WEBVTT 00:00:00.000 --> 00:00:01.915 align:middle line:90% [SQUEAKING] 00:00:01.915 --> 00:00:02.415 align:middle line:90% 00:00:02.415 --> 00:00:03.847 align:middle line:90% [RUSTLING] 00:00:03.847 --> 00:00:04.347 align:middle line:90% 00:00:04.347 --> 00:00:07.728 align:middle line:90% [CLICKING] 00:00:07.728 --> 00:00:09.970 align:middle line:90% 00:00:09.970 --> 00:00:11.950 align:middle line:84% GIAN PAOLO BERETTA: OK, good afternoon. 00:00:11.950 --> 00:00:12.540 align:middle line:90% Let's start. 00:00:12.540 --> 00:00:15.230 align:middle line:90% 00:00:15.230 --> 00:00:15.730 align:middle line:90% All right. 00:00:15.730 --> 00:00:19.140 align:middle line:84% So we are talking about non-equilibrium. 00:00:19.140 --> 00:00:29.510 align:middle line:84% And this table summarizes a bit of what we have done. 00:00:29.510 --> 00:00:34.870 align:middle line:84% We talked about forces and fluxes. 00:00:34.870 --> 00:00:40.510 align:middle line:84% The forces can be thought of as the degree of disequilibrium 00:00:40.510 --> 00:00:44.350 align:middle line:84% of the various aspects, and the fluxes 00:00:44.350 --> 00:00:51.760 align:middle line:84% are the modes with which molecules 00:00:51.760 --> 00:01:00.470 align:middle line:84% transfer stuff, properties, the extensive properties. 00:01:00.470 --> 00:01:09.050 align:middle line:84% In fact, this fact that molecules transport something 00:01:09.050 --> 00:01:16.430 align:middle line:84% gives rise to this language of phoretic, like thermophoretic, 00:01:16.430 --> 00:01:18.670 align:middle line:90% electrophoresis. 00:01:18.670 --> 00:01:22.700 align:middle line:84% Phoretic means-- the etymology of the word 00:01:22.700 --> 00:01:25.140 align:middle line:90% is transporting something. 00:01:25.140 --> 00:01:28.320 align:middle line:90% 00:01:28.320 --> 00:01:36.210 align:middle line:84% And we want today to introduce a couple of examples 00:01:36.210 --> 00:01:43.740 align:middle line:84% of these transport cross effects that 00:01:43.740 --> 00:01:47.570 align:middle line:84% emerge from this table from the fact 00:01:47.570 --> 00:01:54.910 align:middle line:84% that not only we find things on the diagonal, 00:01:54.910 --> 00:01:58.560 align:middle line:84% so we have processes like heat transfer, that follows Fourier's 00:01:58.560 --> 00:02:03.090 align:middle line:84% law, or mass transfer that follows Fick's law that we 00:02:03.090 --> 00:02:17.470 align:middle line:84% will derive, and electrostatics that provides Ohm's law. 00:02:17.470 --> 00:02:21.100 align:middle line:84% And also fluid mechanics, the law 00:02:21.100 --> 00:02:26.870 align:middle line:84% that connects, with viscosity, the stress and the rate 00:02:26.870 --> 00:02:28.670 align:middle line:90% of strain of fluids. 00:02:28.670 --> 00:02:34.470 align:middle line:84% But there are also off diagonal terms. 00:02:34.470 --> 00:02:37.160 align:middle line:90% These are called cross effects. 00:02:37.160 --> 00:02:43.950 align:middle line:84% And in some cases, some of these cross effects 00:02:43.950 --> 00:02:46.680 align:middle line:84% are forbidden by symmetry reasons. 00:02:46.680 --> 00:02:56.540 align:middle line:84% In particular, if you have an isotropic fluid or continuum, 00:02:56.540 --> 00:03:06.690 align:middle line:84% the Curie principle forbids cross effects 00:03:06.690 --> 00:03:11.680 align:middle line:84% between phenomena of different tensorial character. 00:03:11.680 --> 00:03:14.860 align:middle line:90% 00:03:14.860 --> 00:03:18.370 align:middle line:84% Although there may be other situations, like we said, 00:03:18.370 --> 00:03:24.900 align:middle line:84% for example, interfaces between phases, where there is isotropy 00:03:24.900 --> 00:03:28.560 align:middle line:84% is broken by the interface, and it's 00:03:28.560 --> 00:03:32.010 align:middle line:84% substituted by another kind of isotropy that 00:03:32.010 --> 00:03:35.880 align:middle line:84% will allow some additional coupling that would not 00:03:35.880 --> 00:03:37.250 align:middle line:90% be allowed in the bulk. 00:03:37.250 --> 00:03:41.090 align:middle line:90% 00:03:41.090 --> 00:03:44.530 align:middle line:90% OK, all right. 00:03:44.530 --> 00:03:55.330 align:middle line:84% So let's recall also that we have defined these relations 00:03:55.330 --> 00:04:03.020 align:middle line:84% that, yeah, this mode of interaction that 00:04:03.020 --> 00:04:07.790 align:middle line:84% is the extension of the idea of heat interaction, which we call 00:04:07.790 --> 00:04:13.580 align:middle line:84% the heat and diffusion, which is fundamental in that it brings 00:04:13.580 --> 00:04:18.800 align:middle line:84% in a relation, like the heat Q over T relation. 00:04:18.800 --> 00:04:24.320 align:middle line:84% Here, it becomes a relation between heat 00:04:24.320 --> 00:04:30.500 align:middle line:84% and also additional terms like these ones 00:04:30.500 --> 00:04:35.170 align:middle line:84% in the flux for the energy, and also for the entropy. 00:04:35.170 --> 00:04:38.740 align:middle line:90% 00:04:38.740 --> 00:04:47.000 align:middle line:84% The language that you find in papers and books on this subject 00:04:47.000 --> 00:04:50.800 align:middle line:90% sometimes is varied. 00:04:50.800 --> 00:04:53.880 align:middle line:90% And so you have to be careful. 00:04:53.880 --> 00:05:02.870 align:middle line:84% Often, you find people calling this heat. 00:05:02.870 --> 00:05:04.720 align:middle line:84% And this is the measurable heat flux. 00:05:04.720 --> 00:05:07.520 align:middle line:90% 00:05:07.520 --> 00:05:09.800 align:middle line:84% I've tried to call heat flux only this. 00:05:09.800 --> 00:05:13.255 align:middle line:84% And that, I'll try to call it just the energy flux. 00:05:13.255 --> 00:05:16.130 align:middle line:90% 00:05:16.130 --> 00:05:20.300 align:middle line:84% In addition, in some of the situations in which you 00:05:20.300 --> 00:05:24.800 align:middle line:84% have multiple phenomena occurring like heat and mass 00:05:24.800 --> 00:05:33.580 align:middle line:84% transfer simultaneously, or heat and electrical charge transfer, 00:05:33.580 --> 00:05:39.070 align:middle line:84% there are also languages like the heat of transfer 00:05:39.070 --> 00:05:41.510 align:middle line:90% in some particular situations. 00:05:41.510 --> 00:05:48.580 align:middle line:84% For example, if you have no gradient in temperature, which 00:05:48.580 --> 00:05:52.450 align:middle line:84% doesn't necessarily mean that q double prime is 0, 00:05:52.450 --> 00:05:56.710 align:middle line:84% but in that situation, the overall energy flux 00:05:56.710 --> 00:06:03.770 align:middle line:84% is sometimes called the heat of transport 00:06:03.770 --> 00:06:06.710 align:middle line:84% because it's the heat that is transported 00:06:06.710 --> 00:06:11.120 align:middle line:84% in the absence of the traditional cause of heat 00:06:11.120 --> 00:06:13.710 align:middle line:84% transfer, which is a gradient in temperature. 00:06:13.710 --> 00:06:18.910 align:middle line:84% Nevertheless, it's a combination of these terms. 00:06:18.910 --> 00:06:20.600 align:middle line:90% So beware of the language. 00:06:20.600 --> 00:06:25.220 align:middle line:84% And if you ever have to write on this subject, 00:06:25.220 --> 00:06:27.590 align:middle line:90% be careful and be consistent. 00:06:27.590 --> 00:06:30.720 align:middle line:90% 00:06:30.720 --> 00:06:34.780 align:middle line:84% Now, when we define the heat and diffusion interaction, 00:06:34.780 --> 00:06:43.540 align:middle line:84% we did it for a situation in the absence of fluid flow. 00:06:43.540 --> 00:06:50.210 align:middle line:84% So these are the diffusive fluxes with respect 00:06:50.210 --> 00:06:57.200 align:middle line:90% to a non-moving bulk. 00:06:57.200 --> 00:07:01.110 align:middle line:84% And therefore, in the next slide, 00:07:01.110 --> 00:07:06.020 align:middle line:84% we'll have to do a little bit of additional job 00:07:06.020 --> 00:07:12.110 align:middle line:84% to introduce also the idea that typically, in fluids, 00:07:12.110 --> 00:07:14.180 align:middle line:90% there is flow. 00:07:14.180 --> 00:07:16.560 align:middle line:84% And that, of course, is a complication. 00:07:16.560 --> 00:07:19.050 align:middle line:84% It complicates some of the equations. 00:07:19.050 --> 00:07:22.470 align:middle line:84% But this is the right moment to bring it in. 00:07:22.470 --> 00:07:31.760 align:middle line:90% And also, what else? 00:07:31.760 --> 00:07:33.980 align:middle line:90% It escaped. 00:07:33.980 --> 00:07:34.850 align:middle line:90% I'll come back. 00:07:34.850 --> 00:07:39.760 align:middle line:90% And so let's dig into this. 00:07:39.760 --> 00:07:43.120 align:middle line:90% 00:07:43.120 --> 00:07:46.280 align:middle line:84% Well, it's a little crowded slide, as usual. 00:07:46.280 --> 00:07:50.000 align:middle line:84% But it's full of things that we have already seen. 00:07:50.000 --> 00:07:55.090 align:middle line:84% So just to have a way to remember. 00:07:55.090 --> 00:07:57.850 align:middle line:84% For example, you remember that we, 00:07:57.850 --> 00:08:01.120 align:middle line:84% as part of the local equilibrium assumption, and together 00:08:01.120 --> 00:08:06.490 align:middle line:84% with the simple system model, and like when we described 00:08:06.490 --> 00:08:11.990 align:middle line:84% bulk flow, we say that the energy of our system, 00:08:11.990 --> 00:08:20.780 align:middle line:84% including-- like a fluid parcel, is described by the energy 00:08:20.780 --> 00:08:23.690 align:middle line:84% typical of a stable equilibrium state 00:08:23.690 --> 00:08:29.900 align:middle line:84% with the amounts of entropy, volume, and constituents that 00:08:29.900 --> 00:08:34.940 align:middle line:84% are in that fluid element plus additional terms that 00:08:34.940 --> 00:08:37.760 align:middle line:84% are either mechanical or electrical, 00:08:37.760 --> 00:08:41.250 align:middle line:84% like the kinetic energy, the potential gravitational energy, 00:08:41.250 --> 00:08:43.679 align:middle line:84% the potential electrostatic energy. 00:08:43.679 --> 00:08:56.880 align:middle line:84% And if you have also your fluid static in centrifugal field, 00:08:56.880 --> 00:09:00.505 align:middle line:84% you can add this negative potential energy. 00:09:00.505 --> 00:09:06.170 align:middle line:90% 00:09:06.170 --> 00:09:10.168 align:middle line:84% And of course, you remember that the mass 00:09:10.168 --> 00:09:13.110 align:middle line:84% is related to the concentrations, 00:09:13.110 --> 00:09:14.960 align:middle line:90% via the molecular weights. 00:09:14.960 --> 00:09:18.980 align:middle line:84% And the charge, q, is related to the concentrations 00:09:18.980 --> 00:09:25.790 align:middle line:90% via the individual charges of-- 00:09:25.790 --> 00:09:30.650 align:middle line:90% the z of your molecules. 00:09:30.650 --> 00:09:33.800 align:middle line:84% So we have seen a few times that when 00:09:33.800 --> 00:09:38.240 align:middle line:84% you compute the derivative of the energy that 00:09:38.240 --> 00:09:41.600 align:middle line:84% enters in the conditions for mutual equilibrium, which is not 00:09:41.600 --> 00:09:44.360 align:middle line:84% the chemical potential, but the total potential, 00:09:44.360 --> 00:09:49.310 align:middle line:84% that picks up a contribution from just the stable equilibrium 00:09:49.310 --> 00:09:50.003 align:middle line:90% state property. 00:09:50.003 --> 00:09:51.420 align:middle line:84% And that's the chemical potential. 00:09:51.420 --> 00:09:54.440 align:middle line:84% But then, there are also additional contributions 00:09:54.440 --> 00:09:58.070 align:middle line:84% to the chemical potential, which makes it, for example, if you 00:09:58.070 --> 00:10:01.760 align:middle line:84% add this one, it makes it the electrochemical potential, 00:10:01.760 --> 00:10:05.430 align:middle line:84% the gravitational, the kinetic, and the centrifugal. 00:10:05.430 --> 00:10:11.860 align:middle line:90% 00:10:11.860 --> 00:10:17.310 align:middle line:84% In a moment below, we will see that we can write this. 00:10:17.310 --> 00:10:20.970 align:middle line:84% These terms are highlighted in white 00:10:20.970 --> 00:10:30.220 align:middle line:84% as energy minus internal energy, specific, because indeed, 00:10:30.220 --> 00:10:33.070 align:middle line:90% that is what it turns out to be. 00:10:33.070 --> 00:10:40.920 align:middle line:90% 00:10:40.920 --> 00:10:46.380 align:middle line:84% In fact, you remember that in general, without-- 00:10:46.380 --> 00:10:49.950 align:middle line:84% before we make any assumptions about the mixture being 00:10:49.950 --> 00:10:53.010 align:middle line:84% ideal or non-ideal, in general, we 00:10:53.010 --> 00:10:55.890 align:middle line:90% define the partial properties. 00:10:55.890 --> 00:10:59.690 align:middle line:84% And this is the partial internal energy, 00:10:59.690 --> 00:11:02.930 align:middle line:84% which is related to the total internal energy 00:11:02.930 --> 00:11:07.910 align:middle line:84% by this summation, which you can also cast in terms 00:11:07.910 --> 00:11:12.240 align:middle line:90% of concentrations, c sub i. 00:11:12.240 --> 00:11:15.630 align:middle line:84% So if you substitute-- if you rewrite this expression 00:11:15.630 --> 00:11:19.730 align:middle line:90% for the energy in terms of-- 00:11:19.730 --> 00:11:29.150 align:middle line:84% right here-- specific energy per unit volume times the volume, 00:11:29.150 --> 00:11:32.430 align:middle line:84% and here, you substitute this for the internal energy, 00:11:32.430 --> 00:11:35.150 align:middle line:90% which is the first term here. 00:11:35.150 --> 00:11:37.910 align:middle line:84% So you see that the overall energy 00:11:37.910 --> 00:11:41.990 align:middle line:84% is given by a summation weighted on the concentrations of all 00:11:41.990 --> 00:11:44.340 align:middle line:90% these terms. 00:11:44.340 --> 00:11:48.420 align:middle line:84% All these terms somewhere have the concentration 00:11:48.420 --> 00:11:50.915 align:middle line:84% coming in either through the mass or the charge. 00:11:50.915 --> 00:11:53.570 align:middle line:90% 00:11:53.570 --> 00:11:55.550 align:middle line:84% And so this term in parentheses can 00:11:55.550 --> 00:12:03.470 align:middle line:84% be called the specific energy, total energy. 00:12:03.470 --> 00:12:09.445 align:middle line:84% And it can be split like we have seen here. 00:12:09.445 --> 00:12:18.988 align:middle line:90% 00:12:18.988 --> 00:12:23.030 align:middle line:90% So let's see. 00:12:23.030 --> 00:12:25.890 align:middle line:90% 00:12:25.890 --> 00:12:28.720 align:middle line:84% Yeah, I get lost myself in my slides. 00:12:28.720 --> 00:12:29.730 align:middle line:90% Yes. 00:12:29.730 --> 00:12:31.020 align:middle line:90% It's sad. 00:12:31.020 --> 00:12:33.850 align:middle line:90% All right. 00:12:33.850 --> 00:12:36.230 align:middle line:90% OK, so yeah. 00:12:36.230 --> 00:12:42.410 align:middle line:84% We write it in terms of the e without is per unit volume. 00:12:42.410 --> 00:12:46.050 align:middle line:84% The e with the star is per unit mass. 00:12:46.050 --> 00:12:49.630 align:middle line:84% That is also some kind of source of confusion 00:12:49.630 --> 00:12:53.800 align:middle line:90% somewhere in the literature. 00:12:53.800 --> 00:12:54.820 align:middle line:90% And all right. 00:12:54.820 --> 00:12:57.080 align:middle line:90% So this is one first equation. 00:12:57.080 --> 00:12:59.710 align:middle line:84% The second relation that we need to bring in 00:12:59.710 --> 00:13:06.290 align:middle line:84% is the fact that the fluxes, the fluxes of constituents, 00:13:06.290 --> 00:13:13.000 align:middle line:84% are not all independent, because these-- remember, 00:13:13.000 --> 00:13:16.000 align:middle line:84% they don't have the naught on the superscript. 00:13:16.000 --> 00:13:21.440 align:middle line:84% So these are the diffusive fluxes. 00:13:21.440 --> 00:13:25.400 align:middle line:84% Those-- so it's the flux relative to the center, 00:13:25.400 --> 00:13:30.820 align:middle line:84% the barycenter of the fluid element. 00:13:30.820 --> 00:13:34.810 align:middle line:84% And with respect to that, the mass flux 00:13:34.810 --> 00:13:38.595 align:middle line:84% by definition of barycentric velocity is equal to 0. 00:13:38.595 --> 00:13:43.410 align:middle line:90% 00:13:43.410 --> 00:13:49.055 align:middle line:84% And therefore, the weighted sum of the-- 00:13:49.055 --> 00:13:52.750 align:middle line:90% 00:13:52.750 --> 00:13:55.390 align:middle line:84% molar fluxes of the various constituents 00:13:55.390 --> 00:13:59.540 align:middle line:84% weighted on their molecular weight must be equal to 0. 00:13:59.540 --> 00:14:03.260 align:middle line:84% That makes them not all independent. 00:14:03.260 --> 00:14:07.570 align:middle line:90% 00:14:07.570 --> 00:14:15.310 align:middle line:84% So we can recast the equations by writing, taking, 00:14:15.310 --> 00:14:16.590 align:middle line:90% for example, the last one. 00:14:16.590 --> 00:14:19.690 align:middle line:90% 00:14:19.690 --> 00:14:24.490 align:middle line:84% So the flux of the last constituent, this is the mass, 00:14:24.490 --> 00:14:28.060 align:middle line:84% is equal to the negative of the sum of the fluxes of masses 00:14:28.060 --> 00:14:33.350 align:middle line:90% of the remaining constituents. 00:14:33.350 --> 00:14:41.000 align:middle line:84% And we can introduce this idea into the expression 00:14:41.000 --> 00:14:45.900 align:middle line:84% that we need for the heat and diffusion interaction, 00:14:45.900 --> 00:14:47.580 align:middle line:90% because there, we have the sum. 00:14:47.580 --> 00:14:52.250 align:middle line:84% And in this sum, we have all the fluxes playing. 00:14:52.250 --> 00:14:55.400 align:middle line:84% But we have to also make a system with this other relation 00:14:55.400 --> 00:14:56.170 align:middle line:90% between fluxes. 00:14:56.170 --> 00:15:00.510 align:middle line:90% 00:15:00.510 --> 00:15:15.500 align:middle line:84% So one way is to rewrite by subtracting this equation here. 00:15:15.500 --> 00:15:22.430 align:middle line:84% We first do it with the mass so that this is easier to subtract. 00:15:22.430 --> 00:15:26.750 align:middle line:84% So here, when I see the summation, 00:15:26.750 --> 00:15:30.720 align:middle line:84% it goes from 1 to r to the total amount of constituents. 00:15:30.720 --> 00:15:32.900 align:middle line:84% Now, I substitute it with the summation 00:15:32.900 --> 00:15:37.210 align:middle line:90% that goes from 1 to r minus 1. 00:15:37.210 --> 00:15:41.770 align:middle line:90% And the last one is written-- 00:15:41.770 --> 00:15:45.865 align:middle line:84% it comes inside the summation by virtue of this mass. 00:15:45.865 --> 00:15:48.760 align:middle line:90% 00:15:48.760 --> 00:15:49.270 align:middle line:90% Oops. 00:15:49.270 --> 00:15:49.990 align:middle line:90% What happened? 00:15:49.990 --> 00:15:59.020 align:middle line:84% Yeah, by virtue of this mass balance, so to speak, equation. 00:15:59.020 --> 00:16:00.470 align:middle line:90% No, that's not a mass balance. 00:16:00.470 --> 00:16:03.470 align:middle line:84% It's simply the definition of barycentric velocity. 00:16:03.470 --> 00:16:06.620 align:middle line:90% 00:16:06.620 --> 00:16:10.910 align:middle line:84% OK, so now, once we have this expression here, 00:16:10.910 --> 00:16:15.810 align:middle line:84% it's the difference between the total potentials. 00:16:15.810 --> 00:16:17.370 align:middle line:84% But the difference between the total 00:16:17.370 --> 00:16:24.670 align:middle line:84% potentials, we've seen somewhere above. 00:16:24.670 --> 00:16:28.400 align:middle line:90% 00:16:28.400 --> 00:16:33.260 align:middle line:84% I got the mouse stuck a moment, but somewhere above, you 00:16:33.260 --> 00:16:40.740 align:middle line:84% find the relation that gives you here, this relation 00:16:40.740 --> 00:16:48.840 align:middle line:90% here and that one up there. 00:16:48.840 --> 00:17:01.150 align:middle line:84% So this difference can be substituted with this one where 00:17:01.150 --> 00:17:04.619 align:middle line:90% the z's stand out. 00:17:04.619 --> 00:17:11.390 align:middle line:84% And they bring in the charge, the charge flux. 00:17:11.390 --> 00:17:16.359 align:middle line:90% OK, and OK. 00:17:16.359 --> 00:17:18.579 align:middle line:90% So we've substituted. 00:17:18.579 --> 00:17:21.750 align:middle line:84% So now, this expression has the advantage 00:17:21.750 --> 00:17:26.250 align:middle line:84% of having all the fluxes that are independent because you 00:17:26.250 --> 00:17:28.620 align:middle line:90% are excluding the r. 00:17:28.620 --> 00:17:33.310 align:middle line:84% You go the summation only up to r minus 1. 00:17:33.310 --> 00:17:35.360 align:middle line:90% Then we do the usual trick. 00:17:35.360 --> 00:17:37.270 align:middle line:84% We remember that the chemical potential 00:17:37.270 --> 00:17:40.770 align:middle line:90% can be written as h minus T s. 00:17:40.770 --> 00:17:43.500 align:middle line:84% And this is for the partial properties 00:17:43.500 --> 00:17:47.550 align:middle line:84% so it's partial h minus T partial s, which 00:17:47.550 --> 00:17:50.760 align:middle line:90% is written here. 00:17:50.760 --> 00:17:56.070 align:middle line:84% And so you can split terms in these fluxes 00:17:56.070 --> 00:17:58.470 align:middle line:84% that are multiplied by the temperature and terms 00:17:58.470 --> 00:17:59.180 align:middle line:90% that are not. 00:17:59.180 --> 00:18:03.084 align:middle line:90% 00:18:03.084 --> 00:18:09.380 align:middle line:84% We can also go back to molar quantities, molar 00:18:09.380 --> 00:18:12.800 align:middle line:90% fluxes without the stars. 00:18:12.800 --> 00:18:17.900 align:middle line:84% And eventually, we recognize that this term that 00:18:17.900 --> 00:18:23.060 align:middle line:84% is multiplied by T is what we define the measurable heat 00:18:23.060 --> 00:18:28.570 align:middle line:84% flux, that is the heat flux of heat transfer. 00:18:28.570 --> 00:18:31.178 align:middle line:90% 00:18:31.178 --> 00:18:35.420 align:middle line:84% So, the bottom line is highlighted here 00:18:35.420 --> 00:18:40.770 align:middle line:84% that we finally have written the fluxes of energy and entropy, 00:18:40.770 --> 00:18:43.220 align:middle line:84% which are needed to make the energy and entropy 00:18:43.220 --> 00:18:47.470 align:middle line:84% balances, of course, to the flux of heat transfer. 00:18:47.470 --> 00:18:54.480 align:middle line:84% So the measurable heat flux to the fluxes of constituents. 00:18:54.480 --> 00:18:58.020 align:middle line:84% But not all of them, only up to r minus 1, 00:18:58.020 --> 00:19:03.550 align:middle line:84% and to the diffusive flux of electrical charge. 00:19:03.550 --> 00:19:10.900 align:middle line:84% So this is basic to analyze any of those cross effects 00:19:10.900 --> 00:19:14.230 align:middle line:90% that we need to do. 00:19:14.230 --> 00:19:21.080 align:middle line:84% Now, just a little detour to mention 00:19:21.080 --> 00:19:30.240 align:middle line:84% that since we're talking about fluxes, 00:19:30.240 --> 00:19:35.790 align:middle line:84% and we are talking about heat and diffusion interaction, 00:19:35.790 --> 00:19:38.780 align:middle line:84% a legitimate question is, well, is there 00:19:38.780 --> 00:19:44.190 align:middle line:84% some interaction that is not like the heat and diffusion, 00:19:44.190 --> 00:19:47.450 align:middle line:90% it's not like heat? 00:19:47.450 --> 00:19:52.700 align:middle line:84% And one important example, which also gives me a chance 00:19:52.700 --> 00:19:58.220 align:middle line:90% to mention, is radiation. 00:19:58.220 --> 00:20:02.840 align:middle line:84% Electromagnetic radiation contains not only energy, 00:20:02.840 --> 00:20:07.970 align:middle line:84% the photons that come from the sun for example. 00:20:07.970 --> 00:20:10.362 align:middle line:90% They do not have only energy. 00:20:10.362 --> 00:20:11.320 align:middle line:90% They also have entropy. 00:20:11.320 --> 00:20:14.590 align:middle line:90% 00:20:14.590 --> 00:20:19.210 align:middle line:84% And you can think of actually the-- 00:20:19.210 --> 00:20:25.660 align:middle line:84% if you remember probably from your physics courses, this idea 00:20:25.660 --> 00:20:29.360 align:middle line:84% of the black body radiation in a cavity, in a vacuum cavity, 00:20:29.360 --> 00:20:36.680 align:middle line:84% you have photons bouncing around and exchanging with the walls. 00:20:36.680 --> 00:20:43.040 align:middle line:84% Now, the electromagnetic field is an interesting subject 00:20:43.040 --> 00:20:52.230 align:middle line:84% because if those walls in your cavity were perfect mirrors, 00:20:52.230 --> 00:20:54.190 align:middle line:84% so they wouldn't absorb the photons, 00:20:54.190 --> 00:21:00.780 align:middle line:84% but they would simply bounce them away, these-- 00:21:00.780 --> 00:21:03.240 align:middle line:84% The modes of the electromagnetic field 00:21:03.240 --> 00:21:06.425 align:middle line:84% would not interact with each other. 00:21:06.425 --> 00:21:07.800 align:middle line:84% So in other words, you would have 00:21:07.800 --> 00:21:15.690 align:middle line:84% that photons with a certain frequency will remain-- 00:21:15.690 --> 00:21:18.750 align:middle line:84% will not exchange with photons with another frequency. 00:21:18.750 --> 00:21:22.110 align:middle line:84% It is only through the interaction with matter 00:21:22.110 --> 00:21:27.270 align:middle line:84% that radiation can move energy from one mode, namely one 00:21:27.270 --> 00:21:29.260 align:middle line:90% frequency, to another frequency. 00:21:29.260 --> 00:21:32.040 align:middle line:90% 00:21:32.040 --> 00:21:37.680 align:middle line:84% OK, this is all described by Planck's theory 00:21:37.680 --> 00:21:42.000 align:middle line:90% of electromagnetic radiation. 00:21:42.000 --> 00:21:49.920 align:middle line:84% And these are the well-known spectra of the various items. 00:21:49.920 --> 00:21:57.380 align:middle line:84% One is energy, curve A is for energy, curve B is for entropy, 00:21:57.380 --> 00:22:05.190 align:middle line:84% P, the pressure is C, and then you get also-- 00:22:05.190 --> 00:22:13.880 align:middle line:84% yeah, we plotted here also the functions in terms of instead 00:22:13.880 --> 00:22:18.460 align:middle line:90% of frequency of wavelength. 00:22:18.460 --> 00:22:21.860 align:middle line:90% But what is important is that-- 00:22:21.860 --> 00:22:25.200 align:middle line:84% so the equivalent, if you have two bodies, yes-- 00:22:25.200 --> 00:22:27.990 align:middle line:90% 00:22:27.990 --> 00:22:31.170 align:middle line:84% two bodies at different temperatures, TA and TB, 00:22:31.170 --> 00:22:34.080 align:middle line:84% and there is vacuum in between, they 00:22:34.080 --> 00:22:37.860 align:middle line:84% exchange energy via the electromagnetic radiation 00:22:37.860 --> 00:22:40.200 align:middle line:84% because there are photons in each of the solid. 00:22:40.200 --> 00:22:42.165 align:middle line:90% The solid is not-- 00:22:42.165 --> 00:22:44.885 align:middle line:90% 00:22:44.885 --> 00:22:49.960 align:middle line:84% you think it is full completely, but it's full of empty space. 00:22:49.960 --> 00:22:53.500 align:middle line:84% And in that empty space, the photon go in and out. 00:22:53.500 --> 00:23:00.800 align:middle line:84% OK, so the photon that emerge from one body 00:23:00.800 --> 00:23:05.160 align:middle line:84% go directly, ballistically, so to speak, 00:23:05.160 --> 00:23:12.410 align:middle line:84% in like a ray to enter, or to hit the other solid. 00:23:12.410 --> 00:23:15.440 align:middle line:90% And vice versa. 00:23:15.440 --> 00:23:18.260 align:middle line:84% And if you take-- if the solids are 00:23:18.260 --> 00:23:23.330 align:middle line:84% at stable equilibrium and the electromagnetic radiation that 00:23:23.330 --> 00:23:26.910 align:middle line:84% leaves the solids is in equilibrium with that, 00:23:26.910 --> 00:23:29.580 align:middle line:84% it has the features of that temperature. 00:23:29.580 --> 00:23:35.400 align:middle line:84% So the photons that come out of body A have the temperature TA. 00:23:35.400 --> 00:23:40.650 align:middle line:84% And then they arrive to system B. 00:23:40.650 --> 00:23:44.730 align:middle line:84% And that is a non-equilibrium situation. 00:23:44.730 --> 00:23:47.790 align:middle line:84% They have to interact and thermalize. 00:23:47.790 --> 00:23:52.320 align:middle line:84% And so by interacting, and being absorbed, and re 00:23:52.320 --> 00:23:55.620 align:middle line:84% emitted by the molecules of body B, 00:23:55.620 --> 00:24:00.424 align:middle line:84% there is a remesculation (I meant, redistribution) 00:24:00.424 --> 00:24:01.770 align:middle line:90% of the energy. 00:24:01.770 --> 00:24:04.950 align:middle line:84% It's like the collisions that we talked about 00:24:04.950 --> 00:24:13.200 align:middle line:90% that it is an irreversible fact. 00:24:13.200 --> 00:24:18.300 align:middle line:84% that allows the thermalization of these photons. 00:24:18.300 --> 00:24:23.850 align:middle line:84% So the-- so sigma times T to the 4, 00:24:23.850 --> 00:24:26.100 align:middle line:84% sigma is the Stefan-Boltzmann constant, 00:24:26.100 --> 00:24:30.000 align:middle line:84% it's written down here, I'm sure you remember it. 00:24:30.000 --> 00:24:36.720 align:middle line:84% So this sigma T to the 4 is the amount of energy that goes out 00:24:36.720 --> 00:24:41.250 align:middle line:84% of body A. And this is the one that goes out of body B. 00:24:41.250 --> 00:24:44.380 align:middle line:84% So they exchange-- so the net exchange is the difference. 00:24:44.380 --> 00:24:46.905 align:middle line:90% 00:24:46.905 --> 00:24:51.670 align:middle line:84% And together with the sigma to the T to the 4, 00:24:51.670 --> 00:24:56.730 align:middle line:84% which is very well known, there is also a 4/3 sigma T to the 3 00:24:56.730 --> 00:24:59.690 align:middle line:84% for the entropy that goes with it. 00:24:59.690 --> 00:25:02.520 align:middle line:84% So these photons do not carry only energy, 00:25:02.520 --> 00:25:10.950 align:middle line:84% they also carry entropy like the photons that come from the sun. 00:25:10.950 --> 00:25:15.360 align:middle line:84% The sun has photons that on the surface have a temperature 00:25:15.360 --> 00:25:24.870 align:middle line:84% of the order of, what is it, 5,700, 5,760K. 00:25:24.870 --> 00:25:29.130 align:middle line:84% And those photon leave the plasma 00:25:29.130 --> 00:25:33.120 align:middle line:84% that is on the surface of the sun 00:25:33.120 --> 00:25:36.700 align:middle line:84% that we've seen very well during the eclipse the other day. 00:25:36.700 --> 00:25:45.140 align:middle line:84% And they all the way travel to here, to the Earth. 00:25:45.140 --> 00:25:49.290 align:middle line:84% And they carry the features of energy 00:25:49.290 --> 00:25:52.950 align:middle line:84% and entropy proper of that temperature. 00:25:52.950 --> 00:25:57.170 align:middle line:84% And the fact that they have entropy 00:25:57.170 --> 00:26:01.560 align:middle line:84% limits the amount of work that you can do with that energy. 00:26:01.560 --> 00:26:08.510 align:middle line:84% So it's not entirely-- you cannot extract 100% of energy 00:26:08.510 --> 00:26:10.400 align:middle line:90% because of that entropy. 00:26:10.400 --> 00:26:14.060 align:middle line:84% That's the-- do you remember that when we talked about, 00:26:14.060 --> 00:26:16.560 align:middle line:84% I think, in the allocation chapter, 00:26:16.560 --> 00:26:21.470 align:middle line:84% we talked about the exergy of solar radiation 00:26:21.470 --> 00:26:23.670 align:middle line:90% being around 0.93. 00:26:23.670 --> 00:26:26.660 align:middle line:84% That is where-- part of the story, 00:26:26.660 --> 00:26:30.950 align:middle line:90% comes from this entropy content. 00:26:30.950 --> 00:26:37.130 align:middle line:84% But of course, you can also not only talk about the flux, 00:26:37.130 --> 00:26:40.410 align:middle line:84% overall flux of energy carried by these photons, 00:26:40.410 --> 00:26:43.790 align:middle line:84% but also, you can talk about the flux 00:26:43.790 --> 00:26:50.340 align:middle line:84% of energy carried by each mode of the electromagnetic field. 00:26:50.340 --> 00:26:52.990 align:middle line:90% So you do this per frequency. 00:26:52.990 --> 00:26:57.400 align:middle line:84% And you can do that energy wise, but also entropy wise. 00:26:57.400 --> 00:27:03.960 align:middle line:84% So this is the flux that is exchanged 00:27:03.960 --> 00:27:08.970 align:middle line:84% between body A and body B at temperature T via photons 00:27:08.970 --> 00:27:13.190 align:middle line:90% of that frequency nu. 00:27:13.190 --> 00:27:19.090 align:middle line:84% Or if you want, you can write it in terms of the wavelength. 00:27:19.090 --> 00:27:21.670 align:middle line:84% In this paper that we wrote long ago, 00:27:21.670 --> 00:27:24.640 align:middle line:84% and then it was republished recently, 00:27:24.640 --> 00:27:29.110 align:middle line:84% you find all these formulas that are useful if you are 00:27:29.110 --> 00:27:31.963 align:middle line:90% in the solar energy business. 00:27:31.963 --> 00:27:35.450 align:middle line:90% 00:27:35.450 --> 00:27:44.810 align:middle line:84% So let's go back now to the formula 00:27:44.810 --> 00:27:47.490 align:middle line:84% for the entropy production, which 00:27:47.490 --> 00:27:53.300 align:middle line:84% will be the starting point for our applications. 00:27:53.300 --> 00:28:12.340 align:middle line:84% And we do-- also here, we want to introduce the fact that-- 00:28:12.340 --> 00:28:17.420 align:middle line:84% use the fact that these fluxes are not all independent 00:28:17.420 --> 00:28:22.460 align:middle line:84% and rewrite this expression in which-- that you-- 00:28:22.460 --> 00:28:27.900 align:middle line:84% I don't go over it because we talked about it so many times. 00:28:27.900 --> 00:28:30.140 align:middle line:84% But we now want to concentrate only 00:28:30.140 --> 00:28:37.630 align:middle line:84% on this term, which has to do with the diffusion of particles. 00:28:37.630 --> 00:28:41.830 align:middle line:84% And we want to rewrite it in a form that goes only up to r 00:28:41.830 --> 00:28:44.170 align:middle line:84% to the minus 1, or essentially, that it is written 00:28:44.170 --> 00:28:48.760 align:middle line:90% in terms of independent fluxes. 00:28:48.760 --> 00:28:54.520 align:middle line:84% And one way of doing it is through these steps 00:28:54.520 --> 00:28:59.050 align:middle line:84% because now, we also have to remember 00:28:59.050 --> 00:29:04.860 align:middle line:84% that the gradient in the chemical potentials, 00:29:04.860 --> 00:29:10.840 align:middle line:84% which appears here, this is a gradient done at keeping 00:29:10.840 --> 00:29:15.310 align:middle line:90% the temperature constant, is-- 00:29:15.310 --> 00:29:17.980 align:middle line:84% they are not all independent either 00:29:17.980 --> 00:29:21.020 align:middle line:84% because of the Gibbs-Duhem relation. 00:29:21.020 --> 00:29:23.920 align:middle line:84% Remember, the Gibbs-Duhem relation is written here. 00:29:23.920 --> 00:29:29.800 align:middle line:84% It says that the weighted sum of those gradients of chemical 00:29:29.800 --> 00:29:34.000 align:middle line:84% potentials weighted on the concentrations of the elements-- 00:29:34.000 --> 00:29:38.290 align:middle line:90% of the constituents is related-- 00:29:38.290 --> 00:29:42.570 align:middle line:84% gives the gradient in the pressure. 00:29:42.570 --> 00:29:44.510 align:middle line:84% Remember that the pressure, we managed 00:29:44.510 --> 00:29:47.060 align:middle line:84% to make it disappear somehow from-- 00:29:47.060 --> 00:29:50.310 align:middle line:84% by working in terms of properties per unit volume. 00:29:50.310 --> 00:29:54.960 align:middle line:84% But here, it's where it reappears 00:29:54.960 --> 00:29:56.615 align:middle line:90% via the Gibbs-Duhem relation. 00:29:56.615 --> 00:30:02.150 align:middle line:90% 00:30:02.150 --> 00:30:06.180 align:middle line:84% And now, I'll let you play with the math. 00:30:06.180 --> 00:30:09.110 align:middle line:90% Here, it explains how. 00:30:09.110 --> 00:30:11.550 align:middle line:84% The fact that if you have a relation like this, 00:30:11.550 --> 00:30:15.115 align:middle line:84% you can rewrite it in this other form. 00:30:15.115 --> 00:30:17.810 align:middle line:90% 00:30:17.810 --> 00:30:21.665 align:middle line:84% It's just playing with summing and subtracting 0's. 00:30:21.665 --> 00:30:22.165 align:middle line:90% And. 00:30:22.165 --> 00:30:25.025 align:middle line:90% 00:30:25.025 --> 00:30:28.730 align:middle line:90% the good thing is that if you-- 00:30:28.730 --> 00:30:32.420 align:middle line:84% that now, the right-hand side of this expression 00:30:32.420 --> 00:30:37.160 align:middle line:84% has differences between chemical potentials. 00:30:37.160 --> 00:30:43.020 align:middle line:84% And therefore, this summation, which goes still from 1 to r, 00:30:43.020 --> 00:30:48.470 align:middle line:84% but effectively goes to r minus 1, because when j is equal to i, 00:30:48.470 --> 00:30:49.780 align:middle line:90% this difference is 0. 00:30:49.780 --> 00:30:56.570 align:middle line:90% 00:30:56.570 --> 00:30:59.350 align:middle line:84% So this is the first step that we make. 00:30:59.350 --> 00:31:02.700 align:middle line:90% We substitute this into here. 00:31:02.700 --> 00:31:06.050 align:middle line:90% 00:31:06.050 --> 00:31:11.020 align:middle line:84% And next, once we have this expression like this, 00:31:11.020 --> 00:31:12.890 align:middle line:84% again, I let you play with the math. 00:31:12.890 --> 00:31:18.010 align:middle line:84% It's an expression of this type in which it happens 00:31:18.010 --> 00:31:21.490 align:middle line:84% that the coefficients that multiply here 00:31:21.490 --> 00:31:26.820 align:middle line:84% are invariant upon exchange of i and j. 00:31:26.820 --> 00:31:31.980 align:middle line:84% And that symmetry allows you to add and subtract 00:31:31.980 --> 00:31:35.220 align:middle line:84% or to rewrite this term-- this summation 00:31:35.220 --> 00:31:41.170 align:middle line:84% by exchanging those indexes and then subtracting the two, 00:31:41.170 --> 00:31:45.190 align:middle line:84% and essentially, you can get this identity, 00:31:45.190 --> 00:31:47.690 align:middle line:90% which we apply there. 00:31:47.690 --> 00:31:51.020 align:middle line:84% So now, we have this nice expression 00:31:51.020 --> 00:31:56.310 align:middle line:84% in which we have the summation, it's a double summation. 00:31:56.310 --> 00:32:01.680 align:middle line:84% But it has a number of 0 elements because every time-- 00:32:01.680 --> 00:32:04.890 align:middle line:84% it's all written in terms of difference between fluxes 00:32:04.890 --> 00:32:09.710 align:middle line:84% and differences between chemical potentials, which also has 00:32:09.710 --> 00:32:14.760 align:middle line:84% the advantage that makes it reference frame 00:32:14.760 --> 00:32:19.950 align:middle line:84% independent because those are fluxes with respect 00:32:19.950 --> 00:32:22.380 align:middle line:90% to the barycentric velocity. 00:32:22.380 --> 00:32:25.720 align:middle line:84% So they depend on the reference frame, 00:32:25.720 --> 00:32:28.470 align:middle line:84% which is the barycentric one, the one that 00:32:28.470 --> 00:32:37.330 align:middle line:84% follows-- the Lagrangian local (frame) following 00:32:37.330 --> 00:32:38.140 align:middle line:90% your fluid element. 00:32:38.140 --> 00:32:40.098 align:middle line:84% But if you take the difference between the two, 00:32:40.098 --> 00:32:44.080 align:middle line:84% then the difference is independent of that frame. 00:32:44.080 --> 00:32:48.560 align:middle line:90% And that is also an advantage. 00:32:48.560 --> 00:32:57.910 align:middle line:84% So finally, yes, we bring back the pressure here. 00:32:57.910 --> 00:33:02.570 align:middle line:84% And we-- but we have this summation, 00:33:02.570 --> 00:33:05.670 align:middle line:84% which is expressed in terms of these differences 00:33:05.670 --> 00:33:07.030 align:middle line:90% that we now rename. 00:33:07.030 --> 00:33:13.870 align:middle line:84% So J ij is the flux of particles of species i with respect 00:33:13.870 --> 00:33:16.210 align:middle line:90% to particles of species j. 00:33:16.210 --> 00:33:21.460 align:middle line:84% So for example, if you have the water vapor that comes out 00:33:21.460 --> 00:33:25.030 align:middle line:84% of your shower and so those water molecules 00:33:25.030 --> 00:33:31.160 align:middle line:84% travel in the air and go and get your mirrors wet. 00:33:31.160 --> 00:33:38.395 align:middle line:84% OK, so that's diffusion of particles of water vapor. 00:33:38.395 --> 00:33:44.230 align:middle line:84% And of course, the room that is made empty 00:33:44.230 --> 00:33:47.470 align:middle line:84% by those particles of water vapor that goes away 00:33:47.470 --> 00:33:49.970 align:middle line:90% are filled by the air molecules. 00:33:49.970 --> 00:33:55.190 align:middle line:90% So there is a crossing of flows. 00:33:55.190 --> 00:33:56.930 align:middle line:84% OK, so here, we are talking about-- 00:33:56.930 --> 00:34:05.210 align:middle line:84% so J ij is the flux of particles of the water vapor with respect 00:34:05.210 --> 00:34:09.659 align:middle line:84% to the motion of the air molecules. 00:34:09.659 --> 00:34:12.429 align:middle line:90% So it's a relative flux. 00:34:12.429 --> 00:34:18.250 align:middle line:84% And also, this chemical potential differences 00:34:18.250 --> 00:34:22.350 align:middle line:84% that are also written here, we can think of them 00:34:22.350 --> 00:34:23.800 align:middle line:90% as the conjugate-- 00:34:23.800 --> 00:34:25.039 align:middle line:90% So there are relative forces. 00:34:25.039 --> 00:34:41.600 align:middle line:90% 00:34:41.600 --> 00:34:47.300 align:middle line:84% So now, if we take this expression here, 00:34:47.300 --> 00:34:49.780 align:middle line:90% we substitute what we get here. 00:34:49.780 --> 00:34:53.270 align:middle line:84% So we get a term with the gradient in the pressure. 00:34:53.270 --> 00:34:57.010 align:middle line:84% And we also have the term with the dissipation function 00:34:57.010 --> 00:34:58.130 align:middle line:90% from fluid mechanics. 00:34:58.130 --> 00:34:59.420 align:middle line:90% Let's forget about those. 00:34:59.420 --> 00:35:04.210 align:middle line:84% So let's suppose we take no flow and no-- 00:35:04.210 --> 00:35:05.270 align:middle line:90% yeah, no flow. 00:35:05.270 --> 00:35:06.780 align:middle line:90% So we move with the-- 00:35:06.780 --> 00:35:10.270 align:middle line:90% 00:35:10.270 --> 00:35:14.680 align:middle line:84% perhaps we have a box with particles 00:35:14.680 --> 00:35:19.630 align:middle line:84% that are steady with respect to the laboratory frame. 00:35:19.630 --> 00:35:24.670 align:middle line:84% And let's suppose we are at uniform pressure. 00:35:24.670 --> 00:35:27.000 align:middle line:84% And also, we have no chemical reactions. 00:35:27.000 --> 00:35:29.643 align:middle line:84% Then, we are left with these three terms. 00:35:29.643 --> 00:35:31.060 align:middle line:84% And these are the three terms that 00:35:31.060 --> 00:35:38.560 align:middle line:84% will allow us to define the thermodiffusion 00:35:38.560 --> 00:35:41.710 align:middle line:84% and the thermoelectric phenomena that we 00:35:41.710 --> 00:35:44.560 align:middle line:84% are interested in dealing with as examples 00:35:44.560 --> 00:35:49.000 align:middle line:90% of these cross effects. 00:35:49.000 --> 00:35:51.585 align:middle line:84% So this will be our starting point. 00:35:51.585 --> 00:36:03.360 align:middle line:90% 00:36:03.360 --> 00:36:06.600 align:middle line:84% Let's start with the thermoelectric effects 00:36:06.600 --> 00:36:14.610 align:middle line:84% as the more or less simple example of the procedure 00:36:14.610 --> 00:36:18.790 align:middle line:84% that we need in order to understand it. 00:36:18.790 --> 00:36:23.470 align:middle line:84% We start from this expression of the entropy production 00:36:23.470 --> 00:36:29.040 align:middle line:90% that we just reviewed. 00:36:29.040 --> 00:36:38.340 align:middle line:84% And in this case, there is only one particle, 00:36:38.340 --> 00:36:43.230 align:middle line:84% which is the electrons in your conductor, 00:36:43.230 --> 00:36:49.840 align:middle line:84% or semiconductor, or the holes in the semiconductor. 00:36:49.840 --> 00:36:52.760 align:middle line:90% So charges. 00:36:52.760 --> 00:36:56.570 align:middle line:84% But a single constituent, it means 00:36:56.570 --> 00:37:04.710 align:middle line:84% that there is no diffusive flux with anything else. 00:37:04.710 --> 00:37:11.710 align:middle line:84% So the term that we had a couple of slides ago, so this one, 00:37:11.710 --> 00:37:12.730 align:middle line:90% this one does-- oops. 00:37:12.730 --> 00:37:13.400 align:middle line:90% Sorry. 00:37:13.400 --> 00:37:16.900 align:middle line:84% This one doesn't contribute because there 00:37:16.900 --> 00:37:19.662 align:middle line:84% is no relative motion with anything. 00:37:19.662 --> 00:37:21.120 align:middle line:84% There is only one type of particle. 00:37:21.120 --> 00:37:26.575 align:middle line:90% 00:37:26.575 --> 00:37:33.160 align:middle line:84% And here, I have highlighted in blue and red the flux 00:37:33.160 --> 00:37:38.116 align:middle line:84% and the conjugated entropic force. 00:37:38.116 --> 00:37:45.770 align:middle line:84% You'll see in a moment why I suggest that when you do this, 00:37:45.770 --> 00:37:48.770 align:middle line:84% you always start from this entropic point of view 00:37:48.770 --> 00:37:54.980 align:middle line:84% because that's the only safe place in which you 00:37:54.980 --> 00:37:58.430 align:middle line:84% can apply Onsager relations without making 00:37:58.430 --> 00:38:01.160 align:middle line:90% too many mistakes. 00:38:01.160 --> 00:38:02.210 align:middle line:90% All right. 00:38:02.210 --> 00:38:08.600 align:middle line:84% So the next step, remember, this is the extrinsic relation. 00:38:08.600 --> 00:38:11.400 align:middle line:84% It doesn't contain any feature yet of your material. 00:38:11.400 --> 00:38:14.000 align:middle line:90% 00:38:14.000 --> 00:38:19.140 align:middle line:84% Now, in order to bring in the features of the material, 00:38:19.140 --> 00:38:24.750 align:middle line:84% you have to model the relation between the fluxes 00:38:24.750 --> 00:38:27.700 align:middle line:90% and the forces. 00:38:27.700 --> 00:38:31.180 align:middle line:84% And if we assume a linear force flux relation 00:38:31.180 --> 00:38:35.920 align:middle line:84% because we assume that we are near equilibrium and so on, 00:38:35.920 --> 00:38:41.500 align:middle line:84% this is the most general linear relation between these fluxes 00:38:41.500 --> 00:38:43.680 align:middle line:90% and these forces. 00:38:43.680 --> 00:38:48.890 align:middle line:84% And it has a matrix of coefficients, 00:38:48.890 --> 00:38:50.990 align:middle line:84% which is the Onsager matrix of coefficients. 00:38:50.990 --> 00:38:55.190 align:middle line:84% Actually, in this slide, these are 00:38:55.190 --> 00:39:01.490 align:middle line:84% tensors because as you remember, suppose we didn't have-- 00:39:01.490 --> 00:39:05.930 align:middle line:90% suppose we had only Fourier law. 00:39:05.930 --> 00:39:10.040 align:middle line:84% Fourier law is a linear relation between the heat flux, which 00:39:10.040 --> 00:39:13.280 align:middle line:84% is a vector, and the gradient in the inverse temperature, which 00:39:13.280 --> 00:39:14.630 align:middle line:90% is also a vector. 00:39:14.630 --> 00:39:17.220 align:middle line:84% And the most general relation is through a tensor, 00:39:17.220 --> 00:39:20.420 align:middle line:84% which is the conductivity tensor k 00:39:20.420 --> 00:39:23.300 align:middle line:84% double underline that we have already discussed, 00:39:23.300 --> 00:39:28.010 align:middle line:84% and which is symmetric by the Onsager principle. 00:39:28.010 --> 00:39:32.610 align:middle line:84% Here, we have added also relations with other tensors. 00:39:32.610 --> 00:39:35.600 align:middle line:84% So this would be instead the relation 00:39:35.600 --> 00:39:41.080 align:middle line:84% between electrical current, this is current density, 00:39:41.080 --> 00:39:42.405 align:middle line:90% so it's the flux of charge. 00:39:42.405 --> 00:39:45.270 align:middle line:90% 00:39:45.270 --> 00:39:49.230 align:middle line:84% Yeah, I'm calling it now i double prime. 00:39:49.230 --> 00:39:56.840 align:middle line:84% I should have probably more consistently called it Jq. 00:39:56.840 --> 00:40:00.960 align:middle line:84% So the diffusive flux of charge, because this is still diffusive. 00:40:00.960 --> 00:40:04.640 align:middle line:84% But since our solid is a solid, it is not moving, 00:40:04.640 --> 00:40:07.290 align:middle line:84% there is no convective part of the charge. 00:40:07.290 --> 00:40:10.700 align:middle line:90% And so they are the same. 00:40:10.700 --> 00:40:13.658 align:middle line:84% So just to make it more appealing to the idea 00:40:13.658 --> 00:40:15.950 align:middle line:84% that we're talking about electrical currents, something 00:40:15.950 --> 00:40:17.650 align:middle line:90% that you measure in a wire. 00:40:17.650 --> 00:40:20.210 align:middle line:90% 00:40:20.210 --> 00:40:24.060 align:middle line:90% OK, so this is Ohm's law. 00:40:24.060 --> 00:40:27.390 align:middle line:84% If you take only this last term, the relation 00:40:27.390 --> 00:40:30.930 align:middle line:84% between current and the gradient in potential 00:40:30.930 --> 00:40:34.800 align:middle line:84% is the thing that you measure in volt, with the voltmeter, 00:40:34.800 --> 00:40:36.780 align:middle line:90% in volts. 00:40:36.780 --> 00:40:41.110 align:middle line:90% T is-- tau is 1 over T. 00:40:41.110 --> 00:40:51.990 align:middle line:84% And L is the resistivity of your material, which 00:40:51.990 --> 00:40:59.800 align:middle line:84% you would measure in Ohm per meter, I guess. 00:40:59.800 --> 00:41:01.690 align:middle line:90% And it is a tensor also. 00:41:01.690 --> 00:41:04.280 align:middle line:84% So if your material is isotropic, 00:41:04.280 --> 00:41:09.140 align:middle line:84% that tensor to reduce this to just the one number, one scalar. 00:41:09.140 --> 00:41:11.710 align:middle line:84% But if it is anisotropic, exactly 00:41:11.710 --> 00:41:15.260 align:middle line:84% like we have seen for heat transfer, 00:41:15.260 --> 00:41:21.034 align:middle line:84% you would have different principal directions of-- 00:41:21.034 --> 00:41:22.810 align:middle line:84% where the electrical current likes 00:41:22.810 --> 00:41:27.870 align:middle line:84% to run easier than in other directions. 00:41:27.870 --> 00:41:30.210 align:middle line:90% OK, so this is the first step. 00:41:30.210 --> 00:41:33.630 align:middle line:90% Assume linear flux relations. 00:41:33.630 --> 00:41:42.480 align:middle line:84% Next, what we typically do is to rewrite. 00:41:42.480 --> 00:41:46.800 align:middle line:84% So rearrange these equations by bringing 00:41:46.800 --> 00:41:49.050 align:middle line:90% in more familiar variables. 00:41:49.050 --> 00:41:51.660 align:middle line:84% And in this case, the more familiar variable 00:41:51.660 --> 00:41:55.230 align:middle line:84% is simply the temperature instead of tau. 00:41:55.230 --> 00:42:04.250 align:middle line:84% So this is 1 over T. But we also want to rewrite it 00:42:04.250 --> 00:42:05.190 align:middle line:90% in a different way. 00:42:05.190 --> 00:42:09.770 align:middle line:84% So we want in the first equation to substitute this term here 00:42:09.770 --> 00:42:11.125 align:middle line:90% with the electric current. 00:42:11.125 --> 00:42:13.810 align:middle line:90% 00:42:13.810 --> 00:42:17.140 align:middle line:84% So what we do is take the second equation 00:42:17.140 --> 00:42:22.356 align:middle line:84% and extract the gradient in phi from the second equation 00:42:22.356 --> 00:42:26.380 align:middle line:84% that we will then substitute in the first one. 00:42:26.380 --> 00:42:28.090 align:middle line:90% So this is what we are doing. 00:42:28.090 --> 00:42:32.720 align:middle line:84% So we take this, we solve, of course, these are tensors. 00:42:32.720 --> 00:42:35.710 align:middle line:84% So in order to solve, you have to multiply to the right-- 00:42:35.710 --> 00:42:45.030 align:middle line:84% from the left with the inverse of this tensor, which 00:42:45.030 --> 00:42:47.040 align:middle line:84% exists because the tensor is positive 00:42:47.040 --> 00:42:52.410 align:middle line:84% definite because of the principle 00:42:52.410 --> 00:42:55.790 align:middle line:90% of entropy nondecrease. 00:42:55.790 --> 00:42:59.010 align:middle line:84% So you do that, you get the gradient in phi, 00:42:59.010 --> 00:43:01.410 align:middle line:90% you substitute up there. 00:43:01.410 --> 00:43:04.410 align:middle line:84% And here is what you get as a result. 00:43:04.410 --> 00:43:09.030 align:middle line:84% So you get now a relation between the heat flux, 00:43:09.030 --> 00:43:14.180 align:middle line:84% the gradient in temperature, and the electrical current. 00:43:14.180 --> 00:43:20.220 align:middle line:84% And of course, the substitution that you've 00:43:20.220 --> 00:43:24.610 align:middle line:84% made since the gradient in the electric field-- 00:43:24.610 --> 00:43:29.640 align:middle line:84% I'm sorry, in the electric potential, 00:43:29.640 --> 00:43:35.140 align:middle line:84% depends on the current, but also on the gradient in temperature. 00:43:35.140 --> 00:43:39.610 align:middle line:84% When you substitute it up there, it gives its own contribution 00:43:39.610 --> 00:43:42.640 align:middle line:84% to the term that multiplies the factor, the prefactor, 00:43:42.640 --> 00:43:44.560 align:middle line:90% of the gradient in temperature. 00:43:44.560 --> 00:43:46.540 align:middle line:84% That's why you see this difference. 00:43:46.540 --> 00:43:53.040 align:middle line:84% So this is the direct, say, Fourier like kind of term. 00:43:53.040 --> 00:43:57.835 align:middle line:84% And this is the indirect due to-- 00:43:57.835 --> 00:44:02.440 align:middle line:90% 00:44:02.440 --> 00:44:08.000 align:middle line:84% due to the flow of charges, of electrons. 00:44:08.000 --> 00:44:14.370 align:middle line:90% OK, so and then finally, we-- 00:44:14.370 --> 00:44:17.940 align:middle line:84% instead of carrying all these names with the L's, we just 00:44:17.940 --> 00:44:18.920 align:middle line:90% give different names. 00:44:18.920 --> 00:44:22.280 align:middle line:90% 00:44:22.280 --> 00:44:28.250 align:middle line:84% So the term that multiplies in this equation, 00:44:28.250 --> 00:44:34.530 align:middle line:84% the gradient in temperature, we call it epsilon. 00:44:34.530 --> 00:44:38.256 align:middle line:84% And that's called the Seebeck coefficient. 00:44:38.256 --> 00:44:41.740 align:middle line:90% 00:44:41.740 --> 00:44:45.773 align:middle line:84% This other term, we call it the electrical resistivity 00:44:45.773 --> 00:44:46.440 align:middle line:90% of the material. 00:44:46.440 --> 00:44:52.650 align:middle line:90% 00:44:52.650 --> 00:44:57.780 align:middle line:84% So the L was the conductivity, electrical conductivity. 00:44:57.780 --> 00:44:59.700 align:middle line:90% I misspoke before. 00:44:59.700 --> 00:45:04.920 align:middle line:84% Now, this combination of L's here with the minus-- 00:45:04.920 --> 00:45:10.980 align:middle line:84% the 1 over T squared, we call it thermal conductivity. 00:45:10.980 --> 00:45:16.650 align:middle line:84% And this last one, we call it the Peltier coefficient. 00:45:16.650 --> 00:45:19.865 align:middle line:84% And here, we repeat their definitions just for clarity. 00:45:19.865 --> 00:45:26.380 align:middle line:90% 00:45:26.380 --> 00:45:31.030 align:middle line:84% Next, we're not done yet, because these four 00:45:31.030 --> 00:45:32.950 align:middle line:84% numbers, these four properties, these 00:45:32.950 --> 00:45:34.880 align:middle line:90% are equilibrium properties. 00:45:34.880 --> 00:45:38.380 align:middle line:84% They are-- remember that this is a linearization 00:45:38.380 --> 00:45:39.440 align:middle line:90% around the equilibrium. 00:45:39.440 --> 00:45:44.810 align:middle line:84% So these are measured at equilibrium. 00:45:44.810 --> 00:45:49.670 align:middle line:84% And they are not all independent because there is the Onsager 00:45:49.670 --> 00:45:50.980 align:middle line:90% principle of reciprocity. 00:45:50.980 --> 00:45:55.100 align:middle line:90% So L12 is equal to L21. 00:45:55.100 --> 00:45:57.750 align:middle line:84% More precisely, since we're talking about tensors, 00:45:57.750 --> 00:46:01.270 align:middle line:90% you should think of this big-- 00:46:01.270 --> 00:46:02.690 align:middle line:84% this is a big system in which you 00:46:02.690 --> 00:46:07.670 align:middle line:84% have three equations here for the components of the heat flux 00:46:07.670 --> 00:46:09.890 align:middle line:84% and three equations here for the components 00:46:09.890 --> 00:46:12.423 align:middle line:90% of the current density. 00:46:12.423 --> 00:46:15.700 align:middle line:90% 00:46:15.700 --> 00:46:24.010 align:middle line:84% So you have-- so it's a 6 by 6, which you can 00:46:24.010 --> 00:46:27.040 align:middle line:90% cast in this more compact way. 00:46:27.040 --> 00:46:36.620 align:middle line:84% But this 6-by-6 matrix has to be symmetric by the Onsager 00:46:36.620 --> 00:46:38.840 align:middle line:90% reciprocity. 00:46:38.840 --> 00:46:43.710 align:middle line:84% And so this symmetry requires that each 00:46:43.710 --> 00:46:50.730 align:middle line:84% of the groups, these blocks that we have defined of this matrix, 00:46:50.730 --> 00:46:53.100 align:middle line:84% are symmetric themselves, and also 00:46:53.100 --> 00:47:02.700 align:middle line:84% that these off diagonal are one equal to the other. 00:47:02.700 --> 00:47:09.790 align:middle line:84% So if you apply that symmetry, you may notice-- 00:47:09.790 --> 00:47:12.860 align:middle line:90% let's see. 00:47:12.860 --> 00:47:14.900 align:middle line:84% Yeah, you may notice that there is 00:47:14.900 --> 00:47:20.250 align:middle line:84% a relation between the Peltier tensor and the Seebeck tensor. 00:47:20.250 --> 00:47:25.810 align:middle line:84% If you take the transpose of the Peltier, 00:47:25.810 --> 00:47:29.782 align:middle line:84% the transpose of a product of matrices, 00:47:29.782 --> 00:47:31.240 align:middle line:84% when you do the transpose, you have 00:47:31.240 --> 00:47:37.070 align:middle line:84% to also change the order, the product of the transposed. 00:47:37.070 --> 00:47:42.350 align:middle line:84% And then you apply Onsager symmetry. 00:47:42.350 --> 00:47:45.230 align:middle line:90% So the T goes away. 00:47:45.230 --> 00:47:50.770 align:middle line:84% And here comes something that is contained in Seebeck 00:47:50.770 --> 00:47:52.010 align:middle line:90% except the temperature. 00:47:52.010 --> 00:47:55.420 align:middle line:84% So essentially, you find that the transpose of Peltier 00:47:55.420 --> 00:47:58.190 align:middle line:84% is equal to the temperature times Seebeck. 00:47:58.190 --> 00:48:01.790 align:middle line:84% And of course, if the material is isotropic, 00:48:01.790 --> 00:48:06.010 align:middle line:84% these tensors simplify to simple scalars multiplied 00:48:06.010 --> 00:48:11.080 align:middle line:84% by the identity tensor, which is matrix with all 1's 00:48:11.080 --> 00:48:14.800 align:middle line:90% on the diagonal and 0's outside. 00:48:14.800 --> 00:48:17.770 align:middle line:84% And so the Onsager reciprocity implies 00:48:17.770 --> 00:48:20.080 align:middle line:84% that the Peltier coefficient is equal to temperature 00:48:20.080 --> 00:48:23.580 align:middle line:90% times the Seebeck coefficient. 00:48:23.580 --> 00:48:32.540 align:middle line:84% Moreover, you can also rewrite in terms of these new properties 00:48:32.540 --> 00:48:37.030 align:middle line:90% of this material. 00:48:37.030 --> 00:48:41.986 align:middle line:84% You can rewrite the entropy production. 00:48:41.986 --> 00:48:46.170 align:middle line:90% 00:48:46.170 --> 00:48:50.040 align:middle line:84% And you would like to write it in a canonical form just 00:48:50.040 --> 00:48:53.340 align:middle line:90% to make sure that it is-- 00:48:53.340 --> 00:48:58.800 align:middle line:84% that your model does indeed respect the principle 00:48:58.800 --> 00:48:59.940 align:middle line:90% of entropy non decrease. 00:48:59.940 --> 00:49:04.260 align:middle line:90% So these are quadratic-- 00:49:04.260 --> 00:49:08.850 align:middle line:84% canonical form-- And these are quadratic forms, 00:49:08.850 --> 00:49:10.840 align:middle line:90% which as you see, they are. 00:49:10.840 --> 00:49:15.330 align:middle line:84% And you can write it in this way or also in this other way. 00:49:15.330 --> 00:49:16.910 align:middle line:90% A single quadratic form. 00:49:16.910 --> 00:49:21.600 align:middle line:90% 00:49:21.600 --> 00:49:27.590 align:middle line:84% So let's see an example of where we use the Seebeck 00:49:27.590 --> 00:49:32.926 align:middle line:84% effect every day in the lab, which is the thermocouple. 00:49:32.926 --> 00:49:38.860 align:middle line:90% 00:49:38.860 --> 00:49:43.970 align:middle line:84% You see, we started from the general linear relations. 00:49:43.970 --> 00:49:48.720 align:middle line:84% And we ended up with these other relations. 00:49:48.720 --> 00:49:50.170 align:middle line:90% Now, yeah. 00:49:50.170 --> 00:49:51.520 align:middle line:90% Let me go back one moment. 00:49:51.520 --> 00:49:53.100 align:middle line:90% Sorry. 00:49:53.100 --> 00:49:57.970 align:middle line:90% I want to show you this. 00:49:57.970 --> 00:50:05.932 align:middle line:84% Already at this stage, if I change 1 over T 00:50:05.932 --> 00:50:09.570 align:middle line:84% and write the equations in terms of temperature 00:50:09.570 --> 00:50:14.340 align:middle line:84% and not the inverse temperature, look what happens-- 00:50:14.340 --> 00:50:23.560 align:middle line:84% that-- just that little step makes you lose the symmetry 00:50:23.560 --> 00:50:26.350 align:middle line:90% because L12 is equal to L21. 00:50:26.350 --> 00:50:29.200 align:middle line:84% But this coefficient here is divided by T. 00:50:29.200 --> 00:50:31.600 align:middle line:90% This is divided by T squared. 00:50:31.600 --> 00:50:33.680 align:middle line:84% So they are not equal to one another. 00:50:33.680 --> 00:50:40.860 align:middle line:84% So if you by mistake started from-- 00:50:40.860 --> 00:50:42.400 align:middle line:90% Oh, say, well, who cares? 00:50:42.400 --> 00:50:44.580 align:middle line:84% I mean, temperature or inverse temperature. 00:50:44.580 --> 00:50:47.010 align:middle line:90% So you took the wrong-- 00:50:47.010 --> 00:50:54.825 align:middle line:84% so it's not the temperature, the witness of non-equilibrium, 00:50:54.825 --> 00:50:57.650 align:middle line:90% the degree of disequilibrium. 00:50:57.650 --> 00:51:00.070 align:middle line:84% It's the gradient of the inverse temperature. 00:51:00.070 --> 00:51:03.105 align:middle line:84% So that is minus 1 over T squared 00:51:03.105 --> 00:51:05.195 align:middle line:90% times grad T. So that is the-- 00:51:05.195 --> 00:51:08.240 align:middle line:90% 00:51:08.240 --> 00:51:14.150 align:middle line:84% so this is why I say, you better be careful and start always 00:51:14.150 --> 00:51:19.280 align:middle line:84% from the entropic point of view and the proper degrees 00:51:19.280 --> 00:51:20.540 align:middle line:90% of disequilibrium. 00:51:20.540 --> 00:51:24.950 align:middle line:84% Otherwise, you may end up thinking that there are 00:51:24.950 --> 00:51:29.790 align:middle line:90% symmetries where there are not. 00:51:29.790 --> 00:51:31.635 align:middle line:84% And you get the wrong relations, of course. 00:51:31.635 --> 00:51:38.040 align:middle line:90% 00:51:38.040 --> 00:51:42.065 align:middle line:90% Even more so here. 00:51:42.065 --> 00:51:46.290 align:middle line:90% 00:51:46.290 --> 00:51:51.870 align:middle line:84% And the symmetry between this and that is-- 00:51:51.870 --> 00:51:54.060 align:middle line:84% there is a temperature in between-- 00:51:54.060 --> 00:52:00.030 align:middle line:84% because Peltier is Seebeck times the temperature. 00:52:00.030 --> 00:52:02.000 align:middle line:90% OK, all right. 00:52:02.000 --> 00:52:05.100 align:middle line:90% So what is this example? 00:52:05.100 --> 00:52:11.600 align:middle line:84% So if we apply these equations to an electric current that 00:52:11.600 --> 00:52:20.380 align:middle line:84% runs in a wire in which there is a temperature gradient, 00:52:20.380 --> 00:52:22.870 align:middle line:84% then you have this expression here 00:52:22.870 --> 00:52:27.130 align:middle line:84% that relates the temperature gradient, the electric current, 00:52:27.130 --> 00:52:31.680 align:middle line:84% and the voltage drop along the wire. 00:52:31.680 --> 00:52:37.320 align:middle line:90% 00:52:37.320 --> 00:52:40.780 align:middle line:84% Now, a wire is a one-dimensional thing. 00:52:40.780 --> 00:52:52.520 align:middle line:84% So you can describe it by a set of coordinates, 00:52:52.520 --> 00:52:57.130 align:middle line:90% which is like this. 00:52:57.130 --> 00:52:59.780 align:middle line:84% So a local set of coordinates where 00:52:59.780 --> 00:53:06.770 align:middle line:84% one versor is tangent to the axis of your wire. 00:53:06.770 --> 00:53:08.460 align:middle line:84% And the other two are orthogonal. 00:53:08.460 --> 00:53:13.070 align:middle line:90% 00:53:13.070 --> 00:53:16.250 align:middle line:90% And then you have these vectors. 00:53:16.250 --> 00:53:20.570 align:middle line:84% So these vectors are, let's say, the gradient 00:53:20.570 --> 00:53:25.040 align:middle line:90% in phi that's a vector. 00:53:25.040 --> 00:53:27.770 align:middle line:84% But the important part, if you want 00:53:27.770 --> 00:53:31.070 align:middle line:84% to compute the difference in voltage between point 1 00:53:31.070 --> 00:53:36.410 align:middle line:84% and point 2, you have to integrate 00:53:36.410 --> 00:53:43.631 align:middle line:84% the component of the gradient in phi projected onto the ell, 00:53:43.631 --> 00:53:46.290 align:middle line:84% onto the direction of the wire and integrate. 00:53:46.290 --> 00:53:49.310 align:middle line:84% So you do the integral of d (read, nabla) 00:53:49.310 --> 00:53:55.840 align:middle line:84% phi dotted into d ell from 1 to 2. 00:53:55.840 --> 00:53:59.840 align:middle line:84% And that gives you the difference in voltage. 00:53:59.840 --> 00:54:01.830 align:middle line:90% What is it, V2 minus V1. 00:54:01.830 --> 00:54:06.740 align:middle line:90% 00:54:06.740 --> 00:54:07.365 align:middle line:90% It turns out-- 00:54:07.365 --> 00:54:10.940 align:middle line:90% 00:54:10.940 --> 00:54:14.560 align:middle line:84% I don't know if you have seen this, but this is easy to do. 00:54:14.560 --> 00:54:21.455 align:middle line:84% So the gradient, remember what it is, it's this operator here. 00:54:21.455 --> 00:54:29.810 align:middle line:90% 00:54:29.810 --> 00:54:46.900 align:middle line:84% And d ell is also a vector, which has components dx, dy, dz. 00:54:46.900 --> 00:54:50.230 align:middle line:84% So now, what we are doing here is 00:54:50.230 --> 00:54:54.040 align:middle line:84% taking the scalar product of this 00:54:54.040 --> 00:54:57.040 align:middle line:90% applied onto the scalar phi. 00:54:57.040 --> 00:54:59.560 align:middle line:90% 00:54:59.560 --> 00:55:02.260 align:middle line:84% So when you do the dot product of two vectors, 00:55:02.260 --> 00:55:05.980 align:middle line:84% you get the multiplication of these two. 00:55:05.980 --> 00:55:08.890 align:middle line:84% So the multiplication is going to be, 00:55:08.890 --> 00:55:12.880 align:middle line:84% the derivative of phi with respect to x times 00:55:12.880 --> 00:55:18.910 align:middle line:84% dx from these two components plus the derivative of phi 00:55:18.910 --> 00:55:26.040 align:middle line:90% with respect to y and z. 00:55:26.040 --> 00:55:27.700 align:middle line:90% And what is this? 00:55:27.700 --> 00:55:32.820 align:middle line:84% Well, this is the expression for the differential of phi. 00:55:32.820 --> 00:55:38.460 align:middle line:84% OK, so you can rewrite this term here as d phi. 00:55:38.460 --> 00:55:40.510 align:middle line:84% And therefore, the integral is very easy. 00:55:40.510 --> 00:55:41.945 align:middle line:90% It's just phi 2 minus phi 1. 00:55:41.945 --> 00:55:51.020 align:middle line:90% 00:55:51.020 --> 00:55:54.580 align:middle line:90% Which is what we do here. 00:55:54.580 --> 00:55:56.610 align:middle line:90% Now, we are doing it-- 00:55:56.610 --> 00:55:59.180 align:middle line:90% 00:55:59.180 --> 00:56:08.090 align:middle line:84% the thermocouple has to have two different materials, two 00:56:08.090 --> 00:56:11.390 align:middle line:90% different conductors. 00:56:11.390 --> 00:56:14.120 align:middle line:84% And here we call it A and here we 00:56:14.120 --> 00:56:31.830 align:middle line:84% call it B, that are joined with a good conducting junction 00:56:31.830 --> 00:56:32.810 align:middle line:90% here and here. 00:56:32.810 --> 00:56:38.950 align:middle line:90% 00:56:38.950 --> 00:56:48.030 align:middle line:84% And at the other end in this place, they are not touching. 00:56:48.030 --> 00:56:51.230 align:middle line:84% So they are just-- it's an open circuit so that there 00:56:51.230 --> 00:56:55.390 align:middle line:90% is no current circulating. 00:56:55.390 --> 00:57:02.470 align:middle line:84% And you will measure a voltage difference if there is any. 00:57:02.470 --> 00:57:04.910 align:middle line:84% Now, in order to compute that voltage difference, 00:57:04.910 --> 00:57:14.790 align:middle line:84% you need to do that integral three times, from 1 to 2 00:57:14.790 --> 00:57:23.780 align:middle line:84% with metal A, from 2 to 3 with metal B and from 3 to 4, 00:57:23.780 --> 00:57:31.200 align:middle line:84% again, with metal A. And what are we integrating? 00:57:31.200 --> 00:57:34.290 align:middle line:84% Well, the governing equation is this one. 00:57:34.290 --> 00:57:38.080 align:middle line:84% But since we are at open circuit, there is no current. 00:57:38.080 --> 00:57:40.140 align:middle line:90% So there is no-- 00:57:40.140 --> 00:57:42.090 align:middle line:90% the current is equal to 0. 00:57:42.090 --> 00:57:46.440 align:middle line:84% So the remaining stuff is this simple relation 00:57:46.440 --> 00:57:50.800 align:middle line:84% between the gradient in phi and the gradient in temperature. 00:57:50.800 --> 00:57:54.770 align:middle line:90% 00:57:54.770 --> 00:57:56.706 align:middle line:90% So you do those three integrals. 00:57:56.706 --> 00:58:02.570 align:middle line:90% 00:58:02.570 --> 00:58:07.220 align:middle line:84% You cannot assume-- you should not assume that epsilon, 00:58:07.220 --> 00:58:09.450 align:middle line:84% the Seebeck coefficient, is constant. 00:58:09.450 --> 00:58:11.220 align:middle line:84% And so it goes out of the integral. 00:58:11.220 --> 00:58:14.480 align:middle line:84% Yeah, you could, but that would be an approximation 00:58:14.480 --> 00:58:17.540 align:middle line:84% unless you use the proper average in that temperature 00:58:17.540 --> 00:58:20.210 align:middle line:84% range because Seebeck coefficients are 00:58:20.210 --> 00:58:22.100 align:middle line:90% functions of temperature. 00:58:22.100 --> 00:58:25.010 align:middle line:90% 00:58:25.010 --> 00:58:30.560 align:middle line:84% So let's keep them inside the integrals this way. 00:58:30.560 --> 00:58:34.116 align:middle line:84% And then of course, since at the junctions, 00:58:34.116 --> 00:58:40.620 align:middle line:84% there is electrical continuity, the voltage, 00:58:40.620 --> 00:58:44.510 align:middle line:84% so the electrostatic potential, is always-- 00:58:44.510 --> 00:58:47.310 align:middle line:90% is equal. 00:58:47.310 --> 00:58:54.090 align:middle line:84% And so now, you can add these three equations. 00:58:54.090 --> 00:59:01.580 align:middle line:84% And what you are left with is this, that phi 4, 00:59:01.580 --> 00:59:04.400 align:middle line:84% so voltage at 4 minus voltage at 1 00:59:04.400 --> 00:59:07.350 align:middle line:84% is equal to this integral from 3 to 2. 00:59:07.350 --> 00:59:11.180 align:middle line:90% 00:59:11.180 --> 00:59:13.730 align:middle line:84% Yeah, because you're combining this and that, 00:59:13.730 --> 00:59:15.620 align:middle line:84% which are the integral of epsilon 00:59:15.620 --> 00:59:25.176 align:middle line:84% A into a single integral from 3 to 4 and then from 1 to 2. 00:59:25.176 --> 00:59:25.676 align:middle line:90% Why? 00:59:25.676 --> 00:59:30.840 align:middle line:84% Because at 4 and 1, you are assuming the same temperature. 00:59:30.840 --> 00:59:32.230 align:middle line:84% This is a laboratory temperature, 00:59:32.230 --> 00:59:34.330 align:middle line:84% the place where you put your thermometer (I 00:59:34.330 --> 00:59:35.420 align:middle line:90% meant, voltmeter). 00:59:35.420 --> 00:59:37.545 align:middle line:84% So you assume that T1 is equal to T4. 00:59:37.545 --> 00:59:40.490 align:middle line:90% 00:59:40.490 --> 00:59:43.230 align:middle line:84% And therefore, you get this difference. 00:59:43.230 --> 00:59:47.270 align:middle line:84% And sometimes, you call this difference or actually 00:59:47.270 --> 00:59:52.940 align:middle line:84% each generic integral of this difference between reference 00:59:52.940 --> 00:59:54.710 align:middle line:84% temperature and a generic temperature 00:59:54.710 --> 00:59:58.070 align:middle line:84% is called the characteristic function of the thermocouple, 00:59:58.070 --> 01:00:00.920 align:middle line:90% or the couple of the materials. 01:00:00.920 --> 01:00:06.270 align:middle line:84% And here are some plots of those types of thermocouples. 01:00:06.270 --> 01:00:09.530 align:middle line:90% 01:00:09.530 --> 01:00:15.230 align:middle line:84% That means-- at the end of this calculation, what's 01:00:15.230 --> 01:00:17.270 align:middle line:90% the final result? 01:00:17.270 --> 01:00:20.420 align:middle line:84% That we may have a difference in voltage 01:00:20.420 --> 01:00:23.450 align:middle line:84% measured by our voltmeter if there 01:00:23.450 --> 01:00:31.140 align:middle line:84% is a difference in temperatures between 2 and 3, T2 and T3. 01:00:31.140 --> 01:00:34.420 align:middle line:84% And of course, in order to have this voltage, 01:00:34.420 --> 01:00:40.350 align:middle line:84% you need that the characteristic function of that thermocouple 01:00:40.350 --> 01:00:44.520 align:middle line:84% be different at these two temperatures. 01:00:44.520 --> 01:00:52.380 align:middle line:90% 01:00:52.380 --> 01:01:00.820 align:middle line:84% This function here is the function 01:01:00.820 --> 01:01:06.300 align:middle line:84% that you do when you calibrate in your lab. 01:01:06.300 --> 01:01:11.040 align:middle line:84% When you buy a thermocouple, you should do calibration, 01:01:11.040 --> 01:01:15.330 align:middle line:84% or maybe you buy it certified so it comes with a calibration. 01:01:15.330 --> 01:01:20.730 align:middle line:84% And calibration is that somebody has actually measured this 01:01:20.730 --> 01:01:25.350 align:middle line:90% for your thermocouple because-- 01:01:25.350 --> 01:01:31.440 align:middle line:84% yeah, could do it theoretically, but small variations 01:01:31.440 --> 01:01:34.980 align:middle line:84% in the cross section of your material 01:01:34.980 --> 01:01:39.240 align:middle line:84% or some impurity along the wires, 01:01:39.240 --> 01:01:46.300 align:middle line:84% may result in different features and properties of your wires. 01:01:46.300 --> 01:01:48.990 align:middle line:84% So you always work with-- you always 01:01:48.990 --> 01:01:55.820 align:middle line:84% need to do a calibration for your thermocouple. 01:01:55.820 --> 01:01:58.760 align:middle line:84% Now, the nice thing is that therefore, you 01:01:58.760 --> 01:02:02.810 align:middle line:84% have here something hot, here something cold, 01:02:02.810 --> 01:02:04.405 align:middle line:90% and you generate a voltage. 01:02:04.405 --> 01:02:08.020 align:middle line:90% 01:02:08.020 --> 01:02:12.350 align:middle line:84% So if I have a hot reservoir and a cold reservoir, 01:02:12.350 --> 01:02:16.240 align:middle line:84% and I want to produce electricity, 01:02:16.240 --> 01:02:22.770 align:middle line:84% the thermocouple can be made to function as a generator. 01:02:22.770 --> 01:02:25.455 align:middle line:84% This is the thermoelectric generator, which 01:02:25.455 --> 01:02:28.250 align:middle line:90% is exactly what I just said. 01:02:28.250 --> 01:02:31.410 align:middle line:90% 01:02:31.410 --> 01:02:35.520 align:middle line:84% The difference is that now, it's a generator. 01:02:35.520 --> 01:02:39.180 align:middle line:84% So I don't want us to measure the voltage difference, 01:02:39.180 --> 01:02:43.580 align:middle line:84% but I want to do something useful like run 01:02:43.580 --> 01:02:48.360 align:middle line:84% an electrical engine, light up some light, 01:02:48.360 --> 01:02:52.050 align:middle line:90% or use that electricity. 01:02:52.050 --> 01:02:54.980 align:middle line:90% 01:02:54.980 --> 01:02:58.220 align:middle line:84% So I attach a load, an electrical load, 01:02:58.220 --> 01:03:02.930 align:middle line:84% to the two places where before, I was attaching the voltmeter. 01:03:02.930 --> 01:03:06.650 align:middle line:84% An electric current will start flowing, 01:03:06.650 --> 01:03:14.570 align:middle line:84% because that voltage will cause a flow of electrons. 01:03:14.570 --> 01:03:19.340 align:middle line:90% But then that-- what does it do? 01:03:19.340 --> 01:03:21.810 align:middle line:84% If there is a current that goes from 4 to 1, 01:03:21.810 --> 01:03:25.550 align:middle line:84% there is also a current inside the wires 01:03:25.550 --> 01:03:29.180 align:middle line:90% that goes all the way around. 01:03:29.180 --> 01:03:32.280 align:middle line:84% So when I do the calculation for the voltage, 01:03:32.280 --> 01:03:38.390 align:middle line:84% I also need to take into account now 01:03:38.390 --> 01:03:40.280 align:middle line:84% the other part of the equation, the one that 01:03:40.280 --> 01:03:44.490 align:middle line:84% has to do with the electrical current. 01:03:44.490 --> 01:03:49.550 align:middle line:84% So I have to add up for the voltage drop, Ohm's law. 01:03:49.550 --> 01:03:52.540 align:middle line:90% 01:03:52.540 --> 01:03:56.610 align:middle line:90% So the voltage goes down. 01:03:56.610 --> 01:04:05.130 align:middle line:84% The more current you have, the lower the voltage. 01:04:05.130 --> 01:04:11.990 align:middle line:84% And you could actually go all the way to short. 01:04:11.990 --> 01:04:20.620 align:middle line:84% If you short the thing, you have the highest current 01:04:20.620 --> 01:04:27.710 align:middle line:84% but it's all dissipated in Joule effect, in heating. 01:04:27.710 --> 01:04:31.090 align:middle line:90% 01:04:31.090 --> 01:04:35.080 align:middle line:84% OK, so now, I have a voltage, the actual voltage, 01:04:35.080 --> 01:04:37.150 align:middle line:84% multiplied by the current that goes 01:04:37.150 --> 01:04:41.740 align:middle line:84% through my device, the one that uses that electrical energy. 01:04:41.740 --> 01:04:48.350 align:middle line:84% Voltage times current is the electrical work, the power. 01:04:48.350 --> 01:04:52.040 align:middle line:90% Volts, amperes, watts. 01:04:52.040 --> 01:04:56.370 align:middle line:84% OK, so that's the electrical power 01:04:56.370 --> 01:05:02.600 align:middle line:84% which is given by this expression here. 01:05:02.600 --> 01:05:06.920 align:middle line:84% And as you see, there is a Joule effect dissipation. 01:05:06.920 --> 01:05:15.560 align:middle line:84% So the the higher is this, the more inefficient is my engine. 01:05:15.560 --> 01:05:19.860 align:middle line:84% The problem is that if I want to make this 0, I have to make I 0. 01:05:19.860 --> 01:05:21.175 align:middle line:90% So I also get 0 work. 01:05:21.175 --> 01:05:25.390 align:middle line:90% 01:05:25.390 --> 01:05:26.950 align:middle line:90% So there is a compromise. 01:05:26.950 --> 01:05:29.880 align:middle line:84% And one way is to compute the maximum power. 01:05:29.880 --> 01:05:32.940 align:middle line:90% 01:05:32.940 --> 01:05:35.010 align:middle line:84% And so the maximum power is just take 01:05:35.010 --> 01:05:37.440 align:middle line:84% the derivative of this with respect to I 01:05:37.440 --> 01:05:39.570 align:middle line:90% and set it equal to 0. 01:05:39.570 --> 01:05:45.390 align:middle line:84% That gives you this value for the electric current. 01:05:45.390 --> 01:05:55.640 align:middle line:84% And you substitute in there and you get the maximum power, 01:05:55.640 --> 01:06:01.310 align:middle line:84% which goes, you see, as the square of the difference 01:06:01.310 --> 01:06:04.980 align:middle line:84% between the Seebeck coefficients of your two materials, 01:06:04.980 --> 01:06:06.220 align:middle line:90% these materials pair. 01:06:06.220 --> 01:06:09.360 align:middle line:90% 01:06:09.360 --> 01:06:18.340 align:middle line:84% Notice that also, not only for the thermoelectric generator 01:06:18.340 --> 01:06:20.180 align:middle line:84% in which there is a electrical current, 01:06:20.180 --> 01:06:26.670 align:middle line:84% but also for the thermocouple, let me go back one moment. 01:06:26.670 --> 01:06:30.240 align:middle line:84% We focused a lot on-- in order to do this calculation 01:06:30.240 --> 01:06:34.210 align:middle line:84% on the first equation, but also, the second equation is in place. 01:06:34.210 --> 01:06:37.100 align:middle line:90% 01:06:37.100 --> 01:06:39.920 align:middle line:84% And in addition, we should also remember that this 01:06:39.920 --> 01:06:42.390 align:middle line:90% is an irreversible phenomenon. 01:06:42.390 --> 01:06:48.860 align:middle line:84% It's driven by entropy generation. 01:06:48.860 --> 01:06:59.500 align:middle line:84% And in fact, there is a heat q double prime that runs across, 01:06:59.500 --> 01:07:01.300 align:middle line:90% runs down from-- 01:07:01.300 --> 01:07:05.060 align:middle line:84% along the wire, because the wire is an electric conductor, 01:07:05.060 --> 01:07:06.740 align:middle line:84% but it's also a thermal conductor. 01:07:06.740 --> 01:07:08.830 align:middle line:90% It has a thermal conductivity. 01:07:08.830 --> 01:07:11.320 align:middle line:84% And it has a Peltier coefficient. 01:07:11.320 --> 01:07:18.800 align:middle line:84% So it moves heat for two reasons. 01:07:18.800 --> 01:07:22.060 align:middle line:84% One is the gradient in temperature along the wire, 01:07:22.060 --> 01:07:30.400 align:middle line:84% and the other is the electrical current that, how do you say, 01:07:30.400 --> 01:07:32.960 align:middle line:90% carries with it some heat. 01:07:32.960 --> 01:07:41.240 align:middle line:84% That's the Peltier effect, which is this proportionality here. 01:07:41.240 --> 01:07:49.290 align:middle line:84% So that heat is dissipated and produces-- 01:07:49.290 --> 01:07:51.390 align:middle line:84% of course, it produces the effect. 01:07:51.390 --> 01:07:53.950 align:middle line:84% You can also think of it in the other way around. 01:07:53.950 --> 01:07:57.540 align:middle line:84% It's that the heat that flows drags also 01:07:57.540 --> 01:08:05.130 align:middle line:84% the electrical charges together with the flow of heat. 01:08:05.130 --> 01:08:07.230 align:middle line:84% And that is why electrical charges 01:08:07.230 --> 01:08:09.570 align:middle line:84% accumulate on one side of the thermocouple. 01:08:09.570 --> 01:08:13.470 align:middle line:84% And that's how you measure the voltage. 01:08:13.470 --> 01:08:17.430 align:middle line:90% 01:08:17.430 --> 01:08:19.109 align:middle line:90% So it's the combination of-- 01:08:19.109 --> 01:08:23.020 align:middle line:90% 01:08:23.020 --> 01:08:27.669 align:middle line:84% so this same equation gives rise to the Peltier effect. 01:08:27.669 --> 01:08:30.771 align:middle line:90% 01:08:30.771 --> 01:08:40.609 align:middle line:84% The Peltier effect, you can see it here. 01:08:40.609 --> 01:08:46.689 align:middle line:84% It emerges from this second equation more directly. 01:08:46.689 --> 01:08:48.670 align:middle line:84% And the typical setup is that you 01:08:48.670 --> 01:08:52.390 align:middle line:84% have a junction between two different materials in which you 01:08:52.390 --> 01:08:54.020 align:middle line:90% run an electric current. 01:08:54.020 --> 01:08:58.810 align:middle line:84% So you-- they have to be conductors or semiconductors. 01:08:58.810 --> 01:09:00.910 align:middle line:84% And so you have to have a possibility 01:09:00.910 --> 01:09:05.620 align:middle line:84% to drive a current through this junction from material A 01:09:05.620 --> 01:09:07.170 align:middle line:90% to material B. 01:09:07.170 --> 01:09:13.189 align:middle line:84% So that's the current I. And let's assume for a moment 01:09:13.189 --> 01:09:18.380 align:middle line:84% that you can-- that you are keeping 01:09:18.380 --> 01:09:19.979 align:middle line:90% no gradient in temperature. 01:09:19.979 --> 01:09:24.220 align:middle line:84% So the temperature is uniform throughout here. 01:09:24.220 --> 01:09:26.890 align:middle line:84% So this term in the equation is not there. 01:09:26.890 --> 01:09:29.899 align:middle line:84% And but you still have the equation. 01:09:29.899 --> 01:09:33.520 align:middle line:84% It says that the heat flux is proportional 01:09:33.520 --> 01:09:36.790 align:middle line:90% to the electric current. 01:09:36.790 --> 01:09:40.660 align:middle line:84% But the problem is that this proportionality Peltier 01:09:40.660 --> 01:09:45.279 align:middle line:84% coefficient is different for material A and material B. 01:09:45.279 --> 01:09:49.740 align:middle line:84% So at the junction, whereas the electric current must be 01:09:49.740 --> 01:09:55.080 align:middle line:84% continuous because you cannot create charge, 01:09:55.080 --> 01:09:57.370 align:middle line:84% so at the junction, there is a continuity. 01:09:57.370 --> 01:09:59.610 align:middle line:84% So the electric current that comes from A 01:09:59.610 --> 01:10:06.210 align:middle line:84% is the same that goes into B. But this equation also 01:10:06.210 --> 01:10:11.920 align:middle line:84% says that the heat qA is given by this. 01:10:11.920 --> 01:10:16.700 align:middle line:84% And here, I have another heat which is given by this. 01:10:16.700 --> 01:10:19.840 align:middle line:84% And it's different because the Peltier of B 01:10:19.840 --> 01:10:27.440 align:middle line:84% is different from Peltier of A. So there is a jump here 01:10:27.440 --> 01:10:30.920 align:middle line:90% in the heat flux. 01:10:30.920 --> 01:10:37.040 align:middle line:84% Of course, jump in a heat flux wouldn't satisfy the energy 01:10:37.040 --> 01:10:46.060 align:middle line:84% balance unless there is energy coming from somewhere. 01:10:46.060 --> 01:10:51.820 align:middle line:84% So essentially, this jump calls in for the balance of energy 01:10:51.820 --> 01:10:55.190 align:middle line:90% that is needed for it to exist. 01:10:55.190 --> 01:11:00.430 align:middle line:84% And therefore, this junction transforms into either a cooler 01:11:00.430 --> 01:11:05.730 align:middle line:84% or a heater, depending on the difference of the Peltier 01:11:05.730 --> 01:11:08.500 align:middle line:84% coefficients and the direction of the current. 01:11:08.500 --> 01:11:12.180 align:middle line:84% So if you choose a Peltier pair in which, 01:11:12.180 --> 01:11:18.870 align:middle line:84% say, Pi B is greater than Pi A, then-- 01:11:18.870 --> 01:11:21.660 align:middle line:84% and you run the electric current in this way 01:11:21.660 --> 01:11:27.980 align:middle line:84% so that this is positive, then qB is greater than qA. 01:11:27.980 --> 01:11:31.800 align:middle line:84% So that means you need to provide energy. 01:11:31.800 --> 01:11:35.720 align:middle line:84% So that means this junction becomes a cooler. 01:11:35.720 --> 01:11:38.880 align:middle line:84% This is the Peltier cooling effect. 01:11:38.880 --> 01:11:44.660 align:middle line:84% This is what you obtain when you buy a Peltier cell, something 01:11:44.660 --> 01:11:50.690 align:middle line:84% that you can buy on the internet now for a couple of bucks. 01:11:50.690 --> 01:11:57.720 align:middle line:84% And or you can make it simply if you have a thermocouple. 01:11:57.720 --> 01:12:03.090 align:middle line:84% Also, a thermocouple is where you have a junction between two 01:12:03.090 --> 01:12:04.330 align:middle line:90% different materials. 01:12:04.330 --> 01:12:08.810 align:middle line:84% You have the possibility of using this effect. 01:12:08.810 --> 01:12:14.900 align:middle line:84% A nice aspect of the Peltier effect, and also a nice aspect, 01:12:14.900 --> 01:12:22.500 align:middle line:84% I forgot to say it, about the thermoelectric generator, 01:12:22.500 --> 01:12:31.110 align:middle line:84% is that they convert electrical energy into heat and vice 01:12:31.110 --> 01:12:34.620 align:middle line:90% versa without moving parts. 01:12:34.620 --> 01:12:40.520 align:middle line:84% So these are engines that do not have machinery. 01:12:40.520 --> 01:12:45.750 align:middle line:84% So these are very important features for some applications. 01:12:45.750 --> 01:12:47.930 align:middle line:84% For example, for space applications, 01:12:47.930 --> 01:12:50.990 align:middle line:84% this is fundamental because once you 01:12:50.990 --> 01:12:53.570 align:middle line:84% have sent something in space, then 01:12:53.570 --> 01:12:56.970 align:middle line:84% it's very hard to provide maintenance. 01:12:56.970 --> 01:12:59.540 align:middle line:90% So moving parts are a problem. 01:12:59.540 --> 01:13:09.060 align:middle line:84% And so if you have these kind of devices, they're good. 01:13:09.060 --> 01:13:14.550 align:middle line:84% Plus, also, in space, you have the availability of very large 01:13:14.550 --> 01:13:20.400 align:middle line:84% temperature differences because you have the photons coming from 01:13:20.400 --> 01:13:25.600 align:middle line:84% the sun that we just said, are at 5760 K. 01:13:25.600 --> 01:13:30.430 align:middle line:84% And on the other side, you have the dark space, which is maybe, 01:13:30.430 --> 01:13:34.080 align:middle line:90% what is it, 2.7 or 4K. 01:13:34.080 --> 01:13:38.190 align:middle line:84% So that-- through, of course, you 01:13:38.190 --> 01:13:40.320 align:middle line:84% need electromagnetic radiation because those-- 01:13:40.320 --> 01:13:42.960 align:middle line:90% you don't have a heat exchanger. 01:13:42.960 --> 01:13:46.560 align:middle line:84% The provider of those fluxes are the photons 01:13:46.560 --> 01:13:48.095 align:middle line:90% that we just talked about. 01:13:48.095 --> 01:13:52.350 align:middle line:90% 01:13:52.350 --> 01:13:56.610 align:middle line:84% The problem with the development of devices 01:13:56.610 --> 01:14:03.580 align:middle line:84% based on these effects is the materials 01:14:03.580 --> 01:14:09.700 align:middle line:84% because standard materials have very small values of the Peltier 01:14:09.700 --> 01:14:12.400 align:middle line:90% or Seebeck coefficients. 01:14:12.400 --> 01:14:17.043 align:middle line:84% For example, here, I took the ones for copper. 01:14:17.043 --> 01:14:20.820 align:middle line:90% 01:14:20.820 --> 01:14:28.770 align:middle line:84% Copper has a Seebeck coefficient of 1.5 microvolt 01:14:28.770 --> 01:14:34.265 align:middle line:84% per kelvin against a very high conductivity, k. 01:14:34.265 --> 01:14:37.060 align:middle line:90% 01:14:37.060 --> 01:14:42.700 align:middle line:84% And so the Peltier coefficient, which would mean-- 01:14:42.700 --> 01:14:47.830 align:middle line:84% in this case, if I have 1 millimeter cross section 01:14:47.830 --> 01:14:57.060 align:middle line:84% of my wire, and I run one ampere into it, 01:14:57.060 --> 01:15:01.380 align:middle line:84% and suppose I have on the other side an ideal material that has 01:15:01.380 --> 01:15:06.550 align:middle line:84% 0 Peltier, so the difference is only that of copper, 01:15:06.550 --> 01:15:11.920 align:middle line:84% I would be able to generate 1/2 of a milliwatt, 01:15:11.920 --> 01:15:13.930 align:middle line:90% 1/2 of a milliwatt per ampere. 01:15:13.930 --> 01:15:16.510 align:middle line:90% Ampere is a lot of current. 01:15:16.510 --> 01:15:19.590 align:middle line:90% Milliwatt is pretty little. 01:15:19.590 --> 01:15:31.836 align:middle line:90% 01:15:31.836 --> 01:15:39.005 align:middle line:84% Yeah, you can also read this equation here 01:15:39.005 --> 01:15:41.650 align:middle line:90% in another situation. 01:15:41.650 --> 01:15:45.530 align:middle line:84% Suppose you have no current, sorry-- 01:15:45.530 --> 01:15:48.000 align:middle line:90% no heat flux. 01:15:48.000 --> 01:15:51.360 align:middle line:90% So no heat flux, 0 here. 01:15:51.360 --> 01:15:55.700 align:middle line:84% So that says that if you run an electric current 01:15:55.700 --> 01:15:59.180 align:middle line:84% through your wire with no heat flux, 01:15:59.180 --> 01:16:01.560 align:middle line:84% you have a buildup of temperature. 01:16:01.560 --> 01:16:05.710 align:middle line:90% 01:16:05.710 --> 01:16:09.945 align:middle line:84% This is, we call it also Thomson effect. 01:16:09.945 --> 01:16:12.470 align:middle line:90% 01:16:12.470 --> 01:16:13.550 align:middle line:90% But no, no. 01:16:13.550 --> 01:16:14.570 align:middle line:90% I think it's Peltier. 01:16:14.570 --> 01:16:21.420 align:middle line:84% OK, but so how much temperature gradient does it build up? 01:16:21.420 --> 01:16:25.110 align:middle line:84% So again, see if I take my millimeter wire 01:16:25.110 --> 01:16:30.140 align:middle line:84% and I put 1 ampere into it, it does-- 01:16:30.140 --> 01:16:35.920 align:middle line:84% it builds up 1K, 1.2K over a meter of wire. 01:16:35.920 --> 01:16:39.020 align:middle line:90% 01:16:39.020 --> 01:16:43.460 align:middle line:84% So these are small effects, but people are developing 01:16:43.460 --> 01:16:46.280 align:middle line:90% materials for this purpose. 01:16:46.280 --> 01:16:48.280 align:middle line:84% So there is an entire literature, 01:16:48.280 --> 01:16:50.860 align:middle line:84% that you might be familiar with if you're 01:16:50.860 --> 01:17:01.950 align:middle line:84% working on the field, of scientists working on developing 01:17:01.950 --> 01:17:07.200 align:middle line:84% nanomaterials or special materials designed 01:17:07.200 --> 01:17:13.840 align:middle line:84% to maximize these Peltier and Seebeck coefficients. 01:17:13.840 --> 01:17:17.440 align:middle line:90% 01:17:17.440 --> 01:17:24.540 align:middle line:84% Some of these materials are also semiconductors. 01:17:24.540 --> 01:17:29.670 align:middle line:90% So this is a typical setup. 01:17:29.670 --> 01:17:35.730 align:middle line:84% And you put many of these devices in parallel 01:17:35.730 --> 01:17:42.930 align:middle line:84% so as to produce a sensible amount of power. 01:17:42.930 --> 01:17:44.740 align:middle line:90% Each one contributes a little. 01:17:44.740 --> 01:17:48.420 align:middle line:84% But if you put a lot together, overall, you 01:17:48.420 --> 01:17:51.335 align:middle line:84% may obtain the effect that you need. 01:17:51.335 --> 01:17:56.220 align:middle line:90% 01:17:56.220 --> 01:18:01.405 align:middle line:84% And the typical design of these devices 01:18:01.405 --> 01:18:04.200 align:middle line:84% is the same, whether you're using 01:18:04.200 --> 01:18:12.270 align:middle line:84% the thermoelectric generator or, if you want, 01:18:12.270 --> 01:18:21.470 align:middle line:84% your solid-state thermocouple, or you want the Peltier cell. 01:18:21.470 --> 01:18:25.580 align:middle line:84% So it's the same device that, depending on how you use it, 01:18:25.580 --> 01:18:29.560 align:middle line:90% performs in different ways. 01:18:29.560 --> 01:18:35.540 align:middle line:90% 01:18:35.540 --> 01:18:42.690 align:middle line:84% Now, regarding the thermoelectric generator, 01:18:42.690 --> 01:18:51.440 align:middle line:84% traditionally, people like to characterize the materials 01:18:51.440 --> 01:18:57.710 align:middle line:84% with this parameter here, which is called figure of merit 01:18:57.710 --> 01:19:08.950 align:middle line:84% for the thermoelectric material, which is the epsilon, so Seebeck 01:19:08.950 --> 01:19:10.630 align:middle line:84% squared, times temperature divided 01:19:10.630 --> 01:19:15.420 align:middle line:84% by resistivity and thermal conductivity. 01:19:15.420 --> 01:19:20.421 align:middle line:84% You may-- we may see how this parameter comes out 01:19:20.421 --> 01:19:24.570 align:middle line:84% in our own language, or at least the language 01:19:24.570 --> 01:19:27.510 align:middle line:84% that we have developed in this course. 01:19:27.510 --> 01:19:35.390 align:middle line:84% If you accept that you can rewrite the work-- 01:19:35.390 --> 01:19:38.600 align:middle line:84% the power of the thermoelectric generator 01:19:38.600 --> 01:19:40.970 align:middle line:84% that we have developed in the previous 01:19:40.970 --> 01:19:46.680 align:middle line:84% slides as an integral of what happens locally. 01:19:46.680 --> 01:19:51.650 align:middle line:84% So as if everywhere in the wire, there 01:19:51.650 --> 01:19:56.540 align:middle line:84% is something that is the local contribution 01:19:56.540 --> 01:19:58.050 align:middle line:90% to the overall effect. 01:19:58.050 --> 01:20:01.250 align:middle line:84% So you integrate that local contribution 01:20:01.250 --> 01:20:05.870 align:middle line:84% over the entire distance, and you get the overall effect. 01:20:05.870 --> 01:20:10.900 align:middle line:84% In a sense, this is a power density. 01:20:10.900 --> 01:20:17.210 align:middle line:84% Electrical-- useful power density of your device. 01:20:17.210 --> 01:20:20.510 align:middle line:84% OK, so if you accept that, and in fact, 01:20:20.510 --> 01:20:26.750 align:middle line:84% that power density is the product of the electric current. 01:20:26.750 --> 01:20:32.640 align:middle line:90% And the gradient in the voltage. 01:20:32.640 --> 01:20:35.960 align:middle line:84% So that's-- voltage times current is power. 01:20:35.960 --> 01:20:38.570 align:middle line:90% So this is the density, though. 01:20:38.570 --> 01:20:41.300 align:middle line:90% 01:20:41.300 --> 01:20:44.480 align:middle line:84% So I take this one and then I'm going 01:20:44.480 --> 01:20:46.370 align:middle line:84% to-- in order-- so I want to define 01:20:46.370 --> 01:20:50.600 align:middle line:84% an efficiency, second-law efficiency, which is down here. 01:20:50.600 --> 01:20:56.680 align:middle line:84% So I want to take the work obtained and divide it 01:20:56.680 --> 01:21:00.130 align:middle line:84% by the maximum work I could obtain for that same temperature 01:21:00.130 --> 01:21:01.300 align:middle line:90% difference. 01:21:01.300 --> 01:21:03.340 align:middle line:84% Now, locally, the temperature difference 01:21:03.340 --> 01:21:05.540 align:middle line:84% is just the temperature gradient. 01:21:05.540 --> 01:21:10.420 align:middle line:84% And if you go back in the notes, and maybe this image reminds you 01:21:10.420 --> 01:21:14.440 align:middle line:84% where to go, you remember that when 01:21:14.440 --> 01:21:19.640 align:middle line:84% we did the elements of heat transfer, we said, yeah. 01:21:19.640 --> 01:21:23.990 align:middle line:84% We could substitute our material with a little engine. 01:21:23.990 --> 01:21:27.530 align:middle line:84% And how much work would it be able to do? 01:21:27.530 --> 01:21:33.400 align:middle line:84% If you go back, you find that that maximum work, or work rate, 01:21:33.400 --> 01:21:41.030 align:middle line:84% is given by the heat times the gradient in 1 over T times T. 01:21:41.030 --> 01:21:42.510 align:middle line:90% So this is the expression. 01:21:42.510 --> 01:21:45.130 align:middle line:90% 01:21:45.130 --> 01:21:50.140 align:middle line:84% Therefore, the second law efficiency 01:21:50.140 --> 01:21:54.620 align:middle line:84% of your thermoelectric generator, locally, at least, 01:21:54.620 --> 01:21:59.050 align:middle line:84% the local contribution, is given by the ratio between this 01:21:59.050 --> 01:22:04.550 align:middle line:84% and that, which we can calculate because we have the equations 01:22:04.550 --> 01:22:07.900 align:middle line:90% for the gradient in phi, for q. 01:22:07.900 --> 01:22:12.160 align:middle line:84% It's the same equations that we started with. 01:22:12.160 --> 01:22:19.020 align:middle line:84% So let's do the substitutions and compute the work. 01:22:19.020 --> 01:22:31.020 align:middle line:84% Now again, here, the work can take a maximum value. 01:22:31.020 --> 01:22:32.500 align:middle line:90% I think there is an error here. 01:22:32.500 --> 01:22:37.410 align:middle line:90% This must be a minus sign here. 01:22:37.410 --> 01:22:38.930 align:middle line:84% Yeah, because there's a minus here. 01:22:38.930 --> 01:22:46.030 align:middle line:90% 01:22:46.030 --> 01:22:51.260 align:middle line:84% Yeah-- I don't see it right offhand, but this-- 01:22:51.260 --> 01:22:54.580 align:middle line:84% I think this should be a minus because this 01:22:54.580 --> 01:22:56.560 align:middle line:84% is a Joule effect that should detract 01:22:56.560 --> 01:23:00.120 align:middle line:90% from the work that is done. 01:23:00.120 --> 01:23:04.170 align:middle line:84% So you maximize this by taking the derivative with respect 01:23:04.170 --> 01:23:09.030 align:middle line:84% to the current density and set it to 0, substitute, 01:23:09.030 --> 01:23:15.320 align:middle line:84% and you get the expression for the maximum power density. 01:23:15.320 --> 01:23:19.050 align:middle line:90% 01:23:19.050 --> 01:23:21.740 align:middle line:84% And this is the maximum that you should get. 01:23:21.740 --> 01:23:24.920 align:middle line:90% 01:23:24.920 --> 01:23:26.540 align:middle line:90% Ah no, OK. 01:23:26.540 --> 01:23:28.210 align:middle line:90% I see. 01:23:28.210 --> 01:23:33.160 align:middle line:84% They are different in signs because the gradient is negative 01:23:33.160 --> 01:23:38.980 align:middle line:84% because you get work when the current goes down, 01:23:38.980 --> 01:23:43.720 align:middle line:84% the negative-- you need a decreasing temperature gradient. 01:23:43.720 --> 01:23:44.770 align:middle line:90% Yeah. 01:23:44.770 --> 01:23:52.220 align:middle line:84% In fact, you get the maximum when I is minus gradient of T. 01:23:52.220 --> 01:23:57.860 align:middle line:84% And so we take this, substituting 01:23:57.860 --> 01:24:01.820 align:middle line:84% the expression for the q, and obtain therefore 01:24:01.820 --> 01:24:05.710 align:middle line:90% the maximum reversible work. 01:24:05.710 --> 01:24:12.430 align:middle line:84% So at maximum power, the ratio of this to that, 01:24:12.430 --> 01:24:16.140 align:middle line:84% you see, they both depend on the square of the gradient. 01:24:16.140 --> 01:24:22.540 align:middle line:90% So they disappear. 01:24:22.540 --> 01:24:24.710 align:middle line:90% This cancels with that. 01:24:24.710 --> 01:24:28.240 align:middle line:84% And you are left with just a ratio of coefficients 01:24:28.240 --> 01:24:32.180 align:middle line:84% where this parameter of merit appears, 01:24:32.180 --> 01:24:40.440 align:middle line:84% this Z. This is not the standard way 01:24:40.440 --> 01:24:44.990 align:middle line:84% of presenting this parameter of merit. 01:24:44.990 --> 01:24:48.700 align:middle line:90% 01:24:48.700 --> 01:24:52.920 align:middle line:84% But at least, it has some value of being local. 01:24:52.920 --> 01:24:55.670 align:middle line:84% The problem is that in practical applications, 01:24:55.670 --> 01:25:00.800 align:middle line:84% since the materials that you use are temperature dependent, 01:25:00.800 --> 01:25:08.020 align:middle line:84% and it is important that they be temperature dependent, often, 01:25:08.020 --> 01:25:11.630 align:middle line:84% in these kind of derivations of this parameter, 01:25:11.630 --> 01:25:14.590 align:middle line:84% you assume constant Seebeck coefficient, 01:25:14.590 --> 01:25:16.570 align:middle line:84% or constant thermal conductivity, 01:25:16.570 --> 01:25:24.800 align:middle line:84% or constant Pi which is only for the purpose of the derivation, 01:25:24.800 --> 01:25:27.440 align:middle line:84% in order to arrive to this expression. 01:25:27.440 --> 01:25:38.150 align:middle line:84% OK, again, in the list of thermoelectric effects, 01:25:38.150 --> 01:25:39.115 align:middle line:90% we can also-- 01:25:39.115 --> 01:25:42.700 align:middle line:90% 01:25:42.700 --> 01:25:47.120 align:middle line:84% list this Thomson effect, which is, 01:25:47.120 --> 01:25:50.360 align:middle line:84% though, I think still the same thing, 01:25:50.360 --> 01:25:53.430 align:middle line:84% only said in a different language. 01:25:53.430 --> 01:25:59.590 align:middle line:90% 01:25:59.590 --> 01:26:01.690 align:middle line:84% So we have seen that the interplay 01:26:01.690 --> 01:26:05.680 align:middle line:84% between electric current and temperature gradient 01:26:05.680 --> 01:26:11.796 align:middle line:84% allows you to transform electrical energy 01:26:11.796 --> 01:26:14.035 align:middle line:84% into thermal energy, and vice versa. 01:26:14.035 --> 01:26:19.300 align:middle line:90% 01:26:19.300 --> 01:26:22.365 align:middle line:90% So what we do here is to take-- 01:26:22.365 --> 01:26:24.890 align:middle line:90% 01:26:24.890 --> 01:26:28.880 align:middle line:84% to write the balance equation for the internal energy. 01:26:28.880 --> 01:26:32.970 align:middle line:84% And you can do this also for a flow situation. 01:26:32.970 --> 01:26:37.310 align:middle line:84% We've written-- if you go back where a couple of lectures 01:26:37.310 --> 01:26:41.730 align:middle line:84% ago when we did the general balance equations, 01:26:41.730 --> 01:26:45.040 align:middle line:84% we had an expression for the balance for energy. 01:26:45.040 --> 01:26:48.090 align:middle line:90% 01:26:48.090 --> 01:26:53.020 align:middle line:84% And today, in one of the previous slides, 01:26:53.020 --> 01:26:57.820 align:middle line:84% we've done the difference between the energy 01:26:57.820 --> 01:27:03.910 align:middle line:84% and the internal energy, which is all those terms, kinetic, 01:27:03.910 --> 01:27:06.410 align:middle line:90% potential, and so on. 01:27:06.410 --> 01:27:09.020 align:middle line:84% So in order to obtain a balance equation 01:27:09.020 --> 01:27:10.970 align:middle line:84% for just internal energy, you have 01:27:10.970 --> 01:27:17.330 align:middle line:84% to subtract from the energy balance equation, the balance 01:27:17.330 --> 01:27:20.430 align:middle line:84% equation for the non-thermal aspects. 01:27:20.430 --> 01:27:23.300 align:middle line:84% So for the mechanical aspects, like the kinetic, the potential, 01:27:23.300 --> 01:27:25.350 align:middle line:90% and the electrical. 01:27:25.350 --> 01:27:30.390 align:middle line:90% 01:27:30.390 --> 01:27:34.350 align:middle line:84% And if you do that, this is what-- 01:27:34.350 --> 01:27:36.930 align:middle line:84% using the slide that I mentioned-- 01:27:36.930 --> 01:27:38.105 align:middle line:90% you can obtain this. 01:27:38.105 --> 01:27:40.990 align:middle line:90% 01:27:40.990 --> 01:27:44.500 align:middle line:84% Of course, in the balance for energy, 01:27:44.500 --> 01:27:48.490 align:middle line:90% there appears the energy flux. 01:27:48.490 --> 01:27:51.790 align:middle line:84% That's the overall diffusive energy 01:27:51.790 --> 01:27:57.460 align:middle line:84% flux, which we have broken into a non-work and a work part. 01:27:57.460 --> 01:28:03.220 align:middle line:84% The non-work part is the one that now for a solid, 01:28:03.220 --> 01:28:07.240 align:middle line:84% in the absence therefore of other moving 01:28:07.240 --> 01:28:12.940 align:middle line:84% particles other than just the electrons in the conductor, 01:28:12.940 --> 01:28:15.695 align:middle line:84% is just given by these two terms. 01:28:15.695 --> 01:28:17.320 align:middle line:84% Otherwise, you would have the summation 01:28:17.320 --> 01:28:20.200 align:middle line:90% of the partial enthalpies. 01:28:20.200 --> 01:28:27.100 align:middle line:84% And for the work part, it's given by the stress 01:28:27.100 --> 01:28:28.700 align:middle line:90% tensor times the velocity. 01:28:28.700 --> 01:28:32.540 align:middle line:84% But if we are assuming that there is no velocity here, 01:28:32.540 --> 01:28:34.425 align:middle line:90% this is really going away. 01:28:34.425 --> 01:28:38.320 align:middle line:90% 01:28:38.320 --> 01:28:50.120 align:middle line:84% So if you get to the end, the terms that survive are these. 01:28:50.120 --> 01:28:52.328 align:middle line:84% On the left-hand side of the balance equation, 01:28:52.328 --> 01:28:54.620 align:middle line:84% you have the material derivative of the internal energy 01:28:54.620 --> 01:28:57.080 align:middle line:90% of your fluid element. 01:28:57.080 --> 01:29:00.710 align:middle line:84% So that's the internal energy, that part 01:29:00.710 --> 01:29:02.930 align:middle line:84% of the energy of the fluid element 01:29:02.930 --> 01:29:06.150 align:middle line:90% as you follow its motion. 01:29:06.150 --> 01:29:10.280 align:middle line:90% That gets changed by what? 01:29:10.280 --> 01:29:19.720 align:middle line:90% By heat diffusion. 01:29:19.720 --> 01:29:24.640 align:middle line:84% This is Fourier thermal diffusion. 01:29:24.640 --> 01:29:33.970 align:middle line:84% So you should think of this as the equivalent of the Fourier 01:29:33.970 --> 01:29:39.010 align:middle line:84% law or Fourier equation of conduction. 01:29:39.010 --> 01:29:44.485 align:middle line:84% So this would be rho cv times the DT over Dt, 01:29:44.485 --> 01:29:46.750 align:middle line:90% D of the temperature over Dt. 01:29:46.750 --> 01:29:51.520 align:middle line:84% This is k times the Laplacian of the temperature, 01:29:51.520 --> 01:30:00.220 align:middle line:84% if k is independent of position, and therefore of temperature. 01:30:00.220 --> 01:30:02.010 align:middle line:90% Then you have this term here. 01:30:02.010 --> 01:30:04.300 align:middle line:84% This term here is the Joule effect. 01:30:04.300 --> 01:30:06.170 align:middle line:90% This is like a source term. 01:30:06.170 --> 01:30:08.710 align:middle line:84% It would be-- in heat transfer, you 01:30:08.710 --> 01:30:11.150 align:middle line:84% would put it as a q triple prime. 01:30:11.150 --> 01:30:15.960 align:middle line:84% So it's a source term of thermal energy. 01:30:15.960 --> 01:30:19.200 align:middle line:90% 01:30:19.200 --> 01:30:25.200 align:middle line:84% And it actually is electrical energy 01:30:25.200 --> 01:30:27.840 align:middle line:84% that gets dissipated into thermal energy via the Joule 01:30:27.840 --> 01:30:28.340 align:middle line:90% effect. 01:30:28.340 --> 01:30:31.730 align:middle line:90% 01:30:31.730 --> 01:30:35.870 align:middle line:84% If you had written the balance equation 01:30:35.870 --> 01:30:39.740 align:middle line:84% for the mechanical and electrical parts, 01:30:39.740 --> 01:30:43.320 align:middle line:84% the same term would appear there with a negative sign. 01:30:43.320 --> 01:30:46.820 align:middle line:84% So it would detract, it would destroy, 01:30:46.820 --> 01:30:48.630 align:middle line:84% mechanical and electrical energy. 01:30:48.630 --> 01:30:51.560 align:middle line:84% So it would be a negative sign in those equations, 01:30:51.560 --> 01:30:53.780 align:middle line:84% and appears here with positive sign 01:30:53.780 --> 01:31:00.060 align:middle line:84% because it goes from electrical to thermal. 01:31:00.060 --> 01:31:02.270 align:middle line:90% And it's one way. 01:31:02.270 --> 01:31:05.160 align:middle line:84% It has a definite sign, it cannot be negative. 01:31:05.160 --> 01:31:06.300 align:middle line:90% It goes only in that way. 01:31:06.300 --> 01:31:10.225 align:middle line:84% You can only kill electrical energy with the Joule effect. 01:31:10.225 --> 01:31:13.590 align:middle line:90% 01:31:13.590 --> 01:31:17.660 align:middle line:84% Like-- although we're in a solid, 01:31:17.660 --> 01:31:19.120 align:middle line:84% and this doesn't really make sense. 01:31:19.120 --> 01:31:22.540 align:middle line:84% But I just put it for our discussion. 01:31:22.540 --> 01:31:26.340 align:middle line:84% If you also have flow, the viscous dissipation 01:31:26.340 --> 01:31:28.450 align:middle line:90% also has the same feature. 01:31:28.450 --> 01:31:32.940 align:middle line:84% It appears with a negative sign in the equation 01:31:32.940 --> 01:31:36.600 align:middle line:84% for the kinetic energy, the kinetic plus potential energy, 01:31:36.600 --> 01:31:40.470 align:middle line:84% because it kills that kind of energy 01:31:40.470 --> 01:31:46.320 align:middle line:84% and it appears with the positive sign in the internal energy. 01:31:46.320 --> 01:31:48.340 align:middle line:90% So that transforms. 01:31:48.340 --> 01:31:51.010 align:middle line:90% So this is always positive. 01:31:51.010 --> 01:31:53.590 align:middle line:84% But then there is also this term. 01:31:53.590 --> 01:31:57.420 align:middle line:90% This term is the Thomson effect. 01:31:57.420 --> 01:32:00.620 align:middle line:90% 01:32:00.620 --> 01:32:04.130 align:middle line:84% It's-- you see, there is a scalar product of the electric 01:32:04.130 --> 01:32:07.050 align:middle line:84% current and the gradient in temperature. 01:32:07.050 --> 01:32:12.120 align:middle line:84% This can be all signs, it could be positive and negative. 01:32:12.120 --> 01:32:16.710 align:middle line:90% 01:32:16.710 --> 01:32:20.580 align:middle line:84% And it allows you-- and it is that term which 01:32:20.580 --> 01:32:25.320 align:middle line:84% allows the effects that we explained in transforming 01:32:25.320 --> 01:32:30.730 align:middle line:84% thermal energy into electrical energy 01:32:30.730 --> 01:32:36.380 align:middle line:84% if it is negative because with a minus sign, 01:32:36.380 --> 01:32:41.310 align:middle line:84% I'm sorry, since it has a minus sign, if it is positive, 01:32:41.310 --> 01:32:44.280 align:middle line:90% it consumes internal energy. 01:32:44.280 --> 01:32:46.230 align:middle line:90% And where does that energy go? 01:32:46.230 --> 01:32:48.420 align:middle line:90% It becomes electrical. 01:32:48.420 --> 01:32:51.170 align:middle line:90% 01:32:51.170 --> 01:32:57.290 align:middle line:84% Or vice versa, you use electrical energy 01:32:57.290 --> 01:33:01.660 align:middle line:84% for this becoming thermal energy. 01:33:01.660 --> 01:33:06.375 align:middle line:90% OK, so this adds a term. 01:33:06.375 --> 01:33:17.370 align:middle line:90% 01:33:17.370 --> 01:33:19.920 align:middle line:90% The Thomson coefficient is-- 01:33:19.920 --> 01:33:23.940 align:middle line:90% 01:33:23.940 --> 01:33:28.260 align:middle line:84% if you follow the various steps at a certain stage, we have-- 01:33:28.260 --> 01:33:34.080 align:middle line:84% since there is the divergence of the energy flux, 01:33:34.080 --> 01:33:37.260 align:middle line:84% it brings in the gradient of the Peltier coefficient. 01:33:37.260 --> 01:33:39.250 align:middle line:84% And the gradient of the Peltier coefficients 01:33:39.250 --> 01:33:42.780 align:middle line:84% can be written as the partial, as how 01:33:42.780 --> 01:33:45.960 align:middle line:84% it varies with temperature, partial of Pi 01:33:45.960 --> 01:33:48.370 align:middle line:84% over T times the gradient in temperature. 01:33:48.370 --> 01:33:53.020 align:middle line:90% 01:33:53.020 --> 01:33:58.430 align:middle line:84% And this Thomson coefficient, sometimes it's 01:33:58.430 --> 01:34:04.370 align:middle line:84% called with T. I used this upsilon T-looking Greek symbol. 01:34:04.370 --> 01:34:06.590 align:middle line:84% Just not to confuse it with so many T's 01:34:06.590 --> 01:34:09.950 align:middle line:84% here that one more would be too many. 01:34:09.950 --> 01:34:16.790 align:middle line:84% So this is given by the difference between how 01:34:16.790 --> 01:34:19.580 align:middle line:84% the Peltier coefficient varies with temperature and the Seebeck 01:34:19.580 --> 01:34:20.250 align:middle line:90% coefficient. 01:34:20.250 --> 01:34:24.950 align:middle line:84% But since Peltier and Seebeck are connected via the Onsager 01:34:24.950 --> 01:34:27.800 align:middle line:84% relation, Peltier is epsilon times T, 01:34:27.800 --> 01:34:31.190 align:middle line:84% you can also write this coefficient as T times 01:34:31.190 --> 01:34:36.390 align:middle line:84% how Seebeck varies with temperature. 01:34:36.390 --> 01:34:45.850 align:middle line:84% And recently, there have been people discussing the fact that 01:34:45.850 --> 01:34:48.250 align:middle line:90% there may be-- it may be more-- 01:34:48.250 --> 01:34:51.220 align:middle line:84% or at least it may be a good idea, 01:34:51.220 --> 01:34:58.320 align:middle line:84% to try to build thermoelectric generators, not 01:34:58.320 --> 01:35:02.980 align:middle line:84% so much on the basis of that parameter there, 01:35:02.980 --> 01:35:08.430 align:middle line:84% but on the basis of materials that have very high temperature 01:35:08.430 --> 01:35:12.210 align:middle line:84% dependence of the Seebeck coefficient. 01:35:12.210 --> 01:35:17.510 align:middle line:84% So as to enhance this gamma (I meant, upsilon), 01:35:17.510 --> 01:35:20.180 align:middle line:84% and therefore enhance the capability 01:35:20.180 --> 01:35:22.490 align:middle line:84% for a given gradient in temperature 01:35:22.490 --> 01:35:25.795 align:middle line:90% to produce the highest current. 01:35:25.795 --> 01:35:29.970 align:middle line:90% 01:35:29.970 --> 01:35:33.420 align:middle line:90% OK, let me just mention before-- 01:35:33.420 --> 01:35:35.780 align:middle line:84% yeah, we have a couple of minutes-- 01:35:35.780 --> 01:35:38.190 align:middle line:84% I'm not going into the details here. 01:35:38.190 --> 01:35:46.420 align:middle line:84% Just want to mention that the range, the inventory, 01:35:46.420 --> 01:35:54.700 align:middle line:84% the catalog of effects that we have covered with that matrix 01:35:54.700 --> 01:35:59.950 align:middle line:84% that we started on today with the Curie principle, 01:35:59.950 --> 01:36:06.780 align:middle line:84% is not at all complete because for example, 01:36:06.780 --> 01:36:14.340 align:middle line:84% if you add one item, the magnetic field, 01:36:14.340 --> 01:36:21.070 align:middle line:84% you add another spot in that matrix, 01:36:21.070 --> 01:36:24.105 align:middle line:84% and actually, add another four spots. 01:36:24.105 --> 01:36:27.772 align:middle line:90% 01:36:27.772 --> 01:36:33.010 align:middle line:84% And these are effects that belong to the same category. 01:36:33.010 --> 01:36:36.220 align:middle line:84% One is called-- and they also belong to the history-- 01:36:36.220 --> 01:36:38.475 align:middle line:90% one is called the Hall effect. 01:36:38.475 --> 01:36:41.120 align:middle line:90% 01:36:41.120 --> 01:36:42.640 align:middle line:90% H is the magnetic field. 01:36:42.640 --> 01:36:46.040 align:middle line:90% 01:36:46.040 --> 01:36:47.500 align:middle line:90% And the other is called-- 01:36:47.500 --> 01:36:52.090 align:middle line:84% the converse, or let's say the Onsager reciprocal effect 01:36:52.090 --> 01:36:56.990 align:middle line:90% is called Nernst effect. 01:36:56.990 --> 01:37:03.570 align:middle line:84% And they generate-- if you have an electric field 01:37:03.570 --> 01:37:06.810 align:middle line:84% and you have an electric current in one direction, 01:37:06.810 --> 01:37:10.200 align:middle line:84% the magnetic field produces an electric current also 01:37:10.200 --> 01:37:12.040 align:middle line:90% in the orthogonal direction. 01:37:12.040 --> 01:37:13.640 align:middle line:90% This is the Hall effect. 01:37:13.640 --> 01:37:16.260 align:middle line:90% And Nernst does the same thing. 01:37:16.260 --> 01:37:19.610 align:middle line:84% Instead of having as driving force the electric field, 01:37:19.610 --> 01:37:21.800 align:middle line:84% you have as driving force the gradient 01:37:21.800 --> 01:37:26.780 align:middle line:84% in temperature, which we know that drives 01:37:26.780 --> 01:37:29.440 align:middle line:84% an electric current like we've just seen 01:37:29.440 --> 01:37:30.990 align:middle line:90% in the thermoelectric effect. 01:37:30.990 --> 01:37:33.030 align:middle line:84% But in the presence of magnetic field, 01:37:33.030 --> 01:37:36.565 align:middle line:84% it also drives an orthogonal current like the Hall effect. 01:37:36.565 --> 01:37:39.970 align:middle line:90% 01:37:39.970 --> 01:37:43.960 align:middle line:84% And very recently, as you see, this is just a few years ago, 01:37:43.960 --> 01:37:49.690 align:middle line:84% people have been working on the equivalent of these effects 01:37:49.690 --> 01:37:53.853 align:middle line:90% also on spin currents. 01:37:53.853 --> 01:37:56.650 align:middle line:90% 01:37:56.650 --> 01:37:58.990 align:middle line:84% So they're not moving charges but are 01:37:58.990 --> 01:38:07.270 align:middle line:84% moving the spin that is internal into the atoms, or the nuclei 01:38:07.270 --> 01:38:12.645 align:middle line:84% of the atoms that you move around. 01:38:12.645 --> 01:38:16.510 align:middle line:84% And you look just at the current of spin. 01:38:16.510 --> 01:38:18.300 align:middle line:84% And they have found similar effects. 01:38:18.300 --> 01:38:21.460 align:middle line:90% 01:38:21.460 --> 01:38:26.680 align:middle line:84% And this is, I guess, a similar idea 01:38:26.680 --> 01:38:31.973 align:middle line:84% done with spin currents instead of charge currents. 01:38:31.973 --> 01:38:34.940 align:middle line:90% 01:38:34.940 --> 01:38:37.610 align:middle line:84% And so this was only to say that this 01:38:37.610 --> 01:38:47.630 align:middle line:84% is a topic that is so broad and so full of applications 01:38:47.630 --> 01:38:50.615 align:middle line:84% that it's very difficult to cover 01:38:50.615 --> 01:38:56.210 align:middle line:90% in a sort of a systematic way. 01:38:56.210 --> 01:39:05.170 align:middle line:84% But it's also to show that it can be, it must be, 01:39:05.170 --> 01:39:10.120 align:middle line:84% regularized and understood in a systematic framework 01:39:10.120 --> 01:39:16.210 align:middle line:84% like we have done, starting with the entropic point of view 01:39:16.210 --> 01:39:20.890 align:middle line:84% and developing the linear force flux relations, 01:39:20.890 --> 01:39:23.420 align:middle line:90% and applying Onsager relations. 01:39:23.420 --> 01:39:30.810 align:middle line:84% And often, this way of proceeding 01:39:30.810 --> 01:39:36.420 align:middle line:84% is what drives the search for new effects. 01:39:36.420 --> 01:39:40.580 align:middle line:84% You didn't think about it until you said, oh, I see. 01:39:40.580 --> 01:39:42.840 align:middle line:90% Ah. 01:39:42.840 --> 01:39:46.080 align:middle line:84% So by induction, there must be an effect like that 01:39:46.080 --> 01:39:50.922 align:middle line:84% because there is a spot in that matrix of phenomena 01:39:50.922 --> 01:39:56.740 align:middle line:84% that is related via the Onsager principle. 01:39:56.740 --> 01:39:57.350 align:middle line:90% All right. 01:39:57.350 --> 01:39:59.150 align:middle line:90% So yes. 01:39:59.150 --> 01:40:01.550 align:middle line:90% I think we are done for today. 01:40:01.550 --> 01:40:09.580 align:middle line:84% And next time, we do the thermodiffusive effects. 01:40:09.580 --> 01:40:10.920 align:middle line:90% And then, we will be done. 01:40:10.920 --> 01:40:14.230 align:middle line:84% So next lecture will be the last one. 01:40:14.230 --> 01:40:26.000 align:middle line:90%