 Let us do this simple energy balance for a flow through a pipe, general thermal analysis this is what is called as a general thermal analysis, this general means it does not care about the boundary condition. And this is come from right from your 8th standard 7th standard heat coming in plus heat added is equal to heat that is going out. So, if I take a look for a pipe if this is a fluid, if this is my control volume T m is the mean temperature in this direction this is T m plus d T m. If the fluid is being heated, if this is my r direction and z direction, if fluid is being heated by constant wall heat flux this control volume has dimension z in the direction of flow. Obviously, we know the fluid is going to have a higher temperature given by T m plus d T m. So, energy balance E dot in minus E dot out this is sacred plus E dot generated is equal to E dot stored. We are talking of steady state there is no heat generation E dot in is equal to E dot out. What is E dot in mass flow rate times C p times T m is the energy coming in because of this flow energy is also coming in because of this external heat addition. So, that is heat flux times perimeter times the length associated. So, for heat transfer we are dealing with the surface area. So, perimeter times d z refers to the surface area that is equal to m dot C p T m plus d T m. This implies if I expand this bracket I will cancel off the first term this implies I get q double prime p d z is equal to m dot C p d T m m dot C p T m is going to get cancelled with m dot C p T m on the right hand side. And what is this? This is probably the simplest energy balance equation which most of our students in universities do not know how to do. What have I done? I have done energy coming in two paths one is the fluid itself has an energy water at 25 degrees has an energy associated with it. If I am supplying heat what is the heat content that is given by q double prime pi perimeter times z what are going out of this control volume has a temperature say 25.5 that is the energy content here. So, I am just writing a budget and when I do this budget business I will get q double prime p d z is equal to m dot C p d T m. Therefore, I get d T m by d z is equal to q double prime p by m dot this is a differential equation for variation of bulk mean temperature. And if I am talking m dot is a constant m dot is a constant C p is a constant. Let us say property does not change too much with respect to temperature if the dimensions of the pipe as such that the perimeter is constant circular pipe square pipe whatever it is this is constant. And if I say for the situation that q double prime s is constant wall heat flux case. So, this right hand side entirely is a constant. So, I will write d T by d m is a constant what does this mean slope is a constant it means T m varies linearly with z. So, d y by d x is 2 that means y varies linearly with respect to x. So, T m varies linearly with z for constant wall heat flux case. So, T m is just T m of z flow direction only it is varying very nice in fully developed part this is correct. Now, what else am I saying q double prime how is this heat being transported let us go back here this heat is q is being supplied, but the fluid has to carry away the heat. So, fluid is carrying away the heat means this heat transport from the wall to the bulk fluid is happening by convection is this a correct statement. So, this is logically valid h times T s minus T m at any location is equal to q double prime h we saw few moments ago h is a constant in thermally fully developed region. Remember we went through this whole thing of thermally fully developed and we said that the non dimensional temperature difference is independent of x. Therefore, we showed that it is derivative with respect to r is independent of x which means if the derivative with respect to r is independent of x we are saying that that translates to a Nusselt number or heat transfer coefficient which is going to be independent of x. So, we showed this to be a constant in thermally fully developed if this is a constant in this problem we have assumed constant wall heat flux wall heat flux is constant. So, you are wrapped electrical wire which is providing constant wall heat flux. So, this is also a constant. So, this is a constant this is a constant therefore, this quantity also remains a constant. What does this tell me this gives me a very nice explanation for a lot of things. So, left hand side is constant right hand side one term is constant. So, this is also a constant and we showed that this quantity varies linearly with respect to with respect to the flow direction. That means what if this difference is constant we have shown here and if this is going to increase linearly this also has to increase linearly I will show it mathematically you can understand it very easily q double prime is equal to h T s minus T m. If I differentiate d q by d z is equal to 0 because heat flux is constant this is nothing but constant h is also a constant. So, I will take this out I would get d T s by d z minus d T m by d z equal to 0 which means d T s by d z is equal to d T m by d z and this we showed was a function of z only linear variation in z. How did we show that we showed this moment ago this is a linear variation in z. In fact I should not write is a constant is a linear function of z I have written here linear function of z this is what is important. So, this is a linear function of z. So, if this is a linear function of z that means this also has to be a linear function of z which is parallel because the slopes are the same. So, in thermally fully developed part if my heat is being supplied is heat is being supplied to the fluid the surface is going to be hotter than the fluid. So, this is going to be the fluid temperature variation is like this T m of z linearly increasing with the z direction this is the plot of T versus z this difference we showed was a constant where did I show I showed this in the previous slide this is a constant because these two were constants. So, because this is a constant this also varies with the same slope. So, thermally fully developed laminar flow this is a constant why was this a constant it did not come from magic it came because my heat transfer coefficient was a constant because h was a constant in this part if h was not a constant even if I had a constant wall heat flux I would have a delta T T s minus T m which is varying. So, this is what is given in this part of the notes. So, and if I integrate let me just complete this for if I integrate this expression here this one total heat transfer is going to be this one if I integrate this if I take the length of the pipe complete length of the pipe and integrate d q by d T m by d z is equal to d T m by d z equal to d T m by d z equal to q double prime p by m dot c p if I integrate from inlet to exit I would get T bulk mean at exit minus T bulk mean at inlet is equal to q double prime p by m dot c p and integral of d z with respect to z would be z and integrated from l z exit minus z inlet which is the length of the pipe and this we know without knowing heat transfer we have studied this m dot c p T out minus T in is q double prime times perimeter times length is the area surface area which is the heat gain by the fluid is nothing, but mass flow rate time specific heat times the variation in the temperature between the inlet and exit. So, whatever I knew in high school I have now derived this from heat transfer that is all. So, that is what is shown here all that we have studied entrance length I am just going to tell you in a minute in the entrance length if this equation is still going to be valid whether it is entrance on developing or fully developed h is going q double prime is constant in the entrance length this heat transfer coefficient is going to decrease because the heat transfer coefficient decreases in the entrance length this temperature difference is going to increase. So, this heat transfer coefficient is going to come like this and then become a constant in this part I will just show that to you on the same plot here heat transfer coefficient variation is going to be like this h as a function of z the sorry thanks T s is going to be like this T mean is going to be like this T mean does not even know whether you are in the entrance length or the in the fully developed part it is going to be a linear variation. This thing is going to be parallel to this T m in the in the fully developed part in the entrance region because q double prime is equal to h times T s minus T m because this is decreasing this this difference is going to increase. So, this is the thermal entry region. So, this case is a classic textbook case and most of the undergraduate textbooks cover this very well. So, I am not going to spend any more time on this what we need to know is this part of the graph very well and the reasons for this parallelism of the curves in the T s and T m curve in the fully developed part and the increase in the temperature difference in the entrance region and that comes simply by energy balance. And this we said what is this difference I have what is this slope of the curve I have already shown it to you in this in the transparencies here that I have shown this already d T m by d z is q double prime p by m dot C p by m dot C p this is nothing but d T s by d z in the fully developed part which is equal to a constant because constant wall heat flux case and with the dimensions properties constant this is the constant. So, that is what is given to us in this slide. So, d T m by d x is equal to d T by d x and if I am going and doing this thing I am just going to get the same quantity what it tells me physically is that T break constant wall heat flux ok. What it tells me this temperature profile at this location and another location these are two different location what it tells me fully developed flow in a pipe subject to constant wall heat flux the temperature gradient what is temperature gradient temperature gradient is a measure of the heat flux right heat flux is k d T by d r at r equal to r temperature gradient is a constant it is independent of the axial direction. Therefore, the shape of the temperature profile does not change along the length of the tube. Now, we will understand the so called thermally fully developed a little bit better I am not saying the temperature remains the same I have not made the statement wall temperature will increase let me draw this is one of the things which I draw all the time I want to draw this again this is my temperature distribution at this location. What we are saying by thermally fully developed is because heat transfer coefficient turns out to be constant because of the mathematical condition imposed what is heat transfer coefficient this is the temperature gradient at r equal to r this is not equal to, but is related directly to this. So, this is the slope this is the slope of the temperature profile what we are saying even though this curve will expand even though because of heat transfer the temperature of the fluid will increase because of this increase the shape will become fat I have shown a very fat temperature profile does not matter what I am saying is even though the temperature profile changes this slope here slope at z 1 is equal to slope at z 2 that is what we are saying that thermally fully developed. So, the slopes are equal which means the heat transfer coefficient is the same. So, fully developed flow in a tube subject to constant wall heat flux temperature gradient is independent of x which means the shape does not change now I go back to my white board the surface temperature has changed T s 1 this is thinner this is so much fatter the surface temperature has changed. If I put a thermometer and measure the mixing cup temperature here and this here definitely this is going to be integrated larger volume I mean larger heat content larger temperature this is gotten hotter definitely you put your hand and see it is going to be hot. So, local temperature also has changed same location the fluid has become hot. So, T s is not constant T mean is not a constant T local at the same r and x location is not a constant none of these are constant wall heat flux is constant wall heat flux is constant I have taken in the sketches for simplification, but I get such a nice thing that heat transfer coefficient is a constant which means the temperature distribution non dimensional temperature distribution of the shape of this curve temperature distribution curve though it becomes fat it becomes fat such that this constraint is going to be satisfied this is thermally fully developed. So, I hope I have been able to reach out to all of you this is a concept which is very important how important it is it cannot be over emphasize it is extremely important. Now, next we go to one boundary condition we have studied we will study the other boundary condition we have 15 minutes we have not taken any questions consciously because both these things we will cover and we will take questions completely at 1115. So, if I put in for a circular pipe it is trivial perimeter is 2 pi r I substitute I will get whatever I need does not matter. Now, the other boundary condition of interest is again a very practical boundary condition constant wall temperature when is a constant wall temperature condition established when I have condensation or boiling on a surface when condensation of steam occurs over a surface of a tube that condensation at constant pressure if you neglect the pressure drop the condensation is going to occur at constant temperature. So, if the steam is condensing at constant temperature the fluid is at constant temperature therefore, the wall which is touching which is surrounding which on which the fluid is condensing is also at a constant temperature. So, constant temperature is a very practical condition which occurs in boilers and condensers if you neglect pressure drop if the pressure drop of the flow is taken into account saturation temperature decreases with decrease in pressure therefore, you might not get a almost constant temperature nevertheless piece wise if you take small regions you can assume unique constant temperature very useful whatever be it whether it is constant wall heat flux constant wall temperature does not matter I still have my dear governing equation which is going to be valid q double prime is h times T s minus T m this T m plays the role of T infinity from external flow I am going to reiterate this again from external flow this was T m constant wall heat flux this was constant this was varying with respect to z. So, q double prime is constant implies T s varies with respect to z. Now, I am saying T s is constant fluid is getting heated by steam which is condensing all of us have studied heat exchangers fluid steam is condensing it heats up the cold fluid that means, the T m is going to change if T s is constant T m is going to increase and q is also increase. So, heat is going to get added and all of us have drawn this also from heat exchanger constant wall temperature case T is equal to constant condensation let us say steam is condensing cold fluid is coming in we have drawn this cold bulk mean I am saying m stands for bulk mean or mixing cup T c i let me just call it T m i and T m out or exit we have drawn this cup. What we are saying here is a very important thing in constant wall heat flux case what was my heat flux fully developed part constant irrespective of the region it was constant why because that was an imposed boundary condition which gave me the driving delta T also to be constant it can be either way I can look at it either way delta T if it is constant heat transfer coefficient is constant the heat flux will be constant heat transfer rate will be constant here heat transfer rate has been given to us as constant a priory heat transfer coefficient was proved to be constant therefore, delta T driving the heat transfer. So, whether I take this location this location or this location does not matter to me I am going to have the same heat transfer rate wattage is going to be the same here it is slightly different driving temperature difference locally is going to change with respect to z I have a larger delta T here something smaller here something smaller here. So, this is T s minus T m 1 T s minus T m 2 so on and so forth because my driving temperature difference is going to decrease with respect to flow direction in fully developed flow h is constant we cannot lose sight of that this is constant this is decreasing this is also going to change in the same manner. So, maximum heat transfer occurs when the delta T is maximum progressively the delta T keeps on decreasing heat transfer rate keeps on decreasing. So, here and here I am going to have lower and lower amount of heat transfer. So, this all we can relate because we have studied heat exchangers now let us go back to here arithmetic mean temperature difference should I use this arithmetic mean temperature difference meaning what is the temperature difference which I use to write this h A T delta T should I use let me go back should I use this q double prime is equal to h delta T in where delta T is nothing but inlet T surface minus T bulk in delta T out or exit is T s minus T m each what should I use I do not know. So, we know for a fact that the heat transfer rate calculated using this and calculated using this are not equal because these are not equal temperature differences. So, there is an idea of using an arithmetic temperature mean temperature difference which is essentially delta T i plus delta T e by 2 and this comes out to be T s minus if I use the maths T s minus T m i plus T s minus T m e by 2 which is nothing but this which is T s minus inlet plus exit temperature by 2 this is just for convenience arithmetic mean temperature difference is average of the temperature differences between the surface and the fluid at the inlet and exit of the tube inherent to this definition is the assumption of bulk mean temperature varying linearly along with x. If I have a very steep variation which is of this nature and which is not a linear variation because if this is a linear variation I can use this business of average. If I do not have a linear variation then this becomes a slightly approximate method. So, we can use it for small increments piece wise we can make it linear and use it linear variation is hardly ever the case when wall temperature is constant this simple approximation often gives acceptable results. Therefore, we need a better way to evaluate and all of us know what this better way is because of heat actually is log mean temperature difference. So, we are just quickly doing this energy balance m dot c p d T m is nothing but h T s minus T m times d a s d a s is nothing but perimeter times d x or d z d T m is nothing but d of T s minus T m T s is a constant T m is going to vary I now bring separation of variables d of T s minus T m by T s minus T m is equal to minus this minus has come because of this transformation d T m I am putting it as minus of minus T s minus T m that is all I am doing. So, this is going to be minus h p d x divided by m dot c p. So, this all of us has seen even in lumped capacitance log I mean d theta by theta this is of the same form. So, this is going to give me logarithmic function log of T s I will just do this integration on this here. So, I will get log T s minus T m at exit divided by T s minus T m at inlet is equal to minus h p d x minus T m L divided by m dot c. I have integrated from x is equal to 0 to x is equal to L full length of pipe I have integrated therefore, this is the L and p times L we all know is the surface area. So, p times L is the surface area of the pipe. So, I can write this T s minus T m at exit divided by T s minus T m at inlet is equal to e x p exponential by minus h p L divided by m dot c p. Instead of integrating from inlet to exit if I integrate from inlet to any local location just the local location x for which where I want to find the temperature. If the pipe is 1 meter long and I want to find the temperature at 60 centimeter from inlet I will just have to do the integration from 0 to that location. In other words I would get T s minus T m at that location is nothing but T s minus T m at inlet times e x p minus h p x that x location of interest divided by m dot c p. What is this telling me? This is the maximum delta T that is available in the problem in the flow situation this is the inlet delta T. Remember this plot always draw this plot keep this in mind this is the inlet delta T. This inlet delta T is going to decrease as I move along from inlet to exit. How is this how is this going to decrease? This is going to be decreasing in an exponential decay function. So, this delta T decreases exponentially. This is the delta T local it decreases exponentially as I move from inlet to outlet. So, this is like what you saw in lumped capacitance I am not going to spend more time on this. So, this is the exponential decay and asymptotically the length is infinitely long I am going to approach T surface temperature. This quantity h a s by m dot c p is called as the overall heat transfer or is called as number of transfer units N T u which is there and it is a measure of effectiveness of heat transfer system. What it tells me is the size of the heat transfer unit a s perimeter times length is the size dimension. So, and here we have a table of N T u versus T exit for this kind of temperature distribution it is plotted it is tabulated here N T u is h a s by m dot c p. If I fix m dot c p and inlet temperature of 20 degree centigrade the exit temperature has been calculated like this and what you see is increasing h is fixed m dot is fixed c p is fixed. Therefore, going from N T u of 5 to N T u of 10 indicates that the length has been doubled doubling the length has caused only a 0.5 degree temperature rise in the exit temperature. So, this what it tells me is N T u of 5 indicates the limit is reached for heat transfer and heat transfer does not increase in indefinitely why why does this come this comes from this factor because my driving temperature difference is going to decrease in the flow direction. There is a limit after which you keep adding length the usage or effectiveness of this increased length has gotten reduced significantly. This will N T u will study again in heat exchanger and therefore, what we are saying is this concept has led us to what we call as the log mean temperature difference. So, I am going to recast this thing I will complete this part. So, what we are saying is delta t x local delta t in let me just go back here this is my integration. Now, what is this what am I going to do with this I want q. So, q is nothing but delta h a delta t some log mean temperature difference some reference some temperature difference which I want to use for my calculation that is what is this log mean temperature difference and therefore, coming from this I will write this is that same thing from the previous page m dot c p is nothing but h a s by log of the exit temperature difference to inlet temperature difference. So, q is m dot c p times T e minus T i this energy balance holds good no matter whether it is constant wall heat flux or constant wall temperature condition energy balance amount of heat coming in is m dot c p delta t exit minus inlet that is equivalent to some logarithmic temperature difference which I want to define. So, I will substitute that here and m dot c p is replaced by this h a s divided by this quantity. So, h a s T e minus T i is already there delta t log mean is there if I if I do the algebra if I equate these two it is just going to give me log mean temperature difference is nothing but T i minus T e what have I done I have substituted for m dot c p in this part here. So, this m dot c p term has an h a s. So, h a s T e minus T i divided by this whole thing on the left hand side I use this one h a s delta t log and h a s h a s is going to get cancelled what is left is delta t log mean and this delta t log mean all of us know I think all of us can tell this definition even in our sleep T i minus T e we will write this add and subtract T s. So, I will write this as a T s minus T e minus of T s minus T i that is delta T e let me go back to the board this is delta t inlet this is delta t exit and this log mean temperature difference or l m t d l m t d is also what most of us call this is nothing but delta t inlet minus delta t at exit divided by log of delta t inlet divided by delta t exit. So, this we have seen in heat exchangers we will see it the derivation of essentially the same thing nothing new conceptually there also because of the fact that we do not have a constant temperature difference we had to derive this here also because we do not have a constant temperature difference we have to come up with this pseudo log mean temperature difference and we will write q is equal to u a delta t log mean and this u a is nothing but the overall heat transfer coefficient which we will define in heat exchanger this can be delta t e minus T i accordingly the denominator will change to take care of the sign. So, we will stop here thank you.