WEBVTT 00:00:00.030 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:04.000 Commons license. 00:00:04.000 --> 00:00:06.320 Your support will help MIT OpenCourseWare 00:00:06.320 --> 00:00:10.690 continue to offer high quality educational resources for free. 00:00:10.690 --> 00:00:13.300 To make a donation or view additional materials 00:00:13.300 --> 00:00:17.025 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.025 --> 00:00:17.650 at OCW.MIT.edu. 00:00:26.140 --> 00:00:27.860 PROFESSOR: So what I wanted to do today 00:00:27.860 --> 00:00:31.540 was talk about thermal properties of foams. 00:00:31.540 --> 00:00:34.530 And foams are often used for thermal insulation. 00:00:34.530 --> 00:00:36.150 And that's always closed cell foams 00:00:36.150 --> 00:00:37.691 that are used for thermal insulation. 00:00:37.691 --> 00:00:38.780 And we'll see why. 00:00:38.780 --> 00:00:43.290 And the foams tend to have a low thermal conductivity. 00:00:43.290 --> 00:00:46.620 And that's largely because gases have lower conductivity 00:00:46.620 --> 00:00:47.684 than solids. 00:00:47.684 --> 00:00:49.100 And if you have mostly gas, you're 00:00:49.100 --> 00:00:50.559 going to have a lower conductivity. 00:00:50.559 --> 00:00:53.183 So they have a low conductivity because they have a high volume 00:00:53.183 --> 00:00:54.230 fraction of gas. 00:00:54.230 --> 00:00:57.640 And they've got a low volume fraction of the solid. 00:00:57.640 --> 00:00:59.490 They also have cells. 00:00:59.490 --> 00:01:01.480 And the heat is transferred partly 00:01:01.480 --> 00:01:03.450 by radiation and convection. 00:01:03.450 --> 00:01:06.070 And if you have small cells, you reduce the amount 00:01:06.070 --> 00:01:08.070 of convection and radiation. 00:01:08.070 --> 00:01:09.360 And we'll see that. 00:01:09.360 --> 00:01:12.270 So that, by having a cellular structure, 00:01:12.270 --> 00:01:14.230 and in particular, by having small cells, 00:01:14.230 --> 00:01:16.740 you can decrease the heat transfer. 00:01:16.740 --> 00:01:19.010 OK, so let me write some of the stuff down. 00:01:30.160 --> 00:01:36.355 So closed cell foams are widely used for thermal insulation. 00:01:44.480 --> 00:01:48.190 And the only materials with lower thermal conductivity 00:01:48.190 --> 00:01:50.040 than closed cell foams are aerogels gels. 00:02:07.355 --> 00:02:08.729 And I'll I talk a little bit more 00:02:08.729 --> 00:02:11.230 about aerogels later on today. 00:02:11.230 --> 00:02:12.780 But the difficulty with aerogels is 00:02:12.780 --> 00:02:16.600 that they tend to be very weak and brittle, like ridiculously 00:02:16.600 --> 00:02:17.580 weak and brittle. 00:02:17.580 --> 00:02:21.030 So we had a project on aerogels a couple of years ago. 00:02:21.030 --> 00:02:23.530 And the students who I was collaborating with 00:02:23.530 --> 00:02:24.410 would make aerogels. 00:02:24.410 --> 00:02:26.210 And they'd bring it up to my office. 00:02:26.210 --> 00:02:28.390 And I would pick it up and like-- I would pick it up 00:02:28.390 --> 00:02:29.640 like this, and it would break. 00:02:29.640 --> 00:02:33.010 So they have very low thermal conductivity, 00:02:33.010 --> 00:02:34.770 but they're very brittle. 00:02:34.770 --> 00:02:37.060 And I brought a few of our samples of aerogels, 00:02:37.060 --> 00:02:39.060 just so you can see what they look like. 00:02:39.060 --> 00:02:41.000 And I'll pass them around in that little tube, 00:02:41.000 --> 00:02:42.980 so you can kind of play with them. 00:02:46.710 --> 00:02:49.720 OK, so we're going to focus on foams. 00:02:49.720 --> 00:02:57.830 And whoops-- And we can say the low thermal conductivity 00:02:57.830 --> 00:03:03.620 of foam arises mostly from the high volume fraction of gas 00:03:03.620 --> 00:03:05.200 and that the gas has a low lambda, 00:03:05.200 --> 00:03:06.325 a low thermal conductivity. 00:03:32.384 --> 00:03:33.800 So lambda is thermal conductivity, 00:03:33.800 --> 00:03:35.383 so I'm just going to put lambda there. 00:03:38.470 --> 00:03:40.120 Then it has a small volume fraction 00:03:40.120 --> 00:03:43.494 of solid, which has a higher thermal conductivity. 00:03:50.940 --> 00:03:53.679 And then the foams have a relatively small cell size. 00:03:53.679 --> 00:03:55.470 So one of the things we're going to look at 00:03:55.470 --> 00:03:58.829 is how does the cell size effect the thermal transfer, 00:03:58.829 --> 00:03:59.870 the thermal conductivity. 00:04:26.771 --> 00:04:30.980 OK, so there's lots of applications for foams. 00:04:30.980 --> 00:04:33.940 And I guess one of the main ones is in buildings, 00:04:33.940 --> 00:04:35.690 insulating buildings-- also insulating 00:04:35.690 --> 00:04:39.340 refrigerated vehicles, things like LNG tankers. 00:04:39.340 --> 00:04:42.577 So there's lots of the applications for using foams 00:04:42.577 --> 00:04:43.535 for thermal insulation. 00:05:05.800 --> 00:05:08.330 Foams, in addition to having a low thermal conductivity, 00:05:08.330 --> 00:05:11.210 they also have good thermal shock resistance. 00:05:11.210 --> 00:05:14.250 So thermal shock is if you have a material, you heat it up, 00:05:14.250 --> 00:05:16.750 and then you suddenly cool the surface of it, for example. 00:05:16.750 --> 00:05:18.958 So say you takes something and you quench it in water 00:05:18.958 --> 00:05:22.880 or quench it in some fluid, then the surface, 00:05:22.880 --> 00:05:24.922 it wants to shrink because its temperature drops, 00:05:24.922 --> 00:05:26.838 but it's connected to everything underneath it 00:05:26.838 --> 00:05:28.120 and it can't really shrink. 00:05:28.120 --> 00:05:31.230 And so it's constrained and you can get cracking and spalling. 00:05:31.230 --> 00:05:34.060 And so, it turns out foams have a good resistance 00:05:34.060 --> 00:05:37.600 to that thermal shock kind of loading. 00:05:37.600 --> 00:05:38.925 And we'll see why that is, too. 00:05:57.270 --> 00:06:00.710 Roughly, you can see if the thermal expansion 00:06:00.710 --> 00:06:05.030 strain is the thermal expansion coefficient times the change 00:06:05.030 --> 00:06:06.307 in temperature. 00:06:06.307 --> 00:06:07.890 And the stress that you might generate 00:06:07.890 --> 00:06:11.580 is just going to be related to the modulus times alpha times 00:06:11.580 --> 00:06:12.561 delta-t. 00:06:12.561 --> 00:06:14.810 And because we're going to see that alpha for the foam 00:06:14.810 --> 00:06:16.770 is the same as alpha for the solid, but the E 00:06:16.770 --> 00:06:19.670 foam is going to be a lot less that E of a solid would be. 00:06:19.670 --> 00:06:21.710 So because the modulus is smaller, 00:06:21.710 --> 00:06:26.230 you would get a better thermal shock resistance. 00:06:26.230 --> 00:06:31.010 OK, so I wanted to go over a couple of sort 00:06:31.010 --> 00:06:32.980 of laws of heat conduction, so we 00:06:32.980 --> 00:06:35.860 can talk about what thermal conductivity is 00:06:35.860 --> 00:06:37.750 and how we define it. 00:06:37.750 --> 00:06:40.526 So the first one here-- 00:06:49.300 --> 00:06:52.080 --first one here is for steady state conduction. 00:06:57.860 --> 00:07:00.990 So when we say steady state conduction, what we mean 00:07:00.990 --> 00:07:03.597 is that the temperature is constant with time, 00:07:03.597 --> 00:07:05.305 the temperature doesn't change with time. 00:07:13.360 --> 00:07:16.530 So time's not going to come into the equation here. 00:07:16.530 --> 00:07:20.570 And heat transfer for steady state conduction, 00:07:20.570 --> 00:07:23.480 where there is no change in the temperature with time, 00:07:23.480 --> 00:07:24.962 described by Fourier's Law. 00:07:27.840 --> 00:07:31.970 And that says that the heat flux q 00:07:31.970 --> 00:07:37.720 is equal to minus lambda times the gradient in temperature. 00:07:37.720 --> 00:07:41.450 And if you want to think about just a one diversion of that, 00:07:41.450 --> 00:07:46.280 it's equal to minus lambda times dt by dx. 00:07:46.280 --> 00:07:49.710 So here, q is our heat flux. 00:07:54.290 --> 00:07:56.920 So that would have units of joules 00:07:56.920 --> 00:07:59.120 per meter squared per second. 00:07:59.120 --> 00:08:03.622 So how much heat transfer per unit area per unit time. 00:08:03.622 --> 00:08:05.080 Lambda is the thermal conductivity. 00:08:12.370 --> 00:08:17.880 And it has units of watts per meter k, so degrees kelvin. 00:08:17.880 --> 00:08:22.054 And then delta or-- and then this 00:08:22.054 --> 00:08:23.220 is our temperature gradient. 00:08:31.320 --> 00:08:32.919 OK, so that's Fourier's Law, and we're 00:08:32.919 --> 00:08:37.340 going to use that later on when we talk about the foams. 00:08:37.340 --> 00:08:39.940 And then, just so that we have things a little more complete, 00:08:39.940 --> 00:08:42.770 if you have a non-steady heat conduction, if the temperature 00:08:42.770 --> 00:08:44.987 varies with time, then there's a difference equation 00:08:44.987 --> 00:08:46.570 that involves the thermal diffusivity. 00:08:50.570 --> 00:09:00.190 So if we have non-steady heat conduction-- so t 00:09:00.190 --> 00:09:03.170 varies with time. 00:09:03.170 --> 00:09:04.610 I'm going to call a time tau. 00:09:08.830 --> 00:09:12.810 Then a partial differentiation, the partial derivative 00:09:12.810 --> 00:09:16.290 of temperature with respect to time 00:09:16.290 --> 00:09:17.970 is equal to the diffusivity, that's 00:09:17.970 --> 00:09:21.220 given the symbol a, times the second derivative 00:09:21.220 --> 00:09:24.020 of temperature with respect of distance, 00:09:24.020 --> 00:09:25.870 so with respect x squared. 00:09:25.870 --> 00:09:27.836 So here a is the thermal diffusivity. 00:09:36.760 --> 00:09:40.420 And it is equal to the thermal conductivity 00:09:40.420 --> 00:09:44.740 divided by the density and divided by the specific heat. 00:09:44.740 --> 00:09:50.600 So here, rho is the density and cp is the specific heat. 00:09:54.700 --> 00:09:57.250 The specific heat is the heat required 00:09:57.250 --> 00:09:59.570 to raise the temperature of a unit mass 00:09:59.570 --> 00:10:03.220 by the unit temperature. 00:10:03.220 --> 00:10:07.700 And so, the density times cp is the volumetric heat capacity. 00:10:07.700 --> 00:10:09.570 It's how much energy you would need 00:10:09.570 --> 00:10:13.410 to raise a certain volume by, say, 1 degree 00:10:13.410 --> 00:10:14.750 k instead of a certain mass. 00:10:33.090 --> 00:10:38.110 OK, so on the table here, on the screen, 00:10:38.110 --> 00:10:39.810 we have different materials. 00:10:39.810 --> 00:10:42.350 And we have the thermal conductivity lambda. 00:10:42.350 --> 00:10:44.820 And we have the thermal diffusivity, a. 00:10:44.820 --> 00:10:48.320 And I guess I should also say a has units 00:10:48.320 --> 00:10:51.510 of meters squared per second. 00:10:56.240 --> 00:11:00.230 So this table is arranged in order of decreasing 00:11:00.230 --> 00:11:01.280 thermal conductivity. 00:11:01.280 --> 00:11:08.010 So here's copper at the top, 384, watts per meter k. 00:11:08.010 --> 00:11:09.880 Here's, you know, different metals. 00:11:09.880 --> 00:11:11.020 You've got aluminum. 00:11:11.020 --> 00:11:12.420 Here's a couple of ceramics. 00:11:12.420 --> 00:11:15.230 They're about a factor of 10 less than the metals. 00:11:15.230 --> 00:11:18.580 Here's the polymers, another factor of 10 less than that. 00:11:18.580 --> 00:11:20.540 And here's some gases. 00:11:20.540 --> 00:11:23.030 Air is about 0.025. 00:11:23.030 --> 00:11:25.220 Carbon dioxide is less than that. 00:11:25.220 --> 00:11:27.410 Triclorofluoromethane, which used 00:11:27.410 --> 00:11:30.390 to be used as a gas in foams because it's 00:11:30.390 --> 00:11:33.470 got such low thermal conductivity, is 0.008. 00:11:33.470 --> 00:11:36.830 But it's no longer used because it's a-- what you call it? 00:11:36.830 --> 00:11:39.210 A fluorocarbon. 00:11:39.210 --> 00:11:41.590 Anyway, it decreases our ozone layer. 00:11:41.590 --> 00:11:43.100 So they don't use that anymore. 00:11:43.100 --> 00:11:44.220 Now here's some wood. 00:11:44.220 --> 00:11:46.040 So that's one sort of cellular solid. 00:11:46.040 --> 00:11:48.430 And they're around 0.04-- something like that. 00:11:48.430 --> 00:11:51.150 And here's a group of polymer foams. 00:11:51.150 --> 00:11:53.940 And they're a little over 0.025. 00:11:53.940 --> 00:11:56.030 So if you think of-- if you had the gas, 00:11:56.030 --> 00:11:58.680 air-- if the air was the gas inside the foam, 00:11:58.680 --> 00:12:00.905 0.025 is lambda for the gas. 00:12:00.905 --> 00:12:02.850 So you're not going to get lower than that. 00:12:02.850 --> 00:12:06.440 And you have to use a low conductivity gas to get these 00:12:06.440 --> 00:12:10.690 values, like 0.025 here, 0.020, 0.017. 00:12:10.690 --> 00:12:13.360 And then, hear some other sorts of sort of mineral fibers, 00:12:13.360 --> 00:12:15.140 glass foams, glass wools. 00:12:15.140 --> 00:12:19.280 OK, so that's just a table so that you have some data there. 00:12:19.280 --> 00:12:19.780 All right? 00:12:19.780 --> 00:12:20.720 Yes? 00:12:20.720 --> 00:12:30.310 STUDENT: [INAUDIBLE] --foams, if they are closed cell, 00:12:30.310 --> 00:12:31.540 with a different gas rate. 00:12:31.540 --> 00:12:32.480 Because if they're open cell-- 00:12:32.480 --> 00:12:32.530 PROFESSOR: --Right. 00:12:32.530 --> 00:12:32.660 The gas is going to-- 00:12:32.660 --> 00:12:33.680 STUDENT: --it would just always be air. 00:12:33.680 --> 00:12:34.846 PROFESSOR: It's going to go. 00:12:34.846 --> 00:12:37.420 And in fact, one of the difficulties 00:12:37.420 --> 00:12:39.950 with using the lower conductivity of any gases 00:12:39.950 --> 00:12:42.500 is there's a phenomenon called aging, that if, you know, 00:12:42.500 --> 00:12:44.720 you've got your gas inside your foam, 00:12:44.720 --> 00:12:46.480 it's going to diffuse out into the air. 00:12:46.480 --> 00:12:47.900 And air's going to diffuse in. 00:12:47.900 --> 00:12:49.980 So over time, the thermal conductivity 00:12:49.980 --> 00:12:51.850 tends to increase because you're getting 00:12:51.850 --> 00:12:56.440 air coming and the local-- conductivity gas going out. 00:12:56.440 --> 00:12:58.410 But I think, typically, that process 00:12:58.410 --> 00:13:00.104 takes a number of years. 00:13:00.104 --> 00:13:01.270 It doesn't happen in a week. 00:13:01.270 --> 00:13:02.530 But if you're designing a building 00:13:02.530 --> 00:13:04.488 and want the building to be there for 50 years, 00:13:04.488 --> 00:13:05.990 it occurs faster than that. 00:13:05.990 --> 00:13:08.385 So it's not ideal from that point of view. 00:13:10.900 --> 00:13:12.820 All right. 00:13:12.820 --> 00:13:15.600 So let me talk a little bit more about thermal diffusivity. 00:13:15.600 --> 00:13:17.991 Let me scoot over here. 00:13:49.970 --> 00:13:54.260 So materials with a high value of that thermal diffusivity, a. 00:13:54.260 --> 00:13:56.180 They rapidly adjust their temperature 00:13:56.180 --> 00:13:57.680 to their surroundings. 00:13:57.680 --> 00:14:00.190 So if they have a high value of a, what it really 00:14:00.190 --> 00:14:02.220 means they've got a, say, a high value 00:14:02.220 --> 00:14:04.570 of lamda-- so high thermal conductivity. 00:14:04.570 --> 00:14:07.480 And, say, a low value of this volumetric heat capacity. 00:14:07.480 --> 00:14:10.160 So it doesn't take much energy to change their temperature. 00:14:10.160 --> 00:14:11.570 And they also conduct heat well. 00:14:11.570 --> 00:14:13.940 So they tend to adjust their temperature 00:14:13.940 --> 00:14:15.288 to their surroundings quickly. 00:15:01.570 --> 00:15:05.110 OK, so then, let's talk about the thermal conductivity 00:15:05.110 --> 00:15:05.730 of a foam. 00:15:13.620 --> 00:15:15.630 So I'm going to call that lambda star. 00:15:15.630 --> 00:15:16.870 So the star is the foam. 00:15:16.870 --> 00:15:19.700 And then we'll talk about-- lambda 00:15:19.700 --> 00:15:23.410 s will be the lambda for the solid that it's made from. 00:15:23.410 --> 00:15:26.730 So if you think of the thermal conductivity of the foam, 00:15:26.730 --> 00:15:28.720 there's contributions from different types 00:15:28.720 --> 00:15:30.370 of heat transfer. 00:15:30.370 --> 00:15:33.610 So you could have conduction through the solid. 00:15:33.610 --> 00:15:35.380 I'm going to call that lambda s. 00:15:35.380 --> 00:15:38.440 You could have conduction through the gas. 00:15:38.440 --> 00:15:40.420 You could have convection within the cell. 00:15:40.420 --> 00:15:44.550 So convection has to do with having, say, within the cell, 00:15:44.550 --> 00:15:46.390 it might be a different temperature on one 00:15:46.390 --> 00:15:48.390 side of the cell to the other side of the cell. 00:15:48.390 --> 00:15:50.100 And the warmer side of the cell, the gas 00:15:50.100 --> 00:15:52.060 is going to tend to rise to the warmer side 00:15:52.060 --> 00:15:54.160 and fall to the cooler side. 00:15:54.160 --> 00:15:56.536 And you get a convection current set up. 00:15:56.536 --> 00:15:58.160 So you can get heat transfer from that. 00:15:58.160 --> 00:16:00.530 And you can also get heat transfer by radiation. 00:16:00.530 --> 00:16:03.750 So radiation can cause heat transfer, as well. 00:16:03.750 --> 00:16:11.570 So we're going to have contributions from conduction 00:16:11.570 --> 00:16:12.342 through the solid. 00:16:18.850 --> 00:16:22.780 So the amount of conduction in the foam from the solid-- I'm 00:16:22.780 --> 00:16:25.510 going to call lambda star s. 00:16:25.510 --> 00:16:28.670 So lambda s would be the conductivity of the solid. 00:16:28.670 --> 00:16:32.530 And lambda star s is the thermal conductivity contribution 00:16:32.530 --> 00:16:35.490 from the solid in the foam. 00:16:35.490 --> 00:16:38.830 So we get kind of-- through the solid. 00:16:38.830 --> 00:16:42.490 We have conductivity through the gas. 00:16:42.490 --> 00:16:45.230 So it's lambda star g for gas. 00:16:45.230 --> 00:16:47.372 And then we could have convection within the cells. 00:16:55.530 --> 00:16:58.450 We'll call that lambda star c. 00:16:58.450 --> 00:17:04.590 And then we could get radiation through the cell walls 00:17:04.590 --> 00:17:05.465 and across the voids. 00:17:18.609 --> 00:17:21.890 We'll call that lambda star r. 00:17:21.890 --> 00:17:24.069 And so, the thermal conductivity of the foam 00:17:24.069 --> 00:17:25.908 is just the sum of those four contributions. 00:17:40.447 --> 00:17:42.030 So we're just going to go through each 00:17:42.030 --> 00:17:45.880 of those contributions, in turn, and work out 00:17:45.880 --> 00:17:48.759 how much thermal conductivity you get from each of them. 00:17:48.759 --> 00:17:50.800 And it turns out most of the thermal conductivity 00:17:50.800 --> 00:17:53.280 comes through the gas. 00:17:53.280 --> 00:18:04.740 So if we first look at just conduction through the solid, 00:18:04.740 --> 00:18:10.250 we've got that contribution to the conductivity of the foam 00:18:10.250 --> 00:18:14.730 from the solid, it's just equal to some efficiency factor 00:18:14.730 --> 00:18:17.650 times the thermal conductivity of the solid times 00:18:17.650 --> 00:18:21.710 the volume fraction of the solid or the relative density. 00:18:21.710 --> 00:18:24.323 And here, eta is an efficiency factor. 00:18:30.570 --> 00:18:33.320 And it accounts for that tortuosity in the foam. 00:18:33.320 --> 00:18:35.704 So if you think of the solid in the foam, 00:18:35.704 --> 00:18:37.370 it's not like we have little fibers that 00:18:37.370 --> 00:18:39.203 just go from one side to the other like this 00:18:39.203 --> 00:18:41.460 and the heat just moves along those fibers. 00:18:41.460 --> 00:18:44.320 You know, the foam cells have some complicated geometry 00:18:44.320 --> 00:18:46.020 and the heat has to kind of run along 00:18:46.020 --> 00:18:48.390 that complicated geometry. 00:18:48.390 --> 00:18:52.040 And people have made estimates of what this is. 00:18:52.040 --> 00:18:54.540 And it's roughly a factor of 2/3. 00:18:54.540 --> 00:18:58.610 So I guess it would depend on exactly the foam cell geometry. 00:18:58.610 --> 00:19:00.408 But typically it's around 2/3. 00:19:06.990 --> 00:19:08.590 So that's conduction through a solid. 00:19:08.590 --> 00:19:11.600 That's straight forward. 00:19:11.600 --> 00:19:15.900 Conduction through gas is similarly straightforward. 00:19:15.900 --> 00:19:18.460 It's just the conductivity of the gas 00:19:18.460 --> 00:19:21.220 times the amount of the gas. 00:19:21.220 --> 00:19:23.550 And the volume fraction of the gas is just 1 00:19:23.550 --> 00:19:25.440 minus the volume fraction of the solid. 00:19:25.440 --> 00:19:27.805 So it's just 1 minus the relative density. 00:19:27.805 --> 00:19:29.180 So the conduction through the gas 00:19:29.180 --> 00:19:32.760 is just lambda g times 1 minus the relative density. 00:20:07.822 --> 00:20:09.280 So we can do a little example here. 00:20:09.280 --> 00:20:11.030 And you can see how much of the conduction 00:20:11.030 --> 00:20:12.517 comes from the solid in the gas. 00:20:17.900 --> 00:20:24.230 So for example, if we look at a foam that's 2.5% dense and say 00:20:24.230 --> 00:20:28.670 it's a closed cell poly-- what are we doing-- polystyrene. 00:20:39.320 --> 00:20:43.380 So the total thermal conductivity of the foam 00:20:43.380 --> 00:20:50.410 is about 0.04 watts per meter k. 00:20:50.410 --> 00:20:53.640 And the thermal conductivity of polystyrene 00:20:53.640 --> 00:20:59.010 is 0.15 watts per meter k. 00:20:59.010 --> 00:21:07.500 And the thermal conductivity of air is 0.025. 00:21:07.500 --> 00:21:10.900 So let's assume it's just blown with air. 00:21:10.900 --> 00:21:12.510 And then if I just add up, what's 00:21:12.510 --> 00:21:16.315 the contribution of conduction through the solid 00:21:16.315 --> 00:21:18.999 and conduction through the gas-- so I just 00:21:18.999 --> 00:21:20.790 use those two little equations-- conduction 00:21:20.790 --> 00:21:23.720 through the solid-- it's going to be 2/3 00:21:23.720 --> 00:21:28.300 of this value of lambda s times the amount of the solids-- 00:21:28.300 --> 00:21:34.077 that's 0.025 and then plus lambda g, which 00:21:34.077 --> 00:21:41.850 is 0.025 times the amount of the gas, which is 0.975. 00:21:41.850 --> 00:21:46.360 And if I work those two things out, this is 0.003 00:21:46.360 --> 00:21:52.160 and this is 0.024. 00:21:52.160 --> 00:21:54.730 So that total is 0.027 watts per meter k. 00:21:58.240 --> 00:22:02.010 So you can see if the total is 0.04, most of it's 00:22:02.010 --> 00:22:02.800 come from the gas. 00:22:02.800 --> 00:22:04.330 A little bit's come from the solid. 00:22:04.330 --> 00:22:06.900 And the rest is going to be from convection and radiation. 00:22:09.519 --> 00:22:10.310 And that's typical. 00:22:27.560 --> 00:22:29.810 And that's the reason that they sometimes 00:22:29.810 --> 00:22:32.940 use low thermal conductivity gases to blow foams 00:22:32.940 --> 00:22:35.020 for thermal insulation because the gas makes up 00:22:35.020 --> 00:22:39.034 such a big fraction of the total conductivity. 00:22:39.034 --> 00:22:40.450 If you can reduce that, you reduce 00:22:40.450 --> 00:22:41.491 the overall conductivity. 00:22:46.790 --> 00:22:49.220 So, we'll say foams for insulation are blown 00:22:49.220 --> 00:22:53.247 with low conductivity gases. 00:22:58.100 --> 00:23:00.350 But as I mentioned, you have this problem with aging 00:23:00.350 --> 00:23:02.750 that, over time, that gas is going to diffuse out 00:23:02.750 --> 00:23:04.090 and air is going to diffuse in. 00:23:21.810 --> 00:23:26.135 Then the overall thermal conductivity of the foam 00:23:26.135 --> 00:23:27.010 is going to increase. 00:23:46.570 --> 00:23:50.190 So that's the conduction. 00:23:50.190 --> 00:23:54.580 And then the next contribution is from convection. 00:23:54.580 --> 00:23:59.150 So imagine we have one of our little cells here. 00:23:59.150 --> 00:24:03.260 And it's hotter on that side than it is on that side. 00:24:03.260 --> 00:24:06.050 And hot air is going to rise. 00:24:06.050 --> 00:24:07.430 Cold air is going to fall. 00:24:07.430 --> 00:24:10.380 So you get a convection current set up. 00:24:10.380 --> 00:24:13.160 And because of the density changes, 00:24:13.160 --> 00:24:16.320 you get a buoyancy force in the air. 00:24:16.320 --> 00:24:18.600 So that's kind of driving the convection. 00:24:18.600 --> 00:24:20.520 But you also have a viscous drag. 00:24:20.520 --> 00:24:24.580 So the air is moving past the wall of the foam. 00:24:24.580 --> 00:24:27.580 And there's going to be some viscous drag associated. 00:24:27.580 --> 00:24:29.320 And how much convection you can get 00:24:29.320 --> 00:24:31.740 depends on the balance between this buoyancy force 00:24:31.740 --> 00:24:32.815 and the viscous drag. 00:24:37.700 --> 00:24:44.300 So we'll say the gas rises and falls 00:24:44.300 --> 00:24:46.080 due to density changes with temperature. 00:24:54.871 --> 00:24:57.850 And the density changes give rise to buoyancy forces. 00:25:08.750 --> 00:25:18.490 But we also have these viscous forces 00:25:18.490 --> 00:25:22.590 from the drag of the air against the walls of the cell. 00:25:33.070 --> 00:25:35.700 So air moving past the walls-- this 00:25:35.700 --> 00:25:39.220 is kind of a fluid mechanics thing-- so that air is a fluid. 00:25:39.220 --> 00:25:41.340 And in fluid mechanics, they often 00:25:41.340 --> 00:25:42.894 use dimensionless numbers. 00:25:42.894 --> 00:25:44.310 And there's a dimensionless number 00:25:44.310 --> 00:25:45.524 called the Rayleigh number. 00:25:45.524 --> 00:25:47.440 And the Rayleigh number, you can think of it-- 00:25:47.440 --> 00:25:50.570 it's not quite the balance of the buoyancy force 00:25:50.570 --> 00:25:52.370 against the viscous forces. 00:25:52.370 --> 00:25:54.310 But it involves those forces. 00:25:54.310 --> 00:25:56.530 And convection is important if this Raleigh number's 00:25:56.530 --> 00:25:57.215 over 1,000. 00:26:17.400 --> 00:26:19.211 And here's what the Rayleigh number is. 00:26:22.150 --> 00:26:26.120 It's the density of the fluid times the acceleration 00:26:26.120 --> 00:26:28.810 of gravity times beta. 00:26:28.810 --> 00:26:30.880 Beta's the volume expansion coefficient 00:26:30.880 --> 00:26:34.284 for the gas-- times the temperature change. 00:26:34.284 --> 00:26:35.950 And we're going to look at a temperature 00:26:35.950 --> 00:26:38.200 change across a cell. 00:26:38.200 --> 00:26:39.480 And then, times the length. 00:26:39.480 --> 00:26:42.790 That's going to be the cell size. 00:26:42.790 --> 00:26:46.730 And we divide that by the fluid viscosity 00:26:46.730 --> 00:26:50.380 and the thermal diffusivity. 00:26:50.380 --> 00:26:52.910 So let me write down what all these things are. 00:26:52.910 --> 00:26:54.608 So rho is the density of the gas. 00:26:58.910 --> 00:27:01.820 So the g's gravitational acceleration. 00:27:09.170 --> 00:27:11.945 Beta is the volume expansion of the gas. 00:27:23.010 --> 00:27:24.682 And for a constant pressure that's 00:27:24.682 --> 00:27:26.015 equal to 1 over the temperature. 00:27:31.390 --> 00:27:35.460 Then delta tc is the temperature difference across a cell. 00:27:43.090 --> 00:27:44.410 And l is the cell size. 00:27:46.990 --> 00:27:51.815 Mu is the dynamics viscosity the fluid. 00:27:55.610 --> 00:27:57.270 And a is our thermal diffusivity. 00:28:06.680 --> 00:28:09.300 So what I'm going to do is just work out, 00:28:09.300 --> 00:28:16.450 for a typical example, how big of a cell size 00:28:16.450 --> 00:28:19.420 do you need to get this Rayleigh number to be 1,000. 00:28:19.420 --> 00:28:22.060 And we're going to see that, typically, 00:28:22.060 --> 00:28:23.830 that cell size is big. 00:28:23.830 --> 00:28:24.970 It's like 20 millimeters. 00:28:24.970 --> 00:28:26.660 So in most foams, the convection really 00:28:26.660 --> 00:28:27.970 isn't very important at all. 00:28:27.970 --> 00:28:30.510 So it's typically-- people don't worry about convection. 00:28:30.510 --> 00:28:32.500 And let me just show you how that works. 00:28:53.540 --> 00:28:56.165 So for our Rayleigh number, which 00:28:56.165 --> 00:28:59.030 is ra-- for the Rayleigh number to be 1,000-- 00:28:59.030 --> 00:29:01.760 say we had air in the cells. 00:29:01.760 --> 00:29:04.095 And say the temperature was room temperature. 00:29:07.670 --> 00:29:13.170 Then the volume coefficient of expansion is just 1 over t. 00:29:13.170 --> 00:29:16.150 So it's 1 over 300, say. 00:29:16.150 --> 00:29:18.670 degrees k to the minus 1. 00:29:18.670 --> 00:29:22.700 Let's say our change in temperature across one cell 00:29:22.700 --> 00:29:25.480 was 1 degree k. 00:29:25.480 --> 00:29:26.980 Bless you. 00:29:26.980 --> 00:29:32.400 The viscosity of air is 2 times 10 to the minus 5, 00:29:32.400 --> 00:29:34.280 pascal seconds. 00:29:34.280 --> 00:29:41.850 The density of air is 1.2 kilograms per cubic meter. 00:29:45.680 --> 00:29:47.960 And the thermal diffusivity for air 00:29:47.960 --> 00:29:53.840 is 2 times 10 to the minus 5 meters squared per second. 00:29:53.840 --> 00:29:56.220 And if you plug all of these into that equation 00:29:56.220 --> 00:29:59.470 for the Rayleigh number and you solve for the cell size, 00:29:59.470 --> 00:30:04.280 you find that the cell size, l, is 20 millimeters. 00:30:04.280 --> 00:30:06.740 So that says convection is only important if the cell 00:30:06.740 --> 00:30:08.010 size is bigger than that. 00:30:21.200 --> 00:30:23.752 And so most foams have cells much smaller than that. 00:30:23.752 --> 00:30:24.960 And convection is negligible. 00:30:46.000 --> 00:30:48.420 So I have enclosed cells and the heat's 00:30:48.420 --> 00:30:50.770 not transferred so easily from one cell to another 00:30:50.770 --> 00:30:52.230 by the gas moving. 00:30:52.230 --> 00:30:55.280 And by having small cells the convection drops out. 00:30:55.280 --> 00:30:56.930 So you don't have to worry about that. 00:30:56.930 --> 00:31:00.705 So the last contribution to heat transfer is from radiation. 00:31:10.960 --> 00:31:12.830 And there's something called Stefan's law 00:31:12.830 --> 00:31:16.490 that describes the heat flux for radiated heat 00:31:16.490 --> 00:31:18.920 transfer from a surface at one temperature 00:31:18.920 --> 00:31:21.110 to another surface at a different temperature 00:31:21.110 --> 00:31:22.990 across a vacuum. 00:31:22.990 --> 00:31:29.350 So we can say we have a heat flux 00:31:29.350 --> 00:31:34.484 qr not from a surface of one temperature. 00:31:37.310 --> 00:31:41.180 So I'm going to call that t1-- to one at a lower temperature. 00:31:47.090 --> 00:31:51.730 I'm going to call tnot-- with a vacuum in between them. 00:32:03.340 --> 00:32:08.060 So this is [? Stefan's ?] law so this is the radiative heat 00:32:08.060 --> 00:32:13.850 flux is equal to the emissivity of the surfaces, which 00:32:13.850 --> 00:32:16.130 is beta 1 times a constant called 00:32:16.130 --> 00:32:21.460 Stefan's constant-- sigma times the fourth power 00:32:21.460 --> 00:32:24.800 of temperatures. 00:32:24.800 --> 00:32:32.830 I'm taking the difference of the temperatures 00:32:32.830 --> 00:32:37.874 so here are the Stefan's Constant-- is sigma. 00:32:41.750 --> 00:32:48.760 And that's equal to 5.67 times the 10 to the minus 8. 00:32:48.760 --> 00:32:54.740 And that's in watts per meter squared per k to the fourth. 00:32:54.740 --> 00:32:58.770 And beta is a constant describing 00:32:58.770 --> 00:33:00.260 the emissivity of the surfaces. 00:33:18.980 --> 00:33:21.480 So it gives the radiant heat flux 00:33:21.480 --> 00:33:24.244 per unit area of the sample relative to a black body. 00:33:24.244 --> 00:33:26.160 And that's a characteristic of the emissivity. 00:33:29.830 --> 00:33:33.599 All right, so then, so if we-- yes? 00:33:33.599 --> 00:33:34.432 STUDENT: [INAUDIBLE] 00:33:37.932 --> 00:33:39.890 PROFESSOR: Now-- so right now, forget the foam. 00:33:39.890 --> 00:33:40.540 We have no foam. 00:33:40.540 --> 00:33:42.748 We just have two surfaces with a vacuum between them. 00:33:42.748 --> 00:33:45.140 And now I'm going to stick a foam between the surfaces. 00:33:45.140 --> 00:33:48.320 And we're going to see how that changes the heat flux, OK? 00:33:48.320 --> 00:33:50.680 So the next step is we put the foam between those two 00:33:50.680 --> 00:33:51.604 surfaces. 00:33:51.604 --> 00:33:53.270 And the heat flux is going to be reduced 00:33:53.270 --> 00:33:55.686 because the radiation is going to be absorbed by the solid 00:33:55.686 --> 00:33:57.560 and reflected by the cell walls. 00:33:57.560 --> 00:33:59.040 And so we're going to characterize 00:33:59.040 --> 00:34:00.090 how much it's reduced. 00:34:00.090 --> 00:34:03.440 So there's another law called Beer's Law, which characterizes 00:34:03.440 --> 00:34:07.736 the reduction in the heat flux. 00:34:39.239 --> 00:34:41.480 Piece of chalk's getting to small 00:35:34.550 --> 00:35:38.810 OK, so Beer's Law gives us the attenuation, so the sort 00:35:38.810 --> 00:35:42.510 of reduction in the heat flow. 00:35:42.510 --> 00:35:45.990 So qr is equal to qr not. 00:35:45.990 --> 00:35:49.100 That would be the heat flux, if we just had the vacuum. 00:35:49.100 --> 00:35:51.530 And then there's an exponential law. 00:35:51.530 --> 00:35:56.630 And it's the exponential of minus k star t star. 00:35:56.630 --> 00:35:59.050 And here, k stars in an extinction coefficient 00:35:59.050 --> 00:36:00.254 for the foam. 00:36:00.254 --> 00:36:02.170 Talk a little bit more about that in a minute. 00:36:10.090 --> 00:36:12.200 and t star is just the thickness of the foam. 00:36:18.970 --> 00:36:20.758 And then this thing is called Beer's Law. 00:36:28.800 --> 00:36:31.120 So we have very thin walls and struts. 00:36:31.120 --> 00:36:33.890 And we're just going to consider optically thin walls and struts 00:36:33.890 --> 00:36:35.070 to make life easy. 00:36:35.070 --> 00:36:37.070 Then we can say that, if they're optically thin, 00:36:37.070 --> 00:36:38.906 they're transparent to radiation. 00:36:38.906 --> 00:36:40.280 They're optically thin if they're 00:36:40.280 --> 00:36:42.170 less than about 10 microns. 00:36:42.170 --> 00:36:44.950 Then this extinction coefficient is just the amount 00:36:44.950 --> 00:36:47.420 of solid times the extension coefficient for the solids. 00:36:47.420 --> 00:36:49.670 So it's just the relative density times the extinction 00:36:49.670 --> 00:36:50.936 coefficient for the solid. 00:37:20.500 --> 00:37:25.390 OK, and then I can say, the heat flux by radiation. 00:37:28.320 --> 00:37:31.148 I can use two equations to write that down now. 00:37:34.890 --> 00:37:37.060 And then I'm going to let them be equal to get 00:37:37.060 --> 00:37:38.870 the thermal conductivity. 00:37:38.870 --> 00:37:44.920 I can say qr is going equal to lambda r times dt by dx. 00:37:44.920 --> 00:37:49.800 So that's the Fourier's Law that we started out with. 00:37:49.800 --> 00:37:52.160 And then I've also got the qr that I'm 00:37:52.160 --> 00:37:55.780 going to get by combining the Stefan's Law with the Beer's 00:37:55.780 --> 00:37:57.440 Law up there. 00:37:57.440 --> 00:38:00.130 So if I do that, I get that qr is beta 00:38:00.130 --> 00:38:09.630 1 times sigma times t1 to the fourth minus t not the fourth. 00:38:09.630 --> 00:38:14.730 So that's the qr not up there from down there. 00:38:14.730 --> 00:38:18.370 And then I've got an exponential for the attenuation. 00:38:18.370 --> 00:38:21.500 And instead of k star, I'm going to put the relative density 00:38:21.500 --> 00:38:22.268 of times ks. 00:38:27.610 --> 00:38:32.330 and then I've got the thickness of the foam, t star, as well. 00:38:32.330 --> 00:38:39.900 OK, so that's qr, but that has to equal lambda times dt by dx. 00:38:44.700 --> 00:38:46.402 So I'm going to use some approximations. 00:38:46.402 --> 00:38:48.360 Here and I'm going to end up with an expression 00:38:48.360 --> 00:38:51.162 for the contribution from radiation 00:38:51.162 --> 00:38:52.370 to heat transfer in the foam. 00:38:52.370 --> 00:38:53.134 Yeah? 00:38:53.134 --> 00:38:55.300 STUDENT: So when you say optically thin walls, where 00:38:55.300 --> 00:38:57.050 t is less than 10 microns, you mean 00:38:57.050 --> 00:38:59.585 like the walls of the foam? 00:38:59.585 --> 00:39:00.460 PROFESSOR: Yeah, yea. 00:39:00.460 --> 00:39:02.774 STUDENT: So it's different t than the-- 00:39:02.774 --> 00:39:04.190 PROFESSOR: t star is the thickness 00:39:04.190 --> 00:39:05.850 of the whole thing, yeah. 00:39:05.850 --> 00:39:07.800 So imagine we had our two surfaces. 00:39:07.800 --> 00:39:10.320 And they might be like 100 millimeters apart 00:39:10.320 --> 00:39:12.650 or something. t star is the sort of thickness 00:39:12.650 --> 00:39:15.170 of the foam in between the two surfaces. 00:39:15.170 --> 00:39:18.250 And the optically thin is the cell walls, which 00:39:18.250 --> 00:39:21.687 are microns kind of thickness. 00:39:21.687 --> 00:39:22.186 OK. 00:39:30.570 --> 00:39:32.840 So I'm going to make some approximations here. 00:39:36.560 --> 00:39:39.140 And that's going to allow me to solve for t star. 00:39:39.140 --> 00:39:43.060 So I'm going to say that dt by dx 00:39:43.060 --> 00:39:47.100 x is approximately equal to just t1 minus t 00:39:47.100 --> 00:39:51.570 not over the thickness of the foam 00:39:51.570 --> 00:39:58.220 or I'll call that delta t over t star. 00:39:58.220 --> 00:40:00.440 And then, the other approximation I'm going to use 00:40:00.440 --> 00:40:05.420 is that t1 to the 4th minus t not to the fourth 00:40:05.420 --> 00:40:10.860 is equal to 4 times delta t times 00:40:10.860 --> 00:40:13.240 the average temperature cubed. 00:40:13.240 --> 00:40:18.350 So here t bar is the average temperature, t1 plus t 00:40:18.350 --> 00:40:19.090 not over 2. 00:40:44.070 --> 00:40:47.220 So then, if I use those two approximations, 00:40:47.220 --> 00:40:52.470 I can write that qr, our heat flux from radiative transfer. 00:40:52.470 --> 00:40:53.500 I got the beta 1. 00:40:53.500 --> 00:40:54.710 I've got the sigma. 00:40:54.710 --> 00:40:59.280 And instead of the difference of the fourth power, 00:40:59.280 --> 00:41:05.250 I'm going to write 4 delta t t bar cubed. 00:41:05.250 --> 00:41:06.675 And then I've got my exponential. 00:41:09.780 --> 00:41:12.130 Blah, blah, blah, blah, blah. 00:41:12.130 --> 00:41:17.990 So then, here's the relative density times ks times 00:41:17.990 --> 00:41:20.780 t star, the overall thickness. 00:41:20.780 --> 00:41:23.720 That's going to equal the radiative contribution 00:41:23.720 --> 00:41:26.300 to the thermal conductivity of the foam. 00:41:26.300 --> 00:41:29.230 And instead of dt by dx, I'm going 00:41:29.230 --> 00:41:34.160 to have delta t over t star here. 00:41:34.160 --> 00:41:35.940 So part of the reason for doing these 00:41:35.940 --> 00:41:40.490 approximations I end up with a delta t term on both sides. 00:41:40.490 --> 00:41:42.150 Now I can cancel that out. 00:41:42.150 --> 00:41:45.540 And if I just take this mess here and multiply it by t star, 00:41:45.540 --> 00:41:47.670 then I've got lambda r star. 00:42:03.560 --> 00:42:08.790 That's our thermal conductivity contribution from radiation. 00:42:08.790 --> 00:42:10.840 So one of the things to notice here 00:42:10.840 --> 00:42:15.820 is that, as the relative density goes down, 00:42:15.820 --> 00:42:19.690 then the contribution from radiation 00:42:19.690 --> 00:42:22.802 to the thermal conductivity of the foam goes up. 00:42:49.920 --> 00:42:53.900 OK, so this chart here shows thermal conductivity 00:42:53.900 --> 00:42:55.330 as a function of relative density. 00:42:55.330 --> 00:42:56.830 And it breaks down the contributions 00:42:56.830 --> 00:43:00.770 from the gas, g, the solid, s, and the radiation, r. 00:43:00.770 --> 00:43:03.070 And you kind of see the gas contribution 00:43:03.070 --> 00:43:04.230 doesn't change that much. 00:43:04.230 --> 00:43:06.790 These are relative densities between a little over 2 00:43:06.790 --> 00:43:08.330 and a little less than 5%. 00:43:08.330 --> 00:43:10.570 So the amount of gas-- it's mostly 00:43:10.570 --> 00:43:14.050 gas in all of these things. 00:43:14.050 --> 00:43:18.670 The solid contribution increases as the relative density 00:43:18.670 --> 00:43:19.260 increases. 00:43:19.260 --> 00:43:21.670 So you'd expect that. 00:43:21.670 --> 00:43:26.030 And then, as I just said, as the relative density goes down, 00:43:26.030 --> 00:43:30.030 the amount of radiation contribution goes up. 00:43:30.030 --> 00:43:32.800 And so you can kind of see how that all fits together. 00:43:32.800 --> 00:43:35.654 Another plot that shows the thermal conductivity 00:43:35.654 --> 00:43:36.820 versus the relative density. 00:43:36.820 --> 00:43:39.590 These are for a few different types of foams. 00:43:39.590 --> 00:43:42.450 You can see for this plot here, you 00:43:42.450 --> 00:43:45.067 reach a minimum in the thermal conductivity. 00:43:45.067 --> 00:43:46.900 And that's because you've got this trade off 00:43:46.900 --> 00:43:49.399 between the contribution from the solid and the contribution 00:43:49.399 --> 00:43:50.760 from the radiation. 00:43:50.760 --> 00:43:54.750 And those two kind of trade off and you get to a minimum. 00:43:54.750 --> 00:43:56.545 So let me write some of this down. 00:43:59.950 --> 00:44:03.968 So I'll just say that-- hang on. 00:44:03.968 --> 00:44:05.610 Write this over again. 00:44:13.897 --> 00:44:16.063 This is looking at the overall thermal conductivity. 00:44:19.600 --> 00:44:24.650 And we can see the relative contributions 00:44:24.650 --> 00:44:30.180 of lambda, solid, lambda, gas, lambda, radiation. 00:44:30.180 --> 00:44:31.942 I'll just say this shown in the figure. 00:44:38.490 --> 00:44:49.389 I'm going to say the next figure shows a minimum 00:44:49.389 --> 00:44:50.555 in the thermal conductivity. 00:44:53.990 --> 00:45:00.630 Then I'll just say there's a trade off 00:45:00.630 --> 00:45:03.300 between the conduction through the solid 00:45:03.300 --> 00:45:05.593 and I can direction from the radiation. 00:45:10.610 --> 00:45:12.640 And then we also have a plot here 00:45:12.640 --> 00:45:16.637 that shows the conductivity versus the cell size. 00:45:27.620 --> 00:45:30.710 And you can see that the conductivity increases 00:45:30.710 --> 00:45:31.775 with cell size. 00:45:39.580 --> 00:45:41.960 And the reason for that is the bigger the cells get, 00:45:41.960 --> 00:45:45.040 the radiation is reflected less often. 00:45:58.120 --> 00:46:04.800 And one thing I wanted to mention with the cell size 00:46:04.800 --> 00:46:08.920 is that if you look at aerogels, the way aerogels shells work 00:46:08.920 --> 00:46:11.790 is that they have a very small cell size, a very small pore 00:46:11.790 --> 00:46:12.610 size. 00:46:12.610 --> 00:46:15.200 So typically, it's less than 100 nanometers. 00:46:15.200 --> 00:46:20.230 And the mean free path of air is 68 nanometers. 00:46:20.230 --> 00:46:22.140 So the mean free path is the average distance 00:46:22.140 --> 00:46:25.400 the molecules move before they collide with another molecule. 00:46:25.400 --> 00:46:29.130 And if your pore size is less than the mean free path, 00:46:29.130 --> 00:46:32.360 then that reduces the thermal conductivity. 00:46:32.360 --> 00:46:34.200 It reduces the ability of the atoms 00:46:34.200 --> 00:46:37.810 to pass the heat along between one another. 00:46:37.810 --> 00:46:39.310 So the way the aerogels work is they 00:46:39.310 --> 00:46:40.830 have a very small pore size. 00:46:46.690 --> 00:46:49.120 And what's important is how big the pores 00:46:49.120 --> 00:46:50.900 are relative to the mean free path of air. 00:48:04.380 --> 00:48:06.790 OK, so that's the thermal conductivity. 00:48:06.790 --> 00:48:09.820 I wanted to talk about a few other thermal properties 00:48:09.820 --> 00:48:11.205 of foams, as well, today. 00:48:43.100 --> 00:48:45.100 So one is the specific heat. 00:48:45.100 --> 00:48:47.760 And since the specific heat is the energy required 00:48:47.760 --> 00:48:51.990 to raise the temperature by a unit mass, 00:48:51.990 --> 00:48:57.794 then the mass is the same-- you know, 00:48:57.794 --> 00:48:59.210 if you have a certain mass of foam 00:48:59.210 --> 00:49:01.440 or a certain mass of solid-- the specific heat from the foam 00:49:01.440 --> 00:49:02.481 is the same as the solid. 00:49:32.390 --> 00:49:36.030 So the specific heat for the foam 00:49:36.030 --> 00:49:40.080 is the same as the specific heat for the solid. 00:49:40.080 --> 00:49:41.680 So that would have units of joules 00:49:41.680 --> 00:49:43.610 per kilogram per degree k. 00:49:48.560 --> 00:49:51.490 And the next property is the thermal expansion coefficient. 00:50:02.380 --> 00:50:03.460 And it's a similar thing. 00:50:03.460 --> 00:50:06.150 The thermal expansion coefficient for the foam 00:50:06.150 --> 00:50:07.850 is equal to the thermal coefficient 00:50:07.850 --> 00:50:10.840 of expansion for the solid. 00:50:10.840 --> 00:50:14.250 So imagine you have-- say you had something like a honeycomb. 00:50:14.250 --> 00:50:15.910 If you heat it up a certain amount, 00:50:15.910 --> 00:50:17.879 every member is going to expand by alpha. 00:50:17.879 --> 00:50:19.420 And if every member expands by alpha, 00:50:19.420 --> 00:50:20.794 the whole thing expands by alpha. 00:50:20.794 --> 00:50:22.460 And this is the same. 00:50:22.460 --> 00:50:24.090 And it's the same idea with the foam. 00:50:24.090 --> 00:50:26.590 So if every member just gets longer by alpha, 00:50:26.590 --> 00:50:28.340 then the whole thing gets bigger by alpha. 00:50:44.010 --> 00:50:46.730 OK, so the last topic I wanted to talk about 00:50:46.730 --> 00:50:48.612 was the thermal shock resistance. 00:50:48.612 --> 00:50:51.070 And thermal shock is the idea is that if you have something 00:50:51.070 --> 00:50:54.140 that's hot, and say you quench it in a liquid-- 00:50:54.140 --> 00:50:56.220 so you put it suddenly in a liquid-- 00:50:56.220 --> 00:50:57.630 the surface is going to cool down 00:50:57.630 --> 00:50:59.560 faster than the bulk of it. 00:50:59.560 --> 00:51:00.970 And because the surface is trying 00:51:00.970 --> 00:51:02.785 to contract because it's cooling down, 00:51:02.785 --> 00:51:05.160 but it's attached to the bulk of it and it's constrained, 00:51:05.160 --> 00:51:08.692 it can't really cool down, then you generate stresses. 00:51:08.692 --> 00:51:10.150 And if the stresses are big enough, 00:51:10.150 --> 00:51:12.566 you can cause fracture and have the thing crack and spall. 00:51:17.530 --> 00:51:20.340 So we'll say if the materials is subjected 00:51:20.340 --> 00:51:34.880 to a sudden change in the surface temperature, that 00:51:34.880 --> 00:51:37.060 induces thermal stresses at the surface 00:51:37.060 --> 00:51:39.080 and can induce spalling and cracking. 00:52:00.642 --> 00:52:03.100 So we're going to think about a material at one temperature 00:52:03.100 --> 00:52:04.570 that's dropped into, say, a liquid 00:52:04.570 --> 00:52:05.695 at a different temperature. 00:52:27.912 --> 00:52:29.370 So the surface temperature is going 00:52:29.370 --> 00:52:31.564 to drop to the cooler liquid temperature 00:52:31.564 --> 00:52:33.480 and it's going to contract the surface layers. 00:52:52.560 --> 00:52:55.510 And the fact that they're bound to the layers underneath that 00:52:55.510 --> 00:52:57.710 are not contracting as quickly, it 00:52:57.710 --> 00:53:00.860 means that you generate a thermal strain. 00:53:11.821 --> 00:53:13.320 So the thermal strain is going to be 00:53:13.320 --> 00:53:15.904 the coefficient of thermal expansion times the change 00:53:15.904 --> 00:53:16.528 in temperature. 00:53:52.300 --> 00:53:54.560 So you're going to constrain the surface 00:53:54.560 --> 00:53:56.300 to the original dimensions. 00:53:56.300 --> 00:53:58.105 And then you're going to induce the stress. 00:54:19.450 --> 00:54:23.310 So if it's a plane or thing, it's e alpha delta t. 00:54:23.310 --> 00:54:25.970 And then, there's a factor of 1 minus nu, 00:54:25.970 --> 00:54:29.850 just because it's a plane, in a plane. 00:54:29.850 --> 00:54:31.590 And then you'll get cracking or spalling 00:54:31.590 --> 00:54:34.210 when that stress equals some failure, stress. 00:54:43.120 --> 00:54:47.490 So I can rearrange this and solve for the critical delta t 00:54:47.490 --> 00:54:50.580 that you can withstand without getting cracking. 00:54:50.580 --> 00:54:54.106 So I just rearranged this and say sigma's equal to sigma f. 00:54:54.106 --> 00:55:00.600 That would be sigma f times 1 minus nu over e and over alpha. 00:55:00.600 --> 00:55:02.450 So that's the critical change in temperature 00:55:02.450 --> 00:55:03.660 to just cause cracking. 00:55:13.210 --> 00:55:15.864 So now what I can do is I can substitute in there for what 00:55:15.864 --> 00:55:17.030 you would have for the foam. 00:55:20.294 --> 00:55:22.210 And I'm going to do it just for the open cells 00:55:22.210 --> 00:55:24.210 just because it's easier to write the equations. 00:55:27.580 --> 00:55:32.230 So for the foam, I would have some sort of fracture strength. 00:55:32.230 --> 00:55:34.180 So when we did the modeling of the foams, 00:55:34.180 --> 00:55:37.140 we said that was equal to about 0.2 times the modulus 00:55:37.140 --> 00:55:42.980 of rupture times the relative density to the 3/2's power 00:55:42.980 --> 00:55:46.730 and 1 minus nu. 00:55:46.730 --> 00:55:49.460 And if I divide by the modulus of the foam, 00:55:49.460 --> 00:55:54.460 that's es times the relative density squared. 00:55:54.460 --> 00:55:57.150 And then we just had alpha for the foam 00:55:57.150 --> 00:55:58.575 was the same as alpha s. 00:56:01.130 --> 00:56:03.760 So then, I can rearrange this slightly 00:56:03.760 --> 00:56:07.640 and say it's equal to 0.2 over the relative density 00:56:07.640 --> 00:56:09.240 to the 1/2 power. 00:56:09.240 --> 00:56:12.950 So I'm canceling out these relative densities here. 00:56:12.950 --> 00:56:17.431 And then I can combine all the solid properties together. 00:56:17.431 --> 00:56:19.180 And I'm going to say that nu for the solid 00:56:19.180 --> 00:56:21.263 is about equal to the same as nu for the foam. 00:56:34.650 --> 00:56:37.951 So what I can do here is I can group all the solid properties 00:56:37.951 --> 00:56:38.450 together. 00:56:38.450 --> 00:56:41.880 And this just is delta t critical for the solid, right? 00:56:41.880 --> 00:56:45.930 So this is saying that the critical temperature range 00:56:45.930 --> 00:56:48.890 before you get cracking in the foam 00:56:48.890 --> 00:56:52.050 is equal to the range for the solid, 00:56:52.050 --> 00:56:54.879 but multiplied by this factor of 0.2 00:56:54.879 --> 00:56:57.170 and divided by the square root of the relative density. 00:56:57.170 --> 00:56:58.919 So if the square of-- the relative density 00:56:58.919 --> 00:56:59.959 is going to less than 1. 00:56:59.959 --> 00:57:02.000 So this number here is going to be bigger than 1. 00:57:02.000 --> 00:57:04.830 So it's saying that the temperature range that 00:57:04.830 --> 00:57:06.700 will give you spalling in the foam 00:57:06.700 --> 00:57:09.780 is going to be bigger than the temperature range in the solid. 00:57:09.780 --> 00:57:14.440 So the foam's going to be better than the solid, OK? 00:57:14.440 --> 00:57:16.420 And that uses our little models from before. 00:57:21.217 --> 00:57:23.050 So I think I'm going to stop there, probably 00:57:23.050 --> 00:57:27.030 cause my throat is starting to get too sore. 00:57:27.030 --> 00:57:29.370 There's a little case study in the notes. 00:57:29.370 --> 00:57:31.770 And I'll just put that on the notes on the Stellar site. 00:57:31.770 --> 00:57:34.130 It's like one page and it's really straightforward. 00:57:34.130 --> 00:57:36.171 You can just read that, OK? 00:57:36.171 --> 00:57:38.420 So this is the end of the bit on thermal conductivity. 00:57:38.420 --> 00:57:40.530 That's just this one lecture. 00:57:40.530 --> 00:57:43.200 And this is really the end of the whole section 00:57:43.200 --> 00:57:45.300 on modeling of the honey combs and the foams. 00:57:45.300 --> 00:57:47.633 So that's kind of the first half of the term is modeling 00:57:47.633 --> 00:57:49.317 the honey combs and the foams. 00:57:49.317 --> 00:57:50.900 And the second half of the term, we're 00:57:50.900 --> 00:57:53.850 kind of applying those models to different situations. 00:57:53.850 --> 00:57:56.720 So next week, we'll have the review on Monday, 00:57:56.720 --> 00:57:59.730 have a test on Wednesday, week after that is Spring break. 00:57:59.730 --> 00:58:02.339 I can't believe we're at Spring break already. 00:58:02.339 --> 00:58:04.630 And then after that we'll start we'll do the trabecular 00:58:04.630 --> 00:58:05.850 bone for a week. 00:58:05.850 --> 00:58:08.670 We'll do tissue engineering scaffolds and cell mechanics 00:58:08.670 --> 00:58:10.490 for two or three lectures. 00:58:10.490 --> 00:58:12.830 We'll look at some other applications 00:58:12.830 --> 00:58:15.910 to engineering design, look at energy absorption and sandwich 00:58:15.910 --> 00:58:16.880 panels. 00:58:16.880 --> 00:58:19.088 And then, I'm going to talk about plants a little bit 00:58:19.088 --> 00:58:19.970 at the very end, OK? 00:58:19.970 --> 00:58:23.200 So we've already covered a lot of the kind of deriving 00:58:23.200 --> 00:58:24.910 equations part of the course. 00:58:24.910 --> 00:58:27.340 The rest of the course is more applying the equations 00:58:27.340 --> 00:58:29.894 to lots of different situations, OK? 00:58:29.894 --> 00:58:32.060 So I'm going to stop there just because my throat is 00:58:32.060 --> 00:58:33.860 giving out.