Modeling Evaporating Sources

Almost all volatile organic chemicals get into the air via evaporation or vaporization.   Sometimes they are literally sprayed or ejected into the air.   Prime examples of this are sprayed material (e.g.,  spray paint, hair spray) and drum filling where the empty air volume of the drum being filled is displaced into the ambient room air along with the vapors of the entering liquid.   When sprayed, aerosol particles evaporate rapidly because the surface area-to-volume ratio of the particles is relatively large.  When filling empty containers with liquids the entire volume of the container is injected into the surrounding space as the liquid displaces the air within the drum.   What is also injected are the vapors from the incoming liquid as it splashes or otherwise enters into the container.   The EPA did some tests on this many years ago and determined that this volume has between 50% and 100% saturated vapors.    Thus if you fill 10 – 55 gallon drums you will inject 550 gallons or 2.1 m³ of air with a vapor concentration of between 50 to 100% of saturation of the filling liquid VOC.

Just to review, the saturation concentration (Csat) is what happens in the headspace of a drum.    Given enough time, evaporation takes place until the air in the headspace can hold no more.   At that point as much vapor condenses back into liquid as evaporates from it and the concentration in the drum headspace is the saturation concentration Csat.   Csat is easy to calculate.  

Csat = (VP/ATM)(10⁶)     [units = ppmV]

VP = vapor pressure of the liquid

ATM = Atmospheric pressure

The concentration expressed as ppmV is readily converted to mg/m³:

mg/m³  =  (ppmV)  MW/24.45

MW = molecular weight of the vaporizing liquid in g/mole

I have found that most of the time, when VOCs  are evaporated into the air – say from a small spill - that the resulting concentration is only a small fraction of Csat.   In these situations you can ignore what I call the “backpressure effect” in your model predictions.    If however, the concentration builds to a point where it is say 50% of Csat then the evaporate rate (generation rate G) is literally half of what it was when the initial concentration was zero.   That is because the entire force driving evaporation is the diffusion of the molecules from the liquid to the gaseous state.   If there are already a lot of molecule of the same type in the air then the net evaporation rate becomes lower.     Thus, the evaporation or generation rate (G) is constantly decreasing as the concentration in the volume is increasing.   A simple equation for this is:

G  = G0 (C/Csat)

G0 = the initial or maximum generation rate when the airborne concentration is zero.

C = the concentration in the volume.

As mentioned above I view this effect as “backpressure” and wrote a paper about it some time ago.   Anyone interested in a copy of this paper just ask me at mjayjock@gmail.com

Backpressure is the reason that wet clothes to do not dry very well in humid weather.

We need to consider backpressure in our modeling algorithms when the resulting airborne concentration might be a reasonable fraction of the saturation concentration.   This usually happens when there are large spills or when the vaporizing surfaces are very large within an interior room.   Examples would be offgassing from drying paint or carpets.

IH MOD is a freeware spreadsheet modeling tool available from the American Industrial Hygiene Association.   It addresses and handles backpressure in some of its modules.   In the next blog I will talk about many if not most of the things that IH MOD can do.

How to Characterize Inhalation Model Source Rates

Remember one of the simplest of models?  Its  C = G/Q   where, at equilibrium with a constant rate of G and Q, C is the steady-state concentration (mg/m³),  G is the source or emission rate (mg/min) and Q is the ventilation rate (m³/min)  for the volume (V) into which G is flowing.   In the last few blogs we have been concentrating Q and how to estimate it with tracer gas.   We also went over the first order decay model for use with a tracer gas.    This week I am going to talk about source (G) characterization.

But first and briefly, I just want to go over a much more simple way of estimating Q in a room that has obvious input or output streams of ventilation.   We know that every inhabitable room receives as much air as it exhausts or else the room would explode or the folks within it would die from lack of oxygen.  Thus, if you can measure the amount of air coming into or leaving the room you can determine its ventilation rate (Q).   You simply take the area of the incoming or outgoing stream and multiply it by the velocity of the air going in or coming out.   Q = air velocity x area.    If you have both output and input streams measure them both, the one with the highest calculated Q is the most accurate estimator.   

Back to sources (G).   In general we assume that sources are constant or  that they undergo first order decay of an initial mass.   These are all generalized descriptions of a source and like all models they attempt to portray reality but they are not strictly speaking, reality.    IF they describe reality reasonably well then they are useful.  If they do not then we need a better model.

A constant source means that G does not vary with time; it is constant.    Consider someone spraying an aerosol around their head in the application of hairspray.   The can of hairspray is used for 1 minute and then the spraying is stopped.   Measure the weight of the can at the beginning and at the end of this period and the difference is the rate of application G in units of mg/min.   You can determine the source rate of a certain ingredients within the can by knowing its concentration.  For example, if 10% of the stuff in the can is ethanol then 10% of the weight loss was this alcohol.

Imagine a leaking valve that would also put out a relatively constant mass of vapor per min as a constant source.   Another constant source would be an open and relatively deep container of solvent.    The evaporation would be constant given a constant vaporizing surface area in which the temperature remains relatively constant.   If the temperature remains virtually the same and the resulting airborne concentration does not approach a reasonable fraction of the saturation concentration of that vapor, then the source can be considered constant.   I will go over the details estimating evaporation rates and relevance of saturation concentration and “backpressure effect” in next week’s blog.

A first-order decaying source is completely analogous to the first order decaying concentration of a tracer gas.    Consider a spill of a volatile solvent on the floor.    If you plot the natural log of the weight of the remaining spill versus time you will,  in most cases with thin spills,  get a straight-line; that is, a good fit to the first order decay model.   Go back two blogs to see how this is done with Excel.

Another way of measuring the parameters (initial mass and half-life) of a first-order decaying source is to measure the time course of concentration decay in a volume with well mixed ventilation.   A series of C,t points should follow a first-order decay model and one can then back calculate the generation rate parameters that caused them.   If there is interest in more of the details of this technique I can go over them in a future blog.

There is a significantly more complicated source which is a combination of an increasing AND a first-order decaying source.  This may seem daunting but it is doable.  Consider someone cleaning a surface with a solvent.   The solvent goes on the first little section of the surface and starts to evaporate until there is no more of it left.   This is classical first order decay of G.   But that is not the whole story; indeed, solvent is applied continuously and somewhat constantly until the surface cleaning is complete.  This is a growing rate of generation (G) as the area gets larger.    This type of source has fortunately been described mathematically within the documentation for the EPA Model E-FAST2.     Reference:     http://www.epa.gov/opptintr/exposure/pubs/efast2man.pdf

Both constant and first order sources are presented and coded individually in many of the important models presented in IH MOD which you can get at:

http://www.aiha.org/get-involved/VolunteerGroups/Pages/Exposure-Assessment-Strategies-Committee.aspx.   Unfortunately, the combination of an increasing and first order decaying source types, as described in the above paragraph, has not been coded into IH MOD but it may be in future versions of this very valuable tool.    

I will go over exactly what the freeware IH MOD offers in a future blog.

   

Nothing is Perfect - Especially Models

Teaching always involves learning and I just love to hear from the folks reading this blog.   A very thoughtful reader, who has asked to be identified only as a “retired US Air Force Bioenvironmental Engineer”, read the recent blogs on using tracer gas for ventilation measurements.   He had the following comments which I am including and addressing below:

1.       These computations work for gases and vapors, not particulates.

2.       The assumption that the vapor or gas will occupy all room volume is reasonable for a small  room, but in a large room (i.e., an aircraft hangar) the assumption may not be valid, especially  since most solvents used in paints and coatings are heavier than air.

Let’s take point 1 first.   It is correct that these computations will not work for many if not most particulates.  It is especially true for particulates in which a significant portion of the particulate population has an aerodynamic diameter (AD) greater than 10 microns.  This is primarily because these larger particles settle to the ground in a time frame that is comparable to the tracer testing time.   If the particles are very small – for example, AD less than 0.1 microns (e.g., fumes and  non-aggregated nano-particles) they could work because they will settle very slowly.   

Note:  Aerodynamic diameter (AD) is based on the settling velocity of any particle.   That is a particle with an AD =10 microns will fall at the same rate as a unit density sphere with a diameter of 10 microns.    Thus regardless of the shape of the particle we can characterize its size in an aerosol. 

Regarding number 2 above I have some experience and the point about the model not working in large rooms is right to the mark.   The room has to be reasonably small or enough tracer has to be released and mixed that the concentration is pretty even throughout the entire volume.   This is extremely difficult to do in a very large room (e.g. airplane hangar, large warehouse).   In these volumes it is probably easiest  and best to measure the volume of fresh air coming into or out of the volume by measuring the air velocity (Vel)  and the areas (A) of the opening(s) and using the Q   = Vel  x A equation to figure out how much ventilation is going into and coming out of the room.  Remember the average amount going in will always equal the amount coming out in any reasonable time frame.  

The issue of heavier than air solvents is something that I have thought about quite a bit.   I will use the rest of this blog to present the argument that I believe that it is a non-issue in the vast majority of what we do as exposure assessors.  

It is quite nature to think that all “heavier-than-air” vapors sink and sink quickly.   Indeed, we have all seen “dry ice” vapors pouring or spilling out of a vessel or block of dry ice and dropping quickly downward.    Air is 21% oxygen (MW 32) and 79% nitrogen (MW 28) for an average MW of 28.8.    Carbon dioxide MW is 44 g/mole.  Also, its vapors are much colder than most ambient air and thus even more dense.   Both of these factors cause the CO₂ vapors to drop but perhaps the most important factor that causes the CO₂ vapor cloud to fall quickly is that this emitting cloud is essentially 100% CO₂  (because C0₂ is a gas at normal temperature and pressure) and a 100% CO₂ gas emission is at its maximum density and tends to displace all of the air in its path.    The relative density of CO₂ versus air (not counting temperature effect):   (44/28.8)(100%)  = 1.53  or 153%.

Now consider typical solvents in VOC-based paint used to paint cars.   These materials are liquids at normal temperature and pressure and even under saturation conditions their vapor concentrations only get to a small fraction of 100%.    Let’s look at toluene as an example, even assuming the worst case of a saturated vapor of toluene, it only comprises 5% (40mmHg Vapor Pressure/760mmHg ATM)  of the molecules in any volume at normal temperature and pressure with the rest being air.  

To get some idea of the actual  VOC concentrations around workers during spray painting, we did and published a one year study evaluating worker exposures in a small “bump and paint” auto body shop. Please send me an email (mjayjock@gmail.com) if  you want a copy of this paper.   In doing this work we estimated that the average MW of these paint solvent mixtures was 125g/mole.   I measured the breathing zone of workers spraying cars in a small booth that was turned off (because of cold weather outside) with essentially NO ventilation.   The highest total VOC measured in the worker’s breathing zone was about 1500 ppmV. 

The relative density of vapors with 1500 ppmV total VOC with average MW = 125 (versus pure air) is (125/28.8)(1500/1,000,000) =  0.0065 or less than 1%.   It is worth mentioning that the painters we monitored did not wear PPE and were visibly intoxicated by these exposures.   Getting back to the point of heavier than air vapors, even at 4 times this concentration (i.e., 6000 ppm V) the difference in density and buoyancy between pure air and air highly contaminated with VOC appear to be relatively small.  

In conclusion, I believe that it is fairly safe to say that VOC emissions from evaporating liquid pools, spraying or from evaporating aerosol particles will be en-trained into the normally moving ambient air and not have a strong tendency to sink.

All of this reminds me of the wise statement that:  “All models are wrong but some are useful. “   We simply need to keep engaging our minds and allow the models to tell us something useful.

Excel Regression Analysis for Ventilation Rate

This is a hopefully “bite-sized” lesson in tracer gas ventilation rate analysis.  It gets a little technical but I tried to break it down into little pieces and if you stay with me I think you will find it rewarding. 

In last week’s blog we went through a derivation of the first order kinetic model.   The primary equation is:   C = C0 e^-kt.   Where C is the concentration at any time (t) and C0 is the starting concentration.   Here k is the rate of loss as proportion per unit time.  For example k = 0.5 means that 50% is lost every hour.  If you take the natural log of both sides of this equation you get:  ln(C) = -kt + ln(C0).    You may recognize this form as the straight line equation y = bx + c.  Here y = the point-in-time concentration (C) at any time (t) and c equals C0 or the starting or t=0 concentration.  The x value in this equation is time (t).  The slope b is equal to –k or the loss rate or Q/V.

Given this straight line form, a plot of ln(C) versus t data points should equal a straight line with negative slope k and y-intercept value equal to C0. 

The technique of linear regression is very valuable in that it takes the matched data pairs of values of  (y,x ) or  (ln(C), t) in this case and gives you the best straight-line equation that fits these data.   That is, it gives you k and CO in the equation:  ln(C) = -kt + ln(C0).    Once you have k you have a precise estimate of  the ventilation rate based on our tracer gas data.   The technique also tells you how well your data fit the first order decay model.

So how do we do this mathematical magic?  Doing it by hand can be a real pain but doing it by Excel spreadsheet is relatively easy.   First, you have to turn on “Data Analysis” and the Analysis ToolPak if they are not already activated in your copy of Excel (they are free but NOT activated by default).  You do this in Excel 2010 by going to the File tab then Options then Add-ins.  At that point you will see Analysis ToolPak.   You simply add it to Excel at this point and it shows up in the DATA tab.  If you have a different version of Excel or other problems then go to Google and look for YouTube presentations that show you step by step how to add this important tool.

Open a spreadsheet and let’s use the tracer gas data from the July 22 blog:   Concentration, t pairs:    16 ppm @ 0 hour,  12.5 @ 0.5,  9.7 @ 1, 7.9 @ 1.4 and 4.1 ppm after 2.8 hours.   Make a column in the spreadsheet for all the concentrations.    Thus cell A1 has 16 and A5 has 4.1.    Now go to cell B1 and put in =ln(A1).   This calculates the natural logarithm of 16 ppm or 2.7725.    Copy B1 down to B5 so that B5 now equals the ln(A5) or 1.410987  in this example.   In the C column put in the respective times for these concentrations; that is, 0 for C1 filling the cells in down to 2.8 hours for C5.

Now go to the DATA tab and then click “Data Analysis” to the right.   A small window opens entitled “Data Analysis” with many choices.  Go to “Regression” and hit OK.  Highlight the cells B1 thru B5.   Then go to the “input y range”: click the cursor into this window and then highlight B1 thru B5.   It will show up in the window as $B$1:$B$5.   You can also put this exact string in manually.  Now click the cursor into the Input X Range window and then highlight C1 thru B5.   It should show up as $C$1:$C$5.   At this point just hit OK and the magic happens.  A new worksheet opens up (Sheet 4) with all the analysis data.

Remember the straight line equation form we are using in the regression:  y = bx + c .  In this case straight line equation is ln(C)  = kt + ln(C0).      The calculated intercept is the model-predicted ln(C0).   In the regression model calculated by Excel this value is 2.76455 (in cell B17 intercept coefficient).  Taking e to this power is the model-predicted C0 = 15.9 ppm which is very close to the 16 ppm we actually measured at time equal zero.   The value for k is in cell B18 (X variable coefficient) and is -0.49 which is pretty close to the -0.5 that we estimated from looking at the data and assuming a first-order model without doing the regression analysis.    So we now have a reasonably precise analysis of ventilation from our tracer gas data because we now know that Q/V = 0.49/hr.

One of the neatest benefits from running the model is that it tells you how good the model fits the data using the first-order decay assumption.   In this case the R² value (cell B5) in Sheet4 is 0.9996 which means that 99.96% of the data is explained or accounted for by the model and only 0.04% can be considered random error or noise.   

I will be happy to send a copy of this spreadsheet complete with the sheet4 analysis if you email me at mjayjock@gmail.com.   I will be away this week so it may take some time for me to get back to you with the spreadsheet.

I went a bit deep into the technical woods on this one but I thought it might be worth it because the first order decay model is very important for a number of exposure modeling situations.   This example was for ventilation rate but you can use it for modeling the time-dependent rate of an evaporating spill if you have data on the remaining weight or size of spill versus time.