When we spill a volatile chemical we have a problem relative
to potential inhalation exposure.
Depending on the specific situation we need to answer the following
questions: Do we clean it up
immediately or do we simply stand back and let it evaporate? Do I (or we) need to get out of the room or
the area? What is my exposure/risk
potential and the potential of others in the area if I do not evacuate and stay
to clean it up?

It is such questions that modelers live on! In order to answer them, however, they need
some reasonable input relative to the evaporation rate, size of the spill and
the room, the ventilation rate and the rate of linear air movement within the
room. A very useful model to use for
this scenario is the two-zone model developed and promoted by Dr. Mark Nicas and available
on the slick freeware spreadsheet IH
MOD. Please see some of my previous
blogs on these if you interested in learning more about them. Like all models, the 2 zone model needs to be
fed with the appropriate inputs.

The basic idea around the two zone model is to put both the
source of the exposure and the breathing zone of the exposed person into the
inner or near-field zone. It is the
geometry of the scenario that determines the size and shape of the
near-field. For example, someone
spraying the hair on their head with hair-spray would have a near-field well described
by a sphere of about 0.5 m diameter around their head. Someone cleaning up a spill by hand might have a
near-field described as a hemisphere with a radius equal to an arm’s length.

Some of the inputs needed for the 2 zone model of a spill are
usually readily available. Let’s say
our hypothetical spill of liquid has a volume of 100 cm

^{3 }. If we assume a circular spill of 1 m in diameter, that calculates out to a spill thickness of about 1-2 mm. That seems reasonable assuming it is on a relatively nonporous surface like tile or finished cement. We can also approximate the ventilation rate. If it is laboratory or industrial area then 3-5 mixing air-changes per hour seems reasonable, if it a residential area with doors and windows closed, then about 1/10^{th}this value would be typical. Room size is easy to get. So what are we missing? Ans: evaporation rate.
This brings us to the subject of sub-models or models that
are used to feed other models. Some
evaporating sources are essentially constant within a time frame of any one-day
exposure. A prime example would be an
open drum. Spills, on the other hand,
are typically not constant. That is, they
typically shrink with time as they evaporate.
As they shrink their rate of generation decreases until is ceases
entirely when it is all gone. A model
that seems to do a reasonably good job of mathematically describing this situation is the
first-order decay model. First-order
decay was also discussed in detail in a previous blog if you are interested in going
back. It is described here briefly as a
rate that is dependent on and directly proportional to how much of the original
spill remains. That is, the rate is
maximized in the beginning at 100% when all or 100% of the spill is
available to evaporate. After 10%
evaporates the evaporation rate is 90% of the maximum. After 50% evaporates the rate is half that of the
original. The time it takes to get to
50% is the half-life of this first-order kinetic model. After 7
half-lives less than 1% of the spill remains and the evaporation rate is less
than 1% of the maximum. Theoretically, you
never get to zero while in reality you certainty do. After 7-8 half lives it is essentially gone. In any event, this model appears to be a
credible job of describing the evaporation rate of spills when compared against
real world data.

The basic model is:

Evaporation Rate = (Initial Evaporation Rate)(exp(-α t))

It can be shown that:

Evaporation Rate = (α )(M

_{0}) (exp(-α t))
Where:

- M0 = initial mass of the spill
- α = the evaporation rate constant or the proportion of the mass evaporating per unit time usually per minute. Thus α = 0.10 means that 10% of the initial (or remaining mass) will evaporate every minute.

Drs. Nicas and Keil wrote a seminal paper about α in the context of modeling spills which I will
be happy to send to whomever asks me for it at:
mjayjock@gmail.com.

This paper has a number of values for α for common
laboratory solvents and also forwards the following data fitting relationship for other volatile organic compounds:

α, min-1 = 0.000524 Pv + 0.0108 SA/VOL

Where:

- Pv = saturation vapor pressure in mm Hg at 20 C
- SA/VOL = initial surface area to volume ratio of the spill,
cm
^{-1}

An experimental evaluation of this algorithm with n-pentane is
presented in the book: Mathematical
Model for Estimating Occupational Exposure to Chemicals, 2

^{nd}Edition, AIHA Press.
What is so useful about this model, as implemented by IH MOD, is that it provides
estimates of the PEAK breathing zone concentration along with the time-weight
average for whatever time frame you want (e.g., 15 minutes or 8 hours) to be compared
with whatever exposure limit (Ceiling, 15 minute STEL or 8 or 24 hour TWA) you
would deem appropriate.

In a later blog I will discuss other work done and published
on spill modeling in an industrial setting where the risk was driven by peak
exposures.

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