1 00:00:10,500 --> 00:00:14,410 PROFESSOR: So another important source of convection in an air 2 00:00:14,410 --> 00:00:19,180 filled room is buoyancy due to differences 3 00:00:19,180 --> 00:00:22,510 in the density of the air as the temperature varies. 4 00:00:22,510 --> 00:00:25,120 Even relatively small variations in temperature 5 00:00:25,120 --> 00:00:28,960 can lead to significant flows. 6 00:00:28,960 --> 00:00:30,510 There's another dimensionless number, 7 00:00:30,510 --> 00:00:33,290 which controls the appearance and strength 8 00:00:33,290 --> 00:00:45,010 of such flows, which is the Rayleigh number, written Ra. 9 00:00:45,010 --> 00:00:47,150 And this is also a physical property 10 00:00:47,150 --> 00:00:49,640 of the-- it's a combination of physical properties 11 00:00:49,640 --> 00:00:53,280 of the fluid plus the geometry. 12 00:00:53,280 --> 00:00:56,270 So in this case, the relevant geometrical scale 13 00:00:56,270 --> 00:01:02,180 is the height, because this is a gravitational instability. 14 00:01:02,180 --> 00:01:06,880 So in the Rayleigh number, we have gravity. 15 00:01:06,880 --> 00:01:16,600 I'll just define all these-- a gravitational acceleration, 16 00:01:16,600 --> 00:01:19,500 which is 9.8 meters per second squared. 17 00:01:22,820 --> 00:01:27,620 We have-- well, if we define the change in air density 18 00:01:27,620 --> 00:01:30,080 relative to the initial air density that's 19 00:01:30,080 --> 00:01:32,310 caused by changes in temperature, 20 00:01:32,310 --> 00:01:34,060 if the temperature changes aren't too big, 21 00:01:34,060 --> 00:01:36,110 there is a linear response, which 22 00:01:36,110 --> 00:01:41,240 is defined by the thermal expansion coefficient beta, so 23 00:01:41,240 --> 00:01:43,100 beta delta T. 24 00:01:43,100 --> 00:01:53,360 So beta is the thermal expansion coefficient. 25 00:01:53,360 --> 00:01:58,310 And this, for air, is something like 3.1 times 10 26 00:01:58,310 --> 00:02:01,370 to the minus 3 inverse Kelvins. 27 00:02:04,960 --> 00:02:08,400 OK, so basically we have gravity times the change in density. 28 00:02:08,400 --> 00:02:10,350 That's the buoyancy force per volume. 29 00:02:10,350 --> 00:02:14,370 And so we can write that as beta delta T. Or I should say, 30 00:02:14,370 --> 00:02:19,880 delta T is T maximum minus T minimum. 31 00:02:19,880 --> 00:02:21,660 So there's some temperature change. 32 00:02:21,660 --> 00:02:23,250 So the example we're going to consider 33 00:02:23,250 --> 00:02:28,290 is a cold plate above a hot plate with a fluid in between. 34 00:02:28,290 --> 00:02:31,750 And then we have also some other parameters. 35 00:02:31,750 --> 00:02:35,660 So we have-- the length comes in cubed now. 36 00:02:35,660 --> 00:02:39,310 So it's very sensitive to the size of the system. 37 00:02:39,310 --> 00:02:41,250 And then we also have the kinetic viscosity 38 00:02:41,250 --> 00:02:45,370 of air, which I'll write again here. 39 00:02:45,370 --> 00:02:47,350 This also appeared in the Reynolds number. 40 00:02:49,950 --> 00:03:02,620 And for air, this is 1.5 times 10 to the minus 5 meters 41 00:03:02,620 --> 00:03:08,180 squared per second, roughly. 42 00:03:08,180 --> 00:03:13,600 And then finally, we have alpha T, which 43 00:03:13,600 --> 00:03:15,580 is the thermal diffusivity. 44 00:03:20,310 --> 00:03:25,100 So this parameter gives a sense of how quickly heat energy is 45 00:03:25,100 --> 00:03:29,770 transmitted by conduction and diffusion through the fluid. 46 00:03:29,770 --> 00:03:32,900 And thermal diffusivity, turns out, 47 00:03:32,900 --> 00:03:37,370 is pretty close to the kinematic viscosity for a gas, like air. 48 00:03:37,370 --> 00:03:42,070 And the reason is that the ratio of thermal-- 49 00:03:42,070 --> 00:03:46,640 well, in other words, the ratio of the kinematic viscosity 50 00:03:46,640 --> 00:03:54,050 to the thermal diffusivity for air is around 0.7. 51 00:03:54,050 --> 00:03:56,930 And this ratio, by the way, is called Pr-- 52 00:03:56,930 --> 00:03:58,640 the Prandtl number-- which is also 53 00:03:58,640 --> 00:04:02,590 very important in these sorts of flows. 54 00:04:02,590 --> 00:04:07,330 And for gases, basically kinematic viscosity 55 00:04:07,330 --> 00:04:10,060 refers to the diffusion and momentum in the air, 56 00:04:10,060 --> 00:04:12,640 whereas the thermal diffusivity alpha is 57 00:04:12,640 --> 00:04:14,550 the diffusion of heat energy. 58 00:04:14,550 --> 00:04:16,120 And in a gas, both those processes 59 00:04:16,120 --> 00:04:17,690 occur by collisions of molecules. 60 00:04:17,690 --> 00:04:19,450 And since it's the same mechanism, 61 00:04:19,450 --> 00:04:21,279 you have roughly the same order of magnitude 62 00:04:21,279 --> 00:04:22,520 of those quantities. 63 00:04:22,520 --> 00:04:24,190 So basically, all these quantities 64 00:04:24,190 --> 00:04:26,020 enter the Rayleigh number. 65 00:04:26,020 --> 00:04:29,080 And the way to think about, qualitatively, 66 00:04:29,080 --> 00:04:30,670 what the Rayleigh number is telling us 67 00:04:30,670 --> 00:04:42,180 is the ratio of buoyancy force to viscous stress, which 68 00:04:42,180 --> 00:04:44,340 is trying to fight that motion as we talked 69 00:04:44,340 --> 00:04:52,050 about before, but also heat diffusion or thermal diffusion, 70 00:04:52,050 --> 00:04:53,470 which is also kind of fighting it, 71 00:04:53,470 --> 00:04:56,110 because it spreads out the temperature 72 00:04:56,110 --> 00:04:58,960 gradient because this is a motion that is naturally 73 00:04:58,960 --> 00:05:01,090 driven whenever a temperature gradient exists which 74 00:05:01,090 --> 00:05:02,500 is unstable. 75 00:05:02,500 --> 00:05:06,670 So because this beta is typically positive-- 76 00:05:06,670 --> 00:05:09,550 so when you heat the fluid, it expands. 77 00:05:09,550 --> 00:05:12,400 That's certainly the case for most gases 78 00:05:12,400 --> 00:05:14,020 and even for many liquids. 79 00:05:14,020 --> 00:05:17,740 Then what I've sketched here is an unstable density gradient, 80 00:05:17,740 --> 00:05:19,330 where if the cold is above the hot, 81 00:05:19,330 --> 00:05:22,180 there's a heavy fluid above a light fluid. 82 00:05:22,180 --> 00:05:26,500 And at conditions of low Rayleigh number, 83 00:05:26,500 --> 00:05:27,220 this is stable. 84 00:05:27,220 --> 00:05:29,230 And in particular, if the Rayleigh number 85 00:05:29,230 --> 00:05:31,840 is less-- for this particular case of two fixed 86 00:05:31,840 --> 00:05:35,090 plates and an infinite layer of fluid, 87 00:05:35,090 --> 00:05:38,150 if the Rayleigh number is less than 1708, 88 00:05:38,150 --> 00:05:41,880 then we have a stable situation. 89 00:05:41,880 --> 00:05:43,300 Or it can be at least meta-stable. 90 00:05:43,300 --> 00:05:49,450 It won't go spontaneously unstable, so at least local-- 91 00:05:49,450 --> 00:05:52,450 stable to small perturbations. 92 00:05:52,450 --> 00:05:58,170 But then at this critical Rayleigh number of 1708, 93 00:05:58,170 --> 00:06:01,350 we start to get some spontaneous flows, because what's happening 94 00:06:01,350 --> 00:06:02,940 is that the heavy fluid above, which 95 00:06:02,940 --> 00:06:05,580 I've sketched in blue-- the cold fluid-- it wants 96 00:06:05,580 --> 00:06:09,360 to sink to the bottom, whereas the red, warmer 97 00:06:09,360 --> 00:06:12,220 fluid is lighter and wants to rise to the top. 98 00:06:12,220 --> 00:06:13,840 And so it has to find a way to do that. 99 00:06:13,840 --> 00:06:15,630 And eventually, it breaks symmetry and just 100 00:06:15,630 --> 00:06:16,830 starts forming convection. 101 00:06:16,830 --> 00:06:19,080 And that is so-called natural convection. 102 00:06:19,080 --> 00:06:23,790 So you have plumes of hot fluid rising and cold fluid sinking, 103 00:06:23,790 --> 00:06:27,540 driven at first by fairly regular arrays. 104 00:06:27,540 --> 00:06:31,010 But as you increase the Rayleigh number even further, 105 00:06:31,010 --> 00:06:34,560 and if you increase it a lot, then you eventually 106 00:06:34,560 --> 00:06:36,700 get to a complicated turbulent flow. 107 00:06:36,700 --> 00:06:40,290 So if you increase the Rayleigh number on the order of 10 108 00:06:40,290 --> 00:06:42,840 to the 4, you may have some unsteady situations 109 00:06:42,840 --> 00:06:45,150 as we saw with the vortex shedding. 110 00:06:45,150 --> 00:06:47,520 But here, if you go to a very high Reynolds number-- 111 00:06:47,520 --> 00:06:49,290 Rayleigh number, I should say-- 112 00:06:49,290 --> 00:06:55,860 greater than about 10 to the 9, then you again get turbulence. 113 00:06:55,860 --> 00:06:59,090 So simply these temperature variations are strong enough-- 114 00:06:59,090 --> 00:07:02,010 those buoyancy forces-- to completely destabilize 115 00:07:02,010 --> 00:07:05,340 the fluid and generate a turbulent mixture 116 00:07:05,340 --> 00:07:07,370 where the hot and cold are very quickly mixing. 117 00:07:07,370 --> 00:07:09,000 And I've sketched here the hot and cold 118 00:07:09,000 --> 00:07:10,060 as still being separate. 119 00:07:10,060 --> 00:07:13,200 But in fact, due to diffusion, they will kind of also 120 00:07:13,200 --> 00:07:15,720 be re-equilibrating all the time as well, 121 00:07:15,720 --> 00:07:17,700 although the temperature gradient is needed 122 00:07:17,700 --> 00:07:20,820 to kind of maintain that flow. 123 00:07:20,820 --> 00:07:23,340 So let's see how big the Rayleigh number 124 00:07:23,340 --> 00:07:27,030 is in different situations of interest for indoor air now, 125 00:07:27,030 --> 00:07:27,700 as well. 126 00:07:27,700 --> 00:07:31,540 So if we look, let's say, in a room-- 127 00:07:34,780 --> 00:07:39,670 so let's have-- this is another room. 128 00:07:39,670 --> 00:07:43,980 And let's imagine again, we have our heating and ventilation air 129 00:07:43,980 --> 00:07:44,530 conditioning. 130 00:07:44,530 --> 00:07:46,190 Let's say, it's an air conditioning unit 131 00:07:46,190 --> 00:07:49,990 on the top, which is dumping in some cold air. 132 00:07:49,990 --> 00:07:52,760 And it's giving it some velocity. 133 00:07:52,760 --> 00:07:55,659 So the velocity is associated with the Reynolds number. 134 00:07:55,659 --> 00:08:00,220 That inertia will lead to destabilization and vortices 135 00:08:00,220 --> 00:08:01,790 and mixing by itself. 136 00:08:01,790 --> 00:08:04,880 But let's see what the effect is of the temperature difference, 137 00:08:04,880 --> 00:08:05,380 OK? 138 00:08:05,380 --> 00:08:06,540 So we're injecting cold air. 139 00:08:06,540 --> 00:08:08,590 And we usually put it on top, because we actually 140 00:08:08,590 --> 00:08:09,880 want good mixing in the room. 141 00:08:09,880 --> 00:08:11,460 That's how these systems are designed. 142 00:08:11,460 --> 00:08:14,590 We also inject heat normally from below, for example, 143 00:08:14,590 --> 00:08:18,520 from the lower sections of the wall or from the floor. 144 00:08:18,520 --> 00:08:22,430 And so if we were-- and in fact, I could just mention, 145 00:08:22,430 --> 00:08:26,290 if we were to heat, we would do that. 146 00:08:26,290 --> 00:08:28,450 And if we're doing air conditioning or cooling, 147 00:08:28,450 --> 00:08:29,960 we would do it from above. 148 00:08:29,960 --> 00:08:33,730 And so basically, there is an unstable gradient like this. 149 00:08:33,730 --> 00:08:35,600 And let's actually put some numbers in here. 150 00:08:35,600 --> 00:08:38,260 So what if we say that the temperature difference 151 00:08:38,260 --> 00:08:40,539 between the fluid we're injecting, whether it's 152 00:08:40,539 --> 00:08:44,020 heating or cooling, relative to the sort of background 153 00:08:44,020 --> 00:08:46,600 air in the room, which is, let's say, closer 154 00:08:46,600 --> 00:08:48,790 to the target temperature-- 155 00:08:48,790 --> 00:08:50,710 is approaching target temperature-- let's say, 156 00:08:50,710 --> 00:08:52,990 it's only 10 degrees C. So that seems 157 00:08:52,990 --> 00:08:55,790 like not a very big difference. 158 00:08:55,790 --> 00:08:57,830 But then we go back to our height, 159 00:08:57,830 --> 00:08:59,440 which is our length scale. 160 00:08:59,440 --> 00:09:04,240 And we say it's 2.7 meters, just as sort of a standard ceiling 161 00:09:04,240 --> 00:09:05,500 height. 162 00:09:05,500 --> 00:09:09,850 If you plug in the properties of air with these numbers, 163 00:09:09,850 --> 00:09:14,590 the Rayleigh number is actually 10 to the 10. 164 00:09:14,590 --> 00:09:16,450 It's enormous. 165 00:09:16,450 --> 00:09:18,390 So that tells you that if you-- 166 00:09:18,390 --> 00:09:20,730 and that's partly because of the huge scale here, right? 167 00:09:20,730 --> 00:09:22,020 So H comes in cubed. 168 00:09:22,020 --> 00:09:25,680 So if you have 2 or 3 meters of height, that's a lot of height. 169 00:09:25,680 --> 00:09:27,990 And so if you are maintaining that kind of temperature 170 00:09:27,990 --> 00:09:29,670 difference across such a height, you're 171 00:09:29,670 --> 00:09:31,700 going to be generating very serious convection 172 00:09:31,700 --> 00:09:32,370 in that system. 173 00:09:32,370 --> 00:09:34,710 So what's happening is that besides the fact that you're 174 00:09:34,710 --> 00:09:37,650 blowing and generating flows by inertia, 175 00:09:37,650 --> 00:09:43,380 you also have these thermal flows going on, 176 00:09:43,380 --> 00:09:48,000 which can be very, very strong in a system 177 00:09:48,000 --> 00:09:50,400 when there's even just a few degrees of temperature 178 00:09:50,400 --> 00:09:51,990 difference. 179 00:09:51,990 --> 00:09:54,840 You've probably seen dust in the air 180 00:09:54,840 --> 00:09:56,460 near a sunny window, which allows 181 00:09:56,460 --> 00:09:57,560 you to visualize the flow. 182 00:09:57,560 --> 00:09:59,110 And even if nobody's moving-- the air 183 00:09:59,110 --> 00:10:01,200 is fairly still-- you might see plumes 184 00:10:01,200 --> 00:10:03,940 of rising air in one location or sinking in another air. 185 00:10:03,940 --> 00:10:05,490 And if you look closely, those plumes 186 00:10:05,490 --> 00:10:10,890 may actually have very complex convective instabilities 187 00:10:10,890 --> 00:10:15,150 and turbulence even, even when the temperature differences are 188 00:10:15,150 --> 00:10:16,320 not so great. 189 00:10:16,320 --> 00:10:18,100 And in fact, we can see such things. 190 00:10:18,100 --> 00:10:21,300 For example, if we have, let's say, a window-- 191 00:10:21,300 --> 00:10:26,880 and let's say that it's cold outside. 192 00:10:26,880 --> 00:10:28,680 And it's warm inside. 193 00:10:28,680 --> 00:10:30,480 Then just simply that temperature gradient 194 00:10:30,480 --> 00:10:33,210 means that there's a colder air near the surface that 195 00:10:33,210 --> 00:10:34,770 wants to sink. 196 00:10:34,770 --> 00:10:37,500 And so the flow rate is going to look-- 197 00:10:37,500 --> 00:10:40,860 or the flow is going to look something like a boundary layer 198 00:10:40,860 --> 00:10:45,730 flow of fluid that is sort of falling near the surface. 199 00:10:45,730 --> 00:10:47,910 And if the Reynolds number gets high enough, 200 00:10:47,910 --> 00:10:51,270 this flow can actually become itself unstable 201 00:10:51,270 --> 00:10:53,050 as it kind of goes down the surface. 202 00:10:53,050 --> 00:10:56,010 So you can see that these rising or falling 203 00:10:56,010 --> 00:11:00,360 plumes of natural convection near vertical services 204 00:11:00,360 --> 00:11:03,090 that are heated or cooled relative to the environment 205 00:11:03,090 --> 00:11:06,650 can also lead to complex flows. 206 00:11:06,650 --> 00:11:08,700 And actually, a good example of that 207 00:11:08,700 --> 00:11:11,850 is the flow that occurs around a person 208 00:11:11,850 --> 00:11:14,650 just simply due to the temperature. 209 00:11:14,650 --> 00:11:18,860 So if you look very closely at a person-- 210 00:11:18,860 --> 00:11:20,530 I'm not going to draw it very well here. 211 00:11:20,530 --> 00:11:22,710 But let's just say, we have a person. 212 00:11:22,710 --> 00:11:26,340 And that's supposed to be a head, OK? 213 00:11:26,340 --> 00:11:28,650 If you look very closely, the body 214 00:11:28,650 --> 00:11:30,540 has a temperature which is usually higher 215 00:11:30,540 --> 00:11:34,800 than the ambient by at least 10 degrees if not more. 216 00:11:34,800 --> 00:11:37,920 And if you now plug in a little bit smaller size-- let's say, 217 00:11:37,920 --> 00:11:42,270 we plug in 30 or 40 centimeters. 218 00:11:42,270 --> 00:11:45,840 And we-- so let's do that actually. 219 00:11:45,840 --> 00:11:50,100 So if we say that here, maybe, H would be on the order of-- 220 00:11:50,100 --> 00:11:54,010 well, since this was 3 meters, we'll go down to 0.3 meters. 221 00:11:54,010 --> 00:12:00,000 So we'll drop the size roughly by a factor of 10. 222 00:12:00,000 --> 00:12:01,200 But it comes in cubed . 223 00:12:01,200 --> 00:12:04,040 So that drops the Rayleigh number by a factor of 1,000. 224 00:12:04,040 --> 00:12:06,360 So if we still keep our delta T at 10 degrees-- 225 00:12:06,360 --> 00:12:08,430 and it might actually be much more than that-- 226 00:12:08,430 --> 00:12:11,670 the Rayleigh number around this person's head, 227 00:12:11,670 --> 00:12:16,740 just simply by virtue of the heat generated by the body, 228 00:12:16,740 --> 00:12:18,220 can be of order 10 to the 7. 229 00:12:18,220 --> 00:12:20,610 So it may not be quite into the turbulent regime. 230 00:12:20,610 --> 00:12:22,560 But it's certainly in the regime where there'll 231 00:12:22,560 --> 00:12:25,260 be some unsteady complicated flows due only 232 00:12:25,260 --> 00:12:26,200 to natural convection. 233 00:12:26,200 --> 00:12:28,410 And we're not even talking about the person moving, 234 00:12:28,410 --> 00:12:31,030 which gives you even more flow. 235 00:12:31,030 --> 00:12:33,370 And so what it actually looks like if you look closely-- 236 00:12:33,370 --> 00:12:35,220 the air around a person is actually 237 00:12:35,220 --> 00:12:41,090 rising almost like a chimney, driven by these thermal flows. 238 00:12:41,090 --> 00:12:43,970 And those flows even can go turbulent or at least generate 239 00:12:43,970 --> 00:12:46,670 some vortical structures. 240 00:12:46,670 --> 00:12:48,320 And of course, these kinds of flows 241 00:12:48,320 --> 00:12:49,490 are just due to temperature. 242 00:12:49,490 --> 00:12:51,350 In all these cases due to the HVAC 243 00:12:51,350 --> 00:12:52,730 and also due to the person, there 244 00:12:52,730 --> 00:12:54,560 are these convective flows that we've 245 00:12:54,560 --> 00:12:57,590 talked about from inertia, which also contribute to mixing. 246 00:12:57,590 --> 00:13:00,050 So as we'll talk about shortly, we also 247 00:13:00,050 --> 00:13:01,940 know that this person is breathing. 248 00:13:01,940 --> 00:13:04,070 Let's say, they're just breathing through the nose, 249 00:13:04,070 --> 00:13:04,570 even. 250 00:13:04,570 --> 00:13:10,170 Then there's some puffs that are generated. 251 00:13:10,170 --> 00:13:11,920 And you've got the thermal stuff going on. 252 00:13:11,920 --> 00:13:14,210 Also, the air that you're breathing out 253 00:13:14,210 --> 00:13:15,410 is warmer than the ambient. 254 00:13:15,410 --> 00:13:19,170 So it tends to want to rise as well. 255 00:13:19,170 --> 00:13:23,180 So I hope I've convinced you here that-- 256 00:13:23,180 --> 00:13:28,540 I'll write this as, these could be buoyant respiratory jets 257 00:13:28,540 --> 00:13:29,040 and puffs. 258 00:13:33,410 --> 00:13:35,360 So I guess the first part of this section 259 00:13:35,360 --> 00:13:38,090 is just to convince you that the conditions in a room 260 00:13:38,090 --> 00:13:40,610 are such that we have good reason 261 00:13:40,610 --> 00:13:44,030 to believe that there is significant mixing of the air, 262 00:13:44,030 --> 00:13:46,610 either due to inertial effects from movement, 263 00:13:46,610 --> 00:13:51,860 from ventilation air flows, or from thermal effects, 264 00:13:51,860 --> 00:13:53,410 as we have sketched here. 265 00:13:53,410 --> 00:13:55,160 And so at least, that gives us a beginning 266 00:13:55,160 --> 00:13:58,490 of a justification for our assumption 267 00:13:58,490 --> 00:14:01,170 of a well-mixed room. 268 00:14:01,170 --> 00:14:05,850 So in this video from Linden's group at the University 269 00:14:05,850 --> 00:14:10,590 of Cambridge, we can see a visualization of the airflow 270 00:14:10,590 --> 00:14:15,090 around a person who is speaking or just breathing. 271 00:14:15,090 --> 00:14:17,940 And the videos are taken by a differential synthetic 272 00:14:17,940 --> 00:14:20,730 schlieren imaging method, which allows 273 00:14:20,730 --> 00:14:25,890 you to see basically the changes in density in the flow. 274 00:14:25,890 --> 00:14:28,830 And what we see are these thermal plumes 275 00:14:28,830 --> 00:14:33,480 of warm air rising past the body due to the difference 276 00:14:33,480 --> 00:14:36,390 in the body temperature and the ambient air temperature. 277 00:14:36,390 --> 00:14:39,780 And we also see, on top of that, repeated puffs 278 00:14:39,780 --> 00:14:42,960 coming from the breathing, which interact with those plumes 279 00:14:42,960 --> 00:14:48,850 and also themselves have buoyant and turbulent flows therein. 280 00:14:48,850 --> 00:14:52,280 In the next video, we see how different the flows 281 00:14:52,280 --> 00:14:54,050 are when masks are worn. 282 00:14:54,050 --> 00:14:57,830 So we can still see the thermal body plume rising vertically 283 00:14:57,830 --> 00:14:59,480 past the person's face. 284 00:14:59,480 --> 00:15:02,060 But now the mask is preventing the transfer of momentum 285 00:15:02,060 --> 00:15:04,550 to the fluid to push forward these puffs. 286 00:15:04,550 --> 00:15:06,920 And instead, we see the leaking of some 287 00:15:06,920 --> 00:15:11,450 of the breathed, exhaled air rising, almost 288 00:15:11,450 --> 00:15:15,620 entrained in the turbulent thermal plume rising upwards 289 00:15:15,620 --> 00:15:17,970 rather than being ejected forward. 290 00:15:17,970 --> 00:15:21,950 And so this helps to eliminate short-range transmission 291 00:15:21,950 --> 00:15:24,470 due to those puffs and really brings us closer 292 00:15:24,470 --> 00:15:29,840 to an airborne model of a well-mixed room.