Week 7: Lecture 32: Thermally Fully Developed and Bulk Mean temperature

Welcome back. In the last module, we gave a very brief introduction to uh internal flows. We set up how hydrodnamics of flow through a pipe is going to be directly relevant to how heat transfer in when you have flow through a pipe happens. We said that most of the real life applications will involve one or the other boundary condition which is constant wall heat flux or constant wall temperature. Okay. So we are going to now see how the temperature distribution of the fluid is going to be there for when I have flow through a pipe. Let us say uh I look at constant wall heat flux case. So I'm going to introduce this uh to you in this manner. So let us say constant wall heat flux. I'm just using this figure only. I will show both how the cases are going to be depicted and then I will draw a proper diagram on the next page where uh we can show the profile for both constant wall heat flux or constant wall temperature case. So what is heat heat flux? Qp prime is nothing but minus k dt by dx at the wall. This is what we know. Okay. So if heat flux is constant that means dt by dx is constant. Right. So this is a mathematical import of heat flux constant. That means if I'm going to have constant wall heat flux the gradient of temperature. So if I'm going to have a temperature distribution inside so heat is being supplied from outside. So this is QP prime W. So it's basically a electrical wire which is wrapped around a pipe. So I have a resistance heating where the wire is wrapped around this pipe and current and volt I mean you have a potential drop current is supplied heat is generated and that is what is going in. That's what is shown here by this you know arrows pointing inwards which means wall temperature T of wall is greater than T of fluid at any location. Am I right? So we have to understand that wall temperature always is going to be greater than the fluid temperature. Now which fluid temperature we have not talked about that part. We'll see that now. So just as we had no slip condition in fluid mechanics. No slip means what? Hydrodnamic no slip means velocity at wall is equal to mean velocity of the fluid at the wall is same as velocity of the wall. Velocity of the fluid at the wall. Fluid velocity at the wall is same as velocity of the wall. So thermal no slip also is just like that. The temperature of the fluid at the wall is equal to temperature of the wall. Fair enough. This is called thermal no slip condition. Okay. Now these two cases which I talked about constant wall temperature case is very easy to draw. I'll tell that is very easy. Constant wall heat flux case is a little bit uh non-intuitive when I have to draw. Okay. Now let us understand step by step. Wall is going to be at a higher temperature. Heat is being supplied from the wall into the fluid. That means every layer of fluid is going to progressively be cooler than what the fluid at the wall is because the wall is where the heat is supplied. So T wall is going to be the maximum temperature that the fluid sees in a given section. So let me draw this slightly and enlarged. So this is T wall one number. So wall temperature is 80. So this is your 80° line of 80° centigrade. The fluid which is in contact with the wall will be at 80. So this also will be 80. This wall around the circumference everywhere will be 80. Next layer of fluid which is there is going to be slightly cooler, slightly cooler, slightly cooler. Just as you had the velocity you would have we saw in external flow thermal boundary layer, hydrodnamic boundary layer delta less than deltat t which means parental number is less than one. Correct? Delta greater than delta t parental number greater than one. Delta is of the same order as delta t. Pral number is of the order one. Let us say this is of pantal number of the order one case. Which means I will have a progressively decreasing temperature inside and something which is going to be symmetrical. So I uh not good. So let us say this is my center line. Yeah, something like this. So at this section A, this is my temperature profile. I hope it's very clear. This is my x direction. This is my r direction. Now we need to understand what this thing is. This is the gradient of the temperature dt by dx. Right? So this one uh qp prime is dt by dx we have said but heat is flowing in this direction. So this is essentially qp prime is - k dt by d y. y is equal to r - r. So dt by dy becomes dt by d. This will become so qp prime is k of the fluid dt by dr. This is your 4 year's law. So when qp prime is constant constant wall heat flux case dt by dr is equal to constant. Which means at please understand this very carefully the fluid when I say fluid is coming in at 30° centigrade I don't know what this 30 is but it's a measurable mixing cup temperature when the fluid goes out of the pipe let us say it is at 60° centigrade okay heat has been added to the fluid the heat addition will cause the temperature profile. This what I have drawn here is T of R comm X. This called temperature profile or temperature distribution or profile. This temperature profile is going to vary along this direction of flow. Make sense or no? So I have this fluid coming at 30° and at once it is here the temperature profile is something like this. This is say wall is some number T wall one. Next location the temperature profile would be such that T- wall 2 and then third location it is some other number T- wall three T- wall 3 greater than T- wall 2 greater than T- wall one because as I'm h adding heat to the fluid as it flows through the pipe this is my flow direction m dot CP deltat T is heat added Correct. This is from our high school physics. Okay. So the change in temperature. Okay. This is what is happening when the flow is coming from inlet to outlet. The same heat added when I do over the entire length which will become H A delta T T wall minus T bulk mean. So this as the fluid is getting hotter and hotter this temperature is going to go on increasing. That means the length of the lines which I'm showing here inside this temperature profile are an indicator of the heat content. Temperature is a measure of the heat content and the profile as it goes this way is going to become thicker and thicker. Okay. Now I have drawn very arbitrary temperature profile. Why I cannot draw this arbitrary temperature profile is because we have one of the two conditions constant wall heat flux or constant wall temperature. Constant wall heat flux means this gradient dt by dr will be constant. That means every profile profile at location 1 2 and three. This is location one. This is location 2. Location three. We'll have dt by dr is equal to constant. Which means this is QB prime equal to constant case. That means what? If I drew this temperature profiles, all of them would have the same curvature. So this is one of the profiles which I have drawn. At the next location, I would have the same shape but slightly thicker indicating larger amount of heat. So this is location A. This is location B or two. The gradient at the wall is exactly the same as what it was here. Same. But how do I indicate more heat content for the fluid? Larger line, longer lines are indicating of higher temperature of the bulk fluid. So QP prime equal to constant. The temperature profiles are constrained by DT by DR equal to constant. The shape will be same but the thickness would go on changing. This is for constant wall heat flux case. Constant wall temperature case is actually nice. What does it mean? Nice. So if this is my T- wall one, T- wall 2 also should be having the same temperature. T- wall 2 equal to T- wall one. Oho. Okay. So this is my temperature profile. Let me draw this at this location. So flow is happening here. M dot this is my R direction. This is my X direction. Next location A is here. location B I have to draw the temperature profile. This temperature profile will be constrained by the same numerical value of T wall one. But because heat is coming in to the bulk fluid because Q is coming in fluid temperature will increase as the flow progresses along the X direction. If I took the fluid at location A and put a temperature measurement, I would get one value. At B location, it will be higher because heat has been dumped into the fluid. But the constraint is we have T- wall one equal to T- wall 2. And therefore, I will see a slightly thicker profile with the constraint that T- wall one equal to TW wall 2. But the center thing which you're seeing here now these would go on becoming thicker and thicker. So the if this is like a C this is going to be slightly like this. What does this mean? The gradients are not constant. This gradient is this. This gradient is this. DT by dr at location A. DT by DR at location B. They are not equal. But T wall at A is equal to T wall at B. Okay, this is for constant wall temperature case. Constant wall heat flux case. We saw here this DT by DR has to be constant and this is arbitrarily drawn. We cannot have profiles which are arbitrarily drawn. We either have to have a profile which is which is like this. This is uh constant wall heat flux case or we need to have a profile which is like this constant wall temperature case. We cannot have a profile which is drawn randomly. Okay. So that's what we need to understand. Now I have been telling the fluid fluid temperature goes on increasing etc. energy content goes on increasing so on and so forth. Very good. How do I quantify this? So the quantification comes from if I look at a temperature profile at a location, the temperature profile is like this. We introduce this concept called bulk mean or average temperature. This is a Tbulk mean or average or mixing cup temperature which is nothing but the energy content of the fluid inside this black line. So if I want to do this, how do I get this? Remember you got your velocity average velocity by doing mass conservation. Now this is energy conservation. So, m dot cp t b b b b b b b b b b b b b b b b b b b b bulk mean is equal to integral row u of r da this is d m into cp into t of r comma x. This is your m dot. This is your CP and this is your temperature. Essentially what I'm doing I am taking a ring element of the fluid. This ring element has a local velocity which is U of R has a temperature T of R comma X. Why is this X? Because because heat is being added to the fluid at every location whether it is constant wall heat flux case or constant wall temperature case heat addition or heat removal is taking place. So temperature of the fluid will change as a function of x also. Okay. Therefore this is a local temperature. This is local velocity. local temperature and the energy content is essentially integration from center line to the wall. This black boundary is my pipe wall. Okay. So this is what we are trying to do. Let's do that very quickly. So m dot m dot cp into t bulk mean is equal to row into cross-sectional area into bulk mean velocity cp bulk mean. This is nothing but integral row u of r d a into cp into t of r comma x. So this area if you're talking of a pipe flow area is p r² d a is 2 pi r d r. So row into p r² into u bulk mean into cp into t bulk mean is equal to integral 0 to capital r row u of r cp t of r comma x into 2 pi r. So t bulk mean is equal to let's cancel off stuff. We know density is constant. So this can come out. Uh CP is constant here. That is also going to get cancelled. Uh so then you have 2 pi 2 pi. Pi gets cancelled. Pi get cancelled. So this is going to be essentially tbulk mean is 1 / u bar m into r² integral 0 to r 2 r. So this can be two r will stay inside r u of r t of r comma x dr. Okay, this is my bulk mean temperature definition. What is the use of it? If I knew the velocity profile and the temperature profile, I would be able to substitute this and get the bulk mean temperature. What is my bulk mean velocity? M dot is equal to row a c u bar m is equal to row u of r into 2i r d r which will give me 2 / capital r² integral 0 to r u of r comma x. So this one you calculate from your fluid mechanics velocity profile. get a number substitute that here. So this one is coming from here. So fluid mechanics is here. The heat transfer also involves the fluid mechanics. This one is called as a bulk mean temperature. bulk mean temperature or average temperature or mixing cup temperature. So when I say water at 30° centigrade, I'm giving you this bulk mean temperature. I also know that inside the pipe the temperature profile is like this. Fluid is going to have a temperature profile which is varying from T wall at wall to T center line which is going to be different from the bulk mean temperature. The bulk mean temperature is the pseudo value. Okay. I have region near the wall where the temperature is going to be higher than the bulk mean. I have region in the central core where the temperature is going to be lower than the bulk mean. That's okay. The average is such that the energy is conserved. Okay. Let's quickly look at the slides because we have gone quite detailed into whatever we are doing. This is the basic introduction. This is what we looked at bulk mean velocity. This has been done in fluid mechanics. So I'm not going to go through this. This is your average or bulk mean temperature. This is what we just did. And energy balance is what it is. M dot CPT bulk mean is is equal to this particular quantity. And you go ahead and do the maths here. Normally a profile for velocity would be given lamina 1 - 1x4 mu dp by dz 1 minus small r by capital r the whole square. That would be your velocity profile. you would have a similar profile for the temperature. Product of true profiles has to be done and the integration has to be carried out. And this is the formula for the bulk mean temperature. So you what you see is a pseudo temperature which is not the true temperature at all because this is how the fluid temperature is going to be. Somewhere the temperature will be equal to the bulk mean temperature one location along the radius. Okay. But for all engineering applications because I cannot do uh definitions based on local values. T at this point is different from T here. So I need a bulk mean or a uh number which I can use which is representative of the energy content at that location. That's the bulk mean temperature. Okay. So hydraulic diameter again as we have seen in fluid mechanics is 4 a cross-section by wetted perimeter for a circular pipe it is d square duct of side a it is a rectangular duct so on and so forth okay we also looked at what this velocity boundary layer and the concept of hydrodnamically fully developed was the velocity profile doesn't change along the direction of the flow after the boundary layers have merged that is the flow entire flow has become completely viscous. In this part the hydrodnamic entry length we had a region where this is partly viscous and the central core which is invisit. Now similarly when the thermal boundary layers have merged whatever happens in fluid mechanics. Same thing is happening in heat transfer also. The thermal boundary layers which are formed along the wall is now going to merge at a particular location. We want to understand the concept of thermally fully developed and what it is. Okay. So let's just go back to our notes. What is the idea of thermally fully developed? So just again I have to draw the boundary layer. So this is my center line of the pipe. This is where the fluid boundary layers are merging. This is my thermal boundary layer. dt of x all the ideas that we had in external flow continue. Okay. So this is your thermal boundary layer. The temperature profile looks like this. This is T wall and this is T infinity and this is local T. Correct? So same thing here T- wall wall temperature local temperature here this is T wall this is local T of X Y and similarly remember this is circular okay so you have something like this this portion is probably where it is constant and you have this part Okay. So this portion this portion which I have shaded in red is where you don't have the effect of the wall seen by the fluid here. Of course when I go when the boundary layers have merged this is how the temperature profile is going to look like. So this is at location A it is like this. Next location B. This is where fluid mechanics starts to I mean heat transfer starts to differ from fluid mechanics. In fluid mechanics the velocity profile at A would be exactly the same as velocity profile at B. Now I'm talking of temperature profile. This is my temperature profile at region A. The temperature profile is shown like this. T- wall at A. Tbulk mean at A also I can draw T bulb mean at A. So this is water at some temperature has come in. It has now gotten heated to 45° at this location. Next location which is at some distance away from A. More heat is added into the fluid. Right? So as more heat is added, the energy content which is manifested by the bulk mean temperature would be higher. So T bulk mean at A has to be less than T bulk mean at B because fluid has gained more energy as it moves along the flow direction. Therefore whether my constraint is T wall is constant or QP prime is constant. That means whether I have the gradient slope equal or the value of TWW equal, I don't care. All I care is that this profile would have a larger bulk mean temperature TM at B. The profile may be constrained by DT by DR is constant or T wall is constant. I don't care about it. But T bulk mean the thickness of the yellow line which is shown here will be lesser than the thickness of the green line which is shown here because bulk fluid temperature has increased and inherently this is black line which I'm drawing could be my temperature profile at location B. Okay. So this part I have to understand that the fluid is gaining more and more heat. So unlike the velocity distribution which remains constant the temperature profile can never never never remain constant. If the temperature profile remained constant which means at any location it is the same at the subsequent location there is no further heat transfer. Okay that is not what we are studying. We are studying heat transfer. So we want heat to flow from the surrounding to the fluid or from the fluid to the surrounding doesn't matter. Okay. Now all this I have shown you for wall temperature greater than fluid. Suppose it was the other way then your profile would look like this. T wall is smaller than the T bulk mean of the fluid. I'm just showing for completeness because sometimes you'll have a situation where the wall is going to be cooler than the fluid. So this is your T bulk mean at that given location. So tb bulk mean is always a function of x unlike u bulk mean was a constant. What was it? m dot by row into cross-sectional area. It was a constant. This this is always a function of x. That means as I am moving along the flow direction. So, dt m by dx is greater than zero. If qp prime into the fluid that means heat is added to the fluid dtm by dx is greater than zero. It is less than zero. dt m by dx is less than zero if qp prime is given away by fluid common sense right the fluid temperature increases along the flow direction if I'm continuing to heat the flu heat the fluid fluid temperature is going to decrease if I'm if the fluid is giving away the heat oh so dt m by dx is going to be not equal to zero unlike velocity where bulk mean velocity was constant unlike that we have tm which is going to be conveniently varying with respect to x. So does this concept of fully developed flow even exist or no? That is something which is very interesting. We'll try to see that. Unlike the fact that hydrodnamic tells me du by dx is not uh is equal to zero fully developed condition in heat transfer what we say d by dx of this nondimensional quantity which is t wall of x minus t local uh r comma X divided by T wall of X minus T bulk mean of X that is equal to zero. This is the condition for thermally fully developed very very very very important case. This is the condition for thermally fully developed. What do I mean by this? So many brackets, so many things are there. This local temperature is a function of R and X, right? Temperature here is different from temperature here. So it's a function of R. Next location temperature profile would be different. So therefore, for the same X, I mean for the same R at a different X, you have a different temperature. So this is a function of R and X. wall temperature and bulk mean are all functions of location. If I am adding heat and it is constant wall heat flux case, if it is constant wall heat flux case, then the temperature at the wall will go on increasing but the slope will remain constant because I have dumped more heat. T- wall may have become now here may have become here 82° centigrade from 80° but the growth of the profile will be such that the gradient remains the same. Of course when T- wall is held constant that means you don't have to worry about anything. T- wall will not be a function of X. This would be a constant. So either way this is a generic representation T wall as a function of X. The number could be a constant but it could at best be a function of location not a function of r. We also saw this is not a function of r. This is integrated with respect to r. Integrated with respect to r. Therefore it will be only a function of x. Okay. So this part is something we need to remember. Commit to memory condition for fully developed flow. thermally fully developed condition is this. Okay. So this is what we need to do for this part. Next module on we are going to see what this translates to in terms of we'll show that heat transfer coefficient is not a function of X for laminar thermally fully developed part etc. We will do the analysis for laminina flow uh for constant wall heat flux case constant wall temperature case. Thank you.