WEBVTT 00:00:00.000 --> 00:00:01.485 align:middle line:90% [SQUEAKING] 00:00:01.485 --> 00:00:03.465 align:middle line:90% [RUSTLING] 00:00:03.465 --> 00:00:06.930 align:middle line:90% [CLICKING] 00:00:06.930 --> 00:00:09.710 align:middle line:90% 00:00:09.710 --> 00:00:11.910 align:middle line:90% GIAN PAOLO BERETTA: OK. 00:00:11.910 --> 00:00:13.380 align:middle line:90% Good morning. 00:00:13.380 --> 00:00:16.450 align:middle line:84% It will take us a couple of lectures, 00:00:16.450 --> 00:00:23.580 align:middle line:84% I guess, to cover this in some details. 00:00:23.580 --> 00:00:31.210 align:middle line:84% And it is-- we want to extend what we know about heat transfer 00:00:31.210 --> 00:00:34.930 align:middle line:84% to heat and mass transfer and charge transfer. 00:00:34.930 --> 00:00:40.090 align:middle line:84% And we want to do it in a way that exhibits 00:00:40.090 --> 00:00:42.920 align:middle line:90% the logic of the construction. 00:00:42.920 --> 00:00:48.560 align:middle line:84% So this is why I am going to restart-- 00:00:48.560 --> 00:00:51.910 align:middle line:84% I mean, start again from what we have already seen 00:00:51.910 --> 00:00:57.680 align:middle line:84% because it already contains that logic. 00:00:57.680 --> 00:01:00.410 align:middle line:84% Actually, parallel logic between what 00:01:00.410 --> 00:01:02.480 align:middle line:84% we've done in chemical kinetics and what 00:01:02.480 --> 00:01:09.850 align:middle line:84% we've done in our simple review of the heat transfer basics. 00:01:09.850 --> 00:01:14.860 align:middle line:84% And because this contains what we want to develop item 00:01:14.860 --> 00:01:22.210 align:middle line:84% by item with some care and in other details in the rest 00:01:22.210 --> 00:01:26.300 align:middle line:90% of this lecture or two. 00:01:26.300 --> 00:01:28.730 align:middle line:84% So in chemical kinetics, let's start 00:01:28.730 --> 00:01:30.680 align:middle line:84% with-- because we did this first, so let's 00:01:30.680 --> 00:01:31.690 align:middle line:90% start with this first. 00:01:31.690 --> 00:01:35.130 align:middle line:90% 00:01:35.130 --> 00:01:39.980 align:middle line:84% When we considered the closed systems, 00:01:39.980 --> 00:01:43.180 align:middle line:84% we considered the energy balance. 00:01:43.180 --> 00:01:46.130 align:middle line:84% It was implicit that the energy remains constant 00:01:46.130 --> 00:01:48.830 align:middle line:90% if the system is isolated. 00:01:48.830 --> 00:01:51.730 align:middle line:84% And this is the entropy balance that 00:01:51.730 --> 00:01:55.070 align:middle line:84% says that the rate of entropy change 00:01:55.070 --> 00:02:01.940 align:middle line:84% is equal to the entropy produced by irreversibility. 00:02:01.940 --> 00:02:10.699 align:middle line:84% Then we-- this is not a balance of particles, 00:02:10.699 --> 00:02:13.810 align:middle line:84% but you could view the proportionality relations 00:02:13.810 --> 00:02:16.480 align:middle line:84% as sort of a balance in the sense 00:02:16.480 --> 00:02:20.800 align:middle line:84% that they account for the rate of change 00:02:20.800 --> 00:02:25.180 align:middle line:84% of the number of particles in your mixture. 00:02:25.180 --> 00:02:30.350 align:middle line:84% And so that is another ingredient. 00:02:30.350 --> 00:02:34.990 align:middle line:84% Plus, we have assumed something about equilibrium-- 00:02:34.990 --> 00:02:40.380 align:middle line:84% the near-equilibrium situation in which we-- 00:02:40.380 --> 00:02:44.230 align:middle line:84% if it weren't for the chemical reactions that 00:02:44.230 --> 00:02:51.580 align:middle line:84% are changing the composition, our system in our model, 00:02:51.580 --> 00:02:57.500 align:middle line:84% so far, is very close to what it would 00:02:57.500 --> 00:03:01.280 align:middle line:84% be if it were in equilibrium-- a stable equilibrium 00:03:01.280 --> 00:03:06.460 align:middle line:90% state without the reactions on. 00:03:06.460 --> 00:03:11.200 align:middle line:84% And that allows us-- so this relation here 00:03:11.200 --> 00:03:17.140 align:middle line:84% allows us to evaluate the rate of change 00:03:17.140 --> 00:03:21.950 align:middle line:84% of the entropy which is needed for the entropy balance. 00:03:21.950 --> 00:03:24.140 align:middle line:90% So we've to evaluate that. 00:03:24.140 --> 00:03:29.520 align:middle line:84% And when we do that rate of change, since this depends-- 00:03:29.520 --> 00:03:31.130 align:middle line:84% the only thing that is changing here 00:03:31.130 --> 00:03:33.650 align:middle line:84% is n through the proportionality relation-- 00:03:33.650 --> 00:03:35.810 align:middle line:84% so you have to take the partial of S 00:03:35.810 --> 00:03:41.140 align:middle line:84% with respect to n times the rate of change of n. 00:03:41.140 --> 00:03:45.790 align:middle line:84% And then you keep going and you single out 00:03:45.790 --> 00:03:51.460 align:middle line:84% the affinities of the reaction, and you 00:03:51.460 --> 00:03:56.230 align:middle line:90% bring in the reaction rates. 00:03:56.230 --> 00:04:00.850 align:middle line:84% So you combine them, this balance equation, 00:04:00.850 --> 00:04:04.070 align:middle line:84% with the energy balance, which is trivial in this case, 00:04:04.070 --> 00:04:07.270 align:middle line:84% and you obtain an expression for the entropy produced 00:04:07.270 --> 00:04:13.180 align:middle line:84% by irreversibility, which is this product of rates 00:04:13.180 --> 00:04:15.370 align:middle line:84% and affinities that we interpreted 00:04:15.370 --> 00:04:17.600 align:middle line:90% as degrees of disequilibrium. 00:04:17.600 --> 00:04:21.370 align:middle line:90% 00:04:21.370 --> 00:04:30.020 align:middle line:84% And then we assumed a law that relates how the rates-- 00:04:30.020 --> 00:04:34.600 align:middle line:84% relates the rates to the affinities. 00:04:34.600 --> 00:04:39.340 align:middle line:84% And this law is a nonlinear law, but 00:04:39.340 --> 00:04:44.230 align:middle line:84% in the near-equilibrium limit, we have linearized it. 00:04:44.230 --> 00:04:48.750 align:middle line:90% 00:04:48.750 --> 00:04:52.080 align:middle line:84% And here's the parallel structure and parallel logic 00:04:52.080 --> 00:04:55.320 align:middle line:84% that we followed when we derived-- 00:04:55.320 --> 00:04:57.720 align:middle line:84% when we did the heat transfer basis. 00:04:57.720 --> 00:05:00.430 align:middle line:90% We write the energy balance. 00:05:00.430 --> 00:05:01.660 align:middle line:90% It takes this form. 00:05:01.660 --> 00:05:04.570 align:middle line:84% We will return to why it takes this form. 00:05:04.570 --> 00:05:10.090 align:middle line:90% For the moment, just believe it. 00:05:10.090 --> 00:05:12.340 align:middle line:90% So that's the energy balance. 00:05:12.340 --> 00:05:14.630 align:middle line:90% This is the entropy balance. 00:05:14.630 --> 00:05:18.930 align:middle line:84% Also, of course, we will return on this. 00:05:18.930 --> 00:05:22.203 align:middle line:84% There is-- we are talking about a solid, a pure solid, 00:05:22.203 --> 00:05:23.620 align:middle line:84% there is no change in composition, 00:05:23.620 --> 00:05:29.390 align:middle line:84% so the rate of change of composition is 0. 00:05:29.390 --> 00:05:32.160 align:middle line:84% Then we assume, like we did here, 00:05:32.160 --> 00:05:35.140 align:middle line:84% we assume something about the local equilibrium. 00:05:35.140 --> 00:05:36.780 align:middle line:90% We assume local equilibrium. 00:05:36.780 --> 00:05:40.100 align:middle line:90% 00:05:40.100 --> 00:05:42.500 align:middle line:84% So that there is a fundamental relation, 00:05:42.500 --> 00:05:47.910 align:middle line:84% we can write the Gibbs relation between the properties. 00:05:47.910 --> 00:05:53.860 align:middle line:84% And this is a way to write the Gibbs relation in order to-- 00:05:53.860 --> 00:05:56.820 align:middle line:90% 00:05:56.820 --> 00:06:01.590 align:middle line:84% we will also see this in more detail later. 00:06:01.590 --> 00:06:05.220 align:middle line:84% This is-- in order to get rid of the pressure 00:06:05.220 --> 00:06:11.185 align:middle line:84% here and push the pressure in the Gibbs-Duhem independent 00:06:11.185 --> 00:06:11.685 align:middle line:90% equation. 00:06:11.685 --> 00:06:14.840 align:middle line:90% 00:06:14.840 --> 00:06:22.180 align:middle line:84% The Gibbs relation can be used to combine the left-hand sides 00:06:22.180 --> 00:06:25.030 align:middle line:90% of the balance equations. 00:06:25.030 --> 00:06:27.310 align:middle line:84% So you substitute the various balance equations 00:06:27.310 --> 00:06:30.600 align:middle line:90% into this Gibbs relation. 00:06:30.600 --> 00:06:34.560 align:middle line:84% And obtain an expression, like here, 00:06:34.560 --> 00:06:37.860 align:middle line:84% for the entropy produced by irreversibility. 00:06:37.860 --> 00:06:40.020 align:middle line:84% Of course, we will do the details. 00:06:40.020 --> 00:06:45.220 align:middle line:84% Also, because here, I've changed the notation a bit, 00:06:45.220 --> 00:06:48.210 align:middle line:84% but as you know, for a heat transfer, 00:06:48.210 --> 00:06:52.800 align:middle line:84% the energy flux is just q double prime, 00:06:52.800 --> 00:06:56.460 align:middle line:84% and the entropy flux is q double prime over T. 00:06:56.460 --> 00:06:59.600 align:middle line:90% That's the heat interaction. 00:06:59.600 --> 00:07:02.730 align:middle line:84% And when we do these substitutions, 00:07:02.730 --> 00:07:11.390 align:middle line:84% you can rewrite the divergence of JS as the sum of two terms. 00:07:11.390 --> 00:07:14.420 align:middle line:84% The first one cancels with the one 00:07:14.420 --> 00:07:19.380 align:middle line:84% that comes from the divergence of the energy flux. 00:07:19.380 --> 00:07:23.300 align:middle line:84% And you are left with this final result that says that 00:07:23.300 --> 00:07:25.650 align:middle line:84% the entropy produced by irreversibility-- and here, 00:07:25.650 --> 00:07:32.010 align:middle line:84% we are talking about entropy produced per unit volume-- 00:07:32.010 --> 00:07:40.210 align:middle line:84% is equal to the heat flux times the gradient 00:07:40.210 --> 00:07:42.880 align:middle line:84% in the inverse temperature, which 00:07:42.880 --> 00:07:47.215 align:middle line:84% we can interpret as the degree of disequilibrium for the heat 00:07:47.215 --> 00:07:47.715 align:middle line:90% transfer. 00:07:47.715 --> 00:07:50.800 align:middle line:90% 00:07:50.800 --> 00:07:58.160 align:middle line:84% And then if we assume a linear law, then we get Fourier. 00:07:58.160 --> 00:08:01.970 align:middle line:84% In other words, if you-- by looking at this expression here, 00:08:01.970 --> 00:08:03.275 align:middle line:90% if-- 00:08:03.275 --> 00:08:06.620 align:middle line:84% and knowing that the entropy produced by irreversibility must 00:08:06.620 --> 00:08:14.375 align:middle line:84% be non-negative, the simple way is to assume that q is-- 00:08:14.375 --> 00:08:19.460 align:middle line:84% that the flux is proportional to the gradient 00:08:19.460 --> 00:08:24.350 align:middle line:84% through a positive constant of proportionality, which 00:08:24.350 --> 00:08:26.830 align:middle line:84% could end-- actually, since it is a vector, 00:08:26.830 --> 00:08:31.460 align:middle line:84% the most general linear relation between two vectors 00:08:31.460 --> 00:08:33.360 align:middle line:90% is through a tensor. 00:08:33.360 --> 00:08:37.080 align:middle line:84% So we take a positive tensor in between here, 00:08:37.080 --> 00:08:44.290 align:middle line:84% which assures that the sigma is going to be non-negative. 00:08:44.290 --> 00:08:47.920 align:middle line:84% And then-- so we get Fourier law. 00:08:47.920 --> 00:08:56.170 align:middle line:84% And also, we have seen that if we accept Onsager's reciprocity 00:08:56.170 --> 00:09:02.680 align:middle line:84% theorem, that tensor not only needs 00:09:02.680 --> 00:09:07.040 align:middle line:84% to be positive definite-- or non-negative definite, 00:09:07.040 --> 00:09:12.090 align:middle line:84% but it also needs to be symmetric. 00:09:12.090 --> 00:09:14.280 align:middle line:90% You can also invert it. 00:09:14.280 --> 00:09:19.450 align:middle line:84% And therefore, the entropy production 00:09:19.450 --> 00:09:25.730 align:middle line:84% becomes a quadratic form in the degrees of disequilibrium, 00:09:25.730 --> 00:09:30.600 align:middle line:84% or equivalently, a quadratic form in the fluxes. 00:09:30.600 --> 00:09:33.390 align:middle line:90% 00:09:33.390 --> 00:09:35.030 align:middle line:90% OK. 00:09:35.030 --> 00:09:42.700 align:middle line:84% So the ingredients are write the energy balance, 00:09:42.700 --> 00:09:46.630 align:middle line:84% do something about the Gibbs relation-- 00:09:46.630 --> 00:09:49.250 align:middle line:84% or write the Gibbs relation in this form 00:09:49.250 --> 00:09:54.530 align:middle line:84% so that you can combine these balances in a proper way. 00:09:54.530 --> 00:09:58.940 align:middle line:84% Make an assumption about the kind of interaction 00:09:58.940 --> 00:10:04.400 align:middle line:84% that is occurring between the various parcels 00:10:04.400 --> 00:10:08.800 align:middle line:90% or elements of your continuum. 00:10:08.800 --> 00:10:12.830 align:middle line:84% And then assume-- if you want to be near equilibrium, 00:10:12.830 --> 00:10:17.110 align:middle line:84% you can assume a linear relation. 00:10:17.110 --> 00:10:24.020 align:middle line:84% And if you want to discuss more non-equilibrium-- 00:10:24.020 --> 00:10:29.880 align:middle line:84% I mean, not near-equilibrium-- far from equilibrium, 00:10:29.880 --> 00:10:35.423 align:middle line:84% then you need to develop some more elaborate theories. 00:10:35.423 --> 00:10:40.140 align:middle line:90% 00:10:40.140 --> 00:10:44.820 align:middle line:84% And also, here, there is space, as we 00:10:44.820 --> 00:10:51.750 align:middle line:84% have seen for the Cattaneo correction to Fourier law, 00:10:51.750 --> 00:10:55.830 align:middle line:90% there is space to account-- 00:10:55.830 --> 00:11:00.750 align:middle line:84% or to solve some of the problems of the Fourier 00:11:00.750 --> 00:11:05.340 align:middle line:84% law, the infinite speed of propagation that 00:11:05.340 --> 00:11:12.280 align:middle line:90% is implied by this expression. 00:11:12.280 --> 00:11:14.410 align:middle line:90% So let's start. 00:11:14.410 --> 00:11:16.300 align:middle line:90% We take them one by one. 00:11:16.300 --> 00:11:23.200 align:middle line:84% So we have to be patient, but this 00:11:23.200 --> 00:11:25.090 align:middle line:84% will give us also some opportunity 00:11:25.090 --> 00:11:31.460 align:middle line:84% to highlight a few of the details. 00:11:31.460 --> 00:11:37.002 align:middle line:84% This procedure that I am describing 00:11:37.002 --> 00:11:42.270 align:middle line:84% varies depending on the field of application. 00:11:42.270 --> 00:11:44.380 align:middle line:84% In particular, I need to mention, 00:11:44.380 --> 00:11:48.270 align:middle line:84% but I'm not going to follow precisely or-- 00:11:48.270 --> 00:11:53.400 align:middle line:84% but there is strong connections with a famous procedure called-- 00:11:53.400 --> 00:11:57.630 align:middle line:84% under the names of Coleman and Noll, 00:11:57.630 --> 00:12:08.910 align:middle line:84% which is used more by the community of solid mechanics 00:12:08.910 --> 00:12:13.970 align:middle line:84% people that study plasticity, viscoelasticity. 00:12:13.970 --> 00:12:17.190 align:middle line:90% 00:12:17.190 --> 00:12:23.460 align:middle line:84% For example, not to go too far, if you go to Professor Lallit 00:12:23.460 --> 00:12:27.990 align:middle line:84% Anand in this department, he has written books 00:12:27.990 --> 00:12:30.356 align:middle line:84% and he is renowned for that procedure. 00:12:30.356 --> 00:12:33.460 align:middle line:90% 00:12:33.460 --> 00:12:37.210 align:middle line:84% I am more inclined, since I have studied 00:12:37.210 --> 00:12:46.210 align:middle line:84% with people in the fluid mechanics area, 00:12:46.210 --> 00:12:51.430 align:middle line:84% I am more inclined to describe the procedure that 00:12:51.430 --> 00:12:55.870 align:middle line:84% applies for fluid mechanics kind of applications. 00:12:55.870 --> 00:13:01.090 align:middle line:90% But the logics are very similar. 00:13:01.090 --> 00:13:06.190 align:middle line:84% Except when it comes to this generalization of the heat 00:13:06.190 --> 00:13:09.070 align:middle line:84% interaction, that, instead, is peculiar, 00:13:09.070 --> 00:13:15.030 align:middle line:84% of course, of our course here because also, 00:13:15.030 --> 00:13:19.680 align:middle line:84% the definition of heat is peculiar to our course. 00:13:19.680 --> 00:13:26.220 align:middle line:84% Not yet very widespread in textbooks 00:13:26.220 --> 00:13:29.670 align:middle line:90% or other scientific papers. 00:13:29.670 --> 00:13:36.760 align:middle line:84% OK, so let's start with this simple. 00:13:36.760 --> 00:13:42.400 align:middle line:84% Take a piece of material, fluid element. 00:13:42.400 --> 00:13:44.680 align:middle line:84% So already, the fact that I call it fluid 00:13:44.680 --> 00:13:48.036 align:middle line:84% means it shows you my inclination. 00:13:48.036 --> 00:13:52.786 align:middle line:90% 00:13:52.786 --> 00:13:55.040 align:middle line:84% And as you know, this department has 00:13:55.040 --> 00:14:04.220 align:middle line:84% had very influential professors in fluid mechanics. 00:14:04.220 --> 00:14:05.630 align:middle line:90% And heat transfer. 00:14:05.630 --> 00:14:15.900 align:middle line:84% So Rosenow, Shapiro, Sonin, just to mention a few. 00:14:15.900 --> 00:14:23.750 align:middle line:84% So I have these one-dimensional piece of fluid. 00:14:23.750 --> 00:14:28.700 align:middle line:84% And it's subjected to interactions on one side 00:14:28.700 --> 00:14:29.665 align:middle line:90% and on the other side. 00:14:29.665 --> 00:14:33.240 align:middle line:90% 00:14:33.240 --> 00:14:35.340 align:middle line:84% And when we did heat transfer, we 00:14:35.340 --> 00:14:39.660 align:middle line:84% were considering only energy and entropy. 00:14:39.660 --> 00:14:42.990 align:middle line:84% Now here, we're going to introduce also 00:14:42.990 --> 00:14:46.126 align:middle line:90% a flux of particles. 00:14:46.126 --> 00:14:49.370 align:middle line:90% 00:14:49.370 --> 00:14:58.160 align:middle line:84% And also, here, you have the flux. 00:14:58.160 --> 00:15:02.630 align:middle line:84% And this is a component-- the x-component because we are 00:15:02.630 --> 00:15:04.040 align:middle line:90% one-dimensional-- 00:15:04.040 --> 00:15:07.340 align:middle line:84% of the flux, and it changes as you 00:15:07.340 --> 00:15:12.950 align:middle line:84% move from the face at x and the face at x plus dx. 00:15:12.950 --> 00:15:18.650 align:middle line:84% And the area of exchange between the adjacent-- 00:15:18.650 --> 00:15:25.145 align:middle line:84% between these fluid element and the one to its right is A. 00:15:25.145 --> 00:15:36.280 align:middle line:84% And also, the area on the other side, of course, is, again, A. 00:15:36.280 --> 00:15:38.855 align:middle line:84% We want to make a general energy balance, 00:15:38.855 --> 00:15:40.605 align:middle line:84% so we're not going to assume steady state. 00:15:40.605 --> 00:15:45.420 align:middle line:90% 00:15:45.420 --> 00:15:56.300 align:middle line:84% So we call the energy of the fluid in this lamina delta 00:15:56.300 --> 00:15:58.290 align:middle line:90% E. Why delta? 00:15:58.290 --> 00:16:01.650 align:middle line:90% Because it's infinitesimal. 00:16:01.650 --> 00:16:03.820 align:middle line:84% Since the thickness is infinitesimal, 00:16:03.820 --> 00:16:06.690 align:middle line:90% it's from x to x plus dx. 00:16:06.690 --> 00:16:12.510 align:middle line:84% So that is an infinitesimal amount of energy. 00:16:12.510 --> 00:16:15.630 align:middle line:84% Related to the fact that it has an infinitesimal amount 00:16:15.630 --> 00:16:19.230 align:middle line:84% of volume, which is given by the area times dx. 00:16:19.230 --> 00:16:22.390 align:middle line:90% 00:16:22.390 --> 00:16:27.810 align:middle line:84% And the specific energy per unit volume, 00:16:27.810 --> 00:16:32.130 align:middle line:84% you can write it, as we know, like rho times u, 00:16:32.130 --> 00:16:36.630 align:middle line:84% and u here is really u star, what we 00:16:36.630 --> 00:16:38.990 align:middle line:90% call the energy per unit mass. 00:16:38.990 --> 00:16:42.990 align:middle line:90% 00:16:42.990 --> 00:16:43.980 align:middle line:90% All right. 00:16:43.980 --> 00:16:47.140 align:middle line:84% So you write it-- you need volume, 00:16:47.140 --> 00:16:53.350 align:middle line:84% and then we can also substitute the volume for A times dx. 00:16:53.350 --> 00:17:01.710 align:middle line:84% The same-- I mean, this system has also entropy. 00:17:01.710 --> 00:17:03.540 align:middle line:90% And here's the entropy. 00:17:03.540 --> 00:17:07.349 align:middle line:84% Also infinitesimal because it's an infinitesimal amount 00:17:07.349 --> 00:17:09.690 align:middle line:90% of material. 00:17:09.690 --> 00:17:12.550 align:middle line:90% And it stays like that. 00:17:12.550 --> 00:17:17.640 align:middle line:84% So it's the volume times the specific entropy 00:17:17.640 --> 00:17:20.449 align:middle line:84% per unit volume, which is rho times s star. 00:17:20.449 --> 00:17:24.760 align:middle line:90% 00:17:24.760 --> 00:17:29.650 align:middle line:84% And also the amounts, they are infinitesimal. 00:17:29.650 --> 00:17:36.340 align:middle line:84% And the amount per unit volume is the concentration. 00:17:36.340 --> 00:17:40.990 align:middle line:84% So you can write that as concentration times the volume. 00:17:40.990 --> 00:17:43.465 align:middle line:90% OK. 00:17:43.465 --> 00:17:53.400 align:middle line:84% Now inside these element, we have also some volume effects. 00:17:53.400 --> 00:17:57.060 align:middle line:84% For example, chemical reactions that are occurring inside-- 00:17:57.060 --> 00:18:00.340 align:middle line:90% in the interior. 00:18:00.340 --> 00:18:05.440 align:middle line:84% And therefore, the proportionality relations 00:18:05.440 --> 00:18:09.830 align:middle line:84% are governed by this epsilon dot, 00:18:09.830 --> 00:18:13.180 align:middle line:84% which are the rates of advancement 00:18:13.180 --> 00:18:16.770 align:middle line:90% of the reaction coordinates. 00:18:16.770 --> 00:18:21.690 align:middle line:84% We can write also that as the volume times the reaction 00:18:21.690 --> 00:18:24.420 align:middle line:84% rate per unit volume, which is typically 00:18:24.420 --> 00:18:28.030 align:middle line:90% called r without the dot. 00:18:28.030 --> 00:18:30.940 align:middle line:90% 00:18:30.940 --> 00:18:34.210 align:middle line:84% And then the other is the entropy production 00:18:34.210 --> 00:18:41.050 align:middle line:84% due to irreversibility, which occurs throughout the volume 00:18:41.050 --> 00:18:44.330 align:middle line:90% of this system. 00:18:44.330 --> 00:18:49.380 align:middle line:84% And that, too, we like to write in terms of per-unit volume. 00:18:49.380 --> 00:18:55.200 align:middle line:84% So we have defined this symbol s dot rate of entropy production. 00:18:55.200 --> 00:19:00.440 align:middle line:84% Triple means per unit volume, and soon, we 00:19:00.440 --> 00:19:03.940 align:middle line:84% have already started calling this thing sigma. 00:19:03.940 --> 00:19:07.730 align:middle line:90% 00:19:07.730 --> 00:19:08.950 align:middle line:90% All right. 00:19:08.950 --> 00:19:13.760 align:middle line:90% So here is the energy balance. 00:19:13.760 --> 00:19:19.760 align:middle line:84% It says the rate of change of the energy of our system 00:19:19.760 --> 00:19:26.990 align:middle line:84% is equal to the net energy it exchanges 00:19:26.990 --> 00:19:30.140 align:middle line:90% through the interactions. 00:19:30.140 --> 00:19:35.610 align:middle line:84% And here, it has interactions on the left and on the right. 00:19:35.610 --> 00:19:39.150 align:middle line:90% And the net is the difference. 00:19:39.150 --> 00:19:45.120 align:middle line:84% And you have to take it with a minus sign if it goes out. 00:19:45.120 --> 00:19:49.470 align:middle line:84% So if this one is greater than that one, 00:19:49.470 --> 00:19:53.940 align:middle line:84% the difference is this term here. 00:19:53.940 --> 00:19:55.660 align:middle line:90% And that is per unit area. 00:19:55.660 --> 00:19:58.865 align:middle line:84% And then you have to multiply by the area. 00:19:58.865 --> 00:20:03.940 align:middle line:90% 00:20:03.940 --> 00:20:06.890 align:middle line:84% And that is it, because energy is conserved. 00:20:06.890 --> 00:20:09.500 align:middle line:90% 00:20:09.500 --> 00:20:10.940 align:middle line:90% Let's go to entropy. 00:20:10.940 --> 00:20:12.860 align:middle line:84% The entropy balance says the rate 00:20:12.860 --> 00:20:15.920 align:middle line:84% of change of the entropy of our fluid element 00:20:15.920 --> 00:20:24.550 align:middle line:84% is equal to the net entropy flux, via interaction. 00:20:24.550 --> 00:20:28.330 align:middle line:84% So here, I have, again, left and right. 00:20:28.330 --> 00:20:34.700 align:middle line:84% The difference is this term here. 00:20:34.700 --> 00:20:39.340 align:middle line:84% And with the negative sign because if it is positive, 00:20:39.340 --> 00:20:43.180 align:middle line:84% you have more entropy going out than coming in, 00:20:43.180 --> 00:20:45.450 align:middle line:90% and that's this term here. 00:20:45.450 --> 00:20:48.110 align:middle line:90% 00:20:48.110 --> 00:20:55.700 align:middle line:84% I think there is an error here because I need also an area. 00:20:55.700 --> 00:20:57.200 align:middle line:90% Like here. 00:20:57.200 --> 00:21:01.830 align:middle line:84% I need to multiply by A. I need to correct that. 00:21:01.830 --> 00:21:06.020 align:middle line:90% 00:21:06.020 --> 00:21:08.880 align:middle line:84% That doesn't finish the entropy balance 00:21:08.880 --> 00:21:12.500 align:middle line:84% because you also have to account for the entropy generation 00:21:12.500 --> 00:21:17.930 align:middle line:84% inside the element due to irreversibility. 00:21:17.930 --> 00:21:19.870 align:middle line:90% Now what do we do? 00:21:19.870 --> 00:21:29.210 align:middle line:84% We simply divide by the volume delta V left and right 00:21:29.210 --> 00:21:32.620 align:middle line:84% of these equations, and here is what you get. 00:21:32.620 --> 00:21:40.160 align:middle line:90% 00:21:40.160 --> 00:21:42.180 align:middle line:84% So these are the balance equations. 00:21:42.180 --> 00:21:45.070 align:middle line:90% 00:21:45.070 --> 00:21:56.180 align:middle line:84% Now I want to open a little digression that is nevertheless 00:21:56.180 --> 00:22:04.930 align:middle line:84% important regarding how we describe 00:22:04.930 --> 00:22:07.700 align:middle line:84% the properties of a continuum-- in particular, 00:22:07.700 --> 00:22:12.140 align:middle line:84% the extensive properties like energy, 00:22:12.140 --> 00:22:17.680 align:middle line:84% mass, entropy, and a bunch of others 00:22:17.680 --> 00:22:22.360 align:middle line:84% that I'll have in a summary table. 00:22:22.360 --> 00:22:33.110 align:middle line:84% So let's consider, like we typically do in fluid mechanics, 00:22:33.110 --> 00:22:37.990 align:middle line:90% we consider a volume. 00:22:37.990 --> 00:22:41.120 align:middle line:84% Call it sigma M. So it's a material volume. 00:22:41.120 --> 00:22:44.880 align:middle line:84% So I select at a certain instant of time 00:22:44.880 --> 00:22:51.830 align:middle line:84% a certain bunch of particles contained inside this volume. 00:22:51.830 --> 00:23:02.150 align:middle line:84% And I denote the surface of that as partial of sigma M. M 00:23:02.150 --> 00:23:03.823 align:middle line:90% stands for material. 00:23:03.823 --> 00:23:07.180 align:middle line:90% 00:23:07.180 --> 00:23:16.230 align:middle line:84% Of course, when the fluid moves, you need to-- 00:23:16.230 --> 00:23:19.320 align:middle line:84% if you want to follow the material volume in all 00:23:19.320 --> 00:23:25.710 align:middle line:84% this problem, is that you need to track all the particles that 00:23:25.710 --> 00:23:29.610 align:middle line:84% are inside here at time, say, t1. 00:23:29.610 --> 00:23:34.140 align:middle line:84% And then at time t2, you need to see where all those particles 00:23:34.140 --> 00:23:37.140 align:middle line:84% have gone and consider another volume that 00:23:37.140 --> 00:23:40.270 align:middle line:90% contains only those particles. 00:23:40.270 --> 00:23:43.180 align:middle line:90% 00:23:43.180 --> 00:23:47.930 align:middle line:84% And so these would be a Lagrangian point of view, 00:23:47.930 --> 00:23:52.480 align:middle line:84% and it's complicated, so that's why in fluid mechanics, 00:23:52.480 --> 00:23:58.290 align:middle line:84% you introduce control volumes and the Eulerian point of view. 00:23:58.290 --> 00:24:03.340 align:middle line:84% But that's not-- we're not doing control volumes here. 00:24:03.340 --> 00:24:05.260 align:middle line:90% They do them in fluid mechanics. 00:24:05.260 --> 00:24:10.240 align:middle line:84% But what I'm doing here is just the balance equations, 00:24:10.240 --> 00:24:15.960 align:middle line:84% and in particular, I want to obtain from the balance 00:24:15.960 --> 00:24:20.490 align:middle line:84% equation written for the material volume, the balance 00:24:20.490 --> 00:24:24.720 align:middle line:84% equation written for the infinitesimal element 00:24:24.720 --> 00:24:25.540 align:middle line:90% of that volume. 00:24:25.540 --> 00:24:28.050 align:middle line:90% So in other words, something-- 00:24:28.050 --> 00:24:33.300 align:middle line:84% what we usually call the indefinite balance 00:24:33.300 --> 00:24:42.430 align:middle line:84% equations that are what holds point by point in our continuum. 00:24:42.430 --> 00:24:46.190 align:middle line:90% 00:24:46.190 --> 00:24:50.210 align:middle line:84% So first of all, let's call the generic property-- think 00:24:50.210 --> 00:24:53.930 align:middle line:84% of energy, but then it could be also entropy, let's call it A, 00:24:53.930 --> 00:24:56.440 align:middle line:90% capital A. Or even mass. 00:24:56.440 --> 00:24:59.870 align:middle line:90% 00:24:59.870 --> 00:25:00.730 align:middle line:90% Even volume. 00:25:00.730 --> 00:25:03.280 align:middle line:90% 00:25:03.280 --> 00:25:08.515 align:middle line:84% And we call it extensive if you can say-- 00:25:08.515 --> 00:25:16.140 align:middle line:84% if you can assign to that property a local field little a. 00:25:16.140 --> 00:25:20.617 align:middle line:84% And field means it's defined at every point in space. 00:25:20.617 --> 00:25:24.840 align:middle line:90% x is x, y, z, and t is time. 00:25:24.840 --> 00:25:27.900 align:middle line:90% So this is the field. 00:25:27.900 --> 00:25:28.760 align:middle line:90% Like density. 00:25:28.760 --> 00:25:33.010 align:middle line:90% 00:25:33.010 --> 00:25:35.830 align:middle line:84% So you want to-- you are assuming 00:25:35.830 --> 00:25:44.080 align:middle line:84% that you can define a density field as a function of position 00:25:44.080 --> 00:25:46.660 align:middle line:90% and time. 00:25:46.660 --> 00:25:52.120 align:middle line:84% So that you can write the overall property 00:25:52.120 --> 00:25:58.240 align:middle line:84% for the material volume as the sum of contributions 00:25:58.240 --> 00:26:00.160 align:middle line:90% for all the parts. 00:26:00.160 --> 00:26:06.670 align:middle line:84% So in other words, the integral of that property per unit volume 00:26:06.670 --> 00:26:10.130 align:middle line:90% times the volume. 00:26:10.130 --> 00:26:15.895 align:middle line:84% OK, so rho times a is property A per unit volume. 00:26:15.895 --> 00:26:20.790 align:middle line:90% 00:26:20.790 --> 00:26:22.895 align:middle line:90% OK. 00:26:22.895 --> 00:26:28.220 align:middle line:84% Once you have that, you have seen in the balance equations-- 00:26:28.220 --> 00:26:30.460 align:middle line:84% for example, the left-hand side of the energy balance 00:26:30.460 --> 00:26:35.200 align:middle line:84% or the entropy balance, you need to take the rate of change 00:26:35.200 --> 00:26:41.480 align:middle line:84% of the energy of your system or of the entropy of your system. 00:26:41.480 --> 00:26:45.200 align:middle line:84% So we need to take the rate of change, d by dt, 00:26:45.200 --> 00:26:52.200 align:middle line:84% of this integral to compute the left-hand side of the balance 00:26:52.200 --> 00:26:54.800 align:middle line:90% equations. 00:26:54.800 --> 00:27:04.170 align:middle line:84% And since the material volume moves, 00:27:04.170 --> 00:27:09.420 align:middle line:84% you cannot just take the derivative inside the integral. 00:27:09.420 --> 00:27:15.370 align:middle line:84% Yes, you can, that does part of the contribution. 00:27:15.370 --> 00:27:18.070 align:middle line:84% But there is another part which is due to the fact 00:27:18.070 --> 00:27:20.260 align:middle line:84% that the limits of integration, namely 00:27:20.260 --> 00:27:25.970 align:middle line:84% this surface that follows the particles as they move, moves. 00:27:25.970 --> 00:27:32.150 align:middle line:84% And therefore, you have to write a surface integral-- 00:27:32.150 --> 00:27:34.840 align:middle line:84% so an integral over the surface-- 00:27:34.840 --> 00:27:38.560 align:middle line:90% of-- that takes into account-- 00:27:38.560 --> 00:27:45.180 align:middle line:84% for each of these surface positions, 00:27:45.180 --> 00:27:47.700 align:middle line:90% you have to take into account-- 00:27:47.700 --> 00:27:54.700 align:middle line:84% first of all, you assign a unit vector. 00:27:54.700 --> 00:27:58.900 align:middle line:84% And by convention, usually I typically 00:27:58.900 --> 00:28:01.770 align:middle line:90% choose positive outward. 00:28:01.770 --> 00:28:04.530 align:middle line:90% 00:28:04.530 --> 00:28:10.920 align:middle line:84% And then I have to see how the molecules on this point 00:28:10.920 --> 00:28:13.640 align:middle line:90% of the surface move. 00:28:13.640 --> 00:28:19.040 align:middle line:84% And that I call the velocity, v sub m. 00:28:19.040 --> 00:28:23.370 align:middle line:90% 00:28:23.370 --> 00:28:28.466 align:middle line:84% Which we will call soon the barycentric velocity. 00:28:28.466 --> 00:28:35.800 align:middle line:90% 00:28:35.800 --> 00:28:39.580 align:middle line:84% Because, in fact, before I call it v sub m, 00:28:39.580 --> 00:28:44.370 align:middle line:84% it simply should be the velocity assigned to the surface 00:28:44.370 --> 00:28:47.140 align:middle line:90% of the material volume. 00:28:47.140 --> 00:28:50.470 align:middle line:84% And then it turns out that since the volume is a material volume, 00:28:50.470 --> 00:28:55.380 align:middle line:84% so it's made of particles, that part of the surface 00:28:55.380 --> 00:28:59.570 align:middle line:84% moves with the velocity of the particles that are there. 00:28:59.570 --> 00:29:02.450 align:middle line:84% And the velocity of the particles is vm. 00:29:02.450 --> 00:29:08.000 align:middle line:90% 00:29:08.000 --> 00:29:13.640 align:middle line:84% So the net-- so since the velocity may have a different 00:29:13.640 --> 00:29:17.060 align:middle line:84% direction than the normal to the surface, 00:29:17.060 --> 00:29:27.830 align:middle line:84% what brings the particles, say, out of this volume 00:29:27.830 --> 00:29:31.580 align:middle line:84% and contribute to that surface integral-- 00:29:31.580 --> 00:29:34.850 align:middle line:84% or contribute to changing the surface integral 00:29:34.850 --> 00:29:38.460 align:middle line:84% is only the component that is in the direction of the normal. 00:29:38.460 --> 00:29:42.680 align:middle line:84% So that's why you take the scalar product of v and n 00:29:42.680 --> 00:29:50.210 align:middle line:84% to get the normal component that contributes to moving stuff out 00:29:50.210 --> 00:29:52.510 align:middle line:90% of that-- 00:29:52.510 --> 00:29:54.805 align:middle line:90% say energy out of that surface. 00:29:54.805 --> 00:29:58.200 align:middle line:90% 00:29:58.200 --> 00:30:03.370 align:middle line:84% And then you convert by using the divergence 00:30:03.370 --> 00:30:08.830 align:middle line:84% theorem, a surface integral of this form something, 00:30:08.830 --> 00:30:15.590 align:middle line:84% dotted into n, is equal to the integral of the volume-- 00:30:15.590 --> 00:30:17.890 align:middle line:84% the volume integral of the divergence 00:30:17.890 --> 00:30:22.300 align:middle line:90% of whatever is dotted into n. 00:30:22.300 --> 00:30:23.325 align:middle line:90% And so you get this. 00:30:23.325 --> 00:30:31.790 align:middle line:90% 00:30:31.790 --> 00:30:34.700 align:middle line:84% So we transformed this, the rate of change 00:30:34.700 --> 00:30:40.230 align:middle line:84% in an integral-- in a volume integral of two contributions, 00:30:40.230 --> 00:30:48.370 align:middle line:84% the partial with respect to t and the divergence of rho a vm. 00:30:48.370 --> 00:30:55.575 align:middle line:84% These will be called the convective flux in a moment. 00:30:55.575 --> 00:31:02.240 align:middle line:90% 00:31:02.240 --> 00:31:07.346 align:middle line:84% Because that property that is exchanged-- 00:31:07.346 --> 00:31:10.185 align:middle line:90% 00:31:10.185 --> 00:31:10.685 align:middle line:90% OK. 00:31:10.685 --> 00:31:15.200 align:middle line:90% 00:31:15.200 --> 00:31:20.100 align:middle line:84% OK, we are talking about property A. So property A, 00:31:20.100 --> 00:31:23.930 align:middle line:90% like, say, heat-- 00:31:23.930 --> 00:31:31.500 align:middle line:84% I mean energy, if you have a heat interaction happening 00:31:31.500 --> 00:31:35.910 align:middle line:84% on the surface of this control volume-- 00:31:35.910 --> 00:31:42.530 align:middle line:84% so you have, say, another vector here, 00:31:42.530 --> 00:31:51.880 align:middle line:84% which would be q double prime, that is important in your energy 00:31:51.880 --> 00:31:58.390 align:middle line:84% balance because even if it is true that the surface follows 00:31:58.390 --> 00:32:01.155 align:middle line:84% the particles-- so the particles don't go out. 00:32:01.155 --> 00:32:05.480 align:middle line:90% 00:32:05.480 --> 00:32:09.230 align:middle line:84% But if there is a heat interaction 00:32:09.230 --> 00:32:11.990 align:middle line:84% between those particles inside your material 00:32:11.990 --> 00:32:14.000 align:middle line:84% volume and those outside, then there 00:32:14.000 --> 00:32:15.780 align:middle line:84% is energy transfer between the two, 00:32:15.780 --> 00:32:19.890 align:middle line:84% and the energy transfer is represented by this vector here. 00:32:19.890 --> 00:32:28.860 align:middle line:84% And there is also, of course, an entropy transfer 00:32:28.860 --> 00:32:31.840 align:middle line:84% in the same direction if it is a heat interaction. 00:32:31.840 --> 00:32:40.480 align:middle line:90% 00:32:40.480 --> 00:32:46.940 align:middle line:90% So what we define as flux-- 00:32:46.940 --> 00:32:49.200 align:middle line:90% because these are fluxes. 00:32:49.200 --> 00:32:55.610 align:middle line:84% So it's energy per unit area per unit time. 00:32:55.610 --> 00:33:02.240 align:middle line:84% Energy exchanged per unit area per unit time. 00:33:02.240 --> 00:33:04.970 align:middle line:90% And we give that-- 00:33:04.970 --> 00:33:08.570 align:middle line:90% the genetic symbol J sub A-- 00:33:08.570 --> 00:33:20.930 align:middle line:84% or actually, J 0 sub A. And it's defined this way. 00:33:20.930 --> 00:33:25.010 align:middle line:84% And once I have defined that, I can also 00:33:25.010 --> 00:33:31.360 align:middle line:84% assign a velocity of transfer of that quantity. 00:33:31.360 --> 00:33:36.980 align:middle line:90% 00:33:36.980 --> 00:33:41.070 align:middle line:84% So I have the flux, to which is assigned 00:33:41.070 --> 00:33:45.680 align:middle line:84% the velocity of transfer, by dividing the flux 00:33:45.680 --> 00:33:49.050 align:middle line:90% by the property per unit volume. 00:33:49.050 --> 00:33:53.240 align:middle line:84% So if I take q double prime and divide it by rho u-- 00:33:53.240 --> 00:34:03.640 align:middle line:84% q double prime divided by rho u, that gives me a velocity. 00:34:03.640 --> 00:34:10.050 align:middle line:84% We could call that velocity the transport-- 00:34:10.050 --> 00:34:12.070 align:middle line:90% the heat transport velocity. 00:34:12.070 --> 00:34:14.770 align:middle line:90% 00:34:14.770 --> 00:34:19.389 align:middle line:84% And also, you could have q double prime 00:34:19.389 --> 00:34:24.130 align:middle line:84% over T, which is the entropy divided by rho S, 00:34:24.130 --> 00:34:33.790 align:middle line:84% And that would be the transport velocity for entropy. 00:34:33.790 --> 00:34:41.969 align:middle line:90% 00:34:41.969 --> 00:34:47.090 align:middle line:84% The transport velocity for momentum (I meant, mass) 00:34:47.090 --> 00:34:51.920 align:middle line:84% is the barycentric velocity, the velocity that you typically 00:34:51.920 --> 00:34:53.810 align:middle line:90% use in fluid mechanics. 00:34:53.810 --> 00:34:58.160 align:middle line:84% If you say, the fluid has a velocity field v, 00:34:58.160 --> 00:35:05.420 align:middle line:84% you mean this velocity is the mass transport velocity. 00:35:05.420 --> 00:35:24.940 align:middle line:84% So it's Jm, which is the flux of mass, divided by rho, 00:35:24.940 --> 00:35:29.410 align:middle line:84% because rho is the mass per unit volume, 00:35:29.410 --> 00:35:35.030 align:middle line:84% and that is the barycentric velocity of fluid mechanics. 00:35:35.030 --> 00:35:51.980 align:middle line:90% 00:35:51.980 --> 00:35:53.684 align:middle line:90% OK. 00:35:53.684 --> 00:36:03.150 align:middle line:84% Now-- so this velocity-- this flux with the 0 on top is, 00:36:03.150 --> 00:36:09.320 align:middle line:90% in terms of a reference frame-- 00:36:09.320 --> 00:36:12.610 align:middle line:90% 00:36:12.610 --> 00:36:13.570 align:middle line:90% yeah. 00:36:13.570 --> 00:36:14.070 align:middle line:90% OK. 00:36:14.070 --> 00:36:17.020 align:middle line:84% Reference frame the laboratory reference frame 00:36:17.020 --> 00:36:19.090 align:middle line:84% with respect to which you measure velocity 00:36:19.090 --> 00:36:21.340 align:middle line:90% of your fluid elements. 00:36:21.340 --> 00:36:28.360 align:middle line:84% However, you could also consider a local reference frame 00:36:28.360 --> 00:36:34.150 align:middle line:84% to your fluid element that moves with the velocity vm. 00:36:34.150 --> 00:36:36.840 align:middle line:90% 00:36:36.840 --> 00:36:40.430 align:middle line:84% So you move with your fluid element. 00:36:40.430 --> 00:36:43.790 align:middle line:90% 00:36:43.790 --> 00:36:48.600 align:middle line:84% Well actually, the fluid element is made up of many particles. 00:36:48.600 --> 00:36:51.770 align:middle line:84% So you have-- you're not moving with the velocity of one 00:36:51.770 --> 00:36:55.610 align:middle line:84% particular particle, but with a weighted average 00:36:55.610 --> 00:36:59.910 align:middle line:84% of the velocity, which is the barycentric velocity 00:36:59.910 --> 00:37:01.180 align:middle line:90% of those particles. 00:37:01.180 --> 00:37:07.250 align:middle line:90% 00:37:07.250 --> 00:37:13.340 align:middle line:84% So relative to that, if I take the velocity of transport 00:37:13.340 --> 00:37:21.240 align:middle line:84% of this property, and I view it from that reference frame-- 00:37:21.240 --> 00:37:24.140 align:middle line:84% so the relative velocity is vA minus vm. 00:37:24.140 --> 00:37:26.710 align:middle line:90% 00:37:26.710 --> 00:37:30.030 align:middle line:90% So I define these flux. 00:37:30.030 --> 00:37:34.520 align:middle line:90% 00:37:34.520 --> 00:37:39.510 align:middle line:90% And we call it diffusive flux. 00:37:39.510 --> 00:37:42.710 align:middle line:90% 00:37:42.710 --> 00:37:47.660 align:middle line:84% So it's the flux relative to the local barycentric velocity. 00:37:47.660 --> 00:37:53.560 align:middle line:84% This is the most practically used 00:37:53.560 --> 00:37:57.830 align:middle line:84% and what appears typically in the balance equations, 00:37:57.830 --> 00:38:00.380 align:middle line:84% but also, this other one appears. 00:38:00.380 --> 00:38:03.850 align:middle line:84% So you have to be careful if you are in this business 00:38:03.850 --> 00:38:12.690 align:middle line:84% and you see balance equations, you have to be careful whether-- 00:38:12.690 --> 00:38:15.330 align:middle line:84% because sometimes people don't put the naught. 00:38:15.330 --> 00:38:19.960 align:middle line:84% Actually, almost never put the distinction. 00:38:19.960 --> 00:38:24.360 align:middle line:84% They just pick one way of describing fluxes 00:38:24.360 --> 00:38:27.780 align:middle line:84% and then keep going, and presume that 00:38:27.780 --> 00:38:33.600 align:middle line:84% from the context of the paper, you can understand which one-- 00:38:33.600 --> 00:38:35.730 align:middle line:90% which is what. 00:38:35.730 --> 00:38:40.875 align:middle line:84% Now the same concept of flux applies also, as you know, 00:38:40.875 --> 00:38:42.130 align:middle line:90% to vectors. 00:38:42.130 --> 00:38:43.535 align:middle line:90% So you can transfer momentum. 00:38:43.535 --> 00:38:46.430 align:middle line:90% 00:38:46.430 --> 00:38:55.390 align:middle line:84% And so if you define the diffusive flux where 00:38:55.390 --> 00:38:59.770 align:middle line:90% the property A is momentum-- 00:38:59.770 --> 00:39:05.637 align:middle line:84% actually-- so little a is the barycentric velocity itself. 00:39:05.637 --> 00:39:10.390 align:middle line:90% 00:39:10.390 --> 00:39:15.970 align:middle line:90% Then you obtain the diffusion-- 00:39:15.970 --> 00:39:21.340 align:middle line:84% momentum diffusion flux, which is the negative of the stress 00:39:21.340 --> 00:39:21.840 align:middle line:90% tensor. 00:39:21.840 --> 00:39:25.480 align:middle line:90% 00:39:25.480 --> 00:39:26.985 align:middle line:90% Also called the pressure tensor. 00:39:26.985 --> 00:39:33.060 align:middle line:90% 00:39:33.060 --> 00:39:37.300 align:middle line:84% And typically, the stress tensor is 00:39:37.300 --> 00:39:43.546 align:middle line:84% split into two parts, an isotropic part with-- 00:39:43.546 --> 00:39:48.180 align:middle line:90% 00:39:48.180 --> 00:39:52.110 align:middle line:84% delta is like a tensor isotropic. 00:39:52.110 --> 00:39:55.210 align:middle line:84% So 1, 1, 1 on the diagonal and 0 off the diagonal. 00:39:55.210 --> 00:40:02.096 align:middle line:84% And that-- this is the trace of-- 00:40:02.096 --> 00:40:09.340 align:middle line:90% 00:40:09.340 --> 00:40:12.910 align:middle line:84% pm is the trace of the stress tensor-- 00:40:12.910 --> 00:40:16.280 align:middle line:84% I'm sorry, of the pressure tensor, and divided by 3, 00:40:16.280 --> 00:40:20.330 align:middle line:84% and it is called the mechanical energy (I meant, 00:40:20.330 --> 00:40:22.480 align:middle line:84% mechanical pressure), which is not 00:40:22.480 --> 00:40:29.025 align:middle line:84% necessarily equal to the thermodynamic pressure. 00:40:29.025 --> 00:40:38.100 align:middle line:90% 00:40:38.100 --> 00:40:40.338 align:middle line:90% OK. 00:40:40.338 --> 00:40:50.240 align:middle line:84% So what we instead call convective flux 00:40:50.240 --> 00:40:52.970 align:middle line:84% is the flux of that property which 00:40:52.970 --> 00:40:57.620 align:middle line:84% is transported by the simple translation 00:40:57.620 --> 00:40:59.330 align:middle line:90% of the fluid element. 00:40:59.330 --> 00:41:07.610 align:middle line:90% 00:41:07.610 --> 00:41:12.590 align:middle line:84% The fact that the fluid element moves, 00:41:12.590 --> 00:41:16.650 align:middle line:84% it brings with itself the properties that it has inside. 00:41:16.650 --> 00:41:20.550 align:middle line:84% So that produces a flux movement of the property. 00:41:20.550 --> 00:41:25.830 align:middle line:84% If there is energy in that box and you move the box, 00:41:25.830 --> 00:41:27.460 align:middle line:90% you're moving energy. 00:41:27.460 --> 00:41:30.170 align:middle line:90% So that is convection. 00:41:30.170 --> 00:41:40.050 align:middle line:84% And it's the difference between this and the diffusive. 00:41:40.050 --> 00:41:44.050 align:middle line:90% 00:41:44.050 --> 00:41:50.030 align:middle line:84% Or, if you want, you can write the flux with respect 00:41:50.030 --> 00:41:55.880 align:middle line:84% to the reference frame as a sum of the diffusive 00:41:55.880 --> 00:41:57.805 align:middle line:90% plus the convective flux. 00:41:57.805 --> 00:42:03.430 align:middle line:90% 00:42:03.430 --> 00:42:04.906 align:middle line:90% OK. 00:42:04.906 --> 00:42:17.020 align:middle line:84% So the convective momentum flux is the fact 00:42:17.020 --> 00:42:23.790 align:middle line:84% that this fluid element, since it's moving, it has momentum. 00:42:23.790 --> 00:42:26.070 align:middle line:84% And the momentum per unit volume is 00:42:26.070 --> 00:42:29.910 align:middle line:84% rho v. And since it's moving with velocity v, 00:42:29.910 --> 00:42:35.250 align:middle line:84% you're transporting momentum by simple motion. 00:42:35.250 --> 00:42:38.810 align:middle line:84% So it's not diffusion, it's just due to the fact 00:42:38.810 --> 00:42:43.610 align:middle line:84% that the thing is moving, and that is convective momentum 00:42:43.610 --> 00:42:44.590 align:middle line:90% transport. 00:42:44.590 --> 00:43:05.120 align:middle line:90% 00:43:05.120 --> 00:43:14.080 align:middle line:84% Now inside your material volume, inside that fluid element, 00:43:14.080 --> 00:43:18.140 align:middle line:84% there are certain things that happen. 00:43:18.140 --> 00:43:20.210 align:middle line:84% For example, per unit volume-- for example, 00:43:20.210 --> 00:43:22.030 align:middle line:84% the entropy produced by irreversibility 00:43:22.030 --> 00:43:23.840 align:middle line:90% is locally produced. 00:43:23.840 --> 00:43:33.450 align:middle line:84% So it's something that when you want to account for it, 00:43:33.450 --> 00:43:38.300 align:middle line:84% you have to take it, multiply it by the volume, 00:43:38.300 --> 00:43:41.910 align:middle line:90% and then integrate. 00:43:41.910 --> 00:43:45.310 align:middle line:84% So this you can view it as a source term. 00:43:45.310 --> 00:43:47.500 align:middle line:84% I mean, for the entropy balance, it's a source term. 00:43:47.500 --> 00:43:51.090 align:middle line:84% It cannot be a sink because it would violate the principle 00:43:51.090 --> 00:43:53.970 align:middle line:90% of entropy non-decrease. 00:43:53.970 --> 00:44:05.250 align:middle line:84% But in other models, for example, often in heat transfer, 00:44:05.250 --> 00:44:09.930 align:middle line:90% you use terms like-- 00:44:09.930 --> 00:44:12.700 align:middle line:84% you add a term to the balance equation, 00:44:12.700 --> 00:44:15.000 align:middle line:90% you call it triple prime q. 00:44:15.000 --> 00:44:18.210 align:middle line:90% It's the entropy-- 00:44:18.210 --> 00:44:21.660 align:middle line:90% I'm sorry, the energy-- 00:44:21.660 --> 00:44:25.780 align:middle line:84% internal energy deposited locally, for example, 00:44:25.780 --> 00:44:26.620 align:middle line:90% by joule effect. 00:44:26.620 --> 00:44:29.520 align:middle line:90% If you have a conductor-- 00:44:29.520 --> 00:44:31.690 align:middle line:84% electricity conductor, by joule effect, 00:44:31.690 --> 00:44:36.180 align:middle line:84% that dissipates electrical energy 00:44:36.180 --> 00:44:38.800 align:middle line:84% and transforms it into internal energy. 00:44:38.800 --> 00:44:42.220 align:middle line:84% So if you make a balance of internal energy, 00:44:42.220 --> 00:44:45.780 align:middle line:84% you will find a source term, and that source term 00:44:45.780 --> 00:44:52.460 align:middle line:84% is a source-per-unit volume, so it's like a sigma of energy. 00:44:52.460 --> 00:44:55.140 align:middle line:90% 00:44:55.140 --> 00:44:57.780 align:middle line:84% And usually you can distinguish between those 00:44:57.780 --> 00:45:03.820 align:middle line:84% that are produced by local effects-- 00:45:03.820 --> 00:45:06.190 align:middle line:84% like the joule effect is a local effect. 00:45:06.190 --> 00:45:19.030 align:middle line:84% Or a collision-produced effects, including chemical reactions, 00:45:19.030 --> 00:45:23.430 align:middle line:84% that essentially, chemical reactions, 00:45:23.430 --> 00:45:28.190 align:middle line:84% you can consider it as a source of products 00:45:28.190 --> 00:45:32.330 align:middle line:84% of the reaction and the sink for the reactants 00:45:32.330 --> 00:45:40.150 align:middle line:84% of the reaction, which occurs locally in your volume. 00:45:40.150 --> 00:45:43.810 align:middle line:84% And so these-- you would call it with a superscript i. 00:45:43.810 --> 00:45:47.660 align:middle line:84% To be distinguished with the externally generated-- 00:45:47.660 --> 00:45:55.210 align:middle line:84% for example, if this fluid element lives 00:45:55.210 --> 00:46:03.280 align:middle line:90% in a gravitational field g. 00:46:03.280 --> 00:46:06.880 align:middle line:90% And, of course, it has mass. 00:46:06.880 --> 00:46:09.520 align:middle line:84% So it also has a potential energy. 00:46:09.520 --> 00:46:17.650 align:middle line:84% And the fact that it moves makes that potential energy change. 00:46:17.650 --> 00:46:20.150 align:middle line:84% And that's not due to a local effect. 00:46:20.150 --> 00:46:22.820 align:middle line:84% It's due to an externally applied field. 00:46:22.820 --> 00:46:25.450 align:middle line:84% The same, if you had charges there 00:46:25.450 --> 00:46:30.970 align:middle line:84% and you had an electric field, say, an electric field 00:46:30.970 --> 00:46:33.350 align:middle line:90% like this. 00:46:33.350 --> 00:46:41.040 align:middle line:90% 00:46:41.040 --> 00:46:42.420 align:middle line:90% And then you move the charges-- 00:46:42.420 --> 00:46:45.310 align:middle line:90% 00:46:45.310 --> 00:46:49.260 align:middle line:84% so the electrical-- the electrostatic energy 00:46:49.260 --> 00:46:55.920 align:middle line:84% of your fluid element changes, and that 00:46:55.920 --> 00:46:59.070 align:middle line:90% gets into this volume term. 00:46:59.070 --> 00:47:03.245 align:middle line:84% It could be up or down, so it could be a sink or a source. 00:47:03.245 --> 00:47:06.070 align:middle line:90% 00:47:06.070 --> 00:47:08.830 align:middle line:84% So these in principle for various properties, 00:47:08.830 --> 00:47:12.000 align:middle line:90% could be positive or negative. 00:47:12.000 --> 00:47:13.920 align:middle line:84% It's only for the entropy production 00:47:13.920 --> 00:47:18.590 align:middle line:84% that you know precisely which sign it must have. 00:47:18.590 --> 00:47:22.350 align:middle line:90% 00:47:22.350 --> 00:47:27.780 align:middle line:84% So finally, with all those ingredients, 00:47:27.780 --> 00:47:29.930 align:middle line:84% you can write the balance equations 00:47:29.930 --> 00:47:33.140 align:middle line:84% for this extensive property A. And it 00:47:33.140 --> 00:47:40.860 align:middle line:84% says the rate of change of the property inside your material 00:47:40.860 --> 00:47:45.490 align:middle line:84% volume is equal to the amount of-- 00:47:45.490 --> 00:47:53.160 align:middle line:84% the amount of-- the contribution of these sink source terms 00:47:53.160 --> 00:47:55.155 align:middle line:84% integrated, of course, over the volume. 00:47:55.155 --> 00:48:00.230 align:middle line:90% 00:48:00.230 --> 00:48:03.550 align:middle line:84% Minus the amount that leaves the volume through the surface. 00:48:03.550 --> 00:48:07.640 align:middle line:90% 00:48:07.640 --> 00:48:10.340 align:middle line:84% And so that amount that leaves is 00:48:10.340 --> 00:48:16.280 align:middle line:84% equal to the velocity associated with the flux of that property 00:48:16.280 --> 00:48:29.665 align:middle line:84% relative to the velocity of the surface of your material volume. 00:48:29.665 --> 00:48:33.225 align:middle line:90% 00:48:33.225 --> 00:48:33.725 align:middle line:90% OK. 00:48:33.725 --> 00:48:39.660 align:middle line:90% 00:48:39.660 --> 00:48:44.040 align:middle line:84% Times the property per unit volume. 00:48:44.040 --> 00:48:48.300 align:middle line:84% And also here, you see something dotted into n 00:48:48.300 --> 00:48:50.310 align:middle line:84% and integrated over a surface, you 00:48:50.310 --> 00:48:54.180 align:middle line:84% can use the divergence theorem and transform it 00:48:54.180 --> 00:49:02.200 align:middle line:84% into the divergence of whatever is 00:49:02.200 --> 00:49:06.830 align:middle line:84% dotted into the outward normal to the surface. 00:49:06.830 --> 00:49:14.340 align:middle line:84% And you may recognize here that since this velocity is vm-- 00:49:14.340 --> 00:49:23.010 align:middle line:84% vA minus vm dot n times rho a is the diffusive flux we defined it 00:49:23.010 --> 00:49:24.810 align:middle line:90% in the previous slide. 00:49:24.810 --> 00:49:26.800 align:middle line:90% That's why I get that. 00:49:26.800 --> 00:49:32.360 align:middle line:90% 00:49:32.360 --> 00:49:40.880 align:middle line:84% So this is, the right-hand side of the balance equation. 00:49:40.880 --> 00:49:46.420 align:middle line:90% 00:49:46.420 --> 00:49:49.240 align:middle line:84% And the left-hand side we already wrote 00:49:49.240 --> 00:49:53.090 align:middle line:90% in a couple of slides ago. 00:49:53.090 --> 00:49:54.445 align:middle line:90% We wrote it in this form. 00:49:54.445 --> 00:49:57.790 align:middle line:90% 00:49:57.790 --> 00:50:03.620 align:middle line:84% So this is the general form of a balance equation for property 00:50:03.620 --> 00:50:07.760 align:middle line:84% A. For an extensive property A. And I 00:50:07.760 --> 00:50:11.420 align:middle line:84% think I'll repeat it in the next slide 00:50:11.420 --> 00:50:13.195 align:middle line:90% because we need to-- here it is. 00:50:13.195 --> 00:50:15.810 align:middle line:90% 00:50:15.810 --> 00:50:17.482 align:middle line:90% OK. 00:50:17.482 --> 00:50:25.300 align:middle line:84% Now, since that equation-- balance equation 00:50:25.300 --> 00:50:31.930 align:middle line:84% must hold for whatever material volume, large, small, 00:50:31.930 --> 00:50:35.680 align:middle line:84% and so on, arbitrary, the only way 00:50:35.680 --> 00:50:40.450 align:middle line:84% that this equation can hold for arbitrary volumes 00:50:40.450 --> 00:50:44.380 align:middle line:84% is that the arguments of the integrals be equal, 00:50:44.380 --> 00:50:50.800 align:middle line:84% and therefore, you can cross away the integrals 00:50:50.800 --> 00:50:54.440 align:middle line:84% and keep the equality between the integrands, 00:50:54.440 --> 00:51:10.280 align:middle line:84% and that's your indefinite balance equation for property A. 00:51:10.280 --> 00:51:16.800 align:middle line:84% Remember that we said that the diffusive flux is 00:51:16.800 --> 00:51:20.370 align:middle line:84% the Lagrangian flux minus the convective flux 00:51:20.370 --> 00:51:21.930 align:middle line:90% of that property. 00:51:21.930 --> 00:51:31.100 align:middle line:84% So we can combine these two equations, 00:51:31.100 --> 00:51:36.540 align:middle line:84% and therefore, rewrite it in either this form or that form. 00:51:36.540 --> 00:51:39.985 align:middle line:84% These are two equivalent ways to write the balance equation. 00:51:39.985 --> 00:51:42.670 align:middle line:90% 00:51:42.670 --> 00:51:47.710 align:middle line:84% But the important thing is that in this left-hand side, 00:51:47.710 --> 00:51:54.710 align:middle line:84% you get J 0, whereas here, you get J. 00:51:54.710 --> 00:51:58.640 align:middle line:84% And these combination of partial derivative 00:51:58.640 --> 00:52:02.750 align:middle line:84% and the barycentric velocity times the gradient of property 00:52:02.750 --> 00:52:05.240 align:middle line:84% A, which comes in when you substitute these 00:52:05.240 --> 00:52:11.740 align:middle line:84% into there, as you know, it's called the material derivative. 00:52:11.740 --> 00:52:14.540 align:middle line:90% 00:52:14.540 --> 00:52:20.050 align:middle line:84% And it represents how things change as a function of time 00:52:20.050 --> 00:52:26.080 align:middle line:84% as viewed from the fluid element by moving together 00:52:26.080 --> 00:52:29.000 align:middle line:84% with the fluid element at the barycentric velocity. 00:52:29.000 --> 00:52:31.950 align:middle line:90% 00:52:31.950 --> 00:52:34.110 align:middle line:84% And following, therefore, the fluid element. 00:52:34.110 --> 00:52:38.160 align:middle line:84% That's why it follows the trajectory that 00:52:38.160 --> 00:52:41.500 align:middle line:84% is called the path line in fluid mechanics. 00:52:41.500 --> 00:52:46.600 align:middle line:90% 00:52:46.600 --> 00:52:48.310 align:middle line:90% OK. 00:52:48.310 --> 00:52:50.230 align:middle line:84% So these are the balance equations. 00:52:50.230 --> 00:52:53.170 align:middle line:90% We are going to-- 00:52:53.170 --> 00:52:56.760 align:middle line:90% I think we are going to use-- 00:52:56.760 --> 00:52:58.680 align:middle line:90% well, either one. 00:52:58.680 --> 00:53:00.200 align:middle line:90% Probably more this one. 00:53:00.200 --> 00:53:09.610 align:middle line:90% 00:53:09.610 --> 00:53:13.480 align:middle line:84% So if you do that, what we have done for the one-dimensional 00:53:13.480 --> 00:53:16.750 align:middle line:84% problem, if you do it for the three-dimensional-- 00:53:16.750 --> 00:53:24.310 align:middle line:84% I mean for the full-dimensional stuff, 00:53:24.310 --> 00:53:28.275 align:middle line:84% you can write the mass balance equation in this way. 00:53:28.275 --> 00:53:31.490 align:middle line:90% 00:53:31.490 --> 00:53:37.370 align:middle line:84% You can write the energy balance in this way in terms 00:53:37.370 --> 00:53:40.760 align:middle line:90% of the diffusive fluxes. 00:53:40.760 --> 00:53:41.920 align:middle line:90% The entropy balance. 00:53:41.920 --> 00:53:45.750 align:middle line:90% 00:53:45.750 --> 00:53:50.250 align:middle line:84% But also, you can write the momentum balance. 00:53:50.250 --> 00:53:52.396 align:middle line:90% And, of course, the material-- 00:53:52.396 --> 00:53:55.610 align:middle line:90% 00:53:55.610 --> 00:54:03.115 align:middle line:84% actually, these two are related to one another. 00:54:03.115 --> 00:54:06.310 align:middle line:90% 00:54:06.310 --> 00:54:08.680 align:middle line:84% Because what you would need to do 00:54:08.680 --> 00:54:11.900 align:middle line:84% is to write a balance equation for each kind of particle, 00:54:11.900 --> 00:54:12.730 align:middle line:90% each type-- 00:54:12.730 --> 00:54:14.980 align:middle line:90% like each i. 00:54:14.980 --> 00:54:17.800 align:middle line:84% And then once you have it, you can express it 00:54:17.800 --> 00:54:22.490 align:middle line:84% in terms of mole-specific or mass-specific and so on. 00:54:22.490 --> 00:54:24.020 align:middle line:90% And you can write it-- 00:54:24.020 --> 00:54:33.310 align:middle line:84% for example, here, this symbol xi is the mass fraction. 00:54:33.310 --> 00:54:36.730 align:middle line:84% In other places-- actually, probably in the next slide, 00:54:36.730 --> 00:54:40.365 align:middle line:84% I call that same mass fraction with the symbol x. 00:54:40.365 --> 00:54:44.900 align:middle line:90% 00:54:44.900 --> 00:54:47.330 align:middle line:84% So here is the definition of the barycentric velocity 00:54:47.330 --> 00:54:53.150 align:middle line:84% that I mentioned, and here is the expression for the flux of-- 00:54:53.150 --> 00:55:01.990 align:middle line:84% the mass flux in laboratory frame. 00:55:01.990 --> 00:55:08.170 align:middle line:84% It's rho times v. And it is related to the mass 00:55:08.170 --> 00:55:10.440 align:middle line:84% and to the velocities of the various (constituents) 00:55:10.440 --> 00:55:14.560 align:middle line:84% and this table, I'm not describing it, 00:55:14.560 --> 00:55:16.210 align:middle line:90% but simply is there. 00:55:16.210 --> 00:55:19.770 align:middle line:84% It's useful just to mention how many different ways you have 00:55:19.770 --> 00:55:22.800 align:middle line:84% to do the accounting, and depending 00:55:22.800 --> 00:55:26.790 align:middle line:84% on what your application or depending on your community, 00:55:26.790 --> 00:55:36.480 align:middle line:84% people use mole fractions, mass fractions, concentrations, 00:55:36.480 --> 00:55:37.480 align:middle line:90% and so on. 00:55:37.480 --> 00:55:39.990 align:middle line:84% And so these are the interrelations 00:55:39.990 --> 00:55:47.990 align:middle line:84% that intercur between these various properties. 00:55:47.990 --> 00:55:50.600 align:middle line:84% So I'll leave it there just for reference. 00:55:50.600 --> 00:55:57.830 align:middle line:84% And also, this slide provides a list of examples-- 00:55:57.830 --> 00:56:02.620 align:middle line:84% the main examples for those extensive properties 00:56:02.620 --> 00:56:07.140 align:middle line:84% that we have written the balance equations for. 00:56:07.140 --> 00:56:11.450 align:middle line:84% And for which we can define convective fluxes and diffusive 00:56:11.450 --> 00:56:13.610 align:middle line:90% fluxes. 00:56:13.610 --> 00:56:17.540 align:middle line:84% And so the number of moles, the total mole, 00:56:17.540 --> 00:56:26.150 align:middle line:84% the mass of particles of a certain type, the total mass, 00:56:26.150 --> 00:56:31.200 align:middle line:84% the charge of particles of a certain type, the total charge, 00:56:31.200 --> 00:56:35.340 align:middle line:84% the energy, the entropy, the kinetic energy, 00:56:35.340 --> 00:56:37.200 align:middle line:84% the potential gravitational energy, 00:56:37.200 --> 00:56:42.040 align:middle line:84% the potential electrical energy, and momentum. 00:56:42.040 --> 00:56:45.550 align:middle line:84% For each one of these, you can write 00:56:45.550 --> 00:56:48.430 align:middle line:90% balance equations of the form-- 00:56:48.430 --> 00:56:52.590 align:middle line:84% of the generic form that we have discussed. 00:56:52.590 --> 00:56:56.670 align:middle line:90% 00:56:56.670 --> 00:56:58.020 align:middle line:90% OK. 00:56:58.020 --> 00:57:02.860 align:middle line:90% So that is the balance equation. 00:57:02.860 --> 00:57:08.830 align:middle line:84% Now, just a small digression about the fact 00:57:08.830 --> 00:57:11.810 align:middle line:84% that we are dealing, in this model 00:57:11.810 --> 00:57:15.490 align:middle line:84% so far, and all that we have done, 00:57:15.490 --> 00:57:18.280 align:middle line:84% we are dealing with the idea of a continuum. 00:57:18.280 --> 00:57:25.990 align:middle line:84% The fact that you can assign to a position in space and time a-- 00:57:25.990 --> 00:57:32.080 align:middle line:84% for example, a density, it means that you 00:57:32.080 --> 00:57:42.890 align:middle line:84% have enough particles in your volume so that you can-- 00:57:42.890 --> 00:57:46.320 align:middle line:90% 00:57:46.320 --> 00:57:50.590 align:middle line:84% so that you can assume that all the particles are-- 00:57:50.590 --> 00:57:53.650 align:middle line:90% 00:57:53.650 --> 00:57:57.220 align:middle line:84% so that you can take the mass of the particles and consider 00:57:57.220 --> 00:58:02.740 align:middle line:84% it as uniformly spread in that volume. 00:58:02.740 --> 00:58:04.603 align:middle line:90% And that you can do only-- 00:58:04.603 --> 00:58:08.330 align:middle line:90% 00:58:08.330 --> 00:58:13.530 align:middle line:84% of course, that is a limit of the traditional fluid mechanics. 00:58:13.530 --> 00:58:16.150 align:middle line:90% 00:58:16.150 --> 00:58:22.200 align:middle line:84% And you can do it only if the Knudsen number-- 00:58:22.200 --> 00:58:29.540 align:middle line:84% the Knudsen number is the ratio between the mean-free path-- 00:58:29.540 --> 00:58:36.190 align:middle line:84% mean-free path is the travel distance 00:58:36.190 --> 00:58:44.520 align:middle line:84% that, on average, particles travel between one 00:58:44.520 --> 00:58:45.515 align:middle line:90% collision and the next. 00:58:45.515 --> 00:58:48.830 align:middle line:90% 00:58:48.830 --> 00:58:50.690 align:middle line:90% So that is lambda. 00:58:50.690 --> 00:58:56.030 align:middle line:84% And in the denominator is L. L is the typical size 00:58:56.030 --> 00:58:58.610 align:middle line:90% of your fluid mechanics-- 00:58:58.610 --> 00:59:04.280 align:middle line:84% it's the length scale of your fluid mechanics problem. 00:59:04.280 --> 00:59:07.970 align:middle line:84% Could be the boundary layer thickness 00:59:07.970 --> 00:59:13.230 align:middle line:84% or could be the diameter of your pipe if you are in a pipe flow. 00:59:13.230 --> 00:59:15.830 align:middle line:90% 00:59:15.830 --> 00:59:28.760 align:middle line:84% So not always the Knudsen number is smaller than 1. 00:59:28.760 --> 00:59:32.390 align:middle line:84% Smaller than-- much smaller than 1. 00:59:32.390 --> 00:59:39.860 align:middle line:84% So typically, one would say, if it is less than 1%, then 00:59:39.860 --> 00:59:43.950 align:middle line:84% you can apply the continuum hypothesis. 00:59:43.950 --> 00:59:47.750 align:middle line:84% If it is not, then you have to be careful 00:59:47.750 --> 00:59:50.000 align:middle line:84% because the results that you are obtaining 00:59:50.000 --> 00:59:53.730 align:middle line:84% may not correspond to the physical reality. 00:59:53.730 --> 00:59:56.270 align:middle line:84% So you can always do that in the model, 00:59:56.270 --> 00:59:59.870 align:middle line:84% but the results are not providing 00:59:59.870 --> 01:00:05.720 align:middle line:90% valid descriptions of reality. 01:00:05.720 --> 01:00:09.340 align:middle line:84% For example, this table gives for air. 01:00:09.340 --> 01:00:14.600 align:middle line:84% At sea level for dry air at 25 C, 1 atmosphere. 01:00:14.600 --> 01:00:24.606 align:middle line:84% The mean-free path, it's about 67 nanometers. 01:00:24.606 --> 01:00:31.890 align:middle line:90% 01:00:31.890 --> 01:00:36.580 align:middle line:84% So this condition is guaranteed when you see the green. 01:00:36.580 --> 01:00:43.110 align:middle line:84% So if I have a device that is a large, small-- 01:00:43.110 --> 01:00:51.370 align:middle line:84% even pretty small, like 70 microns, it's pretty small, 01:00:51.370 --> 01:00:55.090 align:middle line:84% you still can apply fluid mechanics. 01:00:55.090 --> 01:00:59.530 align:middle line:90% When your device gets smaller-- 01:00:59.530 --> 01:01:03.710 align:middle line:84% see, the denominator of the Knudsen number 01:01:03.710 --> 01:01:07.400 align:middle line:84% gets smaller, so the Knudsen number becomes larger. 01:01:07.400 --> 01:01:14.120 align:middle line:84% And so for small devices, you cannot-- 01:01:14.120 --> 01:01:22.030 align:middle line:84% we cannot use fluid mechanics or the continuum assumption. 01:01:22.030 --> 01:01:27.480 align:middle line:84% But even for air, if you go at this position, 01:01:27.480 --> 01:01:31.460 align:middle line:84% 100 kilometers altitude-- it's called the Karman line, 01:01:31.460 --> 01:01:36.500 align:middle line:84% there, the temperature is minus 80 degrees C 01:01:36.500 --> 01:01:41.480 align:middle line:84% and the pressure is hundreds of atmosphere. 01:01:41.480 --> 01:01:46.173 align:middle line:90% The mean-free path is about-- 01:01:46.173 --> 01:01:49.640 align:middle line:90% 01:01:49.640 --> 01:01:50.870 align:middle line:90% what is it? 01:01:50.870 --> 01:01:57.945 align:middle line:84% 100 times-- 50 times larger than at sea level. 01:01:57.945 --> 01:02:01.840 align:middle line:90% 01:02:01.840 --> 01:02:08.150 align:middle line:84% So there, it's the lambda that is high. 01:02:08.150 --> 01:02:10.990 align:middle line:90% So even for standard devices-- 01:02:10.990 --> 01:02:18.940 align:middle line:84% for example, a typical problem where the Knudsen number fails 01:02:18.940 --> 01:02:20.740 align:middle line:84% and you cannot apply fluid mechanics, 01:02:20.740 --> 01:02:28.420 align:middle line:84% it's in the boundary layer of the space vehicles that have 01:02:28.420 --> 01:02:32.590 align:middle line:84% to come back to the atmosphere, in the upper layers 01:02:32.590 --> 01:02:38.380 align:middle line:84% of the atmosphere, the lambda is so small (I meant, 01:02:38.380 --> 01:02:46.720 align:middle line:84% large) that the Knudsen number is of the order, say, 01:02:46.720 --> 01:02:58.080 align:middle line:84% 1 even for relatively large thicknesses. 01:02:58.080 --> 01:03:03.450 align:middle line:84% And therefore, that was, in fact, one of the problems 01:03:03.450 --> 01:03:08.220 align:middle line:84% when they were studying the reentry problem of the Space 01:03:08.220 --> 01:03:10.290 align:middle line:90% Shuttle. 01:03:10.290 --> 01:03:12.395 align:middle line:90% And not only-- also the-- 01:03:12.395 --> 01:03:17.070 align:middle line:90% 01:03:17.070 --> 01:03:22.130 align:middle line:84% the other vehicles that have re-enter through the atmosphere. 01:03:22.130 --> 01:03:23.994 align:middle line:90% OK. 01:03:23.994 --> 01:03:30.140 align:middle line:84% So remember that what we are doing refers to a continuum, 01:03:30.140 --> 01:03:32.340 align:middle line:84% so there are limitations to this. 01:03:32.340 --> 01:03:37.980 align:middle line:90% 01:03:37.980 --> 01:03:40.410 align:middle line:84% At the end, one of the final lectures 01:03:40.410 --> 01:03:45.005 align:middle line:84% we'll mention what we need to do when this fails. 01:03:45.005 --> 01:03:51.300 align:middle line:90% 01:03:51.300 --> 01:03:52.512 align:middle line:90% OK. 01:03:52.512 --> 01:03:57.020 align:middle line:84% Now we are in the part that we have 01:03:57.020 --> 01:04:00.090 align:middle line:84% to-- in order to assemble the energy balance, 01:04:00.090 --> 01:04:02.340 align:middle line:84% entropy balance, constituents' balance, 01:04:02.340 --> 01:04:09.430 align:middle line:84% we need to use a Gibbs relation to combine them, which 01:04:09.430 --> 01:04:14.770 align:middle line:84% is an assumption of local equilibrium, which goes together 01:04:14.770 --> 01:04:17.090 align:middle line:90% with the continuum assumption. 01:04:17.090 --> 01:04:20.370 align:middle line:90% 01:04:20.370 --> 01:04:25.510 align:middle line:84% So remember, you have the Gibbs relation. 01:04:25.510 --> 01:04:29.720 align:middle line:84% We are talking about large number of particles, 01:04:29.720 --> 01:04:35.260 align:middle line:84% so we assume the Euler relation, and also from the two, 01:04:35.260 --> 01:04:36.780 align:middle line:90% we get the Gibbs-Duhem relation. 01:04:36.780 --> 01:04:41.330 align:middle line:90% 01:04:41.330 --> 01:04:43.550 align:middle line:84% Now here, there are a couple of-- 01:04:43.550 --> 01:04:47.680 align:middle line:90% OK, so that is the basics. 01:04:47.680 --> 01:04:51.700 align:middle line:84% Then, of course, we need to remember 01:04:51.700 --> 01:04:54.796 align:middle line:84% what the definitions are, chemical potentials. 01:04:54.796 --> 01:04:58.000 align:middle line:90% 01:04:58.000 --> 01:04:59.660 align:middle line:90% We've seen it so many times. 01:04:59.660 --> 01:05:03.500 align:middle line:84% The differential of the chemical potential, 01:05:03.500 --> 01:05:06.810 align:middle line:84% if you remember when we started doing mixtures, 01:05:06.810 --> 01:05:08.335 align:middle line:90% we wrote this expression here. 01:05:08.335 --> 01:05:11.010 align:middle line:90% 01:05:11.010 --> 01:05:17.330 align:middle line:84% A new way that is used often is this one. 01:05:17.330 --> 01:05:20.220 align:middle line:90% 01:05:20.220 --> 01:05:24.990 align:middle line:84% You collect the part of the differential that is not related 01:05:24.990 --> 01:05:27.780 align:middle line:90% to changes in temperature. 01:05:27.780 --> 01:05:33.400 align:middle line:84% And instead of writing it all the times explicitly, 01:05:33.400 --> 01:05:37.800 align:middle line:84% you write it as the differential-- 01:05:37.800 --> 01:05:39.960 align:middle line:84% the part of the differential of mu 01:05:39.960 --> 01:05:43.320 align:middle line:84% that would survive if you keep the temperature 01:05:43.320 --> 01:05:45.990 align:middle line:90% constant with this symbol here. 01:05:45.990 --> 01:05:50.160 align:middle line:84% So you change the chemical potential 01:05:50.160 --> 01:05:53.690 align:middle line:90% by changing the temperature. 01:05:53.690 --> 01:05:55.430 align:middle line:90% And so the specific-- 01:05:55.430 --> 01:06:01.980 align:middle line:84% I'm sorry, the partial entropy appears here. 01:06:01.980 --> 01:06:08.180 align:middle line:84% And you call the rest, which relates 01:06:08.180 --> 01:06:12.460 align:middle line:84% to changing pressure and composition, 01:06:12.460 --> 01:06:13.980 align:middle line:90% you put it together in here. 01:06:13.980 --> 01:06:18.400 align:middle line:90% 01:06:18.400 --> 01:06:24.300 align:middle line:84% Then, another item that we do because we are making 01:06:24.300 --> 01:06:27.270 align:middle line:84% balances, for example, of energy-- 01:06:27.270 --> 01:06:29.440 align:middle line:90% not of internal energy. 01:06:29.440 --> 01:06:31.900 align:middle line:84% Yes, we could also make a balance of internal energy, 01:06:31.900 --> 01:06:37.060 align:middle line:84% but usually, the meaningful one is the energy balance. 01:06:37.060 --> 01:06:40.710 align:middle line:84% And so in the energy balance, we typically 01:06:40.710 --> 01:06:47.040 align:middle line:84% use this bulk flow-like expression 01:06:47.040 --> 01:06:50.850 align:middle line:84% of the energy in which we recognize an internal energy, 01:06:50.850 --> 01:06:56.200 align:middle line:84% and then some mechanical addenda that 01:06:56.200 --> 01:07:02.020 align:middle line:84% bring in the kinetic energy, the gravitational potential energy, 01:07:02.020 --> 01:07:09.490 align:middle line:84% the electrostatic potential energy, and even 01:07:09.490 --> 01:07:12.550 align:middle line:84% the centrifugal potential energy with the negative sign 01:07:12.550 --> 01:07:18.010 align:middle line:84% that we have seen when we did the stratification. 01:07:18.010 --> 01:07:22.000 align:middle line:84% And you remember that, therefore, the chemical 01:07:22.000 --> 01:07:27.550 align:middle line:90% potential gets in-- 01:07:27.550 --> 01:07:29.512 align:middle line:84% it's only a part of the expression 01:07:29.512 --> 01:07:31.220 align:middle line:84% of the partial of the energy with respect 01:07:31.220 --> 01:07:34.200 align:middle line:84% to the amounts, which we have called the total potential. 01:07:34.200 --> 01:07:36.200 align:middle line:84% And there are these other contributions 01:07:36.200 --> 01:07:39.175 align:middle line:90% that need to be taken care of. 01:07:39.175 --> 01:07:42.310 align:middle line:90% 01:07:42.310 --> 01:07:46.110 align:middle line:84% There is some-- there is some fishy-- 01:07:46.110 --> 01:07:54.285 align:middle line:84% this dq-- this dq shouldn't be here. 01:07:54.285 --> 01:07:56.830 align:middle line:90% 01:07:56.830 --> 01:08:01.240 align:middle line:84% Yeah, it's from cut and paste from this expression 01:08:01.240 --> 01:08:08.240 align:middle line:84% where I take this one, I take the differential here, 01:08:08.240 --> 01:08:14.620 align:middle line:84% and then you take the total potential is 01:08:14.620 --> 01:08:15.970 align:middle line:90% the partial of the energy. 01:08:15.970 --> 01:08:19.270 align:middle line:84% While you keep fixed the entropy, the volume, 01:08:19.270 --> 01:08:22.240 align:middle line:84% the other amounts, and then you have 01:08:22.240 --> 01:08:27.109 align:middle line:84% to keep fixed, for example, the barycentric velocity, 01:08:27.109 --> 01:08:32.620 align:middle line:84% the height in the potential, the value of the gravitational-- 01:08:32.620 --> 01:08:40.060 align:middle line:84% I'm sorry, of the electrostatic field, the angular speed, 01:08:40.060 --> 01:08:43.750 align:middle line:90% and the location. 01:08:43.750 --> 01:08:50.170 align:middle line:84% So you keep all these fixed and change only the amount 01:08:50.170 --> 01:08:51.819 align:middle line:84% of that particular constituent in order 01:08:51.819 --> 01:08:53.960 align:middle line:90% to get the total potential. 01:08:53.960 --> 01:08:57.399 align:middle line:90% 01:08:57.399 --> 01:09:00.640 align:middle line:84% And these last lines are just a summary of the fact 01:09:00.640 --> 01:09:04.420 align:middle line:90% that you can write properties-- 01:09:04.420 --> 01:09:10.540 align:middle line:84% specific properties per mole, per unit mass with the star, 01:09:10.540 --> 01:09:14.140 align:middle line:90% or per unit volume. 01:09:14.140 --> 01:09:16.779 align:middle line:84% Per unit volume, you could write it as rho u star, 01:09:16.779 --> 01:09:19.380 align:middle line:90% or also c times u. 01:09:19.380 --> 01:09:23.350 align:middle line:90% 01:09:23.350 --> 01:09:30.200 align:middle line:84% So this is a summary of what's needed. 01:09:30.200 --> 01:09:37.926 align:middle line:84% So we have worked so far using molar specific properties. 01:09:37.926 --> 01:09:41.779 align:middle line:90% 01:09:41.779 --> 01:09:48.229 align:middle line:84% And the Gibbs-Duhem relation has been always written this way. 01:09:48.229 --> 01:09:52.760 align:middle line:84% And with this new notation of the differential 01:09:52.760 --> 01:09:56.870 align:middle line:84% of the chemical potential at fixed temperature, 01:09:56.870 --> 01:09:58.465 align:middle line:90% it takes this form. 01:09:58.465 --> 01:10:01.400 align:middle line:90% 01:10:01.400 --> 01:10:06.500 align:middle line:84% And since that differential means any changes 01:10:06.500 --> 01:10:11.720 align:middle line:84% of the property-- if you change only the property by moving 01:10:11.720 --> 01:10:17.950 align:middle line:84% in space at a given instant of time, that becomes-- 01:10:17.950 --> 01:10:21.715 align:middle line:84% you can substitute, essentially, the d with the nabla, 01:10:21.715 --> 01:10:22.465 align:middle line:90% with the gradient. 01:10:22.465 --> 01:10:25.120 align:middle line:90% 01:10:25.120 --> 01:10:26.950 align:middle line:84% And so you have an expression here 01:10:26.950 --> 01:10:31.570 align:middle line:84% that comes from the Gibbs-Duhem and gives you 01:10:31.570 --> 01:10:33.580 align:middle line:84% the gradient in pressure and how it 01:10:33.580 --> 01:10:38.256 align:middle line:84% is related to the gradients in the chemical potentials. 01:10:38.256 --> 01:10:42.280 align:middle line:90% 01:10:42.280 --> 01:10:45.680 align:middle line:90% So they are not independent. 01:10:45.680 --> 01:10:51.900 align:middle line:84% It's just another face of the Gibbs phase rule. 01:10:51.900 --> 01:10:56.920 align:middle line:84% You may also write it in terms of mass-specific properties. 01:10:56.920 --> 01:10:59.320 align:middle line:84% I'm not staying too much on this. 01:10:59.320 --> 01:11:01.680 align:middle line:84% You can just-- but I'm staying more 01:11:01.680 --> 01:11:07.640 align:middle line:84% on this other one, volume-specific properties. 01:11:07.640 --> 01:11:12.620 align:middle line:84% Because if I write the Gibbs relation, the Euler relation, 01:11:12.620 --> 01:11:14.420 align:middle line:84% and the Gibbs-Duhem relation in terms 01:11:14.420 --> 01:11:25.200 align:middle line:84% of these volume-specific properties, what I already said 01:11:25.200 --> 01:11:32.670 align:middle line:84% is that you can show that the pressure disappears 01:11:32.670 --> 01:11:36.390 align:middle line:90% from the Gibbs relation. 01:11:36.390 --> 01:11:41.340 align:middle line:84% Not that you get rid of it, but in terms 01:11:41.340 --> 01:11:43.180 align:middle line:84% of these volume-specific properties, 01:11:43.180 --> 01:11:49.240 align:middle line:84% you decouple the Gibbs relation from the Gibbs-Duhem relation. 01:11:49.240 --> 01:11:52.360 align:middle line:84% And you can so forget about pressure, 01:11:52.360 --> 01:11:55.320 align:middle line:84% and then when you need it, you can use the Gibbs-Duhem relation 01:11:55.320 --> 01:12:02.700 align:middle line:84% as an additional independent equation that 01:12:02.700 --> 01:12:07.050 align:middle line:90% gives you the pressure. 01:12:07.050 --> 01:12:11.460 align:middle line:84% So the gradient in pressure is the weighted sum 01:12:11.460 --> 01:12:13.990 align:middle line:84% of the gradients in the chemical potentials, 01:12:13.990 --> 01:12:20.250 align:middle line:84% the constant temperature part, weighted on the concentrations 01:12:20.250 --> 01:12:21.230 align:middle line:90% of the various things. 01:12:21.230 --> 01:12:27.690 align:middle line:90% 01:12:27.690 --> 01:12:36.070 align:middle line:84% So this Gibbs relation, you can write it-- 01:12:36.070 --> 01:12:39.700 align:middle line:84% this time, we are interested in the change with respect 01:12:39.700 --> 01:12:48.460 align:middle line:84% to the time variable while keeping the position variables 01:12:48.460 --> 01:12:50.450 align:middle line:90% of these fields constant. 01:12:50.450 --> 01:12:54.440 align:middle line:84% So you may substitute the d with the partial with respect to t. 01:12:54.440 --> 01:12:57.130 align:middle line:90% 01:12:57.130 --> 01:13:02.770 align:middle line:84% So that Gibbs relation becomes this one. 01:13:02.770 --> 01:13:05.560 align:middle line:84% Or gives this relation between the partial properties 01:13:05.560 --> 01:13:07.130 align:middle line:90% (I meant, partial derivatives). 01:13:07.130 --> 01:13:11.680 align:middle line:84% And as you notice, these are the left-hand sides 01:13:11.680 --> 01:13:13.990 align:middle line:90% of the balance equations. 01:13:13.990 --> 01:13:16.410 align:middle line:84% This is the left-hand side of the energy balance, 01:13:16.410 --> 01:13:18.430 align:middle line:84% left-hand side of the entropy balance, 01:13:18.430 --> 01:13:21.935 align:middle line:84% left-hand side of the constituent's balance. 01:13:21.935 --> 01:13:32.590 align:middle line:90% 01:13:32.590 --> 01:13:45.540 align:middle line:84% Now we need to generalize the idea of heat interaction. 01:13:45.540 --> 01:13:55.540 align:middle line:90% 01:13:55.540 --> 01:14:00.580 align:middle line:84% And we do it in a way similar to the way we 01:14:00.580 --> 01:14:04.030 align:middle line:90% have done the heat interaction. 01:14:04.030 --> 01:14:11.020 align:middle line:84% So simply here, it's drawn in a different way. 01:14:11.020 --> 01:14:14.340 align:middle line:84% Remember, we took two systems that 01:14:14.340 --> 01:14:17.520 align:middle line:84% are at different temperature, and we 01:14:17.520 --> 01:14:20.200 align:middle line:84% allow them to exchange energy and entropy, 01:14:20.200 --> 01:14:24.210 align:middle line:90% and then we run a test. 01:14:24.210 --> 01:14:30.330 align:middle line:84% Can we identify part of that energy that is transferred 01:14:30.330 --> 01:14:34.420 align:middle line:90% between the two as work? 01:14:34.420 --> 01:14:37.240 align:middle line:90% If yes, we don't call it heat. 01:14:37.240 --> 01:14:43.510 align:middle line:84% If no, yes, that's the condition for a heat interaction. 01:14:43.510 --> 01:14:46.340 align:middle line:84% So in order to analyze this problem, 01:14:46.340 --> 01:14:49.360 align:middle line:84% you run an energy balance and entropy balance 01:14:49.360 --> 01:14:56.990 align:middle line:84% for system A. Same for system B. Same for the machinery here. 01:14:56.990 --> 01:14:59.770 align:middle line:90% 01:14:59.770 --> 01:15:03.550 align:middle line:84% We use the Gibbs relation for system A. 01:15:03.550 --> 01:15:11.830 align:middle line:84% We use the Gibbs relation for system B. 01:15:11.830 --> 01:15:14.320 align:middle line:90% And outcomes that the work-- 01:15:14.320 --> 01:15:18.460 align:middle line:84% the maximum work-- so we set the entropy generation equal 01:15:18.460 --> 01:15:20.260 align:middle line:84% to 0-- the maximum work you can get 01:15:20.260 --> 01:15:26.080 align:middle line:84% is given by the Caront coefficient multiplied 01:15:26.080 --> 01:15:30.290 align:middle line:84% by the energy that is being transferred. 01:15:30.290 --> 01:15:36.230 align:middle line:84% And so that says that, yes, a good fraction of that energy 01:15:36.230 --> 01:15:41.000 align:middle line:84% can be recognized as work unless the Carnot coefficient is 01:15:41.000 --> 01:15:42.150 align:middle line:90% equal to 0. 01:15:42.150 --> 01:15:45.140 align:middle line:84% And that is why we define heat only 01:15:45.140 --> 01:15:49.220 align:middle line:84% when the two temperatures, TA and TB, are-- 01:15:49.220 --> 01:15:53.230 align:middle line:84% their difference vanishes, so they are approaching each other. 01:15:53.230 --> 01:15:55.860 align:middle line:90% 01:15:55.860 --> 01:16:00.250 align:middle line:84% And also, in addition to this result, 01:16:00.250 --> 01:16:05.380 align:middle line:84% you remember that we get a set of inequalities that 01:16:05.380 --> 01:16:10.210 align:middle line:84% says that the entropy exchanged is bound, 01:16:10.210 --> 01:16:12.940 align:middle line:84% has an upper bound and a lower bound 01:16:12.940 --> 01:16:16.040 align:middle line:84% related to the initial temperatures of the two objects, 01:16:16.040 --> 01:16:20.380 align:middle line:84% and only in the limit in which the two temperatures are 01:16:20.380 --> 01:16:26.620 align:middle line:84% equal to one another, that range of values results 01:16:26.620 --> 01:16:34.240 align:middle line:84% in a single definite relation between the entropy transfer 01:16:34.240 --> 01:16:35.460 align:middle line:90% and the energy transfer. 01:16:35.460 --> 01:16:39.670 align:middle line:90% 01:16:39.670 --> 01:16:43.020 align:middle line:90% Which is this relation-- 01:16:43.020 --> 01:16:50.530 align:middle line:84% q, or delta q, because then we call that heat, 01:16:50.530 --> 01:16:55.240 align:middle line:84% is equal to T times the entropy exchanged. 01:16:55.240 --> 01:17:07.290 align:middle line:84% Now if we divide by the area of contact between the two 01:17:07.290 --> 01:17:09.190 align:middle line:90% objects-- 01:17:09.190 --> 01:17:12.850 align:middle line:84% so A here represents the area, not the system A, 01:17:12.850 --> 01:17:20.620 align:middle line:84% and dt is the amount of time that the interaction occurred, 01:17:20.620 --> 01:17:23.680 align:middle line:90% you obtain the flux. 01:17:23.680 --> 01:17:26.230 align:middle line:84% So you may rewrite this expression 01:17:26.230 --> 01:17:29.140 align:middle line:84% as saying that for a heat interaction, 01:17:29.140 --> 01:17:32.140 align:middle line:84% the energy flux and the entropy flux 01:17:32.140 --> 01:17:38.020 align:middle line:84% are related by the temperature, which is this, this expression 01:17:38.020 --> 01:17:40.310 align:middle line:90% here. 01:17:40.310 --> 01:17:42.960 align:middle line:84% And actually, for this particular one, 01:17:42.960 --> 01:17:46.530 align:middle line:84% we could even write q double prime, 01:17:46.530 --> 01:17:49.720 align:middle line:84% the symbol q double prime for heat. 01:17:49.720 --> 01:17:50.540 align:middle line:90% OK. 01:17:50.540 --> 01:18:02.290 align:middle line:84% Now I do the same, but with a new fact that the two systems, 01:18:02.290 --> 01:18:06.430 align:middle line:84% the two-- think of them as the adjacent fluid elements 01:18:06.430 --> 01:18:09.560 align:middle line:84% that for which we are making a balance. 01:18:09.560 --> 01:18:12.850 align:middle line:84% But in general, these two systems 01:18:12.850 --> 01:18:18.490 align:middle line:84% exchange not only energy and entropy, but also a given amount 01:18:18.490 --> 01:18:21.230 align:middle line:90% of each of the constituents. 01:18:21.230 --> 01:18:24.674 align:middle line:90% 01:18:24.674 --> 01:18:27.162 align:middle line:90% The procedure is the same. 01:18:27.162 --> 01:18:31.180 align:middle line:84% You have to write energy and entropy balances for system A, 01:18:31.180 --> 01:18:35.540 align:middle line:84% for system B, and for the machinery here. 01:18:35.540 --> 01:18:38.330 align:middle line:84% And you need to use the Gibbs relation for system A 01:18:38.330 --> 01:18:40.310 align:middle line:84% and the Gibbs relation for system 01:18:40.310 --> 01:19:01.310 align:middle line:84% B. It's a bit improper how I'm doing it here 01:19:01.310 --> 01:19:04.985 align:middle line:90% because the pressure-- 01:19:04.985 --> 01:19:13.430 align:middle line:84% there should be a pressure term there if we are not using 01:19:13.430 --> 01:19:15.170 align:middle line:90% properties per unit volume. 01:19:15.170 --> 01:19:22.620 align:middle line:84% But anyway, the idea is still OK. 01:19:22.620 --> 01:19:25.860 align:middle line:84% And then as a result of these balance equations, 01:19:25.860 --> 01:19:30.890 align:middle line:90% you get this expression. 01:19:30.890 --> 01:19:33.620 align:middle line:84% So the question is always the same. 01:19:33.620 --> 01:19:38.270 align:middle line:84% Can we recognize as part of the energy that is being exchanged 01:19:38.270 --> 01:19:40.830 align:middle line:90% here, part is work? 01:19:40.830 --> 01:19:44.530 align:middle line:84% Well now, we have two ways in which we could. 01:19:44.530 --> 01:19:47.080 align:middle line:90% 01:19:47.080 --> 01:19:49.870 align:middle line:84% If the Carnot coefficient is different than 0, 01:19:49.870 --> 01:19:53.560 align:middle line:84% then a good fraction, yes, could be work. 01:19:53.560 --> 01:20:01.620 align:middle line:84% But also, if these other coefficients-- 01:20:01.620 --> 01:20:07.240 align:middle line:84% I don't know what the name is, but if it is different than 0, 01:20:07.240 --> 01:20:12.430 align:middle line:84% then even if the temperatures are equal, 01:20:12.430 --> 01:20:15.180 align:middle line:90% you could have some work. 01:20:15.180 --> 01:20:18.980 align:middle line:90% So in order to have-- 01:20:18.980 --> 01:20:21.270 align:middle line:84% to extend the idea of heat interaction-- 01:20:21.270 --> 01:20:23.780 align:middle line:84% and we are going to call this new interaction heat 01:20:23.780 --> 01:20:31.070 align:middle line:84% and diffusion interaction with the ampersand that joins them-- 01:20:31.070 --> 01:20:37.010 align:middle line:84% for lack of better names, we call it heat and diffusion, 01:20:37.010 --> 01:20:42.540 align:middle line:84% you need the limit that the temperatures be equal, 01:20:42.540 --> 01:20:44.240 align:middle line:84% but also the chemical potentials of each 01:20:44.240 --> 01:20:48.085 align:middle line:84% of the constituents of the two interacting systems be equal. 01:20:48.085 --> 01:20:52.640 align:middle line:90% 01:20:52.640 --> 01:20:58.530 align:middle line:84% And of course, by the same way that we prove that inequality, 01:20:58.530 --> 01:21:00.350 align:middle line:84% you could prove this inequality here, 01:21:00.350 --> 01:21:06.260 align:middle line:84% that the entropy exchanged is, again, limited by two-- 01:21:06.260 --> 01:21:17.070 align:middle line:84% an upper bound and a lower bound, which, in that limit, 01:21:17.070 --> 01:21:19.415 align:middle line:84% become the same value, and in that limit-- 01:21:19.415 --> 01:21:22.560 align:middle line:90% 01:21:22.560 --> 01:21:24.880 align:middle line:84% so for a heat and diffusion interaction, 01:21:24.880 --> 01:21:27.780 align:middle line:84% you have a definite relation between the energy that 01:21:27.780 --> 01:21:30.340 align:middle line:84% is transferred, the entropy that is transferred, 01:21:30.340 --> 01:21:33.190 align:middle line:84% and the amounts of constituents that are transferred. 01:21:33.190 --> 01:21:40.350 align:middle line:90% 01:21:40.350 --> 01:21:47.380 align:middle line:84% And if you divide by dt and the area, the various transfers, 01:21:47.380 --> 01:21:49.650 align:middle line:90% you get the fluxes. 01:21:49.650 --> 01:21:53.790 align:middle line:84% So it's the transfer per unit area per unit time. 01:21:53.790 --> 01:21:57.960 align:middle line:84% And so this is the relation that characterizes-- 01:21:57.960 --> 01:22:04.890 align:middle line:84% it's the equivalent of q over T, but you see here 01:22:04.890 --> 01:22:07.680 align:middle line:90% that it contains also this part. 01:22:07.680 --> 01:22:10.550 align:middle line:90% 01:22:10.550 --> 01:22:18.680 align:middle line:84% So this is a very important new independent equation 01:22:18.680 --> 01:22:21.500 align:middle line:90% that we need in order to-- 01:22:21.500 --> 01:22:23.780 align:middle line:84% like we needed the q over T in order 01:22:23.780 --> 01:22:26.745 align:middle line:84% to combine the energy and entropy balances. 01:22:26.745 --> 01:22:32.530 align:middle line:90% 01:22:32.530 --> 01:22:34.550 align:middle line:90% Here it is. 01:22:34.550 --> 01:22:37.000 align:middle line:90% So we start from that. 01:22:37.000 --> 01:22:40.560 align:middle line:90% The question now is-- 01:22:40.560 --> 01:22:45.300 align:middle line:84% I see something fishy with this slide, but in any case, 01:22:45.300 --> 01:22:46.535 align:middle line:90% the question is-- 01:22:46.535 --> 01:22:50.130 align:middle line:90% 01:22:50.130 --> 01:22:53.020 align:middle line:90% I'm sorry. 01:22:53.020 --> 01:23:00.360 align:middle line:84% The question is, can we still call heat 01:23:00.360 --> 01:23:04.380 align:middle line:84% the energy that is transferred in a heat and diffusion 01:23:04.380 --> 01:23:05.790 align:middle line:90% interaction? 01:23:05.790 --> 01:23:07.290 align:middle line:90% The answer is no. 01:23:07.290 --> 01:23:11.735 align:middle line:84% Only part of the energy transfer can be called heat. 01:23:11.735 --> 01:23:15.380 align:middle line:90% 01:23:15.380 --> 01:23:18.562 align:middle line:90% How do we identify that? 01:23:18.562 --> 01:23:22.130 align:middle line:84% OK, remember-- OK, we start from this relation 01:23:22.130 --> 01:23:26.870 align:middle line:84% between the fluxes that characterizes the heat 01:23:26.870 --> 01:23:30.160 align:middle line:90% and diffusion interaction. 01:23:30.160 --> 01:23:32.480 align:middle line:84% The chemical potential appears in it. 01:23:32.480 --> 01:23:34.750 align:middle line:84% Remember that the chemical potential can always 01:23:34.750 --> 01:23:41.520 align:middle line:84% be written as the partial enthalpy minus T 01:23:41.520 --> 01:23:46.390 align:middle line:84% times the partial entropy for each of the constituents. 01:23:46.390 --> 01:23:47.400 align:middle line:90% This is general. 01:23:47.400 --> 01:23:50.380 align:middle line:90% 01:23:50.380 --> 01:23:53.090 align:middle line:90% So substitute that. 01:23:53.090 --> 01:23:56.900 align:middle line:84% And then take, on the left-hand side, 01:23:56.900 --> 01:23:59.000 align:middle line:84% the stuff that is not multiplied by T, 01:23:59.000 --> 01:24:03.580 align:middle line:84% and on the right-hand side, the stuff that is multiplied by T. 01:24:03.580 --> 01:24:08.560 align:middle line:84% So now you have found some part of the energy transfer 01:24:08.560 --> 01:24:09.660 align:middle line:90% of the energy flux-- 01:24:09.660 --> 01:24:12.390 align:middle line:90% 01:24:12.390 --> 01:24:14.940 align:middle line:90% the total minus this part-- 01:24:14.940 --> 01:24:20.130 align:middle line:84% which is related to a part of the entropy flux, 01:24:20.130 --> 01:24:21.340 align:middle line:90% via the temperature. 01:24:21.340 --> 01:24:24.150 align:middle line:90% 01:24:24.150 --> 01:24:29.440 align:middle line:84% So we are going to call that part of the energy flux 01:24:29.440 --> 01:24:31.420 align:middle line:90% q double prime. 01:24:31.420 --> 01:24:35.450 align:middle line:84% That is what we call the heat flux. 01:24:35.450 --> 01:24:41.900 align:middle line:84% So we call heat, only the difference between this 01:24:41.900 --> 01:24:50.520 align:middle line:84% and the flux of constituents, each multiplied 01:24:50.520 --> 01:24:52.303 align:middle line:90% by the partial enthalpy. 01:24:52.303 --> 01:24:55.620 align:middle line:90% 01:24:55.620 --> 01:25:01.050 align:middle line:84% You can think of this term that we are subtracting away 01:25:01.050 --> 01:25:08.520 align:middle line:84% as the energy that is carried by the particles that 01:25:08.520 --> 01:25:13.440 align:middle line:84% diffuse because Jn is the diffusion (flux) 01:25:13.440 --> 01:25:17.650 align:middle line:90% of particles of type i. 01:25:17.650 --> 01:25:26.860 align:middle line:84% These particles diffuse, and therefore, carry with them-- 01:25:26.860 --> 01:25:31.900 align:middle line:84% like in their backpack, they have the partial enthalpy 01:25:31.900 --> 01:25:34.060 align:middle line:84% and the partial enthalpy-- they actually 01:25:34.060 --> 01:25:39.576 align:middle line:84% have the partial energy and the partial entropy. 01:25:39.576 --> 01:25:43.696 align:middle line:90% But in order to overcome-- 01:25:43.696 --> 01:25:48.940 align:middle line:84% to move in a pressure field, they have to-- 01:25:48.940 --> 01:25:52.135 align:middle line:84% that's why get the enthalpy and not the energy, 01:25:52.135 --> 01:25:56.860 align:middle line:84% it's like in bulk flow, you have the pulsion work. 01:25:56.860 --> 01:26:00.140 align:middle line:84% This is pulsion work against the partial pressure, 01:26:00.140 --> 01:26:03.910 align:middle line:84% but it's still something that takes you 01:26:03.910 --> 01:26:10.550 align:middle line:84% from the energy plus the work that 01:26:10.550 --> 01:26:15.080 align:middle line:90% is needed to overcome this-- 01:26:15.080 --> 01:26:16.610 align:middle line:90% the pulsion work. 01:26:16.610 --> 01:26:18.668 align:middle line:90% The partial pulsion work. 01:26:18.668 --> 01:26:24.329 align:middle line:90% 01:26:24.329 --> 01:26:29.120 align:middle line:84% So the overall energy flux is, therefore, 01:26:29.120 --> 01:26:33.290 align:middle line:84% now given by this that we will call heat, 01:26:33.290 --> 01:26:43.250 align:middle line:84% plus this transported enthalpy due to the flux of particles. 01:26:43.250 --> 01:26:49.750 align:middle line:84% And also, the entropy flux is given by q over T. So 01:26:49.750 --> 01:26:52.240 align:middle line:90% that's proper of a heat-- 01:26:52.240 --> 01:26:54.970 align:middle line:84% characteristic of the heat interaction. 01:26:54.970 --> 01:26:58.630 align:middle line:84% Plus the entropy that is carried in the backpack by these 01:26:58.630 --> 01:27:01.750 align:middle line:90% particles that are diffused. 01:27:01.750 --> 01:27:06.910 align:middle line:84% So in other words, by calling this q double prime, 01:27:06.910 --> 01:27:13.770 align:middle line:84% essentially we are transforming this single equation 01:27:13.770 --> 01:27:18.570 align:middle line:84% into two equations with the q double prime in it. 01:27:18.570 --> 01:27:20.045 align:middle line:90% And so we have these two sets. 01:27:20.045 --> 01:27:23.540 align:middle line:90% 01:27:23.540 --> 01:27:27.425 align:middle line:84% In heat transfer, we call it q double prime. 01:27:27.425 --> 01:27:33.880 align:middle line:84% In this typical business of non-equilibrium thermodynamics, 01:27:33.880 --> 01:27:38.490 align:middle line:84% this is called the measurable heat flux, particularly 01:27:38.490 --> 01:27:44.020 align:middle line:84% introduced by this famous bible of the subject, 01:27:44.020 --> 01:27:47.090 align:middle line:84% which is the book by De Groot and Mazur. 01:27:47.090 --> 01:28:01.450 align:middle line:90% 01:28:01.450 --> 01:28:07.870 align:middle line:84% If the particles carry also charge, 01:28:07.870 --> 01:28:11.380 align:middle line:90% then we do exactly the same. 01:28:11.380 --> 01:28:23.650 align:middle line:84% So the relation that defines the heat and diffusion interaction 01:28:23.650 --> 01:28:26.960 align:middle line:84% is like this, except that you have 01:28:26.960 --> 01:28:29.900 align:middle line:84% to substitute the chemical potential 01:28:29.900 --> 01:28:34.990 align:middle line:84% with the total potential, so it becomes this one. 01:28:34.990 --> 01:28:37.300 align:middle line:84% And remember that the total potential when you have 01:28:37.300 --> 01:28:40.510 align:middle line:90% particles is given by this. 01:28:40.510 --> 01:28:44.710 align:middle line:84% It's the chemical potential, plus the electrostatic potential 01:28:44.710 --> 01:28:48.760 align:middle line:90% times the charge of-- 01:28:48.760 --> 01:28:53.540 align:middle line:84% the unit charge of your particles. 01:28:53.540 --> 01:28:56.940 align:middle line:90% 01:28:56.940 --> 01:28:59.370 align:middle line:90% So you can write it this way. 01:28:59.370 --> 01:29:01.270 align:middle line:84% Still in terms of the chemical potential. 01:29:01.270 --> 01:29:07.880 align:middle line:84% So now you can still use the fact 01:29:07.880 --> 01:29:12.240 align:middle line:84% that the chemical potential is hi minus Tsi. 01:29:12.240 --> 01:29:14.830 align:middle line:90% And then you can also-- 01:29:14.830 --> 01:29:19.850 align:middle line:90% remember that-- or define. 01:29:19.850 --> 01:29:23.600 align:middle line:84% If you take the flux of particles-- and these particles 01:29:23.600 --> 01:29:26.190 align:middle line:90% carry this much charge. 01:29:26.190 --> 01:29:32.780 align:middle line:84% This flux of charge is what we call current density-- 01:29:32.780 --> 01:29:38.785 align:middle line:84% charge current or electrical charge current density. 01:29:38.785 --> 01:29:43.460 align:middle line:90% 01:29:43.460 --> 01:29:51.020 align:middle line:84% So take this one or this one and substitute as before and-- not 01:29:51.020 --> 01:29:55.010 align:middle line:84% substitute-- and move left-hand side the stuff that is not 01:29:55.010 --> 01:29:57.890 align:middle line:84% multiplied by T and on the right-hand side the stuff that 01:29:57.890 --> 01:30:01.116 align:middle line:84% is multiplied by T, then you recognize-- 01:30:01.116 --> 01:30:03.650 align:middle line:90% 01:30:03.650 --> 01:30:09.920 align:middle line:84% then you call-- define q double prime, the measurable heat flux, 01:30:09.920 --> 01:30:15.970 align:middle line:84% as this term here, and you get two equations. 01:30:15.970 --> 01:30:19.860 align:middle line:84% So now, the energy flux is due to heat. 01:30:19.860 --> 01:30:23.100 align:middle line:84% The energy transported by the diffusion of particles 01:30:23.100 --> 01:30:26.820 align:middle line:84% through the enthalpy, the internal energy plus the pulsion 01:30:26.820 --> 01:30:36.920 align:middle line:84% work, and the electrical energy carried by the charge carriers. 01:30:36.920 --> 01:30:41.030 align:middle line:84% And for the entropy flux, the same. 01:30:41.030 --> 01:30:46.310 align:middle line:84% There is no contribution by the electric charge current. 01:30:46.310 --> 01:30:49.798 align:middle line:90% 01:30:49.798 --> 01:30:52.460 align:middle line:90% So with these ingredients-- 01:30:52.460 --> 01:30:55.200 align:middle line:90% 01:30:55.200 --> 01:31:00.755 align:middle line:84% Yeah we're almost out of time, but yeah, we can-- 01:31:00.755 --> 01:31:08.270 align:middle line:90% 01:31:08.270 --> 01:31:14.010 align:middle line:84% well, yeah, one-- let me just say-- 01:31:14.010 --> 01:31:16.190 align:middle line:84% one thing from which we will start next time 01:31:16.190 --> 01:31:19.490 align:middle line:84% is that you can rewrite that expression that 01:31:19.490 --> 01:31:25.520 align:middle line:84% defines the relation between energy, entropy, 01:31:25.520 --> 01:31:34.010 align:middle line:84% and materials flux in an entropy form instead of an energy form 01:31:34.010 --> 01:31:37.460 align:middle line:84% in terms of new variables, like the inverse of the temperature 01:31:37.460 --> 01:31:40.340 align:middle line:84% and the chemical potential, the total potential divided 01:31:40.340 --> 01:31:42.700 align:middle line:90% by the temperature. 01:31:42.700 --> 01:31:47.190 align:middle line:84% But let's do that next time because we're out of time now. 01:31:47.190 --> 01:31:51.000 align:middle line:90%