1 Heat Transfer Prof. Dr. Ale Kumar Ghosal Department of Chemical Engineering Indian Institute of Technology, Guwahati Module No. # 04 Convective Heat Transfer Lecture No. # 03 Heat Transfer Correlation for Laminar and Internal Flows Welcome to lecture three of module four, till last classes couple of lectsures what we have discussed is that how to get different dimensionless groups to define assistance, and we have shown two cases for a force convection case, and for free convection case we have shown, and we have found out different dimensionless group. Now, we will carry forward our discussions to see that using those dimensionless group what are the various types of empirical relations, correlations that do exist, that being suggested, and reported by various researchers, and which are being used in practice to calculated the values of heat transfer coefficient, or convective heat transfer coefficient, at the same time to calculate the values of total rate of heat transfer through a particular system. So Now, as we have discuss in the last class, last lecture also that there are various possibilities for the systems like, it can be internal flows, it can be external flows, it can be flow through packed beds. So, we will be distributing like this way, and now today a will be first discussing the internal flows that means,
2 (Refer Slide Time: 01:42) So, correlations various correlations which are available for internal flow, internal flow means flow of fluid through a device, and heat transfer is accompanied with that. So, that kind of situations we are going to discuss, but before we discuss this various correlations that is there, we will discuss a few terminologies, which are very important for understanding these correlations, which is very important to understand the heat transfer phenomena. First of all is the characteristic length, in many a time we have seen many a time we have seen the characteristic length is used. Now, what is this characteristic length? It is defined based on the geometry of the system. So, characteristic length is, it is defined based on geometry of the system. Now, if we take an example, say so flow through a pipe, for flow through, or heat and mass transfer in a pipe, the pipe diameter is taken as characteristic length. Now similarly, when we have flat plate for flat plate is the length of plate from the leading edge. Similarly, the flow through for packed bed it may be particle diameter is used as characteristic length. So, this way we can have different characteristic lengths for different situations.
3 (Refer Slide Time: 05:15) Now, next terminology what I am going to discuss is that bulk temperature or mixing cup temperature. So, if I say it to be say T b, we can assume, we can understand that it is the average temperature, flow temperature in a cross section. So, if you consider like a tube is there this is the central line, and this is the cross section, so in this cross section the bulk temperature mixing cup temperature is the average temperature in this cross section, in a cross section sorry is T b. So, we can have in a pipe line, we can have that bulk temperature calculated like this way, say T b will be equal to for a pipe flow, say zero to r, r is the radius of the pipe, zero to r is the Two pie r into d r into V z, if this is the z direction V z, this is the Two pie r d r is the differential area into V z is the flow rate, volumetric flow rate, this becomes volumetric flow rate, and then multiplied by rho, that becomes the density of the fluid that is flowing through this cross section, and c p is the heat capacity of the material, and t is a temperature at that position. So, if I consider any cross sectional region, any cross sectional region and in that region, say Two pie r into d r if you take a smaller element of this Two pie r into d r Two pie r into d r into V z, V z is the velocity at that particular point, so Two pie r d r into V z this become in the volumetric flow rate into the rho, that means it is become the mass flow rate into c p, that means heat capacity flow rate into the time, temperature that is
4 becoming the total energy flow, that is the total energy flow, this becoming m c p del t t minus t inference to reference is taken as zero divided by heat capacity, that is again r 2 by r d r V z into again rho and c p. So, this is the expression that is gives us the value of bulk temperature, or mixing cup temperature. So, it is like this mixing cup temperature means in a cup there is a fluid, and we are whole fluid it is mixed up, so the temperature of the fluid within the cup becomes like uniform, and it is everywhere in the cup is the same. That mean, here everywhere in this cross section the temperature is to be T b, which is constant, which is fixed, that is called mixing cup temperature, or bulk temperature. It is basically the average temperature in a cross section that is very important things that we should know. And then many of the correlations are being based on average bulk temperature, so it is nothing, but if we have a pipe, and say here it is say, this is the input, and here it is say T b i, and here it is T b o this is output in out. So, that means T b i is the bulk temperature at the input, T b o is the bulk temperature of the output. Therefore, then average bulk temperature if these T b average is usually at the arithmetic average, and that is equal to T b i plus T b o by 2, that is also one important information that we should know. Then another thing is that wall temperature that is also very important. So, wall temperature we usually will be denoting as t w, wall temperature is the temperature of the wall that is from the name itself we can understand. Now many a situation we may find that correlations are used for constant wall temperature, sometimes it is used for constant wall flux, when there is a constant wall temperature then it is becoming t w is constant all throughout the surface, and all throughout the wall, where as when there is a constant wall flux, then t w is may not be constant, t w can be a variable quantity, so this is called wall temperature. Then we have another temperature is called film temperature. In the film temperature we have we u i will be usually denoting as T f, T f is given as T w, that is the wall temperature plus T b the bulk temperature at any location i am sorry it will be T infinity that free stream temperature, I will come to that what is free stream temperature by two, so T infinity is called free stream temperature, so there is another temperature that comes into picture is that free stream temperature, and that is T infinity.
5 So, film temperature is the arithmetic average of wall temperature and free stream temperature. Now, this free stream temperature from the name itself we should understand, that the stream of fluid is free from some kind of interaction, that means it is the free from the effect of the wall, that is why is called free stream temperature. When we will be discussing the boundary level theories, there will be discussing little more details about these free stream conditions. And then there is another one (Refer Slide Time: 12:06) that is called logarithmic log mean temperature difference, which are it is called to be LMTD, sometime we will refer to is like this delta 2 l m that means long mean temperature difference. It is like this for two streams that is flowing, say this is in cocurrent direction, co-current flow, co-current means that both the streams, stream one and this is stream two there flowing in the same direction. So, this is the temperature difference for this two is delta T one, in this side station one, and this is station two, that this is say temperature difference between this two stream, this is say delta t two. So, this is temperature different in the left station and station one, and that extra temperature define relating to in the right station will station two, then delta T l m is the log mean temperature difference, and that is becoming equal to delta T one minus delta T two by del l m Lon delta t one by delta t Two
6 So, this becoming the log mean temperature difference, this is also very frequently being used in many situations. So, will be discussing even details about this log mean temperature difference, from discussing about that performance of heat exchangers and all this. Now, we have got some ideas about that various terminologies, now what we are going to do is that we are going to see the various correlations those are being used particularly for different kind of flow situations to start with, as I started with that will be discussing first the internal flow situation. Internal flow means we will be discussing the flow through a device, now this device can be a circular dart, it may not be a circular dart. So this also, this device can be a circular dart, circular dart slash, we can say it be pipe or tube, and either it can be non circular dart, and also this flow through a device can be your laminar flow, and it can be turbulent flow. Usually, what is being done that flow, when there is a flow through a device it is expressed it is flow characteristics is being decided by the dimensionless group, already we have discussed in some places that it is decided by 8 dimensionless group, let us call Reynolds s number. And we will say that in case of flow through a pipe, flow of diameter d we will say that it denote like this, r a d is equal to d v bar rho by mu, what is this? Now d is the diameter of the pipe, not only it is the inner diameter of the pipe, v bar is the average velocity through which it is flowing, average velocity by which the fluid is flowing, rho is the density of the fluid, and mu is the viscosity of the fluid. So, d is pipe diameter, then v bar is velocity, average velocity of fluid, and rho and mu are density and viscosity of a fluid. so when we have flow through a non-circular dart so when we have a flow through a non-circular dart then it will be little bit changed
7 (Refer Slide Time: 17:24) So, when there is a flow through a non-circular, then it will be a little bit changed, will have Reynolds s will be written as R ed e, and that will be d e rho v bar rho by mu. Now, this d e is called equivalent diameter, equivalent diameter and it is written as 4 into r h, this is r h is called hydraulic radius, and this is equal to, this is defined as flow area by weighted perimeter. this is defined by flow area by weighted perimeter. Now, let us consider a channel flow through a rectangular channel of sides a and b, then what will happen? r h is equal to flow area will be a into b by weighted perimeter will be 2 into a plus b, then d e will be equal to, equivalent diameter will be equal to 4 a b by 2 into a plus b, and that is equal to 2 a b by a plus b. Now, there are certain situations that this equivalent diameter can be different when there is a fluid flow situation, can be different when there is a heat transfer situation. So, for fluid flow and heat transfer equivalent diameter may not be the same. Typical examples I will tell for example, a flow through annular if we consider this, so we have this, say this is the outer pipe, and this is the say inner pipe, now this is the diameter of the say inner pipe, and this is the diameter of the say outer pipe. So, this diameter is the outer diameter, so d i if we say that d i is outer diameter of inner pipe, and d o equals to inner diameter of outer pipe.
8 Now, what we see in this case, when the fluid flow in the annular that means, the fluid is flowing like this in the annular region the fluid is flowing, in that case the fluid, is flow area is flow area is we can understand that pie by four into d o square minus d i square outer this is the flow area of the pipe, flow area of the fluid though the annular region. Now, weighted perimeter, because the fluid is flowing through the annular, it is weighing in both the side, so weighted perimeter is pie into d o plus d i, because both the tube surfaces, the outer surface of the inner pipe, as well as inner surface of the outer pipe both are getting weighted by the fluid flowing. Therefore, the equivalent diameter becomes, so for this case that equivalent diameter will be becoming. If we do the calculations, we will get that flow area by weighted perimeter, so it is actually now d o minus d i, so this equivalent diameter is used d o minus d i, is used for calculation of hydrodynamic situation, that means for calculation of pressure drop. So, when we have non-circular darts, when there is a flow through a non-circular dart, and if we are interested to know about the hydrodynamic situation, if know want to know about the pressure drop, then we may have to use the diameter, and that diameter which is used is called equivalent diameter, and here it will be d naught minus d i. But, when we consider about the heat transfer situation then that case, if we see carefully the heat transfer is taking place between the inner fluid of the inner tube, and the fluid in the annular. So basically, this is the region where the heat transfer is taking place, so only one surface is being involve for the heat transfer situation, so it is not both the surfaces, it is not the outer surface of the inner tube, as well as the inner surface of the outer tube, so outer surface tube is no way involve in any such operation. Therefore, we will be having weighted perimeter for this situations will be for heat transfer.
9 (Refer Slide Time: 23:47) Weighted perimeter is changed, thou flow area remains the same, but weighted perimeter is pie d i. So therefore, for heat transfer d e equivalent diameter becomes, for heat transfer it becomes 1 by d i into d o square minus d i square. So, this is what that we should know in case of non-circular dart, we can use the equivalent diameter for calculating the heat transfer coefficients, and for applying, or for calculating the Reynolds s number, and then for calculating the heat transfer coefficients by using the Reynolds s number expressions in some correlations. Now, what we will see is that internal flow will first consider the laminar flow, now there is a question of laminar flow and turbulent flow. When there is a laminar flow, when there is a turbulent flow, we have some idea from the fluid mechanics, but still just as a reminder we can say that when Reynolds s number which is based on the diameter, if Reynolds s, when the flow is internal flow through a device, and Reynolds s number is expressed as R ed, we have to be very careful about that Reynolds s number expressing is R ed is based on the diameter of the inner diameter of the pipe, or tube, so which is flowing or equivalent diameter, or R ede in under this situation if this is less than 2100, then we call it to be, when the we call the flow to be a laminar flow, then if R ed or R ede, d means based on the equivalent diameter is greater than 4200, we call it to be turbulent flow, and between 2100 and 4200 is the situation when transition from
10 laminar to turbulent flow takes place. So, that is why that this situation which is transition region, in that region the mechanism of flow is for the complicated, anyway. (Refer Slide Time: 26:56) So, will now see that internal flow, laminar flow, so laminar flow, internal flow, then flow through smooth tubes slash pipes. So, when we have flow through smooth tube or pipe, laminar flow and internal flow, under that situation for fully developed flow at constant wall temperature, at constant wall temperature. Housen 1943, has recommended that N ud, that is Nusselt number, Nusselt number based on diameter. N ud is equal to 3.66 plus into d by l into R ed, Reynolds s number based on diameter into Prandtl divided by one plus zero point zero 4 d by l into Reynolds s based on diameter into Prandtl, this whole to the power 2 by 3. Now, here already as I told Nusselt number is based on diameter and it is h d by k, and this h is basically that average heat transfer coefficient throughout the length of the tube, so h bar and R ed is d v bar rho by mu, and we know that Prandtl is equal to mu c p by k. All this things are known to us, so if we say that h bar d is the diameter of the pipe, l is the length of the pipe, and h bar is average heat transfer coefficient over the length of the, or entire length of the pipe. So, and rho and muare are the viscosity density, k is a thermal conductivity, these are usual meanings, v bar c b is the specific heat, and v bar is the average velocity through the pipe.
11 Now, one thing we can see that for very long tube, what we have that Nusselt based on diameter is becoming now only 3.66, because the rest of the parts becomes negligible, so for a very long tube Nusselt based on diameter is more or less a constant quantity, which is equal to Now, it is being also, similar thing has been reported by (Refer Slide Time: 31:15) Sellers, try bus, and Klein in 1956, what they have said that for fully developed laminar flow for uniform wall temperature that is t w, in that case Nusselt d based on diameter is equal to 4.36, which is a constant quantity, and they also have said that for I am sorry this is for the first one is for initial Nusselt is the this is 3.66, and for uniform wall heat flux at the wall N ud is equal to 4.36, that is also a constant quantity. So, this relationship has been given by sellers and try bus in 1956, so we can see that for when there is an uniform wall temperature, then Nusselt number is 3.66 point that length is very high, but Housen has given some extended form of that, so heat has generalize this and when l is very high, then it is becoming Now, if you think we can see here that this in the previous expression of Housen, you have seen that there is a terminology that R ed into Prandtl, so this things also can be heat in many places it may be seen as the pack let based on diameter, and this packlet based on diameter is equal to pack let number. So, what we get is for pack let number is packlet number expression, we can say it is or if you say that it is based on diameter, we can say that it is like this d v bar rho by mu into mu c p by k, so this gives us d v bar rho
12 c p by k. So, if we rewrite by some rearrangements then it becomes rho v bar c p delta T by k delta T by d. Now, if you see the top and bottom portions, numerator and denominator, we see that numerator is nothing but heat removal by bulk flow and denominator is heat removal by conduction, conductive flow. So, we can say that heat transport we can say it like we can say that this is also can be written as v bar into delta T by alpha delta T by d. So, this is nothing but like a Fourier law, so heat transport by thermal diffusion, it is by conduction. So, we will say that it can be written as heat transfer by bulk flow due to the velocity and heat transfer by thermal diffusion, or you can say that conductive or heat transfer by conduction, whatever you say, so it can be said like that. Thus, we can say that similar kind of things can be there even, in case of mass transfer. So therefore, will be to designate whether the it is a packlet number for heat transfer, or pack let number for mass transfer, we can have two pack let numbers actually, one for heat transfer, and one for mass transfer, so we will write it to be that P e d and that is for heat transfer. Similarly, we can have pack let number based on diameter for mass transfer, this is for mass transfer, that also is possible. So, there are two terminologies, one is pack let number and that is for heat transfer,that is for mass transfer. In the present situation for this particular discussions, we will be concentrating on the pack let number for heat transfer, T d and d e indicates that it is the based on the diameter of the or the characteristic length of the flow device through which the fluid is flowing.
13 (Refer Slide Time: 37:12) Also, we can see that if you just see the product p pack let into d by l, this is also called grazed number, grazed number this is defined as grazed number g z grazed number. So, this grazed number is nothing, but it is just some modification of the pack let number y d by l that aspect ratio is being taken care of to take care of the entrance effect. So, d by l this is multiply to take care of the entrance effect, so this is called grazed number. Now, we have seen that one typical expression for laminar flow situation, and we have seen that how the equation comes that correlation there is a Housen correlation, now there is a better or simpler correlation, not better a simple correlation, which is very frequently used is given by sidarantarktic in 1936, it is also called as sidarantarktic equation. It is like this one point sorry Nusselt based on diameter is equal to 1.86 into R ed into Prandtl to the power one-third into d by l to the power one-third into mu by mu w to the power Now, what we can see this also we can write as that 1.86, and already we have seen Reynolds into Prandtl is pack let, and into d by l is called grazed number. So, you can write it to be V z to the power one-third into mu by mu w to the power So, this mu and mu w, both are viscosities of the fluid at different temperatures. Now, what are the restrictions for this cases, this relationship is applicable it is applicable when as I said the grazed number is greater than 10, and all fluid properties are evaluated at the main bulk temperature of the fluid, as I told previously that it is say T b bar, except mu w
14 which is evaluated at t equals to t w wall temperature. So, mu w is the viscosity of the fluid, which is evaluated at wall temperature. Now, in addition to whatever I have discussed little thing is being is supposed to be discussed, which is called there is another terminology that is call Stanton number, which is very frequently used in many situations of heat transfer characteristic. So, Stanton number it is actually is defines as Nusselt by Reynolds into Prandtl. Now if we see that Stanton number that expression if we just put all values of Nusselt Reynolds and Prandtl, we will see that it is h say l by k in this case it is d h d by k into d v bar rho by mu into mu c p by k. So it is becoming h into v bar into rho into c p. So, then we can again write that Stanton. if we rewrite again h into delta T by v bar rho c p into delta t, we can understand that the 1st one, if we can multiply by the area a h, or rather h a delta T by rho into v bar into a, this is the volumetric flow rate into rho the mass flow rate into c p into delta t, so heat transfer into bulk flow. So, we can say that convective heat transfer by heat transfer by bulk flow. So, we can see the differences between these, now here it is the Stanton number and previously we have discussed about the pack let number. In case of pack let number, we have seen that it is heat transfer by bulk fluid by heat transfer by thermal division or conductive heat transfer, but Stanton numbers says it is convective heat transfer divided by heat transfer by bulk flow. So, we have some relationship between that bulk flow heat transfer, to convective heat transfer, or conductive heat transfer, either by pack let number, or by Stanton number. So, Stanton number is very commonly being seen, and similar to pack let number, Stanton number also can be for mass, can be for heat. Because, as we shown here that it is convective heat transfer and the heat transfer by bulk flow similarly, we can have convective mass transfer, and mass transfer by bulk flow. Therefore, we will write here Stanton number can be Stanton heat or Stanton mass. In the present case it is Stanton heat that is Stanton number for heat transfer.
15 (Refer Slide Time: 43:58) Now, in addition to whatever I have discussed, there are some correlations particularly, in case of liquid metals. As we know that flow of the liquid metals would be ah laminar region, where here it is Prandtl number is low Prandtl number, under that situation Nusselt is, Nusselt based on diameter is approximated as 8.0, it is a constant value for uniform heat flux, and Nusselt based on diameter is equal to 5.75 for uniform wall temperature. Now, in addition to this whatever when we started, we have started with a case of flow through a smooth tube, now when the tube is rough so when the tube is rough that means the tube surface is rough, when tube surface is rough there is all this correlations, whatever I have told regarding the flow through a tube or pipe and laminar flow, this correlations are not very useful. The correlations are not very useful and many a times we have to depend on, these are not very useful because, they may not give predictive results, so many a times we have to depend on analogical questions between fluid flow, or fluid friction and heat transfer. What I was trying to emphasize over here is that as the rough surface of this increases, the friction of the fluid with the surface increases, and that can be a huge amount of pressure drop, because of this friction there can be huge amount of change in the velocity profile,there can be a change in the pattern of the flow, and as a result the mechanism of heat transport also which is going to change, the convective heat transfer to heat transfer
16 coefficient, which is a flow property it depends upon the flow rate and all. So, types of mechanism, mechanism of flow, so many things it depends upon as we have seen. So therefore, the h value prediction of the h value may not be an accurate prediction, and we may land up with some false estimation. Now, what will do is, we will take off a problem that will discuss to some extent on the on the demonstration of the various dimensional equations that we have discussed. So, the problem is, say first problem is say (Refer Slide Time: 47:43) it is a desired to heat water, it is desired to heat water through a flowing, water flowing through a tube sorry flowing through a tube, water enters at 50 degree centigrade in the tube of diameter or 2 centimeter, and length 3 meter. The average velocity of water is of water, so the tube is 2 centimeter percent second, and the wall temperature of the tube is maintained at constant temperature equal to say 80 degree centigrade. Now, what we have to do? Calculate the exit water temperature, now given at 50 degree centigrade mu is equal to 5 point zero into 10 to the power minus 4 k g per meter, per second, then the rho is equal to 990 k g per meter cube, density is equal to 4.18 kilojoules per k g, per Kelvin, and then thermal conductive is equal to 0.65 watt per meter per Kelvin, and mu w is equal to, that means at 80 degree centigrade 3.5 into 10 to the power minus 4 k g per meter per second.
17 So, we can see that for this problem, we have to calculate the exit water temperature a fluid is flowing through a tube, and it is surface is at is a constant wall temperature condition. Now, what we will do is? If we see that Prandtl number, Prandtl number is equal to mu c p by k, and that is equal to5 into 10 to the power, now if we working on the solution part of this is 5 into 10 to the power 4 in minus 4 into, then and we have to be sorry 4.180, we have to very careful about the units it is given as kilojoules, so we have to make it to multiplied by 1000 by 0.65, so this becoming so Prandtl number is becoming And let us check the Reynolds number and try to understand whether, it is lying in the Reynolds laminar region, or it is lying in the turbulent region, so that is what is very important to know. So, accordingly we have to use a particular correlation, so we will write now d v bar rho by mu, and one thing we are calculating these Prandtl number, and Reynolds number all based on inlet temperature property, so at all the calculations we are doing at inlet conditions, because we are given the values at inlet conditions. So, Reynolds number is becoming 0.02 into 0.02 into 990 d v rho by mu mu 5 into 10 to the power minus 4, and this is become equal to 792, and which is well below to 100, so we can say it is a laminar flow. (Refer Slide Time: 53:20) So, when it is becoming a laminar flow, we will see that, we will try to use sidertratic equation, but when we have to use sidertratic equation, we have to check that whether a grazed number is greater than 10 or not, that to be check. So, grazed number is we know that it is Reynolds into Prandtl into d by l and, if we put the value this 792 into 3.22 into
18 d 0.02 by 3, so it is becoming 17 which is greater than 10, so sidertatic equation is applicable, so it is applicable. So, now once it is applicable, now let us try to find out what is Nusselt number. Nusselt number is 1.86 into Reynolds diameter i am sorry why should I write all this? So we know already it is grazed to the power one-third into mu by mu w to the power 0.14, and if you put values here 1.86 into 17 to the power one-third into mu by mu w is equal to, that already we know it is given 3.5 five by 5.0. I am sorry, it is just reverse of that, it is 5.0 by 3.55 whole to the power So, this is approximately equal to 4.5 and then 4.55 approximately equal to this, and then from here we can find out what is the h bar or average heat transfer coefficient, which comes about it is 4.55 into d by k, and this is becoming now So, we have got a value of heat transfer coefficient, now we have to find out that actually that temperature, (Refer Slide Time: 55:42) so if we find out the temperature that should be found out temperature, we should be found out from energy balance. That mean, the heat transfer is taken, that is the sensibly taken away by the fluid, so we can get it is like this, that h bar pie d l into t w minus T b 2 plus sorry T b i plus T b o 2 bulk temperatures by 2, so this is h delta t h a into delta t, that will be equal to m dot c p into T b o minus T b i.
19 So, this is that sensibly heat that is being taken away by the fluid, m dot is the mass flow rate of the fluid. So, if we put these values here we will get that into pie into 0.02 into l is 3meter into 80 minus 50 plus T b o, T b o is of important to us it is not known, I would like to find out that is equal to this m dot is can be calculated there as 6.22 into 10 to the power minus 3 into T b o minus 50, and we are writing all in Kelvin, because we are finding that differences in the temperature. So therefore, Kelvin and centigrade does not matter, so we are writing all same phrases in terms of centigrade into c p is equal to 4180, so this m dot is the mass transfer or mass flow rate, that can be calculated as pie d square by 4 into v bar into rho, so m dot is rho v bar into pie d square by 4, and that value is 6.22 into 10 to the power minus 3 k g per second. So, then from this we can find out that T b o from this relation, previous relations we can find out T b o as 60.5 degree centigrade. Now, this T b o is the exit temperature of fluid which is being required, but what we have done is, all the calculations we have done using the temperature of the inlet using the properties at inlet condition. So, we have to now upgrade the properties based on the actual exit conditions, because we have to grade the properties and mean bulk temperature, so mean bulk temperature now, T b mean or T b average is becoming now T b i plus T b o by i am sorry this value will be or nearly equal to 71 degree centigrade, so this will be becoming now T b i plus T b o by 2 and this is 60.5 degree centigrade. So, we have to now redo all calculations, all the previous calculations whatever we have done, we have to redo the all the calculations and see till, or rather is called trial and error, we have to do trial and error method, we have to continue this method till that old T b o and new T b o matches. So, we have to find we have found out T b bar, then repeat all calculations to get new T b o. So, like that we will be getting a new T b o, so repeat all calculation means, we have to calculate. What is this repeat of all calculation means? We have to get all the fluid properties at T b bar get, and then repeat the calculation, and then get new T b o, and continue the search, or continue the work till the old T b o and new T b o matches, then that is called the external temperature of the fluid. And h, the corresponding h, h bar is the heat transfer average coefficient.
20 So, basically this is becoming repetitive and through this repetitions will be marginally improving the values that were calculating, so this is a demonstration of the problem of that flow through internal tube in laminar flow situation, flow through a pipe in the laminar flow situations. Now, in the next class we will see the flow through a pipe, or flow through any devices under turbulent flow situation, what are the correlations available, and we will take off some problem and try to demonstrate, how the problem to be solve. Thank you very much.