Simple Approaches to Common Modeling Problems

Previous blogs have been pretty technical – this one is intended to provide more practical advice on simple approaches to common modeling problems.

I am going to stick with the theme of the modeling of the concentration of volatile organic contaminants (VOCs) in the breathing zone.   As discussed in previous blogs we have the models and the software tools to make these estimations but we need to feed them.   Traditionally the most difficult inputs for modelers to get are the ventilation rate and the emission rate.   We spent a lot of ink talking about estimation of the ventilation rate (Q) and I think that between the two, it is the easier to estimate.   The generation or emission rate (G) is more challenging but certainly doable.

Consider a small spill of a mixture of aromatic volatile organic compounds (VOCs).   The first task is to figure out which VOC(s) need to be evaluated.   This goes to the concept of controlling health hazard.   That is, which compound or compounds will present the highest risk such that if it (they) are controlled then the other components will also not present an untoward risk to human health. If all the VOCs have similar vapor pressures then the simplest first crack at this problem is to take the ratios of percentage within the mixture of each chemical over its exposure limit.   For example let us consider an example we used earlier of a mixture with 1% benzene, 50% toluene and 49% xylene.  I am not certain of the current TLVs but for the sack of this example let’s say they are 0.1, 50 and 50 ppmV, respectively.   The ratios would be 1%/.1 = 10 for benzene and about 1 for toluene and xylene.   Clearly, benzene is the controlling health hazard in this example.     If it were just a 50/50 mixture of xylene and toluene (without the benzene) with the above supposed TLVs then you should probably look at both, at least initially.   Toluene would be expected to be more volatile but with the techniques we are using, xylene may be a close enough second that you need to look at both.  Plus, they most likely will have similar adverse health end-points to each other and thus need to be considered additive in their exposure and risk.  On the other hand, at this point in our understanding benzene presents a dramatically different and more dreaded risk compared to the other two aromatics and concurrent exposure to these two would not be considered to add to this risk.

The next important task is to figure out how much of the chemical you are interested in is evaporating.  One simple way of getting at this value is to weigh the loss of the delivering vessel (Paint can, aerosol can, pump spray, etc.) figure out how much is the evaporating chemical of interest and divide by the drying time.  These are rough estimates but can provide good information.

   

So at this point we have determined that benzene is the controlling health hazard in our spill and we have figured out approximately how much has been released and evaporated.   Now we have to figure out a rate of emission.   If you know how long the spill takes to evaporate you can estimate G but simply taking 1% of the total mass of the spill and divide it by the time of evaporation.   This gives you a rough average for G. This would be modeled with a constant G (using IH MOD) for that time period with G = 0 at the end of the evaporation period.   No matter how long the evaporation takes you need to compare the modeled concentration that occurs as an 8 hr time-weighted average with the 8 hr TLV-TWA of the chemical.    This assumes the worker is only exposed for 8 hours. 

If the spill is actually shrinking in area during the evaporation then you can use a first order assumption.  In this case we would assume that the spill took 8 half-lives to essentially disappear (e.g., it starts at 640 grams and go to 2.5 grams in 8 half-lives).   Thus as a rough estimate the half live is the evaporation time divided by 8 and a derivation in the previous blog will provide the decay constant; that is, k = 0.693/half-live.   If the drying time is less than 8 hours, you will get essentially the same 8 hr time weight average room volume concentration as with the constant G but it will affect the 15-minutes time-weight average for comparison to a Short Term Exposure Limit (STEL).  The highest 15 minute STEL during the evaporation of the spill will be significantly higher with the first order assumption versus an assumed constant rate of evaporation.

Another way to get an evaporation rate is to put a relatively thin layer (simulating the spill) evaporating material into a open dish, put it on a open scale under the same conditions as the spill and record the weight loss with time.    The thickness of the material in the watch glass should be similar to the thickness in the actual exposure scenario.    Consider the data set below for a product that has about 50% solids and 50% solvent.   The solvent is the one from our example:  1% benzene, 50% toluene and 49% xylene.  The only measurement is the total weight versus time. The difference in weight over time is the solvent evaporation and we are assuming that the benzene is 1% of that number.

  

You are pretty sure you have the vast majority of weight loss when the weight does not change significantly over a period of a few hours.   In this case less than 1% of the product mass is lost between 140 and160 minutes.  The time it took to lose half the solvent weight is roughly the half-life and again k= 0.693/half-life.  In this case the half-life is 20 minutes.   To be more precise you could model the weight loss data as a first order decay as has been shown in previous blogs.  Remember we are only interested in the portion of the evaporating mass that is the chemical of interest.  So if benzene is 1% of the mass, G is 1/100^th of the overall evaporation rate as indicated in the data.

Remember that this scale experiment only gets you the half-life and k values – you still need to know how much was actually spilled to estimate the airborne concentration with a model.   Note also that the average emission rate in the scale experiment is roughly 320mg/160 minutes for 64 grams of product.  If you spill 640 grams of product it would be 10 times higher.   Also the first order evaporation rate at the beginning of a 640 gram spill would be 80 mg/minute over the first 20 minutes after the spill.

If the evaporating surface area does not shrink with time (e.g., an open vessel) then you can do the above experiment with a deep source so that the measured rate should be relatively constant.

I am doing all this without peer review - which is usually a bad idea; however, this is a simple educational blog.  If you find some errors or have some issues with the advice please let me know and I will issue corrections and expand the discussion.