 Good morning everybody, welcome to this 10 day workshop on heat transfer. On behalf of professor Prabhu and myself, we welcome all of you. Hope you have a fruitful interaction through the next 10 days. What we are planning to do is, we are going to alternate between the two of us. So, roughly 1 hour or 45 minutes of lecture would be delivered by one person and then there will be a changeover. We would like to take questions in that intermediate period of changeover, so that the lecture is not interrupted. So, with that background and when we have questions, we request you to say over to you, so that we know the question is complete. So, this workshop essentially on heat transfer, we are just going to cover whatever is taught in an typical undergraduate institution in a very relaxed manner hopefully and going to present lot of numerical examples as a part of the course. We are going to have tutorial sessions almost every day except today and the last day, where you would be assigned problems and we will be there, your coordinators will be there and you can solve the problems in and there and we will have feedback on that also. So, what is heat transfer? Heat transfer all of us are familiar anybody who is done it does not have to be engineering even from high school, we know what heat transfer is. Essentially a transfer of energy, thermal energy due to a finite temperature difference. So, when we have a temperature difference, you are going to have some kind of flow of heat to from the higher temperature to the lower temperature, so that after some time the temperatures become equal and what we learnt in thermodynamics, thermodynamic equilibrium or thermal equilibrium first is established. So, heat transfer exists primarily because of temperature difference and our bottom line is if there is no temperature difference, we are not going to have any form of heat transfer be it any mode conduction, convection or radiation. So, how is heat transfer? Well, heat can be transferred by one or a combination of more than one, two or all the three modes all of us are familiar with the different modes of heat transfer that is conduction, convection and radiation and our high school physics books give us very simple definitions of conduction, convection and radiation, we will go through those definitions and write the typical governing equations that are there for each of these modes of heat transfer. Why do we have to study heat transfer? Well, why do we have to study anything for that matter? We have to study heat transfer because a large number of real life applications have some component of heat transfer associated with it and our analysis of any real life problem will involve some aspect and we want to do it in a logical educated way. So, unfortunately engineering education or any education has been compartmentalized as electronics, communications, electrical, mechanical so on and so forth. But if you see electronic chip cooling for example, involves some kind of a design aspect related to electronics, electrical circuitry, it involves material science because you need to look at the appropriate materials that you will be using for designing the micro channels. Then you have to have heat transfer and fluid mechanics associated with us, so that you can effectively transfer or remove the heat generated by the electronic components. So, even a non-mechanical engineering discipline student will have to use heat transfer in some form or the other and this is true for any discipline, a person sitting in mechanical engineering will also probably will have to use electrical, electronic circuitry etcetera for his or her applications later on. All of us have studied thermodynamics and in fact in the engineering mechanical engineering curriculum, it is taught as a precursor to heat transfer and fluid mechanics. So, how is heat transfer different from thermodynamics? Well in thermodynamics we say this much of heat is transferred out from the turbine to the environment. We give initial conditions, final conditions of the working fluid in a turbine or a compressor or in a refrigerating system etcetera. We do not really look at how this heat has gone out. We say a turbine, steam enters a turbine at 40 bar and 520 degree centigrade, exists at 95 percent quality 2 bar and whatever. So, with this information we say please calculate the and if the turbine work output is so many kilowatts omega, what is please calculate the heat loss by the turbine or many times we say if the turbine is assumed to be insulated, calculate the work interaction so on and so forth. So, in thermodynamics though heat transfer was there, it was just the black box given to us and number was associated with the heat transfer rate. Heat transfer rate essentially any term with the word rate associated with it would mean it is that quantity per unit time. So, joules per second or kilo joules per second which we call as watts or kilo watts. So, from thermodynamics we know this much amount of heat has to be transferred to have the first law energy balance, but now we do not know how that heat has been transferred whether it was convection, whether it was conduction through the insulation of the turbine, whether there was radiation associated. So, now heat transfer will say let us take the same turbine, now let us analyze the turbine and say this is much amount of heat is going out, what are the various ways by which it is going out of the turbine. So, that my basic first law of thermodynamics of governing equation for conservation of energy is balanced. In fact, later on much later in the course in convective heat transfer we will derive what we call as law of conservation of energy. Essentially that is your first law of thermodynamics it is exactly the same thing with a different flavor associated with it. So, you have seen one of the forms of conservation of energy, we in heat transfer would derive the same thing in a different form. So, that it is more useful from a heat transfer point of view. I mentioned this already due to temperature difference we will have energy transfer, three modes first one is conduction. Conduction of course, all of us know it happens because of closely packed molecules. So, conduction happens in a solid typically or a stationary fluid ok, stationary fluid means a fluid which is not in motion. So, typically what happens in conduction is you have you have the surface which is maintained at a particular temperature T 1, you have another surface which is maintained at a temperature T 2 and if T 1 is greater than T 2 heat is going to flow from the direction of higher temperature T 1 to the lower temperature T 2 and the direction of heat transfer is normal. I mean normal it is perpendicular to the surface. So, if this surface into the plane of the board is into the plane of the screen is a T 1, heat will flow perpendicular to that surface come out through the surface. Now, that is the main direction of heat flow, it does not mean heat is not going to flow in the other direction. We are here in during the definition phase of conduction we are talking about or talking of one dimensional conduction which means heat is flowing primarily in a single direction that direction is the direction along which the temperature is going to decrease. So, temperature decreases from T 1 to T 2 as I move from the left hand side to the right hand side heat is going to flow from left to right here ok. So, this is conduction will also be there in case of stationary fluid, fluid would mean liquid as well as gas. So, if you have an enclosure which is filled with gas eventually conduction will take over between the two bounding surfaces and the gas will act as a medium for conduction. Convection is energy transfer from a surface to a moving fluid. So, convection will necessarily need to have a fluid associated with it meaning it can be a stationary flow I mean it can be a stationary fluid or sorry naturally moving fluid or it can be a forced flow. Naturally moving by that I mean the fluid for example, in this room there is air in this room. The air is not stationary though here for most practical application you will say the air is not moving that much, but the air is moving a little bit which you are not able to perceive by naked eye, but we know from experience that there is some kind of a circulation and we say this is called as natural circulation where it is happening on its own by virtue of a finite temperature difference between one part of the air which is near the air conditioning units which is cold another part of the air which is near the lights in the room which is hot. So, there is some kind of motion and you will see natural convective phenomena. When you are boiling water for a cup of tea what happens you keep the vessel on the gas and you turn on the gas the water initially is at uniform temperature you do not have any flow water is confined to the vessel, but what happens over time you will see that the whole water volume has become hot or warm. The if you have kept the gas quite low and you periodically insert your finger and see you will see that the bottom is relatively hotter the top is cooler and over time there is a good amount of mixing of the fluid. So, that the entire unit entire volume of water is at uniform temperature hopefully. So, this also involved a mixing a convection convective current which we have used many times this word as teachers we would have used convective current means a motion which is set up because of a difference in temperatures cause difference in temperature leading to a difference in density this is natural convection. When we have force flow for example, all of us are familiar with your water heater. So, you have water coming in there is some kind of a coil which is wrapped around some kind of a pipe which is there inside the unit you turn on the electricity it is going to supply heat to the water and you get hot or warm water as per the desired flow. So, every unit every geyser unit associated with as a certain wattage and higher power you use for applications where you need larger flow and larger heat transfer. So, in this case you have a flow a fluid which is flowing and heat is being transferred by some external means it could be another fluid which is exchanging heat or it could be electric current which is contributing to the heat transfer. Both these cases are convective heat transfer processes convective heat transfer the basic governing equation again is what we call as Newton's law of cooling. It essentially relates the heat flux we call this q double prime q with a double lines on top of it as heat flux it is essentially heat transfer per unit area. If you go back here conduction this is the direction of heat transfer and this q double prime is the heat flux associated. So, total heat transfer divided by the normal area associated with. So, q double prime is nothing but h times T infinity minus T s or T s minus T infinity depending on how the nature of heat is going to flow is going to be that is if you have heat coming out from the heated surface to the environment in this case T s is greater than T infinity you would have heat going to the surrounding medium if the surrounding fluid is hotter it will go to the surface whatever be the case this is the governing equation. And if we write it in terms of wattage q is equal to h a T minus T infinity T s minus T infinity where this area would represent the area associated with heat transfer. So, for example, if I am having this plate. So, which is being heated from down the area associated would be the area occupied by the plate. So, this length times the breadth. So, q divided by this area would represent the heat flux associated with the heat transfer process. The h represents the heat transfer coefficient which is not a constant it is not a material property it depends on the nature of flow it depends on the fluid which is flowing it depends on various boundary conditions that are associated with the problem. Next would be radiative heat transfer or radiation where all of us have used this equation associated with radiation very frequently when two surfaces are at different temperatures whether it is high or low we do not care even in this room. There is a panel of lights on top we are sitting here down there is radiation heat transfer between the light bulb there and us here or the table here and the human being here or another table there. Whenever any surface is at a temperature greater than absolute 0 there will be energy emitted in the form of radiation there will be interaction. So, this surface will emit energy there will be another surface which is receiving this energy part of it that will also emit energy by virtue of its temperature and there will be a net interaction between the two surfaces and that is what is shown here in the diagram T 1 and T 2 are the temperatures associated with the two surfaces q 1 double prime q 2 double prime are the heat fluxes which are leaving surface 1 and surface 2 and if I want to write the balance between the two it would be essentially what is leaving 1 which is being received by surface 2 and all of us have used this equation q is equal to sigma epsilon 1 a T 1 raise to 4 minus T 2 raise to 4 so on and so forth. We will come to that little bit later. So, what is conduction? So, conduction at a we have defined it is something where you have the molecules participating in a significant manner. What it means is that it does not mean that in other modes molecules are not participating it just means that the primary mode of heat transfer is because of a molecule with a higher energy being interacting with a molecule with a lower energy and transferring it. It is like a set of balls which you place on a table and flick one ball all of them are going to move and the motion associated with the last ball would be much less compared to what is associated with the fifth ball or the sixth ball, but eventually everything is going to move. So, a molecule with a higher temperature will pass on the energy to a molecule at a lower temperature next to it and that will continue all along. So, that you have a heat transfer by conduction there is no bulk motion of the fluid the molecules are not moving the fluid I mean there is no fluid the solid is not expanding or anything solid is not moving and our experience of conduction heat transfer essentially if you see is most of us will use a spoon to stir a hot cup of tea and you just keep a spoon there or if you are making tea you just make something and leave the spoon there and you are talking on the phone after a while you come and touch the spoon has become hot. Unless the spoon has a wooden handle it becomes hot essentially what is happening is the heat is transferred from the spoon portion that is the portion which is used for serving which is near the hot portion of the gas to the handle which is much away from it and that is primarily because of conduction heat transfer. So, this figure shows schematic basically the red molecules essentially are molecules which are at a higher energy level and these are interacting with molecules which are at a lower energy level and this graph shows the temperature change with respect to the direction of heat flow. Heat is flowing from high temperature to low temperature which is from this to the surface and you have temperature decrease the basic governing equation associated with heat conduction all of us are familiar again as I said we are talking of one dimensional conduction which means heat transfer in general is three dimension it can happen in the x y z direction, but what we mean by one dimensional is one of the dimensions of the one of the directions of heat transfer is of greater importance than the other two what do you mean by greater importance it means that relative to the amount of heat flowing in the other two directions say if this is a plain wall it is a wall of a thickness this much there can be heat transfer towards me and in this vertical direction also because of a temperature difference between these two plates. What we are saying is the amount of heat which flows because of this surface at say 500 degree centigrade and this surface at 30 degree centigrade is maximum in this direction and much lesser in this and in this direction. So, we are going to neglect the energy transfer in the other two directions. So, we say heat transfer is essentially one dimensional and the mode is conduction. So, we say one d conduction the basic governing equation is what we call as Fourier's law Fourier's law essentially relates q to the temperature gradient. Now, how is this Fourier's law come Fourier's law has not is not direct it is not a derivation which is there it is essentially a observation a set of observations which have been put together in a form of a form of an governing equation. So, q or q double prime let us say q is essentially given to us. So, we will just say the Fourier's law right now and your set of transparencies have heat flux there I am just going to say we will write a slightly different form which is having the area associated with it also. Let us say we have two surfaces which are at T 1 and T 2 and T 1 is greater than T 2. So, heat is going to flow in this direction from left to right hand side and let us say this is the coordinate direction x. So, it was been found or several experimental results observation people found that q was directly proportional to the temperature difference T 1 minus T 2 and I think all of us intuitively will agree with this also larger the temperature difference larger is the heat transfer. So, if you have a room which is a 30 degree centigrade and the outside is a 25 degree centigrade there is hardly any difference in the temperature you do not feel lot of heat loss. Whereas, if you are in a cold country your room is a 25 degree centigrade and outside is at minus or even y minus 5 degree centigrade you definitely will feel that there is sufficient amount of heat loss and eventually the room starts to become cold because of the effect of the outside air therefore, you say the house has to be insulated. So, you prevent some kind of heat transfer. So, larger the temperature difference have a larger amount of heat transfer also it was found that q is directly proportional to the area associated with the heat transfer by area I mean the area normal to the direction of heat transfer. So, if this is a set of parallel surfaces this area is the area associated with heat transfer a normal to the direction of flow this is the direction of flow perpendicular to the direction we have the shaded area. So, q is directly proportional and we can obviously visualize this a small very small surface area is going to have less amount of heat transfer. If I have a house where only one wall is exposed to the outside and other three sides of the house are connect have common wall with another house you are not going to have that much of heat loss from those other three walls you will have from the wall which is exposed to the air the most. So, area associated and therefore, you will say in that also if I have windows there will be a different rate of heat transfer because the material is different if I have brick wall it is different if I have wooden door it is different so on and so forth all these are common sensical observation, but we are saying area associated is very very important then we say larger the thickness we have a smaller amount of heat transfer that is why when you put insulation you say you will decrease the heat transfer. So, the room remains at 25 degrees centigrade outside is 5 degrees there is some amount of insulation the amount of heat transfer from the room to the outside is reduced and you keep the room warm. So, it is inversely proportional to the thickness or the distance between the two walls delta x I am calling it and if you combine all this q was found to be proportional to a t 1 minus t 2 by delta x which I can write as t 1 minus t a delta t by delta x q directly proportional to a times delta t by delta x and this proportionality will get replaced by an equality if I say that q is now equal to some k which is a constant we will see what that constant is and a d t by d x or delta t by delta x. This k we have arbitrarily introduced as we call it any constant this constant is not anything we said if I have a brick wall amount of heat transfer is going to be different if I have a glass window the heat transfer is different. So, between the same temperature difference. So, if this is t 1 this is t 2 if this medium is brick you have a different heat transfer rate if the medium is glass it is a different heat transfer rate if the medium is wood it is a different heat transfer rate. And this proportionality constant takes is incorporating all these aspects related to the material and we call it as the thermal conductivity. So, all of us are familiar with this word thermal conductivity it essentially is a material property which is related to the heat carrying capacity of the material. So, that is why we say there are insulators there are conductors so on and so forth copper aluminum brass bronze all these are good conductors we say whereas, plastic rubber etcetera are poor conductors or insulating materials. So, this k is absorbing all those things associated with the material and the s i units associated with this quantity is watt per meter Kelvin q is in watts we do not know the units of k area is in meter square d t by d x or delta t by delta x is degree centigrade or degree Kelvin does not matter or Kelvin per meter this is meter square I do not know the dimensions of k and this is q which is in watts. So, dimensions of k will come out to be watt per meter Kelvin or watt per meter degree centigrade does not matter it is the same thing. So, this thermal conductivity is strictly a material property it may be a constant it may be a function of temperature it may vary with temperature in different way whatever we are not concerned with that at this definition stage we say it is a material property and this is not enough for Fourier's law we say that for Fourier's law. So, we say this Fourier's law is not over as yet we always like to keep the magnitude of heat transfer to be positive and we say from the beginning we said heat flows in the direction of decreasing temperature from a high temperature surface to a low temperature surface. So, we said that heat flows from higher temperature to lower temperature and remember in the first figure we had put x as this direction and q is going from left to right. So, this delta t by delta x if I write from this algebra slope related definition this will be what t 2 minus t 1 divided by L and this is 0 L minus 0 and this will be t 2 minus t 1 by L t 2 is going to be smaller than t 1. So, this quantity the slope is always negative because this is always negative q using this definition k a delta t by delta x k is a positive quantity area is a positive quantity d t delta t by delta x is always a negative quantity q is going to come out to be negative. Therefore, Fourier's law says we put in a minus sign in front this negative sign takes care of the fact that the slope is always negative emphasizing that heat flow direction is always from higher temperature to a lower temperature. So, t 1 is the higher temperature t 2 is the lower temperature from higher to the lower temperature along the coordinate direction. So, that is why if you are asked to write Fourier's law if you ask your students to write Fourier's law and explain it this minus sign definitely has to be put and explained very very correctly saying that this minus sign is included in the formulation or in the mathematical representation of the Fourier's law to take care of the fact that heat flows from a higher temperature to the lower temperature along the coordinate direction. So, that is what is given to us here by this Fourier's law of conduction and this when I divide in the derivation which I showed I mean in the equation that was shown in the written part we had the a. So, if I divide through by a I would have the heat flux associated with the x direction that is why the subscript x is here q double prime is the flux. So, it is q divided by area essentially is k times t 1 minus t 2 by l and now you see here the previous equation is minus k t 2 minus t 1 by l this becomes t 1 minus t 2 by l. So, we are saying when I write the magnitude of heat it is essentially the temperature difference higher minus lower divided by the thickness associated with that object. We have a small problem here numerical exercise very very straight forward. So, says a wall of an industrial furnace is constructed from a point 1 5 meter thick fire clay brick having a thermal conductivity of 1.7 watt per meter Kelvin measurements are made during steady state operation and it reveals temperature of 1400 Kelvin and 1150 Kelvin on the inner and outer surfaces respectively what is the rate of heat loss through a wall that is 0.5 meter by 3 meter on its side. So, very simple straight forward application of Fourier's law of heat conduction we have a wall of a furnace which is 0.5 meters thick. So, the green shaded portion represents the thickness the left portion is at 1400 Kelvin the right portion this has to be t 2 not t 1 t 2 is 1150 Kelvin the direction coordinate direction shown x here from left to right. The yellow line here I do not know whether it is visible it shows yellow arrow is here which shows the direction of heat transfer q x double prime dimensions are 0.5 meters this side and 3 meters this side does not matter you can take 3 meters on the longer side and 0.5 meters this way area is what matters not the individual dimensions at this point. So, every problem that we solve every problem that you should solve every problem that you expect your students to solve should always always always follow a particular methodology and to the point of being like a high school teacher, but I want to emphasize this again and professor Prabhu will also do it we have to state what is known that does not mean we reproduce the text of the problem that is not the aim we want to say what is known means what is the given input information. So, we are known certain temperatures we know that it is a material which is of called as fire clay brick. So, fire clay brick wall wall temperatures are known some thermal conductivity value is known and boundary temperatures are known what are we asked to find we are asked to find the heat loss. So, we always should write any problem that we are given we are asked to solve. So, this is going beyond heat transfer. So, you write what is given. So, given say fire clay wall T 1 1400 Kelvin T 2 is 1150 Kelvin dimensions or area is 1 meter by 3 into 0.5 the dimensions. So, this is the dimensions given to us this is 0.5 meters and this is 3 meters thickness or T I will call it is 0.15 meters. So, some such description has to be given then what you are supposed to find you are supposed to find heat loss or Q. Then wherever possible we are engineers we understand the language of a diagram drawing. So, we have to try to draw a schematic or a sketch for this problem and we will need not be with scale and pencil it can just be with a pen which shows what the problem statement is. So, look it is not very good, but nevertheless it is going to convey what we are going to say this is T this is the area associated with it heat is going to flow from left to right this is the coordinate axis X and T 1 is on this side T 2 is on this side this is more than enough then we say every person will solve a problem differently. So, I will say then which solution is correct each solution is correct provided it is substantiated by a set of assumptions that the person makes. So, as I said conduction can be multi dimensional I can solve this very same problem using a sophisticated computer program which gives me temperature distribution etcetera etcetera, but now for a class work exercise a one dimensional solution is more than enough. So, I have to list a few assumptions. So, after given find and schematic I will say let me list the assumptions which are there. So, list of assumptions Now, this is where each individual solution will be different hopefully, but in a classroom environment in a typical exam setting most of us will know a typical set of assumptions that we make to solve the problem. So, we expect a unique closed ended solution, but in real life problems when you can have a different solution from each student or each person assumptions is what is going to matter. So, I will write steady state that means nothing is changing temperatures are not changing with respect to time these temperature values that have been given to you are at a given instant and for this set of values of temperatures what is the heat transfer rate. So, first one is steady state second one known constant thermal conductivity it is given to us some value is given to us for the fire clay material because I said it could be a varying property with vary with temperature vary with space etcetera, but we are saying it is a known constant and also you can make other assumptions that it is homogeneous material so on and so forth. We are not going to make all those things at this point because we have not gone that far in the course, but a list of assumptions that the student will have to make to be able to use a particular equation directly. After having put this we will say according to Fourier's law. So, this is where the actual solution is going to start till here is only communicating to the reader or to the teacher if it is a student communicating to another person what he or she has understood about the problem what is being asked and his mind set or the set of assumptions that he makes to solve a problem because that is a individual based thing. So, according to Fourier's law golden rule of thumb please do not put in numbers at the first step we do not we are not looking at calculators in the form of human beings we are looking at engineers who are able to solve problems. So, Fourier's law has to be applied means what you do not say q is equal to k A d t by d x put k put a put d t by d x and get the answer that that anybody can do what we are saying is q is equal to minus k A d t by d x we will write this in this form and then we will substitute for the values in the next step. This problem is very straight forward because it involves only one step of calculation, but in subsequent examples later on you will see that a fundamental equation can be simplified to be used in the given problem. So, those set of simplifications can be shown in the initial steps because otherwise what will happen is you will have a set of formulae for each type of problem. For example, in thermodynamics you will have seen p v raise to n is equal to constant. So, you can have a formula for work done for constant pressure process formula for work done for constant volume process constant temperature process isentropic process whereas, actually you do not have to remember any of these things all of these come from the fundamental equation of w equal to integral p d v same thing here you have a fundamental equation cancel terms put in assumptions and make something is equal to 0 and then you come up with this specific form of the equation for that problem there is no need to memorize anything. In this case if I use k which is given to us area which is also known d t by d x I have put the minus sign here. So, I have to follow the coordinate system or if I want just the magnitude I can write this is k a t 1 minus t 2 by thickness this is also fine as long as I have taken care of the minus sign and realize that this minus sign is there because d t by d x is negative and I have made this thing as a positive quantity 1400 minus 1150 divided by thickness it is a positive number this minus sign is absorbed there actually it would have been minus k a t 2 minus t 1 divided by l which is absorbed the minus sign is absorbed here I get this and then I substitute for all the values it is nothing but very straight forward very straight forward exercise it is there here for us and I have q is equal to this solution is been done using heat flux q double prime is equal to k delta t by l it is not taken the area here both are correct. If I take the area here it is been taken in the next step h times w I can do the same problem by having it as q is equal to k a delta t by l substitute the numbers I can get heat flux and multiplied by the area will give you the heat transfer rate. So, assumptions you have a steady state one dimensional conduction say I forgot to write that. So, you had one dimensional conduction in the wall this is very important and after you get an answer as this problem of course, there is not much to write in the form of comments but as engineer as a person who is going to communicate any work that we do is going to be reviewed by somebody either from an academic performance point of view or from information gathering point of view. So, you want to communicate to the person what has been done what whether the results are correct or wrong that is secondary it with this problem tells me that about 4.25 kilowatts of heat is being lost through this wall it does not mean much but 4.25 gives a feel for an engineer what it is and if it is more than permissible you are she will try to do something to reduce it as I said in this problem there is not much to write in form of comments but in subsequent example problems you will have things where you can write and say the temperature associated is of the order of so and so this represents a very specific case or something like that so on and so forth. Now, convection energy transfer as we said is due to molecular motion random molecular motion and advection now these are two very specific terms most of us understand convection as where there is a flow there is heat transfer we call it convection but what we are forgetting or what is implied when we say convection is that there is an element of conduction also associated with it meaning you have conduction between the molecules. The molecules do not stop vibrating the molecules do not lose their energies there is a higher energy molecule there is a lower energy molecule. So, if we say here this is a flat plate which is say at a higher temperature T s compared to the surrounding fluid the molecules of fluid near the hot surface are always going to be at a higher temperature compared to the molecule let us say somewhere here and energy is going to be transferred from this molecule to the next one to the next one to this. So, that molecular activity is also there always there in addition to that there is a bulk fluid motion that means the fluid is being pushed by either a pump or a blower or some other mechanism which we call as advection. So, conduction or diffusion because of molecular motion and advection we call that as convection. So, diffusion plus advection this should not look like a minus sign it essentially is an equal to sign diffusion plus advection equal to convection where diffusion we say is within the velocity boundary layer and advection is outside the boundary layer. Now, we are not going to come to boundary layers at this point sufficient for us to say that fluid mechanics is indispensable during the study of convection. What is the boundary layer why is it formed all of us know that, but we will take a recap when we come to convection boundary layer essentially represents a region over the surface where the viscous effects are dominant compared to the surface compared to the free stream where viscous effects are negligibly small. So, this boundary layer development etcetera we will study later. So, bulk motion as well as molecular motion coupled together contributes to convection when the bulk motion is removed that is advection is brought to 0 essentially you have a conduction problem. So, when you see later on derivation of the conservation of energy equation you will definitely have the conduction term associated with it and terms which have u v and w components of the velocity. So, in fact as we did in thermodynamics we studied closed system first then we did the open system or control volume and then we say look at the open system equation take the velocities to 0 you come back to the closed system equation. Here also we do the heat diffusion equation for conduction first which we will see today and then when we go to convection we derive the complete energy equation which has the conduction or the diffusion part the advection part then we say by magic I put u v w is to 0 I get back what is studied in the first or second lecture which was the heat diffusion equation. Because of the shear complexity associated with convection we do it at a later stage. So, we already studied this force convection where we have bulk motion of the fluid because of external means there is a fan or a blower natural convection where because of just a temperature difference there is heat transfer and not because of any bulk motion. What it means is that this is always there natural convection is always there even when you have force convection what do I mean by that if I have a hot cup of tea which has to be cooled it can be cooled on its own you just leave the cup of tea on the table for 1 hour it will get cooled that is natural convection. If I put a fan to cool it I am only accelerating the cooling process it does not mean that its own ability to cool is lost. So, natural convection is always there force convection is over and above and force convection dominates natural convection so much that the presence of the effect or natural convection is completely neglected rather that is what we should say. But both of these are natural convection is always there force convection may or may not be there and depending on the magnitude of the force convection component we may neglect the natural convection component of heat transfer. Boiling and condensation of course are 2 very specific situations of convection both these involve phase change that is transformation from one phase to another boiling is from liquid to vapor phase condensation is from vapor to liquid phase and these are very very specialized specific kind of topics. We will study about for about 2 to 3 hours on boiling and condensation to give you a idea of the processes that are involved. This is Newton's law of cooling we already said that it can be written as q is equal to h A T s minus T infinity or T infinity minus T s or write it in the form of heat flux does not matter whatever be the form we say depending on the sign associated with q double prime if q double prime is positive according to this definition T infinity is greater than T s it is transferred from there is a small typographical error here this should be T s minus T infinity then I can say that so this slide number 13 q double prime is h times T s minus T infinity and q double prime is positive heat is transferred from the surface to the surrounding fluid q double prime is negative heat is transferred from the fluid to the surface. So, make a change in this equation it is h times T s minus T infinity and as I told earlier h is a function of the surface geometry the fluid nature of flow. So, you can have flow over a flat plate as we had in fluid mechanics we can have flow around a cylinder we can have flow around a sphere so on and so forth whatever be the geometry that is going to be important type of fluid whether it is water whether it is air or oil or whatever fluid is important nature of flow laminar turbulent or mixed mode transition all these are also important. So, please make a change here it is T s minus T infinity if these two statements are going to be valid typical values of this heat transfer coefficient now as engineers we deal with numerical values. So, just to give a feel free convection for gases is the worst in terms of the heat transfer coefficient typically 2 to 25 these are all ranges this does not mean 26 is going to be force convection or 1 is going to be something rarefied gas or something like that it just means that these are typical numbers order of magnitude liquids about 50 to 1000 force convection for gases is about 25 to 250 and boiling and condensation is very very huge values of heat transfer coefficient typically of the order of 10s of 1000 20,000 watt per meter square kelvin is a very common heat transfer coefficient what is the units of heat transfer coefficient this is heat flux watt per meter square per degree centigrade that is delta T or per unit temperature change kelvin. So, watt per meter square degree centigrade or watt per meter square kelvin is a unit of H. So, K was watt per meter kelvin H is watt per meter square radiation energy emitted by matter at any finite temperature. So, radiation can as all of us know does not need a medium for propagation. So, it is transported by means of electromagnetic waves radiation can be there from a solid can be there even on liquid as well as gas and the basic governing equation which is going to dictate or which is going to characterize the radiation heat transfer is what we call as it is given on the next page this one is equal to epsilon sigma T s to the power 4. This Stephen Boltzmann law is applied to a black body we will see what a black body is e b is the emissive power of a black body the subscript b is for a black body e is called as a emissive power this because it is e b it should be emissive power of the black body. The amount of energy emitted in all directions by a surface at a given finite temperature is nothing but what is given by this equation e is equal to sigma T to the power 4 times epsilon which is called as the emissivity associated with that surface. So, if the surface was a so called black body it would emit energy according to this equation sigma T to the power 4 if it is a real surface or what we call as a gray surface if you use gray surface we will have sigma T to the power 4 times a factor which is between 0 and 1 which is called as emissivity all of us are familiar with these terms we will study them in detail later on black surface we have Stephen Boltzmann law and the sigma is 5.67 10 to the power minus 8 Stephen Boltzmann constant any real surface we have emissive power is lesser than the emissive power of the black body at that same temperature. Now in reality in most real applications we will have edge convection we will have radiation as long as the medium between two surfaces is not evacuated you cannot prevent convection from happening it is just that convection may be small compared to radiation therefore neglected in many situations just as in this room radiation is there, but if you have to analyze this heat transfer processes in this room you would neglect radiation because by and large the temperature differences are very small. So, I will one of the assumptions which I will make in most conduction convection problem is negligible radiation heat transfer it does not mean radiation is not there it just means that radiation is small much smaller in comparison to the primary mode of heat transfer which is conduction or convection therefore neglecting it does not affect the final results of the solution to the problem too much. So, G is irradiation this again is the definition will come later is essentially the amount of energy coming from another surface to the surface of interest. So, out of this G which is coming in so that if there is a tube light on top. So, there is energy coming from the tube light to this table where we are sitting there is another tube light which is at a different location there is a different amount of energy just because it is at a physically different location. So, this G which is called as irradiation depends on various things one amongst them is the relative position of the two surfaces of interest and out of this G some amount of energy is absorbed some may be reflected transmitted so on and so forth. And now we just quickly in a few minutes we will relate heat transfer to thermodynamics I have already told you this we do not in thermodynamics we do not talk about any temperature gradient we just talk about equilibrium states and amount of energy required in the form of heat to pass from one equilibrium state to another typically in the turbine problem which I talked about you know 40 bar 520 degree centigrade and expands to 2 bar 0.95 quality does it happen over you know at an instant no it is not going the steam is not going to happen expand so quickly it is a process continuous process during which the temperature of this steam is going to vary continuously the heat transfer from the fluid to the environment is going to change progressively when the steam is at 520 to when it is at 2 bar 0.95 quality. But thermodynamics says I do not care about what is happening in between I just care about the end states which are equilibrium states heat transfer we say no that is not what we want when the steam temperature is 520 and outside is 30 the amount of heat transferred I know by intuition is much more than when the steam temperature is closer to its saturation temperature at 2 bar definitely because the delta T driving temperature difference is larger. So, I want to calculate the heat transfer at every instant when the steam flows to that turbine. So, heat transfer is essentially a thermodynamic non equilibrium process that is a primary fundamental difference. The more you just looked at the heat transfer work interaction as a black box we took it as a number we did not care how it came we did not care about the modes what was done we say between these two things for my energy equation or first law to be valid this much heat has to go out how much heat went out in the beginning how much went out in the intermediate time how much went out later we did not care about that is the primary difference here. And heat transfer of course, we look at the rate at which it has been transferred at every instant this is a complete summary of the governing equation and the mode and the mechanism conduction rate equation is by Fourier's law convection molecular motion as well as bulk motion of the fluid H times T s minus T infinity radiation energy transfer by electromagnetic waves q double prime is sigma epsilon T s to the power 4 all these are written in terms of heat fluxes or q per unit area if you want the total heat transfer you have to multiply by the appropriate area. So, q is minus k a dT by dx H a T s minus T infinity sigma epsilon a T s to the power 4. Now, we come to one of the most would say probably the most important part of this course I should not say the course I think of any kind of engineering thing any engineering problem or engineering subject where it is a budget. So, why engineering it can be even in finance it can be in any other field also we always like to budget things correctly one of the common examples that I give all the time related to budget is your bank account if all of us are salaried people. So, there is some amount of finite amount of money coming into our account every month this is through your salary there is various forms of expenditure that you will have you will withdraw money through an ATM there will be a credit card bill there will be a telephone bill there will be some other bill which is directly debited so on and so forth. There is also some kind of a penalty probably because of some overdraft check bouncing so on and so forth. There is also bank pays you some 4 percent 3 and half percent as interest for whatever money is there in your account on a given day that is calculated nobody is giving you that bank is giving you. So, it is something which is there because there is some money in the account it is being generated every day and between the first of every month and the last day of every month if you look at your account a bunch of transactions are listed day one day two so on and so forth. And then you say I got say 50,000 rupees in the beginning of the month there was so much of expenditure etcetera and if I have 25,000 left in the account I say well I have saved 25,000 rupees in this month if I spent 40,000 there in the saving is only 10,000 if I spent 60,000 well you will have to borrow from somebody and repay him or her later. So, we have all of us have done this kind of budget every time and hopefully we will do for other things also, but this bank is a very very classic example where we use this budgeting all the time and you go to a bank employee and you say give me 1 rupee more if I have written check for 4000 rupees give me 4001 never you can stand upside down they are not going to give you because that 1 rupee which they are dealing in crores, but if that end of the day that alley is not correct they are in for big trouble. So, wherever we are some kind of a budget accountability account keeping book keeping is important so why not in heat transfer why not in energy. So, what we are saying is we studied first law first law of thermodynamics was an energy budget what is first law of thermodynamics let us say all of us will write down the first law of thermodynamics first law of thermodynamics if I write down it is let us write for a closed system or for the full control volume actually D e by D t all of you have seen this probably in a slightly different form something on the left side right side does not matter D e C v by D t is equal to q dot C v minus W dot C v plus summation of m dot i h i plus v i square by 2 plus g z i minus summation of m dot e h e plus v square by 2 plus g z e what was this how was this taught to us when we were in our second year some equation was derived and ultimately we said this equation has to be valid and various simplifications were made for a turbine we said this would come out to be W dot C v is equal to m dot h i minus h e h i would be greater than h e so turbine work positive for a compressor h i would be smaller than h e so the work would be negative so on and so forth. But all the time we had this parent equation and left hand side had to be made equal to right hand side. So, this W dot C v was calculated as a result of dropping of this term this term kinetic potential energy terms so on and so forth and writing m dot i equal to m dot e we came up with this. Suppose there was heat loss from the turbine very poorly insulated turbine or very good well insulated turbine which has been working for a year or so insulation has started to wear out and heat was going to be lost to the environment definitely if I cancelled this and say this is equal to 0 this budget would not be correct because you would see that you are not generating so much amount of work though on equation terms you are generating this much amount of work actually you are not generating that much amount of work because the budget is governed by this parent equation. If this budget has to be satisfied then as the turbine is working I have to write q minus W is equal to or q minus W plus m dot h i minus h e equal to 0 or I will say work output from this turbine which is changing with time because of heat transfer is m dot h i minus h e which I would have got if heat transfer rate was 0. But I have lost some amount of heat because of poor insulation and this quantity has to be subtracted. If I do not subtract this I will calculate a number which is different from what I actually get in the power plant the budget is incorrect. So, what we are trying to say is whatever we do budget is essential first law of thermodynamics was an energy budget. But it was very very specific to thermodynamics related problems how this q was lost I did not care whether it was convection radiation I do not care I just had to account for this q whereas, now in heat transfer we will say this q what are the ways this is going out whether it is conduction write the appropriate equation whether it is convection please tell me how what are the what are the temperatures associated what is the heat transfer coefficient associated. So, this calculation becomes a key for heat transfer the budget again I have not written anything new we have looked at first law of thermodynamics something which all the students would have studied. Now unfortunately again this is fallacy of the education system we are in and all over the world we compartmentalize subject so much that there is hardly any connection between one subject and another students say I have forgotten what I have studied fluid mechanics especially when they come to heat transfer you know my fluid mechanics is really bad so I do not I am not able to understand it is because of the way we do not try to correlate one thing to the other. So, if I am able to correlate and say look when you study heat transfer you will look at this equation again and when you look at this equation this q which was a black box which we are getting a number now you will find ways and means to calculate this equation does not change why should it change conservation of energy right we said this was conservation of energy. So, whether it is thermodynamics or electrical engineering conservation of energy is the same so in heat transfer also it is the same only thing we will write it in a slightly different manner. So, budget there in thermodynamics this q was a black box we still did a budget. So, in heat transfer also we will do a budget very nicely we will do actually very elegant way and one basic equation which all students will have to emphasize learn or literally say that this has to be known and in your tests or quizzes or whatever form of feedback that you take from the students in the form of exams please ensure that this question is featured somewhere sometime during the semester. So, I will write this equation it is already given here in a convenient form I will write it this is a word equation which is given I will write it in the appropriate form mathematical form e dot in minus and I want all of you to take this down and if you have not taught this to your students so far please ensure that you spend at least 15 20 minutes on this during the introductory first or second lecture of heat transfer. I cannot over emphasize the importance of this equation dv by dt does this look familiar yes it should look familiar because from sorry from the first law of thermodynamics we had this d e cv by dt essentially the same thing d e cv by dt I have put and instead of writing q w potential kinetic energy enthalpy I have just said d e by dt is there other things I am writing in terms of some kind of terms this is e dot in minus e dot out plus e dot generated equal to e dot stored what is this e dot stored first let me draw a quick diagram this is my control volume I hope all students are taught what a control volume is in thermodynamics if not please emphasize what a control volume is. So, this is my control volume and there is some energy coming in it need not be only one arrow it can be multiple arrows coming in e in 1 e in 2 there is energy out 1. So, this is as general as it can be this is your bank account now. So, e in your salary which is whatever x rupees 50,000 rupees a month there has been some function people are given your gift that is e in 2. So, that also goes into the account so multiple parts for energy into the control volume. So, e in you have a number beginning of the month you have put in some money in the account because of salary and whatever other ways e out multiple ways a credit card bill credit card 1 some other credit card bill your electricity bill or your telephone bill which you pay online this let us say some penalty that you have incurred because of some overdraft all said and done this is also a summation of all the money that is flowing out what is this e dot g this e dot g as I said was the interest that my money in the account generates. So, that is a positive contribution. So, you had 50,000 rupees in the account for one month at 4 percent if you did not withdraw anything that 50,000 would generate an interest. So, at the end of the month you would see in your pass book in your electronic statement interest and that would be a quantity which comes to you essentially free because of having the money in the account. And on the last day of the month you would have a final number for the amount of money and between the first day and the last day if you see this d e by d t or d rupees how much is the amount changed over the month if it is a positive number very good you have saved money if this is a negative number well you have borrowed some money and need to repay if this is 0 whatever has come in has gone out. So, this analogy if you explain in this way and now come bring it to heat transfer energy coming in by whatever means I do not care thermodynamics did not care. So, energy coming in whatever means energy going out whatever means let me go back to that slide energy coming in whatever means q plus w we had a sign convention when said that q is positive if it goes out and w is positive if it is generated q is positive if it is coming in this is energy coming in by virtue of a flow of an incoming fluid. This is energy which is going out by virtue of a flow of a fluid at a different set of conditions. So, e in e out these are also e in e out terms only additional term that I have is this e generated which is the most general form of this equation. So, e dot in minus e dot out plus e dot generated is nothing but d e c v by d t which I have already written here. So, I have not written a different form I have just clubbed all this and said let us write a general equation which all of us should know e dot in minus e dot out plus e dot generated equal to e dot stored or rate of change of energy of the control volume. And if I integrate it with respect to time that means this is this much of money has come into my account in 30 days. So, that is a rate equation if I say what is the overall. So, this dot is lost when I integrate with respect to time energy in in joules minus energy out in joules plus energy generated in joules is equal to delta e c v change of energy of the control volume sorry here I have to write e dot e dot in when I integrate with respect to time I will get e in e dot out when I get in integrate with respect to time I will get total energy out which is in joules or kilo joules e dot generated when I integrate with respect to time I will get e gen which is again in joules or kilo joules and the right hand side on integration would essentially give you the change of energy of the control volume which is again in joules or kilo joules. What is this tell me this tell me it tells me that this much energy this much amount of energy has come in I do not care about how fast or how slow it has come in 200 joules has come in into this control volume 50 joules has gone out 50 20 joules has been generated and balance is essentially 200 minus 50 plus 20 which is 170 so there has been a change of 170 joules between that instant and this instant over which this observation was made what is this energy generation term where does it come from this energy generation term especially in heat transfer is very pertinent you can have a chemical reaction in a vessel is going exothermic endothermic reaction exothermic means energy is coming out heat is coming out endothermic it is taking in heat nuclear fission products are going to generate energy which is going to contribute to some kind of heating. So, e generated is not a quantity which we can neglect in heat transfer it will be there how we are going to put it mathematically there will be some kind of a functional form given to it or a value would be given to it we need not worry about it if logically we can say that is equal to 0 then we should eliminate it, but otherwise the most general form which is student of heat transfer should remember should appreciate is this equation. And this equation has been given to us in word form here amount of energy entering a control volume amount of energy generated within the control volume amount of energy leaving the control volume if you just look here there is a plus sign for everything, but what it means leaving a control volume by itself has a negative connotation to it. So, you can write it as amount of energy entering a control volume minus amount of energy leaving the control volume plus amount of energy generated in the control volume and we are specifying thermal and mechanical energy entering thermal mechanical energy leaving and thermal energy generated. And this balance is equal to the amount of increase increase in the amount of energy stored in the given control volume and this is a representation e dot in e dot out e dot generated e dot stored and entering and exit are surface phenomena they have to pass through a surface. So, conduction convection all these are surface phenomena that is boundaries are taking part in the interaction whereas generation is a volumetric phenomenon meaning you have a chemical reaction per unit volume you have a chemical which is only occupying a small flask that only that much amount of energy would be generated. If it is a large reactor vessel larger amount of energy would be generated. So, these are essentially volumetric phenomena energy storage term again is a volumetric phenomena. If I take a small container flask in a laboratory that is a very small control volume whereas industrial size furnace with a very large control volume of interest it would have a substantially larger energy storage term.