Recent blogs have covered the subject of surprises. During some of my early work on modeling I
got a surprise relative to the calculated airborne saturation concentration of an evaporating
source. Like almost all scientific surprises, this
one was very useful and lead to the development of a modified model for estimating airborne concentrations above and around large evaporating
sources which I termed the backpressure model.
If you put an evaporating source into a closed vessel and have
enough of the material within the vessel, ultimately the saturation concentration
(Csat) will be obtained in the headspace air volume of the vessel. Csat is the highest concentration of vapors
possible from the evaporating source. Csat is directly related to the vapor pressure as
follows:
Csat = (vapor pressure/atmospheric pressure)(1,000,000)
The units of Csat are ppm volume/volume or ppmv.
Now take a relatively small sample of the material of
interest and put it on a watch-glass on an open scale and allow it to evaporate
so that you can measure its evaporation rate in mg/hour. This should equal the generation rate of an
evaporating source (G).
We know from previous work that the concentration in any volume
with a constant generation rate (G) and ventilation rate (Q) is:
Ceq = G/Q
Remember our watch glass experiment above. We now have a value for G in mg/hr. Say the evaporating surface area was about
5 cm2. So now normalize this rate per unit area so we can express this G
rate as mg/((cm2)(hr)).
So we used this G to estimate the Ceq (=G/Q) of a room with a large (10 m2)
spill of toluene in a reasonably well ventilated room. We calculate G by multiplying our experimental G rate per cm2 by the area of the spill which is 100,000 square centimeters.
When we calculate Ceq we get a value that is
MUCH larger than Csat which is physically impossible! Surprise surprise!
What is going on here?
The answer is backpressure. The
entire driving force for evaporation is the diffusion of the molecules from
high concentration in and immediately above an evaporating liquid and the
relatively low concentration of the same molecules in the receiving air
volume. When we release relatively few
molecules from the source compared to the receiving volume, this driving force
is maintained and G is constant. However, when the
receiving volume starts to contain a built up concentration of the evaporating
molecules the driving force is diminished and the evaporation rate (G) decreases.
Think of it this way.
In the desert with very low humidity in the air, wet clothes dry very
quickly. In more humid environs when
the relatively humidity is 50% clothes dry only half as fast and at 100%
humidity they do dry at all.
In a large spill the initial generation rate (G) is
maximized because there are no molecules of the evaporating liquid in the
receiving air; however, relatively soon thereafter the molecules build up in
the air and begin to retard the generation rate (G). Thus, the generation rate is not a constant
but is variable with time:
G = Gmax (1-C/Csat)
Gmax is the initial generation rate at the beginning (t = 0)
C is the concentration in the receiving volume (which is a variable
function of time until it reaches Ceq)
Csat is defined above
If C in any volumne ever gets to the point where it equals Csat (Ceq = Csat), as it does in a closed vessel,
then G becomes equal to zero.
Backpressure is always present over evaporating or
vaporizing sources even in ventilated volumes; however, we usually do not have to account for it because its
effect is relatively small when we have evaporating sources with small area-to-receiving volume ratios. When it
becomes important is when we have LARGE evaporating sources such as a large
area spill indoors or large vaporizing sources like off-gassing wall paint or
carpets.
There is a module in IH MOD for considering backpressure
(The Well-Mixed Room Model with Backpressure). Because backpressure shows itself mostly in
indoor sources with large areas, this is the correct model in which to evaluate
its effects.
The original paper on backpressure has a lot more detail and
I would be happy to send it to whomever is interested and writes to me at mjayjock@gmail.com
ERATA: I mention a "watch glass" in the above blog. Perry Logan reminds me that because of the geometry of a watch glass (a kind of concave dish) that the surface area changes during evaporation and this is an artifact of the watch-glass and not the area-normalized evaporation we are attempting to study. Thus, it would be better to use a flat bottom plate like a petri dish; however, a petri dish has relatively high sides which can disturb the airflow. I still think a thin layer of evaporating VOC in a petri dish might give reasonable results. 3M actually machined a block of aluminum with a shallow wall for their experiments and this is perhaps the best approach if you are going to do a lot of these experiments.
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