 Good morning everybody, good to see you back in time, I was worried that whether we will be able to start 9 o'clock or not but I think my worries were unfounded, fine. So we were in transient conduction and today's plan is that we will start off with transient conduction maybe within couple of maybe one half an hour or 45 minutes we will be through with transient conduction and we will get started with convection, okay. So now let us, so we just, we will just take a recap, if temperature is a function of only time it is transient, it is lumped system and we put our lumped body into a hot water and then we found the temperature distribution as T minus T infinity upon Ti minus T infinity is equal to e to the power of minus T by tau, tau is the time constant. So here 1 by HA is the resistance and rho V Cp is the capacitance, one can think like that, so this is the temperature distribution and these are for different tau's and I had quoted several examples or two examples I guess at least how one can utilize this equation for the measurement of heat transfer coefficient while I was going through your transient response characteristics you are doing that means you are almost I see in your lab experiments you have temperature versus time plot for a thermocouple bead, is that right? I can use that data, I can use that data and complete the heat transfer coefficient, isn't it? But that heat transfer coefficient will be for natural convection, will be for natural convection. So just when you go back home if your guys are doing the experiment or you have the setup in your hand I would request you to take the data for temperature versus time and plot that if you have any problems you mail that excel sheet to me okay, so we will analyze it together. It is possible in the same setup whatever you are doing not only get the time constant with that time constant you can now go back and infer the heat transfer coefficient but only problem here is little bit of it is not as straightforward as it appears because the key is here again rho and Cp, thermocouple bead is not going to be of one material, it is a combination of two materials, so we have to be careful by taking the data in the literature which thermocouple I am using, chromyl alumil means there will be a combination again if you see the problem here there are the values here are funny, they are not of neither representative of copper nor anything, here of course they are not told what is the material you see k is quite less, k is quite less, so point here is so we need to be little careful in getting the properties of my thermocouple bead okay and if you are doing the experiment the thermocouple bead you change the thermocouple bead size I guess you are changing the thermocouple bead size one or two sizes no, so I would recommend you please do that for two three bead sizes heat transfer coefficient should be independent of bead sizes do I get that, do I get that I can do with the same setup whatever I have okay that was the whole idea of welling or spending so much time on this little equation okay I love this equation okay so many setup so many experiments we can do with this little innocuously appearing equation okay fine, so now let us this is the energy transfer once I know the temperature distribution I can get the energy transfer pretty straight forward manner, so I know that it is h A s t minus t infinity and we have put this temperature distribution in the form of theta and I know theta is theta by theta I know the temperature distribution if I plug in the temperature distribution here I am going to get heat transfer that is the total energy transfer occurring up to some time t okay, so I do not think I need to spend time on this I only want to spend time when do I say so far I have not told when can I tell all that I told is if the temperature is uniform everywhere my body is lumped but when can I when a body will be lumped how do I know apriery whether my body is lumped or not, so it depends on biot number biot number h l by k typically not always we have to take this number biot number less than point one as a word of caution it is not true all the time it is true may be when we have convective heat transfer boundary condition all sides if convective boundary condition is there on one side and insulated boundary condition is there on the other side this number may not be right I have to do some analysis separately and figure out what is that biot number for that case for that case which case I took an yesterday an example in which I took a lumped body in which a copper plates which were interspersed between with wooden pieces were put there the boundary condition is one side you have h another side it is insulated isn't it, so there biot number may not be point one but still you try to take care thumb rule I would say if you want to be doubly triply sure or infinitely sure that your body is truly lumped you go one order lesser than try to keep it less than or equal to or around point zero one so then may be my body is okay, so that is how we we decide whether my body is lumped or not okay, so now of course I would skip all this because there are no temperature gradients definitely it is going to be lumped I have told this already in yesterday's class. Now let us focus on spatial effects, so that is here spatial effects is we are taking convective boundary conditions maybe it is plain wall maybe it is long cylinder maybe it is sphere of course maybe many of you are not interested in this spatial effects because in your syllabi I see that many of your syllabi doesn't contain but I think it nevertheless it is quite interesting to see the spatial effects and here again we do not there are two approaches to solve this problem I see everyone most of the syllabi are involving these less charge and grow but but it is a good idea to see how those charge are generated okay, so if you really see what is the governing equation I have del square t by del x squared equal to one by alpha del t del t why because del square t by del y squared and del square t by del z squared I have thrown out because there is no temperature variation in y and z direction okay, the important thing is to make our students appreciate when will this occur how can I give an example to say that when will this occur I will take a plate as big as this room as big as this room and its thickness is let us say 10 mm is that okay, so will that follow this and I apply convective boundary condition on the top and the bottom on all sides I can even insulate or it can be left because it is so big perhaps it won't matter, so these are one of few the examples which we can quote to say when does 1D transient conduction occur, so what are the boundary conditions, so here I have tx, 0 equal to ti that is initial temperature I need one initial condition and two boundary conditions because my equation is second order in space and first order in time, so del t by del x at x equal to 0 is 0 that means it is what does that mean physically insulated but actually in this case it is symmetrical, mathematically it intends it is symmetrical and on the other side it is convective boundary condition okay, so nevertheless the solution what we are going to use can be applied for the case what you said one side insulated and the other side convective boundary condition okay, so the same thing now these are the temperature distributions I am not going to spend time on that what I am going to spend time is how do I non dimensionalize this, so these are the equations which are in the left hand side and these are the equations which are on the right hand side, the important thing is we need to drive home the students the importance of non dimensionalization okay, so I think I will postpone this discussion on non dimensionalization when we come for dimensionless parameters in the reference to Reynolds number and Prandtl number here I will postpone that okay why because it is quite easy to see there the influence of or the power of non dimensionalization okay, so here dash square t by del x square equal to 1 upon alpha, so these are the boundary conditions what are we trying to do this t we will express in terms of theta that is t of x, t minus t infinity upon t i minus t infinity that is theta and the characteristic length or distance x is divided by characteristic length l and Bayard number is hl by k and Fourier number I prefer to use fo but because we have taken the charge from jungle, so his notation is tau, so I have taken tau, tau equal to alpha t by l square, so now Bayard number we have understood Fourier number I will postpone, I will in couple of transparencies later I am going to get what is Fourier number, so what we see here is if I were to see this temperature is a function of x and t and h but here if I non dimensionalize that I get theta is a function of capital X Fourier number and Bayard number, so my number of variables by number of variables have come down, number of variables have come down, so of course as I said we will not study how we solve this but this is the series solution what I get for plane wall, that is you see here there are three terms a n even if it teach through charts how are those charts generated earlier excel sheet was not there today computer is there, so I do not think actually we need to teach through charts although we write Keisler and Grober chart I think we can insist them nowadays calculators are almost computers and the other day I had given integration same problem we had given integration problem with radiative boundary condition for lumped body in fact that you can take as an exercise okay and integration we get with e to the power of 4 numerical integration we intended that he will solve do the integration and come up with the relation but there were 4 guys who had solved through calculator I was not knowing that numerical integration is possible with the present day calculators present day calculators are as good as computers okay, so my request or our opinion would be to go for numerical calculations and we do not have to do series we will show that in fact what is the first series if I take that is enough first series of this here you see here n equal to 1 to infinity first shot we will get we would get bogged down come on here we are having infinite series how will I get if I go on doing with hand I will go mad that is the first impression one would get but it is not so if I take just the first term of the series that is enough how do I get here there are two parameters a n and lambda n how do I get a n a n is 4 sin lambda n upon blah blah blah but how do I get lambda n lambda n is if I the roots of this equation so the roots are a function of biot number we do not have to solve because it is iterative you will not get either I have to use secant method or Newton Raphson method or iterative method or bisection method to get this but let us not do that or insist my our students to do that they are there in our notes that is biot number first 4 roots are given in fact I do not even need 4 roots I need only first root because I said first term is enough first root is enough so if I take lambda 1 plug in that here I am going to get what am I going to get a1 so if I plug in that a1 and lambda 1 I get my theta pretty straight forward do you agree with me this will not take me more than 2 minutes to do this calculation and it is much simpler than chart you can still go ahead with chart compare the value with the chart and this in chart we are rounding off so many things isn't it in fact I think I have a problem cooked up yeah if you see here maybe I will show you the chart and then come back so same thing is so this is for exact solution for plane wall similar solutions you can see this is for plane wall this is for cylinder this is for sphere okay so same thing there also only thing is that first series would suffice first term of the series would suffice as long as my Fourier number is greater than 0.2 if it is less than 0.2 I may have to take 1 or 2 more terms I do not have to take n series 2 or 3 terms extra would be sufficient but rarely we would come across Fourier number less than 0.2 anyway we are cooking problems and giving this isn't it I can cook a problem such that my Fourier number all the time is greater than otherwise students may get an impression if we do this they do not get an impression that it is coming experiment because charts and tables means we have a notion that they are all coming from experiment it is not sure these were like nomograms we have studied I think still we teach nomograms in drawing I think do we yeah this nomogram concept was there when there was no computer or calculator when we were using perhaps logarithmic tables to solve in my plus 2 I used to use logarithmic tables to do all the multiplication and division during that time it was relevant but not today's world I will take your question I am not going to stop my flow I will intend to take the questions at the end okay why because this is a quality time and I want to utilize this fully okay so lambda n a n lambda n if I plug in I am going to get the solution so now same way in fact lambda 1 and a 1 for a given biot number are also given that is also calculated if you do not want to calculate a 1 are you with me instead of calculating a 1 both lambda 1 and a 1 are listed for a given biot number for all cases plane wall cylinder and sphere okay this is incorpore and David's and Chungles and I think in other textbooks it is not there it is there only in these two textbooks see if we if we give this table if I have that relation I can get the temperature distribution so that is and center line all that is for center line this term is going to become 1 this term is going to become 1 because cos 0 x equal to 0 so with that these are charts now I do not think if I have told what it is I do not have to here you have Bessel functions that is what professor is telling for cylinder you have Bessel function but you do not have to be worried about the Bessel function at all because you have been given the roots here J0 lambda is already given for J0 and J1 are given and you do not even need J0 you need only the J0 okay so with this I guess we do not have to deal much with the charge this is theta not versus tau okay and this is theta versus 1 upon biot number for different x by L so if I multiply these two I will get theta at any location and this is Q by Q maximum is that okay now I can perhaps take questions yeah you can ask them to do it in excel sheet and ask them to check whether indeed first no no I said what did I say what was my conclusion if tau is greater than 0.2 in the class if I demonstrate that in the class I can demonstrate or I can give that as an assignment saying that you take first term only you take two terms you take three term you take four terms you take five terms now see the difference they will see themselves they will they will see themselves through excel sheet that first term is good in the exam you can take only first term of the series so we have to give the particular equation to the students or not you can give because we see in fact that is a good question it is opening something different in fact we are of the opinion that we do not intend here all our exams are open book exams all our exams are open book exams that is a good question you opened up anyway in the morning we should not be discussing but anyway we have opened we shall take it I would we would think that we are not checking them their remembrance we are checking them their thought process the way they are applying their ideas not remembering how will I cannot remember this equation how will I expect my student to remember this equation that is the question I ask myself can I remember this whenever I am giving a question for my student to the exam professor Arun had questioned me on the other day when I give transient conduction for lumped body or you are giving such a huge equation which involves integration for me it is difficult but my students are much smarter than me okay last time when I had made my problem very simple and goofed up the problem students came back and told sir you could have given the integration that integration is straight forward for us that is why I give so that is how I consult Arun saying that okay these guys are smarter than me they will do it point is point is we have to ask ourselves a question can I do without any assistance if the answer is honest I know here know whether it is yes or no if the answer is no here I should not be giving that too much so to answer your question I can give the solution to them I can give this relation it is only typing I can even give this chart to him in the question paper at the back end let the question paper be four pages who cares it is logistics who will take care but while preparing the question paper I should not be worried about the logistics logic logic should not decide my method of making the question paper I think you will all agree with me on that note in real world actually a person who is going to apply this will have these equations in front of it all he needs to know how to use it if in the process how difficult as should I option may show that this is the logic the fellow uses I am going to leave this an option because I cannot remember this material you have killed everything he is not going to learn transient conduction he is never going to be able to apply these equations so the idea is these equations if you are in real life you will have the sources available nowadays everything is online if I do not know something I search I get all these equations also you do not even have to literally type another thing yes I think there might be certain diffidence from all of you because we cannot change things overnight in a university system I can understand that I can understand that but at least we should start at least in your assignments you should give because I see most of your you have in house assignment quite a bit at least there you can give this type of that is in your hand right that is in your hand let us inculcate that and as you grow older you are going to sit in your committees now you are all young as you grow older you are going to sit in your committees in your university system try to bring a change try to bring a change we can do away with this data handbook and chart business whatever you have to learn for a given problem we can give for that problem then and there. See when for machine design you accept such a huge data handbook for exam so I think for heat transfer all core courses I think an equation sheet or whatever you know formula should be given because if you cannot remember those equations I do not think you should be expected to remember equations of this kind also correct that is what I am saying why is data book accepted because a person is not expected to remember this this this this thing who decides is that one of us only right you tell me I will stop that is the logic anyway there is a problem with the technical snack is there so we can continue to discuss this you have any doubt no they are not recording because of some problem so they have asked us to stop but we can take questions for questions I do not think there is any problem so that is the that is the philosophy that is the philosophy of using these charts you have any other questions I can take because we have 10 more minutes time what all of you will be setting examination paper either for your course or as a part of a university team so what is the philosophy that you have in mind when you set an examination paper this is something which I think as teachers it is very very important to you sorry sorry in fact we had thought that we should well upon maybe we will try to see still how to set a question paper I think as teachers we should be spending more time don't worry if there is any technical skill we will model it that is no issue so I think we need to sit down and well upon that issue more seriously because that will make or break a student's interest towards a course because if a question paper is too lengthy and too too much of remembering I will just throw it out of my mind honestly you most universities colleges yesterday I have seen you teach boiling curve and write short notes on pool boiling curve one of the favorite that is a stock question that is a stock question what what is the student going to gain by writing a short note on boiling curve or best question I have seen as university write short notes on psychrometric chart it is a tool yeah it is a graph whatever how is that is what so easily the chart yeah so we should we should we should avoid these type of questions yeah as we go along we will maybe we will try somewhere during our course or at the end maybe we will try to spend some time how to set questions how to set questions okay so we will spend time on that okay fine so now see this is just to show the chart if you just use the chart these are the numbers you are going to get but if you do not use the chart you see you are going to get theta of 0.01516 but with the chart you will end up getting 0.0, 0.0135 so much difference you get with the chart and without chart sorry with equation and with chart so that is the reason I think it is a good idea to go ahead with we can continue to use chart for historical reasons or for whatever reasons but we can always compare we can always compare okay so with this I guess I will this is about q maximum only one point I want to stress here is 1 upon yeah yeah so one stock question with our students is that see in all these charts we have bio number defined in a particular fashion if you see the bio number what is the bio number here h l by k for plane wall h r0 by k for cylinder and sphere so but then in lumped capacitance we have defined r0 as our characteristic length as volume by surface area so students come back this is a stock question we have out of 128 guys this time at least 25 students have asked this question separately that is in spite of telling so many times so the characteristic length which is used for lumped body continues to be volume by area but the bio number definition in exact solutions continues to be l for plane wall r0 for cylinder and sphere because we have generated solutions with this definition we have cast the solution this way it is not that we cannot cast it other way we can we can but we want to generalize everyone has to use it in a particular fashion so that is the reason it has been done this way so this is the stock question our students get confused this is one question where students get confused so we have to be careful or we have to reiterate twice or thrice about this question another thing in this chart you would have seen that there is a situation where in which 1 upon bio number is very high sorry the question is why am I plotting with 1 upon bio number instead of bio number why am I doing this because for 1 upon bio if the bio number is very high what will happen to that what will that case come nearer to let us say bio number infinity when can bio number be infinity hl by k physically my h is very very high when h is very very large very very large what will happen to my wall temperature I have a plane wall how will I apply h, t infinity there is a fluid flowing I have a plane wall I have a plane wall and fluid is flowing along this I have told that h is bio number is very large I can maintain that by infinity transfer coefficient then in that case what is the convective resistance 0 so if a convective resistance is 0 what will be my wall temperature compared to t infinity see so what is the boundary condition I have solved here if my bio number is infinite my boundary condition will be constant wall temperature that is precisely why we have plotted this chart for 1 upon bio number as opposed to bio number this insight we have to tell this student will keep wondering why this guy is taking 1 upon bio number this is the reason this is the reason ok so perhaps I can take 1 upon bio number here 100 cases infinity and just get the solution that will be nearing my constant wall temperature boundary condition ok how many of you are paper setter in the couple of you are there then you can bring a change I implore upon you to bring a change to bring a change all of you I see at least 20 percent of paper setter here I am sure few of your colleagues are paper setter if when you say open book there are problems also students get to think that ok I will bring the notes to the exam and I will get through but then what we should do is we should not allow them to bring this printed material what we do is we try to say that you write hand written you bring reams of paper we do not mind but at least while writing it is told that we all appreciate once written is equivalent to 5 times reams we all know that at least we have ensured that he has read 5 times by making sure that he has written and he does not want to write twice so he will be attentive in the class he will write down all the notes what you write on the board and extra material what is there in the power point he is going to write that before coming to the exam so do not allow printed material or xerox material he will xerox his friends notes and bring so you have to avoid ok and question papers are supposed to be thought about that means I have to think that means I have to spend more time ok so I may have to spend almost one or two days first setting one question it is not taking answers from some other book questions from some other book and simply putting it sometimes I can formulate myself or take incorporate and do you sit down and solve solution manual all of us know it can be downloadable from the internet all of us know that and students also know that no problems but let me as a teacher honestly try without seeing the solution then we are done I mean our job is done then we have made them think the idea is to make them think we have to make them think means we have to think hundred times more than you are through so everything I keep saying this everything is proportional to time the more I invest time the better I am that is all that is all it is as simple as that so that is what I have intended here ok so of course energy transfer whatever energy transfer is mcp delta t I think we have done this right from my school so I do not have to spend time on this but only thing is that this q whatever q is there energy transfer from the boss is non dimensionalized with q maximum the maximum heat transfer would be mcp t infinity minus t that is the maximum temperature gradient I have so that is non dimensionalized with q maximum so that is what is plotted here in the chart q by q maximum versus biot squared into tau for various biot numbers again so I think the relations for that also are there here q by q maximum you do not have to rely on charts again so if you know the lambda you can get this is not at all involved only thing you need to get theta not that is the central line temperature and lambda which you anyway have which you have completed so I think of course this is the history that is Hisler has come up with these charts way back in 1947 and Grober in 1961 so that is what we study as Hisler charts but actually we never say Grober charts or I think we do say Grober chart so that is that is about this and Fourier number so in fact professor Arun was telling when he was telling about alpha that is Fourier number is alpha t by L squared it is the ratio of how much is being conducted to how much is being stored if a Fourier number is large means what heat conducted is more compared to so that is all that is all the idea here is. So here I have again I have given this example to you already what constitutes a infinitely long plate and infinitely long skeleton this is an egg problem I think I can skip this problem because this is a problem in which we have just tried to measure the central line temperature of the egg this is taken from chungal or one of the two so I do not think I will spend time so I will spend time on semi infinite medium okay so first thing is we need to understand it is not infinite this is the first question students keep asking us there is nothing infinite about semi infinite okay so any finite body can also be approximated as a semi infinite let us see how do we do that now let us say earth is a very good example what do you visualize or how do you visualize a semi infinite medium let me get the feel of it how do you visualize a semi infinite medium through that finger I am having the feel of the very hot plate that is the limit up to particular sense particular portion of the finger so one of the boundary condition for this finger will be taken as a finite boundary condition and second finger this is you are modeling finger yeah but then how long it will be semi infinite okay that is why it is approximated as a semi infinite body for all times because perhaps we are having the very high temperature so one of the boundary condition is taken as finite 0 okay and the infinite direction is through the body body temperature you are taking okay any other any other interpretations do not worry I kept telling yesterday also those who never make mistakes never make anything I tell this in the class also we are not launching a rocket we are not answerable for hundreds of crores nothing wrong in going wrong okay so please please throw yourself to make mistakes otherwise we will never learn yes ma'am maybe you go ahead and try to interpret it no problem okay here we are not trying to test you maybe you will give a new interpretation so suppose suppose you have a very hot object and expose to ambient condition so during the initial condition the surface the outer surface will be will have the temperature variation okay and it will take some time to reach that constant temperature very nice so for that time period small time period for that small time period we model that as a semi very right very right interpretation anyone else want to give yes sir you are also teaching at least in one direction it will be finite so I can do it as a semi infinite but then plane wall is also finite no in one wall for example the case which we solved now so when will that plane wall become semi infinite if I ask that question otherwise why should if it is again plane wall why again come up with one more case which is semi infinite okay no problem oh yeah you have yes sir please there will be at least one point in the object where the temperature continues to be in the initial temperature that is with infinite time okay infinite time I could say almost infinite time it takes infinite time for the temperature to change when you expose the surface to a sudden change in temperature yes sir yes I am writing about it so the point is what we understood with this discussion is that another thing is to encourage your students to ask questions that is another thing we keep bothering our questions to ask questions our students to ask questions if they are silent means that means we have not reached them and you can see in their eyes a blankness you can you can feel it I think you are all teachers you can feel what I am telling so if you see a blank face that means I have not reached yesterday Arun was saying I see blank faces yes I see a sense of blankness then when I see a sense of blankness maybe I am uttering words but that is not reaching them that happens so please encourage your students to ask as many questions as possible and do not feel bad about saying I do not know that is one thing we all should inculcate everyone cannot know everything at any point of time that is knowledge how can I know everything at at my fingertips it is just not possible it is just not possible so we need to admit in the heart of my mind that okay I do not know many things okay so but then let me carry back that question and answer that question think or discuss with my colleagues no harm no no harm so in fact we would like to sit in each other's class why because we will learn faster learning from a colleague is much faster because I am learning faster okay so anyway I am digressing here and there but these are few things which are important okay so same thing same interpretations whatever you all told reasonably correct what we understand is any plate see for example a professor said that one of the points should be at initial condition for example if I take soil or rock or earth material or any insulating medium whose thermal conductivity is very less see if I impose heat on one side if I if this is made of very highly insulating material of the order of 0.001 or 0.05 let us see if I apply heat flux it takes long long time for example soil let us say you take a bowl of soil you apply constant heat flux boundary condition on the top how much time do you think that it will take for the temperature at the bottom most layer to increase to feel that that fellow has been heated can you tell me or to reach steady state it will take days that is why for people try to come up with mechanisms to measure thermal conductivity on the basis of transient measurements for insulating materials because to reach steady state it takes long long time so this can be considered as semi infinite because the bottom most layer has not reached the is still as professor said it is still at the initial condition it is still at the initial condition other case let it be a plate of finite thermal I mean high thermal conductivity it is a metal it is a metal but when can I apply it as semi infinite medium for that short time during which my bottom most layer has not felt the heat what I have given on the top layer till that time I can treat it as that that is that is what professor said professor said that one of the points should be at the initial condition within my time frame whatever I am considering but for metal it is very obvious to see that it will be semi infinite for very short time it can be sometimes milliseconds practical use because here why am I using semi infinite medium because I can get the closed form solution I can get the practical use I can use several examples can I postpone this question for a minute when I get the solution in fact this is the methodology one adapts for measuring the heat transfer coefficient using liquid crystal liquid crystals I will come to that I will come to that is but to answer your question in simplistic terms we get a closed form solution for that period of time as long as it is semi infinite medium I get a closed form solution one time solution if I solve it and keep it is possible to get the closed form solution not only for constant wall temperature closed form solution so far we got for only constant wall temperature if you take constant heat flux boundary conditions things get complicated so but it is not so that way for semi infinite medium you can get closed form solutions for not only constant wall temperature constant heat convective boundary conditions all cases you can get okay so that is the answer for your question now let us see the nitty gritty I think before we solve semi infinite medium please make sure that you spend 5 to 10 minutes in the class to define or to make them understand what is semi infinite medium and these are the boundary conditions so this is this is the boundary condition what we have okay so this is the boundary condition that is I am taking constant boundary constant temperature boundary condition here as x tends to infinity is ti that is what professor said and tx, 0 that is a t equal to 0 we have ti I have taken eta equal to x upon square root of 4 alpha t for a minute I am not going to tell you why I have taken this form okay I will postpone that for a minute if I take this eta and transform this equation in terms of eta rather than x and t which is what it is done here is that okay if I transform that equation if I transform that equation I get what is the difference between this equation and this equation one is PDE another one is ODE what happened what trick I played to convert ODE sorry PDE to ODE that who is doing this trick only question unanswered is only question unanswered is how did I know that eta equal to x that I will answer so why did I do this PDE conversion to ODE because I can get the closed form solution solving PDE is difficult compared to ODE it is as simple as that okay so then now that I have ODE I can go ahead and apply appropriate boundary condition and do all the mathematics full algebra is there I am not going to do that I am not going to do that I get the temperature distribution in terms of error function in terms of error function in terms of error function this error function is already listed if you just see error function is already listed that is this integral is called the error function 0 to eta e to the power of minus eta square d eta into 2 by square root of pi is called error function it has been integrated and kept can you tell me why I have to keep moving this so this temperature distribution you see T s what is T s here constant wall temperature that is the boundary condition I apply T i is my initial temperature and this is the error function so likewise of course I can do the heat flux also minus k dT dS and I get the heat flux also now let me take up the question how did I get this eta equal to x upon square root of 4 alpha t so for that we will take the recourse of Professor Bejan's method scale analysis ok so what is this if I take a body if I take a body and take a core region small core region if I apply that temperature if I just plot the temperature if I apply constant temperature wall temperature all of a sudden what will happen my temperature is what is happening here what am I doing here ok hot body I am yes it is not it is a hot body I am putting all of a sudden into a cold fluid cold fluid which is T infinity fluid temperature the it will take a while for my temperature to seep and reach the center is that right so for just a short time it has reached this is my surface temperature which is higher than the fluid temperature and it will take it will this is the initial temperature this is the initial temperature only for a small thickness delta this temperature gradient is felt is that right are you with me so far fine so now let me take del square T by del x squared equal to 1 upon alpha del T del T ok so if I take del square this can be written del square T by del x squared as del T by del x extending to delta sorry at x of the order of delta minus del T by del x at x equal to 0 or of the order of order of I am using this term order of this tilde represents order of order of means whether it is in centimeters meters are in so I give this example always in the class if I lose 10 rupees in today's world perhaps I will not bother so much if I lose 100 rupees I will bother if I lose 1000 rupees definitely I will be worried I will lose my sleep overnight so 10 rupees order if I lose the money of the order of 10 rupees I will not bother if it is that means 10 rupees 10 rupees when I say it is of the order of 10 it can be 20 it can be 25 it can be 15 they are not equal but they are of the order of tens of rupees but if it is 100s they can be 100 110 120 130 so this example if we give they will understand what is the order of or mm and meter if it is of the order of mm and meter if you compare which can be neglected mm can always be neglected 1 mm need not be equal to 2 mm when I say of the order of it can be 2 mm 3 mm 4 mm 5 mm but definitely 5 mm to 1 mm is much much smaller than my 1 meter it is much smaller than my 1 meter okay so here okay so what will happen to my del square t by del x square and what will happen to this so at x tending to delta at x tending to delta do you see any temperature gradient do you see any temperature gradient no no but at x tending to 0 yes what is that initial temperature minus surface temperature upon delta is that right and that is that that if I substitute that if I substitute in the numerator what do I get what do I get ti minus t naught upon delta whole divided by delta okay so I get ti minus t naught upon delta square what is there on my right hand side how what will I get del t by del t please write down along with me let us not lose time because of this technical snack del t by del t what is the scale of del t by del t t naught minus t i it was initially t naught it came down to ti from here to here within a time span of t okay t naught minus t i by t now I have to equate the scales of the left hand side and the right hand side what did I get what do I get delta squared is of the order of alpha t so delta is of the order of square root of alpha t so that means what delta means what what is delta here x x upon square root of alpha x upon square root of alpha t so eta what did we take eta as what did we take eta as x upon square root of 4 alpha 4 is our contrivance 4 if you throw out also you will get the solution we just want to do not want to have tools and force in my final solution so that is why I have taken 4 even if you do not take 4 you are going to get you can convert PDE to ODE this is the idea this is the scale analysis this is what professor bhejan proposes it is quite difficult to teach a little bit to be guys but I think we should otherwise we will not otherwise student will anyway going to ask you why is eta equal to x upon square root of 4 alpha t how did I know that I should be taking like that yes the trick is it converts PDE to ODE but how do I know that to answer that there is no other way except to take the recourse of scale analysis scale analysis this is very important even if see emphasis can be more on these insights rather than on the algebra algebra they will anyway work out they are very far if you put that in the notes if they get doubt they can figure it out okay so these insights are something which we need to focus more on okay so now I have shown for constant surface temperature constant heat flux okay one can derive maybe it is not straight forward on the other day Saturday a student came saying that sir I am getting only this term I am not getting I have to sit down and derive I am not at derived yet I said I am not derived I will have to sit down and derive so we can ask them to solve or derive for these equations it appears that it is straight forward to come back to your question I had postponed your question you see now in this you see now in this what do they do in liquid crystal thermography you said what are the applications I am trying to answer liquid crystal thermography what is a liquid crystal liquid crystal is that crystal whose color changes with the temperature now if I have a CCD camera and capture that temperature I am going to get the with the color if you how do I get color to temperature in today's matlab if you just give that image to matlab it will give you any picture is consisting of R, G and B red, green and blue so if you take it as a monochromatic image that is black and white image R will become equal to G will become equal to B if you give it that to matlab so it will give R equal to G equal to B between 0 to 2052 based on number of bits 2 to the power of it is converting analog to digital so number based on the number of the bits there will be number based either 2052 or 2 to the power of 6 based on the number of bits what was I trying to say yeah now you can convert this bits to temperature you got the temperature now this liquid crystal paint I will put on a perspex plate why perspex plate or a plexiglass and an acrylic because it is an insulating material so that I can apply semi-infinite medium assumption I can apply semi-infinite medium assumption this side is when I all of a sudden apply hot air or hot water into my test section which is made of a plexiglass in which I have painted with liquid crystal other side is not going to feel the heat for that short time so within a plexiglass I can apply semi-infinite medium assumption I get temperature as a function of time now you see this equation t at a given x as a function of time I have recorded now I will iteratively play with this h until at every time step my computed time and computed temperature and the measured temperature so I can get my h point is you can design experiments based on semi-infinite medium based on semi-infinite medium people measure k and cp why it is fast measurement it is over in thick of a time only thing I need to have a data logger for measuring it so semi-infinite medium that was a very good question what you are in fact we have not covered measurements in fact measuring k and cp itself is a ocean by itself but I think we can there are very good incorpora and David problems where in which the design of experiment to measure k and cp is there that time we will put in tutorial so we can design the experiment by solving the tutorial problem okay what am I there is a mistake here this is x by 2 square root of alpha t plus whole of this term should be inside the bracket we will correct that yeah please correct please note this down CD has been burnt already we will correct only in the later iterations see there is a problem in fact the quality of our transparencies are not so great because we are not projecting power points we are projecting PDFs on the last day as usual nth moment Yiddha Kalesh Ashtarabhyasaha so we realized in the last minute that there have to be margins on all sides so we converted ppt axis to PDFs in that process all our figures are which are looking very bright and catchy are not so catchy now okay but if we we will reduce those ppt's in a proper fashion so we have burnt PDFs which are stones okay this is a problem which is cooked very rice problem that is where should I keep the pipe in an underground layer so that my water which is going through my underground pipe doesn't freeze so I can apply semi infinite medium assumption I think I should not be taking more time now I will not solve this problem so you can solve it and figure out that 0.8 meters you will have to dig down to lay your pipe so that the water in the pipe never freezes this is the problem in cold country not for us but for cold country this is the problem so that's why this problem has been given because anyway people who write jungle and improper they are all from cold country so that's why that problem has been cooked okay so now multi-dimensional conduction I will simply touch and go so if you know plane wall and infinite cylinder if you take the multiplication the product of the two you get for short cylinder it is not that the solution is always product sometimes it can be division also okay it depends on the nature of the problem it's quite difficult one of the students asked this the question how can I physically feel this okay how can I physically feel that plane wall into infinite cylinder is going to give me the finite cylinder solution that was the question asked so I could not answer the question so but then later on Venu told me that Professor Vedula while teaching used to teach that and we realized that we cannot always say that the product of a solution only is product of two solutions is only going to give me the solution it can be quotient also it depends it depends from problem to problem intuitively it's quite difficult to answer that question is that okay so with this I think there is a problem I would like you to solve it yourself there is a problem here with this I am going to solve stop multi-dimensional sorry the transient conduction okay