A Simple Derivation of the First Order Kinetic Model for Ventilation

Last week I promised to really break down for you the details of the derivation of the equation for first order decay.   You will remember that this means that the amount or rate of decay is proportional to the amount remaining.   The basic equation describing this situation is below:

C is the concentration at any time (t),  C0 is the initial concentration and k is the rate of decay.  The units are mg/m³ for C and C0, hour for t and 1/hours for k.

A Simple Tracer Gas Technique to Measure Ventilation

A simple tracer gas technique can determine the ventilation rate in any room or volume of air.   Last week I discussed natural logarithms with base e.    One of the most useful equations involving natural logs is the equation for first-order decay.  This sounds complicated but all it really means that the rate of loss or decay of a concentration is directly proportional to the amount remaining.   Thus, for example, the rate of decay (or material loss) for a ventilated room volume concentration that is 10 ppm will be twice as high as the loss rate for a  room concentration of 5 ppm under the same ventilation conditions.  More important, the time it takes to get from 10 ppm to 5 ppm is the half-life.   Indeed, it is the same amount of time (i.e., the half-life) that it will take decay or go from 5 ppm to 2.5 ppm, 2.5 ppm to 1.25 ppm, etc.   That is really all there is to first-order kinetics.   Theoretically you never get to zero but after 7 half-lives the concentration is less than 1% of what you started with and after 10 or so it is vanishingly small.

The decay rate of a gas concentration put into a room typically follows a first order decay process driven by the ventilation rate.

The equation is:   - ln (0.5)/(Q/V) =  (0.693)(V/Q) = the half-life  = t_1/2

Thus the ventilation rate (Q) = (0.693)(V/ t_1/2)

Consider a room in a home that has a volume (V) of 20 m³.   We put a tracer gas into this room and mix it up for a few minutes with portable fans so that the average concentration is 16 ppm measured with a real time monitor.  We get the following data over the next few hours:  12.5 ppm at 0.5 hours, 9.7 at 1 hour, 7.9 at 1.4 hours and 4.1 at 2.8 hours.    You would typically have an automatic data-logger to take a reading every 5 minutes or so but the above data points tell the story.   The initial concentration goes in half after about 1.4 hours and in half again in the next 1.4 hours; the half-life is about 1.4 hours.   Put 20 m³ and 1.4 hours in the above equation and  Q = 9.9 m³/hr.   Another way of expressing this is the ratio Q/V or 0.50 air changes per hour.   This is a typical ventilation rate for a home with doors and windows closed in the winter.   Crack the windows open and this can go up 10 fold.   Industrial rooms are typically higher and in “hot” industrial settings or rooms with a lot of local exhaust ventilation it could be much higher.

What are good tracer gases to use?   Freon R-134a used in car air conditioners is easy to get and it works quite well with a portable flame ionization detector (FID).   If you are very careful not to release too much (say 10 ppm max), carbon monoxide (CO) could work well with a suitably sensitive real-time analyzer that can continuously output the concentration in ppm.   In industrial rooms with active gas fired fork lifts there is often enough ambient CO in the room that it represents a “built-in” tracer simply by having the folk lift operators stop for an hour or so (e.g. waiting for lunch or the end of the day) and measuring the CO concentration fall off will give you the data you need.

Next week I will show you more about “e” and natural logarithms; specifically, we will go over how the above equations and similarly useful algorithms are derived.  We will also get into how one can use an Excel spreadsheet to get a more precise estimate of half-life from tracer gas studies.

In the meantime, my friend and colleague Dr. Mark Nicas, who during his career has developed some really useful exposure models and also teaches at Berkeley, has kindly sent me a document which has 18 pages packed with quantitative math and science review notes.  I would be happy to send it to anyone who sends me an email request to mjayjock@gmail.com.    The stuff in this pdf is pretty concentrated with useful information and perhaps it would spawn some questions that I could address in future blogs.

A Simple Refresher of Logarithms

Some folks are intimidated by the mathematics of modeling.  I am here to tell you that by taking a little time to really understand a few basic concepts in math (which I am going to help you with) you are ready to approach and understand the mathematical underpinning of many models.  

When and where many of us went to school, math was not very well taught.  Indeed, many of us were taught to manipulate numbers to get answers but we were not schooled in exactly what the process meant in a larger sense.  We soldiered through math course with a good portion of us hating it but never really seeing what it was good for.   In this blog I am hoping to refresh our understanding of logarithms so that we can clearly understand and use them in models. 

The first thing to understand about logarithms is that they are just another way of expressing a number.  All logarithms have a BASE.   The one we see much of the time is BASE 10.    So the “power” or exponent that 10 is raised to is its logarithm or log.    So the log of 100 is 2 since base 10² is equal to 100.    The base10 log of 10 is equal to 1 for the same reason.    Similarly, if we know that the base 10 log is 3 we know the number is 10³ or 1000.  Calculating logs are easy for simple integers but considerably more difficult for those numbers in between.   In the old days before calculators we had “log tables” to give us these numbers.  Today we just put the number in our calculator; for example, enter 15 and hit LOG to get the log of 15 or 1.1761.  Thus 10^1.1761 = 15.   

 Numbers less than 1 have negative logs. Log of 0.5 =   -0.3010 or   10^-0.3010 = 0.5.   

Try putting in -0.3010 for x and hit the 10^x button and wa-lah!    0.500 is calculated which is the  "antiLOG" of -0.3010.   

Logs are handy in that adding the logs of numbers is the same as multiplying the numbers.    Add -0.3010 to 1.1761 = 0.8751.  The antiLOG of 0.8751 =  10^0.8751 = 7.5 or the same as 0.5 x 15.    Similarly subtracting logs is the same as dividing the numbers.    Subtract -0.3010 from 1.1761 = 1.4771.   10^1.4771= 30 = 15/0.5.   Flip it around and Subtract 1.1761 from  -0.3010 =  -1.4771.   10^-1.47711 = 0.03333 or the same as 0.5/15.   For us folks who have been around since before calculations,  this is how a slide-rule works because it has logarithmically-spaced scales, you are adding or subtracting logs by sliding the scales to do multiplication or division on the slide-rule.

You will see a lot of logs expressed as base 10 because that is our basic number system (a system based on 12 might be better) and that is how many fingers we have.   Logarithmic scales are convenient for compressing large ranges of numbers because in base10 the distance between each integer represents 10x.   For example, the difference in shaking amplitude between a Richter magnitude 7 earthquake is 10 times higher than a magnitude 6 because the scale is base 10 logarithmic.

Now that we understand algorithms in base 10 it is time to understand logarithms with the NATURAL base number.   This natural BASE number is 2.718281... or the italicized lower case letter “e”.   The only thing you need to know about “e”  is that it is an important number in mathematics and actually much more “natural” than 10.   Thus it is called the NATURAL log (sometimes called LN or ln).    The same rules apply, that is, the natural log of 10 is 2.3026 (it’s also on your calculator - sometimes with key named LN) which means that  e^2.3026= 10.

Next week I am going to talk about the subject of integration and how it fits into modeling.   We are going to see “e” again and we will walk through a practical model using a tracer gas that lets us estimate the ventilation rate in any workplace room or any residential volume. 

  

Well mixed model vs Models that consider air concentrations close to the source

Well mixed box models are relatively easy to understand.  You start with a room (a box) and the same amount of fresh air goes into it as comes out of it.  If not it would explode or the folks inside would suffocate. You put a contaminant into the room air (emission rate G) and take it out with ventilation (Q).   Last week we saw that given a steady G and Q that the steady-state concentration (C) in the box will eventually equal G/Q.   There are other more complicated equations that determine C as it approaches steady-state but that is not important for this discussion.   In later blogs I will get into how you can determine or estimate G and Q but for now it is enough to know that this is a simple model and quite useful in many cases.  It is important to understand that the C determined by this model is the AVERAGE concentration within the room.  If the source G is relatively spread out within the room then this model works quite well.  An example of a spread out source would be paint emissions into a room with painted walls and ceiling.  If, however, the source is localized then there typically will be a much higher concentration (C) of the contaminant near the source than away from it.   An example would be someone using a volatile degreaser spray with TCE on a workbench to clean a part.  The larger the room the more the gradient or difference in concentration near the source versus the far corner of the room.  The well mixed box model does not work very well here since the average concentration in the room will be significantly lower than the breathing zone concentration of the person doing the cleaning.

Many years ago I approached this problem by assuming a virtual box within the real box of the room.  For example, I assumed that most of the contaminant would be in an area of an 8 foot cube around the source.   That is a volume of 512 ft3.   It did not matter how big the room was as long as it was larger than this virtual box.  The only thing I needed to know about the actual room was its volume (V) and its ventilation rate (Q).   Once I knew Q for the large room I would figure out the air changes per hour in that large room; that is, Q/V.   This has the units of 1/hr.   Once I had Q/V for the large room I assumed this relative exchange rate would be the same for the virtual box.   So that (Q/V)(512ft3) = Qb or ventilation rate in the virtual box in units of ft3/hr.   I'll let you do the conversion to m3/hr.  The predicted steady-state C  in this box = G/Qb.   If you want a copy of the original paper just email me (mjayjock@gmail.com) and I will send it to you.

Years later Mark Nicas came up with a much more elegant model called the 2 zone model (that considered the virtual box or NEARFIELD and the FARFIELD or rest of the room volume).  This model calculates the C in the virtual volume around the source AND the average concentration in the rest of the room.  This model and lots more are available in the freeware Excel spreadsheet IH MOD from the AIHA web site.
http://www.aiha.org/get-involved/VolunteerGroups/Pages/Exposure-Assessment-Strategies-Committee.aspx

There is another near source model known as the eddy diffusivity model which actually calculates a continuous gradient of exposure from a point source of emission (G) to any location within the volume.  To run this model one needs G and the eddy diffusivity coefficient (D).  Until recently D was hard to come by, but the very smart researchers at Stanford recently published a paper that allows one to estimate D from a room's dimensions and its ventilation rate.  Kai-Chung Cheng, et al,  Modeling Exposure Close to Air Pollution Sources in Naturally Ventilated Residences: Association of Turbulent Diffusion Coefficient with Air Change Rate.  Environ. Sci. Technol. 2011, 45, 4016-4022.  The calculation engine for the eddy diffusivity model is also available in IH MOD.

I appreciate the comments I receive on this blog and it helps me to determine what I am going to cover next.

A Simple Exposure Modeling Example as a way of Getting Started

One of the really nice things about doing a blog is the connection with colleagues.  As a prime example, I received the following note from Dr. Gurumurthy Ramachandran (Ram) about last week’s blog. 

“You hit the nail on the head when you said that all hygienists need to become explicit modelers, not subliminal ones.  Just the process of thinking about each input parameter to even a simple model will lead to a much better understanding of the workplace and the limitations in that understanding. I remember reading somewhere that these two elements correspond to knowledge and self-knowledge.”

Ram is a brilliant teacher and researcher at the University of Minnesota so I consider these words to be very heartening. Indeed they, along with the other comments I received last week, are enough to encourage me to hopefully provide more insight this week to newbies into the modeling process. 

  

The basic elements of all inhalation models are relatively simple.  In every case, you have a volume of air and a rate of contaminant input to that volume.  The model is simply using these elements to predict the concentration in the air that might be inhaled by a worker. 

The model I am going to discuss this week is one of the simplest but still very useful; namely, the well mixed box model at equilibrium.   Any room (or box of air) which is receiving a steady inflow of contaminant will reach a steady (or equilibrium) concentration of contaminant as the amount of contaminant that is put into the room is balanced with the amount that is leaving via ventilation removal.  All volumes or spaces in which we exist have some fresh air ventilation – even our well-insulated homes in winter exchange air with the outside via infiltration/ex-filtration through cracks and other openings.   This exchange typically occurs in the range of 30 to 60% per hour in the rooms of homes and is often well above 100-300% in industrial rooms.

At equilibrium the airborne concentration (C) in the box (room) is equal to the generation or emission rate (G) of contaminant into the box divided by the ventilation rate (Q).                 C = G/Q    G = wt/time.    Q = volume/time     C = wt/volume.   Pretty simple uh?

So how does one estimate G?   Let’s consider an example of a small (20m³) bedroom in which someone is painting with a water-based paint that has 0.5% ethylene glycol (EG) as a drying agent.  If they use 4,000 gram (4,000,000 mg) of paint, that is 20 grams (20,000 mg) of EG.   If it is assumed to take 8 hr to dry and that all of the EG comes out that is an average of 2,500 mg of EG being emitted into the room air per hour.   There are quite a few ways of estimating G and I can go over these in future blogs.

Estimating Q:  The 20m³ room with 60% ventilation/hr is (20 m3)(0.6/hr) = Q = 12 m³/hr.   There are a number of ways of getting a Q which could also be a topic of a future blog.

Using C = G/Q = 2,500/12 = 208 mg/m³ of EG at equilibrium.

That wasn't too bad was it?

This is a simple model – like all models it is a portrayal of reality but NOT reality.  The generation rate is most likely not constant but this exercise does give one some reasonable insight into the process and into the magnitude of the exposure potential.   If the EG comes out of the paint more quickly it could have a peak concentration higher than 200 mg/m³ but perhaps not very much higher.  The time-weight average concentration would always be lower than 200 mg/m³ if our assumption about all of the EG being vaporized in 8 hr is correct. 

If the EG takes much longer to come out it changes a lot of things.  Indeed, rework the above so that it takes 24 hours for all of the EG to come out and again assume that it comes out evenly.  Then the estimated equilibrium concentration for that 24 hour day would be about 70 mg/m³ and if one was only exposed during 8 hrs of that day their exposure could never be higher than about 25 mg/m³.  

There are a number of assumptions here in this simple model but I hope you can see how it might be helpful and how it might encourage you to go and manipulate the inputs to gain more insight or go to more sophisticated models.    IH MOD is a freeware Excel spreadsheet available on the AIHA web site that can do all of the math with ease and it provides a graphical output so that you can better see what is happening.   

I am again at a point where I need some feedback.   Do you good folks need or want information on:

• Specifically where to get IH MOD and exactly what does it do?
• Some of the math background that you may want to brush up on to help you with modeling (Note: It’s not a lot)
• The difference between equilibrium, point-in-time and time-weight average airborne concentrations
• Well mixed models versus models that consider air concentrations close to the source
• How to do source estimation (G)
• How to measure or estimate ventilation rates (Q)
• All or none of the above

Send your wishes/comments to me at mjayjock@gmail.com or in comments to this blog.  Absent any feedback I will go back to talking in generalities about risk assessment, risk management and modeling which many or most of you may find to be preferable.