WEBVTT 00:00:00.000 --> 00:00:02.425 align:middle line:90% [SQUEAKING] 00:00:02.425 --> 00:00:04.365 align:middle line:90% [RUSTLING] 00:00:04.365 --> 00:00:07.760 align:middle line:90% [CLICKING] 00:00:07.760 --> 00:00:11.280 align:middle line:90% 00:00:11.280 --> 00:00:13.265 align:middle line:84% GIAN PAOLO BERETTA: So good morning again. 00:00:13.265 --> 00:00:18.840 align:middle line:90% 00:00:18.840 --> 00:00:22.090 align:middle line:90% Lecture number 8. 00:00:22.090 --> 00:00:28.450 align:middle line:84% Today, we finally get to the macroscopic world and models. 00:00:28.450 --> 00:00:31.660 align:middle line:84% But let me just remind you what we've done-- 00:00:31.660 --> 00:00:33.985 align:middle line:90% the last topic. 00:00:33.985 --> 00:00:38.950 align:middle line:84% We've seen the LeChatelier-Braun principle as a consequence 00:00:38.950 --> 00:00:45.840 align:middle line:84% of some interesting relations that emerge from the fact that 00:00:45.840 --> 00:00:50.760 align:middle line:84% stability conditions involve quadratic forms of some 00:00:50.760 --> 00:00:51.900 align:middle line:90% of the fundamental-of-- 00:00:51.900 --> 00:00:59.040 align:middle line:84% where the matrix is second derivatives 00:00:59.040 --> 00:01:02.880 align:middle line:84% of the fundamental relation or of some other characteristic 00:01:02.880 --> 00:01:06.360 align:middle line:84% function, like the Gibbs free energy 00:01:06.360 --> 00:01:08.540 align:middle line:90% or the Helmholtz free energy. 00:01:08.540 --> 00:01:13.220 align:middle line:84% And as a result of the mathematics 00:01:13.220 --> 00:01:21.120 align:middle line:84% of putting a quadratic form in canonical form, 00:01:21.120 --> 00:01:29.360 align:middle line:84% we obtain these inequalities that we have also 00:01:29.360 --> 00:01:38.340 align:middle line:84% interpreted in the sense of a theorem, 00:01:38.340 --> 00:01:41.750 align:middle line:84% because after all, this is a theorem. 00:01:41.750 --> 00:01:44.220 align:middle line:84% We could call it LeChatelier-Braun theorem that 00:01:44.220 --> 00:01:48.500 align:middle line:84% has two assertions-- namely, the fact that the system responds 00:01:48.500 --> 00:01:49.130 align:middle line:90% to-- 00:01:49.130 --> 00:01:54.230 align:middle line:84% when you move it away from where it is comfortable-- so 00:01:54.230 --> 00:01:59.120 align:middle line:84% when it is mutual equilibrium, for example, with a reservoir. 00:01:59.120 --> 00:02:03.380 align:middle line:84% And we have seen that there are different types of reservoirs 00:02:03.380 --> 00:02:06.090 align:middle line:84% with which the system can be in mutual equilibrium. 00:02:06.090 --> 00:02:10.070 align:middle line:84% So the same idea apply for mutual equilibrium 00:02:10.070 --> 00:02:12.710 align:middle line:84% with all those kinds of reservoirs. 00:02:12.710 --> 00:02:16.370 align:middle line:84% But the idea is that the system, if you do something 00:02:16.370 --> 00:02:19.760 align:middle line:90% to it from outside-- 00:02:19.760 --> 00:02:23.240 align:middle line:84% for example, put in energy, or put in particles, 00:02:23.240 --> 00:02:38.480 align:middle line:84% or change the volume, the system reacts 00:02:38.480 --> 00:02:41.690 align:middle line:84% trying to go back to where it was comfortable. 00:02:41.690 --> 00:02:45.540 align:middle line:90% That's the first assertion. 00:02:45.540 --> 00:02:49.020 align:middle line:84% So if you put in energy, it will increase its temperature 00:02:49.020 --> 00:02:51.090 align:middle line:84% so that it increases its willingness 00:02:51.090 --> 00:02:54.540 align:middle line:84% to give away energy and return to equilibrium 00:02:54.540 --> 00:02:56.760 align:middle line:90% with the reservoir. 00:02:56.760 --> 00:03:00.330 align:middle line:84% But the second assertion, which is 00:03:00.330 --> 00:03:04.680 align:middle line:84% a consequence of this additional math 00:03:04.680 --> 00:03:10.510 align:middle line:84% from the canonical form of the quadratic forms, 00:03:10.510 --> 00:03:14.130 align:middle line:90% is that it depends-- 00:03:14.130 --> 00:03:18.210 align:middle line:84% the reaction of the system depends on which direction 00:03:18.210 --> 00:03:22.710 align:middle line:84% you take it away from equilibrium, 00:03:22.710 --> 00:03:25.590 align:middle line:84% and in particular, if you break conditions 00:03:25.590 --> 00:03:27.600 align:middle line:90% for mutual equilibrium. 00:03:27.600 --> 00:03:32.023 align:middle line:90% So for example, if you have a-- 00:03:32.023 --> 00:03:34.620 align:middle line:90% 00:03:34.620 --> 00:03:37.000 align:middle line:84% if the system can exchange volume-- 00:03:37.000 --> 00:03:41.190 align:middle line:84% like in the example of the air springs, 00:03:41.190 --> 00:03:43.180 align:middle line:90% the system can exchange volume. 00:03:43.180 --> 00:03:49.860 align:middle line:84% And suppose that it is in mutual equilibrium 00:03:49.860 --> 00:03:54.500 align:middle line:84% also with the thermal reservoir, so at the same temperature. 00:03:54.500 --> 00:03:59.180 align:middle line:84% So there are two conditions for mutual equilibrium-- equality 00:03:59.180 --> 00:04:03.700 align:middle line:84% of pressure and equality of temperature. 00:04:03.700 --> 00:04:09.360 align:middle line:84% If you change the volume while breaking both conditions, 00:04:09.360 --> 00:04:16.190 align:middle line:84% the system will react stronger than if you just 00:04:16.190 --> 00:04:17.819 align:middle line:90% break one condition. 00:04:17.819 --> 00:04:20.519 align:middle line:90% 00:04:20.519 --> 00:04:24.590 align:middle line:90% And that's the idea. 00:04:24.590 --> 00:04:27.630 align:middle line:90% And the difference is important. 00:04:27.630 --> 00:04:34.530 align:middle line:84% For example, in the case of the air spring, 00:04:34.530 --> 00:04:36.480 align:middle line:90% the difference in the-- 00:04:36.480 --> 00:04:43.740 align:middle line:84% you get the reaction makes this volume act as a standard spring. 00:04:43.740 --> 00:04:50.430 align:middle line:84% But the spring constant depends on how many conditions 00:04:50.430 --> 00:04:53.190 align:middle line:84% for mutual equilibrium you've broken 00:04:53.190 --> 00:04:56.560 align:middle line:90% while changing that volume. 00:04:56.560 --> 00:04:58.870 align:middle line:84% And the change may be of the order of-- 00:04:58.870 --> 00:05:02.410 align:middle line:84% I mean, it's 40% if you do it slowly so 00:05:02.410 --> 00:05:10.660 align:middle line:84% that the system has the time to remain in the same temperature. 00:05:10.660 --> 00:05:14.030 align:middle line:84% So by exchanging energy with the reservoir, 00:05:14.030 --> 00:05:16.350 align:middle line:84% it remains in mutual equilibrium. 00:05:16.350 --> 00:05:25.520 align:middle line:84% Then you get a milder reaction than if you do it fast. 00:05:25.520 --> 00:05:28.940 align:middle line:84% And if you do it fast, you change both the pressure 00:05:28.940 --> 00:05:29.850 align:middle line:90% and the temperature. 00:05:29.850 --> 00:05:32.450 align:middle line:84% And so the system wants to-- has two reasons 00:05:32.450 --> 00:05:35.965 align:middle line:90% to go back to equilibrium. 00:05:35.965 --> 00:05:45.110 align:middle line:84% So now we're finally to the question-- 00:05:45.110 --> 00:05:50.180 align:middle line:84% well, what's the difference between microscopic systems 00:05:50.180 --> 00:05:51.730 align:middle line:90% and macroscopic systems? 00:05:51.730 --> 00:05:56.610 align:middle line:90% 00:05:56.610 --> 00:06:00.970 align:middle line:84% What do we gain if we make some additional assumptions? 00:06:00.970 --> 00:06:06.260 align:middle line:84% It will involve having many particles. 00:06:06.260 --> 00:06:16.100 align:middle line:84% So let me start with a few pictures, but also even a more 00:06:16.100 --> 00:06:27.110 align:middle line:84% simple idea, because there are several ways in which we 00:06:27.110 --> 00:06:29.570 align:middle line:84% can see the effect of what happens 00:06:29.570 --> 00:06:31.360 align:middle line:90% when we have few particles. 00:06:31.360 --> 00:06:35.000 align:middle line:84% For example, consider a single particle, 00:06:35.000 --> 00:06:40.130 align:middle line:84% and consider the most basic description, which is one-- 00:06:40.130 --> 00:06:42.800 align:middle line:84% the quantum mechanics description. 00:06:42.800 --> 00:06:53.525 align:middle line:84% In quantum mechanics, you have that the particle at-- 00:06:53.525 --> 00:06:56.420 align:middle line:90% 00:06:56.420 --> 00:06:59.570 align:middle line:84% the position of the particle is described 00:06:59.570 --> 00:07:05.450 align:middle line:84% by a probability density function, which tells you that-- 00:07:05.450 --> 00:07:10.200 align:middle line:84% so it tells you what is the probability to find, 00:07:10.200 --> 00:07:14.670 align:middle line:84% for example, the particle when you make a position 00:07:14.670 --> 00:07:19.590 align:middle line:84% experiment in this small interval between x and x 00:07:19.590 --> 00:07:21.540 align:middle line:90% plus dx. 00:07:21.540 --> 00:07:24.120 align:middle line:84% And in fact, according to quantum mechanics, 00:07:24.120 --> 00:07:28.590 align:middle line:84% all you can predict is the probability. 00:07:28.590 --> 00:07:32.360 align:middle line:84% You cannot predict where the particle will show up 00:07:32.360 --> 00:07:37.735 align:middle line:84% in the next time you call for, where are you? 00:07:37.735 --> 00:07:42.740 align:middle line:84% So essentially, the state in quantum mechanics 00:07:42.740 --> 00:07:45.290 align:middle line:84% is the probability distribution, or the set 00:07:45.290 --> 00:07:48.080 align:middle line:84% of probability distributions that 00:07:48.080 --> 00:07:54.920 align:middle line:84% allow you to describe all these measurement 00:07:54.920 --> 00:07:57.310 align:middle line:84% acts, but in a statistical sense. 00:07:57.310 --> 00:08:01.208 align:middle line:90% 00:08:01.208 --> 00:08:03.140 align:middle line:84% You see, the important thing that I 00:08:03.140 --> 00:08:08.000 align:middle line:84% want to emphasize today, that is important for our discussion 00:08:08.000 --> 00:08:12.170 align:middle line:84% today, is the fact that you find-- 00:08:12.170 --> 00:08:15.330 align:middle line:84% in order to solve, in quantum mechanics, this problem, 00:08:15.330 --> 00:08:19.550 align:middle line:84% you write the Hamiltonian of your particle, or system, 00:08:19.550 --> 00:08:20.750 align:middle line:90% or particles. 00:08:20.750 --> 00:08:24.350 align:middle line:84% And you have to solve an eigenvalue problem 00:08:24.350 --> 00:08:30.600 align:middle line:84% to find the eigenfunctions of that Hamiltonian. 00:08:30.600 --> 00:08:34.740 align:middle line:84% And you have to face the boundary conditions. 00:08:34.740 --> 00:08:42.240 align:middle line:84% So the walls of your system, the things 00:08:42.240 --> 00:08:46.530 align:middle line:84% that we call external forces, are represented 00:08:46.530 --> 00:08:50.200 align:middle line:84% by an infinite potential here on both sides, 00:08:50.200 --> 00:09:00.130 align:middle line:84% which forces the functions to go to 0 at the wall. 00:09:00.130 --> 00:09:05.450 align:middle line:90% 00:09:05.450 --> 00:09:09.190 align:middle line:84% Then, of course, if you solve that eigenvalue problem, 00:09:09.190 --> 00:09:12.830 align:middle line:90% you find functions that-- 00:09:12.830 --> 00:09:17.940 align:middle line:84% you can look it up in any Wikipedia page. 00:09:17.940 --> 00:09:21.780 align:middle line:84% There is the various solutions and various eigenfunctions. 00:09:21.780 --> 00:09:28.100 align:middle line:84% And then when the system is in a stable equilibrium state, 00:09:28.100 --> 00:09:36.340 align:middle line:84% the overall probability is a mixture of those eigenfunctions. 00:09:36.340 --> 00:09:43.580 align:middle line:84% For example, here, I took a paper in which there is not 00:09:43.580 --> 00:09:50.360 align:middle line:84% one particle, but four particles confined in a box like that 00:09:50.360 --> 00:09:52.220 align:middle line:90% and at very low temperature. 00:09:52.220 --> 00:09:54.260 align:middle line:90% Now, very low temperature-- 00:09:54.260 --> 00:10:01.280 align:middle line:84% that mixture of eigenfunctions makes the lower eigenvalue 00:10:01.280 --> 00:10:03.550 align:middle line:90% ones prevail. 00:10:03.550 --> 00:10:08.070 align:middle line:90% 00:10:08.070 --> 00:10:14.400 align:middle line:84% And you see here that-- so these are the distribution 00:10:14.400 --> 00:10:18.540 align:middle line:84% of probability to find the particles for various cases, 00:10:18.540 --> 00:10:20.820 align:middle line:90% going from a boson-- 00:10:20.820 --> 00:10:26.585 align:middle line:84% black line-- to a fermion, to things in between. 00:10:26.585 --> 00:10:29.820 align:middle line:90% 00:10:29.820 --> 00:10:34.390 align:middle line:84% The common feature that is important to us 00:10:34.390 --> 00:10:40.010 align:middle line:84% is that at the wall, those go to 0. 00:10:40.010 --> 00:10:41.600 align:middle line:90% So there is a-- 00:10:41.600 --> 00:10:44.320 align:middle line:84% that means-- I would call it the rarefaction effect. 00:10:44.320 --> 00:10:47.150 align:middle line:90% 00:10:47.150 --> 00:10:51.920 align:middle line:84% You can think of it as a gas, a probability gas, 00:10:51.920 --> 00:10:53.800 align:middle line:90% but it's rarefied at the walls. 00:10:53.800 --> 00:10:57.390 align:middle line:90% 00:10:57.390 --> 00:10:59.000 align:middle line:90% Here is another paper-- 00:10:59.000 --> 00:11:02.280 align:middle line:90% 00:11:02.280 --> 00:11:04.920 align:middle line:84% the review written by Professor Hadjiconstantinou, who 00:11:04.920 --> 00:11:07.565 align:middle line:90% is the mentor of this class. 00:11:07.565 --> 00:11:11.150 align:middle line:90% 00:11:11.150 --> 00:11:14.940 align:middle line:90% And in another realm-- 00:11:14.940 --> 00:11:17.850 align:middle line:90% this is not quantum mechanics. 00:11:17.850 --> 00:11:26.710 align:middle line:84% It's molecular dynamics-- the molecular fluid mechanics. 00:11:26.710 --> 00:11:33.103 align:middle line:84% So there is a solid wall probably made of carbon. 00:11:33.103 --> 00:11:39.050 align:middle line:84% And what happens is that the particles-- 00:11:39.050 --> 00:11:44.740 align:middle line:84% the interaction of the particles due to Van der Waals forces-- 00:11:44.740 --> 00:11:47.220 align:middle line:84% so this is repulsive and attractive forces, 00:11:47.220 --> 00:11:48.360 align:middle line:90% like Lennard-Jones. 00:11:48.360 --> 00:11:54.700 align:middle line:84% And we will talk again about this, but for the moment, 00:11:54.700 --> 00:11:59.835 align:middle line:84% I presume that you have some ideas about that. 00:11:59.835 --> 00:12:02.910 align:middle line:90% 00:12:02.910 --> 00:12:06.930 align:middle line:84% Due to these forces, and due to the fact 00:12:06.930 --> 00:12:12.795 align:middle line:84% that the wall is a fixed layer, and it imposes-- 00:12:12.795 --> 00:12:16.350 align:middle line:90% 00:12:16.350 --> 00:12:19.200 align:middle line:84% forces that do not move, whereas on the side 00:12:19.200 --> 00:12:22.570 align:middle line:84% of the bulk of the fluid, the particles move. 00:12:22.570 --> 00:12:25.980 align:middle line:84% And so there is more random action of forces. 00:12:25.980 --> 00:12:31.830 align:middle line:84% The ordering imposed by the wall implies an ordering-- 00:12:31.830 --> 00:12:36.250 align:middle line:84% a layering of particles at various distances. 00:12:36.250 --> 00:12:38.710 align:middle line:84% There is a first preferred distance, 00:12:38.710 --> 00:12:40.380 align:middle line:84% and then there is a second, third, 00:12:40.380 --> 00:12:46.830 align:middle line:84% and so on fading out as the randomness become more 00:12:46.830 --> 00:12:49.770 align:middle line:90% important away from the wall. 00:12:49.770 --> 00:12:56.470 align:middle line:90% So this is also a sort of wall. 00:12:56.470 --> 00:12:58.980 align:middle line:84% I would still call it the rarefaction effect 00:12:58.980 --> 00:13:01.020 align:middle line:90% because, yeah, in-- 00:13:01.020 --> 00:13:05.100 align:middle line:84% very close to between the first layer and the wall, 00:13:05.100 --> 00:13:06.280 align:middle line:90% there is nothing. 00:13:06.280 --> 00:13:08.045 align:middle line:84% The particles don't like to stay there. 00:13:08.045 --> 00:13:10.630 align:middle line:90% 00:13:10.630 --> 00:13:12.370 align:middle line:84% These in fluid mechanics is important 00:13:12.370 --> 00:13:16.030 align:middle line:84% because it is related to the problem 00:13:16.030 --> 00:13:19.640 align:middle line:90% of the no-slip condition. 00:13:19.640 --> 00:13:22.770 align:middle line:84% If you are a fluid mechanicist, when you have few particles, 00:13:22.770 --> 00:13:24.930 align:middle line:84% you have to deal with this problem. 00:13:24.930 --> 00:13:34.140 align:middle line:90% And this is another example. 00:13:34.140 --> 00:13:45.380 align:middle line:84% Again, watching the density distributions of a gas. 00:13:45.380 --> 00:13:48.330 align:middle line:84% Now, here, it's not few particles. 00:13:48.330 --> 00:13:52.040 align:middle line:90% It's, like, 10 to the 5th atoms. 00:13:52.040 --> 00:13:55.430 align:middle line:90% It's not very few. 00:13:55.430 --> 00:14:02.190 align:middle line:84% Still in these cases, you see density-- 00:14:02.190 --> 00:14:08.940 align:middle line:84% wall effects with a rarefaction very near the wall. 00:14:08.940 --> 00:14:12.350 align:middle line:84% And then there are peaks where the particles 00:14:12.350 --> 00:14:15.505 align:middle line:84% like to stay near the wall more than in the bulk. 00:14:15.505 --> 00:14:18.370 align:middle line:90% 00:14:18.370 --> 00:14:24.525 align:middle line:84% Now, 10 to the 5th is not small, but it's not large. 00:14:24.525 --> 00:14:36.160 align:middle line:90% 00:14:36.160 --> 00:14:40.550 align:middle line:90% Why is this important? 00:14:40.550 --> 00:14:44.570 align:middle line:90% And so let's start with-- 00:14:44.570 --> 00:14:47.500 align:middle line:84% Suppose I have two particles like this. 00:14:47.500 --> 00:14:51.900 align:middle line:90% 00:14:51.900 --> 00:14:56.530 align:middle line:84% So I have the wall, but on the other side, 00:14:56.530 --> 00:15:01.070 align:middle line:84% I have a specular situation in which 00:15:01.070 --> 00:15:08.160 align:middle line:84% I have another particle in the same stable equilibrium state. 00:15:08.160 --> 00:15:12.912 align:middle line:84% This is the meaning of this part of the figure. 00:15:12.912 --> 00:15:15.070 align:middle line:84% Think of just one particle on one side 00:15:15.070 --> 00:15:16.850 align:middle line:84% and one particle on the other side. 00:15:16.850 --> 00:15:17.490 align:middle line:90% That's OK. 00:15:17.490 --> 00:15:20.530 align:middle line:90% 00:15:20.530 --> 00:15:29.280 align:middle line:84% Now, suppose I find a way to make that wall that separates 00:15:29.280 --> 00:15:31.236 align:middle line:90% the two particles disappear. 00:15:31.236 --> 00:15:37.802 align:middle line:90% 00:15:37.802 --> 00:15:48.060 align:middle line:84% So the moment it disappears, particles don't know yet. 00:15:48.060 --> 00:15:50.310 align:middle line:90% We just removed it. 00:15:50.310 --> 00:15:51.610 align:middle line:90% They're still there. 00:15:51.610 --> 00:15:55.410 align:middle line:84% Let's say, oh, hey, there's more room available. 00:15:55.410 --> 00:15:59.310 align:middle line:84% There is more volume we can fill. 00:15:59.310 --> 00:16:06.270 align:middle line:84% So there will be a spontaneous process, whereby the particles 00:16:06.270 --> 00:16:08.010 align:middle line:90% will fill that volume. 00:16:08.010 --> 00:16:15.090 align:middle line:84% And eventually, we'll reach a new stable equilibrium state-- 00:16:15.090 --> 00:16:20.600 align:middle line:90% let's say something like that. 00:16:20.600 --> 00:16:26.800 align:middle line:90% 00:16:26.800 --> 00:16:33.740 align:middle line:84% If we look at it with the help of the E versus S diagram, 00:16:33.740 --> 00:16:37.780 align:middle line:90% which is very useful here-- 00:16:37.780 --> 00:16:42.120 align:middle line:90% I have drawn the E versus S-- 00:16:42.120 --> 00:16:45.060 align:middle line:84% the stable equilibrium state curves-- 00:16:45.060 --> 00:16:53.120 align:middle line:84% fundamental relation of the two systems that we have considered. 00:16:53.120 --> 00:16:57.460 align:middle line:84% One is the system in which there is a partition that divides it 00:16:57.460 --> 00:16:59.480 align:middle line:90% into two. 00:16:59.480 --> 00:17:02.960 align:middle line:84% The other is the system without the partition. 00:17:02.960 --> 00:17:07.859 align:middle line:84% And that system without the partition, for that one, 00:17:07.859 --> 00:17:10.160 align:middle line:90% we are considering two states. 00:17:10.160 --> 00:17:13.400 align:middle line:90% The state we started with. 00:17:13.400 --> 00:17:15.780 align:middle line:84% When we just removed the partition, 00:17:15.780 --> 00:17:18.200 align:middle line:84% it's a non-equilibrium state, so much 00:17:18.200 --> 00:17:22.410 align:middle line:84% that we know that we'll evolve towards the stable equilibrium 00:17:22.410 --> 00:17:22.950 align:middle line:90% state. 00:17:22.950 --> 00:17:35.430 align:middle line:84% So this point here represents for the system with the-- 00:17:35.430 --> 00:17:40.570 align:middle line:84% partition system as the initial stable equilibrium state. 00:17:40.570 --> 00:17:48.190 align:middle line:84% But it also represents the state of the system without partition, 00:17:48.190 --> 00:17:51.460 align:middle line:84% and it's the non-equilibrium state for that system. 00:17:51.460 --> 00:17:55.900 align:middle line:84% So here, we call system A the one without the partition. 00:17:55.900 --> 00:17:59.830 align:middle line:90% And system 2 lambda-- 00:17:59.830 --> 00:18:04.810 align:middle line:84% I mean, lambda plus lambda, so it's 2 lambda-- 00:18:04.810 --> 00:18:07.600 align:middle line:90% the system with the partitions. 00:18:07.600 --> 00:18:13.450 align:middle line:84% The reason I draw them on this same spot, 00:18:13.450 --> 00:18:20.470 align:middle line:84% meaning that they have the same energy and entropy 00:18:20.470 --> 00:18:24.310 align:middle line:84% is a consequence of the idea that I 00:18:24.310 --> 00:18:27.220 align:middle line:84% have found a way to remove a partition without perturbing 00:18:27.220 --> 00:18:29.510 align:middle line:84% the system, without putting energy in, 00:18:29.510 --> 00:18:40.470 align:middle line:84% without any external exchange in no time 00:18:40.470 --> 00:18:45.580 align:middle line:84% so that, essentially, I don't change any of the properties. 00:18:45.580 --> 00:18:46.910 align:middle line:90% Ideally, you could do it. 00:18:46.910 --> 00:18:49.950 align:middle line:90% 00:18:49.950 --> 00:18:52.410 align:middle line:84% And the fact that I have put this state 00:18:52.410 --> 00:18:57.420 align:middle line:84% to the left of the final stable equilibrium state is that-- 00:18:57.420 --> 00:19:03.240 align:middle line:84% yeah, intuitively, we know that the particles will evolve 00:19:03.240 --> 00:19:05.550 align:middle line:90% by filling the entire space. 00:19:05.550 --> 00:19:09.270 align:middle line:84% So that's the direction in which the system 00:19:09.270 --> 00:19:13.750 align:middle line:90% will evolve, not vice versa. 00:19:13.750 --> 00:19:18.790 align:middle line:84% If I put this state on the other side, 00:19:18.790 --> 00:19:25.170 align:middle line:84% it would mean that the system would separate spontaneously, 00:19:25.170 --> 00:19:26.270 align:middle line:90% which doesn't happen. 00:19:26.270 --> 00:19:37.000 align:middle line:90% 00:19:37.000 --> 00:19:40.690 align:middle line:84% So the first consequence of this is 00:19:40.690 --> 00:19:48.220 align:middle line:84% that the entropy of the stable equilibrium state of the two 00:19:48.220 --> 00:19:53.220 align:middle line:84% particles that have the same initial energy 00:19:53.220 --> 00:19:55.570 align:middle line:90% because this system is-- 00:19:55.570 --> 00:20:00.870 align:middle line:84% it all happened in an isolated system. 00:20:00.870 --> 00:20:05.910 align:middle line:84% The entropy of the final stable equilibrium state 00:20:05.910 --> 00:20:11.250 align:middle line:84% is greater than the initial entropy of this state 00:20:11.250 --> 00:20:14.730 align:middle line:84% or also of this non-equilibrium state. 00:20:14.730 --> 00:20:17.160 align:middle line:84% That is by the principle of entropy nondecrease. 00:20:17.160 --> 00:20:21.330 align:middle line:84% A spontaneous process goes only in the direction of increasing 00:20:21.330 --> 00:20:23.830 align:middle line:90% entropy for an isolated system. 00:20:23.830 --> 00:20:32.890 align:middle line:84% So what is that changes when you have many particles? 00:20:32.890 --> 00:20:37.900 align:middle line:84% Somehow, I have already used this idea here. 00:20:37.900 --> 00:20:39.920 align:middle line:90% When you have many particles-- 00:20:39.920 --> 00:20:45.430 align:middle line:84% because here, I go from 1 to 2 particles-- 00:20:45.430 --> 00:20:49.000 align:middle line:84% the slope of this probability distribution at the wall 00:20:49.000 --> 00:20:50.330 align:middle line:90% becomes steeper. 00:20:50.330 --> 00:20:55.240 align:middle line:84% So in other words, the more particles you have, 00:20:55.240 --> 00:21:04.940 align:middle line:84% the smaller is the rarefaction layer next to the wall. 00:21:04.940 --> 00:21:09.880 align:middle line:90% 00:21:09.880 --> 00:21:18.600 align:middle line:90% And as a result, you see here-- 00:21:18.600 --> 00:21:24.930 align:middle line:90% 00:21:24.930 --> 00:21:26.480 align:middle line:90% you see the difference with-- 00:21:26.480 --> 00:21:29.260 align:middle line:90% 00:21:29.260 --> 00:21:31.220 align:middle line:84% let me highlight where the differences are 00:21:31.220 --> 00:21:40.700 align:middle line:84% between the probability distributions of the two 00:21:40.700 --> 00:21:44.130 align:middle line:90% stable equilibrium states. 00:21:44.130 --> 00:21:47.570 align:middle line:90% Here, they're remarkable. 00:21:47.570 --> 00:21:53.930 align:middle line:84% But as this distribution becomes sharp-- 00:21:53.930 --> 00:21:59.720 align:middle line:84% so eventually, you have an almost flat distribution 00:21:59.720 --> 00:22:07.650 align:middle line:84% in the bulk of your system and a sharp, very thin area-- 00:22:07.650 --> 00:22:10.365 align:middle line:84% we call it boundary layer, or wall layer, 00:22:10.365 --> 00:22:13.350 align:middle line:90% or rarefraction layer. 00:22:13.350 --> 00:22:17.500 align:middle line:84% So even at the center, this region is very small. 00:22:17.500 --> 00:22:21.320 align:middle line:90% 00:22:21.320 --> 00:22:28.430 align:middle line:84% That means that the two stable equilibrium state curves 00:22:28.430 --> 00:22:30.180 align:middle line:90% get closer to one another. 00:22:30.180 --> 00:22:33.240 align:middle line:90% 00:22:33.240 --> 00:22:35.070 align:middle line:84% And in the limit of many particles, 00:22:35.070 --> 00:22:39.200 align:middle line:90% you can neglect this difference. 00:22:39.200 --> 00:22:41.740 align:middle line:90% So you can neglect this fact. 00:22:41.740 --> 00:22:47.560 align:middle line:84% You can approximate the properties 00:22:47.560 --> 00:22:53.170 align:middle line:84% of a system with the partitions with the properties 00:22:53.170 --> 00:22:55.540 align:middle line:84% of the system without the partition. 00:22:55.540 --> 00:23:00.250 align:middle line:84% Or if you are interested, you can model the system 00:23:00.250 --> 00:23:06.310 align:middle line:84% without the partitions as if it had the partitions. 00:23:06.310 --> 00:23:07.740 align:middle line:90% This is very important. 00:23:07.740 --> 00:23:11.560 align:middle line:84% We will return to this when we go through the continuum 00:23:11.560 --> 00:23:13.450 align:middle line:90% hypothesis. 00:23:13.450 --> 00:23:17.350 align:middle line:84% It is very important for example in fluid mechanics-- 00:23:17.350 --> 00:23:26.120 align:middle line:84% because you have a big box, or boundary for whatever flow 00:23:26.120 --> 00:23:28.990 align:middle line:90% you are considering. 00:23:28.990 --> 00:23:36.880 align:middle line:84% And we want to work in terms of fluid elements, fluid parcels 00:23:36.880 --> 00:23:37.950 align:middle line:90% that are small. 00:23:37.950 --> 00:23:41.690 align:middle line:90% 00:23:41.690 --> 00:23:49.130 align:middle line:84% Small enough with respect to the big system, 00:23:49.130 --> 00:23:55.580 align:middle line:84% but not small enough to have this kind of effects. 00:23:55.580 --> 00:23:58.490 align:middle line:84% Meaning you have enough particles inside here so 00:23:58.490 --> 00:24:00.380 align:middle line:90% that you can consider it-- 00:24:00.380 --> 00:24:02.060 align:middle line:84% consider the continuum hypothesis valid. 00:24:02.060 --> 00:24:04.050 align:middle line:84% We will return to the continuum hypothesis. 00:24:04.050 --> 00:24:04.550 align:middle line:90% But-- 00:24:04.550 --> 00:24:10.030 align:middle line:90% 00:24:10.030 --> 00:24:11.830 align:middle line:84% What I'm interested now in saying 00:24:11.830 --> 00:24:18.450 align:middle line:84% is that when I consider, my fluid element here, 00:24:18.450 --> 00:24:23.450 align:middle line:84% I am going to apply models, like the local equilibrium model. 00:24:23.450 --> 00:24:26.860 align:middle line:90% 00:24:26.860 --> 00:24:29.810 align:middle line:90% And we will assign to this-- 00:24:29.810 --> 00:24:34.730 align:middle line:84% the properties of a system in which the particles 00:24:34.730 --> 00:24:41.570 align:middle line:90% are closed by physical walls. 00:24:41.570 --> 00:24:44.330 align:middle line:84% So we approximate this fluid element 00:24:44.330 --> 00:24:48.275 align:middle line:84% as if locally it has the properties 00:24:48.275 --> 00:24:52.070 align:middle line:84% of a fluid element that is separated from the rest 00:24:52.070 --> 00:24:55.530 align:middle line:90% by physical walls. 00:24:55.530 --> 00:24:56.910 align:middle line:90% How do you do that? 00:24:56.910 --> 00:25:00.990 align:middle line:90% Well, I insert a partition-- 00:25:00.990 --> 00:25:04.710 align:middle line:84% three-dimensional-- make this small three-dimensional box. 00:25:04.710 --> 00:25:08.654 align:middle line:90% Take this system outside. 00:25:08.654 --> 00:25:14.460 align:middle line:84% Study how it behaves when it has some temperature, some pressure, 00:25:14.460 --> 00:25:15.480 align:middle line:90% some energy, and so on. 00:25:15.480 --> 00:25:17.820 align:middle line:90% And then I assign-- 00:25:17.820 --> 00:25:19.560 align:middle line:84% I say, well, the properties of this 00:25:19.560 --> 00:25:28.545 align:middle line:84% are the same as that, forgetting about the walls' effect. 00:25:28.545 --> 00:25:32.580 align:middle line:84% But this idea of physically inserting or removing 00:25:32.580 --> 00:25:37.630 align:middle line:84% partitions and the fact that it doesn't change 00:25:37.630 --> 00:25:41.330 align:middle line:84% the properties of your system is fundamental in our description 00:25:41.330 --> 00:25:46.870 align:middle line:84% of continua at the macroscopic level. 00:25:46.870 --> 00:25:54.835 align:middle line:90% 00:25:54.835 --> 00:25:59.310 align:middle line:84% But in principle, we should know that if you-- 00:25:59.310 --> 00:26:04.780 align:middle line:84% in that fluid parcel, or fluid element, 00:26:04.780 --> 00:26:10.850 align:middle line:84% you have too few particles, then you may be in trouble 00:26:10.850 --> 00:26:12.920 align:middle line:90% playing this game. 00:26:12.920 --> 00:26:23.130 align:middle line:84% And so your assumption that you usually make, 00:26:23.130 --> 00:26:29.190 align:middle line:84% which involves local equilibrium and the continuum hypothesis, 00:26:29.190 --> 00:26:34.900 align:middle line:84% also should take care about the many particles hypothesis. 00:26:34.900 --> 00:26:41.870 align:middle line:90% 00:26:41.870 --> 00:26:47.030 align:middle line:84% So let's see what we gain in the limit of many particles. 00:26:47.030 --> 00:26:52.140 align:middle line:84% Then we'll go back toward computing this effect 00:26:52.140 --> 00:26:55.560 align:middle line:90% we got within partitions. 00:26:55.560 --> 00:26:57.960 align:middle line:90% Let's see what we gain. 00:26:57.960 --> 00:27:03.990 align:middle line:84% So here is a more schematic idea. 00:27:03.990 --> 00:27:06.030 align:middle line:84% So suppose we have our system A is 00:27:06.030 --> 00:27:14.115 align:middle line:84% this one, where we have whatever particles, a given volume. 00:27:14.115 --> 00:27:17.590 align:middle line:84% Volume here is important because, of course, 00:27:17.590 --> 00:27:23.910 align:middle line:84% it has to be-- we have to be able to partition the system. 00:27:23.910 --> 00:27:34.720 align:middle line:84% Also, you see, when we are in this limit, 00:27:34.720 --> 00:27:38.310 align:middle line:84% since the rarefaction effects become 00:27:38.310 --> 00:27:43.410 align:middle line:84% unimportant to the boundary, essentially, 00:27:43.410 --> 00:27:50.570 align:middle line:84% the geometry of your container loses importance. 00:27:50.570 --> 00:27:54.566 align:middle line:84% For example, here, where in this-- 00:27:54.566 --> 00:27:57.050 align:middle line:84% this picture here shows very well 00:27:57.050 --> 00:28:04.850 align:middle line:84% the geometry of your container does play a big role. 00:28:04.850 --> 00:28:07.250 align:middle line:84% So if you go to this paper, there 00:28:07.250 --> 00:28:11.900 align:middle line:84% are other geometries that are shown in nice pictures. 00:28:11.900 --> 00:28:15.770 align:middle line:84% You see that the way particles are layered out, 00:28:15.770 --> 00:28:20.550 align:middle line:84% they form layers near the walls depends, of course, 00:28:20.550 --> 00:28:22.670 align:middle line:90% on the shape of the container. 00:28:22.670 --> 00:28:26.840 align:middle line:84% And therefore, the properties of stable equilibrium 00:28:26.840 --> 00:28:29.030 align:middle line:90% depend on the shape. 00:28:29.030 --> 00:28:39.900 align:middle line:84% But in the limit in which those wall effects become unimportant, 00:28:39.900 --> 00:28:42.630 align:middle line:84% only the overall volume of the container, 00:28:42.630 --> 00:28:46.730 align:middle line:90% not the details are important. 00:28:46.730 --> 00:28:53.050 align:middle line:84% Because in the bulk, the probability distribution 00:28:53.050 --> 00:28:55.140 align:middle line:90% becomes uniform. 00:28:55.140 --> 00:28:56.962 align:middle line:90% So it's uniform everywhere. 00:28:56.962 --> 00:28:59.860 align:middle line:90% 00:28:59.860 --> 00:29:02.680 align:middle line:84% So here, suppose we have our system, 00:29:02.680 --> 00:29:09.160 align:middle line:84% and suppose we partition it into lambda partitions. 00:29:09.160 --> 00:29:13.390 align:middle line:90% 00:29:13.390 --> 00:29:14.760 align:middle line:90% Here, we call it lambda. 00:29:14.760 --> 00:29:15.930 align:middle line:90% Later, we'll call it n. 00:29:15.930 --> 00:29:21.110 align:middle line:90% 00:29:21.110 --> 00:29:26.890 align:middle line:84% So that means that on this side, this is a composite system. 00:29:26.890 --> 00:29:29.900 align:middle line:90% 00:29:29.900 --> 00:29:32.120 align:middle line:84% It's still in a stable equilibrium state. 00:29:32.120 --> 00:29:34.930 align:middle line:84% Each one is in a stable equilibrium state. 00:29:34.930 --> 00:29:38.334 align:middle line:84% Like the system we started on here on the blackboard. 00:29:38.334 --> 00:29:41.120 align:middle line:90% 00:29:41.120 --> 00:29:45.005 align:middle line:84% And we assume that we have partitioned it 00:29:45.005 --> 00:29:47.150 align:middle line:84% in such a way that not only the volume, 00:29:47.150 --> 00:29:54.310 align:middle line:84% but also number of particles is exactly 1 over lambda. 00:29:54.310 --> 00:29:58.150 align:middle line:84% The volume is 1 over lambda of the overall, 00:29:58.150 --> 00:29:59.690 align:middle line:90% and the energy of 1 over lambda. 00:29:59.690 --> 00:30:03.120 align:middle line:90% 00:30:03.120 --> 00:30:10.750 align:middle line:84% So now, remember what the fundamental relation means. 00:30:10.750 --> 00:30:13.060 align:middle line:84% For example, here, fundamental relation 00:30:13.060 --> 00:30:15.880 align:middle line:84% means that it gives us the properties 00:30:15.880 --> 00:30:21.470 align:middle line:84% of all the stable equilibrium states as a function of energy, 00:30:21.470 --> 00:30:22.590 align:middle line:90% volume, and the amounts. 00:30:22.590 --> 00:30:25.090 align:middle line:90% So it is valid. 00:30:25.090 --> 00:30:29.700 align:middle line:84% It is a function-- the same functional form-- 00:30:29.700 --> 00:30:33.750 align:middle line:84% maybe E squared, E cubed, log of n, things like that. 00:30:33.750 --> 00:30:44.400 align:middle line:84% It's the same function for all values E, V, and n. 00:30:44.400 --> 00:30:49.760 align:middle line:84% I can compute it also for E, but also for E over lambda. 00:30:49.760 --> 00:30:56.100 align:middle line:84% The same function, if it is the same mixture of substances, 00:30:56.100 --> 00:30:56.600 align:middle line:90% for example. 00:30:56.600 --> 00:31:00.030 align:middle line:90% 00:31:00.030 --> 00:31:03.420 align:middle line:84% For a lot of energy or small energy, 00:31:03.420 --> 00:31:08.185 align:middle line:84% large volume, small volume, many particles, few particles. 00:31:08.185 --> 00:31:11.440 align:middle line:90% 00:31:11.440 --> 00:31:17.860 align:middle line:84% As long as this few doesn't get too few, it's OK. 00:31:17.860 --> 00:31:19.720 align:middle line:84% I mean, it's OK what we are going to say. 00:31:19.720 --> 00:31:22.595 align:middle line:84% So at this stage, it's OK in general. 00:31:22.595 --> 00:31:25.260 align:middle line:90% 00:31:25.260 --> 00:31:28.445 align:middle line:84% But what we said here is that these curves-- 00:31:28.445 --> 00:31:31.510 align:middle line:90% 00:31:31.510 --> 00:31:33.430 align:middle line:90% this is the partition curve. 00:31:33.430 --> 00:31:35.780 align:middle line:84% It's the summation-- it's this side, 00:31:35.780 --> 00:31:38.990 align:middle line:84% so it's this much for each subsystem. 00:31:38.990 --> 00:31:42.550 align:middle line:84% So if you have lambda of those, you multiply it by lambda, 00:31:42.550 --> 00:31:47.740 align:middle line:84% so lambda times S, because S is an additive property. 00:31:47.740 --> 00:31:50.230 align:middle line:84% The entropy of the overall is the sum 00:31:50.230 --> 00:31:52.270 align:middle line:90% of the entropies of the parts. 00:31:52.270 --> 00:31:59.915 align:middle line:84% And so this composite system has an entropy lambda S. 00:31:59.915 --> 00:32:02.530 align:middle line:90% And this S sub lambda. 00:32:02.530 --> 00:32:05.075 align:middle line:90% 00:32:05.075 --> 00:32:13.240 align:middle line:84% Whereas system A has an entropy S sub A. In general, 00:32:13.240 --> 00:32:16.160 align:middle line:84% this has bigger entropy than that. 00:32:16.160 --> 00:32:20.290 align:middle line:84% But in the limit of many particles, 00:32:20.290 --> 00:32:22.180 align:middle line:84% this difference becomes unimportant. 00:32:22.180 --> 00:32:29.150 align:middle line:84% And so we can approximate that the two are equal. 00:32:29.150 --> 00:32:33.015 align:middle line:84% That's what we call the "simple system model." 00:32:33.015 --> 00:32:37.710 align:middle line:90% 00:32:37.710 --> 00:32:42.360 align:middle line:84% You don't find it around with this name except in our book. 00:32:42.360 --> 00:32:52.590 align:middle line:84% Perhaps the choice of the name wasn't so fortunate, 00:32:52.590 --> 00:32:57.420 align:middle line:84% but it expresses what you find under the title 00:32:57.420 --> 00:33:01.390 align:middle line:90% of "thermodynamic limit." 00:33:01.390 --> 00:33:04.190 align:middle line:84% In physics, that's how it's called-- this idea. 00:33:04.190 --> 00:33:10.720 align:middle line:90% 00:33:10.720 --> 00:33:14.495 align:middle line:84% Indeed, it is a limit because now look what we're going to do. 00:33:14.495 --> 00:33:17.741 align:middle line:90% 00:33:17.741 --> 00:33:20.880 align:middle line:84% Here, we choose a finite value for lambda. 00:33:20.880 --> 00:33:24.800 align:middle line:84% So we partition the system in a finite number of partitions. 00:33:24.800 --> 00:33:30.110 align:middle line:84% And it has to be finite in order for the number of particles 00:33:30.110 --> 00:33:35.680 align:middle line:84% inside each to become not too small. 00:33:35.680 --> 00:33:46.455 align:middle line:84% Yet if lambda can be large enough for practical purposes-- 00:33:46.455 --> 00:33:50.225 align:middle line:90% 00:33:50.225 --> 00:33:52.070 align:middle line:90% You have many particles-- 00:33:52.070 --> 00:33:53.510 align:middle line:90% 10 to the 23-- 00:33:53.510 --> 00:33:58.040 align:middle line:84% that means that lambda could be, say, a million. 00:33:58.040 --> 00:34:06.950 align:middle line:84% And still, in each partition, you have 10 to the what, 17? 00:34:06.950 --> 00:34:11.810 align:middle line:84% So that means still a lot of particles. 00:34:11.810 --> 00:34:15.659 align:middle line:84% So from the practical point of view, 00:34:15.659 --> 00:34:18.260 align:middle line:84% when you have so many particles, you 00:34:18.260 --> 00:34:25.983 align:middle line:84% can think that this lambda can be any number. 00:34:25.983 --> 00:34:30.630 align:middle line:84% Oh, yes, could be integer, but I mean, who cares? 00:34:30.630 --> 00:34:33.580 align:middle line:84% Between the integers, there are the reals, 00:34:33.580 --> 00:34:35.050 align:middle line:90% so let's just make it real. 00:34:35.050 --> 00:34:38.429 align:middle line:90% 00:34:38.429 --> 00:34:42.420 align:middle line:84% So you make this mathematical assumption. 00:34:42.420 --> 00:34:49.889 align:middle line:84% Essentially, take this formula here and make it valid-- 00:34:49.889 --> 00:34:57.265 align:middle line:84% make it as valid for any lambda, integer or real decimal. 00:34:57.265 --> 00:35:00.470 align:middle line:90% 00:35:00.470 --> 00:35:05.640 align:middle line:84% So you formally assume that the fundamental relation 00:35:05.640 --> 00:35:07.830 align:middle line:90% has this property. 00:35:07.830 --> 00:35:11.800 align:middle line:84% And this property for a function in mathematics, 00:35:11.800 --> 00:35:14.390 align:middle line:84% is called the homogeneity of first degree. 00:35:14.390 --> 00:35:18.010 align:middle line:90% 00:35:18.010 --> 00:35:20.800 align:middle line:84% The fundamental relation is the homogeneous function 00:35:20.800 --> 00:35:27.810 align:middle line:84% of first degree in all of its independent variables. 00:35:27.810 --> 00:35:31.100 align:middle line:90% 00:35:31.100 --> 00:35:39.420 align:middle line:84% Divide each one by the same lambda, 00:35:39.420 --> 00:35:41.290 align:middle line:84% then multiply the result by lambda. 00:35:41.290 --> 00:35:48.390 align:middle line:84% You get the value of the function for the original values 00:35:48.390 --> 00:35:50.790 align:middle line:90% of the independent properties. 00:35:50.790 --> 00:35:55.910 align:middle line:84% And this is an important assumption, of course. 00:35:55.910 --> 00:36:02.630 align:middle line:84% It's a heavy assumption which has this physics to support it. 00:36:02.630 --> 00:36:11.660 align:middle line:84% And when you do it, we gain one relation-- 00:36:11.660 --> 00:36:13.510 align:middle line:84% We're going to prove it in a moment-- 00:36:13.510 --> 00:36:18.130 align:middle line:84% which is called the Euler relation. 00:36:18.130 --> 00:36:27.570 align:middle line:84% And notice also-- I changed the letter for energy from E to U. 00:36:27.570 --> 00:36:34.530 align:middle line:84% And we're also going to call it internal energy for distinction. 00:36:34.530 --> 00:36:38.890 align:middle line:84% So as soon as we say internal energy, we say, 00:36:38.890 --> 00:36:42.628 align:middle line:84% aha, we are within the simple system model approximation. 00:36:42.628 --> 00:36:51.040 align:middle line:90% 00:36:51.040 --> 00:36:53.120 align:middle line:84% So how do we prove that relation? 00:36:53.120 --> 00:36:55.640 align:middle line:90% 00:36:55.640 --> 00:36:57.260 align:middle line:90% Well, it's simple. 00:36:57.260 --> 00:37:05.890 align:middle line:84% One way is-since this has to be true for all lambdas. 00:37:05.890 --> 00:37:15.660 align:middle line:84% Take the partial derivative of this with respect to lambda. 00:37:15.660 --> 00:37:29.100 align:middle line:90% 00:37:29.100 --> 00:37:31.830 align:middle line:90% Now, is it-- yeah, OK. 00:37:31.830 --> 00:37:40.790 align:middle line:84% I think there is something fishy in this-- 00:37:40.790 --> 00:37:46.490 align:middle line:90% 00:37:46.490 --> 00:37:51.620 align:middle line:84% there's something fishy in this viewgraph-- something missing. 00:37:51.620 --> 00:37:53.085 align:middle line:90% Sorry about that. 00:37:53.085 --> 00:37:55.790 align:middle line:90% 00:37:55.790 --> 00:37:58.360 align:middle line:84% So when you take the partial derivative with respect 00:37:58.360 --> 00:38:07.310 align:middle line:84% to lambda of this thing, you get U because of this lambda here. 00:38:07.310 --> 00:38:08.490 align:middle line:90% And then you have to take-- 00:38:08.490 --> 00:38:13.280 align:middle line:84% you have lambda that multiplies the derivative of U 00:38:13.280 --> 00:38:17.870 align:middle line:84% with respect to S times the derivative of this with respect 00:38:17.870 --> 00:38:18.420 align:middle line:90% to lambda. 00:38:18.420 --> 00:38:21.365 align:middle line:84% So it's minus 1 over lambda squared and so on. 00:38:21.365 --> 00:38:26.230 align:middle line:84% So yeah, this is wrong viewgraph. 00:38:26.230 --> 00:38:26.730 align:middle line:90% Sorry. 00:38:26.730 --> 00:38:29.310 align:middle line:90% I'll correct it. 00:38:29.310 --> 00:38:32.390 align:middle line:84% And then in the end of this derivative, 00:38:32.390 --> 00:38:39.130 align:middle line:84% you take lambda equals to 1, and here comes this relation. 00:38:39.130 --> 00:38:42.315 align:middle line:84% It's better done in the book than in this viewgraph. 00:38:42.315 --> 00:38:42.815 align:middle line:90% Sorry. 00:38:42.815 --> 00:38:47.010 align:middle line:90% 00:38:47.010 --> 00:38:51.630 align:middle line:84% And what is another consequence of these derivatives? 00:38:51.630 --> 00:38:54.520 align:middle line:90% These are OK now. 00:38:54.520 --> 00:39:03.305 align:middle line:84% The main consequence of the Euler relation 00:39:03.305 --> 00:39:07.680 align:middle line:90% is that we can define-- 00:39:07.680 --> 00:39:13.020 align:middle line:84% it makes sense to define specific properties. 00:39:13.020 --> 00:39:15.710 align:middle line:90% For example, I have the energy. 00:39:15.710 --> 00:39:21.240 align:middle line:84% I'm not going to call it E. I'll call it U. 00:39:21.240 --> 00:39:27.560 align:middle line:90% And suppose I have n particles. 00:39:27.560 --> 00:39:34.470 align:middle line:84% Then n is the sum of all the particles 00:39:34.470 --> 00:39:38.500 align:middle line:84% that are present of different type. 00:39:38.500 --> 00:39:41.340 align:middle line:90% Those are the amounts. 00:39:41.340 --> 00:39:51.470 align:middle line:84% So this I am going to call it specific molar energy. 00:39:51.470 --> 00:39:54.180 align:middle line:90% 00:39:54.180 --> 00:40:03.750 align:middle line:84% I do the same for the entropy, for the volume 00:40:03.750 --> 00:40:07.330 align:middle line:90% and for the amounts. 00:40:07.330 --> 00:40:11.140 align:middle line:84% Well, the amount of substance i divided by n 00:40:11.140 --> 00:40:15.245 align:middle line:84% is yi, the mole fraction of that constituent. 00:40:15.245 --> 00:40:20.140 align:middle line:90% 00:40:20.140 --> 00:40:26.110 align:middle line:90% So now, I'm going to express. 00:40:26.110 --> 00:40:34.720 align:middle line:84% I want to see how the specific molar energy that is-- that 00:40:34.720 --> 00:40:37.490 align:middle line:84% represents how much energy I have 00:40:37.490 --> 00:40:42.810 align:middle line:90% per unit number of particles. 00:40:42.810 --> 00:40:49.810 align:middle line:84% Instead of n, I put at the denominator the m, the mass. 00:40:49.810 --> 00:40:53.640 align:middle line:84% That would be the specific energy 00:40:53.640 --> 00:40:56.290 align:middle line:84% per unit mass, the amount of energy 00:40:56.290 --> 00:40:59.650 align:middle line:84% I have per kilogram of substance. 00:40:59.650 --> 00:41:07.540 align:middle line:90% 00:41:07.540 --> 00:41:17.005 align:middle line:84% So from the definition, U is a function of S, V, and n. 00:41:17.005 --> 00:41:24.280 align:middle line:84% But S can be written here as small s, specific entropy 00:41:24.280 --> 00:41:28.480 align:middle line:90% times the amount, n times s. 00:41:28.480 --> 00:41:32.300 align:middle line:90% 00:41:32.300 --> 00:41:37.340 align:middle line:84% Also the volume can be written as n times lowercase v, 00:41:37.340 --> 00:41:41.520 align:middle line:84% and the amounts can be written as n times the mole fractions. 00:41:41.520 --> 00:41:44.490 align:middle line:90% 00:41:44.490 --> 00:41:47.020 align:middle line:84% So if I look at this expression here, 00:41:47.020 --> 00:41:55.340 align:middle line:84% I see that the specific energy does 00:41:55.340 --> 00:41:58.590 align:middle line:84% depend on the specific entropy, specific volume, 00:41:58.590 --> 00:42:02.780 align:middle line:84% the mole fraction, but it also depends on n 00:42:02.780 --> 00:42:07.690 align:middle line:84% because n appears in several places here. 00:42:07.690 --> 00:42:13.124 align:middle line:90% 00:42:13.124 --> 00:42:18.170 align:middle line:84% So that means, for example, if I have one kilogram, a given 00:42:18.170 --> 00:42:26.610 align:middle line:84% amount of substance, that has a certain specific energy. 00:42:26.610 --> 00:42:33.670 align:middle line:84% But if have 2 kilograms, the energy per kilogram 00:42:33.670 --> 00:42:38.270 align:middle line:84% may not be the same, if it depends on n. 00:42:38.270 --> 00:42:39.770 align:middle line:90% It's like here. 00:42:39.770 --> 00:42:45.728 align:middle line:84% I have-- If I have one particle, it has this probability 00:42:45.728 --> 00:42:46.270 align:middle line:90% distribution. 00:42:46.270 --> 00:42:53.120 align:middle line:90% 00:42:53.120 --> 00:43:00.710 align:middle line:84% If I have two particles, their properties 00:43:00.710 --> 00:43:04.890 align:middle line:84% will not scale down to the same probability 00:43:04.890 --> 00:43:06.510 align:middle line:90% distribution per unit particle. 00:43:06.510 --> 00:43:14.970 align:middle line:90% 00:43:14.970 --> 00:43:19.650 align:middle line:84% So for small systems, this dependence on n is there, 00:43:19.650 --> 00:43:22.620 align:middle line:84% and it is represented by the fundamental relation. 00:43:22.620 --> 00:43:26.390 align:middle line:90% So you need to measure it. 00:43:26.390 --> 00:43:29.460 align:middle line:84% But for small systems, you have to measure-- 00:43:29.460 --> 00:43:32.850 align:middle line:84% it's not enough to measure the unit of a substance, 00:43:32.850 --> 00:43:34.150 align:middle line:90% like 1 kilogram. 00:43:34.150 --> 00:43:37.890 align:middle line:84% When you measure the properties of water for the steam tables, 00:43:37.890 --> 00:43:40.150 align:middle line:84% you measure 1 kilogram, and you're done. 00:43:40.150 --> 00:43:41.710 align:middle line:84% Because when you have 2 kilograms, 00:43:41.710 --> 00:43:43.710 align:middle line:90% you just multiply by 2. 00:43:43.710 --> 00:43:46.290 align:middle line:84% That's good in the macroscopic limit. 00:43:46.290 --> 00:43:53.480 align:middle line:84% But when you have 200 molecules of water, 00:43:53.480 --> 00:44:00.860 align:middle line:84% it's not just twice the properties of 100 molecules. 00:44:00.860 --> 00:44:04.730 align:middle line:84% You have to see how those properties depend also 00:44:04.730 --> 00:44:07.030 align:middle line:90% on the amount. 00:44:07.030 --> 00:44:11.440 align:middle line:84% But in the limit of many particles, 00:44:11.440 --> 00:44:14.170 align:middle line:90% that dependence vanishes. 00:44:14.170 --> 00:44:18.360 align:middle line:84% So this proof, which takes again the partial derivative of U 00:44:18.360 --> 00:44:19.620 align:middle line:90% with respect to n. 00:44:19.620 --> 00:44:24.650 align:middle line:84% So you want to see how much is it. 00:44:24.650 --> 00:44:32.480 align:middle line:84% Well, that dependence is equal to-- if you do the derivatives-- 00:44:32.480 --> 00:44:35.155 align:middle line:90% and let you do it-- 00:44:35.155 --> 00:44:38.080 align:middle line:90% 00:44:38.080 --> 00:44:41.916 align:middle line:84% it's minus 1 over n squared times-- 00:44:41.916 --> 00:44:45.930 align:middle line:90% you remember what was this? 00:44:45.930 --> 00:44:53.000 align:middle line:84% You should add there it's the Euler characteristic function. 00:44:53.000 --> 00:44:55.610 align:middle line:84% That term in parentheses is what we called the Euler 00:44:55.610 --> 00:44:56.810 align:middle line:90% characteristic function. 00:44:56.810 --> 00:45:07.360 align:middle line:84% But here, what we say is that the way the specific energy 00:45:07.360 --> 00:45:09.100 align:middle line:84% of the system at stable equilibrium 00:45:09.100 --> 00:45:11.535 align:middle line:84% depends on the number of particles. 00:45:11.535 --> 00:45:18.210 align:middle line:90% 00:45:18.210 --> 00:45:23.550 align:middle line:84% It is measured by the Euler characteristic function. 00:45:23.550 --> 00:45:26.820 align:middle line:84% But in the limit as n goes to infinity-- 00:45:26.820 --> 00:45:28.950 align:middle line:90% so that's the simple system-- 00:45:28.950 --> 00:45:35.320 align:middle line:84% it goes to 0, because the Euler relation just 00:45:35.320 --> 00:45:39.955 align:middle line:90% says that this is equal to 0. 00:45:39.955 --> 00:45:45.430 align:middle line:84% That relation here, or what we call the Euler relation, 00:45:45.430 --> 00:45:49.740 align:middle line:84% is equivalent to saying Euler characteristic function equals 00:45:49.740 --> 00:45:50.240 align:middle line:90% 0. 00:45:50.240 --> 00:45:54.040 align:middle line:90% 00:45:54.040 --> 00:45:56.457 align:middle line:90% So that's what we gain. 00:45:56.457 --> 00:45:57.540 align:middle line:90% We gain one more relation. 00:45:57.540 --> 00:46:00.120 align:middle line:90% 00:46:00.120 --> 00:46:02.100 align:middle line:84% And you know-- we have obtained already 00:46:02.100 --> 00:46:05.190 align:middle line:84% so many things from just watching 00:46:05.190 --> 00:46:07.780 align:middle line:90% the fundamental relation. 00:46:07.780 --> 00:46:09.180 align:middle line:90% Now we have another one. 00:46:09.180 --> 00:46:11.790 align:middle line:84% You can imagine that you can expect also 00:46:11.790 --> 00:46:14.830 align:middle line:90% a lot of additional information. 00:46:14.830 --> 00:46:20.640 align:middle line:84% For example, one way which is now equivalent to the Euler 00:46:20.640 --> 00:46:26.570 align:middle line:84% relation is to take its differential 00:46:26.570 --> 00:46:30.420 align:middle line:84% and subtract from it the Gibbs relation, which 00:46:30.420 --> 00:46:33.820 align:middle line:84% is the differential of the fundamental relation. 00:46:33.820 --> 00:46:41.615 align:middle line:84% The result is called Gibbs-Duhem relation, and it is this one. 00:46:41.615 --> 00:46:44.260 align:middle line:90% It's a relation between-- 00:46:44.260 --> 00:46:50.940 align:middle line:84% in which you see the independent differentials 00:46:50.940 --> 00:46:55.210 align:middle line:84% are temperature, pressure, and the chemical potentials. 00:46:55.210 --> 00:47:01.670 align:middle line:84% So these are the properties that enter in the conditions 00:47:01.670 --> 00:47:03.990 align:middle line:90% for mutual equilibrium. 00:47:03.990 --> 00:47:29.050 align:middle line:90% 00:47:29.050 --> 00:47:35.870 align:middle line:84% Once we have this idea, we can also 00:47:35.870 --> 00:47:41.380 align:middle line:84% proceed with defining another kind of interaction, which 00:47:41.380 --> 00:47:43.580 align:middle line:90% is the bulk flow interaction. 00:47:43.580 --> 00:47:48.314 align:middle line:90% 00:47:48.314 --> 00:47:51.040 align:middle line:84% Which uses this idea that now that I 00:47:51.040 --> 00:47:59.530 align:middle line:84% have a system of many particles, and I am within that model, 00:47:59.530 --> 00:48:03.070 align:middle line:84% I can consider a fluid element, and I 00:48:03.070 --> 00:48:06.190 align:middle line:84% can assign to it properties as if it were 00:48:06.190 --> 00:48:08.190 align:middle line:90% separated by physical walls. 00:48:08.190 --> 00:48:11.360 align:middle line:90% 00:48:11.360 --> 00:48:16.250 align:middle line:84% Instead of doing it like a cube, I consider a sphere, 00:48:16.250 --> 00:48:18.490 align:middle line:90% but that's the same idea. 00:48:18.490 --> 00:48:26.040 align:middle line:84% So suppose that locally I have a fluid element, 00:48:26.040 --> 00:48:28.840 align:middle line:84% and it is in a stable equilibrium state. 00:48:28.840 --> 00:48:32.260 align:middle line:90% 00:48:32.260 --> 00:48:37.310 align:middle line:84% This little box, it contains enough particles 00:48:37.310 --> 00:48:41.170 align:middle line:84% so that the wall rarefaction effects are not important. 00:48:41.170 --> 00:48:47.880 align:middle line:84% This little box is in a stable equilibrium. 00:48:47.880 --> 00:48:55.440 align:middle line:84% Now, if I look at that fluid element 00:48:55.440 --> 00:49:01.940 align:middle line:84% from a different frame of reference, 00:49:01.940 --> 00:49:06.490 align:middle line:84% for example one that moves, I look at it 00:49:06.490 --> 00:49:10.190 align:middle line:84% from the frame of reference that moves this way. 00:49:10.190 --> 00:49:12.940 align:middle line:90% It goes the other way. 00:49:12.940 --> 00:49:14.800 align:middle line:84% It acquires a velocity with respect 00:49:14.800 --> 00:49:18.320 align:middle line:90% to the new frame of reference. 00:49:18.320 --> 00:49:22.975 align:middle line:84% And associated with that velocity is a kinetic energy. 00:49:22.975 --> 00:49:28.210 align:middle line:90% 00:49:28.210 --> 00:49:33.550 align:middle line:84% In addition, if I look at it from an accelerated reference 00:49:33.550 --> 00:49:45.180 align:middle line:84% frame that has the same acceleration as the gravity, g, 00:49:45.180 --> 00:49:51.580 align:middle line:84% then this applies also a potential energy. 00:49:51.580 --> 00:49:53.500 align:middle line:90% This is Mechanics. 00:49:53.500 --> 00:49:57.660 align:middle line:90% 00:49:57.660 --> 00:49:59.850 align:middle line:90% So the idea is this. 00:49:59.850 --> 00:50:06.050 align:middle line:84% Now, in fluid mechanics we consider interactions 00:50:06.050 --> 00:50:10.860 align:middle line:84% in which particles cross the boundary between two systems, 00:50:10.860 --> 00:50:13.428 align:middle line:90% there is movement. 00:50:13.428 --> 00:50:18.030 align:middle line:84% If we choose as reference frame that 00:50:18.030 --> 00:50:24.100 align:middle line:84% of the boundary where the particles are crossing, 00:50:24.100 --> 00:50:27.750 align:middle line:90% then they have velocity. 00:50:27.750 --> 00:50:29.630 align:middle line:84% But for each of these particles, I 00:50:29.630 --> 00:50:32.920 align:middle line:84% can always find the reference frame in which it is not moving. 00:50:32.920 --> 00:50:36.900 align:middle line:90% 00:50:36.900 --> 00:50:40.500 align:middle line:84% And even if it is in presence of a gravitational field, 00:50:40.500 --> 00:50:45.750 align:middle line:84% for each of these particles, you can find an accelerated frame 00:50:45.750 --> 00:50:48.090 align:middle line:90% where it is not moving. 00:50:48.090 --> 00:50:50.970 align:middle line:90% 00:50:50.970 --> 00:50:53.310 align:middle line:84% With respect to those frames there 00:50:53.310 --> 00:50:57.040 align:middle line:84% is no kinetic energy and no potential energy. 00:50:57.040 --> 00:51:00.600 align:middle line:84% So somehow, this kinetic energy and potential energy 00:51:00.600 --> 00:51:05.010 align:middle line:84% are effects that can be added or removed 00:51:05.010 --> 00:51:07.120 align:middle line:84% by just changing the reference frame. 00:51:07.120 --> 00:51:12.955 align:middle line:90% 00:51:12.955 --> 00:51:15.860 align:middle line:84% Clearly, that doesn't change the internal properties 00:51:15.860 --> 00:51:17.930 align:middle line:90% of the stuff. 00:51:17.930 --> 00:51:20.060 align:middle line:90% It doesn't change the energy. 00:51:20.060 --> 00:51:22.340 align:middle line:84% Yes, it does change the energy, but it 00:51:22.340 --> 00:51:27.994 align:middle line:84% doesn't change the other properties like entropy. 00:51:27.994 --> 00:51:30.585 align:middle line:90% 00:51:30.585 --> 00:51:33.870 align:middle line:84% So this is the model that we make-- 00:51:33.870 --> 00:51:38.760 align:middle line:84% that we call bulk-flow state-- bulk-flow local-equilibrium 00:51:38.760 --> 00:51:41.640 align:middle line:90% state. 00:51:41.640 --> 00:51:47.460 align:middle line:84% We assign to the fluid element, the properties 00:51:47.460 --> 00:51:50.800 align:middle line:90% of a local stable equilibrium. 00:51:50.800 --> 00:51:55.030 align:middle line:90% 00:51:55.030 --> 00:51:57.730 align:middle line:84% Or we assume that there is some reference frame with respect 00:51:57.730 --> 00:52:06.940 align:middle line:84% to which it is not moving and has no potential energy. 00:52:06.940 --> 00:52:10.360 align:middle line:84% We subtract, in a sense, the kinetic energy 00:52:10.360 --> 00:52:13.240 align:middle line:84% and the potential energy with respect to the laboratory 00:52:13.240 --> 00:52:20.080 align:middle line:84% reference frame, so that we get an energy which 00:52:20.080 --> 00:52:25.220 align:middle line:84% satisfies the fundamental relation of stable equilibrium. 00:52:25.220 --> 00:52:31.470 align:middle line:84% In that sense, it has temperature. 00:52:31.470 --> 00:52:34.230 align:middle line:90% It has a pressure. 00:52:34.230 --> 00:52:38.218 align:middle line:84% It has all the slopes of the fundamental relation. 00:52:38.218 --> 00:52:42.420 align:middle line:90% 00:52:42.420 --> 00:52:46.390 align:middle line:90% So that's step one-- 00:52:46.390 --> 00:52:51.570 align:middle line:84% defines a local-equilibrium model. 00:52:51.570 --> 00:52:56.620 align:middle line:84% Step two is that if you have a boundary between two systems-- 00:52:56.620 --> 00:52:58.260 align:middle line:84% for example, you have a pipe that 00:52:58.260 --> 00:53:01.945 align:middle line:84% brings fluid elements inside your system-- 00:53:01.945 --> 00:53:05.950 align:middle line:90% 00:53:05.950 --> 00:53:08.680 align:middle line:84% well, usually, how do we-- this is an open system 00:53:08.680 --> 00:53:11.940 align:middle line:90% we call it in open system. 00:53:11.940 --> 00:53:25.610 align:middle line:84% And we usually represent as a boundary-- a formal boundary-- 00:53:25.610 --> 00:53:28.850 align:middle line:84% we imagine-- in our mind-- there is 00:53:28.850 --> 00:53:32.570 align:middle line:84% a boundary between the pipe and the system, 00:53:32.570 --> 00:53:35.220 align:middle line:84% but this boundary isn't physical. 00:53:35.220 --> 00:53:40.210 align:middle line:84% Yet in order to model the interaction, 00:53:40.210 --> 00:53:42.350 align:middle line:90% we suppose it is physical. 00:53:42.350 --> 00:53:46.950 align:middle line:84% Suppose we can put a physical partition, which 00:53:46.950 --> 00:53:49.690 align:middle line:84% we are allowed because we are in the simple system model. 00:53:49.690 --> 00:53:53.550 align:middle line:90% 00:53:53.550 --> 00:54:02.450 align:middle line:84% So let me put at time instant t-- 00:54:02.450 --> 00:54:11.420 align:middle line:84% let me put a partition that separates that amount of fluid 00:54:11.420 --> 00:54:17.600 align:middle line:84% that is going to cross this boundary in the next time 00:54:17.600 --> 00:54:21.230 align:middle line:90% interval delta t, or dt. 00:54:21.230 --> 00:54:23.482 align:middle line:90% So now it's here. 00:54:23.482 --> 00:54:25.810 align:middle line:90% Later, it's there. 00:54:25.810 --> 00:54:31.310 align:middle line:84% And it's physically separated by a partition. 00:54:31.310 --> 00:54:34.680 align:middle line:84% At no cost, I have put in that partition. 00:54:34.680 --> 00:54:40.500 align:middle line:90% 00:54:40.500 --> 00:54:43.130 align:middle line:90% Then what happens? 00:54:43.130 --> 00:54:51.910 align:middle line:84% If I look from the fluid element, the fluid parcel that 00:54:51.910 --> 00:54:55.330 align:middle line:84% is crossing here, from a reference frame that 00:54:55.330 --> 00:55:00.520 align:middle line:84% moves with its velocity, I am assuming for simplicity 00:55:00.520 --> 00:55:03.540 align:middle line:84% here that there is a flat velocity profile, 00:55:03.540 --> 00:55:07.030 align:middle line:84% all the particles move with the same velocity. 00:55:07.030 --> 00:55:10.600 align:middle line:84% Or if not, you should make it for smaller parcels 00:55:10.600 --> 00:55:15.260 align:middle line:84% to also single out the velocity profile. 00:55:15.260 --> 00:55:16.940 align:middle line:84% But for simplicity, let's suppose 00:55:16.940 --> 00:55:19.080 align:middle line:90% we look at just one element. 00:55:19.080 --> 00:55:21.560 align:middle line:90% And so that has its velocity. 00:55:21.560 --> 00:55:26.440 align:middle line:84% And we look at it from the reference frame from which 00:55:26.440 --> 00:55:28.515 align:middle line:90% you see it as not moving. 00:55:28.515 --> 00:55:34.330 align:middle line:90% 00:55:34.330 --> 00:55:39.420 align:middle line:84% This partition that you put in is a partition 00:55:39.420 --> 00:55:48.680 align:middle line:84% that separates two systems that are in mutual equilibrium. 00:55:48.680 --> 00:55:54.020 align:middle line:84% They have the same pressure and temperature. 00:55:54.020 --> 00:55:58.430 align:middle line:84% In particular, the pressure is the same at partition. 00:55:58.430 --> 00:56:00.680 align:middle line:84% Even if it could move, it wouldn't 00:56:00.680 --> 00:56:04.760 align:middle line:84% move because all the fluids that are coming in the pipe here 00:56:04.760 --> 00:56:06.625 align:middle line:90% have the same pressure. 00:56:06.625 --> 00:56:09.960 align:middle line:90% 00:56:09.960 --> 00:56:17.230 align:middle line:84% But if you now make the energy balance-- 00:56:17.230 --> 00:56:20.490 align:middle line:84% and here is the details of making the energy balance 00:56:20.490 --> 00:56:24.750 align:middle line:90% for this composite system of-- 00:56:24.750 --> 00:56:28.650 align:middle line:84% the system which is receiving that amount of fluid 00:56:28.650 --> 00:56:35.760 align:middle line:84% plus this fluid that will eventually get in. 00:56:35.760 --> 00:56:38.510 align:middle line:90% 00:56:38.510 --> 00:56:42.510 align:middle line:84% So now that we have separated it with a physical partition, 00:56:42.510 --> 00:56:47.160 align:middle line:84% this partition makes it a legitimate system. 00:56:47.160 --> 00:56:51.500 align:middle line:84% So the composite system, A plus B prime, 00:56:51.500 --> 00:56:57.120 align:middle line:84% is well-separated from the rest because it has a wall. 00:56:57.120 --> 00:57:00.710 align:middle line:84% And it's always satisfied our definition of system 00:57:00.710 --> 00:57:07.290 align:middle line:84% in which we get external forces with certain kind. 00:57:07.290 --> 00:57:12.632 align:middle line:84% So for these kind of systems, which are closed system now, 00:57:12.632 --> 00:57:17.380 align:middle line:84% we have already introduced the energy balance. 00:57:17.380 --> 00:57:20.880 align:middle line:84% We can do the energy balance, which 00:57:20.880 --> 00:57:25.980 align:middle line:84% involves saying that the final energy minus the initial energy 00:57:25.980 --> 00:57:30.670 align:middle line:84% is equal to the energy exchanged via interactions. 00:57:30.670 --> 00:57:34.230 align:middle line:84% And here, there is only one interaction. 00:57:34.230 --> 00:57:38.640 align:middle line:84% It's the fact that there is a pressure acting 00:57:38.640 --> 00:57:41.490 align:middle line:90% on a moving piston-- 00:57:41.490 --> 00:57:44.240 align:middle line:90% moving boundary. 00:57:44.240 --> 00:57:51.324 align:middle line:84% And that is a work, but this is the work. 00:57:51.324 --> 00:57:56.170 align:middle line:90% 00:57:56.170 --> 00:58:00.505 align:middle line:84% That means that now we play additivity. 00:58:00.505 --> 00:58:03.760 align:middle line:90% 00:58:03.760 --> 00:58:10.000 align:middle line:84% In the final state, we note that in system B prime, 00:58:10.000 --> 00:58:10.980 align:middle line:90% there is nothing left. 00:58:10.980 --> 00:58:14.100 align:middle line:90% 00:58:14.100 --> 00:58:17.760 align:middle line:84% The system B prime is now between this fixed boundary 00:58:17.760 --> 00:58:18.800 align:middle line:90% and the moving boundary. 00:58:18.800 --> 00:58:23.280 align:middle line:90% 00:58:23.280 --> 00:58:28.840 align:middle line:90% If you want, we could say that-- 00:58:28.840 --> 00:58:32.170 align:middle line:84% to be more precise, I should say that at the initial time, 00:58:32.170 --> 00:58:38.370 align:middle line:84% at time t, in order for system B prime to be a system, 00:58:38.370 --> 00:58:41.590 align:middle line:84% I have to have introduced physical boundary here. 00:58:41.590 --> 00:58:44.290 align:middle line:90% 00:58:44.290 --> 00:58:46.840 align:middle line:90% Then I remove it. 00:58:46.840 --> 00:58:49.350 align:middle line:90% I let things go. 00:58:49.350 --> 00:58:56.230 align:middle line:84% And in the end, I will put, again, a physical boundary here. 00:58:56.230 --> 00:58:58.740 align:middle line:84% But there is nothing between the two boundaries. 00:58:58.740 --> 00:59:01.100 align:middle line:84% So the energy and entropy there are 0. 00:59:01.100 --> 00:59:05.000 align:middle line:90% 00:59:05.000 --> 00:59:10.520 align:middle line:84% And again, in order to do the adding or removing partitions, 00:59:10.520 --> 00:59:12.800 align:middle line:84% I need to be within the simple system model. 00:59:12.800 --> 00:59:17.830 align:middle line:90% 00:59:17.830 --> 00:59:21.290 align:middle line:84% So to make the long story shorter, 00:59:21.290 --> 00:59:26.060 align:middle line:84% here is the expression for the work. 00:59:26.060 --> 00:59:34.890 align:middle line:84% It's pressure times the area times the displacement. 00:59:34.890 --> 00:59:38.340 align:middle line:84% And you can show that it is proportional 00:59:38.340 --> 00:59:42.510 align:middle line:84% to the specific volume times the pressure 00:59:42.510 --> 00:59:50.870 align:middle line:84% times the amount of the fluid parcel that is being exchanged. 00:59:50.870 --> 00:59:53.240 align:middle line:84% And in the end of the story, you see 00:59:53.240 --> 00:59:58.970 align:middle line:84% that this work, which we usually call the pulsion work, 00:59:58.970 --> 01:00:03.840 align:middle line:84% adds a contribution to the energy that 01:00:03.840 --> 01:00:06.150 align:middle line:90% is brought into system A. 01:00:06.150 --> 01:00:12.480 align:middle line:84% It's not only the internal energy of the fluid-- 01:00:12.480 --> 01:00:16.030 align:middle line:84% not only its kinetic energy and potential energy, 01:00:16.030 --> 01:00:18.810 align:middle line:90% but also this pulsion work. 01:00:18.810 --> 01:00:25.710 align:middle line:84% And that is why in a bulk-flow interaction, it's u, 01:00:25.710 --> 01:00:29.490 align:middle line:84% internal energy, plus pv, which, if we recall, 01:00:29.490 --> 01:00:31.860 align:middle line:84% it's called it corresponds to the definition 01:00:31.860 --> 01:00:36.210 align:middle line:84% of that characteristic function that we called enthalpy. 01:00:36.210 --> 01:00:40.830 align:middle line:84% That enters into the expression that 01:00:40.830 --> 01:00:45.800 align:middle line:84% says how much energy is transferred 01:00:45.800 --> 01:00:48.830 align:middle line:90% in this kind of interaction. 01:00:48.830 --> 01:00:52.340 align:middle line:84% Of course, you can do the same when the parcel is 01:00:52.340 --> 01:00:54.505 align:middle line:90% going from inside to outside. 01:00:54.505 --> 01:00:57.870 align:middle line:90% 01:00:57.870 --> 01:01:07.220 align:middle line:84% And therefore, we get this summary of defining properties 01:01:07.220 --> 01:01:10.750 align:middle line:90% for the bulk flow interaction. 01:01:10.750 --> 01:01:16.030 align:middle line:84% It contributes mass to the system. 01:01:16.030 --> 01:01:23.010 align:middle line:84% And by the mass balance, you can write it this way. 01:01:23.010 --> 01:01:27.390 align:middle line:84% Energy gives the enthalpy and also entropy. 01:01:27.390 --> 01:01:35.580 align:middle line:84% The entropy is just the entropy that the fluid element simply 01:01:35.580 --> 01:01:38.135 align:middle line:84% translates from outside to inside-- 01:01:38.135 --> 01:01:40.700 align:middle line:90% 01:01:40.700 --> 01:01:41.470 align:middle line:90% translates. 01:01:41.470 --> 01:01:51.490 align:middle line:90% 01:01:51.490 --> 01:01:52.960 align:middle line:90% OK, in/out. 01:01:52.960 --> 01:01:55.410 align:middle line:84% You can also do this per unit time. 01:01:55.410 --> 01:02:02.930 align:middle line:90% 01:02:02.930 --> 01:02:08.450 align:middle line:84% Therefore, now with these additional type of interaction, 01:02:08.450 --> 01:02:14.590 align:middle line:84% you can write balance equations for a system 01:02:14.590 --> 01:02:18.500 align:middle line:84% that is subject to all sorts of interactions. 01:02:18.500 --> 01:02:24.280 align:middle line:84% For example, here, we have heat at temperature Ta. 01:02:24.280 --> 01:02:27.430 align:middle line:84% Typically, we use the letter A or the letter 0 01:02:27.430 --> 01:02:31.550 align:middle line:84% for the environmental conditions. 01:02:31.550 --> 01:02:37.070 align:middle line:84% Here, we have a heat interaction at some other temperature-- 01:02:37.070 --> 01:02:40.220 align:middle line:90% maybe cold. 01:02:40.220 --> 01:02:44.840 align:middle line:90% This one at another temperature. 01:02:44.840 --> 01:02:49.320 align:middle line:84% We may have work interactions of various types, 01:02:49.320 --> 01:02:53.300 align:middle line:84% like a shaft bringing in mechanical energy, 01:02:53.300 --> 01:02:56.550 align:middle line:84% electrical wires bringing in electrical energy. 01:02:56.550 --> 01:02:59.836 align:middle line:90% These are all work equivalent. 01:02:59.836 --> 01:03:06.650 align:middle line:84% Or a moving boundary against the pressure of the atmosphere, 01:03:06.650 --> 01:03:09.810 align:middle line:90% that is also a work interaction. 01:03:09.810 --> 01:03:16.850 align:middle line:84% And plus we may have many bulk-flow interactions. 01:03:16.850 --> 01:03:22.490 align:middle line:84% And as we have seen, in order to characterize a bulk flow 01:03:22.490 --> 01:03:24.380 align:middle line:84% interaction, we need to characterize 01:03:24.380 --> 01:03:27.230 align:middle line:84% the bulk-flow states of the fluid elements that 01:03:27.230 --> 01:03:28.920 align:middle line:90% are carried in and out. 01:03:28.920 --> 01:03:32.570 align:middle line:84% So you need the enthalpy, the local enthalpy, 01:03:32.570 --> 01:03:37.730 align:middle line:84% the local entropy, the velocity of the fluid elements, 01:03:37.730 --> 01:03:40.400 align:middle line:84% and the local height in the gravitational field 01:03:40.400 --> 01:03:43.560 align:middle line:84% so that you can compute the potential energy. 01:03:43.560 --> 01:03:46.520 align:middle line:90% 01:03:46.520 --> 01:03:51.850 align:middle line:84% Of course, subject to all these interactions, 01:03:51.850 --> 01:03:52.725 align:middle line:90% you're going to be-- 01:03:52.725 --> 01:03:59.570 align:middle line:90% 01:03:59.570 --> 01:04:02.780 align:middle line:84% it's not going to be easy to balance them 01:04:02.780 --> 01:04:07.020 align:middle line:84% so that you keep the energy the same, the entropy the same, 01:04:07.020 --> 01:04:09.750 align:middle line:84% the volume the same, the mass the same, 01:04:09.750 --> 01:04:14.200 align:middle line:84% which would be required to keep the system in a steady state. 01:04:14.200 --> 01:04:17.238 align:middle line:90% 01:04:17.238 --> 01:04:21.650 align:middle line:84% But very often, that's exactly what you do. 01:04:21.650 --> 01:04:23.570 align:middle line:84% For example, this is a power plant, 01:04:23.570 --> 01:04:26.690 align:middle line:84% and you want to operate it in such a way 01:04:26.690 --> 01:04:29.060 align:middle line:90% that it doesn't consume itself. 01:04:29.060 --> 01:04:30.540 align:middle line:90% It simply processes stuff. 01:04:30.540 --> 01:04:35.850 align:middle line:84% So you need to balance what goes in with what goes out. 01:04:35.850 --> 01:04:41.910 align:middle line:84% And while you do that, definitely, most of the times, 01:04:41.910 --> 01:04:46.700 align:middle line:84% produce entropy by irreversibility. 01:04:46.700 --> 01:04:51.790 align:middle line:84% These are the balance equations of mass. 01:04:51.790 --> 01:04:57.330 align:middle line:84% Rate of change of the mass is the net mass flow 01:04:57.330 --> 01:05:00.180 align:middle line:90% rates that go in and out. 01:05:00.180 --> 01:05:07.050 align:middle line:84% Here, to simplify the notation, I consider m dot with the arrow 01:05:07.050 --> 01:05:08.430 align:middle line:90% just in. 01:05:08.430 --> 01:05:10.410 align:middle line:90% So it's positive if it's in. 01:05:10.410 --> 01:05:16.110 align:middle line:84% That means that if I want to represent something going out, 01:05:16.110 --> 01:05:19.740 align:middle line:90% I do need that negative. 01:05:19.740 --> 01:05:27.410 align:middle line:84% Or you could split the summation into two sums and put on one sum 01:05:27.410 --> 01:05:31.390 align:middle line:84% the masses that go in and minus the summation of the masses that 01:05:31.390 --> 01:05:33.190 align:middle line:90% go out. 01:05:33.190 --> 01:05:35.410 align:middle line:90% So also this for the energy-- 01:05:35.410 --> 01:05:36.970 align:middle line:84% the rate of change of the energy is 01:05:36.970 --> 01:05:40.240 align:middle line:84% equal to the net amount of energy 01:05:40.240 --> 01:05:42.730 align:middle line:90% transferred by interactions. 01:05:42.730 --> 01:05:45.400 align:middle line:90% Here are the heats, works. 01:05:45.400 --> 01:05:48.510 align:middle line:90% Here are the bulk flows. 01:05:48.510 --> 01:05:49.890 align:middle line:90% And for the entropy-- 01:05:49.890 --> 01:05:54.240 align:middle line:84% the rate of change of the entropy of the system is-- 01:05:54.240 --> 01:05:57.940 align:middle line:84% first term here is the entropy exchanged 01:05:57.940 --> 01:06:03.120 align:middle line:84% either via the bulk flow interactions or via the heats. 01:06:03.120 --> 01:06:05.770 align:middle line:90% 01:06:05.770 --> 01:06:12.965 align:middle line:84% And then, well, how about this sign? 01:06:12.965 --> 01:06:15.490 align:middle line:90% 01:06:15.490 --> 01:06:16.250 align:middle line:90% What do you think? 01:06:16.250 --> 01:06:16.960 align:middle line:90% That's an error! 01:06:16.960 --> 01:06:19.780 align:middle line:90% 01:06:19.780 --> 01:06:24.950 align:middle line:84% The entropy-- plus the entropy produced by irreversibility. 01:06:24.950 --> 01:06:26.420 align:middle line:90% That needs to be corrected. 01:06:26.420 --> 01:06:29.190 align:middle line:90% 01:06:29.190 --> 01:06:36.630 align:middle line:84% I probably left it there to see if I'm awake enough 01:06:36.630 --> 01:06:40.956 align:middle line:84% and you are awake enough to spot it. 01:06:40.956 --> 01:06:43.710 align:middle line:90% So luckily we did. 01:06:43.710 --> 01:06:48.350 align:middle line:84% And once we have these balance equations, 01:06:48.350 --> 01:06:51.250 align:middle line:84% we can manipulate them, solve problems. 01:06:51.250 --> 01:06:54.330 align:middle line:90% 01:06:54.330 --> 01:07:04.830 align:middle line:84% In a large community of engineers 01:07:04.830 --> 01:07:09.425 align:middle line:84% that have played with these equations, 01:07:09.425 --> 01:07:16.790 align:middle line:84% they developed this idea of exergy and exergy analysis 01:07:16.790 --> 01:07:17.905 align:middle line:90% of energy systems. 01:07:17.905 --> 01:07:22.930 align:middle line:90% 01:07:22.930 --> 01:07:28.100 align:middle line:84% It adds nothing to these equations 01:07:28.100 --> 01:07:34.300 align:middle line:84% except perhaps it helps in reading them. 01:07:34.300 --> 01:07:36.820 align:middle line:84% It adds nothing because it simply-- 01:07:36.820 --> 01:07:40.430 align:middle line:84% it is based on a linear combination of these equations. 01:07:40.430 --> 01:07:42.640 align:middle line:84% So mathematically, all the information 01:07:42.640 --> 01:07:45.070 align:middle line:90% is contained in here. 01:07:45.070 --> 01:07:46.180 align:middle line:90% So this is it, basically. 01:07:46.180 --> 01:07:47.830 align:middle line:84% Definitely, we need to make an energy 01:07:47.830 --> 01:07:50.020 align:middle line:90% balance and an entropy balance. 01:07:50.020 --> 01:07:52.390 align:middle line:84% If you don't like working with entropy, 01:07:52.390 --> 01:07:55.750 align:middle line:84% and you prefer to work with a linear combination of energy 01:07:55.750 --> 01:07:58.010 align:middle line:90% and entropy, fine. 01:07:58.010 --> 01:08:00.650 align:middle line:90% Then you call it exergy balance. 01:08:00.650 --> 01:08:03.910 align:middle line:84% And so one way to combine these equations 01:08:03.910 --> 01:08:13.750 align:middle line:84% is to take one of these forms that is still-- 01:08:13.750 --> 01:08:17.130 align:middle line:84% take the second equation, the entropy balance equation, 01:08:17.130 --> 01:08:20.330 align:middle line:90% multiply it by some temperature. 01:08:20.330 --> 01:08:23.350 align:middle line:90% 01:08:23.350 --> 01:08:27.779 align:middle line:84% And it is convenient to choose the temperature of one 01:08:27.779 --> 01:08:32.500 align:middle line:90% of your heat interactions. 01:08:32.500 --> 01:08:35.050 align:middle line:84% And it's even more convenient to choose 01:08:35.050 --> 01:08:37.660 align:middle line:84% the temperature of the heat interaction 01:08:37.660 --> 01:08:42.370 align:middle line:84% that is considered free of charge. 01:08:42.370 --> 01:08:48.220 align:middle line:84% For example, if your machinery operates in an environment 01:08:48.220 --> 01:08:51.760 align:middle line:90% like air-- 01:08:51.760 --> 01:08:58.529 align:middle line:84% your air conditioning at home works in the air and dumps 01:08:58.529 --> 01:09:02.500 align:middle line:90% heat in the environment. 01:09:02.500 --> 01:09:08.370 align:middle line:84% I would use that temperature as the temperature 01:09:08.370 --> 01:09:11.510 align:middle line:84% that I use in this linear combination. 01:09:11.510 --> 01:09:15.080 align:middle line:84% If instead your power plant is cooled 01:09:15.080 --> 01:09:19.651 align:middle line:84% by the water of the river, I would use 01:09:19.651 --> 01:09:20.859 align:middle line:90% the temperature of the river. 01:09:20.859 --> 01:09:24.170 align:middle line:90% 01:09:24.170 --> 01:09:28.560 align:middle line:84% In any case, you take that, multiply, and then subtract it 01:09:28.560 --> 01:09:29.659 align:middle line:90% from the energy balance. 01:09:29.659 --> 01:09:39.519 align:middle line:90% 01:09:39.519 --> 01:09:40.439 align:middle line:90% Yeah, see? 01:09:40.439 --> 01:09:44.702 align:middle line:84% Here also, you have to be a little-- 01:09:44.702 --> 01:09:48.479 align:middle line:90% 01:09:48.479 --> 01:09:50.609 align:middle line:84% I should have been a little more careful 01:09:50.609 --> 01:09:54.605 align:middle line:90% in making explicitly the work-- 01:09:54.605 --> 01:09:57.540 align:middle line:90% 01:09:57.540 --> 01:10:01.800 align:middle line:84% write separately the work related to the p times 01:10:01.800 --> 01:10:03.210 align:middle line:90% rate of change of volume. 01:10:03.210 --> 01:10:07.770 align:middle line:84% The pa times the rate of change of volume 01:10:07.770 --> 01:10:11.740 align:middle line:84% should have been singled out here. 01:10:11.740 --> 01:10:19.110 align:middle line:84% Maybe I'll do it later on the slide-- just for the slides, 01:10:19.110 --> 01:10:23.460 align:middle line:84% when I correct also the minus sign here-- 01:10:23.460 --> 01:10:30.450 align:middle line:84% so that that work term can go on this side of the equation 01:10:30.450 --> 01:10:34.210 align:middle line:84% and make this pa times dV over dt. 01:10:34.210 --> 01:10:40.590 align:middle line:90% 01:10:40.590 --> 01:10:46.980 align:middle line:84% So here is an exergy balance equation. 01:10:46.980 --> 01:10:55.142 align:middle line:84% And the exergy is a topic that we need to discuss. 01:10:55.142 --> 01:10:57.970 align:middle line:84% I will not make the story very long, 01:10:57.970 --> 01:11:04.920 align:middle line:84% but it's part of thermodynamics, so it is a good idea 01:11:04.920 --> 01:11:06.060 align:middle line:90% to discuss it-- 01:11:06.060 --> 01:11:09.150 align:middle line:84% also because it brings several useful ideas 01:11:09.150 --> 01:11:16.090 align:middle line:84% about how to define efficiencies of energy conversion devices. 01:11:16.090 --> 01:11:22.520 align:middle line:84% And so it is important in a mechanical engineering 01:11:22.520 --> 01:11:23.250 align:middle line:90% department. 01:11:23.250 --> 01:11:31.910 align:middle line:90% 01:11:31.910 --> 01:11:36.320 align:middle line:84% We already have seen that combinations like that-- 01:11:36.320 --> 01:11:39.275 align:middle line:84% I think we use the letter capital Psi to represent 01:11:39.275 --> 01:11:42.710 align:middle line:90% this availability function-- 01:11:42.710 --> 01:11:46.100 align:middle line:90% E minus TRS sub plus PRV. 01:11:46.100 --> 01:11:49.220 align:middle line:90% 01:11:49.220 --> 01:11:54.305 align:middle line:84% So this says that's the rate of change of the availability. 01:11:54.305 --> 01:11:58.490 align:middle line:84% But exergy and availability are the same idea, same concept. 01:11:58.490 --> 01:12:00.055 align:middle line:84% There's two words for the same idea. 01:12:00.055 --> 01:12:04.990 align:middle line:90% 01:12:04.990 --> 01:12:08.830 align:middle line:90% Heat times Carnot coefficient-- 01:12:08.830 --> 01:12:11.960 align:middle line:90% 01:12:11.960 --> 01:12:15.680 align:middle line:84% and the Carnot coefficient is between temperature 01:12:15.680 --> 01:12:25.890 align:middle line:84% of what I've chosen as my environment and the temperature 01:12:25.890 --> 01:12:29.810 align:middle line:90% at which the heat is exchanged. 01:12:29.810 --> 01:12:36.000 align:middle line:84% This represents, for a given Carnot engine, 01:12:36.000 --> 01:12:39.950 align:middle line:84% the maximum work that you could extract by exploiting that heat 01:12:39.950 --> 01:12:45.680 align:middle line:84% interaction, having available an environment at temperatures T0. 01:12:45.680 --> 01:12:52.190 align:middle line:84% And so this is the exergy associated with that heat 01:12:52.190 --> 01:12:56.800 align:middle line:90% interaction at temperature Tk. 01:12:56.800 --> 01:13:01.320 align:middle line:84% Work appears directly-- not multiplied, not corrected-- 01:13:01.320 --> 01:13:05.070 align:middle line:84% because work is already available to-- 01:13:05.070 --> 01:13:07.350 align:middle line:84% let's say if you have some work, you 01:13:07.350 --> 01:13:09.330 align:middle line:90% can do useful things directly. 01:13:09.330 --> 01:13:10.780 align:middle line:90% You don't have to convert. 01:13:10.780 --> 01:13:16.680 align:middle line:84% The work is already, in a sense, exergy, 01:13:16.680 --> 01:13:24.580 align:middle line:84% because the word "exergy" really means the essential part 01:13:24.580 --> 01:13:25.610 align:middle line:90% of the energy. 01:13:25.610 --> 01:13:32.890 align:middle line:84% So the useful fraction of the energy that-- 01:13:32.890 --> 01:13:42.930 align:middle line:84% it's either within your system or an energy transferred. 01:13:42.930 --> 01:13:47.250 align:middle line:84% So the concept applies both for the additive properties 01:13:47.250 --> 01:13:53.450 align:middle line:84% of the system, like energy is, or the modes of energy 01:13:53.450 --> 01:13:54.240 align:middle line:90% transfer-- 01:13:54.240 --> 01:13:56.800 align:middle line:90% 01:13:56.800 --> 01:14:03.590 align:middle line:84% heat, work, and here is also the exergy expression 01:14:03.590 --> 01:14:06.780 align:middle line:90% for the bulk-flow interactions. 01:14:06.780 --> 01:14:09.705 align:middle line:90% 01:14:09.705 --> 01:14:11.790 align:middle line:90% And out of this-- 01:14:11.790 --> 01:14:13.500 align:middle line:90% do the proper sign here-- 01:14:13.500 --> 01:14:15.060 align:middle line:90% you get a minus-- 01:14:15.060 --> 01:14:20.310 align:middle line:84% and here's correct-- minus T0 times S dot irreversible. 01:14:20.310 --> 01:14:29.820 align:middle line:84% This is the dissipation, the terms that represents 01:14:29.820 --> 01:14:32.250 align:middle line:90% the destruction of exergy. 01:14:32.250 --> 01:14:36.240 align:middle line:90% 01:14:36.240 --> 01:14:46.815 align:middle line:84% And like entropy cannot be destroyed, 01:14:46.815 --> 01:14:50.520 align:middle line:90% exergy cannot be created. 01:14:50.520 --> 01:14:55.720 align:middle line:90% You can only destroy it. 01:14:55.720 --> 01:15:00.190 align:middle line:84% So they are related, so irreversibility. 01:15:00.190 --> 01:15:04.630 align:middle line:84% If you do things reversibly, you don't create entropy, 01:15:04.630 --> 01:15:08.340 align:middle line:90% and you don't destroy exergy. 01:15:08.340 --> 01:15:11.820 align:middle line:84% If you do things irreversibly, you create entropy 01:15:11.820 --> 01:15:14.640 align:middle line:90% and destroy some exergy. 01:15:14.640 --> 01:15:15.520 align:middle line:90% That's the idea. 01:15:15.520 --> 01:15:23.140 align:middle line:90% 01:15:23.140 --> 01:15:30.230 align:middle line:84% And so since we are at the business of exergy, 01:15:30.230 --> 01:15:31.620 align:middle line:90% let's just do it. 01:15:31.620 --> 01:15:41.110 align:middle line:84% We have already done, a few lectures ago, 01:15:41.110 --> 01:15:43.300 align:middle line:84% the first-law and second-law efficiencies 01:15:43.300 --> 01:15:48.250 align:middle line:84% for heat engines, refrigerator units, and heat pumps. 01:15:48.250 --> 01:15:50.500 align:middle line:84% This is just a summary of the same equations 01:15:50.500 --> 01:15:52.250 align:middle line:84% that we have already seen up there. 01:15:52.250 --> 01:15:58.070 align:middle line:84% Just notice that the first-law efficiencies 01:15:58.070 --> 01:16:00.230 align:middle line:84% are a ratio where you put at the numerator 01:16:00.230 --> 01:16:03.830 align:middle line:84% an energy, and at the denominator another energy. 01:16:03.830 --> 01:16:07.400 align:middle line:84% Typically the numerator is the useful effect that you want, 01:16:07.400 --> 01:16:10.460 align:middle line:84% and the denominator is the resource 01:16:10.460 --> 01:16:15.100 align:middle line:90% that you are using, energy-wise. 01:16:15.100 --> 01:16:20.350 align:middle line:84% Instead, second-law efficiency, you 01:16:20.350 --> 01:16:23.410 align:middle line:84% put at the numerator, what you want to achieve, 01:16:23.410 --> 01:16:34.230 align:middle line:84% and at the denominator, the minimum expenditure that you 01:16:34.230 --> 01:16:39.530 align:middle line:84% need to pay in order to achieve that, which is typically 01:16:39.530 --> 01:16:42.600 align:middle line:84% done-- minimum in the sense of thermodynamics, of course. 01:16:42.600 --> 01:16:49.560 align:middle line:90% And so it's the reversible case. 01:16:49.560 --> 01:16:54.400 align:middle line:84% Not saying this is the practical optimum. 01:16:54.400 --> 01:17:02.510 align:middle line:84% Because in practice, you don't want to operate reversibly. 01:17:02.510 --> 01:17:06.830 align:middle line:84% Your objective is rarely to operate reversibly. 01:17:06.830 --> 01:17:11.390 align:middle line:84% Because if you operate reversibly, most of the times, 01:17:11.390 --> 01:17:15.125 align:middle line:84% you have to do it in such limiting conditions 01:17:15.125 --> 01:17:16.000 align:middle line:90% that you get nothing. 01:17:16.000 --> 01:17:21.555 align:middle line:90% 01:17:21.555 --> 01:17:25.380 align:middle line:84% So in practice, what you want to optimize 01:17:25.380 --> 01:17:36.290 align:middle line:84% is power, or a combination of power and efficiency, 01:17:36.290 --> 01:17:42.750 align:middle line:84% depending on the cost of equipment. 01:17:42.750 --> 01:17:45.930 align:middle line:84% Because of course, the more you put in, 01:17:45.930 --> 01:17:49.540 align:middle line:84% in terms of sophisticated equipment, 01:17:49.540 --> 01:17:54.610 align:middle line:84% you may achieve higher efficiencies. 01:17:54.610 --> 01:18:05.660 align:middle line:84% But the cost may kill you, so you have to compromise. 01:18:05.660 --> 01:18:08.740 align:middle line:90% 01:18:08.740 --> 01:18:20.040 align:middle line:84% Therefore, there are other terms in the equation of optimization 01:18:20.040 --> 01:18:22.570 align:middle line:90% in terms of practicality. 01:18:22.570 --> 01:18:25.455 align:middle line:84% So here, we are just talking about the second-- 01:18:25.455 --> 01:18:27.810 align:middle line:84% this is the limitation of the second-law efficiencies 01:18:27.810 --> 01:18:31.170 align:middle line:84% that it gives in denominator, which 01:18:31.170 --> 01:18:37.050 align:middle line:84% is your term of comparison, the reversible process, which may be 01:18:37.050 --> 01:18:40.810 align:middle line:90% too far away from practicality. 01:18:40.810 --> 01:18:46.950 align:middle line:84% One sort of halfway solution would 01:18:46.950 --> 01:18:52.230 align:middle line:84% be to put there the work required 01:18:52.230 --> 01:18:58.140 align:middle line:84% by the best available technology to do that particular job. 01:18:58.140 --> 01:19:04.750 align:middle line:84% In that way, at least you compare the efficiency 01:19:04.750 --> 01:19:07.810 align:middle line:84% of your device with the efficiency of the best available 01:19:07.810 --> 01:19:08.480 align:middle line:90% technology. 01:19:08.480 --> 01:19:11.660 align:middle line:90% 01:19:11.660 --> 01:19:13.860 align:middle line:84% And of course, if your device is better, 01:19:13.860 --> 01:19:18.670 align:middle line:84% your efficiency will be higher, higher than 1. 01:19:18.670 --> 01:19:24.320 align:middle line:84% And eventually, your technology will impact 01:19:24.320 --> 01:19:27.330 align:middle line:84% and will become the new best available technology. 01:19:27.330 --> 01:19:31.940 align:middle line:90% 01:19:31.940 --> 01:19:39.160 align:middle line:84% So this viewgraph simply summarizes again 01:19:39.160 --> 01:19:42.970 align:middle line:84% the idea of first law, where energy associated 01:19:42.970 --> 01:19:47.980 align:middle line:84% with the useful effects divided by the energy associated 01:19:47.980 --> 01:19:51.670 align:middle line:84% with the resource that you have used, 01:19:51.670 --> 01:19:56.835 align:middle line:84% which could be burning a fuel or using solar energy. 01:19:56.835 --> 01:19:59.460 align:middle line:90% 01:19:59.460 --> 01:20:05.860 align:middle line:84% And here, for the second law, for the second law efficiency, 01:20:05.860 --> 01:20:09.620 align:middle line:84% you can cast it by saying that I put the-- 01:20:09.620 --> 01:20:18.030 align:middle line:84% that also the exergy associated with the useful effect-- 01:20:18.030 --> 01:20:26.490 align:middle line:84% because the exergy represents the work equivalent 01:20:26.490 --> 01:20:28.290 align:middle line:90% of that useful effect. 01:20:28.290 --> 01:20:31.540 align:middle line:84% And then you put at the denominator 01:20:31.540 --> 01:20:36.090 align:middle line:84% the exergy associated with the resource that you've used. 01:20:36.090 --> 01:20:38.980 align:middle line:90% If you reason it out. 01:20:38.980 --> 01:20:47.700 align:middle line:84% You realize that this is equivalent to putting here 01:20:47.700 --> 01:20:57.500 align:middle line:84% the energy of the useful effect, and below, the minimum 01:20:57.500 --> 01:21:01.280 align:middle line:84% resource you should use in order to obtain that, which 01:21:01.280 --> 01:21:05.300 align:middle line:90% is, I think, written here. 01:21:05.300 --> 01:21:12.320 align:middle line:84% See, this form of the second law efficiency 01:21:12.320 --> 01:21:17.270 align:middle line:84% is also equivalent-- once you do the balance equations and obtain 01:21:17.270 --> 01:21:19.130 align:middle line:84% the expression for the reversible work-- 01:21:19.130 --> 01:21:21.580 align:middle line:90% as a ratio of exergies. 01:21:21.580 --> 01:21:23.510 align:middle line:84% So for example, for the heat engine, 01:21:23.510 --> 01:21:26.970 align:middle line:84% it's the ratio of the exergy of your product-- 01:21:26.970 --> 01:21:29.060 align:middle line:84% which is work, so just pure exergy-- 01:21:29.060 --> 01:21:32.900 align:middle line:84% divided by the exergy of the resource, which 01:21:32.900 --> 01:21:37.870 align:middle line:84% is the heat times the Carnot coefficient. 01:21:37.870 --> 01:21:39.850 align:middle line:84% And notice that the Carnot coefficient 01:21:39.850 --> 01:21:45.710 align:middle line:84% has to be computed using the temperature-- as environment 01:21:45.710 --> 01:21:48.160 align:middle line:84% the temperature of the cold reservoir, which you are using 01:21:48.160 --> 01:21:51.520 align:middle line:84% as an environment or as the thing where 01:21:51.520 --> 01:21:52.875 align:middle line:90% you dump your entropy. 01:21:52.875 --> 01:21:57.350 align:middle line:90% 01:21:57.350 --> 01:22:00.440 align:middle line:84% And similarly, for a refrigerator, 01:22:00.440 --> 01:22:04.440 align:middle line:84% this definition is equivalent to the definition 01:22:04.440 --> 01:22:12.090 align:middle line:84% in which you have exergies numerator and denominator. 01:22:12.090 --> 01:22:17.850 align:middle line:84% Now, for a refrigerator, this is the exergy 01:22:17.850 --> 01:22:24.838 align:middle line:84% of cooling with a cooling load QB times-- this 01:22:24.838 --> 01:22:26.380 align:middle line:84% is equivalent to a Carnot coefficient 01:22:26.380 --> 01:22:29.560 align:middle line:84% because minus signs becomes a Carnot coefficient. 01:22:29.560 --> 01:22:33.080 align:middle line:90% 01:22:33.080 --> 01:22:35.900 align:middle line:84% But here, now you see the Carnot coefficient 01:22:35.900 --> 01:22:38.040 align:middle line:90% with the T at the numerator. 01:22:38.040 --> 01:22:41.630 align:middle line:84% So the T of the environment now is the hot, 01:22:41.630 --> 01:22:48.010 align:middle line:84% because in the refrigerator, you are pumping entropy 01:22:48.010 --> 01:22:51.925 align:middle line:84% to the environment, which is hotter than the refrigerator 01:22:51.925 --> 01:22:56.499 align:middle line:84% cell, and similarly here for the heat pump. 01:22:56.499 --> 01:23:01.190 align:middle line:90% 01:23:01.190 --> 01:23:05.590 align:middle line:84% So going back to this idea of exergy-- 01:23:05.590 --> 01:23:08.160 align:middle line:90% 01:23:08.160 --> 01:23:13.690 align:middle line:84% so the word "exergy," or the word "availability--" 01:23:13.690 --> 01:23:15.610 align:middle line:84% in the old times, there were fights-- 01:23:15.610 --> 01:23:17.310 align:middle line:90% well, how should we call it? 01:23:17.310 --> 01:23:21.580 align:middle line:90% 01:23:21.580 --> 01:23:26.755 align:middle line:84% I participated in conferences where people would really fight. 01:23:26.755 --> 01:23:30.700 align:middle line:90% 01:23:30.700 --> 01:23:37.210 align:middle line:84% Nowadays, I think everybody is more loose about the idea, 01:23:37.210 --> 01:23:40.530 align:middle line:84% and just use exergy or availability-- doesn't really 01:23:40.530 --> 01:23:41.730 align:middle line:90% matter. 01:23:41.730 --> 01:23:43.690 align:middle line:90% It's a generic concept, though. 01:23:43.690 --> 01:23:47.080 align:middle line:90% 01:23:47.080 --> 01:23:53.150 align:middle line:84% So exergy does not have one definition. 01:23:53.150 --> 01:23:56.990 align:middle line:84% You have different forms of exergy. 01:23:56.990 --> 01:24:00.170 align:middle line:84% For example, we encountered many already. 01:24:00.170 --> 01:24:01.970 align:middle line:84% For example, the automatic availability 01:24:01.970 --> 01:24:04.970 align:middle line:90% is a form-- is one exergy. 01:24:04.970 --> 01:24:09.740 align:middle line:84% It's the exergy associated with when you restrict 01:24:09.740 --> 01:24:11.670 align:middle line:90% yourself to weight processes. 01:24:11.670 --> 01:24:14.190 align:middle line:90% 01:24:14.190 --> 01:24:16.480 align:middle line:84% Well, you can call that an adiabatic system. 01:24:16.480 --> 01:24:18.343 align:middle line:84% A system that undergoes only weight 01:24:18.343 --> 01:24:20.010 align:middle line:84% processes, so you can call it adiabatic. 01:24:20.010 --> 01:24:26.110 align:middle line:90% 01:24:26.110 --> 01:24:35.100 align:middle line:84% Or we have seen that if your system is in combination 01:24:35.100 --> 01:24:43.500 align:middle line:84% with a thermal reservoir, the maximum 01:24:43.500 --> 01:24:47.130 align:middle line:84% you can do-- the maximum work you can do 01:24:47.130 --> 01:24:49.942 align:middle line:84% depends on the type of reservoir. 01:24:49.942 --> 01:24:51.940 align:middle line:84% If the reservoir has fixed volume, 01:24:51.940 --> 01:24:57.360 align:middle line:84% we have this that we call available energy, omega R. 01:24:57.360 --> 01:25:00.840 align:middle line:84% But then we have defined, also, available energy with respect 01:25:00.840 --> 01:25:06.830 align:middle line:84% to other kinds of reservoirs, where the reservoir can change 01:25:06.830 --> 01:25:09.910 align:middle line:84% also volume; or maybe the osmotic reservoir, where 01:25:09.910 --> 01:25:13.190 align:middle line:84% it can exchange through a semi-permeable membrane, 01:25:13.190 --> 01:25:18.370 align:middle line:84% only one type of substance; or a completely open exchange 01:25:18.370 --> 01:25:22.780 align:middle line:84% in which your system can exchange 01:25:22.780 --> 01:25:24.495 align:middle line:90% volume and any of the amounts. 01:25:24.495 --> 01:25:27.390 align:middle line:90% 01:25:27.390 --> 01:25:29.600 align:middle line:90% These are all exergies. 01:25:29.600 --> 01:25:33.690 align:middle line:90% Different formulas, same word. 01:25:33.690 --> 01:25:35.720 align:middle line:90% Work is just pure exergy. 01:25:35.720 --> 01:25:39.574 align:middle line:90% 01:25:39.574 --> 01:25:43.010 align:middle line:84% And heat interaction-- just said. 01:25:43.010 --> 01:25:46.250 align:middle line:84% So the exergy associated with the heat interaction 01:25:46.250 --> 01:25:52.940 align:middle line:84% at temperature TQ is given by the heat times the Carnot 01:25:52.940 --> 01:25:53.910 align:middle line:90% coefficient. 01:25:53.910 --> 01:25:56.730 align:middle line:90% 01:25:56.730 --> 01:26:04.610 align:middle line:84% Exergy requires as before, here, requires a reservoir 01:26:04.610 --> 01:26:08.750 align:middle line:90% as reference in some cases. 01:26:08.750 --> 01:26:09.670 align:middle line:90% Not in here. 01:26:09.670 --> 01:26:13.440 align:middle line:90% 01:26:13.440 --> 01:26:17.770 align:middle line:84% For heat interaction, it does require that. 01:26:17.770 --> 01:26:21.870 align:middle line:90% 01:26:21.870 --> 01:26:24.720 align:middle line:84% And you need to specify what kind of-- 01:26:24.720 --> 01:26:27.510 align:middle line:90% 01:26:27.510 --> 01:26:28.860 align:middle line:90% so you need to specify-- 01:26:28.860 --> 01:26:30.250 align:middle line:90% you choose a temperature. 01:26:30.250 --> 01:26:35.270 align:middle line:90% 01:26:35.270 --> 01:26:38.822 align:middle line:84% Exergy is relative to a choice of a particular reference 01:26:38.822 --> 01:26:40.030 align:middle line:90% or environmental temperature. 01:26:40.030 --> 01:26:44.580 align:middle line:90% 01:26:44.580 --> 01:26:48.630 align:middle line:84% Well, even that makes it a broad concept. 01:26:48.630 --> 01:26:57.030 align:middle line:84% So you can define the exergy for a heating/cooling interaction. 01:26:57.030 --> 01:27:01.570 align:middle line:90% 01:27:01.570 --> 01:27:14.360 align:middle line:84% And, for example, suppose you have a machinery that 01:27:14.360 --> 01:27:17.150 align:middle line:90% operates like a Carnot engine. 01:27:17.150 --> 01:27:21.390 align:middle line:84% And this is simple case in which we have two reservoirs. 01:27:21.390 --> 01:27:23.980 align:middle line:90% 01:27:23.980 --> 01:27:31.180 align:middle line:84% But suppose that neither the hot reservoir nor the cold reservoir 01:27:31.180 --> 01:27:33.265 align:middle line:90% are environmental reservoirs. 01:27:33.265 --> 01:27:38.020 align:middle line:90% 01:27:38.020 --> 01:27:46.090 align:middle line:84% For example, this cold reservoir could be your house-- 01:27:46.090 --> 01:27:50.020 align:middle line:84% the house that you want to warm up. 01:27:50.020 --> 01:28:01.040 align:middle line:84% You want to be at 70 Fahrenheit or 21 C in your house. 01:28:01.040 --> 01:28:04.860 align:middle line:84% And one way to heat up your house-- 01:28:04.860 --> 01:28:09.420 align:middle line:84% it's actually one of the best ways from the point of view 01:28:09.420 --> 01:28:11.880 align:middle line:90% of thermodynamics-- 01:28:11.880 --> 01:28:16.310 align:middle line:90% is to use cogeneration. 01:28:16.310 --> 01:28:23.055 align:middle line:84% You use some costly heat source-- 01:28:23.055 --> 01:28:25.750 align:middle line:90% 01:28:25.750 --> 01:28:30.740 align:middle line:84% for example, ok, this is a reservoir, 01:28:30.740 --> 01:28:34.360 align:middle line:84% so it's not exactly a proper example. 01:28:34.360 --> 01:28:47.860 align:middle line:84% But usually, you buy gas to heat up water, 01:28:47.860 --> 01:28:53.710 align:middle line:84% and then you have your water circulating like here-- 01:28:53.710 --> 01:28:55.210 align:middle line:90% heating elements. 01:28:55.210 --> 01:29:00.465 align:middle line:84% And that hot water is going to heat up the air in your room. 01:29:00.465 --> 01:29:03.650 align:middle line:90% 01:29:03.650 --> 01:29:07.750 align:middle line:84% So you're using gas with which you could 01:29:07.750 --> 01:29:10.580 align:middle line:90% reach very high temperatures. 01:29:10.580 --> 01:29:13.470 align:middle line:90% 01:29:13.470 --> 01:29:19.365 align:middle line:84% And all you do is, by a series of heat transfers, 01:29:19.365 --> 01:29:23.280 align:middle line:84% use it to produce something at the temperature which 01:29:23.280 --> 01:29:26.820 align:middle line:84% is very close to the environmental temperature. 01:29:26.820 --> 01:29:31.260 align:middle line:84% That's a very dissipative way of doing it. 01:29:31.260 --> 01:29:35.350 align:middle line:84% We have been doing it for a long time like that. 01:29:35.350 --> 01:29:36.580 align:middle line:90% We are still doing it. 01:29:36.580 --> 01:29:40.090 align:middle line:90% 01:29:40.090 --> 01:29:43.480 align:middle line:84% That's the way it's done in my apartment building 01:29:43.480 --> 01:29:46.330 align:middle line:90% where I live right now. 01:29:46.330 --> 01:29:50.670 align:middle line:90% But that's not the good way. 01:29:50.670 --> 01:29:52.090 align:middle line:90% It's a very inefficient way. 01:29:52.090 --> 01:29:54.750 align:middle line:90% 01:29:54.750 --> 01:30:04.420 align:middle line:84% Cogeneration is the idea that you take that high level energy, 01:30:04.420 --> 01:30:12.610 align:middle line:84% and before you and you do as much as you can with it. 01:30:12.610 --> 01:30:16.480 align:middle line:84% For example, to run a power plant, or run a plant, 01:30:16.480 --> 01:30:18.525 align:middle line:90% or run an engine-- 01:30:18.525 --> 01:30:22.570 align:middle line:90% 01:30:22.570 --> 01:30:25.985 align:middle line:84% for example, you could run a car engine-- 01:30:25.985 --> 01:30:31.940 align:middle line:84% this is our cold total energy modules, like car, 01:30:31.940 --> 01:30:37.690 align:middle line:84% or truck engines, or maybe ship engines, depending on the size-- 01:30:37.690 --> 01:30:44.230 align:middle line:84% where you burn your fuel, make mechanical energy. 01:30:44.230 --> 01:30:50.360 align:middle line:84% With that mechanical energy, you produce electricity. 01:30:50.360 --> 01:30:57.370 align:middle line:84% And then since your engine needs to dump entropy somewhere, 01:30:57.370 --> 01:31:00.900 align:middle line:84% instead of dumping it directly into the environment, 01:31:00.900 --> 01:31:05.550 align:middle line:84% you dump it into the heating system of your house. 01:31:05.550 --> 01:31:11.580 align:middle line:84% And then your house takes care of dumping the energy that 01:31:11.580 --> 01:31:14.052 align:middle line:90% leaks to the environment. 01:31:14.052 --> 01:31:17.970 align:middle line:84% So you get this heating as a byproduct, 01:31:17.970 --> 01:31:23.340 align:middle line:84% or as one of the two products of your system, which 01:31:23.340 --> 01:31:28.590 align:middle line:90% is combined heat and power. 01:31:28.590 --> 01:31:33.200 align:middle line:84% So it's called cogeneration, or CHP, solution. 01:31:33.200 --> 01:31:42.805 align:middle line:84% And in this way, you have used that energy at best. 01:31:42.805 --> 01:31:48.880 align:middle line:84% So this would be, say, the best-available technology. 01:31:48.880 --> 01:31:59.020 align:middle line:84% For example, in the town where I worked 30 years or longer, 01:31:59.020 --> 01:32:03.110 align:middle line:84% which is Brescia in Italy, this idea 01:32:03.110 --> 01:32:09.290 align:middle line:84% has been taken very seriously starting in 1973. 01:32:09.290 --> 01:32:14.270 align:middle line:84% And now about 95% of the volume of the town-- 01:32:14.270 --> 01:32:16.110 align:middle line:84% and it's a town with the size of Boston-- 01:32:16.110 --> 01:32:21.230 align:middle line:90% 01:32:21.230 --> 01:32:26.570 align:middle line:84% it's heated in this way with cogeneration power plants 01:32:26.570 --> 01:32:30.950 align:middle line:84% that dump energy into a district heating system. 01:32:30.950 --> 01:32:36.200 align:middle line:84% So you get this hot water going around the town under the roads 01:32:36.200 --> 01:32:37.560 align:middle line:90% and reaching all the houses. 01:32:37.560 --> 01:32:40.165 align:middle line:84% And then with heat exchangers, you do the heating. 01:32:40.165 --> 01:32:43.395 align:middle line:90% 01:32:43.395 --> 01:32:49.260 align:middle line:90% More recently-- 20 years ago-- 01:32:49.260 --> 01:32:55.080 align:middle line:84% they also added a more complex system in which you also-- 01:32:55.080 --> 01:33:01.740 align:middle line:84% not only use fossil fuels, but you use also waste. 01:33:01.740 --> 01:33:06.510 align:middle line:84% You burn the trash that is not recyclable. 01:33:06.510 --> 01:33:14.020 align:middle line:84% And they also integrated with, in a hybrid way with some solar. 01:33:14.020 --> 01:33:20.200 align:middle line:84% So it's an entire system aimed to efficiency 01:33:20.200 --> 01:33:25.080 align:middle line:90% for heating the homes. 01:33:25.080 --> 01:33:29.070 align:middle line:84% I know that many are thinking about doing these things also 01:33:29.070 --> 01:33:29.920 align:middle line:90% in the States. 01:33:29.920 --> 01:33:33.630 align:middle line:84% But here, the problem has been that-- 01:33:33.630 --> 01:33:37.920 align:middle line:84% the problem-- the reason that these things have not evolved 01:33:37.920 --> 01:33:43.280 align:middle line:84% is the cheaper cost of energy than it should. 01:33:43.280 --> 01:33:46.710 align:middle line:84% One incentive for improving efficiency 01:33:46.710 --> 01:33:48.930 align:middle line:90% is the cost of the energy. 01:33:48.930 --> 01:33:52.330 align:middle line:84% Higher the costs, people will invest to spend less. 01:33:52.330 --> 01:33:52.830 align:middle line:90% Yeah. 01:33:52.830 --> 01:34:02.365 align:middle line:90% 01:34:02.365 --> 01:34:03.140 align:middle line:90% OK. 01:34:03.140 --> 01:34:03.640 align:middle line:90% Good. 01:34:03.640 --> 01:34:16.700 align:middle line:90% 01:34:16.700 --> 01:34:17.400 align:middle line:90% Glad to know it. 01:34:17.400 --> 01:34:19.910 align:middle line:90% 01:34:19.910 --> 01:34:22.610 align:middle line:84% Yeah, I thought that somebody was thinking 01:34:22.610 --> 01:34:27.470 align:middle line:90% of also doing it for the town. 01:34:27.470 --> 01:34:33.470 align:middle line:84% The problem is that it takes a lot of investment. 01:34:33.470 --> 01:34:36.890 align:middle line:84% And the way it got started in Brescia 01:34:36.890 --> 01:34:42.950 align:middle line:84% was that the municipal company was-- 01:34:42.950 --> 01:34:46.610 align:middle line:84% the company that would run also the roads 01:34:46.610 --> 01:34:51.960 align:middle line:84% would run a lot of public services, 01:34:51.960 --> 01:34:53.630 align:middle line:90% including transportation. 01:34:53.630 --> 01:35:01.650 align:middle line:84% And so they could decide and move quickly about doing it, 01:35:01.650 --> 01:35:04.680 align:middle line:84% because they were the people working on the roads. 01:35:04.680 --> 01:35:09.345 align:middle line:84% So put in pipes because you need to put very large pipes 01:35:09.345 --> 01:35:10.571 align:middle line:90% under the roads. 01:35:10.571 --> 01:35:14.130 align:middle line:90% 01:35:14.130 --> 01:35:16.900 align:middle line:84% If you do it internally, it is easier. 01:35:16.900 --> 01:35:21.330 align:middle line:84% And also, they did it in such a way that-- 01:35:21.330 --> 01:35:23.850 align:middle line:84% because another issue is people would 01:35:23.850 --> 01:35:28.270 align:middle line:84% want to get connected to the district heating system, 01:35:28.270 --> 01:35:34.310 align:middle line:84% but sometimes it's more efficient to start 01:35:34.310 --> 01:35:38.300 align:middle line:84% with big buildings and then go to smaller ones. 01:35:38.300 --> 01:35:41.780 align:middle line:84% So these kind of decisions, it's political. 01:35:41.780 --> 01:35:45.672 align:middle line:90% And so that is an issue. 01:35:45.672 --> 01:35:47.148 align:middle line:90% OK. 01:35:47.148 --> 01:35:49.290 align:middle line:90% Sorry, we are late. 01:35:49.290 --> 01:35:51.530 align:middle line:90% So I'll see you next time. 01:35:51.530 --> 01:35:58.000 align:middle line:90%