Evaporation Rate of Small Spills

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 cm³ .   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 )(M₀) (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.