WEBVTT 00:00:00.000 --> 00:00:09.398 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:09.398 --> 00:00:10.940 align:middle line:84% GIAN PAOLO BERETTA: So today, I would 00:00:10.940 --> 00:00:18.890 align:middle line:84% like to finish the first part by saying a few more 00:00:18.890 --> 00:00:23.780 align:middle line:84% words about exergies and the allocation problems. 00:00:23.780 --> 00:00:26.573 align:middle line:84% That's going to be also useful for the homework. 00:00:26.573 --> 00:00:29.870 align:middle line:90% 00:00:29.870 --> 00:00:38.270 align:middle line:84% And remember that one of the things that we have touched upon 00:00:38.270 --> 00:00:41.950 align:middle line:90% is heating a bulk flow-- 00:00:41.950 --> 00:00:44.900 align:middle line:90% 00:00:44.900 --> 00:00:50.920 align:middle line:84% for example, heating water from condition 1 to condition 2, 00:00:50.920 --> 00:00:56.810 align:middle line:84% which therefore requires, by the energy balance, an energy, 00:00:56.810 --> 00:01:01.420 align:middle line:84% which is given by the flow rate, times the change 00:01:01.420 --> 00:01:05.770 align:middle line:90% in the enthalpy of the flow. 00:01:05.770 --> 00:01:12.260 align:middle line:84% That may be understood or called heat, 00:01:12.260 --> 00:01:17.380 align:middle line:84% even if it is a bit improper, because that is not really 00:01:17.380 --> 00:01:18.250 align:middle line:90% a heat interaction. 00:01:18.250 --> 00:01:20.080 align:middle line:84% Or at least, it is a heat interaction 00:01:20.080 --> 00:01:23.810 align:middle line:84% at multiple temperatures as the temperature changes from 1 to 2 00:01:23.810 --> 00:01:26.350 align:middle line:90% here. 00:01:26.350 --> 00:01:32.050 align:middle line:84% Nevertheless, if we agree on the meaning, we can call it heat. 00:01:32.050 --> 00:01:39.020 align:middle line:84% And then the coefficient of performance of, for example, 00:01:39.020 --> 00:01:46.840 align:middle line:84% in this case a heat pump, that provides that heat by taking 00:01:46.840 --> 00:01:50.710 align:middle line:84% the necessary entropy from the environment 00:01:50.710 --> 00:01:56.470 align:middle line:84% because, in order to heat up, I also need to provide entropy. 00:01:56.470 --> 00:02:00.440 align:middle line:84% And if I take it from the environment, 00:02:00.440 --> 00:02:04.760 align:middle line:84% I avoid having a generation by irreversibility. 00:02:04.760 --> 00:02:08.070 align:middle line:90% 00:02:08.070 --> 00:02:13.020 align:middle line:84% The machine needs some work to operate. 00:02:13.020 --> 00:02:19.290 align:middle line:84% And the ratio between the heat and the work that are provided 00:02:19.290 --> 00:02:20.330 align:middle line:90% is called the COP. 00:02:20.330 --> 00:02:24.640 align:middle line:90% 00:02:24.640 --> 00:02:27.590 align:middle line:84% So the COP plays the role of the first-law efficiency. 00:02:27.590 --> 00:02:29.570 align:middle line:90% But it can be higher than 1. 00:02:29.570 --> 00:02:30.620 align:middle line:90% So we call it COP. 00:02:30.620 --> 00:02:32.030 align:middle line:90% We don't call it efficiency. 00:02:32.030 --> 00:02:35.330 align:middle line:90% 00:02:35.330 --> 00:02:38.600 align:middle line:84% It's a ratio of two energies, whereas 00:02:38.600 --> 00:02:42.060 align:middle line:84% the second-law efficiency is the ratio. 00:02:42.060 --> 00:02:47.780 align:middle line:84% You can cast it as either the actual COP 00:02:47.780 --> 00:02:51.530 align:middle line:84% over the maximum COP, which you would 00:02:51.530 --> 00:02:54.830 align:middle line:84% obtain if the machinery was reversible. 00:02:54.830 --> 00:03:01.310 align:middle line:84% Or you can cast it as the ratio of the exergy change 00:03:01.310 --> 00:03:03.440 align:middle line:84% or the exergy that you have provided 00:03:03.440 --> 00:03:07.610 align:middle line:84% to your bulk flow over the exergy of the resource 00:03:07.610 --> 00:03:10.465 align:middle line:84% that you have used-- in this case, work. 00:03:10.465 --> 00:03:15.140 align:middle line:90% 00:03:15.140 --> 00:03:21.830 align:middle line:84% We have also seen that if I take h2 minus h1 times 00:03:21.830 --> 00:03:26.510 align:middle line:84% m dot outside of the parentheses and I recognize 00:03:26.510 --> 00:03:30.410 align:middle line:84% that we called it QA, the remainder 00:03:30.410 --> 00:03:36.920 align:middle line:84% is a Carnot coefficient-looking combination 00:03:36.920 --> 00:03:44.570 align:middle line:84% that if we define h2 minus h1 over s2 00:03:44.570 --> 00:03:49.700 align:middle line:84% minus s1 as the logarithmic mean temperature 00:03:49.700 --> 00:03:53.060 align:middle line:84% or the equivalent heat temperature that provides 00:03:53.060 --> 00:03:56.090 align:middle line:84% the same ratio of energy and entropy, 00:03:56.090 --> 00:04:00.560 align:middle line:90% then you can write it this way. 00:04:00.560 --> 00:04:04.370 align:middle line:84% And here, you recognize that QA over W is COP. 00:04:04.370 --> 00:04:07.580 align:middle line:84% So the second-law efficiency is given by COP times the Carnot 00:04:07.580 --> 00:04:10.750 align:middle line:90% coefficient. 00:04:10.750 --> 00:04:13.540 align:middle line:90% And what is the maximum COP? 00:04:13.540 --> 00:04:19.029 align:middle line:84% Well, the maximum COP is when the second-law efficiency 00:04:19.029 --> 00:04:25.180 align:middle line:84% is equal to 100%, reversible machinery. 00:04:25.180 --> 00:04:27.340 align:middle line:90% So you set this equal to 1. 00:04:27.340 --> 00:04:29.890 align:middle line:84% And therefore, COP max is the inverse 00:04:29.890 --> 00:04:32.360 align:middle line:90% of that Carnot coefficient. 00:04:32.360 --> 00:04:37.410 align:middle line:90% 00:04:37.410 --> 00:04:39.390 align:middle line:84% So we also played with some numbers 00:04:39.390 --> 00:04:41.970 align:middle line:84% because this is an important topic. 00:04:41.970 --> 00:04:46.240 align:middle line:84% And it is a way in which we consume a lot of energy. 00:04:46.240 --> 00:04:53.220 align:middle line:84% So if we want to improve our ways of using fuels 00:04:53.220 --> 00:05:00.120 align:middle line:90% or any other exergy resource-- 00:05:00.120 --> 00:05:07.620 align:middle line:84% natural resource, for example, nuclear and solar and so on-- 00:05:07.620 --> 00:05:14.140 align:middle line:84% we want to know the efficiencies. 00:05:14.140 --> 00:05:18.780 align:middle line:84% And so here we gave an example of heating water 00:05:18.780 --> 00:05:22.440 align:middle line:90% in an industrial setup. 00:05:22.440 --> 00:05:26.010 align:middle line:84% Typical numbers could be from 60 to 120 degrees 00:05:26.010 --> 00:05:32.940 align:middle line:84% C. And then if we compute that Carnot ratio-- 00:05:32.940 --> 00:05:36.000 align:middle line:84% and remember that here also we need 00:05:36.000 --> 00:05:40.300 align:middle line:84% to make explicit the assumption of the environment 00:05:40.300 --> 00:05:44.510 align:middle line:84% that we have available when we compute that because that makes 00:05:44.510 --> 00:05:47.680 align:middle line:84% a lot of difference, especially for small temperatures 00:05:47.680 --> 00:05:50.450 align:middle line:90% and temperature differences. 00:05:50.450 --> 00:05:55.970 align:middle line:84% So here, for example, if I take 300 K, here's the number. 00:05:55.970 --> 00:06:01.450 align:middle line:90% The Carnot ratio is 17.2%. 00:06:01.450 --> 00:06:09.560 align:middle line:84% And its inverse, which is the maximum COP for the heat pump 00:06:09.560 --> 00:06:14.860 align:middle line:90% therefore is 5.8. 00:06:14.860 --> 00:06:23.100 align:middle line:84% So 5.8 means that I spend 1 kilowatthour of energy and I 00:06:23.100 --> 00:06:27.785 align:middle line:84% deliver 5.8 kilowatthours of energy to my stream. 00:06:27.785 --> 00:06:32.020 align:middle line:90% 00:06:32.020 --> 00:06:34.810 align:middle line:84% If I go to lower temperatures, like in residential heating 00:06:34.810 --> 00:06:41.170 align:middle line:84% from 20 to 60 degrees at the same environment, 300 K, which 00:06:41.170 --> 00:06:49.030 align:middle line:84% is like summertime because it's 27 degrees C, in that case, 00:06:49.030 --> 00:06:53.080 align:middle line:90% the Carnot coefficient is 4%. 00:06:53.080 --> 00:06:57.880 align:middle line:84% So you take the inverse of 4%, it gives 25. 00:06:57.880 --> 00:07:00.500 align:middle line:84% So that means that, in principle, 00:07:00.500 --> 00:07:04.150 align:middle line:84% if you had a reversible heat pump with 1 kilowatthour 00:07:04.150 --> 00:07:08.200 align:middle line:84% of electrical energy, you provide 25 kilowatthours 00:07:08.200 --> 00:07:14.350 align:middle line:84% of heat, whereas, in an electric boiler, 00:07:14.350 --> 00:07:18.520 align:middle line:84% for 1 kilowatthour of energy, you provide 1 kilowatthour 00:07:18.520 --> 00:07:21.880 align:middle line:84% of heat or maybe a little less because maybe some is lost 00:07:21.880 --> 00:07:24.540 align:middle line:90% to the insulation of the boiler. 00:07:24.540 --> 00:07:31.020 align:middle line:90% 00:07:31.020 --> 00:07:38.930 align:middle line:84% Then we also argued that the current way in which we 00:07:38.930 --> 00:07:43.580 align:middle line:84% use fuels, namely burning them in a flame 00:07:43.580 --> 00:07:47.480 align:middle line:84% and then using the hot products of combustion 00:07:47.480 --> 00:07:53.930 align:middle line:84% to heat up a working fluid of some thermal cycle, 00:07:53.930 --> 00:07:59.000 align:middle line:84% if we want to improve on that, the efficiency of these 00:07:59.000 --> 00:08:05.540 align:middle line:84% and go beyond the current, say, 64%, which 00:08:05.540 --> 00:08:10.550 align:middle line:84% is obtained as the product of an about 75% 00:08:10.550 --> 00:08:13.160 align:middle line:84% efficiency of the combustion of the flame 00:08:13.160 --> 00:08:19.550 align:middle line:84% and an 85% efficiency of the thermal cycle, 00:08:19.550 --> 00:08:23.030 align:middle line:84% you need to improve on these two efficiencies. 00:08:23.030 --> 00:08:26.670 align:middle line:84% But even if you improve the thermal cycle to 100%, 00:08:26.670 --> 00:08:33.710 align:middle line:84% you're still stuck to the limit of 75%, say 80% for the flame. 00:08:33.710 --> 00:08:37.610 align:middle line:84% And we've said we've seen that history tells us 00:08:37.610 --> 00:08:41.960 align:middle line:84% that we are, on our learning process, 00:08:41.960 --> 00:08:48.410 align:middle line:84% heading towards higher efficiencies than 80%. 00:08:48.410 --> 00:08:53.900 align:middle line:84% And so the only way we can reach them is to find alternatives 00:08:53.900 --> 00:09:01.280 align:middle line:84% to the combustion in flames, the oxidation of the fuel in flames. 00:09:01.280 --> 00:09:06.480 align:middle line:84% And we also said that when we do chemical reactions 00:09:06.480 --> 00:09:08.740 align:middle line:84% and combustion in a little more detail, 00:09:08.740 --> 00:09:10.380 align:middle line:90% we will see this diagram. 00:09:10.380 --> 00:09:13.230 align:middle line:84% And we will see that the flame goes this way. 00:09:13.230 --> 00:09:16.530 align:middle line:84% And that generates entropy by irreversibility, 00:09:16.530 --> 00:09:26.400 align:middle line:84% whereas if you are in a fuel cell or more 00:09:26.400 --> 00:09:31.410 align:middle line:84% able to control this process, and while the system 00:09:31.410 --> 00:09:35.680 align:middle line:84% spontaneously wants to go from fuel to products of combustion, 00:09:35.680 --> 00:09:41.770 align:middle line:84% you manage to extract also some energy, like in the fuel cell, 00:09:41.770 --> 00:09:45.990 align:middle line:84% then you may reduce the entropy produced by reversibility 00:09:45.990 --> 00:09:51.840 align:middle line:84% and your efficiencies can be higher, like in some examples 00:09:51.840 --> 00:09:54.000 align:middle line:90% here. 00:09:54.000 --> 00:09:59.680 align:middle line:84% So usually, separate production of electricity-- 00:09:59.680 --> 00:10:06.480 align:middle line:84% we produce electricity in facilities, power plants, 00:10:06.480 --> 00:10:09.690 align:middle line:90% that use our fuel. 00:10:09.690 --> 00:10:13.120 align:middle line:84% And remember, we said that, for the moment, 00:10:13.120 --> 00:10:17.190 align:middle line:84% but also for typical applications, 00:10:17.190 --> 00:10:23.880 align:middle line:84% it is fair to approximate the exergy of the fuel 00:10:23.880 --> 00:10:26.850 align:middle line:90% by its heating volume. 00:10:26.850 --> 00:10:31.140 align:middle line:84% So energy and exergy content are very similar, 00:10:31.140 --> 00:10:33.990 align:middle line:84% of the order of plus or minus 2%, depending 00:10:33.990 --> 00:10:35.470 align:middle line:90% on the hydrocarbons. 00:10:35.470 --> 00:10:41.670 align:middle line:84% So this is a fair approximation in practice. 00:10:41.670 --> 00:10:43.350 align:middle line:90% So you use this fuel. 00:10:43.350 --> 00:10:44.350 align:middle line:90% You burn it. 00:10:44.350 --> 00:10:46.630 align:middle line:84% You obtain these products of combustion. 00:10:46.630 --> 00:10:50.670 align:middle line:84% And then you transfer the heat from the products of combustion 00:10:50.670 --> 00:10:57.030 align:middle line:84% to a working fluid of the thermal cycle. 00:10:57.030 --> 00:10:58.950 align:middle line:84% Then, of course, the products combustion 00:10:58.950 --> 00:11:04.430 align:middle line:84% must be exhausted to the atmosphere. 00:11:04.430 --> 00:11:09.410 align:middle line:84% And we are all concerned about pollution 00:11:09.410 --> 00:11:25.750 align:middle line:84% that comes from these exhausts, not only the global pollutant, 00:11:25.750 --> 00:11:32.650 align:middle line:84% CO2, carbon dioxide, which is a pollutant only in terms 00:11:32.650 --> 00:11:40.280 align:middle line:84% of playing a role in greenhouse effect, as we all know, 00:11:40.280 --> 00:11:48.070 align:middle line:84% but also other products of combustion, like NOx, 00:11:48.070 --> 00:11:53.830 align:middle line:84% so the various oxides of nitrogen, 00:11:53.830 --> 00:11:57.840 align:middle line:84% which can produce acid rain and other various effects. 00:11:57.840 --> 00:12:01.020 align:middle line:90% 00:12:01.020 --> 00:12:05.440 align:middle line:84% So you need to control what gets out there. 00:12:05.440 --> 00:12:11.380 align:middle line:84% And typically, what gets out there 00:12:11.380 --> 00:12:16.810 align:middle line:84% is at relatively high temperature, 00:12:16.810 --> 00:12:19.976 align:middle line:84% say maybe 110, 140 degrees Celsius. 00:12:19.976 --> 00:12:23.650 align:middle line:90% 00:12:23.650 --> 00:12:28.630 align:middle line:84% This is when you don't want the water that 00:12:28.630 --> 00:12:32.350 align:middle line:84% is in the products of combustion to condense. 00:12:32.350 --> 00:12:37.640 align:middle line:84% That is to avoid-- because some fuels have some sulfur content. 00:12:37.640 --> 00:12:43.030 align:middle line:84% And water with sulfur makes sulphuric acid. 00:12:43.030 --> 00:12:46.180 align:middle line:90% And that's corrosive. 00:12:46.180 --> 00:12:49.990 align:middle line:90% And you want to avoid that. 00:12:49.990 --> 00:12:58.360 align:middle line:84% But you can also have condensation. 00:12:58.360 --> 00:13:04.470 align:middle line:84% So if you can use higher level materials that 00:13:04.470 --> 00:13:07.950 align:middle line:84% can sustain corrosion, you can also 00:13:07.950 --> 00:13:11.730 align:middle line:84% condense part of the water that is produced by the combustion 00:13:11.730 --> 00:13:12.430 align:middle line:90% process. 00:13:12.430 --> 00:13:18.335 align:middle line:84% In that case, you extract a little more energy. 00:13:18.335 --> 00:13:20.870 align:middle line:90% 00:13:20.870 --> 00:13:23.730 align:middle line:84% And then we talk instead of a lower heating value, 00:13:23.730 --> 00:13:26.660 align:middle line:84% it's actually you're exploiting, getting closer 00:13:26.660 --> 00:13:31.910 align:middle line:84% to the higher heating value or somewhere in between. 00:13:31.910 --> 00:13:34.270 align:middle line:84% We'll see some details about this. 00:13:34.270 --> 00:13:36.530 align:middle line:84% But here, the idea is that this energy 00:13:36.530 --> 00:13:40.860 align:middle line:84% is transferred to this thing and to this working fluid. 00:13:40.860 --> 00:13:48.290 align:middle line:84% And then the thermal cycle provides work. 00:13:48.290 --> 00:13:52.950 align:middle line:84% And of course, since you're extracting not only energy, 00:13:52.950 --> 00:13:55.250 align:middle line:84% but also entropy, the thermal cycle 00:13:55.250 --> 00:14:00.470 align:middle line:84% must discharge that entropy to the environment, which 00:14:00.470 --> 00:14:13.220 align:middle line:84% is, at the moment, still considered a free way of getting 00:14:13.220 --> 00:14:14.905 align:middle line:90% rid of entropy. 00:14:14.905 --> 00:14:17.540 align:middle line:90% 00:14:17.540 --> 00:14:20.805 align:middle line:84% Although, some places we are talking about, 00:14:20.805 --> 00:14:22.680 align:middle line:84% of course, you need authorization to do that. 00:14:22.680 --> 00:14:29.270 align:middle line:84% So that is equivalent to not being exactly for free, 00:14:29.270 --> 00:14:30.830 align:middle line:90% not anymore. 00:14:30.830 --> 00:14:35.390 align:middle line:84% In other words, we cannot use the environment as a thermal 00:14:35.390 --> 00:14:44.060 align:middle line:90% dump for free anymore. 00:14:44.060 --> 00:14:44.780 align:middle line:90% OK. 00:14:44.780 --> 00:14:46.730 align:middle line:84% First of all, efficiency here, we 00:14:46.730 --> 00:14:50.360 align:middle line:84% have only one product, whose energy content 00:14:50.360 --> 00:15:01.170 align:middle line:84% is W and only one energy source, which is the fuel. 00:15:01.170 --> 00:15:08.800 align:middle line:84% Its energy content is Q. So that's the first-law efficiency. 00:15:08.800 --> 00:15:12.190 align:middle line:84% And typical first law efficiencies, as we said, 00:15:12.190 --> 00:15:14.950 align:middle line:84% could be, yes, the best available technology 00:15:14.950 --> 00:15:18.820 align:middle line:90% could be maybe 65%. 00:15:18.820 --> 00:15:31.690 align:middle line:84% But the average efficiency of a national lot of power plants, 00:15:31.690 --> 00:15:35.020 align:middle line:84% well, it varies from country to country, of course. 00:15:35.020 --> 00:15:40.430 align:middle line:84% For example, I think that in Italy is between 42% and 45%. 00:15:40.430 --> 00:15:44.360 align:middle line:90% 00:15:44.360 --> 00:15:47.570 align:middle line:84% The second-law efficiency is the ratio 00:15:47.570 --> 00:15:56.720 align:middle line:84% of the exergy of the product to the exergy of your fuel. 00:15:56.720 --> 00:16:00.320 align:middle line:84% Now, considering that the exergy can be approximated 00:16:00.320 --> 00:16:03.440 align:middle line:84% to the energy of the fuel, you see 00:16:03.440 --> 00:16:08.330 align:middle line:84% that, for this particular case, the second-law efficiency 00:16:08.330 --> 00:16:12.030 align:middle line:84% is equal to the first-law efficiency. 00:16:12.030 --> 00:16:17.410 align:middle line:84% So indeed, if you are at 65%, the 65%, 00:16:17.410 --> 00:16:20.910 align:middle line:84% the margin for improvement is 35%. 00:16:20.910 --> 00:16:25.480 align:middle line:84% We only have 35% to go to get 100%. 00:16:25.480 --> 00:16:32.810 align:middle line:84% And remember that second-law wise 100% is not impossible. 00:16:32.810 --> 00:16:34.850 align:middle line:90% It's the upper limit. 00:16:34.850 --> 00:16:37.640 align:middle line:84% It's what you would have for reversible machinery. 00:16:37.640 --> 00:16:43.901 align:middle line:90% 00:16:43.901 --> 00:16:47.200 align:middle line:84% So the message is here that, for this particular case, 00:16:47.200 --> 00:16:50.220 align:middle line:84% eta 2 is equal to eta 1 approximately. 00:16:50.220 --> 00:16:54.630 align:middle line:90% 00:16:54.630 --> 00:16:57.400 align:middle line:84% Now, when instead of producing electricity, 00:16:57.400 --> 00:16:59.950 align:middle line:84% we want to heat up something-- for example, 00:16:59.950 --> 00:17:08.010 align:middle line:84% here, a simple case of heating up a room at some fixed 00:17:08.010 --> 00:17:12.930 align:middle line:84% temperature or maybe heating up water to make it 00:17:12.930 --> 00:17:18.550 align:middle line:84% boil at some temperature, for example, here, 120 degrees, 00:17:18.550 --> 00:17:22.420 align:middle line:84% some pressurized steam production, 00:17:22.420 --> 00:17:26.609 align:middle line:84% which is very common in industrial processes 00:17:26.609 --> 00:17:30.360 align:middle line:84% for material processing and so on. 00:17:30.360 --> 00:17:36.990 align:middle line:84% So now, the first-law efficiency is the ratio of the heat 00:17:36.990 --> 00:17:41.220 align:middle line:84% that you produce, that you deliver. 00:17:41.220 --> 00:17:48.150 align:middle line:84% So this is your final product over the energy of the fuel. 00:17:48.150 --> 00:17:50.460 align:middle line:84% The second-law efficiency instead 00:17:50.460 --> 00:17:54.540 align:middle line:84% is the ratio of the exergy of that heat. 00:17:54.540 --> 00:17:59.430 align:middle line:84% So it's the heat multiplied by the Carnot coefficient divided 00:17:59.430 --> 00:18:01.350 align:middle line:90% by the exergy. 00:18:01.350 --> 00:18:03.330 align:middle line:84% Then again, we use the approximation 00:18:03.330 --> 00:18:07.660 align:middle line:84% that the exergy is equal to the energy content of the fuel. 00:18:07.660 --> 00:18:11.830 align:middle line:84% And therefore, we recognize this ratio here is eta 1. 00:18:11.830 --> 00:18:15.660 align:middle line:84% So in this case, the second-law efficiency 00:18:15.660 --> 00:18:18.780 align:middle line:84% is equal to the first-law efficiency times the Carnot 00:18:18.780 --> 00:18:19.506 align:middle line:90% coefficient. 00:18:19.506 --> 00:18:23.690 align:middle line:90% 00:18:23.690 --> 00:18:28.115 align:middle line:84% So for example, if you want to produce steam at 120 degrees 00:18:28.115 --> 00:18:32.240 align:middle line:84% and you're in an environment at 283, 00:18:32.240 --> 00:18:42.120 align:middle line:84% you compute and you have a boiler that consumes, 00:18:42.120 --> 00:18:44.040 align:middle line:90% for example, natural gas. 00:18:44.040 --> 00:18:48.580 align:middle line:84% And it is 95% first-law efficient. 00:18:48.580 --> 00:18:51.200 align:middle line:90% So eta 1 is 95%. 00:18:51.200 --> 00:18:56.570 align:middle line:84% It means that only 5% is lost to the flue gases 00:18:56.570 --> 00:18:58.400 align:middle line:90% through the stack. 00:18:58.400 --> 00:19:05.460 align:middle line:84% And that means, so you've got the 95% times 00:19:05.460 --> 00:19:09.163 align:middle line:84% the Carnot coefficient here makes about 27%. 00:19:09.163 --> 00:19:12.370 align:middle line:90% 00:19:12.370 --> 00:19:18.130 align:middle line:84% So in that case, the margin for improvement 00:19:18.130 --> 00:19:23.960 align:middle line:84% is much higher, like 73%, to go to 100%. 00:19:23.960 --> 00:19:28.070 align:middle line:90% 00:19:28.070 --> 00:19:32.810 align:middle line:84% In some sense, we should say that if this is representative 00:19:32.810 --> 00:19:40.280 align:middle line:84% of our typical way of heating up water in industrial processes 00:19:40.280 --> 00:19:46.670 align:middle line:84% by using fuels, the fact that, for this technology 00:19:46.670 --> 00:19:51.950 align:middle line:84% we are only 27%, whereas for power production we are 65% 00:19:51.950 --> 00:19:56.360 align:middle line:84% means that we are at different levels of maturity 00:19:56.360 --> 00:20:01.520 align:middle line:84% of these uses of natural resources for these two 00:20:01.520 --> 00:20:06.085 align:middle line:84% different technologies and products. 00:20:06.085 --> 00:20:11.410 align:middle line:90% 00:20:11.410 --> 00:20:14.140 align:middle line:90% Most of the times, we-- 00:20:14.140 --> 00:20:17.260 align:middle line:84% many times, the burner doesn't have to heat up something 00:20:17.260 --> 00:20:18.700 align:middle line:84% at constant temperature, like when 00:20:18.700 --> 00:20:25.030 align:middle line:84% you boil something, but needs to heat up, for example, water. 00:20:25.030 --> 00:20:29.170 align:middle line:84% Again, for residential purposes, typical numbers 00:20:29.170 --> 00:20:31.130 align:middle line:90% are from 20 to 60. 00:20:31.130 --> 00:20:33.780 align:middle line:90% 00:20:33.780 --> 00:20:36.850 align:middle line:90% And so here are these again. 00:20:36.850 --> 00:20:40.390 align:middle line:84% We have that, since the temperature is changing here, 00:20:40.390 --> 00:20:45.510 align:middle line:84% we can define a logarithmic mean temperature, 00:20:45.510 --> 00:20:48.510 align:middle line:84% which is the temperature that enters in the Carnot 00:20:48.510 --> 00:20:50.100 align:middle line:90% coefficient. 00:20:50.100 --> 00:20:52.320 align:middle line:84% And again, the second-law efficiency 00:20:52.320 --> 00:20:54.690 align:middle line:84% is equal to the first-law efficiency times the Carnot 00:20:54.690 --> 00:20:55.430 align:middle line:90% coefficient. 00:20:55.430 --> 00:21:00.290 align:middle line:90% 00:21:00.290 --> 00:21:03.200 align:middle line:84% The typical number varies depending 00:21:03.200 --> 00:21:07.040 align:middle line:84% on what temperature T0 you choose, so 00:21:07.040 --> 00:21:11.840 align:middle line:84% depending on whether you are winter or summer. 00:21:11.840 --> 00:21:15.950 align:middle line:84% So for example, if you are in the summer, 00:21:15.950 --> 00:21:26.630 align:middle line:84% 25 degrees C is to 298 K. A residential domestic boiler, 00:21:26.630 --> 00:21:33.740 align:middle line:84% which has an efficiency of about 85%, first law, 00:21:33.740 --> 00:21:36.543 align:middle line:84% gets a second-law efficiency of just 4%. 00:21:36.543 --> 00:21:39.480 align:middle line:90% 00:21:39.480 --> 00:21:47.580 align:middle line:84% Wintertime, if outside is at 0 degrees Celsius, it's freezing, 00:21:47.580 --> 00:21:51.780 align:middle line:90% then it's 11%. 00:21:51.780 --> 00:21:56.080 align:middle line:84% For an industrial boiler, the efficiency is a little better. 00:21:56.080 --> 00:21:59.280 align:middle line:84% First-law efficiency, let's say 95%. 00:21:59.280 --> 00:22:06.920 align:middle line:90% And if you go from-- 00:22:06.920 --> 00:22:08.130 align:middle line:90% yeah, OK. 00:22:08.130 --> 00:22:09.770 align:middle line:90% Maybe this is not true. 00:22:09.770 --> 00:22:11.190 align:middle line:90% This should be a 16%. 00:22:11.190 --> 00:22:17.030 align:middle line:84% Probably these numbers I think I did that from 60 to 120. 00:22:17.030 --> 00:22:19.710 align:middle line:84% Well, check the numbers if you are interested. 00:22:19.710 --> 00:22:26.360 align:middle line:84% And it gives a 12% in the summer and 19% 00:22:26.360 --> 00:22:29.600 align:middle line:84% for second-law efficiency during the winter. 00:22:29.600 --> 00:22:33.900 align:middle line:84% So there are ample margins of improvement. 00:22:33.900 --> 00:22:37.670 align:middle line:90% 00:22:37.670 --> 00:22:42.830 align:middle line:84% And one of them is cogeneration, as we said. 00:22:42.830 --> 00:22:46.800 align:middle line:84% Produce with that fuel electricity, 00:22:46.800 --> 00:22:53.780 align:middle line:84% and then use your low temperature need-- 00:22:53.780 --> 00:23:00.115 align:middle line:84% a low temperature heat that is needed as a dump for entropy. 00:23:00.115 --> 00:23:03.700 align:middle line:90% 00:23:03.700 --> 00:23:10.870 align:middle line:84% And therefore, this becomes a second product, or a byproduct. 00:23:10.870 --> 00:23:14.890 align:middle line:84% The second-law efficiency then has, as the numerator, 00:23:14.890 --> 00:23:18.760 align:middle line:84% the exegies of the two products-- 00:23:18.760 --> 00:23:22.600 align:middle line:84% work for the exergy of the work, of the electricity, 00:23:22.600 --> 00:23:26.560 align:middle line:84% and heat times its Carnot coefficient for the exergy 00:23:26.560 --> 00:23:29.220 align:middle line:90% of the heat divided by the fuel. 00:23:29.220 --> 00:23:35.600 align:middle line:90% 00:23:35.600 --> 00:23:41.600 align:middle line:84% And again, this is the same thing, 00:23:41.600 --> 00:23:47.210 align:middle line:84% where we highlight that the temperature-- 00:23:47.210 --> 00:23:50.120 align:middle line:84% if the temperature of what you're heating is changing, 00:23:50.120 --> 00:23:53.760 align:middle line:84% you need to put in the logarithmic mean temperature. 00:23:53.760 --> 00:24:00.130 align:middle line:90% 00:24:00.130 --> 00:24:10.270 align:middle line:84% You can also do the same kind of calculations for refrigeration 00:24:10.270 --> 00:24:14.150 align:middle line:84% because, often in industrial processes, 00:24:14.150 --> 00:24:18.080 align:middle line:84% but also in air conditioning, residential air conditioning, 00:24:18.080 --> 00:24:21.610 align:middle line:90% you need to cool down. 00:24:21.610 --> 00:24:23.110 align:middle line:84% For example, air conditioning, you 00:24:23.110 --> 00:24:27.880 align:middle line:84% need to cool down a working fluid between 12 and 7, 00:24:27.880 --> 00:24:30.660 align:middle line:90% from 12 to 7 degrees C, typical. 00:24:30.660 --> 00:24:34.480 align:middle line:90% 00:24:34.480 --> 00:24:41.320 align:middle line:84% So you can define a COP of the refrigeration, which 00:24:41.320 --> 00:24:44.170 align:middle line:84% is the heating load, so the heat that it takes away 00:24:44.170 --> 00:24:49.690 align:middle line:84% from the working fluid divided by the work 00:24:49.690 --> 00:24:52.840 align:middle line:84% that you have spent in order to run the machinery, 00:24:52.840 --> 00:25:00.440 align:middle line:84% typically compression cycle or refrigeration cycle, 00:25:00.440 --> 00:25:02.096 align:middle line:90% but not necessarily. 00:25:02.096 --> 00:25:05.290 align:middle line:90% 00:25:05.290 --> 00:25:06.790 align:middle line:90% So this is the COP. 00:25:06.790 --> 00:25:12.400 align:middle line:90% The second-law efficiency is-- 00:25:12.400 --> 00:25:18.460 align:middle line:84% well, you can view this as minus QC times the Carnot coefficient. 00:25:18.460 --> 00:25:21.010 align:middle line:84% And again, the second-law efficiency, therefore, 00:25:21.010 --> 00:25:25.640 align:middle line:84% is the COP times the Carnot coefficient. 00:25:25.640 --> 00:25:30.470 align:middle line:84% And for typical values, I mean, for an industrial refrigerator 00:25:30.470 --> 00:25:36.650 align:middle line:84% for which the COP is about 5, which is relatively high, 00:25:36.650 --> 00:25:39.410 align:middle line:84% it means that you spend 1 kilowatt hour 00:25:39.410 --> 00:25:45.170 align:middle line:84% and extract 5 kilowatt hours of heat 00:25:45.170 --> 00:25:51.650 align:middle line:84% from the cooling-- as the cooling effect. 00:25:51.650 --> 00:25:56.390 align:middle line:84% So 5, you multiply by the Carnot coefficient 00:25:56.390 --> 00:26:02.480 align:middle line:84% for those temperatures at an outside temperature of-- 00:26:02.480 --> 00:26:08.030 align:middle line:84% what is it-- 30 degrees C. And you get a second-law efficiency 00:26:08.030 --> 00:26:08.633 align:middle line:90% of 36%. 00:26:08.633 --> 00:26:13.390 align:middle line:90% 00:26:13.390 --> 00:26:14.000 align:middle line:90% Not bad. 00:26:14.000 --> 00:26:16.840 align:middle line:90% 00:26:16.840 --> 00:26:20.440 align:middle line:84% At least compared with heating, it 00:26:20.440 --> 00:26:24.870 align:middle line:84% seems that, with cooling, we're doing better. 00:26:24.870 --> 00:26:28.290 align:middle line:84% In fact, the machinery for cooling is, as you know, 00:26:28.290 --> 00:26:31.860 align:middle line:84% the same as the machinery that we use for heat pumps. 00:26:31.860 --> 00:26:37.150 align:middle line:84% It requires more investment, more materials. 00:26:37.150 --> 00:26:41.340 align:middle line:84% And it's more sophisticated than just plain burning something 00:26:41.340 --> 00:26:44.110 align:middle line:90% and using a heat exchanger. 00:26:44.110 --> 00:26:48.730 align:middle line:90% 00:26:48.730 --> 00:26:57.070 align:middle line:84% Now, people have tried also the simultaneous combined production 00:26:57.070 --> 00:27:01.510 align:middle line:90% of power, heat, and cooling. 00:27:01.510 --> 00:27:03.230 align:middle line:90% It's called trigeneration. 00:27:03.230 --> 00:27:05.860 align:middle line:90% You have three products. 00:27:05.860 --> 00:27:12.820 align:middle line:84% And actually, this considers also the fact 00:27:12.820 --> 00:27:14.440 align:middle line:90% that you may have-- 00:27:14.440 --> 00:27:22.190 align:middle line:84% that you may use, as a possible resource-- 00:27:22.190 --> 00:27:25.632 align:middle line:84% sorry, this is the cooling effect. 00:27:25.632 --> 00:27:28.940 align:middle line:90% This is the cooling effect. 00:27:28.940 --> 00:27:31.555 align:middle line:84% So this uses as a resource only the fuel. 00:27:31.555 --> 00:27:38.010 align:middle line:90% 00:27:38.010 --> 00:27:46.350 align:middle line:84% And therefore, you have three products. 00:27:46.350 --> 00:27:49.780 align:middle line:90% And here is one of the problems. 00:27:49.780 --> 00:27:57.030 align:middle line:84% It's not so easy to define a first-law efficiency 00:27:57.030 --> 00:28:03.520 align:middle line:84% because you have to put a numerator apple and oranges. 00:28:03.520 --> 00:28:10.410 align:middle line:84% So you have to put heat, work, and cooling. 00:28:10.410 --> 00:28:14.310 align:middle line:90% So they are different animals. 00:28:14.310 --> 00:28:18.130 align:middle line:84% And they shouldn't stay together in a numerator. 00:28:18.130 --> 00:28:20.950 align:middle line:90% 00:28:20.950 --> 00:28:27.472 align:middle line:84% But you could define separate efficiencies, 00:28:27.472 --> 00:28:33.020 align:middle line:84% provided we have a way to allocate the fuel consumption 00:28:33.020 --> 00:28:41.080 align:middle line:84% and assign a fraction of it for each of the products, which 00:28:41.080 --> 00:28:48.610 align:middle line:84% is the idea of this allocation that we're now 00:28:48.610 --> 00:28:52.185 align:middle line:90% getting to refresh again. 00:28:52.185 --> 00:28:57.290 align:middle line:90% 00:28:57.290 --> 00:29:05.300 align:middle line:84% So you see, every local area that could be a nation-- 00:29:05.300 --> 00:29:09.570 align:middle line:84% for example, here, I've put some numbers and the flag for Italy. 00:29:09.570 --> 00:29:12.780 align:middle line:84% And if you're interested in-if you have a preferred country, 00:29:12.780 --> 00:29:14.700 align:middle line:90% you can do this table. 00:29:14.700 --> 00:29:17.000 align:middle line:84% It would be interesting, in any case, 00:29:17.000 --> 00:29:24.560 align:middle line:84% to search and to see, for example, 00:29:24.560 --> 00:29:28.880 align:middle line:84% for separate production, one thing 00:29:28.880 --> 00:29:35.536 align:middle line:84% is to consider what is the average national efficiency. 00:29:35.536 --> 00:29:40.180 align:middle line:90% 00:29:40.180 --> 00:29:43.596 align:middle line:84% And one is the efficiency of the best available technology. 00:29:43.596 --> 00:29:46.730 align:middle line:90% 00:29:46.730 --> 00:29:52.130 align:middle line:84% And then you have the same for combined heat and power. 00:29:52.130 --> 00:29:55.880 align:middle line:84% And you can do also the same for when 00:29:55.880 --> 00:30:01.880 align:middle line:84% you hybridize your use of resources 00:30:01.880 --> 00:30:05.390 align:middle line:84% by using some that are fossil fuels and some 00:30:05.390 --> 00:30:11.840 align:middle line:90% that are renewable resources. 00:30:11.840 --> 00:30:15.230 align:middle line:84% And of course, each line is for what product 00:30:15.230 --> 00:30:19.490 align:middle line:84% you are interested in-- so electric power, 00:30:19.490 --> 00:30:27.300 align:middle line:84% residential heat, winter, summer, industrial heat, 00:30:27.300 --> 00:30:31.520 align:middle line:90% cooling, and so on. 00:30:31.520 --> 00:30:34.310 align:middle line:84% The list is long because energy products-- 00:30:34.310 --> 00:30:39.330 align:middle line:84% you could also consider the production of materials. 00:30:39.330 --> 00:30:44.120 align:middle line:84% For example, aluminum from the ore-- 00:30:44.120 --> 00:30:46.760 align:middle line:90% bauxite or whatever it is-- 00:30:46.760 --> 00:30:52.350 align:middle line:84% is a very energy consuming process. 00:30:52.350 --> 00:30:56.610 align:middle line:84% And so talking about efficiencies is important. 00:30:56.610 --> 00:31:02.270 align:middle line:84% And also doing cogeneration is also 00:31:02.270 --> 00:31:09.960 align:middle line:84% a way to save energy, so use better our natural resources. 00:31:09.960 --> 00:31:16.350 align:middle line:84% And for each one of these technologies or products, energy 00:31:16.350 --> 00:31:23.190 align:middle line:84% products, we can also study where 00:31:23.190 --> 00:31:25.550 align:middle line:90% we are in the learning curve. 00:31:25.550 --> 00:31:28.630 align:middle line:90% 00:31:28.630 --> 00:31:31.570 align:middle line:84% And as we've seen, this is the learning curve 00:31:31.570 --> 00:31:33.820 align:middle line:90% for power production. 00:31:33.820 --> 00:31:41.290 align:middle line:84% If we did a learning curve for heat, we would be just about 00:31:41.290 --> 00:31:43.955 align:middle line:90% here between 10% and 20%. 00:31:43.955 --> 00:31:46.960 align:middle line:90% 00:31:46.960 --> 00:31:49.990 align:middle line:90% Cooling is between 20% and 30%. 00:31:49.990 --> 00:31:53.290 align:middle line:90% So the curves would be shifted. 00:31:53.290 --> 00:32:00.700 align:middle line:84% And probably also the characteristic time 00:32:00.700 --> 00:32:02.330 align:middle line:90% could be different. 00:32:02.330 --> 00:32:05.290 align:middle line:84% But in any case, you see, it's not 00:32:05.290 --> 00:32:10.750 align:middle line:84% in the matter of just a few years. 00:32:10.750 --> 00:32:15.950 align:middle line:84% It requires many years to mature a technology. 00:32:15.950 --> 00:32:21.130 align:middle line:90% 00:32:21.130 --> 00:32:24.910 align:middle line:84% So that also carries the message that if you 00:32:24.910 --> 00:32:29.380 align:middle line:84% are in the energy business, or if you 00:32:29.380 --> 00:32:34.840 align:middle line:84% want to listen to a politician, you 00:32:34.840 --> 00:32:40.690 align:middle line:84% have to be careful because nobody can promise results 00:32:40.690 --> 00:32:42.180 align:middle line:90% in short times. 00:32:42.180 --> 00:32:48.110 align:middle line:90% 00:32:48.110 --> 00:32:51.920 align:middle line:84% So that's another interesting story. 00:32:51.920 --> 00:32:58.640 align:middle line:84% So that brings us to the problem of allocation. 00:32:58.640 --> 00:33:05.390 align:middle line:84% As we have already discussed, in a cogeneration facility, 00:33:05.390 --> 00:33:11.540 align:middle line:84% it amounts to defining these coefficients alpha 00:33:11.540 --> 00:33:13.340 align:middle line:84% that are the fractions of the fuel 00:33:13.340 --> 00:33:16.820 align:middle line:84% that we think are used for the two products, 00:33:16.820 --> 00:33:18.900 align:middle line:84% that we assign to the two products. 00:33:18.900 --> 00:33:24.060 align:middle line:84% And we've seen that this assignment is a bit arbitrary. 00:33:24.060 --> 00:33:28.130 align:middle line:84% There is no law of physics that tells us how to do it. 00:33:28.130 --> 00:33:31.190 align:middle line:84% And each one has its pros and cons. 00:33:31.190 --> 00:33:37.340 align:middle line:90% 00:33:37.340 --> 00:33:39.920 align:middle line:84% In particular, in order to evaluate these, 00:33:39.920 --> 00:33:45.410 align:middle line:84% it is interesting to see these primary energy savings 00:33:45.410 --> 00:33:54.290 align:middle line:84% because cogeneration does allow you to reduce the overall fuel 00:33:54.290 --> 00:33:58.460 align:middle line:84% consumption than if you did it in separate production 00:33:58.460 --> 00:33:59.310 align:middle line:90% facilities. 00:33:59.310 --> 00:34:01.520 align:middle line:90% So there is a savings. 00:34:01.520 --> 00:34:04.160 align:middle line:84% And the problem of these alphas is 00:34:04.160 --> 00:34:12.020 align:middle line:84% how you allocate those savings to the two products, heat 00:34:12.020 --> 00:34:12.819 align:middle line:90% and work. 00:34:12.819 --> 00:34:15.409 align:middle line:90% 00:34:15.409 --> 00:34:17.969 align:middle line:84% And once you have chosen those alphas, 00:34:17.969 --> 00:34:25.130 align:middle line:84% you can define the COPs and the partial efficiency of your work 00:34:25.130 --> 00:34:27.949 align:middle line:90% and of your heat production. 00:34:27.949 --> 00:34:33.830 align:middle line:84% And as we said, this topic all started 00:34:33.830 --> 00:34:37.190 align:middle line:84% when the COP of heat production became 00:34:37.190 --> 00:34:44.060 align:middle line:84% important in determining the energy efficiency of buildings, 00:34:44.060 --> 00:34:49.730 align:middle line:84% which, in turn, determines the value of these buildings. 00:34:49.730 --> 00:34:52.613 align:middle line:90% So this has an economic impact. 00:34:52.613 --> 00:34:56.409 align:middle line:90% 00:34:56.409 --> 00:35:00.340 align:middle line:84% And this incremental electricity-centered allocation 00:35:00.340 --> 00:35:06.730 align:middle line:84% method, which was assigning all the energy savings 00:35:06.730 --> 00:35:16.870 align:middle line:84% to the production of heat, is what triggered the need 00:35:16.870 --> 00:35:24.100 align:middle line:84% to do something better because that is unfair with respect 00:35:24.100 --> 00:35:27.970 align:middle line:84% to other people who would produce heat 00:35:27.970 --> 00:35:29.970 align:middle line:90% by just burning the fuel. 00:35:29.970 --> 00:35:35.410 align:middle line:90% 00:35:35.410 --> 00:35:43.270 align:middle line:84% So typically, one would define these reference efficiencies 00:35:43.270 --> 00:35:49.960 align:middle line:84% for the separate production facilities, defined 00:35:49.960 --> 00:35:56.590 align:middle line:84% by some authority, and then define this coefficient 00:35:56.590 --> 00:36:00.880 align:middle line:84% of allocation as the weighted sum of what I would consume 00:36:00.880 --> 00:36:04.210 align:middle line:84% to produce that same amount of electricity 00:36:04.210 --> 00:36:08.916 align:middle line:84% in separate production facilities. 00:36:08.916 --> 00:36:12.770 align:middle line:90% 00:36:12.770 --> 00:36:15.290 align:middle line:84% At least, this method has the advantage 00:36:15.290 --> 00:36:24.680 align:middle line:84% to split evenly the primary energy savings between the two 00:36:24.680 --> 00:36:26.180 align:middle line:90% products. 00:36:26.180 --> 00:36:34.590 align:middle line:84% So if I say 20%, I assign a 20% improvement 00:36:34.590 --> 00:36:38.250 align:middle line:84% to the efficiency of work and a 20% improvement 00:36:38.250 --> 00:36:40.070 align:middle line:90% to the efficiency of heat. 00:36:40.070 --> 00:36:52.080 align:middle line:90% 00:36:52.080 --> 00:36:55.290 align:middle line:84% The people that are really fond of thermodynamics 00:36:55.290 --> 00:37:00.900 align:middle line:84% and would like to solve every problem with thermodynamics 00:37:00.900 --> 00:37:03.150 align:middle line:84% insisted for a while that you should 00:37:03.150 --> 00:37:06.990 align:middle line:84% use the exergy method, which takes 00:37:06.990 --> 00:37:11.070 align:middle line:84% as reference the reversible process because, 00:37:11.070 --> 00:37:16.470 align:middle line:84% after all, that is the ultimate target that we 00:37:16.470 --> 00:37:21.480 align:middle line:90% should be heading towards. 00:37:21.480 --> 00:37:26.400 align:middle line:84% However, the problem is that, as we've seen in the learning 00:37:26.400 --> 00:37:32.940 align:middle line:84% curve, heat and work technologies have 00:37:32.940 --> 00:37:34.620 align:middle line:90% not the same level of maturity. 00:37:34.620 --> 00:37:40.320 align:middle line:84% And therefore, the distance of where 00:37:40.320 --> 00:37:44.040 align:middle line:84% we are from where we could be with reversible machinery, which 00:37:44.040 --> 00:37:50.280 align:middle line:84% is the exergy target, is very different, 00:37:50.280 --> 00:37:57.940 align:middle line:84% like maybe 35% for electricity and maybe 80% or even more 00:37:57.940 --> 00:38:00.360 align:middle line:90% for heat. 00:38:00.360 --> 00:38:12.420 align:middle line:84% And this gives again too much advantage to the heat 00:38:12.420 --> 00:38:16.950 align:middle line:84% because the weight with which the energy savings would 00:38:16.950 --> 00:38:20.400 align:middle line:84% be split between work and heat would be proportional 00:38:20.400 --> 00:38:24.600 align:middle line:90% to that margin of improvement. 00:38:24.600 --> 00:38:32.760 align:middle line:84% And in fact, if we look at the numbers for this example here, 00:38:32.760 --> 00:38:44.610 align:middle line:84% we find that we would have a COP of 200, 00:38:44.610 --> 00:38:51.390 align:middle line:84% which is better than the 350 of the electricity-centered method. 00:38:51.390 --> 00:39:04.410 align:middle line:84% But with more realistic reference efficiencies, 00:39:04.410 --> 00:39:11.470 align:middle line:84% you would have much more reasonable numbers-- 00:39:11.470 --> 00:39:18.510 align:middle line:84% I mean, lower numbers, like 92, maybe 96% for the COP 00:39:18.510 --> 00:39:22.410 align:middle line:84% of residential district heating, which 00:39:22.410 --> 00:39:25.780 align:middle line:84% is good because it's better than, say, the typical 85%. 00:39:25.780 --> 00:39:32.490 align:middle line:90% 00:39:32.490 --> 00:39:43.800 align:middle line:84% And also, it gives a more fair split between work and heat. 00:39:43.800 --> 00:39:51.150 align:middle line:84% Now, the homework doesn't call you to do anything 00:39:51.150 --> 00:39:56.190 align:middle line:84% about this method of local area because that has not yet 00:39:56.190 --> 00:40:00.910 align:middle line:84% been accepted or incorporated in any of the standards. 00:40:00.910 --> 00:40:02.090 align:middle line:90% It's just an idea. 00:40:02.090 --> 00:40:05.640 align:middle line:90% 00:40:05.640 --> 00:40:13.700 align:middle line:84% And nevertheless, it's interesting to say that 00:40:13.700 --> 00:40:19.560 align:middle line:84% essentially what the local authority should use, 00:40:19.560 --> 00:40:25.800 align:middle line:84% according to our suggestion, is as reference efficiencies 00:40:25.800 --> 00:40:28.630 align:middle line:84% the average efficiencies of the local area, 00:40:28.630 --> 00:40:32.996 align:middle line:84% including the combined heat and power production facilities. 00:40:32.996 --> 00:40:36.150 align:middle line:90% 00:40:36.150 --> 00:40:50.040 align:middle line:84% That way, if your combined facility is, like in this case, 00:40:50.040 --> 00:40:59.100 align:middle line:84% more efficient than the reference, 00:40:59.100 --> 00:41:07.710 align:middle line:84% then it deserves some incentive, or at least a clap, 00:41:07.710 --> 00:41:12.090 align:middle line:84% because they are doing better than the average 00:41:12.090 --> 00:41:13.380 align:middle line:90% in the local area. 00:41:13.380 --> 00:41:19.800 align:middle line:84% So one criterion also for either giving incentives or giving 00:41:19.800 --> 00:41:23.550 align:middle line:84% authorization to make a new power plant 00:41:23.550 --> 00:41:29.850 align:middle line:84% is so that it meets a higher efficiency. 00:41:29.850 --> 00:41:31.840 align:middle line:84% It has a higher efficiency than the average. 00:41:31.840 --> 00:41:37.500 align:middle line:84% And therefore, in turn, it will higher the average efficiency 00:41:37.500 --> 00:41:39.120 align:middle line:90% of the area. 00:41:39.120 --> 00:41:43.300 align:middle line:84% So that would be beneficial for everybody. 00:41:43.300 --> 00:41:50.320 align:middle line:90% 00:41:50.320 --> 00:41:56.870 align:middle line:84% The converse is when we do hybrid power plants. 00:41:56.870 --> 00:42:03.410 align:middle line:84% And in this case, we are using, in this paper, 00:42:03.410 --> 00:42:08.840 align:middle line:84% not cogeneration, but just power production, only power 00:42:08.840 --> 00:42:11.630 align:middle line:90% production. 00:42:11.630 --> 00:42:13.790 align:middle line:84% And instead of using just fuel, you 00:42:13.790 --> 00:42:18.320 align:middle line:84% use fuel and integrate that fuel with some renewable, 00:42:18.320 --> 00:42:21.530 align:middle line:90% like solar energy or vice versa. 00:42:21.530 --> 00:42:25.190 align:middle line:84% Instead of having just a renewable, just solar energy 00:42:25.190 --> 00:42:29.480 align:middle line:84% heating a power plant, like those power plants 00:42:29.480 --> 00:42:34.710 align:middle line:84% with mirrors that we have said the other day, 00:42:34.710 --> 00:42:37.460 align:middle line:84% and you can look it up, there are 00:42:37.460 --> 00:42:40.710 align:middle line:84% a few prototypes in California, in Spain, 00:42:40.710 --> 00:42:43.470 align:middle line:84% in Italy, and probably other places, too. 00:42:43.470 --> 00:42:48.290 align:middle line:90% 00:42:48.290 --> 00:42:53.370 align:middle line:84% Nevertheless, the fact that, of course, 00:42:53.370 --> 00:42:56.240 align:middle line:84% solar energy is not always available, 00:42:56.240 --> 00:43:01.880 align:middle line:84% and therefore you need to either make some storage in order 00:43:01.880 --> 00:43:04.920 align:middle line:84% to have it function continuously, 00:43:04.920 --> 00:43:09.710 align:middle line:84% or you could integrate with fuel for when-- 00:43:09.710 --> 00:43:14.490 align:middle line:84% to compensate, when there are clouds or at nighttime, 00:43:14.490 --> 00:43:18.050 align:middle line:84% so there is no solar energy available. 00:43:18.050 --> 00:43:21.890 align:middle line:84% In this way, you may have advantages, 00:43:21.890 --> 00:43:27.990 align:middle line:84% both from the point of view of efficiency-- 00:43:27.990 --> 00:43:35.250 align:middle line:84% so it's more stable input for your power plant-- 00:43:35.250 --> 00:43:39.060 align:middle line:84% and also for the cost because the investment 00:43:39.060 --> 00:43:47.530 align:middle line:84% is going to be diluted over more hours of operation. 00:43:47.530 --> 00:43:51.820 align:middle line:84% So it is important, in this case, 00:43:51.820 --> 00:44:00.450 align:middle line:84% also to determine these fractions, the betas, 00:44:00.450 --> 00:44:07.920 align:middle line:84% the fractions of the product that can be understood 00:44:07.920 --> 00:44:13.110 align:middle line:84% as being the product of the fossil 00:44:13.110 --> 00:44:17.160 align:middle line:84% or the product of the renewable part. 00:44:17.160 --> 00:44:19.950 align:middle line:84% So if I could identify what fraction 00:44:19.950 --> 00:44:23.980 align:middle line:84% of the electricity produced is considered renewable, 00:44:23.980 --> 00:44:27.690 align:middle line:84% this would be at the same level as, say, 00:44:27.690 --> 00:44:37.650 align:middle line:84% photovoltaic energy because photovoltaic energy is 00:44:37.650 --> 00:44:43.596 align:middle line:84% like a separate production from solar to electricity. 00:44:43.596 --> 00:44:46.580 align:middle line:90% 00:44:46.580 --> 00:44:50.790 align:middle line:84% And so there is no reason why, if the country 00:44:50.790 --> 00:44:58.202 align:middle line:84% decides to give an incentive to photovoltaics-- 00:44:58.202 --> 00:45:00.160 align:middle line:84% there is no reason why they shouldn't give also 00:45:00.160 --> 00:45:03.170 align:middle line:84% an incentive to a hybrid facility, 00:45:03.170 --> 00:45:12.040 align:middle line:84% at least in as far as the fraction of electricity that 00:45:12.040 --> 00:45:23.500 align:middle line:84% is recognized as renewable because, in the end, 00:45:23.500 --> 00:45:27.940 align:middle line:84% it turns out that these facilities, hybrid facilities, 00:45:27.940 --> 00:45:33.430 align:middle line:84% have higher, slightly higher efficiencies than photovoltaics. 00:45:33.430 --> 00:45:38.020 align:middle line:84% So if photovoltaics is like 15% efficient, 00:45:38.020 --> 00:45:41.770 align:middle line:84% these facilities may go over 15%. 00:45:41.770 --> 00:45:47.590 align:middle line:84% And so it may be a better way of using solar energy 00:45:47.590 --> 00:45:51.260 align:middle line:90% for electricity production. 00:45:51.260 --> 00:46:01.700 align:middle line:84% So also here, at the beginning, people 00:46:01.700 --> 00:46:06.680 align:middle line:84% thought of assigning all the benefit of the hybridization 00:46:06.680 --> 00:46:08.980 align:middle line:90% to the renewable aspect. 00:46:08.980 --> 00:46:12.650 align:middle line:90% 00:46:12.650 --> 00:46:22.870 align:middle line:84% But then this would be unfair because these power facilities, 00:46:22.870 --> 00:46:28.720 align:middle line:84% hybrid facilities, would get the incentives even 00:46:28.720 --> 00:46:31.280 align:middle line:90% if they don't deserve it. 00:46:31.280 --> 00:46:35.500 align:middle line:90% 00:46:35.500 --> 00:46:40.120 align:middle line:84% Because part of the savings made by the hybridization, 00:46:40.120 --> 00:46:45.820 align:middle line:84% should also go to the fossil line of the power plant. 00:46:45.820 --> 00:46:50.440 align:middle line:84% So also here, a reasonable way is 00:46:50.440 --> 00:46:53.650 align:middle line:90% to use a reference production. 00:46:53.650 --> 00:47:00.710 align:middle line:90% So it would be the weighted-- 00:47:00.710 --> 00:47:04.580 align:middle line:90% 00:47:04.580 --> 00:47:13.760 align:middle line:84% the fraction of how much electricity 00:47:13.760 --> 00:47:18.590 align:middle line:84% you would produce with that fuel in a separate production 00:47:18.590 --> 00:47:19.700 align:middle line:90% facility. 00:47:19.700 --> 00:47:23.540 align:middle line:84% And the other is how much electricity 00:47:23.540 --> 00:47:29.450 align:middle line:84% you would produce in a solar facility. 00:47:29.450 --> 00:47:33.630 align:middle line:84% And these weighted ratios are balanced. 00:47:33.630 --> 00:47:36.800 align:middle line:84% But of course, somebody has to define what 00:47:36.800 --> 00:47:38.905 align:middle line:90% reference efficiencies to use. 00:47:38.905 --> 00:47:43.800 align:middle line:90% 00:47:43.800 --> 00:47:48.930 align:middle line:84% And again, there is this consideration 00:47:48.930 --> 00:47:58.060 align:middle line:84% of the exergy method that would provide 00:47:58.060 --> 00:48:09.810 align:middle line:84% 100% as the target, the reversible target, and 93% 00:48:09.810 --> 00:48:13.650 align:middle line:84% for the reversible target in the conversion of solar energy 00:48:13.650 --> 00:48:17.580 align:middle line:90% into electricity. 00:48:17.580 --> 00:48:25.020 align:middle line:84% But these targets are very far from the actual level 00:48:25.020 --> 00:48:29.370 align:middle line:90% of average efficiencies. 00:48:29.370 --> 00:48:36.750 align:middle line:84% And therefore, it is dangerous to use those as references. 00:48:36.750 --> 00:48:42.180 align:middle line:84% It is better to use as references something fixed 00:48:42.180 --> 00:48:47.910 align:middle line:84% or, again, something based on a local area, so 00:48:47.910 --> 00:48:55.440 align:middle line:84% the average values of the partial efficiencies 00:48:55.440 --> 00:49:02.610 align:middle line:84% in the local area, so that, again, also here you 00:49:02.610 --> 00:49:06.520 align:middle line:84% could argue that who deserves the incentives, 00:49:06.520 --> 00:49:08.130 align:middle line:84% the economic incentives, are only 00:49:08.130 --> 00:49:15.690 align:middle line:84% those who improve somewhat the average usage of those resources 00:49:15.690 --> 00:49:17.675 align:middle line:90% in the local area. 00:49:17.675 --> 00:49:28.330 align:middle line:90% 00:49:28.330 --> 00:49:32.813 align:middle line:84% So for the numbers, here, there is an example. 00:49:32.813 --> 00:49:37.400 align:middle line:90% 00:49:37.400 --> 00:49:39.840 align:middle line:90% Yeah, I think I skipped one. 00:49:39.840 --> 00:49:44.220 align:middle line:84% I removed one slide in which it was saying-- yeah, here it is. 00:49:44.220 --> 00:49:53.170 align:middle line:84% The separate production for solar, that is typically, 00:49:53.170 --> 00:50:00.750 align:middle line:84% say, average of today for photovoltaics is 15.3%. 00:50:00.750 --> 00:50:03.670 align:middle line:90% And here, we use the 38% map. 00:50:03.670 --> 00:50:06.610 align:middle line:84% Maybe we could have put also 42%, maybe also 00:50:06.610 --> 00:50:11.830 align:middle line:84% 45% today for the separate production of electricity 00:50:11.830 --> 00:50:13.430 align:middle line:90% from fuels. 00:50:13.430 --> 00:50:17.706 align:middle line:84% Now, here is the hybrid facility. 00:50:17.706 --> 00:50:20.380 align:middle line:90% 00:50:20.380 --> 00:50:26.350 align:middle line:84% And these numbers define the relative role 00:50:26.350 --> 00:50:30.790 align:middle line:84% of hybrid production in the local area. 00:50:30.790 --> 00:50:34.850 align:middle line:84% For example, this says that, of the overall electricity, 00:50:34.850 --> 00:50:43.035 align:middle line:84% 45% is hybrid and 10% is separate solar, 00:50:43.035 --> 00:50:44.396 align:middle line:90% so like photovoltaics. 00:50:44.396 --> 00:50:50.350 align:middle line:90% 00:50:50.350 --> 00:51:05.950 align:middle line:84% And then by using these self-tuned average 00:51:05.950 --> 00:51:14.990 align:middle line:84% separate local production references, here is the numbers. 00:51:14.990 --> 00:51:21.170 align:middle line:84% For example, it would say that this hybrid facility improves 00:51:21.170 --> 00:51:31.340 align:middle line:84% the efficiency of production from solar from 15% to 18.7%. 00:51:31.340 --> 00:51:34.580 align:middle line:84% The exergy method would have said that the improvement is 00:51:34.580 --> 00:51:39.140 align:middle line:84% to 35%, giving therefore a lot of credit 00:51:39.140 --> 00:51:42.150 align:middle line:84% and therefore very high incentives. 00:51:42.150 --> 00:51:46.220 align:middle line:84% And the fuel centered even higher. 00:51:46.220 --> 00:51:46.930 align:middle line:90% It's nonsense. 00:51:46.930 --> 00:51:51.620 align:middle line:90% 00:51:51.620 --> 00:51:58.610 align:middle line:84% So if you happen to end up working in one of these agencies 00:51:58.610 --> 00:52:05.450 align:middle line:84% that define regulations and that keep following 00:52:05.450 --> 00:52:13.400 align:middle line:84% the evolution of local areas and the improvements, 00:52:13.400 --> 00:52:29.060 align:middle line:84% you have to be aware of avoiding wrong numbers that would change 00:52:29.060 --> 00:52:36.140 align:middle line:84% the direction of technologies or investments 00:52:36.140 --> 00:52:41.540 align:middle line:84% not where you want it to be going. 00:52:41.540 --> 00:52:43.830 align:middle line:84% Because these numbers are in particular, 00:52:43.830 --> 00:52:48.020 align:middle line:84% the laws that determine incentives 00:52:48.020 --> 00:52:52.980 align:middle line:84% based on these numbers determine the choices of the companies 00:52:52.980 --> 00:52:55.370 align:middle line:90% where to invest in what. 00:52:55.370 --> 00:52:58.200 align:middle line:90% 00:52:58.200 --> 00:53:01.080 align:middle line:84% All right, so with this, I think I 00:53:01.080 --> 00:53:07.840 align:middle line:84% have said a little more about this problem of allocation. 00:53:07.840 --> 00:53:10.540 align:middle line:84% And if you still have some questions, 00:53:10.540 --> 00:53:13.570 align:middle line:84% maybe this is the time-- for the homework, for example, 00:53:13.570 --> 00:53:15.815 align:middle line:90% maybe this is the time to ask. 00:53:15.815 --> 00:53:18.470 align:middle line:90% 00:53:18.470 --> 00:53:22.010 align:middle line:90% If it is all clear, then OK. 00:53:22.010 --> 00:53:27.720 align:middle line:84% In any case, if you need help, I'm available in the office. 00:53:27.720 --> 00:53:28.580 align:middle line:90% Just email me. 00:53:28.580 --> 00:53:33.055 align:middle line:84% And we can set up an appointment or just chat by email. 00:53:33.055 --> 00:53:36.000 align:middle line:90% 00:53:36.000 --> 00:53:36.500 align:middle line:90% OK. 00:53:36.500 --> 00:53:40.760 align:middle line:90% 00:53:40.760 --> 00:53:45.387 align:middle line:84% Yeah, because now I want to move to the second part 00:53:45.387 --> 00:53:45.970 align:middle line:90% of the course. 00:53:45.970 --> 00:53:49.880 align:middle line:90% 00:53:49.880 --> 00:53:52.790 align:middle line:84% This was an initial slide, you remember, 00:53:52.790 --> 00:53:55.770 align:middle line:84% where we said, in the second part of the course, 00:53:55.770 --> 00:53:58.130 align:middle line:84% we're going to deal with chemical potentials 00:53:58.130 --> 00:54:02.023 align:middle line:90% and multi-constituent systems. 00:54:02.023 --> 00:54:06.390 align:middle line:90% 00:54:06.390 --> 00:54:10.273 align:middle line:84% In part, we have already discussed-- 00:54:10.273 --> 00:54:15.020 align:middle line:90% 00:54:15.020 --> 00:54:16.360 align:middle line:90% these are stable-- 00:54:16.360 --> 00:54:21.180 align:middle line:84% A lot of this is played with stable equilibrium states. 00:54:21.180 --> 00:54:24.010 align:middle line:84% And we said a lot about the properties of stable equilibrium 00:54:24.010 --> 00:54:24.510 align:middle line:90% states. 00:54:24.510 --> 00:54:31.280 align:middle line:84% And in fact, in the next few slides, 00:54:31.280 --> 00:54:36.980 align:middle line:84% which are full of formulas as usual, 00:54:36.980 --> 00:54:43.390 align:middle line:84% I'll just review quickly the things that we've already 00:54:43.390 --> 00:54:48.040 align:middle line:84% said that are relevant to the applications 00:54:48.040 --> 00:54:54.695 align:middle line:90% that we want to focus on next. 00:54:54.695 --> 00:55:00.390 align:middle line:90% 00:55:00.390 --> 00:55:15.150 align:middle line:84% So this part is divided into three other parts. 00:55:15.150 --> 00:55:18.630 align:middle line:84% The first one is defining properties 00:55:18.630 --> 00:55:25.520 align:middle line:90% of mixtures in a single phase. 00:55:25.520 --> 00:55:28.320 align:middle line:90% 00:55:28.320 --> 00:55:30.970 align:middle line:84% So we'll have to talk about partial properties, 00:55:30.970 --> 00:55:37.290 align:middle line:84% define what they are, define some ideal mixture. 00:55:37.290 --> 00:55:40.410 align:middle line:84% And here, I should also add another topic, which 00:55:40.410 --> 00:55:45.270 align:middle line:90% is ideal solution behavior. 00:55:45.270 --> 00:55:47.910 align:middle line:84% We will talk about osmotic pressure. 00:55:47.910 --> 00:55:57.480 align:middle line:84% We will talk about the minimum work for separation of a mixture 00:55:57.480 --> 00:56:01.530 align:middle line:84% into its components or the maximum work 00:56:01.530 --> 00:56:09.550 align:middle line:84% that you can extract from mixing if you don't do it irreversibly. 00:56:09.550 --> 00:56:13.900 align:middle line:84% And then yeah, osmotic pressure and the so-called blue energy, 00:56:13.900 --> 00:56:17.980 align:middle line:84% and then also the problem of stratification-- 00:56:17.980 --> 00:56:24.460 align:middle line:84% so what's the composition in the air at various heights. 00:56:24.460 --> 00:56:26.170 align:middle line:84% Then there will be a second part that 00:56:26.170 --> 00:56:30.610 align:middle line:84% will deal with liquid vapor equilibrium 00:56:30.610 --> 00:56:35.140 align:middle line:84% or, say, two-phase states for mixtures. 00:56:35.140 --> 00:56:39.280 align:middle line:84% And the third part, which deals with systems or mixtures 00:56:39.280 --> 00:56:41.420 align:middle line:90% with chemical reactions. 00:56:41.420 --> 00:56:46.930 align:middle line:90% 00:56:46.930 --> 00:56:49.420 align:middle line:90% Most of it is covered-- 00:56:49.420 --> 00:56:53.115 align:middle line:84% is still covered-- sorry, it's covered by the book here. 00:56:53.115 --> 00:57:00.230 align:middle line:90% 00:57:00.230 --> 00:57:11.120 align:middle line:84% So here is a dense summary of things that we have seen. 00:57:11.120 --> 00:57:16.910 align:middle line:84% You remember, we defined these characteristic functions 00:57:16.910 --> 00:57:20.450 align:middle line:84% starting from the fundamental relation 00:57:20.450 --> 00:57:25.520 align:middle line:84% of the stable equilibrium states and performing the Legendre 00:57:25.520 --> 00:57:27.740 align:middle line:84% transform of some of its variables 00:57:27.740 --> 00:57:29.260 align:middle line:90% or all of its variables. 00:57:29.260 --> 00:57:35.210 align:middle line:84% We defined alternative ways of describing the properties 00:57:35.210 --> 00:57:36.870 align:middle line:90% of stable equilibrium. 00:57:36.870 --> 00:57:41.980 align:middle line:90% 00:57:41.980 --> 00:57:47.980 align:middle line:84% And among these Legendre transforms, 00:57:47.980 --> 00:57:51.820 align:middle line:84% we defined the F, which is the Helmholtz free energy; 00:57:51.820 --> 00:57:54.040 align:middle line:84% G, which is the Gibbs free energy; 00:57:54.040 --> 00:57:57.250 align:middle line:84% H, which is the enthalpy; and Eu, 00:57:57.250 --> 00:57:59.680 align:middle line:90% which is the Euler free energy. 00:57:59.680 --> 00:58:04.560 align:middle line:90% They are all related like this. 00:58:04.560 --> 00:58:17.680 align:middle line:84% Of course, where you see this, that 00:58:17.680 --> 00:58:29.030 align:middle line:84% means the summation of the products 00:58:29.030 --> 00:58:30.930 align:middle line:90% of the various constituents. 00:58:30.930 --> 00:58:37.065 align:middle line:84% So we already set up for dealing with mixtures. 00:58:37.065 --> 00:58:44.630 align:middle line:90% 00:58:44.630 --> 00:58:51.650 align:middle line:84% Also, this Euler free energy is a function 00:58:51.650 --> 00:58:53.850 align:middle line:84% of temperature, pressure, and chemical potentials. 00:58:53.850 --> 00:59:01.050 align:middle line:84% So that means that you can measure it by appropriate-- 00:59:01.050 --> 00:59:05.060 align:middle line:90% 00:59:05.060 --> 00:59:07.400 align:middle line:84% mutual equilibrium of your mixture 00:59:07.400 --> 00:59:10.315 align:middle line:84% with an appropriate environment of which you 00:59:10.315 --> 00:59:12.440 align:middle line:84% know the temperature, the pressure and the chemical 00:59:12.440 --> 00:59:19.910 align:middle line:84% potentials because essentially the value 00:59:19.910 --> 00:59:31.460 align:middle line:84% of the Euler free energy of your mixture depends only on that. 00:59:31.460 --> 00:59:36.670 align:middle line:84% So once you've fixed your reservoir 00:59:36.670 --> 00:59:41.770 align:middle line:84% in a completely open situation in which the system can exchange 00:59:41.770 --> 00:59:43.930 align:middle line:84% energy, therefore temperature equality 00:59:43.930 --> 00:59:46.870 align:middle line:90% with the reservoir, pressure-- 00:59:46.870 --> 00:59:50.140 align:middle line:84% I'm sorry, volume, therefore pressure equality 00:59:50.140 --> 00:59:53.260 align:middle line:84% with the reservoir and it can exchange also 00:59:53.260 --> 00:59:55.660 align:middle line:84% all constituents with your reservoir, 00:59:55.660 --> 00:59:59.770 align:middle line:84% so all the chemical potentials of your system 00:59:59.770 --> 01:00:04.930 align:middle line:84% for each substance must be equal to the chemical potential 01:00:04.930 --> 01:00:10.506 align:middle line:84% of the reservoir of the corresponding substance. 01:00:10.506 --> 01:00:14.080 align:middle line:90% 01:00:14.080 --> 01:00:19.750 align:middle line:84% And so that provides ways to do these measurements of this Euler 01:00:19.750 --> 01:00:20.720 align:middle line:90% function. 01:00:20.720 --> 01:00:25.490 align:middle line:84% Of course, the Euler function is important, as we emphasized, 01:00:25.490 --> 01:00:26.565 align:middle line:90% for small systems. 01:00:26.565 --> 01:00:29.270 align:middle line:90% 01:00:29.270 --> 01:00:33.140 align:middle line:84% It's also important for large systems 01:00:33.140 --> 01:00:37.400 align:middle line:84% because, for large systems, it is identically equal to 0 01:00:37.400 --> 01:00:46.160 align:middle line:84% as part of the result of the simple system model in which we 01:00:46.160 --> 01:00:54.110 align:middle line:84% have seen that the refraction effects at walls 01:00:54.110 --> 01:00:56.990 align:middle line:90% are not important. 01:00:56.990 --> 01:01:02.000 align:middle line:84% And therefore, for example, the work 01:01:02.000 --> 01:01:07.490 align:middle line:84% associated in placing a partition 01:01:07.490 --> 01:01:13.080 align:middle line:84% in your system or the entropy produced by irreversibility 01:01:13.080 --> 01:01:16.640 align:middle line:84% if you remove a partition, become negligible 01:01:16.640 --> 01:01:19.190 align:middle line:90% for large amounts. 01:01:19.190 --> 01:01:28.010 align:middle line:84% And so in that case, since the Euler free energy is related, 01:01:28.010 --> 01:01:32.090 align:middle line:84% as we have seen, to that work and to that irreversibility 01:01:32.090 --> 01:01:38.630 align:middle line:84% of placing or removing partitions, 01:01:38.630 --> 01:01:44.480 align:middle line:84% in the limit of large number of particles, that goes to 0. 01:01:44.480 --> 01:01:46.940 align:middle line:84% And that gives rise to the so-called-- 01:01:46.940 --> 01:01:49.195 align:middle line:84% I mean, what is known as the Euler relation. 01:01:49.195 --> 01:01:53.980 align:middle line:90% 01:01:53.980 --> 01:01:59.780 align:middle line:84% And also, if you subtract the differential of the Euler 01:01:59.780 --> 01:02:02.570 align:middle line:84% relation from the Gibbs relation, which 01:02:02.570 --> 01:02:04.650 align:middle line:84% is the differential of the fundamental relation, 01:02:04.650 --> 01:02:10.070 align:middle line:84% you get this other relation between differentials 01:02:10.070 --> 01:02:18.460 align:middle line:84% that is called the Gibbs-Duhem relation, which we already 01:02:18.460 --> 01:02:21.900 align:middle line:84% used when we did, for example, the Gibbs phase rule. 01:02:21.900 --> 01:02:31.790 align:middle line:90% 01:02:31.790 --> 01:02:38.780 align:middle line:84% Now, the various forms, the various differentials 01:02:38.780 --> 01:02:48.350 align:middle line:84% of the fundamental relation or the Legendre transforms of it-- 01:02:48.350 --> 01:02:52.400 align:middle line:84% Helmholtz, Gibbs, enthalpy-- so this expression 01:02:52.400 --> 01:02:54.500 align:middle line:90% for the differentials-- 01:02:54.500 --> 01:02:58.010 align:middle line:84% are useful in order to identify, just 01:02:58.010 --> 01:03:02.390 align:middle line:84% by inspection, what are the natural variables 01:03:02.390 --> 01:03:05.840 align:middle line:90% of these functions. 01:03:05.840 --> 01:03:12.090 align:middle line:84% So for the energy form of the fundamental relation, 01:03:12.090 --> 01:03:13.610 align:middle line:84% the natural variables, as we know, 01:03:13.610 --> 01:03:18.540 align:middle line:84% are S, entropy, volume, and the amounts of constituents. 01:03:18.540 --> 01:03:21.110 align:middle line:90% 01:03:21.110 --> 01:03:23.930 align:middle line:84% And the amounts of constituents can 01:03:23.930 --> 01:03:30.950 align:middle line:84% be written in terms of mole fractions, this way 01:03:30.950 --> 01:03:35.000 align:middle line:84% and also, again by inspection, you 01:03:35.000 --> 01:03:41.090 align:middle line:84% can infer immediately what the partial derivatives 01:03:41.090 --> 01:03:42.740 align:middle line:84% of this function are with respect 01:03:42.740 --> 01:03:44.490 align:middle line:84% to the various natural variables. 01:03:44.490 --> 01:03:47.390 align:middle line:84% For example, T is the partial of E with respect to S. 01:03:47.390 --> 01:03:50.630 align:middle line:84% Minus p is the partial of E with respect to V. 01:03:50.630 --> 01:03:55.280 align:middle line:84% The chemical potential-- and this is important-- 01:03:55.280 --> 01:04:01.310 align:middle line:84% when viewed this way is the partial of E with respect to n. 01:04:01.310 --> 01:04:07.790 align:middle line:84% And being a derivative of this function here, 01:04:07.790 --> 01:04:13.481 align:middle line:84% it is viewed as a function of the same variables, S, V, and n. 01:04:13.481 --> 01:04:17.630 align:middle line:90% 01:04:17.630 --> 01:04:19.440 align:middle line:84% Then you do the same also, for example, 01:04:19.440 --> 01:04:23.980 align:middle line:84% for the same-- let's go down here to the Gibbs free energy. 01:04:23.980 --> 01:04:27.750 align:middle line:84% The Gibbs free energy has this form of the differential. 01:04:27.750 --> 01:04:30.470 align:middle line:84% So you identify its natural variables 01:04:30.470 --> 01:04:32.845 align:middle line:84% as being temperature, pressure, and the amounts. 01:04:32.845 --> 01:04:35.430 align:middle line:90% 01:04:35.430 --> 01:04:38.880 align:middle line:84% And the partial derivatives, minus S 01:04:38.880 --> 01:04:42.580 align:middle line:84% is the partial of G with respect to T. 01:04:42.580 --> 01:04:47.430 align:middle line:84% Volume is the partial of G with respect to pressure. 01:04:47.430 --> 01:04:50.880 align:middle line:84% The chemical potential is the partial of G with respect to n. 01:04:50.880 --> 01:04:55.950 align:middle line:84% And this way, the chemical potential 01:04:55.950 --> 01:05:00.460 align:middle line:84% is viewed as a function of T, p, and n. 01:05:00.460 --> 01:05:02.510 align:middle line:84% It's the same chemical potential. 01:05:02.510 --> 01:05:05.080 align:middle line:84% We're talking about the same system in a stable equilibrium 01:05:05.080 --> 01:05:06.310 align:middle line:90% state. 01:05:06.310 --> 01:05:10.960 align:middle line:84% But you can view the chemical potential 01:05:10.960 --> 01:05:14.980 align:middle line:84% as a function of different variables 01:05:14.980 --> 01:05:17.650 align:middle line:84% because the game of the Legendre transformation 01:05:17.650 --> 01:05:26.110 align:middle line:84% is to change from not readily measurable variables, 01:05:26.110 --> 01:05:30.670 align:middle line:84% like the entropy, to more readily measurable variables, 01:05:30.670 --> 01:05:35.270 align:middle line:84% like temperature, pressure, or for the Helmholtz free energy 01:05:35.270 --> 01:05:36.255 align:middle line:90% temperature and volume. 01:05:36.255 --> 01:05:39.980 align:middle line:90% 01:05:39.980 --> 01:05:44.450 align:middle line:84% The enthalpy doesn't gain much because it still 01:05:44.450 --> 01:05:45.720 align:middle line:90% has the entropy there. 01:05:45.720 --> 01:05:48.900 align:middle line:84% But the enthalpy is important for bulk flow interactions. 01:05:48.900 --> 01:05:52.420 align:middle line:84% And that's why we keep it in play. 01:05:52.420 --> 01:05:57.180 align:middle line:90% 01:05:57.180 --> 01:06:01.710 align:middle line:84% Now, of course, this is that summation. 01:06:01.710 --> 01:06:05.350 align:middle line:84% But when we have a single substance, 01:06:05.350 --> 01:06:07.420 align:middle line:84% so when we have a pure substance, 01:06:07.420 --> 01:06:10.960 align:middle line:84% we have this same set of equations. 01:06:10.960 --> 01:06:13.680 align:middle line:90% 01:06:13.680 --> 01:06:16.610 align:middle line:84% This is valid, in general, so including small systems. 01:06:16.610 --> 01:06:19.260 align:middle line:90% 01:06:19.260 --> 01:06:23.050 align:middle line:84% It's the same as two slides before, 01:06:23.050 --> 01:06:32.460 align:middle line:84% only that we've written this as simply mu and n, only one 01:06:32.460 --> 01:06:35.130 align:middle line:90% substance. 01:06:35.130 --> 01:06:38.640 align:middle line:90% Or you could call it one. 01:06:38.640 --> 01:06:45.060 align:middle line:84% Or actually here, we are going to use the double subscript 01:06:45.060 --> 01:06:51.150 align:middle line:84% as a convention to denote pure substance. 01:06:51.150 --> 01:06:53.470 align:middle line:84% So if you see a double subscript, 01:06:53.470 --> 01:06:57.520 align:middle line:84% it means we are referring to pure constituent. 01:06:57.520 --> 01:07:06.560 align:middle line:90% 01:07:06.560 --> 01:07:11.770 align:middle line:84% So in the large number of particle limit, 01:07:11.770 --> 01:07:14.300 align:middle line:84% so the simple-system approximation applies. 01:07:14.300 --> 01:07:21.310 align:middle line:84% And we still have the Euler relation. 01:07:21.310 --> 01:07:24.730 align:middle line:84% The Gibbs-Duhem relation simplifies 01:07:24.730 --> 01:07:29.590 align:middle line:84% and can be viewed as essentially the differential 01:07:29.590 --> 01:07:36.040 align:middle line:84% of the chemical potential of the only substance that we have. 01:07:36.040 --> 01:07:45.810 align:middle line:84% So you see that the chemical potential is a function of T 01:07:45.810 --> 01:07:50.090 align:middle line:90% and P. Let's see. 01:07:50.090 --> 01:07:55.283 align:middle line:84% Yeah, here, I think I made a mistake in these-- 01:07:55.283 --> 01:07:57.830 align:middle line:90% 01:07:57.830 --> 01:08:01.320 align:middle line:90% let's see-- In the limit-- 01:08:01.320 --> 01:08:07.100 align:middle line:84% In the large n limit, the chemical potential 01:08:07.100 --> 01:08:12.150 align:middle line:84% loses its dependence on the amounts. 01:08:12.150 --> 01:08:14.160 align:middle line:84% OK, here it is written correctly. 01:08:14.160 --> 01:08:17.890 align:middle line:90% 01:08:17.890 --> 01:08:21.950 align:middle line:84% Oh yeah, OK, here, I did take it away. 01:08:21.950 --> 01:08:22.450 align:middle line:90% Fine. 01:08:22.450 --> 01:08:28.100 align:middle line:90% 01:08:28.100 --> 01:08:38.069 align:middle line:84% So the chemical potential is not an independent property. 01:08:38.069 --> 01:08:40.930 align:middle line:84% Once you fix the temperature and the pressure, 01:08:40.930 --> 01:08:43.300 align:middle line:84% chemical potential is a function of T and P. 01:08:43.300 --> 01:08:47.600 align:middle line:84% So that fixes also the chemical potential, whereas, 01:08:47.600 --> 01:08:55.120 align:middle line:84% for a mixture, the phase rule, depending 01:08:55.120 --> 01:09:01.720 align:middle line:84% on how many phases you have coexisting and so on, 01:09:01.720 --> 01:09:07.060 align:middle line:84% some chemical potentials can be changed independently, 01:09:07.060 --> 01:09:10.100 align:middle line:90% not for the pure substance. 01:09:10.100 --> 01:09:15.899 align:middle line:90% 01:09:15.899 --> 01:09:19.960 align:middle line:84% And in fact, if you have, for a pure substance, two phases, 01:09:19.960 --> 01:09:24.300 align:middle line:84% you have one relation like this for one phase, like liquid, 01:09:24.300 --> 01:09:27.819 align:middle line:84% and another relation, like this, for the vapor. 01:09:27.819 --> 01:09:35.240 align:middle line:84% And if they coexist, the two must be equal. 01:09:35.240 --> 01:09:38.450 align:middle line:84% And therefore, temperature and pressure 01:09:38.450 --> 01:09:43.660 align:middle line:84% are not even anymore independently variable, 01:09:43.660 --> 01:09:45.249 align:middle line:84% like when you have a single phase. 01:09:45.249 --> 01:09:51.270 align:middle line:90% 01:09:51.270 --> 01:09:54.960 align:middle line:84% We will see that chemical potentials play 01:09:54.960 --> 01:10:00.930 align:middle line:84% an important role for mixtures because they 01:10:00.930 --> 01:10:04.230 align:middle line:90% determine the properties. 01:10:04.230 --> 01:10:07.140 align:middle line:84% And they also play an important role 01:10:07.140 --> 01:10:10.800 align:middle line:84% in determining the condition for mutual equilibrium. 01:10:10.800 --> 01:10:15.540 align:middle line:90% So they are really our focus. 01:10:15.540 --> 01:10:26.100 align:middle line:84% For the pure substance, notice that, due to the Euler relation, 01:10:26.100 --> 01:10:31.020 align:middle line:84% the chemical potential is also equal to the specific Gibbs 01:10:31.020 --> 01:10:31.720 align:middle line:90% free energy. 01:10:31.720 --> 01:10:41.580 align:middle line:84% So the Gibbs free energy per unit mole, 01:10:41.580 --> 01:10:49.960 align:middle line:90% g, is capital G over n. 01:10:49.960 --> 01:10:54.710 align:middle line:90% 01:10:54.710 --> 01:11:02.370 align:middle line:84% Now, in the simple-system model, for a mixture-- 01:11:02.370 --> 01:11:05.060 align:middle line:84% so this is the simple-system model-- 01:11:05.060 --> 01:11:07.830 align:middle line:90% G is equal to that summation. 01:11:07.830 --> 01:11:11.260 align:middle line:90% 01:11:11.260 --> 01:11:18.780 align:middle line:84% When you have only one, that summation becomes just one term. 01:11:18.780 --> 01:11:27.440 align:middle line:84% And therefore, you see that, if you take g-- 01:11:27.440 --> 01:11:33.170 align:middle line:90% so it's mu11 n11 divided by-- 01:11:33.170 --> 01:11:36.200 align:middle line:90% let me put 11 here. 01:11:36.200 --> 01:11:38.000 align:middle line:90% So that's mu. 01:11:38.000 --> 01:11:41.850 align:middle line:84% So that proves that, in that case, 01:11:41.850 --> 01:11:45.900 align:middle line:90% the Gibbs free energy molar. 01:11:45.900 --> 01:11:48.140 align:middle line:90% So that's g11 is equal to mu11. 01:11:48.140 --> 01:11:51.320 align:middle line:90% 01:11:51.320 --> 01:11:52.920 align:middle line:90% Only for a pure substance. 01:11:52.920 --> 01:12:01.570 align:middle line:90% 01:12:01.570 --> 01:12:03.606 align:middle line:90% So how do we measure properties? 01:12:03.606 --> 01:12:08.990 align:middle line:90% 01:12:08.990 --> 01:12:13.220 align:middle line:84% Well, in order to measure, we said 01:12:13.220 --> 01:12:16.280 align:middle line:84% that the chemical potentials are central because they determine, 01:12:16.280 --> 01:12:20.210 align:middle line:84% for example, the Gibbs free energy, which 01:12:20.210 --> 01:12:24.240 align:middle line:84% is a Legendre transform of the fundamental relation. 01:12:24.240 --> 01:12:26.330 align:middle line:84% And therefore, you can extract all the information 01:12:26.330 --> 01:12:31.740 align:middle line:84% from it, of course if you know it as a function. 01:12:31.740 --> 01:12:34.190 align:middle line:84% So you need to know not just one value, 01:12:34.190 --> 01:12:38.825 align:middle line:84% you need to know how it varies with temperature and pressure. 01:12:38.825 --> 01:12:40.700 align:middle line:84% You need to know all these functions in order 01:12:40.700 --> 01:12:43.135 align:middle line:90% to characterize your mixture. 01:12:43.135 --> 01:12:47.680 align:middle line:90% 01:12:47.680 --> 01:12:55.060 align:middle line:84% And the chemical potentials are the key of the game. 01:12:55.060 --> 01:13:03.970 align:middle line:84% And the way you can measure them is by a manometer, 01:13:03.970 --> 01:13:05.745 align:middle line:84% by measuring the partial pressure. 01:13:05.745 --> 01:13:10.330 align:middle line:90% 01:13:10.330 --> 01:13:17.170 align:middle line:84% So you need to develop rigid, semi-permeable membranes capable 01:13:17.170 --> 01:13:21.730 align:middle line:90% of keeping the molecules-- 01:13:21.730 --> 01:13:27.190 align:middle line:84% all molecules in the mixture except one inside the mixture 01:13:27.190 --> 01:13:32.440 align:middle line:84% and allowing only the molecules of one type to cross. 01:13:32.440 --> 01:13:36.535 align:middle line:84% Once you have identified a membrane like that, 01:13:36.535 --> 01:13:42.880 align:middle line:84% you just wait for mutual equilibrium 01:13:42.880 --> 01:13:47.270 align:middle line:84% to occur, and then measure this partial pressure. 01:13:47.270 --> 01:13:52.390 align:middle line:84% And then as you remember, the condition for mutual equilibrium 01:13:52.390 --> 01:13:57.130 align:middle line:84% is that the chemical potential of constituent i in the mixture 01:13:57.130 --> 01:14:01.660 align:middle line:84% must be equal to the chemical potential of the constituent 01:14:01.660 --> 01:14:06.040 align:middle line:84% on the other side of the membrane through which 01:14:06.040 --> 01:14:09.170 align:middle line:90% that substance can flow. 01:14:09.170 --> 01:14:12.600 align:middle line:90% 01:14:12.600 --> 01:14:15.420 align:middle line:84% And we call partial pressure that pressure at which 01:14:15.420 --> 01:14:19.350 align:middle line:84% that occurs, the pressure of the pure system 01:14:19.350 --> 01:14:23.670 align:middle line:84% so that that's the meaning of the double i. 01:14:23.670 --> 01:14:36.680 align:middle line:90% And so this equality makes the-- 01:14:36.680 --> 01:14:37.620 align:middle line:90% yeah, OK. 01:14:37.620 --> 01:14:39.800 align:middle line:84% Here, I kept because, for small systems, 01:14:39.800 --> 01:14:45.800 align:middle line:84% you also have this dependence, even for the pure substance. 01:14:45.800 --> 01:14:52.490 align:middle line:84% But for large amounts, that drops. 01:14:52.490 --> 01:14:55.730 align:middle line:84% And then you see that you can solve this equation 01:14:55.730 --> 01:14:58.640 align:middle line:84% and evaluate the partial pressure 01:14:58.640 --> 01:15:01.380 align:middle line:84% as a function of the other variables, which is temperature, 01:15:01.380 --> 01:15:07.670 align:middle line:84% pressure, and the amounts in the mixture, 01:15:07.670 --> 01:15:13.940 align:middle line:84% and the amount in this pure system on the other side 01:15:13.940 --> 01:15:15.140 align:middle line:90% of the membrane. 01:15:15.140 --> 01:15:20.240 align:middle line:84% But the absolute amounts would drop out for large n. 01:15:20.240 --> 01:15:26.570 align:middle line:84% So that dependence is only through the relative composition 01:15:26.570 --> 01:15:29.510 align:middle line:84% of your mixture, so the percentages 01:15:29.510 --> 01:15:35.365 align:middle line:84% of how many molecules of each type I have in my mixture. 01:15:35.365 --> 01:15:42.910 align:middle line:90% 01:15:42.910 --> 01:15:49.370 align:middle line:84% And again, here, I use this relation, 01:15:49.370 --> 01:15:55.810 align:middle line:84% which is also important to remember, namely that therefore, 01:15:55.810 --> 01:15:59.440 align:middle line:84% for large n, the chemical potential of the constituent 01:15:59.440 --> 01:16:02.260 align:middle line:84% in the mixture is equal to the chemical potential 01:16:02.260 --> 01:16:06.490 align:middle line:84% of the pure component evaluated at the temperature 01:16:06.490 --> 01:16:08.620 align:middle line:84% and the partial pressure of the mixture. 01:16:08.620 --> 01:16:11.470 align:middle line:84% And the chemical potential of the pure component 01:16:11.470 --> 01:16:13.990 align:middle line:84% is given by the Gibbs free energy. 01:16:13.990 --> 01:16:17.470 align:middle line:84% And the Gibbs free energy, if you remember, 01:16:17.470 --> 01:16:21.610 align:middle line:90% is also written as H minus TS. 01:16:21.610 --> 01:16:26.650 align:middle line:84% So what we are saying is that chemical potential 01:16:26.650 --> 01:16:32.700 align:middle line:84% of a component in the mixture can be written as this. 01:16:32.700 --> 01:16:36.275 align:middle line:84% But since this is g, it can also be written as h-- 01:16:36.275 --> 01:16:43.300 align:middle line:90% 01:16:43.300 --> 01:16:43.800 align:middle line:90% I'm sorry. 01:16:43.800 --> 01:16:50.150 align:middle line:84% This is muii-- hii at T and the partial pressure minus T 01:16:50.150 --> 01:16:52.225 align:middle line:90% sii at T and pii. 01:16:52.225 --> 01:16:58.385 align:middle line:90% 01:16:58.385 --> 01:17:02.320 align:middle line:84% And these, with double i, are the properties 01:17:02.320 --> 01:17:03.680 align:middle line:90% of the pure substance. 01:17:03.680 --> 01:17:07.970 align:middle line:90% 01:17:07.970 --> 01:17:16.330 align:middle line:84% So a good part, but not all of the properties of the mixture 01:17:16.330 --> 01:17:21.500 align:middle line:84% are determined by the properties of the pure components. 01:17:21.500 --> 01:17:25.080 align:middle line:90% 01:17:25.080 --> 01:17:30.890 align:middle line:84% In fact, we will define some behaviors 01:17:30.890 --> 01:17:36.610 align:middle line:84% that we will call ideal, which are simplified versions, 01:17:36.610 --> 01:17:44.860 align:middle line:84% simplified models in which all of the properties of the mixture 01:17:44.860 --> 01:17:50.620 align:middle line:84% are determined by the properties of the pure components. 01:17:50.620 --> 01:17:58.660 align:middle line:84% That is not to say that most real mixtures are not 01:17:58.660 --> 01:18:03.640 align:middle line:84% ideal in the sense that we will try to define. 01:18:03.640 --> 01:18:10.270 align:middle line:84% But the ideal mixture model, or the ideal solution model, 01:18:10.270 --> 01:18:11.205 align:middle line:90% are a good start. 01:18:11.205 --> 01:18:13.920 align:middle line:90% 01:18:13.920 --> 01:18:17.760 align:middle line:84% And then the typical procedure is just 01:18:17.760 --> 01:18:19.790 align:middle line:84% to start with an ideal description. 01:18:19.790 --> 01:18:26.700 align:middle line:84% And then since my actual mixture departs from ideal, 01:18:26.700 --> 01:18:32.170 align:middle line:84% you describe the departure from ideal. 01:18:32.170 --> 01:18:36.006 align:middle line:84% We are going to call those excess properties. 01:18:36.006 --> 01:18:48.440 align:middle line:90% 01:18:48.440 --> 01:18:53.570 align:middle line:84% Now, a first step towards the construction of properties 01:18:53.570 --> 01:18:57.590 align:middle line:84% of a mixture from the properties of the pure components 01:18:57.590 --> 01:19:01.500 align:middle line:90% is to define-- 01:19:01.500 --> 01:19:05.050 align:middle line:90% 01:19:05.050 --> 01:19:06.300 align:middle line:90% I think we've already done it. 01:19:06.300 --> 01:19:09.930 align:middle line:84% Yeah, we defined already the partial properties. 01:19:09.930 --> 01:19:13.060 align:middle line:90% 01:19:13.060 --> 01:19:26.320 align:middle line:84% The way we did it was to write the differential 01:19:26.320 --> 01:19:32.480 align:middle line:90% of the expression that-- 01:19:32.480 --> 01:19:40.220 align:middle line:84% you see, this mu i is also, as we said, a function of T, P 01:19:40.220 --> 01:19:43.840 align:middle line:90% and the amounts. 01:19:43.840 --> 01:19:47.820 align:middle line:90% 01:19:47.820 --> 01:19:51.545 align:middle line:84% So we write the differential of that. 01:19:51.545 --> 01:19:55.810 align:middle line:90% 01:19:55.810 --> 01:20:00.340 align:middle line:84% And then we use those Maxwell relations. 01:20:00.340 --> 01:20:02.030 align:middle line:90% That allows us to-- 01:20:02.030 --> 01:20:04.540 align:middle line:84% for example, this partial derivative of u 01:20:04.540 --> 01:20:07.690 align:middle line:84% with respect to T, which is the coefficient of dT 01:20:07.690 --> 01:20:13.150 align:middle line:84% in this differential, we recognize 01:20:13.150 --> 01:20:16.720 align:middle line:84% that, except for the minus sign, it 01:20:16.720 --> 01:20:23.210 align:middle line:84% is equal to the second derivative of G, 01:20:23.210 --> 01:20:29.560 align:middle line:84% of the capital G, of the Gibbs free energy with respect 01:20:29.560 --> 01:20:32.050 align:middle line:84% to T and with respect to n because the chemical 01:20:32.050 --> 01:20:35.530 align:middle line:84% potential is the partial of G with respect to n. 01:20:35.530 --> 01:20:40.110 align:middle line:84% So this is the second derivative. 01:20:40.110 --> 01:20:47.740 align:middle line:84% But then due to the reciprocity or Schwarz theorem 01:20:47.740 --> 01:20:51.370 align:middle line:84% about second partial derivatives, 01:20:51.370 --> 01:20:55.540 align:middle line:84% this can also be viewed that you first 01:20:55.540 --> 01:20:58.450 align:middle line:84% do the partial of G with respect to T, and then 01:20:58.450 --> 01:20:59.560 align:middle line:90% with respect to n. 01:20:59.560 --> 01:21:01.450 align:middle line:84% So the partial of G with respect to T 01:21:01.450 --> 01:21:04.670 align:middle line:84% is minus S, as we can see from here. 01:21:04.670 --> 01:21:11.350 align:middle line:84% Partial of G with respect to T is minus S. 01:21:11.350 --> 01:21:17.050 align:middle line:84% So by writing the differentials, we go and watch by inspection 01:21:17.050 --> 01:21:19.420 align:middle line:84% what these partial derivatives are. 01:21:19.420 --> 01:21:24.000 align:middle line:84% But also, we should remember, I mean, 01:21:24.000 --> 01:21:29.270 align:middle line:84% we could remember by heart these partial derivatives, 01:21:29.270 --> 01:21:32.020 align:middle line:90% at least this and that. 01:21:32.020 --> 01:21:39.110 align:middle line:84% In any case, this says how much the entropy 01:21:39.110 --> 01:21:42.650 align:middle line:84% changes while keeping temperature and pressure fixed 01:21:42.650 --> 01:21:45.290 align:middle line:90% and also the various amounts. 01:21:45.290 --> 01:21:49.130 align:middle line:84% I add a little bit of one substance. 01:21:49.130 --> 01:21:52.280 align:middle line:84% This is why it takes the name partial entropy. 01:21:52.280 --> 01:21:57.110 align:middle line:84% It's the contribution of that constituent 01:21:57.110 --> 01:22:02.470 align:middle line:90% to the property of the mixture. 01:22:02.470 --> 01:22:07.640 align:middle line:90% 01:22:07.640 --> 01:22:08.790 align:middle line:90% Then we do the same. 01:22:08.790 --> 01:22:14.040 align:middle line:84% We did the same for these other partial derivatives, 01:22:14.040 --> 01:22:20.340 align:middle line:84% which is the coefficient of the pressure, which is here. 01:22:20.340 --> 01:22:25.140 align:middle line:84% It can also be viewed as a second partial derivative of G. 01:22:25.140 --> 01:22:28.890 align:middle line:84% And therefore, in this case, it's 01:22:28.890 --> 01:22:31.710 align:middle line:84% the partial of the volume with respect to n. 01:22:31.710 --> 01:22:36.430 align:middle line:84% So it says how much the volume changes, increases, 01:22:36.430 --> 01:22:39.690 align:middle line:84% if I add a little bit of substance i 01:22:39.690 --> 01:22:41.530 align:middle line:84% while keeping temperature, pressure, 01:22:41.530 --> 01:22:43.853 align:middle line:84% and the other constituents fixed. 01:22:43.853 --> 01:22:45.395 align:middle line:84% And that's called the partial volume. 01:22:45.395 --> 01:22:53.310 align:middle line:90% 01:22:53.310 --> 01:23:01.380 align:middle line:84% Also here, we have the expression for the differential. 01:23:01.380 --> 01:23:06.535 align:middle line:84% I have to take care also of the differentials of the n's here. 01:23:06.535 --> 01:23:09.600 align:middle line:90% 01:23:09.600 --> 01:23:13.380 align:middle line:84% So I have a summation because I have many n's. 01:23:13.380 --> 01:23:16.500 align:middle line:84% I have these partial derivatives of the chemical potential 01:23:16.500 --> 01:23:21.310 align:middle line:84% of one substance with respect to the amount of another substance. 01:23:21.310 --> 01:23:25.380 align:middle line:90% 01:23:25.380 --> 01:23:32.620 align:middle line:84% And the reciprocity of second order derivatives 01:23:32.620 --> 01:23:37.420 align:middle line:84% tells us that there is a reciprocity in this relation. 01:23:37.420 --> 01:23:41.120 align:middle line:84% And for shorthand in what follows, 01:23:41.120 --> 01:23:44.680 align:middle line:84% I'm going to denote these partial derivatives as mu i 01:23:44.680 --> 01:23:46.910 align:middle line:90% comma j or mu j comma i. 01:23:46.910 --> 01:23:51.880 align:middle line:84% So the comma i means partial with respect to ni. 01:23:51.880 --> 01:24:00.460 align:middle line:90% 01:24:00.460 --> 01:24:03.070 align:middle line:84% For example, this is the Gibbs relation, the differential 01:24:03.070 --> 01:24:04.335 align:middle line:90% of the fundamental relation. 01:24:04.335 --> 01:24:09.930 align:middle line:90% 01:24:09.930 --> 01:24:17.780 align:middle line:84% I can compute the partial energy of constituent i 01:24:17.780 --> 01:24:20.887 align:middle line:90% because I can take formally-- 01:24:20.887 --> 01:24:22.470 align:middle line:84% I mean, don't tell the mathematicians. 01:24:22.470 --> 01:24:26.360 align:middle line:84% But if I take these expressions here 01:24:26.360 --> 01:24:34.810 align:middle line:84% and I divide by delta partial n, partial ni, 01:24:34.810 --> 01:24:37.270 align:middle line:90% so I divide through-- 01:24:37.270 --> 01:24:41.860 align:middle line:84% so that becomes that d divided by dni 01:24:41.860 --> 01:24:47.290 align:middle line:84% is equal to T times dS divided by dni. 01:24:47.290 --> 01:24:53.240 align:middle line:84% But that dS divided by dni is the partial entropy. 01:24:53.240 --> 01:24:55.700 align:middle line:90% Then I have dV divided by dni. 01:24:55.700 --> 01:24:58.210 align:middle line:90% That's the partial volume. 01:24:58.210 --> 01:25:04.735 align:middle line:90% And dn divided by dni is 1. 01:25:04.735 --> 01:25:07.840 align:middle line:90% dni divided by dni is 1. 01:25:07.840 --> 01:25:12.770 align:middle line:84% So that leaves me with just the chemical potential. 01:25:12.770 --> 01:25:20.120 align:middle line:84% I mean, it's 1 if the i I'm dividing is the same as the n. 01:25:20.120 --> 01:25:24.250 align:middle line:84% It's 0 if I'm dividing one by the other 01:25:24.250 --> 01:25:25.760 align:middle line:90% because they are independent. 01:25:25.760 --> 01:25:29.830 align:middle line:90% And so this is how we prove-- 01:25:29.830 --> 01:25:31.130 align:middle line:90% I mean, you can do it better. 01:25:31.130 --> 01:25:37.100 align:middle line:84% But this was just a game because sometimes proceeding formally 01:25:37.100 --> 01:25:41.750 align:middle line:84% is helpful to go fast and do some checking. 01:25:41.750 --> 01:25:44.240 align:middle line:84% In any case, so this is the expression 01:25:44.240 --> 01:25:49.170 align:middle line:84% for the partial energy of a constituent. 01:25:49.170 --> 01:25:52.670 align:middle line:84% And you can play the same game using the differential 01:25:52.670 --> 01:25:56.420 align:middle line:84% for the enthalpy to express the partial enthalpy 01:25:56.420 --> 01:25:58.850 align:middle line:90% of a constituent. 01:25:58.850 --> 01:26:05.660 align:middle line:84% What is important here is that all these partial properties-- 01:26:05.660 --> 01:26:07.980 align:middle line:84% entropy, volume, energy, enthalpy-- 01:26:07.980 --> 01:26:13.410 align:middle line:90% 01:26:13.410 --> 01:26:23.650 align:middle line:84% are related to the chemical potential-- 01:26:23.650 --> 01:26:26.970 align:middle line:90% you see the chemical potential-- 01:26:26.970 --> 01:26:32.130 align:middle line:84% or partial derivatives of the chemical potential, 01:26:32.130 --> 01:26:37.350 align:middle line:84% either with respect to T or with respect to p. 01:26:37.350 --> 01:26:43.990 align:middle line:84% So that means that, if I have this expression 01:26:43.990 --> 01:26:51.790 align:middle line:84% here, I can determine, by just mathematical manipulation, all 01:26:51.790 --> 01:26:56.530 align:middle line:84% the partial properties of the constituent in the mixture. 01:26:56.530 --> 01:27:10.300 align:middle line:84% So from this one, I get si, vi, hi, ei. 01:27:10.300 --> 01:27:26.740 align:middle line:90% 01:27:26.740 --> 01:27:28.215 align:middle line:90% And what do I do with them? 01:27:28.215 --> 01:27:32.110 align:middle line:90% 01:27:32.110 --> 01:27:42.550 align:middle line:84% Well, this next slide is to show that with them you 01:27:42.550 --> 01:27:49.380 align:middle line:84% can compute the overall properties of the mixture. 01:27:49.380 --> 01:27:59.910 align:middle line:90% 01:27:59.910 --> 01:28:03.550 align:middle line:84% So here, I do it again for the general case. 01:28:03.550 --> 01:28:13.970 align:middle line:84% And then I look at the large number of particles limit. 01:28:13.970 --> 01:28:20.070 align:middle line:90% 01:28:20.070 --> 01:28:36.565 align:middle line:84% Essentially, we can take the expression for the Euler-- 01:28:36.565 --> 01:28:38.190 align:middle line:84% this is one way of doing it, of course. 01:28:38.190 --> 01:28:43.260 align:middle line:84% There are many different ways of getting to the same result. 01:28:43.260 --> 01:28:47.090 align:middle line:84% For example, in the book, there is no discussion 01:28:47.090 --> 01:28:53.150 align:middle line:90% about the Euler free energy. 01:28:53.150 --> 01:28:58.220 align:middle line:84% And so you find a different derivation 01:28:58.220 --> 01:29:02.560 align:middle line:84% starting from just the value in the large limit, 01:29:02.560 --> 01:29:06.560 align:middle line:90% large number of particles. 01:29:06.560 --> 01:29:11.570 align:middle line:84% In any case, I start from the expression 01:29:11.570 --> 01:29:17.810 align:middle line:84% for the Euler relation, which is a function of T, p, and mu. 01:29:17.810 --> 01:29:28.850 align:middle line:84% And then I substitute inside the mu here this expression here 01:29:28.850 --> 01:29:32.790 align:middle line:84% so that it becomes a function of T, p, and n. 01:29:32.790 --> 01:29:36.350 align:middle line:90% 01:29:36.350 --> 01:29:47.030 align:middle line:84% And then I take the derivatives of this with respect 01:29:47.030 --> 01:29:49.200 align:middle line:90% to these independent variables. 01:29:49.200 --> 01:29:55.180 align:middle line:90% 01:29:55.180 --> 01:29:57.730 align:middle line:84% Of course, we need to do some chain rule because here we 01:29:57.730 --> 01:29:59.690 align:middle line:90% have a function of a function. 01:29:59.690 --> 01:30:06.220 align:middle line:84% So for example, the derivative of Eu with respect to T 01:30:06.220 --> 01:30:12.520 align:middle line:84% gets one contribution from the T that is here, this T here. 01:30:12.520 --> 01:30:16.420 align:middle line:84% And that's the derivative with respect to T 01:30:16.420 --> 01:30:22.060 align:middle line:84% done when I think that the other variables-- so let's say, 01:30:22.060 --> 01:30:26.830 align:middle line:84% loosely speaking, at constant p and constant mu. 01:30:26.830 --> 01:30:31.450 align:middle line:90% So I put that p and the mu here. 01:30:31.450 --> 01:30:35.470 align:middle line:84% Plus, there is also a contribution 01:30:35.470 --> 01:30:39.530 align:middle line:84% to changes in temperature through the chemical potentials. 01:30:39.530 --> 01:30:44.560 align:middle line:84% So that adds the partial of u with respect 01:30:44.560 --> 01:30:50.470 align:middle line:84% to the chemical potentials times the derivative 01:30:50.470 --> 01:30:52.910 align:middle line:84% of the chemical potential with respect to temperature. 01:30:52.910 --> 01:30:58.415 align:middle line:90% 01:30:58.415 --> 01:31:02.420 align:middle line:84% Now, it turns out that if we look 01:31:02.420 --> 01:31:07.160 align:middle line:84% at the expression for the differential of the Euler 01:31:07.160 --> 01:31:14.100 align:middle line:84% free energy, which is this one, by inspection, as usual, 01:31:14.100 --> 01:31:18.420 align:middle line:84% we can extract the values of the partial derivatives. 01:31:18.420 --> 01:31:21.110 align:middle line:84% For example, minus S is equal to partial 01:31:21.110 --> 01:31:27.230 align:middle line:90% of Euler over temperature. 01:31:27.230 --> 01:31:31.670 align:middle line:84% Volume is the partial of Euler over pressure. 01:31:31.670 --> 01:31:36.800 align:middle line:84% The amount of constituent is the negative 01:31:36.800 --> 01:31:40.440 align:middle line:84% of the Euler with respect to the chemical potential. 01:31:40.440 --> 01:31:42.530 align:middle line:84% So this partial derivative, partial 01:31:42.530 --> 01:31:46.670 align:middle line:84% of Eu with respect to chemical potential of constituent i, 01:31:46.670 --> 01:31:51.770 align:middle line:84% while I keep fixed temperature, pressure, 01:31:51.770 --> 01:31:53.990 align:middle line:84% and the chemical potentials of all other constituents 01:31:53.990 --> 01:31:57.135 align:middle line:84% is equal to the negative of the amount of constituent i 01:31:57.135 --> 01:31:57.760 align:middle line:90% in the mixture. 01:31:57.760 --> 01:32:00.990 align:middle line:90% 01:32:00.990 --> 01:32:06.973 align:middle line:90% So this thing here is minus ni. 01:32:06.973 --> 01:32:13.750 align:middle line:90% 01:32:13.750 --> 01:32:16.160 align:middle line:84% Then if I look in the previous slide 01:32:16.160 --> 01:32:18.190 align:middle line:84% and I look at the partial of mu with respect 01:32:18.190 --> 01:32:22.070 align:middle line:84% to T at constant p and n, that's minus s. 01:32:22.070 --> 01:32:27.400 align:middle line:84% That's the negative of the partial entropy. 01:32:27.400 --> 01:32:31.300 align:middle line:84% So minus and minus cancel becomes a plus. 01:32:31.300 --> 01:32:37.300 align:middle line:84% And so I have the summation of ni times si, 01:32:37.300 --> 01:32:43.580 align:middle line:84% whereas the first derivative here is here. 01:32:43.580 --> 01:32:48.100 align:middle line:84% It's minus S, capital S. It's the entropy of the mixture. 01:32:48.100 --> 01:32:51.590 align:middle line:90% 01:32:51.590 --> 01:32:56.700 align:middle line:84% So if I now take this expression and rewrite it here, 01:32:56.700 --> 01:33:00.200 align:middle line:90% it says that the entropy of-- 01:33:00.200 --> 01:33:02.570 align:middle line:84% the overall entropy of my mixture 01:33:02.570 --> 01:33:07.610 align:middle line:84% is equal to the weighted sum of the partial entropies weighted 01:33:07.610 --> 01:33:09.830 align:middle line:84% by the amounts with which the constituents are 01:33:09.830 --> 01:33:18.530 align:middle line:84% present in the mixture minus a correcting term that has to do 01:33:18.530 --> 01:33:27.060 align:middle line:90% with the Euler free energy. 01:33:27.060 --> 01:33:30.600 align:middle line:84% But if we are not interested in small systems, 01:33:30.600 --> 01:33:33.790 align:middle line:90% the Euler free energy is 0. 01:33:33.790 --> 01:33:39.990 align:middle line:84% And so in that limit, only the first term remains. 01:33:39.990 --> 01:33:48.530 align:middle line:84% And so we obtain that if I have these Si's, with them, 01:33:48.530 --> 01:33:52.905 align:middle line:84% I can compute the properties of my mixture. 01:33:52.905 --> 01:33:55.890 align:middle line:90% 01:33:55.890 --> 01:33:57.575 align:middle line:84% So I know the entropy of my mixture. 01:33:57.575 --> 01:34:01.440 align:middle line:90% 01:34:01.440 --> 01:34:06.002 align:middle line:84% And here, you can play the same game also for the volume. 01:34:06.002 --> 01:34:07.330 align:middle line:90% We skip this. 01:34:07.330 --> 01:34:11.440 align:middle line:90% 01:34:11.440 --> 01:34:15.680 align:middle line:84% We can play the same game also with respect-- 01:34:15.680 --> 01:34:18.610 align:middle line:84% when I do the partial derivative of Euler with respect 01:34:18.610 --> 01:34:21.680 align:middle line:84% to the amounts of one of the constituents. 01:34:21.680 --> 01:34:27.810 align:middle line:90% 01:34:27.810 --> 01:34:29.940 align:middle line:84% If you do carefully the derivatives, 01:34:29.940 --> 01:34:34.860 align:middle line:84% as we are doing here, this becomes 01:34:34.860 --> 01:34:41.340 align:middle line:84% equal to this summation, which is written here again. 01:34:41.340 --> 01:34:45.100 align:middle line:90% 01:34:45.100 --> 01:34:53.840 align:middle line:84% And since, in the large limit Euler is equal to 0 identically, 01:34:53.840 --> 01:34:58.030 align:middle line:84% it means that this summation is equal to 0 identically. 01:34:58.030 --> 01:35:05.740 align:middle line:84% Actually, there is one for each constituent because, you see, 01:35:05.740 --> 01:35:09.970 align:middle line:84% there is a summation over the constituents i. 01:35:09.970 --> 01:35:14.410 align:middle line:84% So i is a dummy index within the summation. 01:35:14.410 --> 01:35:18.700 align:middle line:84% But then there is a j, which is free. 01:35:18.700 --> 01:35:22.810 align:middle line:84% So this has to be equal to 0 for every j. 01:35:22.810 --> 01:35:28.333 align:middle line:84% This is called a Duhem-Margules relation. 01:35:28.333 --> 01:35:32.090 align:middle line:90% 01:35:32.090 --> 01:35:37.850 align:middle line:84% And it will play some role in making sure 01:35:37.850 --> 01:35:42.380 align:middle line:84% that when you develop models, you're doing it consistently. 01:35:42.380 --> 01:35:44.900 align:middle line:84% For the moment, we leave it there. 01:35:44.900 --> 01:35:51.260 align:middle line:84% Now, in the next slide, I'm going to start from these three. 01:35:51.260 --> 01:36:00.750 align:middle line:84% So these three lines become the first lines here 01:36:00.750 --> 01:36:13.300 align:middle line:84% because, from these relations, I can prove similar relations also 01:36:13.300 --> 01:36:17.500 align:middle line:84% for all the other properties that we have talked 01:36:17.500 --> 01:36:22.045 align:middle line:84% about-- energy, Helmholtz free energy, Gibbs free energy, 01:36:22.045 --> 01:36:22.545 align:middle line:90% enthalpy. 01:36:22.545 --> 01:36:33.470 align:middle line:90% 01:36:33.470 --> 01:36:38.390 align:middle line:84% So that means we can for all these-- 01:36:38.390 --> 01:36:49.130 align:middle line:84% and also, I can go it can go on to f, g, h, 01:36:49.130 --> 01:36:53.270 align:middle line:90% I have relations like that. 01:36:53.270 --> 01:36:56.540 align:middle line:84% If I want them for the mixture, all I need to do 01:36:56.540 --> 01:37:01.280 align:middle line:84% is play some derivatives of the chemical potentials. 01:37:01.280 --> 01:37:04.340 align:middle line:84% And with the proper relation, I get 01:37:04.340 --> 01:37:07.440 align:middle line:84% to the relation for the property of the mixture. 01:37:07.440 --> 01:37:11.540 align:middle line:84% Of course, I need to also know the amounts. 01:37:11.540 --> 01:37:14.780 align:middle line:84% So I need to know the chemical potentials, how 01:37:14.780 --> 01:37:17.430 align:middle line:84% they depend on temperature, pressure, and amounts, 01:37:17.430 --> 01:37:20.695 align:middle line:84% and I need to know how many I have in my mixture. 01:37:20.695 --> 01:37:26.430 align:middle line:90% 01:37:26.430 --> 01:37:28.810 align:middle line:90% Yeah, I think time is up. 01:37:28.810 --> 01:37:32.180 align:middle line:90% So we could stop here. 01:37:32.180 --> 01:37:39.000 align:middle line:90%