 Let us now visit professor Kirchhoff okay, we know Kirchhoff only through Wheatstone bridge sorry not Wheatstone bridge where we apply in electrical circuits okay but the same Kirchhoff has given us a law for emissivity equal to absorptivity already we have stated that. But then this is little involved explanation let us go through this line by line because this is a proof which is based on explanations what heuristics that is based on pure logic there is no derivation as such but interestingly he had first stated without derivation later on he came up with heuristics. So it is not a derivation per se but it is heuristic based off basis of arguing things out and saying that okay it is convincing okay but in true sense it is not derivation okay so Kirchhoff's law what does it say let us take a large isothermal enclosure that is which is maintained at a temperature Ts consider a large isothermal enclosure of surface temperature Ts within which several small bodies are confined okay but these bodies are so small that one is not going to interfere with the other okay. Now this surface at least this cavity is going to this can be visualized as a what can I visualize okay let us go line by line since these bodies are small relate to the enclosure they have negligible influence on the radiation field which is due to the cumulative effect of the emission and reflection by the enclosure surface because enough emission and reflection is happening so one is not getting affected by the other regardless of the radiative properties such as surface forms a black body cavity is that okay fine according to okay accordingly regardless of its orientation what is the irradiation what is the irradiation experienced by any body any body overall because it is going to get irradiated not only from the cavity cavity that is Ts but also other bodies so now what will what will be the irradiation that is the because now we are saying that said that everybody is a black body so irradiation will also be equal to EB at a temperature Ts now under steady state conditions under steady state means all bodies are now going to reach in thermal equilibrium with reference to my enclosure temperature so all bodies are going to sit at one temperature that is Ts so that is what is being told in the first two so if I do energy balance if I do energy balance what do I get that is what in fact professor had already told this is e dot in minus e dot out equal to 0 that is all what is this what is the first term saying the absorptivity of my surface into irradiation into a1 into a1 minus of minus of this is a black body that is it is emitting e1 Ts a1 equal to 0 so what do I get what do I get and of course for a black body irradiation is also equal to emissive power is that right can you see that yes so if I now equate this what do I get alpha equal to alpha 1 equal to e1 Ts upon EB Ts this is what we are derived so what does this say what does this say and I am halfway through I am halfway through this is only saying that it is again the same as what we had said no surface can no real surface can emit more than the black body but that is not the story is not over now if I take total hemispherical emissivity total hemispherical emissivity is also defined as you already defined this as e of t upon emissive power of real surface upon emissive power of black surface so if I equate that if I compare sorry this equation with this equation I am going to get emissivity equal to observe this is not a pure derivation but this is as I said logically we are arguing out and saying that emissivity is equal to absorptivity but the restrictive condition is that this should be at the see in this derivation itself we have said that my body is at they are all in thermal equilibrium so that means the temperatures are all same so this is valid only when temperatures are same that is emissivity is equal to absorptivity at the same temperature so this is one thing and of course there are various things one can one can they say that one can argue out and say that spectral emissivity is also equal to spectral absorptivity and again spectral directional emissivity is also equal to spectral directional absorptivity so this was done by Kirchhoff again as usual my usual tendency is to put the photograph and put something about him so Kirchhoff in fact he has circuits last were done as a part of his seminar exercise okay so and later on he worked on the same thing for his desertation but of course he did this in 1859 and the spectroscopic work whatever I was talking about that was all done along with Kirchhoff along with Bunsen Kirchhoff and Bunsen have worked together and found because with spectroscopy only you can identify as professor was saying you can identify the elements that is what precisely they did and found out cesium and rubidium okay so okay I think we will spend time on only one another thing which is very important that is before we move on to gray surface two things I will I know you are all hungry our energy levels are going down but I think this is reasonably well understood thing what is green house effect I am sure most of you are explaining this in the class so I would like to listen from you rather than me telling you with the help of this figure what is green house effect I always tell this in a car we feed if we park a car in a broad daylight and we when we get into that we always feel very hot unusually hot okay so why is it so yeah you should be explaining with reference to transmissivity I am showing transmissivity with reference to wavelength of course for different thicknesses that does not matter how do you decide whether that is high frequency or low frequency that is solar radiation that is the solar temperature is sitting at 5800 Kelvin so for the first time or so only I am explaining this for the other guys please feel brod please pardon me for that see you have for the low frequencies that is between 0.3 to 3 micrometer 0.3 to 3 micrometer what is the transmissivity it is maximum it is maximum so here you have maximum transmissivity but at but within the car the temperature is temperature is quite low it is almost near about room temperature though that is high frequency the transmissivity there is almost 0 so whatever comes in cannot get out from by radiation that is why people keep their car windows slightly open so that convection some air blows inside and some convection happens and it is little cooler okay this is a very this is I think this we should spend a lot of time in the class why because this is another instance where wavelength and temperature for a given temperature what is the wavelength you will have to go back I am not going back here because you are all you conversion you have to go back to plants distribution you have to in fact we can put plants distribution again here we can we can again copy paste that plants distribution here and say that okay otherwise how will you know 5800 is 0.3 to 3 micrometer it has to come from plants distribution so we have to show the student the plants distribution and make him appreciate that for a given temperature I am in this domain at 5800 and for room temperature I am somewhere here so that is what I need to show so this is what we need to emphasis for the students okay next thing is the gray surface what is a gray surface someone was asking me over coffee so gray surface is that surface which is dependent on this tells this is the summary of everything this is the summary of everything okay real surface is going to be dependent on direction and wavelength diffuse surface is one which is independent of direction direction let us not get confused diffuse diffuse the very fact diffuse the word diffuse means it is independent of direction okay you use you use the word actually my head is little diffuse today you use that word actually means you do not know what you are doing you are directionless you are rudderless that is what you mean actually I mean why I use these examples because when you relate with English it becomes that much easier to understand that that is why I said in the very first lecture you use technical terms in your colloquial language then those terms become very obvious to you okay so epsilon lambda what does this here if a surface is dependent only on wavelength then it is gray surface if it is diffuse and gray if it is diffuse and gray sorry sorry sorry what did I say what did I say that is right no did I make a mistake when I told gray surface what is a gray surface what is a gray surface just to check out whether I told the right thing or not yeah gray surface is that surface in which it is independent of wavelength now it becomes a diffuse and gray means and direction okay I think these are things which we should be very clear about okay I think with this I will not I will skip this commentary because it is little complicated I do not think we should be touching upon that but I think we will okay so before we closed for lunch I think we had spent the whole morning trying to understand some fundamentals definitions related to radiation now having done that I think this part is relatively easy compared to what was done in the morning so we are going to talk about interaction between two surfaces so all this while fundamentals of radiation definitions we were dealing with one surface we were not concerned with its interaction with surrounding surfaces or surrounding objects and we were dealing with a particular surface whether it is with respect to emission irradiation absorption transmission reflection all those things was related to one particular surface and we did not care about what was the source of energy or source of radiation that was incident on that surface we did not bother about it but in real life that is not possible so always we are going to have interacting surfaces and I think these things students will understand very easily you can say if I examples are given here facing the fire from front or back you have different kind of radiation than from the side if I stand directly under the light I feel it will be hotter than if I stand like this so all these examples students will appreciate very easily we do not have to spend too much time on that. So we quantify or we are now we have to give a mathematical form or some kind of a number to this concept of fartherness, nearness, relative position with respect to the parent surface and so on and so forth so mathematically we put together some kind of formulation and we call it as a view factor configuration factor or view factor I think everybody likes this will appreciate it is quite easy and always we like to give a problem also related to view factor calculation why what is it physically telling us it is telling us what is the fraction of energy which is leaving a particular surface which is being intercepted by the receiving surface that is all it is it is a fraction so it has to be between 0 and 1 and mathematically we are going to derive a simple equation for it all all textbooks will have this all of us will do this also and there are some general rules which will be there view factor with respect to itself need not always be equal to 0 depends on the geometry of the surface so points to take over take home would be the geometry of the surface the orientation the relative distances all these three are going to contribute the size of the surface geometry of the surface the orientation the same two plates okay if I think of these two these are two plates if they are one above the other the view factor is different if I move them further away it is going to be different if I move them like this like this like this each time you get a different number for view factor it is primarily because as I move away the amount of radiation which surface to receives from surface one will change because of the nearness or farness or the orientation I think that is quite straightforward so Fij represents a fraction of radiation leaving surface I that strikes surface J 1 to 2 means leaving surface 1 striking surface 2 2 to 1 is the other way and if this is da 1 we always like to deal with infinitesimal areas so this is da 1 and you think of the outward normal N 1 da 2 and this is the outward normal N 2 the angle made by the ray joining the center to this center makes an angle theta 1 with the normal here and makes an angle theta 2 with the normal here and this area da 2 which is the receiving from one subtrends the solid angle d omega 2 1 okay these are all conventions 2 1 okay of 2 with respect to 1 that is what it means and the center to center distance is given by R so with this with this diagram we can say solid angle d omega 2 1 is nothing but da 2 cos theta 2 divided by R 2 square so this is again a definition of solid angle we have and then we say the portion or the fraction of energy which is going to be received q da 1 to da 2 is nothing but what I intensity so we always have to keep that in mind so what is the dimension of this quantity I units what per meter square so I cos theta 1 da 1 d omega 2 2 1 so where did I get this from where did I get this equation from have you seen this equation before definition definition of intensity is nothing but d q divided by d a cos theta d omega d lambda etc etc same thing now I have I have cast it in d q is equal to di times all these things so nothing new so far and how do I know it is from 1 to 2 and not 2 to 1 omega 2 1 not omega 1 to 2 okay so da 1 cos theta 2 things are there to tell you what we are what is the sending surface and what is the receiving surface and here we are dealing with the sending surface by suffix 1 receiving surface by suffix 2 subscript 2 so da 1 cos theta 1 what is this cos theta 1 coming from where is this coming from why is this cos theta normal of the okay like that so it has to it has to be rotated by angle theta 2 so that the ray is perpendicular to that surface okay so now is algebra after we have understood this it is just algebra I will substitute for solid angle definition so it will be da 2 cos theta 2 divided by r 2 sorry r square and total rate of radiation let us write this equation let me write here da 1 da 2 that is energy leaving da 1 reaching da 2 I 1 cos theta 1 da 1 d omega t 1 is da 2 cos theta 2 r square by r square see if I have to make it q dot d a 1 to a 2 what should I do double integral so that is double integral a 1 a 2 I 1 cos theta 1 cos theta 2 upon r square d a 1 d a 2 what is the energy leaving q dot a 1 what is the energy leaving q dot a 1 no not just that something is getting reflected also it is emitting and it is receiving something from some other sources out of which some amount is getting reflected so I have to take not the emissivity emissive power alone I have to take the radiosity so that is j 1 d a 1 which is equal to if it is a diffuse no I do not have to make any no assumption so j 1 is equal to e b 1 e b 1 plus rho 1 g 1 how do I remember this I do not have to do any special effort j is equal to the sum of emission plus reflected component of all incident radiation always I draw this for myself so that is g is coming emission is e e and some amount is getting reflected how much is getting reflected rho times so g equal to j is equal to e plus rho if that is the case that is what is given here in fact from that I will get so this will be e plus rho g times d a 1 so what are we looking at we are looking at a ratio fraction so what it will be q a 1 to a 2 divided by q a 1 correct divided by q a 1 so if I just go back one slide so if you are talking of the total quantity total rate of radiation in all direction that is what we have to make this assumption of diffuse it is going to be j times a 1 which is nothing but pi times i 1 this pi i is equal to j comes from the fact that it is a diffuse surface for diffuse surface for emission plus reflection back again what is diffuse surface something which has no directional preference for anything okay so it can be diffuse for emission but it need not be diffuse for reflection what we are saying is this surface 1 is diffuse both for emission as well as for reflection that means it has no directional preference for both these things so radiosity also is a diffuse it is a it is a diffuse emitter and a diffuse reflector so I can write j is equal to pi times i is what intensity okay it is and pi is introduced because of this integration around the full direction okay so all I have said this is a heat flux j is a heat flux I am replacing j by pi times i I am carrying this pi because of this directional business okay so this is heat flux times the area which is watts is that clear? I think it is easier to remember colleges okay portion of this radiation which strikes this is a if I integrate it in all directions ha em diffuse you go back what was our definition E is equal to E is equal to pi we are not doing black body E is equal to when I say it is diffuse you remember in our first radiation 1 E equal to pi i here that is what is open directly okay so this is nothing but black body E is equal to pi i diffuse if it is independent of the E is for emission j is equal to pi i where the subscript will be E plus R meaning it is diffuse for both instead of talking it will be if I want to write it will be j is equal to pi times i of E plus R that means this is for emission and this is for reflection and this comes because of diffuse nature it is diffuse for both these things okay so fraction is what we are saying now this is Q of A1 to DA2 it was DA1 to DA2 now we are saying A1 to DA2 so integrate over area 1 and double integral will make it integrated from for both the area so one large surface to another large surface and if I go to the next thing the fraction essentially will be obtained by bringing this that is it so I will have F1 to 2 is nothing but 1 over A1 integral A1 to A2 it is just algebra so there is nothing to teach conceptually here cos theta 1 cos theta 2 DA1 DA2 by pi R2 what it tells me is that it depends on size relative sizes of the objects given by DA1 and DA2 it depends on the physical distance between the objects R square which is coming there it depends on the cos theta 1 cos theta 2 or the relative orientation okay how is it oriented that is why I told you the same parallel plates I mean same set of rectangular plates when it it is moved this way this way this way you have the different view factor because the orientation has changed the cos theta will get changed and of course this very same equation will give us F21 and that and if I multiply this here F12 times A1 is equal to F21 times A2 because this integral essentially is the same so that will give me what we call as the famous reciprocity rule and I just quickly go through the other rules also and then we will come back to this reciprocity relation other is summation rule which says energy is conserved so summation of energies going from surface I to all other J surfaces 1 2 3 4 up to N should add up to what it is what is leaving so that is energy balance we cannot violate that if you divide through by energy that is leaving the surface you get view factor of surface I with respect to 1 plus view factor of surface 2 with respect to 1 all the way till N with respect to 1 the sum has to be equal to 1 so again that is what it says so summation of view factors from surface I to often enclosure to all other surfaces of the enclosure including itself must be equal to unity and students will be able to write all this but you ask them what is that is the problem so example 3 surfaces F11 1 2 1 3 summation equal to 1 one of these can be 0 F1 2 1 can be 0 depending on the orientation of the geometry but in general the rule is all the view factors have to be taken into account I do not want to I think we will go back to splitting rule also yeah superposition rule suppose I have to deal with this to this view factor F1 to this long surface and it is split into 2 and 3 I can write this as F1 to 2 plus F1 to 3 many times I will want to find out the view factor of F1 to 3 actually so I will get F of this to this minus F of 1 to 2 that is where this will be useful surface I to a surface J is equal to some of the view factors of surface I to several parts associated with surface J reverse is not true that is not true convulsing is definitely not true that is a typical question which also university where they will ask so I am not going to do this so F1 2 to 3 is equal to 1 to 2 plus 1 to 3 there is F2 to 3 2 3 to 1 is is not equal to F2 1 plus F3 1 okay and then quickly go through this standard okay plane surface 1 to 1 is 0 plane surface view factor to itself is 0 convex surface view factor to itself is 0 concave surface view factor to itself is non-zero okay this is something which we have to keep in mind inner sphere inner cylinder to outer sphere or outer cylinder view factor is 1 outer sphere to inner sphere outer cylinder to inner sphere cylinder view factor is not 1 because part of it is going to see itself now tell me if I reduce the diameter of this inner inner cylinder let us take this cylinder it is easy to visualize cylinder if I reduce the diameter of the inner cylinder to a smaller and smaller value what happens to F2 1 decreases remain same increases F2 1 F2 to 1 F2 to 1 it will tend to 0 because bulk of the energy is seen by the parent surface itself very few very small fraction will go to the other surface now if I bring it closer and closer when I make the gap between the two small then what happens 2 to 2 will almost become 0 2 to 1 will almost become 0.9 whatever okay what is the inherent assumption in this whole thing let us we are dealing with cylinder right so let us deal with cylinder I will draw two cylinders coaxial once this is cylinder 1 and this is cylinder 2 okay longer what am I saying F1 F F1 to 2 etc what is the meaning what is the inherent assumption concentric is correct we are not changing geometry and defects are neglected okay what other major assumption okay right or anything with respect to temperature entire object entire body is at uniform temperature okay why how do you know that assumption which aspect of the derivation gives you that thing implied do not sleep which aspect of the derivation tells you or is where is it implied that the temperature is assumed to be constant which aspect of the derivation tells you about constant temperature sourcing both in here it is somewhere implied what okay conversely what if it was varying temperature what will happen no let us say we are going digressing too much but why why this constant temperature is important I am saying let us say this is at T1 this is at T3 this is at T2 and this is at T4 hypothetically okay what will happen huh intensity amount of energy emitted by this part will be different than what is emitted by this part whereas here I have not even bothered about that I have taken one I intensity is constant right was intensity treated as a function of geometry or R or L or anything nothing was that I was a number okay I was always treated as one whole quantity so temperature being constant is an implicit so can I not do anything with such problems of course you can do what you will do is you have to be careful and treat each isothermal portion as one element reaching out to you so isothermally each portion is one element now continuously if temperature is varying what do you do piecewise linear you can may piecewise constant not piecewise linear piecewise constant so what why am I telling all this even if you impose constant wall heat flux condition or constant yeah constant wall temperature condition or constant wall heat flux many of the times you will not get a constant temperature depending on what is the environment especially if heat is forced to be removed only by radiation and natural convection the heat transfer coefficient is generally very very poor especially in the lower half because it is only natural convection and you see a strong temperature gradient actually so what you will do is you will deal with each thing as a ring something like this each of this is at t2 t1 t3 t4 t5 so on and so forth and correspondingly on the outer cylinder also you will have some kind of such rings with which this is going to talk this is going to talk to this this is going to talk to the next one so on and so forth same thing for everything so t3 is going to talk to the top most string it is going to talk to the second last third last something in front of it directly and like this so this assumption or inherent simplicity of this form can be very deceiving where you have a non isothermal surface things become complicated very quickly because you have multiple channels multiple parts for heat transfer whereas here you have only one part from 1 to 2 you do not care and again here axial conduction will be there that we are neglecting that is okay so this is one and there are formulae available catalogs are there view factor algebra catalogs are there for various geometries for getting the view factor and I think many textbooks carry bulk of these things on the net also you have how else catalog etc which are available parallel discs parallel rectangles perpendicular surfaces with the common edge parallel plates different dimensions inclined plates with the common edge so on and so forth okay so all these people have done and come up with these expressions which are presented and all we need to do is use them correctly that is all and these are also presented in the form of graphs I think nobody is nobody uses these graphs anymore because we can do the algebra also quite easily so I am not going to spend time on this okay so this is again the small problem associated with view factor calculation this is also not needed symmetry rule this is again obvious one if the geometry can be made into a symmetrical geometry then what is going from 1 to 2 should not matter whether it is the surface 2 or 3 it should be identical so that is one thing and associated with the tetrahedron okay so this is a problem 1, 2, 5 plus 4 plus 2 plus 3 all of them add to 1 so if you know what is the view factor from 1 to 2 everything else will be identical 4 times so 1, 1 is 0 and 1, 1 is 0 and all are equal so they do not turn out to be 1 by 4, 1, 2, 5 okay so these are just exercises which are quite simple should we go we will just take it and leave it okay so crossing method is one of the methods which is used was used actually even now it is used to get the view factors for infinitely long surface now what is this infinitely long surface why is this infinite long surface assumption what if it is not infinite end effects are neglected because of this so whatever radiation is there and believe me actually if you have finite length geometry many times is infinite length assumption if you use you do not get correct actually you do not get it so blindly people use infinite length geometry formula for finite length geometry things can be off as much as 100% okay I have two short cylinders of this dimension I cannot use infinite length formula for that is not correct okay so that is something again this is just buried under used because it is available formula we will use Bernoulli equation we will use Kirchhoff's law epsilon equal to alpha we will use whether the conditions are satisfied or no okay so such geometries especially infinite length geometries we can be considered considering them as two dimensional so end effects are neglected and view factor is done by this so called cross strings method what it tells me is f 1 to 2 is nothing but the lengths of the cross strings minus lengths of the uncross strings which is this divided by two times string on surface I that is the source surface so this is two times l 1 okay so these cross string method is generally applicable when two surfaces consider even share a common edge such as a triangle common edge can be treated as an imaginary string of length 0 okay so it is quite a powerful thing but again you have to be careful okay so now fun part is over now we will come to business so okay having studied view factors what are we going to use it for we have to study heat transfer interaction so if heat is all the while before it was one surface now two surfaces are there so let us try to see what is going to be the interaction between the two surface just because surface one is giving out energy surface two is not going to sit idle it is also going to give out energy and part of it is going to be seen by this so if this surface for example is at say 1000 Kelvin and that is at 400 Kelvin over time what is going to happen this is going to lose heat that is going to gain this I think logically we can tell so that means there is a net loss of energy from this surface which manifests itself by a decrease in temperature net gain of energy of that surface which is manifesting by a rise in temperature so what I have to say is E in minus E out is equal to we have never used that equation here but inherently that is there all the time it is under current E in minus E out E generated 0 is equal to DE by DT which is manifesting itself as a change in temperature with respect to time once things have reached steady state you will not have any heat transfer exchange between them so this DE by DT is also there and it is there because of the temperature difference and what happens is this goes down from 1000 to whatever 908 and so on and so forth that energy content changes and so E in E out also will be changing in radiation because of the change in temperature so essentially what we are going to do is we are thinking always in the radiation only the steady state condition yes we are not considering unsteady radiation at all anywhere so also we are going to take only the steady state radiation what is the heat transfer between surface T1 and T2 when they are at this given temperatures after that next instead of time T1 has gone down and T2 has gone up we are not concerned with that with these given values of temperature or at this instant what is there. Somehow I am managing them that they are continuously being monitored such that the temperature is anchored at T1 and T2 so that is a inherent thing we are fixing the temperature and trying to solve in real life it will be a different ball game. For example, if you put things in furnace when you just load the material inside the furnace definitely my wash temperatures are coming down because of my loading I am loading no again it will take a while or otherwise I have to increase the heat flux to overcome this loading because of putting the boulder or whatever I have put inside. Steady state does not come just like that but here in all our analysis throughout we are taking steady state. So Q1 to 2 what does this tell me this is just giving me the net energy transfer from 1 to 2 that means at steady state Q1 represents the amount of energy which is leaving one something is coming on to 2 that something is coming into 1 the difference between these two what is leaving minus what is coming in because of surface 2 here we are going to consider only two surfaces and what is the nature of the surfaces we will deal black body first because black body is easy so two black bodies are there those two black bodies are at finite temperature T1 and T2 they have areas A1 and A2. So what is the net energy transfer logic tells me if T1 is greater than T2 Q1 to 2 is greater than 0 correct higher temperature body there will be a net energy loss energy transfer from 1 to 2 correct if I write Q2 to 1 2 to 1 will be receiving energy from 1 so the signs will be opposite that is all. So what I am saying is now let us use reciprocity A1 F12 is A2 F21 this will be useful little later so Q1 to 2 inwards E in minus E out instead of E in minus E out I am writing radiation leaving the entire surface 1 that strikes to I do not care about other things 1 and 2 there is open space here open space here I am not concerned with that I am dealing with what is between 1 and 2 alone what is between 2 and 1 alone. So what fraction of energy leaving 1 reaches to minus what fraction of energy emitted by 2 or energy leaving I should not use emitted what fraction of radiation leaving to strike surface 1 why should I not use emitted but I corrected myself. Energy is what we have to deal with not just emission because emission happens by virtue of temperature but reflection can happen because of other components of energy coming on to the surface what is coming from a surface 3 here will also get part of it will get reflected in this direction to surface 2 okay so radiation leaving so Q1 to 2 therefore is what. Let us look at one term the other term is identical EB1 is the emissive power watt per meter square heat flux associated with that surface 1 which is the temperature T1 times A1 will give me total heat transfer rate watt that is what is coming out of the surface how much of that reaches surface 2 is multiplied obtained by multiplying this number with F1 to 2 this is what is going from here to here what is coming from there to here is A2 EB2 again the same logic times F2 to 1 and now I use reciprocity theorem which is here this I know I would like to deal with surface 1 so I will use A2 F2 1 is A1 F1 2 so I pull that out common so Q1 to 2 is A1 F1 2 EB1 minus EB2 and we know EB is nothing but sigma T to the power 4 that sigma also comes out that is how you get this famous equation T1 raise to 4 minus T2 raise to 4 times sigma epsilon 1 A1 because it is blackboard if it is not a blackbody what will happen emissive powers will be associated here we will see that as the next steps so Q dot 1 is nothing but Q1 to 2 which is same as minus Q dot 2 just bringing this to perspective if I have n black surfaces same thing can be extrapolated Y2 can have multiple black surface so it is just the summation so the net energy transfer the net energy net radiation heat transfer from surface 1 is nothing but summation of this j goes from 1 to n, n could be 2, 3, etc., etc. Negative value of heat transfer indicates that net radiation heat transfer from surface I that is surface I gains energy instead of losing also net transfer from a surface to itself 0 regardless of the shape of the surface from a surface to itself is 0 that is what should be miss okay so Q I for 1 to 3 surfaces is nothing but something given like this okay okay put that for that then we will go through that okay so what is governing this heat transfer rate from one to the other there was some concept L by K for plane wall 1 by HA for what if there is no temperature difference there is no heat transfer so that we have to keep in mind so temperature difference is there something is going to prevent heat transfer or is going to change the heat transfer even though there is a same temperature difference that is what we called as thermal resistance we change the resistance we get higher heat transfer for the same temperature difference so on and so forth now we are saying okay in radiation also just because there is a temperature difference yes we appreciate that I have 1000 degree Kelvin 400 Kelvin 2 surfaces I bring them closer I can see it much heat transfer if I take it away lesser heat transfer between the two so temperatures are the same so driving delta T is the same but between these two cases there is a difference in heat transfer so something has to govern the heat flow and so there is some so called resistance which is there analogous or similar in concept conceptually at least I can relate saying that just as we had a conduction convection resistance there must be something which is going to drive the amount of energy flow for similar temperature differences and that we will call as space resistance geometry resistance we will see that so this q1 is summation of q1 to 1 plus q1 to 2 plus q1 to 3 q1 to 1 0 because it is a plane wall q1 to 2 is a1 f1 to 2 sigma t1 raise to 4 minus t2 raise to 4 just like expanding this into 3 terms and the third term is for f1 to 3 what can you say by inspection for f1 to 2 and 1 to 3 if this is an equilateral triangle what is f1 to 2 and f1 to 3 same what is the value 0.5 because symmetrical 1 1 is 0 f1 2 plus f1 3 f1 1 is 0 so other two are same so it should be half so this essentially will boil down to sigma t1 raise to 4 minus t2 raise to 4 as I told you before I like to keep everything in the denominator except some temperature difference here instead of temperature difference this sigma t to the power 4 that is kept minimator why because the sigma t to the power 4 is your what is sigma t to the power 4 emissive power of the black body at that temperature so we are dealing not with temperature we are dealing with the emissive powers associated with that temperature so sigma t to the power 4 is like your t sigma t2 t2 to the power 4 is like another t the difference between them that is the driving potential for heat transfer divided by some kind of resistance to heat transfer okay t physically I can bring sigma down does not matter but it does not make sense to bring sigma down because t1 I cannot give a physical interpretation to it sigma t1 to the power 4 I can relate it to eb1 sigma t to the power 4 is eb2 that is why that sigma is stuck there because I had told delta t divided by everything else one I have to take that back physically driving potential to this okay so when I write it in that form I will get this as in this way and for q2 which is from 2 to 2 2 to 1 2 to 3 I can write it in similar form similarly for 3 and this quantity is dependent on view factor what is coming in the denominator these two terms these two terms these two terms depends on the area of the surface also and the view factor you can relate can we not see that the more they are seeing each other if they are if they are just like this will will the interaction will be more no if they are seeing each other then my view factor so it is only representative so the word is space resistance process used it is it is really you can feel that space resistance we can feel actually space resistance you can bring it closer view factor has increased resistance to heat transfer would have decreased therefore the net heat transfer between them you can change for example 1 2 3 if it was 0.5 versus 0.4 this denominator will appropriately change and you will have a different heat transfer rate okay shape factor configuration factor view factor all are the same shape resistance okay okay how is that difference actually as far as I understand all are same in fact if I am correct every name I have put there I think I think in this way picking see here no no as far as our analysis is concerned shape factor configuration factor angle factor all are same pick and I could have specified what he means when he is using with something else as far as this is used for these kind of situations I do not know yeah okay okay we will see that we will see that this is concerned this is what it is in fact if you see any u g book this is what we will see yeah actually coming close you factor will increase geometry distance decreases I increase decreases I was wondering I have when he was telling I could not see that actually I was wondering why should view factory increase or decrease when they are coming closer after professor told that r squared is sitting 9 my view factor so that is how coming close and so that is my space resistance yeah I keep my hand here I feel the heat more yeah I take it away I feel less near the fire it's always dangerous yes view factory is having to say what is the definition of a view factor what is the definition of view factor cos theta 1 cos theta 2 cos theta 1 cos theta 2 double integral divided by r squared r is what you also had the same question when in doubt think of example like this you keep your hand on the gas when it is burning is it possible to keep no no take it up you can keep why view factor the same fire same hand you don't feel the heat that much okay