Point-in-time versus time-weighted average exposure

This week we go back to some of the nuts and bolts technical stuff of modeling.  Specifically, the difference between point-in-time and time-weight average concentrations and exposures.

The following curve shows a typical pattern of airborne concentration versus time for a process that starts at time zero and ends at time = 45 minutes.   Notice that after 45 minutes the source is cut off and the airborne concentration decays to essentially zero in about 80 minutes.

The point-in-time airborne concentration anywhere along the curve is easy to see.  For example, as indicated on the above curve at 7.2 minutes it is 109 mg/m³.   Note that in IH MOD it shows concentrations in both a graphic (the image above is an IH MOD cut and paste of its graphic output from this exposure) and in a table of concentration, time data points.

As it turns out, we are not typically interested in the specific point-in-time concentrations except in situations where short term peaks are important.   Indeed, they are important for exposure limits that have a “C” designation for ceiling concentration.   If the “C” exposure limit for the chemical in the graph was 150 mg/m³ the peak of about 240 show above indicates an overexposure.

There are very few chemicals with “C” exposure limits.  Most occupational exposure limits are either 8 hour (TWA) or 15 minute (STEL) time-weighted averages.   Here we need to understand the worst case time-weighted average exposure in the appropriate averaging time of interest.

Just for general information, the time-weighted average concentration for any time interval is the area under the point-in-time concentration curve, divided by the time interval used. 

For 8 hour TWAs IH MOD makes it relatively easy.    If you check the box on IH MOD that says “TWA on chart” it will calculate the time-weighted average for any time from zero out to the “maximum time for simulation” you put into the model.  Thus if you run the model out to 480 minutes (8 hours) you get the 8 hour TWA exposure which is 23 mg/m³.    If you run it out for 1440 minutes you get a 24 hour time-weighted average exposure from this scenario.

If you are working to a 15 minute short term exposure limit (STEL) it gets a bit more difficult to calculate in this example.  What we want is the time-weighted average exposure during the worst 15 minute period during the scenario.   This is the number we will compare to the 15 minute STEL exposure limit.

Perhaps the easiest way to do it is to find the peak point-in-time exposure and take the point-in-time values at 15 minutes before the peak to calculate the time-weighted average.   We would assume linearity in this relatively short period such the average exposure between each of these intervals represents the time-weighted average during this 15 minute interval.     

If we look at the predicted concentration output in IH MOD we see the peak point-in-time concentration occurs at 45 minutes (when the source dies) with a concentration of 240 mg/m³.   The point-in-time value at 15 minutes before the 45 minute mark (i.e., the 30 minute mark) shows an estimated point-in-time exposure of 225 mg/m3.   The average concentration occurring between the end value of 240 to the beginning point-in-time value of 225 is 232 mg/m³.  This is the time-weighted average concentration over the 15 minutes before the peak.   This is also the 15 minute time-weighted average exposure to be used for comparison to the 15 minute STEL. Notice that it is almost 10 times higher than the 8 hour time-weighted average exposure of 23 m³.

Perhaps someday we can get IH MOD enhanced to provide this calculation more directly.  My guess is that it would not be a trivial task but Daniel Drolet continues to surprise me.  He is a truly gifted programmer and I have no doubt that he can do it.  Indeed, what would be difficult for me could be relatively easy for him.

There is another important lesson in all this; namely, you should never monitor any batch operation for a STEL by starting the monitor at time = 0.   This example clearly shows that it takes time for the concentration to build to a peak.   If the source stops abruptly, start monitoring about 20 minutes before its end and end monitoring at the end of the source or shortly afterward.   You can also get a sense of where to monitor a decreasing evaporative source in a similar manner by first running IH MOD and seeing where the peak is predicted.   Please note that constant rate sources can be monitored for STELs any time after they achieve steady state concentration and IH MOD can help you with that estimation as well.