do and in this particular in the next nine hours or so we will focus on three regions so first we will look at what is called region three okay so let me identify this this is region one right here where your velocity hydrodynamically is growing and also your thermal boundary layer is growing this is your region two where your velocity hydrodynamically has developed but your thermal development is still underway okay now this is your region three where both the boundary layers have merged and both the hydrodynamic and thermal boundary layers are fully developed okay so therefore the first part of the course will be on region three which is fully developed both hydrodynamically and thermal and as you can see that is the simplest to start with because you have constant heat transfer coefficient so you can calculate the velocity profile the temperature profile and for a given boundary condition whether it is a heat flux or wall temperature you can identify what is that constant value of H okay so then slightly more complicated is region two where you have fully developed hydrodynamically and thermal entry length so here your velocity profile is fully developed profile but the equation that you are solving does not have that condition d theta by dx 0 okay and finally the most complicated region is region number one where you have both hydrodynamic and thermal entry so this region you cannot get good close form analytical solution you have to solve the complete Navier-Stokes equation because you cannot neglect any terms the gradient of velocity is not zero the gradient of non-dimensional temperature is not zero so therefore all the terms in the Navier-Stokes equation has to be present you cannot neglect any of the terms and therefore you have to do a complete numerical solution for region one okay so in this particular course we will be focusing on region three and then lot of problems related to region two okay so we will stop here and tomorrow we will look at we will start of course with region three we will look at the solution to first the hydrodynamic fully developed condition get the velocity profiles and then depending on the thermal boundary condition we will also get the fully developed temperature profile.