1 00:00:10,500 --> 00:00:12,020 PROFESSOR: So we've just discussed 2 00:00:12,020 --> 00:00:16,710 how inertia and buoyancy forces can destabilize the fluid 3 00:00:16,710 --> 00:00:19,980 and lead to complex flows of air in a room. 4 00:00:19,980 --> 00:00:23,430 But now, let's think about how those flows will 5 00:00:23,430 --> 00:00:26,160 influence the transport of aerosol droplets 6 00:00:26,160 --> 00:00:28,750 which transmit disease. 7 00:00:28,750 --> 00:00:31,230 So maybe a simpler way to think about it 8 00:00:31,230 --> 00:00:33,720 is imagine we have some object which 9 00:00:33,720 --> 00:00:37,020 is releasing a substance which could 10 00:00:37,020 --> 00:00:39,190 be particles containing virus. 11 00:00:39,190 --> 00:00:44,760 It could also be heat or other chemical release. 12 00:00:44,760 --> 00:00:47,820 And that's in a flow field similar to the types of flow 13 00:00:47,820 --> 00:00:49,440 fields that we've been discussing. 14 00:00:49,440 --> 00:00:54,630 So that's the problem of forced or natural convection, 15 00:00:54,630 --> 00:00:57,340 convective transport of particles. 16 00:00:57,340 --> 00:01:00,290 So here I show the problem of flow past a cylinder. 17 00:01:00,290 --> 00:01:02,610 And let's say the cylinder is a hot cylinder, which 18 00:01:02,610 --> 00:01:05,160 is releasing heat into the fluid which I've kind of sketched 19 00:01:05,160 --> 00:01:07,400 by this red region here. 20 00:01:07,400 --> 00:01:11,250 Now if the flow is fast compared to the diffusion of heat, 21 00:01:11,250 --> 00:01:12,840 you would expect a situation like this 22 00:01:12,840 --> 00:01:15,200 where there's a thin boundary layer of heat 23 00:01:15,200 --> 00:01:17,370 transfer along the front end of the sphere. 24 00:01:17,370 --> 00:01:18,930 But on the trailing end, there might 25 00:01:18,930 --> 00:01:22,080 be a wake of kind of hot fluid that's 26 00:01:22,080 --> 00:01:24,390 been kind of carried away. 27 00:01:24,390 --> 00:01:26,190 And that is indeed what happens when 28 00:01:26,190 --> 00:01:31,060 we're in the regime of dominance of convection over diffusion. 29 00:01:31,060 --> 00:01:34,560 And there is a dimensionless which describes that, 30 00:01:34,560 --> 00:01:38,039 which is the Peclet number. 31 00:01:38,039 --> 00:01:43,140 So the Peclet number is defined as the velocity times a length 32 00:01:43,140 --> 00:01:47,960 scale divided by D where D that is the mass diffusivity. 33 00:01:47,960 --> 00:01:51,640 Now normally, D is the molecular diffusivity. 34 00:01:56,250 --> 00:01:58,950 Or it's the diffusivity or let's say the droplets 35 00:01:58,950 --> 00:02:01,380 or whatever the individual particles are. 36 00:02:01,380 --> 00:02:08,530 Now, if you want to think about diffusion of gas-- 37 00:02:08,530 --> 00:02:11,820 so if we have for gas molecules-- for example, 38 00:02:11,820 --> 00:02:15,270 like for CO2 or oxygen in the air, 39 00:02:15,270 --> 00:02:18,180 then the ratio of kinematic viscosity 40 00:02:18,180 --> 00:02:21,450 to diffusivity of the gas molecules, 41 00:02:21,450 --> 00:02:23,160 that's called the Schmidt number, 42 00:02:23,160 --> 00:02:26,550 is also around 0.7 for air. 43 00:02:26,550 --> 00:02:28,170 So basically molecules of momentum 44 00:02:28,170 --> 00:02:30,790 are diffusing at about the same rate. 45 00:02:30,790 --> 00:02:35,020 And so that tells us that for the molecules of the gas, 46 00:02:35,020 --> 00:02:38,590 the Peclet number is actually the same as the Reynolds number 47 00:02:38,590 --> 00:02:40,380 and, in fact, is very large. 48 00:02:40,380 --> 00:02:49,200 If we have a larger object such as maybe a droplet, 49 00:02:49,200 --> 00:02:52,010 we have, of course, much lower diffusivity than the air 50 00:02:52,010 --> 00:02:52,760 molecules. 51 00:02:52,760 --> 00:02:55,310 And we're not talking and it's a different process 52 00:02:55,310 --> 00:02:56,420 on a collisional. 53 00:02:56,420 --> 00:03:02,200 But we're still dealing with very large Peclet numbers 54 00:03:02,200 --> 00:03:04,120 which will lead to kind of wakes, 55 00:03:04,120 --> 00:03:06,050 as I've just described here. 56 00:03:06,050 --> 00:03:09,190 So it depends on the details and the size of the particles. 57 00:03:09,190 --> 00:03:11,110 But for the gas, I just want to write here 58 00:03:11,110 --> 00:03:14,200 that Peclet number is on the same order as the Reynolds 59 00:03:14,200 --> 00:03:19,140 number and hence is very much larger than 1 in many cases. 60 00:03:19,140 --> 00:03:22,060 Now the thing is that because the Peclet number is 1, 61 00:03:22,060 --> 00:03:24,300 the Peclet number is measuring the importance 62 00:03:24,300 --> 00:03:29,110 of convective transport to diffusion. 63 00:03:31,730 --> 00:03:34,840 And so because the Peclet number is typically large, 64 00:03:34,840 --> 00:03:36,910 we have a dominance of convection 65 00:03:36,910 --> 00:03:37,900 over diffusive process. 66 00:03:37,900 --> 00:03:39,280 So think of our aerosol droplets. 67 00:03:39,280 --> 00:03:41,600 They do diffuse in the air. 68 00:03:41,600 --> 00:03:43,900 And we can calculate that with the same Stokes-Einstein 69 00:03:43,900 --> 00:03:46,600 formula that we've used earlier. 70 00:03:46,600 --> 00:03:48,850 But that diffusion rate is very slow 71 00:03:48,850 --> 00:03:52,570 compared to the sort of convective processes 72 00:03:52,570 --> 00:03:54,110 that are occurring in the room. 73 00:03:54,110 --> 00:03:56,050 And so we typically have a high Peclet number 74 00:03:56,050 --> 00:03:58,230 and we see this kind of behavior. 75 00:03:58,230 --> 00:04:00,730 On the other hand, we're also in the regime of high Reynolds 76 00:04:00,730 --> 00:04:01,600 number. 77 00:04:01,600 --> 00:04:05,350 And that really changes things because then the diffusion 78 00:04:05,350 --> 00:04:08,230 due to sort of random fluctuations and collisions 79 00:04:08,230 --> 00:04:12,580 with other molecules is overwhelmed by the sort 80 00:04:12,580 --> 00:04:15,040 of transport and diffusion light transport 81 00:04:15,040 --> 00:04:19,360 that follows from vortices and turbulence. 82 00:04:19,360 --> 00:04:21,160 So if I take that same cylinder and I 83 00:04:21,160 --> 00:04:23,600 at now at a higher Reynolds number, 84 00:04:23,600 --> 00:04:26,890 then we can see that the wake is not a thin little tail that 85 00:04:26,890 --> 00:04:28,390 extends downstream but it's really 86 00:04:28,390 --> 00:04:32,560 a turbulent wake where the warm fluid in this case 87 00:04:32,560 --> 00:04:34,870 is dispersed everywhere throughout that wake. 88 00:04:34,870 --> 00:04:37,980 In fact, it's fairly uniformly mixed. 89 00:04:37,980 --> 00:04:41,370 A similar situation occurs in the case of breathing, 90 00:04:41,370 --> 00:04:44,250 coughing, sneezing, and other forms of respiration, which 91 00:04:44,250 --> 00:04:46,590 we will come to shortly, where we 92 00:04:46,590 --> 00:04:50,850 have a relatively high Reynolds number flow, often turbulent, 93 00:04:50,850 --> 00:04:54,120 and we are injecting in it some respiratory aerosol particles. 94 00:04:54,120 --> 00:04:58,020 And they're quite well mixed across that jet. 95 00:04:58,020 --> 00:05:00,540 So we can see there's a very strong coupling 96 00:05:00,540 --> 00:05:03,190 between the fluid flow that we've just discussed, 97 00:05:03,190 --> 00:05:07,080 which is often turbulent and containing vortices and eddies, 98 00:05:07,080 --> 00:05:12,090 and the transport of suspended particles and droplets. 99 00:05:12,090 --> 00:05:14,380 So how can we think about this problem? 100 00:05:14,380 --> 00:05:16,920 So the important thing is when we 101 00:05:16,920 --> 00:05:19,080 get into this turbulent regime, the way 102 00:05:19,080 --> 00:05:21,490 an individual particle-- 103 00:05:21,490 --> 00:05:23,490 imagine-- I should think even here in this case. 104 00:05:23,490 --> 00:05:25,410 How does an individual particle move? 105 00:05:25,410 --> 00:05:26,730 It kind of follows the flow. 106 00:05:26,730 --> 00:05:28,530 So it goes first through a little vortices 107 00:05:28,530 --> 00:05:30,090 occasionally around a big vortex, 108 00:05:30,090 --> 00:05:31,930 and then it does some little ones. 109 00:05:31,930 --> 00:05:33,570 And so it's also doing a random walk 110 00:05:33,570 --> 00:05:37,650 but it's one that's driven by the turbulent flow itself. 111 00:05:37,650 --> 00:05:40,110 And the length scale for the sort 112 00:05:40,110 --> 00:05:44,400 of steps of the random walk is actually the vortex size. 113 00:05:44,400 --> 00:05:45,900 And in particular, it's dominated 114 00:05:45,900 --> 00:05:47,430 by the largest vortex. 115 00:05:47,430 --> 00:05:49,380 So whatever the flow is, there's always 116 00:05:49,380 --> 00:05:53,040 a certain scale which sets the size of the largest vortex. 117 00:05:53,040 --> 00:05:59,750 And so that leads to the concept of so-called eddy diffusivity 118 00:05:59,750 --> 00:06:06,190 in turbulent flows, which is also important in air flows. 119 00:06:06,190 --> 00:06:10,960 So the eddy diffusivity is an effective diffusion-like 120 00:06:10,960 --> 00:06:13,750 parameter that describes the mixing and spreading 121 00:06:13,750 --> 00:06:18,730 of suspended particles or droplets in a flow field that 122 00:06:18,730 --> 00:06:20,290 is turbulent. 123 00:06:20,290 --> 00:06:22,970 And in that case, we can write it two ways. 124 00:06:22,970 --> 00:06:27,490 So either we have an imposed velocity u 125 00:06:27,490 --> 00:06:36,630 and we have length scale l, and then our eddy viscosity 126 00:06:36,630 --> 00:06:40,290 might be written as u times l. 127 00:06:40,290 --> 00:06:45,940 So u is distance per time. 128 00:06:45,940 --> 00:06:46,570 l is distance. 129 00:06:46,570 --> 00:06:47,990 So it's distance squared per time. 130 00:06:47,990 --> 00:06:50,460 So that has units of viscosity. 131 00:06:50,460 --> 00:06:52,210 And the way to interpret this is basically 132 00:06:52,210 --> 00:06:55,720 sort of swirling around eddies of a size l 133 00:06:55,720 --> 00:06:57,880 and a characteristic velocity u. 134 00:06:57,880 --> 00:07:02,680 Another way to write this would be that it's l 135 00:07:02,680 --> 00:07:05,740 squared over 2 times a timescale. 136 00:07:05,740 --> 00:07:07,910 Because we often write diffusion, 137 00:07:07,910 --> 00:07:13,260 if there's a time step tau, and a length scale 138 00:07:13,260 --> 00:07:15,520 l for a given step, then l squared over 2 tau 139 00:07:15,520 --> 00:07:16,570 is the diffusivity. 140 00:07:16,570 --> 00:07:19,220 That's how we think about molecular diffusivity as well. 141 00:07:19,220 --> 00:07:21,760 But the question is, what is this timescale. 142 00:07:21,760 --> 00:07:28,720 So we can see here that the timescale is l over u. 143 00:07:34,340 --> 00:07:36,060 So it's a convective times scale. 144 00:07:36,060 --> 00:07:39,820 It's the time to, essentially, go around one of those eddies. 145 00:07:39,820 --> 00:07:43,730 And I'm writing these really just as scaling arguments here. 146 00:07:43,730 --> 00:07:45,840 So if we look at these flows, what 147 00:07:45,840 --> 00:07:47,140 are the relevant length scales? 148 00:07:47,140 --> 00:07:50,310 So at the beginning of this flow here, 149 00:07:50,310 --> 00:07:52,150 it's the length scale is that of the object. 150 00:07:52,150 --> 00:07:54,060 In the case of the breathing, the length scale 151 00:07:54,060 --> 00:07:56,020 is initially that of the mouth opening. 152 00:07:56,020 --> 00:07:59,230 But then we form these turbulent structures that expand. 153 00:07:59,230 --> 00:08:00,730 And a constant we will return to you 154 00:08:00,730 --> 00:08:02,890 shortly is that the relevant length 155 00:08:02,890 --> 00:08:07,780 scale as you continue here is actually something 156 00:08:07,780 --> 00:08:10,440 which depends on position. 157 00:08:10,440 --> 00:08:11,810 So as this thing grows the eddies 158 00:08:11,810 --> 00:08:13,490 are are getting bigger and bigger, 159 00:08:13,490 --> 00:08:16,850 and so also is faster and faster the transport by diffusion, 160 00:08:16,850 --> 00:08:20,020 which sort of maintains a fairly uniform concentration 161 00:08:20,020 --> 00:08:23,350 across that space. 162 00:08:23,350 --> 00:08:25,790 And that brings us then to what happens in the whole room. 163 00:08:25,790 --> 00:08:29,230 So we've been very interested in mixing in the whole room. 164 00:08:29,230 --> 00:08:32,570 And so a natural picture here is to say, well, 165 00:08:32,570 --> 00:08:39,030 if the room has a height H, then the eddy diffusivity, 166 00:08:39,030 --> 00:08:41,250 which aerosol particles in the room 167 00:08:41,250 --> 00:08:44,540 are feeling as long as it's a fairly well mixed turbulent, 168 00:08:44,540 --> 00:08:51,270 more isotropic flow, could be described as the height squared 169 00:08:51,270 --> 00:08:53,660 over 2 times a timescale. 170 00:08:53,660 --> 00:09:00,540 And the timescale should be that of the effective air 171 00:09:00,540 --> 00:09:04,530 change or the total air change time, basically. 172 00:09:04,530 --> 00:09:07,260 So this here, again, I've written 173 00:09:07,260 --> 00:09:10,830 lambda bar a is the outer airflow 174 00:09:10,830 --> 00:09:13,260 plus the re-circulation airflow, which 175 00:09:13,260 --> 00:09:17,160 may be going through a filter, and divided by the total volume 176 00:09:17,160 --> 00:09:18,030 of the room. 177 00:09:18,030 --> 00:09:24,320 So this is the total ACH. 178 00:09:24,320 --> 00:09:29,560 And so what we see here then is that the formula is roughly 179 00:09:29,560 --> 00:09:35,070 that we should have 1/2 H squared lambda a bar. 180 00:09:38,540 --> 00:09:41,650 And so this is a very simple argument based on the largest 181 00:09:41,650 --> 00:09:44,120 eddy is going to be, in a well mixed room, 182 00:09:44,120 --> 00:09:47,090 at the size of the scale of the room. 183 00:09:47,090 --> 00:09:49,730 And sure enough, this relationship 184 00:09:49,730 --> 00:09:54,740 has actually been verified for houses and actual indoor rooms 185 00:09:54,740 --> 00:09:57,320 with all the furniture in them, where 186 00:09:57,320 --> 00:09:59,180 it's been shown that if you release 187 00:09:59,180 --> 00:10:01,450 a passive tracer such as carbon dioxide in the middle 188 00:10:01,450 --> 00:10:03,620 of the room and you have the ventilation 189 00:10:03,620 --> 00:10:11,690 on at a certain ACH, Air Change per Hour, lambda a bar, then 190 00:10:11,690 --> 00:10:13,430 this formula, even with the 1/2 actually, 191 00:10:13,430 --> 00:10:16,610 turns out to be a pretty good approximation for the spreading 192 00:10:16,610 --> 00:10:18,650 in that room as you change the size of the room 193 00:10:18,650 --> 00:10:22,100 and look at different rooms and also look at different air 194 00:10:22,100 --> 00:10:23,440 change rates. 195 00:10:23,440 --> 00:10:25,820 So that's, I think, a good starting point for us as well. 196 00:10:25,820 --> 00:10:27,440 At least when the room is well mixed, 197 00:10:27,440 --> 00:10:30,470 this is a good way to think about transport in the room. 198 00:10:30,470 --> 00:10:34,970 Also, we see from this picture that the timescale for mixing 199 00:10:34,970 --> 00:10:40,320 is the inverse of the air change time. 200 00:10:40,320 --> 00:10:46,100 So the mixing time is also comparable to the residence 201 00:10:46,100 --> 00:10:51,180 time of the air, including re-circulation. 202 00:10:51,180 --> 00:10:52,640 So basically it's the time it takes 203 00:10:52,640 --> 00:10:54,980 for air to typically go through this system 204 00:10:54,980 --> 00:10:59,360 is also the time it takes to fully mix the system, roughly 205 00:10:59,360 --> 00:11:01,010 the same order of magnitude. 206 00:11:01,010 --> 00:11:05,170 And that is the characteristic of a well mixed turbulent room 207 00:11:05,170 --> 00:11:06,920 which could occur by any of the mechanisms 208 00:11:06,920 --> 00:11:08,960 we've just been describing. 209 00:11:08,960 --> 00:11:12,770 Although the same principles also apply to jets or strong, 210 00:11:12,770 --> 00:11:15,650 maybe ventilation flows past an object where you might still 211 00:11:15,650 --> 00:11:17,930 have some heterogeneities as I've sketched here, 212 00:11:17,930 --> 00:11:21,380 that we will need to consider. 213 00:11:21,380 --> 00:11:23,720 The last point I'd like to come back to 214 00:11:23,720 --> 00:11:27,890 also is the question of sedimentation. 215 00:11:27,890 --> 00:11:29,360 So you may have found it surprising 216 00:11:29,360 --> 00:11:32,300 that we describe the flux of sedimenting particles 217 00:11:32,300 --> 00:11:34,040 to the ground in a very simple way 218 00:11:34,040 --> 00:11:38,810 by just using the Stokes velocity, vs, 219 00:11:38,810 --> 00:11:41,280 and multiplying by the area. 220 00:11:41,280 --> 00:11:49,490 So we said the droplet flux out of the room 221 00:11:49,490 --> 00:11:53,300 was just the sedimentation velocity of the droplet, which 222 00:11:53,300 --> 00:12:01,080 was radius dependent, times the area, the floor surface area. 223 00:12:01,080 --> 00:12:03,000 And what's a bit confusing about that at first 224 00:12:03,000 --> 00:12:05,190 is that we know the flows are very complex in the room. 225 00:12:05,190 --> 00:12:07,190 In fact, if you look at dust particles in a room 226 00:12:07,190 --> 00:12:09,510 as we discussed earlier in some cases, for sure, 227 00:12:09,510 --> 00:12:11,920 you will see them actually rising and not settling. 228 00:12:11,920 --> 00:12:14,110 So they may be settling relative to the flow. 229 00:12:14,110 --> 00:12:16,920 But the flow is actually convecting them upwards. 230 00:12:16,920 --> 00:12:18,420 And that's here, for example, if you 231 00:12:18,420 --> 00:12:20,640 look at there's a dominant role in the flow 232 00:12:20,640 --> 00:12:21,990 that I've sketched here. 233 00:12:21,990 --> 00:12:24,000 The particles over here actually might 234 00:12:24,000 --> 00:12:25,760 have a net velocity going up. 235 00:12:25,760 --> 00:12:28,050 And over here, they're going down. 236 00:12:28,050 --> 00:12:31,290 But if you decompose that velocity field, then 237 00:12:31,290 --> 00:12:38,320 in some cases, they're going up due to flow or convection, 238 00:12:38,320 --> 00:12:41,890 while they're sedimenting at the rate vs. 239 00:12:41,890 --> 00:12:43,930 But then necessarily, because the fluid 240 00:12:43,930 --> 00:12:47,960 is approximately incompressible and is returning somewhere-- 241 00:12:47,960 --> 00:12:50,650 wherever it goes up, somewhere else it's coming down, 242 00:12:50,650 --> 00:12:53,800 we find that in other areas you have vs still 243 00:12:53,800 --> 00:12:57,510 pointing down with the same rate and now the flow is going down. 244 00:12:57,510 --> 00:12:58,930 And if you imagine a particle that 245 00:12:58,930 --> 00:13:00,930 is sampling all the different velocity vectors, 246 00:13:00,930 --> 00:13:02,560 the blue vectors, sometimes they're up, 247 00:13:02,560 --> 00:13:04,570 sometimes they're down, on average 248 00:13:04,570 --> 00:13:07,240 the blue velocity vectors of the flow have to average to 0. 249 00:13:07,240 --> 00:13:09,880 Because there's no-- or at least near 0. 250 00:13:09,880 --> 00:13:11,860 There's not, let's say, a very strong 251 00:13:11,860 --> 00:13:14,620 vertical relative motion. 252 00:13:14,620 --> 00:13:16,870 Then it's reasonable to assume that the particles will 253 00:13:16,870 --> 00:13:21,160 sediment out of a well mixed turbulent flow at a rate given 254 00:13:21,160 --> 00:13:22,090 by vsa. 255 00:13:22,090 --> 00:13:23,890 And that is something that has also 256 00:13:23,890 --> 00:13:27,880 been validated experimentally for well mixed chambers 257 00:13:27,880 --> 00:13:29,550 and rooms.