# Lecture 1: Introduction for Direct Solar/Thermal to Electrical Energy Conversion Technologies

the following content is provided under a Creative Commons license your support will help MIT open courseware continue to offer highquality educational resources for free to make a donation or to view additional materials from hundreds of MIT courses visit MIT opencourseware at ocw.mit.edu I have a two set of handouts if you don't get a copy it's all on web okay and if uh I do this well every time uh you should get a hand out and my objective is uh for you to focus on listening and discussion not to uh take notes uh so it's all your handout uh or you go uh it's online Okay um it's an experimental course uh the first time I'm offering it uh so it has a special uh say two uh 997 uh Lum UH 60 units uh I uh if you look at the hand out if you printed it yesterday I Chang something and one of the Chang is very important uh this course will be uh pass speld basis okay and what does it mean is if you are undergraduate student uh for example in course two uh pass field uh could count for unrestricted electric but it does not count for restricted elect and if you are course 2 a it does not come for your uh concentration okay and if you graduate student uh this is a not Edge level course it's not Edge level course so those are the uh just for your uh clarification and uh if you look at the handout uh the syllabus uh what we will have first is once a week at this time and also um I have office hour and I have a one ta uh Daniel is here uh you can contact him if you have questions uh if necessary will ask Daniel also set up office hour and uh uh what else I don't have a copy in my hand take a look thank you uh in terms of the requirement homework uh my uh uh the homework is every week I will assign one reading and you will turn in your report one page summary of the reading uh uh you can uh summarize what do you understand what you do not understand and uh uh that's the um I do not see we given actual Lumber homeworks so it's a reading based uh homework and at the end of the uh semester uh you will write a final uh report project report and the detail of the report uh will laun uh in the middle of the uh semester um any questions okay uh if not uh I will uh start like I said uh all the um uh everything is online and uh uh you can draw down whatever notes you want but uh my purpose is not to have you write too much and uh focus on this I do not have a textbook on this uh so let me uh start so this course if you look at the title is a direct solar thermal uh to electrical energy conversion Technologies there are few uh points I want to uh emphasize one is uh is is a a heat based course solar thermal this is all focused on the thermal aspect the solar energy is uh in this course is a big giant uh heat source and uh the other aspect important in this course is the direct energy conversion so the focus is on how to convert the heat into electricity directly without the moving parts and when I see the technology uh uh you uh realize that we move on those Technologies uh some already existing but clearity is not widely spread used right the it's not widely used so they are still under development and uh uh whether we work out or not not down the road uh is not pretty sure but say that's your opportunities you're here to do research uh and future development and maybe you will uh push some of this technology to real world so uh I will uh start with uh this slide to make a case that is important that we think about heat uh you can start either from the input side look at how we are using all the energy fossil fuel or even nuclear the energy source from the energy source to convert this energy into electricity into mechanical energy you can see more than 90% is VP right the uh uh mostly is a thermal mechanical means to convert the energy uh into either mechanical energy uh or uh um electrical energy and the other aspect is look at the end use how much energy is lost right and you count all the uh say uh lers there you'll find out that more than 60 about 60 to 70% of the energy are wasted so uh what we are looking into is whether we can developable alternative ways to either use the input energy or uh convert some of the wite energy into electricity yes uh 10% missing you have to look at the sum of this end here okay okay uh we're talking about the esources so just a few examples if you look at the the driving efficiency as amaz is less than 20% right and if you want to remember it's easy just say about one3 of the energy goes through the tail pipe exhaust the Ser rly goes to the radiator right those are completed wisd so can we recover some of this energy and in fact uh uh uh this is challenging because it's a mobile you have to Fed into the compact systems and in those cases uh uh in terms of people are looking at the VAR options but say uh Direct the mechanical way of converting this energy into electricity is not very feasible so there are a lot of companies looking to Thermo Electric for example and uh now you look at another way that we use energy is the buildings and uh give an example in New England in the winter all home furnace boilers most of time they are on lot very open the arm right and this a combustion process and I end what you were using it for H shower heat up the room right that's a low quality heat Source once you quench to this temperature you you have high temperature combustion you quench to low temperature just take a shower and you are generating lot of entropy and and uh in some regions there big uh cold fire power plant or oil fire power plant right in this cases people take the Steam and Supply the heat to the community that's a electricity heat co uh Co generation Coen plant right but we don't do this at home the technology is not out there yet people are still people are pushing and in fact there is a uh Honda uh is fine for examp developing home stand alone internal combustion base system uh but it's not still not taking advantage during the compression during the heat of the hot water process we are really generating lot of introp we are wasting useful energy when we de develop uh technologies that can do for home cenation industrial process right that's about another third of the uh energy that's used and uh uh there P sources of different temperature worldwide temperature range once I visited aluminum smelter plant so that's a just converting aluminum oxide into aluminum and and the process is a high temperature electrochemical process so you heat up aluminum oxide to Mel State and then you use electrocam way to extract the aluminum and those plants are typically build just beside power plants electrical power plants the plant that I with is 500 megawatt electrical input okay that's a huge power generation uh say power required just give an example how much is 500 megawatt a typical power plant is about the 1 GW and the word solar installation Sol phot will take installation installation it's about 6 GW now so one power plant uh one aluminum smelter plant takes 500 me and of course there is part of this aluminum oxide to aluminum energy stored as chemical energy but half of this is dumped into environment as he right and there's right now there's no technology to really recoup this into uh anything useful so now you think about this is the waste he right Renewables when we think about the fot will take uh solar we often think about foto Volt take but fundamentally solar is a high temperature P Source uh anyone know the equivalent temperature of the solar radiation coming to close it's about yeah 6,000 K 580 100 5 800 okay and geothermal those are renewable those are heat sources right so uh how we can can we use as a heat and in fact if you look at how people what the people are doing now uh the solar utilization the uh biggest installation is in sort of water not in P just hot water and uh uh I came from China when I grew up there was no hot water so I don't didn't sell them I take a shower no water in the winter and noway when I go back home we have solar hot water and uh it was a really fundamental lifechanging technology very cheap but it's actually Hightech so here I'm showing uh uh the uh sort of how water system that's videly used in China and give say aut magnitude it's about 100 million square meter installation 100 million square meter and if you take a 100 million square meter you take a solar uh energy one square meter is about 1 kilowatt and a Thal efficiency they collect the heat around 60% so you multiply is together how much you get you get you get about uh 60 G right Thal energy being used and uh I said as a Hightech uh uh this is a a evacuated tube vacuum tube okay and so there's a inner layer outer layer and in between is a vacuum to and in the inner layer to there's a there's a a coding a code this coding will absorb the solar radiation but the minimize the seral radiation and it's done very cheaply in a typical a tube like this high is only uh I was the on a factory $1.5 to this all manufacturer so because of the really the low cost it's why in developing countries and of course you talk about the electrical generation uh in uh this is a their installation in US Creer Junction where the t is from the sand is a forur through those parabolic TOS heop the fluids and these fluids are used to drive steam turbines okay and there are also uh the C system that people are developing some are used in Spain and I think also Aria where the mirors focus and heat up the uh Power here and this goes to higher temperature and the conversion is via mechanical systems and the question of here uh in this course we're not as I said we're not looking into mechanical energy conversion we're looking into Alternatives and uh uh I just want uh so some of those Alternatives we're going to discuss one is a Thermo Electric uh in fact say Daniel does that work uh if you can turn it on so it's uh uh you will see that this is a conversion that as long as there's a temperature difference it can generate uh electricity it's by solid if you have used thermal couples those are essentially thermal couples put in together right thermal cou of course we normally just want to generate electrical signal but uh uh in this case uh uh you can also if you make the efficient you convert it into electricity uh the second uh so this is a here the work fluids is electron and in this demo what we have is just a copper here is a heat Source right copper come down to this side the top side is half and there's a this thermoelectric device very thin it's about 2 mm three few mm sandwiched between this copper and the the P Lum block and hopefully I don't see it but uh once the temperature difference uh there's tempat difference you'll start say the uh LED is lighting up and so uh that's direct no movement Parts using electron as working flu and the second example is thermionic energy conversion uh how many of you knows that in our department there's the micro fluid pastos la pastos that's on the second building three second floor hasbl with a professor in the mechanical engineering department 1950s and uh he developed this cionic engine so you can see it lighted up right and uh of course you have to maintain this temperature difference so if we burn it for a long time the other side the cooling is is not there the temperature difference will be become small your power I I saw it before it actually fluctuated before but uh this is also another important aspect it's a Sero system so you have to put the heat in take the heat out and what the converter is small but your Thal uh auxiliary system may be very big so you have to do a good engineering design uh but see I was mentioning uh th energy emission and energy conversion and Professor hos started his PhD on thoa engines and uh uh sping of thermal electron you drive 128 you probably see that compony theral electron now it's called theral feature right and uh uh the original purpose was to develop the thermionic engine in fact you go to read is 1956 or 58 I forgot in the 6 1950s to 1960s the efficiency was about 18% from El but it's very high temperature so there are lot of technological difficulties at the end they didn't pursue uh uh in terms of commercialization uh of course photo will take uh normally we don't think it as a heat engine but it is a heat engine it's limited by the second law of Thermodynamics okay and in fact anytime we can get close to Second LW you'll be really happy and uh but see this is of course using the photon from the Sun and there's another way is using the photon from a teror heat Source right you have home furnace and you can use the photon from there to do put us F take that's called the theral F take so those are all ways some of the direct energy conversion now M part here is using the charge as a fluids and we'll be talking some of those Technologies and the basic principles and of course when we think about the solar the advantage compare potential Advantage I say say it's demonstrated fully yet is that in the case of photo vol only the energy above the bank gap of semiconductor useful but when you think about solar thermal you can think about the full spectrum of the solar energy so that's the uh we we uh this uh uh year uh we started actually uh Department Energy Center looking into how we cance this technology uh what is a solar THM photo Volt take so here the idea is I take this solar energy I absorb it direct not directly on the photo voltic cell but absorb it to heat up this and Reit Photon why you do that because when you Reit the photon if you can do better control you can emit Photon right at the B gap of theable Tel and Theory says you can potenti together a really high efficiency and with a SLE Junction itself but no demonstration so that's one of the direction we're looking uh in this Center and the other direction is looking uh into Thermo elastic just like what we come here the temperature difference instead of the uh burner here using the S anytime you have temperat difference and again the cas is how we can make this efficient how we can make that low cost so when you think about energy the cost is the main concern right the photovoltaic cost right now is just too high so most people are reluctant to put on to their rooftop so that's the uh uh uh the uh motivations for this course and to really uh uh understand those technology we have to dive into for some of us then this for some of us this may be uh UNAM familiar territory particular I think for mechanical engineers because uh uh the uh most of time we are familiar with heat engines using steam using molecules and here we're using we're talking about heat engine using charge electrons and solid so we have to cover some of the basic the background knowledge in this area and understand the technology understand the limitation and uh bring you I want to bring you to the Forefront of research what are people doing maybe you have new ideas in doing new things now just is make another comment this is a very different right you can say the dying down process is slow cmore has a potential to do storage right with a theral mass you can store the energy a photo will take if there's no light we really don't have any power there so that's one potential advantage of thinking thermal okay so now uh I'm going to move into uh really the first uh what I want to use here is to uh reveal uh some thermodynamics and the heat trans transer and if you have taken thermody thermodynamics if you will know some of the material if you are from other department like physics uh you also know some materials uh both group have some material maybe enough materal but uh my advice is just to take it this is I say this is my starting point for future discussions uh so when we think about uh uh first law the thermodynam we think about two laws first law thermodynamic Second Law thermodynamics right we take a system and we Define by defining the boundaries right so the boundary Define our system and across the boundary we have P flow we have workflow and typical uh convention is that the uh uh see he goes in we take that positive and what goes out we think as a positive that's just due to Convention if you get a negative sign means your heat goes out or work goes in and then we have closed system means there's no mass flow across system boundary or open system when there's a mass flow processes in the simplest form of thermodynamic the first law is just theend energy balance right uh on the left hand side I have the energy of the system the energy change and on the right hand side is the heat transfer into the system M World output simple simple uh energy balance uh you can write into differential form or you can write into read form and then you do heat transfer you actually use the read form most often but I say uh some of you may still remember what's important is to recognize on the left hand side is a state property that does not depend on the process on the right hand side is a process dependent so it's actually very amazing you take two process dependent quantity you take a difference you get a quantity that's independent of the process looks very simple but this is a this is truly important aspects and the energy of the system including kinetic energy potential say potential energy and internal energy or if you have other forms of energy like elastic magnetic all those and consider relation that we using uh is specific heat for example and here uh uh the specific heat is the uh I say the time the temperature derivative of the internal energy here the internal energy I use small U it could be either per unit Mass based or per unit volume based okay so that's the first law Second Law Second Law we written into a equation but it's not it's not equation equation it's a inequality and what we have here is the entropy change again this is a state uh property entropy equals the entropy transferred across the system boundary and plus inop generation in the system again on the right hand side it's process dependent and on the left hand side is our state properties that's independent processes right now I think about the heat engines the heat engine we when we think about engine we think about cyclic motion right you don't want the design engine only go One Direction doesn't come back and uh uh uh so when we think about CYCC uh uh motion uh during a cycle any three property right go back to the original it's independent process once it goes back the change is zero so here we just wrote entropy in energy any other state properties so uh if entropy changes zero now what's the maximum efficiency you can get and uh maximum happens when there's no in tropy generation so if there's no entropy gener SG is zero and I look at this is my system the heat in is positive heat out is negative so I have my uh second L written into left hand side is zero entropy generation zero this entropy transfer across system boundary right and uh this gives me relation between the Heat in and heit out and temperature of the two reservoirs and if I use this relation I go to calculate what's in this case what the efficiency of this heat engine you can see the efficiency of course we Define as the work we get versus the price you pay that's the heat in okay efficiency here is the work now the work first law tells you the work equals heus he out because the internal energy during cycle change is zero so you can see you got a Col efficiency right that's the maximum you can get for any heat engines operating between two Conant temperature theral Reservoir people sometimes look down into heat Andes it's a heat engine meaning it's ther ther it's not you can't convert all the heat into electricity that's that's correct unless you have zero Kelvin right but the important is you put in the lamper and they see where we are the heat engine is very respectful if you have even just a 200° temperature heat source and if you can get a color deficiency 40% right you take the sun 95% and the more than thermal power plant about 40% I said that automobile internal combus engines right the driving efficiency less than 20% and the IC engine efficient is probably around 25% so heat engine if you can achieve the thetical efficiency you you you're doing really well the problem is there are a lot of losses that limits us to get to practical say the practically get too close to the col efficiency right so that's the uh what you learn in 25 thermodynamics now I want to go to a little bit microscopic picture because what's the iny between that and uh uh so let's look at the isolated system isolated means there's no uh heat transfer across system boundary now work trans now work transfer no Mass transfer isol and but I'm going to look at the imagine I can I have the Magic Eye I can looking at the individual molecules electrons in the system okay and uh there are many different possible configurations a molec see the molecules in the system be had a different states different position and each atom could have many different possible energy states this is from quantum mechanics right so every possible configuration possibility you can uh uh you can go back and take a two color two different color BS and mix them up that's your experiment on this micro picture and every possible configuration is a micro state and let's suppose we know how to count What's the total L of those possible configuration so the total lber that's Omega micros those are very big lber okay now what the basic principle of this microscope picture says every state is equally possible equal rights okay no discrimination and uh so the this is the uh equal probability principle every micro state is equally possible and uh it's based on this all the statistical description of uh uh say this Str thermodynamics is based on starting from here and uh what B man did is to relate this number of micro state to inter this is actually in bman Tombstone you go to V and uh this KB is the B con 1.3 I don't remember any there people in physics they use Atomic units I always remember things in inter say SI unit so that's a difficult large exponent 10 Theus 3 jar but it's good to keep some lers in mind okay so that's the uh uh like I said a starting point for the microscopic description and uh here we're considering a isolated system and uh now if you think about a l isolated system A system that can Ser talk exchange energy when the environment and because of that each micro state is no longer have equal right in terms of observation the probability of this system if you can observe this system in C specific micro state that has certain energy e and in this case it's longer equal probability and uh we're not going to derivation but you can go from here equal probability to construct isolated system that made of your system and the environment and with all that derivation at the end we get the probability of the system actually in C in one micro State this is a one micro state that has energy e is depending on the temperature of the system and energy here exponent right so you can say higher energy less probable and you if you do coner combustion or a lot of arous SN you probably learn in high school right that's this exponent that's coming from here Forman statistics and so here is a close system what if there is a mass flow po flow across system boundary and in that case this is similar but there's extra temp potential up here in the exponent and this is a uh say constant temperature but open system and mu here is chemical potential what is chemical potential think about temperature temperature is driving force for heat flow chemical potential is the driving Force for Mass diffusion mass flow okay pressure is the driving force for mechanical motion those are all thermodynamic intrinsic thermodynamic quantities pressure C potential temperature the other way to think about the C potential is it's the average energy led to add par into the system or take the par out of the system so first time potential is energy look at this is equivalent here and second uh it's driving force for the mass transform so if you do uh uh say uh chemistry you do Mass diffusion or you do batteries this is a quantity you would be dealing with all the time okay so this is a this is microscopic and uh now I'm going to apply what I just wrote down in the previous slide to molecules right I have bxs of molecules and I equilibrium temperature t i take one molecule it has a kinetic energy I'm thinking about the simple kinetic energy of the molecule right so it's 12 m² of that the W modle is say velocity is the three components VX sare v y Square VZ Square so that's what I have here and this is the energy remember in previous slides we say this is now a constant temperature molecule in thermal equilibri at constant temperature so that's the STA close system I'm using okay and this is the energy that's a kinetic energy and here is probability right probability I have a normalization factor so I put in the probability of this molecu having the velocity VX v y v z and then next step I want to determine a determine that pre factor a right what I do the probability so the molecule probability is the velocity is from minus infinite to Plus I include every possible chance right so if I include every possible chance that's my normal the way I find out this factor a that's all the uh say velocity from negative infinite to positive infinite and they should sum up to one they got to have the Velocity in between this and if you do this integration you find your factor a and this is the F Max real distribution for Moes it should be a minus oh thank you yes there is a minus there's this minus sign here thank you let's see the other one yeah yes please add and and here is the same problem I just cat uh so now if it's a probability once you have a probability you can calculate your observation right what's the average once you have probability so I want to find that the average energy kinetic translational energy motion kinetic energy of this molecule so here is the energy weighed by this probability over all possible velocities right and again it's integration you either can do yourself if you look at table and what you find out is a very simple result e = 2 3 kdt okay for it's three half KBT actually is a very good L to remember KBT but also it's a fundamental there is a fundament principle equal partition principle which says if your microscopic energy is quadratic so if you look at this kinetic energy is Quadra is in velocity MV squ right so if energy is quadratic each of this quadratic term contribute the average energy half KB so if you look at this molecule there because it has three quadratic terms so each contribute one kdt and total so that's the this is the equal partition principle and now I would like you to get the idea what is KBT this is a uh uh really something that uh if you want to have microscopic uh mind you should remember how large is it KT 1.3 10 23r 300° C I think I forgot to correct this again I this should be four five 4.14

right uh 4.14 and uh if you take a 4.14 divided by four that's a big number to hard to remember and people again different field people use different ways different measures and uh one uh uh often used the unit is electron W as energy unit what is the electron W you have a charge you you apply a electric field one volt say under one volt electric potential how much energy it has so one electron Vol is 1.6 temp 19 J so I translate this into a room temperature 1 KBT is 26 near electron volt okay now think about if this is the average energy of the molecule what's how fast they moving this is the molecule moving in this room you can go to estimate right because this energy equal 12 M v² so you can go to actually the energ the velocity of the molecule 12 cancel you do the mass I took the oxygen uh as example and know the uh the number of oxygen Atomic L is eight so it's 16 Neutron proton and is of the proton mass is 1.67 27 right so lambra is about 200 m/s depends on whether it's a light molecule heavy molecules so those are the microscopic molecular fixure right that's about molecules let's think about the electrons Mo there what we say we did not do anything Quantum side Maxwell distribution is a classical cical thermodynamic now when I do electrons the quum mechanics sometimes is very important so think more quum picture uh from quum mechanics we learn you had a physics before basic idea you I'm sure you all heard about energy levels of Comm P right and uh uh this is a some of you may still remember if you have a quantum mechanical state it can only have one electron right powerly Exclusion Principle and uh uh so and the uh quantum mechanics actually uh here I have I listed uh F Einstein relation that gives the energy this actually started for Photon so next slide I have more um energy relate to frequency of the photon and the momentum relate to wance of the photon so that's the uh uh so-called PL Einstein relation and what's see again this is another L if you think about the quum you always see H PL constant at 6.6 10us 34 okay so uh now I'm going to look at one quantum mechanical State and this Quantum mechanic State can have maximum of one electron right you could have zero electron you have one electron and this Quantum state has energy e Quantum mechanically allowable energy when the electron is there this state has energy when it's EMP no electron they actual energy of the system is zero right but I only can have two possibilities so I have this two possibilities I sum this up I want to find this a the difference between the molecule I I did before and here for molecular velocity I see okay from minus infinite to pass infity I include everything but for the electron here I only have zero or one ELR so I sum this up to one I will find a I'm not going I leave the the detail math so from the prev Rel find a and now I give you average again I can give you the average energy because uh if I do observation it's zero or it's e and if I do a lot of observation it's a equilibrium temperature T what's the average average time average energy or average L of electron right each individual observation could be zero one but there's an average uh possibility right so if I look at the average again say this is a n could be01 and you sum up to per distribution this is a uh the you look at that what's the range of this F the average huh 0 to one that's the you can't be about one this maximum is one minimum zero so it's between zero and one and then you PL those depends on the temperature right and uh it depends on whether the energy of this level is above the chemical potential or below the chemical potential the CH potential is the uh say we we said before now it's you can think of this one qualum state that the open system the electron can come in and goes out the chical potential is what's the average energy you need to take the electron in or put the electron take the electron out put the electron right so depends on whether it's above or below you can say it's close to one or close to zero very larly distributed near the tempal potential and in fact if your temperature is zero this will be a shock transition at the chemical potential above is one and here below see above zero and below is one and the temp potential at zero temperature there's special length people okay uh oftenly call that fir Lev but uh in lot of different different fields sometimes people just call fir level for any this terminology convention okay so that's for electron now we're going to Le to uh we talk about molecule talk about electron next I'm going to talk about electromagnetic weight the photon right I mentioned before uh the um FR Einstein relation uh is really looking into Photon and later on was generalized for any material way and uh so what we have is if uh there is electromagnetic W and a certain frequency mu what the PL form is that the energy of that Mo the quantum mechanical State at the time there was no quantum mechanics so electromagnetic Mo of specific mod the energy uh has to be quantized and Basic Energy Quant is H new and this Quant later on was called a f okay and what Einstein came in is saying that the quanta not only just say have the wave but also has a momentum so this is the where the wave partical Duality came from so the momentum of the quantum is related to the wavelength of the wave and again relate via Edge and uh so the uh uh each Quantum State the energy can only be a multiple of the H new normally when we think about the way Contin mechanic can be any value the the Comm difference is that uh the energy could be only multiple so n is now integer and it has one half that's also called this one half nor is not important it's called Z Point Energy and is really uh if you still record some quantum mechanics that's the uh soal Heen certain principal requirement so this is 0 point energy uh normally it's not important so I'm not going to deil much so that's a full okay now I'm going to it turns out the the uh whole Photon you can think this is the electromagnetic wave now I'm going to talk about different wave that's the latest wave the atomical vibration right the atomical vibration now you can think about started with classical mechanics if you have mass spring system the fundamental natural frequency is square root k f right and of course in clal mechanics we see the energy is related to the amplitude of the wave the velocity and once you go to Quantum it turns out this way the energy cannot be any value same as electromagnetic way and the energy can only be multiples same you can say multiples of H and now there was anog for this qu of vibration that anal is full okay electr magnetic way the Basic Energy Quant is the phon that is atomical vibration the energy Basic Energy Quantum is full okay so what's this statistically obey right at a certain Quantum state if you think about electron could be zero and one maximum is one and here at a mo at this frequency it could be Z1 to infinite so that's a difference electron 0 one and here I could be any L integer Lage so I need to find the the distribution average L of phon or fullon if I have equum system okay uh do the same right I have the probability and uh now if I have n phon forong in this micro State the qu mechanical state that has this specific frequency that's the probability I sum this up from Nal Z to infinite that still should give me one right you have either zero or one or infinite number all the in between somewhere in between so sum up I have one I will determine a and this a from this a I go to find the average once I find the probability I find the average Lum right same way as I did for electron and this average L now is the cosine Einstein distribution and this for the Einstein distribution you if you compare with the previous fir distribution the only thing change is a plus sign now goes to minus sign and because of that is no longer between 0 to one between zero to infinite right and in fact for Fon and Photon because the L is not conserved okay it's not like electron is conserve L soal potential concept is not called is not good one so that's a zero we take it a zero and this is the distribution of for Einstein Distributing with reason for setting z uh it's not a conserved quantity it's say uh when we talk about Mass transfer particle that's a fixed lber right total lber either going out or going but it's a that's a conserved L here is not a conserv but there actually see later on uh when we talk about the solar cell you'll find out there's a qu formula will get into you here so it's actually a very deep uh uh thing there it's a this is just a simple argument okay and uh that's about now in this time frame like I said you're probably familiar with classical thermodynamic you're not side like you just take it okay this how the result will be using then what they what get into mind microscopically we have Bine distribution we have per distribution for electron we have both Einstein distribution for foron phon that's what you need to know okay and uh uh what we need uh I'm going to summarize next uh this is the P transfer before was theral dyamic now uh heat transfer again you take 25 26 this is very uh should be uh very familiar if I have uh we have we have three modes of he transfer right conduction combustion and radiation and describing the conduction is a for real law of diffusion so the heat transfer rate here I put the rate in what it's proportional to the area proportion temperature gradient and proportionality constant is the thermal conductivity and that's a material property normally we think it as a material proper and uh uh the negative sign actually take care of the second LW say temperature heed will flow from high temperature to low temperature and you can do that's a per what you can do flux that's a per unit area based so just long noriz the area uh or you WR generalize into three dimension rather than write into one dimension so that's and contion is when you have fluid flow in the surrounding of say a heated or cold surface the heat transfer between the Surface and the fluid is proportional to the temperature difference of the wall and the fluids this fluids actually transition the temperature goes from the wall to gradually to the center of the fluid or ID right and uh the proportionality constant here is H the uh heat transfer coefficient and uh uh but H is a flow dependent is not a mature prop depends on how flu how fast the flu flow and depends on the geometry uh so there this is a big Topic in 25 2006 and we know there's natural conversion Force conversion this just uh uh in terms of the uh terminology there so those are conduction and convetion and of course the theral of transfer is the radiation thermal radiation right seral radiation for black body one single surface or black object the emission is given by the step band as the proportional to temperature for power the proportionality constant is step important my con right so this is for black object the real object is not black so we use the imity to characterize it and we discuss some more down the road now if I have two objects exchange heat in this case it could be vacuum right it's that's the difference of radiation compared to conduction and convetion which requires a media and here the heat transfer is by electromagnetic waves so uh I wrote a simple form of reding the left heat transfer is the compter Force Power and remember here you have to use the kin not degre C look at the units so in kelvin and uh if the object is not black we use the emissivity and here is the effective between two surfaces because the two surface may be different like set this discuss more than the ru so those are three modes of right and uh in this course we'll talk more on conduction we'll talk more on radiation you have learned a lot uh some of you learn a lot of convertion so that that's a probably fear right now let's think a little bit more on conduction uh if I have a one dimensional no generation inside I can the heat flow is constant so cure is constant you go back to the fal law you integrate it you find out the heat transfer between the two points from from this to the other side a steady state is proportional to Thermal conductivity temperature uh area inverse proportional to the length so if you want to do thermal isolation what you do use a long object use a small cross-section right that's thermal isolation lot of times when you design a thermal system not only in h getting in you also want to isolate it and uh the r we can combine all this right into uh thermal resistance just a definition because this way it looks very similar to electrical circuit where your driving voltage is now the temperature difference and then your current is a Heat current and your resistance so you can use the same way as you do Circle analysis if you know the thermal resistance this is for different geometries different and even for converstion you can express you go to do a cal resistance so so you can build uh Su resistor LW work to do some simple transfer okay and another thing that's important to keep in mind is order Manu what are the good conductors what are good insulators right and the best conductor turns out is dond why that's because the bumping the Sprint so in this case is the wave atomical wave that carries the heat so if the bounding between the atoms strong the spring constant is large the velocity is large so that's why diamond is good conductor and metals are not that say metals are by electrons and the insulators you can say the wood material course those po theral conductors because the atomic Arrangement is random and that Randomness uh make the wave harder to propagate in the material okay so this is a again macroscopic now I'm going to also give you the other view microscop of and uh the kic picture so what happens how heat really is conducted let's think about the molecules and the molecules the uh let's say we have a box and one side is hot the other side is cold let's forget about conversion how it goes from one H side we of the molecule to the PO side it's really on the H side the air molecule right collide with the wall and on the wall is hard means the velocity is faster so this molecule will get say faster velocity due to that energy change between the w and air molecule now this harder air molecule will collide with the liant molecule of course they have chance go randomly every mole is moving a few hundred meter per second as we talked before right and they collide with the Coler one and they impart some of that excess energy to the L labor the Le labor go back and the forth again they do this Cascade the energy from the h to the close side a diffusion process and uh typical there is uh the distance of this poistion right you can go to estimate it and then just turns out this distance between the average molec potion is about 100 mic nomer in the in this room the air molecule distance between Collision is about 100 n okay and uh uh so now I give you a a few more definition right the energy prople let's again take energy Perle as e and the average velocity is V and time between colation is to and distance between coition is gamma that's a mean free pass so time is the average time between colation is relaxation time and this distance and the time related by the velocity right so what is the heat transfer when I think about Heat going through this imaginary surface here what's how I calculate it I count I do counting I will say okay only those molecules within one mean free pass VX is velocity the component in the X Direction times the Collision time right so only those molecule can go zip through the interface without Collision because T is the average time between Collision so this part of the molecule is hotter they go from this side to the other side and on this side the molecule is cooler again in the VX tab tall that range they go from this side to the other side right and I take the difference I count all the molecules I take the difference that's my heat flux so let me write down the MTH that's what I do I say on the left hand side this is a per unit volume how many n is the uh particle L density I forgot uh the first uh the second bullets there each particle how much energy has and this is the velocity in the X direction right so I say only in this range from X to vxus to in this range you can go through on the other side is X Plus vx to you can do more detail but that clearly this is very simple argument right so that gives me local P flux you go check a unit is a watt per meter squar he flux just do the expansion of these two terms and uh write into differential form because this is about 100 NM if I do meters or centimeter or millimeters that's essentially continum right very small Delta so I do the derivative and I do T expansion this one half is because there's only half of this go this way half go that way right so that's where the cff came from and after I do the T expansion uh I have e and V and VX VX is random velocity I pull out it is independent of distance it could be depend on this just to make assumption approximation in different distance this is VX s VX Square Is Random so it's 13 of the average velocity 13 v s so that's my 1/3 V sare replacing this VX s t is here and the n * e is per uni volume how much energy internal energy the molecule have right so I have this n * e is U and I take a CH r d DT DT DX okay what do you get you get for right cuz here here is 13 w squ p c and that's the thermal conductivity and this temperature gradient you can do that you can go back and do your fix law and uh new share stress law in fact all the diffusion law you learn in three steps yes you said we could uh move the velocity out of the derivative because it was constant but isn't the velocity related to the temperature yeah good question uh this is a uh uh later on we discuss the Thal electric effect I'll come back to this okay here I'm talking about molecule and I'm just I just want to show you the F they clearly approximation and what you're talking is more like a coule the mass and heat transfer and in fact we uh if you learn once the heat contr you know mass and heat contr couples okay so the thermal connectivity is one specific heat velocity and me fre CL okay and the specific heat is per unit volume if I do Capital C is per unit volume base and uh typically we do per unit mass bity times per unit Mass okay so this is a the a very uh useful relation to keep in mind simple uh the thermal connectivity depends on how many energy stored by the molecu how fast they travel and how far they travel between so if you want the high th connectivity want the power Fall want low th connectivity you reduce that right and uh for most solid this say is about 10 to the six power this doesn't vary by Factor five so when I do on my back of envelop aut magnitude estimation I say say is 10 to the six you go to check and uh velocity in solid is just so you can roughly think the speed of sound few hundred to few thousand met per second the me free pass is short right nomers 100 Nom so that's the range order okay so this is a I I think most of you probably in 2005 haven't seen it it's a very simple but uh uh the picture is really the diffusion picture all similar let me give you I Haven the 12 finished I'm read condition I come back to I said that step bman law right and what I want to show you next is how I can derive stat for from the statistics just I just show so I'm considering uh enclosure let me even imagine the enclosure is perfectly reflecting there are photons and the was is like temp so it's a equilibrium system I want to see how many the electromagnetic weight inside this box how much energy they have like constant temperature equ thermal equ right so uh let's go back to a little bit the basic relations th say electromagnetic weight we know the speed of KN is a proportional it's speed of night is wance times frequency right that's a uh high school we learn and uh definition I say angular velocity is 2 pi mu we vector magnitude is 2 pi or Lambda what is we Vector the we propagates in certain direction right and uh so it's a because of Direction it's a vector 2 pi Lambda Lambda is a WAV length it's a scalar and the directionality GES the vector so the it's KX KY KZ if you do c coordinate and so I can write the C equals mu * Lambda in terms of frequency and we Vector is C time Omega equal c k so this relation between V vector magnitude and the frequency it's called dispersion and then in the speed of in the case of light it's a linear right is 3 10 8 m/s and of course because a vector so it's a square root of the components so that's the dis and now I want to count when I want my question is how much energy Photon has inside this box right I'm going to count how many modes it has remember each mode the B Einstein distribution already tells me each say each mode how much how many photons so how many modes so that's basically I'm cting right a mode because I'm a perfect wall so you can see the mode has to be standing W here either this way or this way that is boundary it's zero why is it zero huh why should otherwise it doesn't get a steady state it keeps propaga okay so this is my wav length component in the X Direction KX right is corresponding wavelength in X Direction and The the LX is the length here right so if I just say this must satisfy the rule then I have to have integer those are the possibilities and that translate in the possibility of my K KX is just 2 pi over Lambda X so this is n to 2 LX okay so each combination of NX NY n z each set gives me one Mo and uh when I want to find out how many how much energy I just sum up all those modes right so I sum that up and remember when I do molecules I do velocity from minus infinite to plus infinite and here when I do this I do discrete summation of NX n y n z right so I do this summation this is a NX N Y and Z each mode has this is the number of photon in this every cell and this is the energy of a photon is H Omega I got a factor of two because electromagnetic wave is transverse wave and has a two compol two polarizations so that factor to come what com okay if you don't look at this that's where my starting point and what I want to do is rather than do the summation I'm going to translate it into integration summation is over n so if you look at this interval between n and K is 2 pi 2 LX so when I have a dkx I divide by 2 pi 2 LX that's n if this is large L but small you have a lot of problem so I translate this summation in NX into this dkx integration I do same for KY k z this is I'm just copying I I wrote the extra step here just for the T later purpose because my function is a even function so I extend to find minus infinite to plus infinite the difference is a factor of two so I don't have this two here anymore because the factor of two is gone okay that's just for my later purpose the end result this is just equivalent and what it means my I can rather than standing wave I can say do two counter propagating one goes left one goes right that's when you do the maximal velocity is minus infinite to plus infinite so that's equivalent okay I think I will probably have to stop here there are two more slides you can look at but I'll discuss what's coming is the FL body radiation in each wavelength how much energy carries with simple this okay

*2024-07-03 22:11*