PROFESSOR: So we've just discussed
how inertia and buoyancy forces can destabilize the fluid
and lead to complex flows of air in a room.
But now, let's think about how those flows will
influence the transport of aerosol droplets
which transmit disease.
So maybe a simpler way to think about it
is imagine we have some object which
is releasing a substance which could
be particles containing virus.
It could also be heat or other chemical release.
And that's in a flow field similar to the types of flow
fields that we've been discussing.
So that's the problem of forced or natural convection,
convective transport of particles.
So here I show the problem of flow past a cylinder.
And let's say the cylinder is a hot cylinder, which
is releasing heat into the fluid which I've kind of sketched
by this red region here.
Now if the flow is fast compared to the diffusion of heat,
you would expect a situation like this
where there's a thin boundary layer of heat
transfer along the front end of the sphere.
But on the trailing end, there might
be a wake of kind of hot fluid that's
been kind of carried away.
And that is indeed what happens when
we're in the regime of dominance of convection over diffusion.
And there is a dimensionless which describes that,
which is the Peclet number.
So the Peclet number is defined as the velocity times a length
scale divided by D where D that is the mass diffusivity.
Now normally, D is the molecular diffusivity.
Or it's the diffusivity or let's say the droplets
or whatever the individual particles are.
Now, if you want to think about diffusion of gas--
so if we have for gas molecules-- for example,
like for CO2 or oxygen in the air,
then the ratio of kinematic viscosity
to diffusivity of the gas molecules,
that's called the Schmidt number,
is also around 0.7 for air.
So basically molecules of momentum
are diffusing at about the same rate.
And so that tells us that for the molecules of the gas,
the Peclet number is actually the same as the Reynolds number
and, in fact, is very large.
If we have a larger object such as maybe a droplet,
we have, of course, much lower diffusivity than the air
molecules.
And we're not talking and it's a different process
on a collisional.
But we're still dealing with very large Peclet numbers
which will lead to kind of wakes,
as I've just described here.
So it depends on the details and the size of the particles.
But for the gas, I just want to write here
that Peclet number is on the same order as the Reynolds
number and hence is very much larger than 1 in many cases.
Now the thing is that because the Peclet number is 1,
the Peclet number is measuring the importance
of convective transport to diffusion.
And so because the Peclet number is typically large,
we have a dominance of convection
over diffusive process.
So think of our aerosol droplets.
They do diffuse in the air.
And we can calculate that with the same Stokes-Einstein
formula that we've used earlier.
But that diffusion rate is very slow
compared to the sort of convective processes
that are occurring in the room.
And so we typically have a high Peclet number
and we see this kind of behavior.
On the other hand, we're also in the regime of high Reynolds
number.
And that really changes things because then the diffusion
due to sort of random fluctuations and collisions
with other molecules is overwhelmed by the sort
of transport and diffusion light transport
that follows from vortices and turbulence.
So if I take that same cylinder and I
at now at a higher Reynolds number,
then we can see that the wake is not a thin little tail that
extends downstream but it's really
a turbulent wake where the warm fluid in this case
is dispersed everywhere throughout that wake.
In fact, it's fairly uniformly mixed.
A similar situation occurs in the case of breathing,
coughing, sneezing, and other forms of respiration, which
we will come to shortly, where we
have a relatively high Reynolds number flow, often turbulent,
and we are injecting in it some respiratory aerosol particles.
And they're quite well mixed across that jet.
So we can see there's a very strong coupling
between the fluid flow that we've just discussed,
which is often turbulent and containing vortices and eddies,
and the transport of suspended particles and droplets.
So how can we think about this problem?
So the important thing is when we
get into this turbulent regime, the way
an individual particle--
imagine-- I should think even here in this case.
How does an individual particle move?
It kind of follows the flow.
So it goes first through a little vortices
occasionally around a big vortex,
and then it does some little ones.
And so it's also doing a random walk
but it's one that's driven by the turbulent flow itself.
And the length scale for the sort
of steps of the random walk is actually the vortex size.
And in particular, it's dominated
by the largest vortex.
So whatever the flow is, there's always
a certain scale which sets the size of the largest vortex.
And so that leads to the concept of so-called eddy diffusivity
in turbulent flows, which is also important in air flows.
So the eddy diffusivity is an effective diffusion-like
parameter that describes the mixing and spreading
of suspended particles or droplets in a flow field that
is turbulent.
And in that case, we can write it two ways.
So either we have an imposed velocity u
and we have length scale l, and then our eddy viscosity
might be written as u times l.
So u is distance per time.
l is distance.
So it's distance squared per time.
So that has units of viscosity.
And the way to interpret this is basically
sort of swirling around eddies of a size l
and a characteristic velocity u.
Another way to write this would be that it's l
squared over 2 times a timescale.
Because we often write diffusion,
if there's a time step tau, and a length scale
l for a given step, then l squared over 2 tau
is the diffusivity.
That's how we think about molecular diffusivity as well.
But the question is, what is this timescale.
So we can see here that the timescale is l over u.
So it's a convective times scale.
It's the time to, essentially, go around one of those eddies.
And I'm writing these really just as scaling arguments here.
So if we look at these flows, what
are the relevant length scales?
So at the beginning of this flow here,
it's the length scale is that of the object.
In the case of the breathing, the length scale
is initially that of the mouth opening.
But then we form these turbulent structures that expand.
And a constant we will return to you
shortly is that the relevant length
scale as you continue here is actually something
which depends on position.
So as this thing grows the eddies
are are getting bigger and bigger,
and so also is faster and faster the transport by diffusion,
which sort of maintains a fairly uniform concentration
across that space.
And that brings us then to what happens in the whole room.
So we've been very interested in mixing in the whole room.
And so a natural picture here is to say, well,
if the room has a height H, then the eddy diffusivity,
which aerosol particles in the room
are feeling as long as it's a fairly well mixed turbulent,
more isotropic flow, could be described as the height squared
over 2 times a timescale.
And the timescale should be that of the effective air
change or the total air change time, basically.
So this here, again, I've written
lambda bar a is the outer airflow
plus the re-circulation airflow, which
may be going through a filter, and divided by the total volume
of the room.
So this is the total ACH.
And so what we see here then is that the formula is roughly
that we should have 1/2 H squared lambda a bar.
And so this is a very simple argument based on the largest
eddy is going to be, in a well mixed room,
at the size of the scale of the room.
And sure enough, this relationship
has actually been verified for houses and actual indoor rooms
with all the furniture in them, where
it's been shown that if you release
a passive tracer such as carbon dioxide in the middle
of the room and you have the ventilation
on at a certain ACH, Air Change per Hour, lambda a bar, then
this formula, even with the 1/2 actually,
turns out to be a pretty good approximation for the spreading
in that room as you change the size of the room
and look at different rooms and also look at different air
change rates.
So that's, I think, a good starting point for us as well.
At least when the room is well mixed,
this is a good way to think about transport in the room.
Also, we see from this picture that the timescale for mixing
is the inverse of the air change time.
So the mixing time is also comparable to the residence
time of the air, including re-circulation.
So basically it's the time it takes
for air to typically go through this system
is also the time it takes to fully mix the system, roughly
the same order of magnitude.
And that is the characteristic of a well mixed turbulent room
which could occur by any of the mechanisms
we've just been describing.
Although the same principles also apply to jets or strong,
maybe ventilation flows past an object where you might still
have some heterogeneities as I've sketched here,
that we will need to consider.
The last point I'd like to come back to
also is the question of sedimentation.
So you may have found it surprising
that we describe the flux of sedimenting particles
to the ground in a very simple way
by just using the Stokes velocity, vs,
and multiplying by the area.
So we said the droplet flux out of the room
was just the sedimentation velocity of the droplet, which
was radius dependent, times the area, the floor surface area.
And what's a bit confusing about that at first
is that we know the flows are very complex in the room.
In fact, if you look at dust particles in a room
as we discussed earlier in some cases, for sure,
you will see them actually rising and not settling.
So they may be settling relative to the flow.
But the flow is actually convecting them upwards.
And that's here, for example, if you
look at there's a dominant role in the flow
that I've sketched here.
The particles over here actually might
have a net velocity going up.
And over here, they're going down.
But if you decompose that velocity field, then
in some cases, they're going up due to flow or convection,
while they're sedimenting at the rate vs.
But then necessarily, because the fluid
is approximately incompressible and is returning somewhere--
wherever it goes up, somewhere else it's coming down,
we find that in other areas you have vs still
pointing down with the same rate and now the flow is going down.
And if you imagine a particle that
is sampling all the different velocity vectors,
the blue vectors, sometimes they're up,
sometimes they're down, on average
the blue velocity vectors of the flow have to average to 0.
Because there's no-- or at least near 0.
There's not, let's say, a very strong
vertical relative motion.
Then it's reasonable to assume that the particles will
sediment out of a well mixed turbulent flow at a rate given
by vsa.
And that is something that has also
been validated experimentally for well mixed chambers
and rooms.