WEBVTT 00:00:10.430 --> 00:00:12.980 PROFESSOR: So now, let's start talking about airborne disease 00:00:12.980 --> 00:00:17.050 transmission, thinking especially of viral diseases 00:00:17.050 --> 00:00:19.330 where we expect the transmission to occur 00:00:19.330 --> 00:00:22.510 through aerosol droplets or at least smaller droplets, 00:00:22.510 --> 00:00:24.880 which may sediment but are going to be 00:00:24.880 --> 00:00:28.820 suspended in the air for a significant amount of time. 00:00:28.820 --> 00:00:31.300 So the first approximation of such a situation 00:00:31.300 --> 00:00:34.750 is to assume a well-mixed room, meaning 00:00:34.750 --> 00:00:36.910 that the air in the room is well mixed. 00:00:36.910 --> 00:00:42.750 And even if there is, let's say, one person who's infected 00:00:42.750 --> 00:00:49.010 and that infected person is breathing, talking, singing, 00:00:49.010 --> 00:00:54.500 respiring, and exhaling infected aerosol droplets, 00:00:54.500 --> 00:00:57.580 then these droplets start to spread around the room. 00:01:01.650 --> 00:01:04.099 And they do so because the room is-- 00:01:04.099 --> 00:01:05.860 and we assume that the room is well mixed. 00:01:05.860 --> 00:01:06.990 So even though, as I've sketched, 00:01:06.990 --> 00:01:09.150 there's certainly going to be a higher concentration 00:01:09.150 --> 00:01:12.330 of the droplets near the infected person, that 00:01:12.330 --> 00:01:15.260 is there are significant airflows in the room, which 00:01:15.260 --> 00:01:16.800 induce mixing. 00:01:16.800 --> 00:01:19.620 That is partly driven by the flow of fresh air 00:01:19.620 --> 00:01:20.820 through the room. 00:01:20.820 --> 00:01:23.730 Typically, we may have some forced ventilation 00:01:23.730 --> 00:01:25.289 coming through ducts with fans. 00:01:25.289 --> 00:01:26.820 There could be open windows. 00:01:26.820 --> 00:01:32.390 And even when a room is not open, 00:01:32.390 --> 00:01:34.350 meaning that the windows are closed, maybe even 00:01:34.350 --> 00:01:35.759 the door is closed, there's always 00:01:35.759 --> 00:01:38.610 some leakage and exchange of air with the outside, which 00:01:38.610 --> 00:01:40.680 typically happens on the order of hours. 00:01:40.680 --> 00:01:43.200 So there's always at least some kind of flow rate. 00:01:43.200 --> 00:01:45.539 Also, there's a movement of people in the room, 00:01:45.539 --> 00:01:47.850 and there's also breathing itself, which basically 00:01:47.850 --> 00:01:51.120 imparts momentum to the fluid and causes 00:01:51.120 --> 00:01:53.289 swirling motions of the fluid. 00:01:53.289 --> 00:01:57.270 So the situation actually is more like this 00:01:57.270 --> 00:02:01.080 where there's these flows that are generated either 00:02:01.080 --> 00:02:05.820 by the ventilation, by the movement 00:02:05.820 --> 00:02:09.280 and respiration of the people, also by thermal flows. 00:02:09.280 --> 00:02:11.940 So when you're breathing, the air coming out of your lungs 00:02:11.940 --> 00:02:14.220 is warmer, and that tends to rise. 00:02:14.220 --> 00:02:17.130 But, also, if it's very humid air and very warm air, 00:02:17.130 --> 00:02:20.370 then actually the weight of all the humidity and droplets 00:02:20.370 --> 00:02:21.750 that come with air could actually 00:02:21.750 --> 00:02:23.530 cause some settling as well. 00:02:23.530 --> 00:02:25.950 So you have all these different processes going on. 00:02:25.950 --> 00:02:27.990 And your first approximation, just 00:02:27.990 --> 00:02:32.110 assume that those lead to a well-mixed situation. 00:02:32.110 --> 00:02:34.829 And so we will proceed to analyze disease transmission 00:02:34.829 --> 00:02:36.390 from that assumption, and then we'll 00:02:36.390 --> 00:02:39.570 come back at the end to consider what 00:02:39.570 --> 00:02:41.640 would happen if there are actually fluctuations 00:02:41.640 --> 00:02:44.730 and think about the distance from an infected person 00:02:44.730 --> 00:02:47.190 and how that might play a role in departures 00:02:47.190 --> 00:02:49.540 from the predictions of a well mixed room. 00:02:49.540 --> 00:02:50.370 OK. 00:02:50.370 --> 00:02:53.250 So defining some variables, then. 00:02:53.250 --> 00:02:57.270 We'll let C of t be the, basically, 00:02:57.270 --> 00:02:59.550 infectiousness of the air, if you will. 00:02:59.550 --> 00:03:01.800 But more specifically, if we think of virus, 00:03:01.800 --> 00:03:07.960 it'll be the virions per air volume. 00:03:07.960 --> 00:03:09.340 So they're contained in droplets. 00:03:09.340 --> 00:03:11.640 And later we will consider what size droplets 00:03:11.640 --> 00:03:13.480 they may be contained in, but for now, let's 00:03:13.480 --> 00:03:15.310 just average over everything and say there's 00:03:15.310 --> 00:03:18.980 some concentration in the air. 00:03:18.980 --> 00:03:24.290 We have a production rate of these infectious droplets, 00:03:24.290 --> 00:03:31.720 which can be broken down into many terms. 00:03:31.720 --> 00:03:33.510 So this is the production rate, so this 00:03:33.510 --> 00:03:38.500 is the number of virions per time that are produced. 00:03:38.500 --> 00:03:43.610 So that would be Qb is our breathing flow rate. 00:03:50.870 --> 00:03:54.070 So that's basically the volume of exhaled air per breath 00:03:54.070 --> 00:03:57.700 cycle, so that's how much air is being pushed out 00:03:57.700 --> 00:03:58.630 in your breathing. 00:03:58.630 --> 00:04:00.490 And this is typically-- 00:04:00.490 --> 00:04:07.790 ranges from 0.5 up to around 3 meters cubed per hour. 00:04:07.790 --> 00:04:11.980 So 0.5 is a typical resting breathing rate. 00:04:11.980 --> 00:04:13.990 And, in fact, even if you're just 00:04:13.990 --> 00:04:17.089 kind of calmly speaking and sitting in a room breathing 00:04:17.089 --> 00:04:19.510 through your nose, that'll be your typical breathing rate. 00:04:19.510 --> 00:04:22.430 But if you start exercising and start exerting yourself, 00:04:22.430 --> 00:04:25.020 then that could go up to maybe around 3. 00:04:25.020 --> 00:04:28.200 And so that's the typical range of the breathing rate. 00:04:28.200 --> 00:04:33.770 The next thing we need is Nd is the droplet concentration, 00:04:33.770 --> 00:04:39.060 so this is the number of drops per air volume. 00:04:39.060 --> 00:04:41.070 So that's sort of the number density of drops. 00:04:41.070 --> 00:04:44.200 So if I have various visualization techniques, 00:04:44.200 --> 00:04:46.620 I can actually see all the droplets, and I can count them. 00:04:46.620 --> 00:04:50.070 I can say how many droplets are in a given volume. 00:04:50.070 --> 00:04:51.990 The next thing is I need the volume of a drop. 00:04:56.270 --> 00:04:57.990 So as I mentioned, of course there 00:04:57.990 --> 00:04:59.540 are different drop volumes, and we'll 00:04:59.540 --> 00:05:03.200 come back to assessing the effects of a droplet size 00:05:03.200 --> 00:05:04.310 distribution. 00:05:04.310 --> 00:05:06.230 But for simplicity, why don't we just take 00:05:06.230 --> 00:05:09.200 these sort of average size drops that we discussed 00:05:09.200 --> 00:05:10.700 coming from respiration, 00:05:10.700 --> 00:05:13.970 And that might be a number around 1 micron in size. 00:05:13.970 --> 00:05:16.790 And there's a volume that corresponds with that, which, 00:05:16.790 --> 00:05:20.840 of course, is the 4/3 pi r cubed where 00:05:20.840 --> 00:05:25.340 we assume for now only one size r for all drops just 00:05:25.340 --> 00:05:27.570 for the moment. 00:05:27.570 --> 00:05:29.930 So now we have the volume here, so now 00:05:29.930 --> 00:05:37.270 we sort of what is the total amount of drops. 00:05:37.270 --> 00:05:40.130 So the next thing we need now is Cv. 00:05:40.130 --> 00:05:47.020 Cv will be the number of virions per liquid volume or per drop 00:05:47.020 --> 00:05:49.650 volume, so there's certain-- so if I could take the pure liquid 00:05:49.650 --> 00:05:51.810 in the drop-- let's say it's mucus-- 00:05:51.810 --> 00:05:53.730 and it's been coming out of your pharynx 00:05:53.730 --> 00:05:56.860 and has sort of fragmented and taken some virions with it, 00:05:56.860 --> 00:05:59.010 then that would be the viral load, essentially. 00:05:59.010 --> 00:06:03.670 Sometimes that's another word for this. 00:06:03.670 --> 00:06:06.800 And the viral load varies with time. 00:06:06.800 --> 00:06:09.820 So when you first get infected, at first the viral load 00:06:09.820 --> 00:06:12.550 is very low in the fluids that you're breathing out. 00:06:12.550 --> 00:06:14.080 And then that raises up. 00:06:14.080 --> 00:06:16.400 And during the period when you're most infectious, 00:06:16.400 --> 00:06:21.820 which for COVID-19 and SARS-CoV-2 virus, 00:06:21.820 --> 00:06:25.600 that time ends up being around a week or so when you-- 00:06:25.600 --> 00:06:31.060 well, within a few days, you reach the peak infectiousness. 00:06:31.060 --> 00:06:34.510 And then when you're at the peak viral load, 00:06:34.510 --> 00:06:39.100 this ends up being about 10 to the ninth virions 00:06:39.100 --> 00:06:42.250 per milliliter of fluid when you're 00:06:42.250 --> 00:06:48.340 at your peak infectiousness for SARS-CoV-2. 00:06:52.110 --> 00:06:52.750 OK. 00:06:52.750 --> 00:06:54.700 So just to give you a sense, but of course, a lot of times 00:06:54.700 --> 00:06:55.659 it might be less than that. 00:06:55.659 --> 00:06:57.280 But if you're very infected individual, 00:06:57.280 --> 00:06:58.450 that's kind of a worst case. 00:06:58.450 --> 00:07:00.410 And we're going to be interested in calculating 00:07:00.410 --> 00:07:02.630 safety guidelines and probabilities of transmission, 00:07:02.630 --> 00:07:04.290 so to be conservative, it's good to have 00:07:04.290 --> 00:07:07.700 an idea of how big this number can actually be. 00:07:07.700 --> 00:07:09.370 So what we just calculate here-- so, 00:07:09.370 --> 00:07:12.570 basically, this Nd, Vd is the amount of drop 00:07:12.570 --> 00:07:15.460 or liquid volume per air volume, so that's essentially 00:07:15.460 --> 00:07:18.070 the volume fraction of liquid. 00:07:18.070 --> 00:07:20.530 When we go times Cv, we're essentially 00:07:20.530 --> 00:07:26.140 getting the number of virions in the air per volume of air. 00:07:26.140 --> 00:07:28.580 And then Qv is the volume per time, 00:07:28.580 --> 00:07:30.730 so this is basically virions per time. 00:07:35.590 --> 00:07:37.510 And then one other factor that we should also 00:07:37.510 --> 00:07:40.120 consider is what if we're actually 00:07:40.120 --> 00:07:42.930 filtering those droplets right at the source? 00:07:42.930 --> 00:07:44.980 And that'll be the case if you're wearing a mask. 00:07:44.980 --> 00:07:48.340 So this is an important quantity we'll come back to. 00:07:48.340 --> 00:07:55.380 This will be the mask penetration probability 00:07:55.380 --> 00:07:56.100 for a droplet. 00:07:56.100 --> 00:07:59.080 So this is-- of course is size dependent, 00:07:59.080 --> 00:08:00.250 but we'll come back to that. 00:08:00.250 --> 00:08:01.930 But for the moment, we're just saying it's one drop size. 00:08:01.930 --> 00:08:04.220 And so for the size of drops of interest, we're asking, 00:08:04.220 --> 00:08:05.670 do they go through the mask? 00:08:05.670 --> 00:08:12.030 So 1 minus Pm is also called the filtration efficiency. 00:08:17.230 --> 00:08:18.520 Yes. 00:08:18.520 --> 00:08:21.590 So a very good mask might be 99% of droplets are filtered. 00:08:21.590 --> 00:08:24.190 A very poor cloth covering might be 00:08:24.190 --> 00:08:26.440 10% of droplets are filtered, and we'll come back 00:08:26.440 --> 00:08:28.190 to that in just a moment. 00:08:28.190 --> 00:08:31.630 So this here is our production rate capital P. 00:08:31.630 --> 00:08:34.390 And already with the variables that we've written down here, 00:08:34.390 --> 00:08:42.510 we can write down a mass balance for the virions. 00:08:42.510 --> 00:08:44.200 So virions are being produced. 00:08:44.200 --> 00:08:45.450 They end up in these droplets. 00:08:45.450 --> 00:08:49.200 The droplets are being swept out of the room at a flow rate Q, 00:08:49.200 --> 00:08:52.120 and the room has a volume V. So if I write down 00:08:52.120 --> 00:08:54.420 just the conservation of mass, making sure that I'm not 00:08:54.420 --> 00:08:56.370 losing any virus yet-- 00:08:56.370 --> 00:08:58.200 I'm not allowing them to stick to the walls 00:08:58.200 --> 00:08:59.880 or do anything else just yet but just 00:08:59.880 --> 00:09:03.400 looking at the mass balance of one infected person breathing 00:09:03.400 --> 00:09:03.900 out. 00:09:03.900 --> 00:09:08.040 This is, I should say, production rate per infector, 00:09:08.040 --> 00:09:09.520 or an infected person. 00:09:09.520 --> 00:09:11.610 So if there are more infected people, 00:09:11.610 --> 00:09:14.420 then you'll have this production rate for each person. 00:09:14.420 --> 00:09:16.630 They also might be at different stage of the disease, 00:09:16.630 --> 00:09:18.290 so maybe the viral load will be a little 00:09:18.290 --> 00:09:19.650 different for each person. 00:09:19.650 --> 00:09:21.720 But let's not worry about such details right now. 00:09:21.720 --> 00:09:24.030 We want to keep things general. 00:09:24.030 --> 00:09:25.620 So for the mass balance, we write down 00:09:25.620 --> 00:09:29.790 the total number of virions in the room 00:09:29.790 --> 00:09:31.180 and how that changes in time. 00:09:31.180 --> 00:09:34.250 So that'll be the concentration per room air volume, which 00:09:34.250 --> 00:09:36.420 is well-mixed times the volume of the room per time. 00:09:36.420 --> 00:09:40.250 So this is the change in the number of virions per time, 00:09:40.250 --> 00:09:42.220 and that can change in two ways. 00:09:42.220 --> 00:09:45.090 One, we have the production P. So for every infected person, 00:09:45.090 --> 00:09:47.650 we have production P, but this will be-- 00:09:47.650 --> 00:09:50.940 I should say virions per air volume per infector. 00:09:50.940 --> 00:09:53.780 So if we want to think about having multiple infected people 00:09:53.780 --> 00:09:56.040 in the room, we can always just basically increase 00:09:56.040 --> 00:09:56.880 this concentration. 00:09:56.880 --> 00:09:59.370 A well-mixed room, it doesn't matter where people are placed. 00:09:59.370 --> 00:10:01.160 You're just getting more and more droplets, 00:10:01.160 --> 00:10:03.030 and it's assumed to be mixed. 00:10:03.030 --> 00:10:06.600 So we produce at a rate P, but then the outdoor flow 00:10:06.600 --> 00:10:08.940 is taking away droplets and, hence, 00:10:08.940 --> 00:10:14.520 removing virions at a rate Q. So we have a Q times 00:10:14.520 --> 00:10:17.910 C removal rate. 00:10:17.910 --> 00:10:19.330 So this is our equation. 00:10:19.330 --> 00:10:20.970 So let's divide through by V. 00:10:20.970 --> 00:10:28.420 So we can write this as dC dt is P/V minus, 00:10:28.420 --> 00:10:32.260 and then Q/V I'll write as lambda a C. 00:10:32.260 --> 00:10:44.140 So lambda a, which is Q/V, this is the outdoor air change 00:10:44.140 --> 00:10:46.670 or exchange rate. 00:10:46.670 --> 00:10:49.540 So that is the rate at which the entire volume of the room 00:10:49.540 --> 00:10:51.510 is replaced with outdoor air. 00:10:51.510 --> 00:10:53.320 So the outdoor air is refreshing the air 00:10:53.320 --> 00:10:57.330 in the room at this rate Q/V. OK. 00:10:57.330 --> 00:10:58.670 And so that's what appears here. 00:10:58.670 --> 00:11:01.210 And if you compare these two, you can see that this is dC dt. 00:11:01.210 --> 00:11:04.630 This is C times lambda, so lambda is units of 1 over time. 00:11:04.630 --> 00:11:05.620 It's a t. 00:11:05.620 --> 00:11:07.450 This is also sometimes called-- 00:11:07.450 --> 00:11:17.100 the ACH is air changes per hour if you write it in per hour. 00:11:17.100 --> 00:11:19.970 So that's a typical way that this is written. 00:11:19.970 --> 00:11:21.850 And, in fact, while we're just talking about, 00:11:21.850 --> 00:11:23.510 this is a very important concept. 00:11:23.510 --> 00:11:33.020 So lambda a is around 0.3 per hour, so roughly every 3 hours 00:11:33.020 --> 00:11:34.580 for a closed room. 00:11:34.580 --> 00:11:41.760 So closed room or what you might call natural ventilation where 00:11:41.760 --> 00:11:45.770 there's no attempt to deliver air to that room. 00:11:45.770 --> 00:11:47.730 Of course, this number depends on the tightness 00:11:47.730 --> 00:11:51.610 of the construction and whether there is cracks in the windows, 00:11:51.610 --> 00:11:53.530 whether doors are being opened to the hallway. 00:11:53.530 --> 00:11:55.320 So, of course, that's not a perfect number, 00:11:55.320 --> 00:11:58.140 but that's a rough estimate how quickly air is escaping 00:11:58.140 --> 00:12:00.640 from typical construction. 00:12:00.640 --> 00:12:02.620 But then it can be, also, in a different range. 00:12:02.620 --> 00:12:04.760 And this a very important parameter for the theory, 00:12:04.760 --> 00:12:08.230 so let's pause just to look at some of the numbers. 00:12:08.230 --> 00:12:11.440 So it's in the range-- it's typically 3 to 8 00:12:11.440 --> 00:12:13.900 per hour for mechanical ventilation. 00:12:22.500 --> 00:12:25.020 So this could be-- 00:12:25.020 --> 00:12:27.690 it could be open windows with fans blowing in and out, 00:12:27.690 --> 00:12:30.840 which might give you 3 or even 6 on this number. 00:12:30.840 --> 00:12:32.520 It could also be a ventilation system, 00:12:32.520 --> 00:12:34.380 which is delivering fresh air to the space. 00:12:34.380 --> 00:12:38.920 And for typical classrooms, offices, and even homes, 00:12:38.920 --> 00:12:41.530 this is a typical range. 00:12:41.530 --> 00:12:44.160 So for example, if it's 3, then that 00:12:44.160 --> 00:12:47.100 would be every 20 minutes the room gets its air 00:12:47.100 --> 00:12:49.110 to be fully exchanged. 00:12:49.110 --> 00:12:51.630 But of course, if you have situations 00:12:51.630 --> 00:12:53.670 where you need to have better air quality 00:12:53.670 --> 00:12:55.920 and you have more risk of, say, transmission 00:12:55.920 --> 00:13:00.030 of disease or passage of pollutants or contaminants, 00:13:00.030 --> 00:13:02.010 then we need higher values. 00:13:02.010 --> 00:13:04.890 A typical number in the United States for hospitals 00:13:04.890 --> 00:13:06.810 is 18 air changes per hour. 00:13:10.730 --> 00:13:12.030 And then it can be even larger. 00:13:12.030 --> 00:13:13.440 So if you have a laboratory, which 00:13:13.440 --> 00:13:16.100 is dealing with toxic chemicals or even, 00:13:16.100 --> 00:13:19.790 let's say, virus and pathogens, then you 00:13:19.790 --> 00:13:21.830 need even higher air changes. 00:13:21.830 --> 00:13:26.600 And typical rates, then, can be as high as 20 to 30 00:13:26.600 --> 00:13:33.180 for labs that are dealing with toxins of various types 00:13:33.180 --> 00:13:33.680 because-- 00:13:33.680 --> 00:13:36.390 and any airborne toxins have to be quickly removed 00:13:36.390 --> 00:13:37.850 so that if they happen to be leaked 00:13:37.850 --> 00:13:41.620 into the air from your experiment or from your hood, 00:13:41.620 --> 00:13:43.970 they need to be quickly sucked out. 00:13:43.970 --> 00:13:45.500 Also, even parking lots where you 00:13:45.500 --> 00:13:47.980 have cars in an enclosed space that are generating carbon 00:13:47.980 --> 00:13:50.570 monoxide and other fumes, which have to be quickly rushed out, 00:13:50.570 --> 00:13:52.640 parking lots tend to have this number around 30. 00:13:52.640 --> 00:13:56.690 So that's a full air change of the entire room in 2 minutes. 00:13:56.690 --> 00:13:58.020 That's a very fast flow rate. 00:13:58.020 --> 00:14:00.350 So this is kind of the range of this lambda a. 00:14:00.350 --> 00:14:03.420 That's, obviously, a very important parameter. 00:14:03.420 --> 00:14:06.300 So let's now solve this equation here. 00:14:06.300 --> 00:14:09.450 So, first of all, you can see when dC dt is 0, 00:14:09.450 --> 00:14:13.190 then the steady state is just P over lambda a V. 00:14:13.190 --> 00:14:16.400 But if I write Q equals lambda a V, 00:14:16.400 --> 00:14:20.350 I can see the steady state is just P/Q. 00:14:20.350 --> 00:14:22.320 So I can write the solution like this-- 00:14:22.320 --> 00:14:27.740 that if we're given time dependence, that the P/Q is 00:14:27.740 --> 00:14:29.230 the steady state. 00:14:29.230 --> 00:14:32.270 And if I started out with my initial condition, 00:14:32.270 --> 00:14:36.260 was that C of 0 equals 0. 00:14:36.260 --> 00:14:39.710 So let's say time equals 0 is when the infected person enters 00:14:39.710 --> 00:14:42.110 the room and starts breathing. 00:14:42.110 --> 00:14:43.720 And then there's a mixing process 00:14:43.720 --> 00:14:45.350 and there's a build-up of concentration 00:14:45.350 --> 00:14:49.700 until there's a balance between the production of virus, 00:14:49.700 --> 00:14:53.660 virions, or infectious air and the removal of infectious air 00:14:53.660 --> 00:14:55.310 by the ventilation. 00:14:55.310 --> 00:14:58.580 And that gives you this ratio P/Q. But the way the relaxation 00:14:58.580 --> 00:14:59.720 happens, though-- 00:14:59.720 --> 00:15:01.700 if you balance these two terms here, 00:15:01.700 --> 00:15:06.500 that's just an exponential decay with a decay rate lambda 00:15:06.500 --> 00:15:10.180 a, so e to the minus lambda a t. 00:15:10.180 --> 00:15:12.560 So this is basically the way the concentration builds up. 00:15:12.560 --> 00:15:17.150 If I plot this, then at a certain time here, which 00:15:17.150 --> 00:15:18.590 is sometimes called Tres-- 00:15:18.590 --> 00:15:21.320 in chemical engineering, these kinds of models 00:15:21.320 --> 00:15:23.480 are commonly used to design chemical reactors. 00:15:23.480 --> 00:15:26.150 In fact, this kind of model is in chemical engineering 00:15:26.150 --> 00:15:29.200 called a continuous stirred-tank reactor. 00:15:29.200 --> 00:15:30.750 We don't worry about the details, 00:15:30.750 --> 00:15:33.110 but we have a flow of some reactants 00:15:33.110 --> 00:15:35.700 and various chemical species going into a tank. 00:15:35.700 --> 00:15:37.880 We assume it's well mixed and then it leaves, 00:15:37.880 --> 00:15:40.690 and this is the mass balance that we use. 00:15:40.690 --> 00:15:45.500 And the residence time, Tres, is the inverse 00:15:45.500 --> 00:15:47.040 of lambda a in this case. 00:15:47.040 --> 00:15:48.270 So that is the-- 00:15:48.270 --> 00:15:50.830 as soon as you know the volume and the ventilation flow rate, 00:15:50.830 --> 00:15:52.640 there's a typical speed going through here. 00:15:52.640 --> 00:15:55.070 And the time that fresh air spends 00:15:55.070 --> 00:15:57.560 in the room interacting with all the droplets and people 00:15:57.560 --> 00:16:01.070 and then leaving, carrying some of those droplets, is this. 00:16:01.070 --> 00:16:03.200 And in a simple model like this, it's just 00:16:03.200 --> 00:16:10.070 an exponential relaxation at that time scale approaching 00:16:10.070 --> 00:16:17.720 the steady state, which is P/Q. So there's always that kind 00:16:17.720 --> 00:16:21.030 of balance which is reached. 00:16:21.030 --> 00:16:23.720 Now let's also ask ourselves, briefly at 00:16:23.720 --> 00:16:27.320 this point, how reasonable is the well-mixed approximation? 00:16:27.320 --> 00:16:29.870 We will come back to this and analyze it much more carefully, 00:16:29.870 --> 00:16:32.320 taking into account all the different processes I described 00:16:32.320 --> 00:16:35.330 at the beginning, including breathing, motion of people. 00:16:35.330 --> 00:16:39.350 But let's just think about the motion caused by the airflow 00:16:39.350 --> 00:16:39.930 itself. 00:16:39.930 --> 00:16:42.830 So in the case of mechanical ventilation 00:16:42.830 --> 00:16:46.280 where that flow rate can be sometimes rather high, 00:16:46.280 --> 00:16:49.150 that can be a significant source of mixing in the system. 00:16:49.150 --> 00:16:50.570 And the way we'll think about that 00:16:50.570 --> 00:16:53.360 is by writing down the typical velocity of the air 00:16:53.360 --> 00:16:57.080 due to the outdoor air flow. 00:16:57.080 --> 00:17:01.230 That is Q. We can write it as either Q divided by the area, 00:17:01.230 --> 00:17:03.260 so some kind of representative area of the room. 00:17:03.260 --> 00:17:04.960 It could be, let's say, the floor area, 00:17:04.960 --> 00:17:06.589 but depending on the shape of the room, 00:17:06.589 --> 00:17:09.810 it might be a little bit different value than that. 00:17:09.810 --> 00:17:14.030 We can also write this as Q times H over V, where 00:17:14.030 --> 00:17:15.680 V is the volume of the room and H 00:17:15.680 --> 00:17:18.140 is some characteristic height, like, for example, a ceiling 00:17:18.140 --> 00:17:19.470 height of the room. 00:17:19.470 --> 00:17:20.240 OK. 00:17:20.240 --> 00:17:26.900 So this is the mean airspeed due to ventilation. 00:17:32.070 --> 00:17:34.530 And we'll come back to a more deeper investigation 00:17:34.530 --> 00:17:37.770 of these fluid mechanics of the room. 00:17:37.770 --> 00:17:40.950 But as a first example of what we'll be interested in is we'd 00:17:40.950 --> 00:17:44.870 like to calculate the Reynolds number due to the airflow. 00:17:44.870 --> 00:17:47.490 So I'll put a subscript a there. 00:17:47.490 --> 00:17:52.380 And that is the typical velocity times a length scale, which 00:17:52.380 --> 00:17:54.270 could be, let's say, the height of the room 00:17:54.270 --> 00:17:56.130 or some linear length scale of the room, 00:17:56.130 --> 00:17:58.290 depending on the direction of the airflow. 00:17:58.290 --> 00:18:02.250 And then the kinematic viscosity of air, 00:18:02.250 --> 00:18:04.500 so I'll write that as new a. 00:18:04.500 --> 00:18:06.000 So this here is the Reynolds number. 00:18:12.540 --> 00:18:17.020 This is basically telling us how important inertia of the fluid 00:18:17.020 --> 00:18:21.760 is compared to viscous stresses that slow the fluid down. 00:18:21.760 --> 00:18:24.540 So, basically, it tells us how quickly-- 00:18:24.540 --> 00:18:26.380 how much of a tendency there is for momentum 00:18:26.380 --> 00:18:27.960 to be carried in the fluid, which then 00:18:27.960 --> 00:18:30.610 leads to sort of swirling motions and complex flows, 00:18:30.610 --> 00:18:32.260 as I've sketched here. 00:18:32.260 --> 00:18:35.650 And this new a is the viscosity of the air 00:18:35.650 --> 00:18:38.560 that we've already talked about when looking at Stokes flow 00:18:38.560 --> 00:18:41.390 but divided by the density of the air. 00:18:41.390 --> 00:18:41.890 OK. 00:18:41.890 --> 00:18:43.660 So that's the kinematic viscosity. 00:18:43.660 --> 00:18:46.840 And for air, the kinematic viscosity 00:18:46.840 --> 00:18:54.360 is 1.5 times 10 to the minus 5 meters squared per second. 00:18:54.360 --> 00:18:57.710 And so if I plug these numbers in and I pick, 00:18:57.710 --> 00:19:03.700 let's say, a typical ceiling height of maybe a few-- 00:19:04.960 --> 00:19:09.070 I think this-- if the H is of order 3 meters-- 00:19:09.070 --> 00:19:12.160 or 2 meters might be a typical scale. 00:19:12.160 --> 00:19:16.120 Just to get an approximate sense of the scale here, 00:19:16.120 --> 00:19:21.670 the Reynolds number will be varying from around 50 or tens, 00:19:21.670 --> 00:19:25.420 up to 5,000, which would be in the case of very fast 00:19:25.420 --> 00:19:28.300 ventilation like the 30 ACH. 00:19:28.300 --> 00:19:37.340 So this would be if we have 0.3 ACH up to around 30 ACH. 00:19:37.340 --> 00:19:39.640 That's just a rough number. 00:19:39.640 --> 00:19:41.740 And from fluid mechanics, we know 00:19:41.740 --> 00:19:44.200 the significance of these large Reynolds numbers 00:19:44.200 --> 00:19:47.940 is that the flows really do look a bit like I've shown here. 00:19:47.940 --> 00:19:50.110 So, basically, when the air is sitting in the room-- 00:19:50.110 --> 00:19:51.730 you know this from looking at the smoke 00:19:51.730 --> 00:19:54.460 from a candle or other flows that you can visualize-- 00:19:54.460 --> 00:19:58.270 it's not just a sort of uniform flow. 00:19:58.270 --> 00:20:00.910 But instead there are all these sort of plumes and swirls 00:20:00.910 --> 00:20:02.260 and vortices. 00:20:02.260 --> 00:20:04.900 And when the Reynolds number is on the order of tens, 00:20:04.900 --> 00:20:07.570 when there is a motion, it tends to lead 00:20:07.570 --> 00:20:11.460 to a shedding of a vortex and to some kind of swirling flows. 00:20:11.460 --> 00:20:13.300 And when the Reynolds number gets up as high 00:20:13.300 --> 00:20:15.940 as several thousand, then in most geometries 00:20:15.940 --> 00:20:18.190 that starts to lead to a transition to turbulent flow. 00:20:18.190 --> 00:20:19.640 And that's when the flow is getting 00:20:19.640 --> 00:20:21.910 so complicated there are eddies of different sizes 00:20:21.910 --> 00:20:23.980 and very rapid mixing. 00:20:23.980 --> 00:20:25.780 So I just wanted to show this right 00:20:25.780 --> 00:20:27.430 at the beginning of the discussion 00:20:27.430 --> 00:20:29.260 to point out that for typical flows 00:20:29.260 --> 00:20:31.120 we would expect that there is going 00:20:31.120 --> 00:20:33.160 to be a decent amount of mixing occurring just 00:20:33.160 --> 00:20:34.680 because of the ventilation. 00:20:34.680 --> 00:20:37.060 And now if you add to the fact that people are breathing, 00:20:37.060 --> 00:20:38.860 imparting momentum to the fluid, we 00:20:38.860 --> 00:20:42.970 have people moving and other kinds of activities in a room, 00:20:42.970 --> 00:20:45.940 that all of those processes lead to giving us 00:20:45.940 --> 00:20:50.400 a reasonable assumption of a well-mixed room.