Forecasting Occupational Exposures Using “Well Mixed Room” Models
In the
, Chris Keil made the case that modeling should be a prominent tool in our collective exposure assessment toolbox. We never have the resources to sample every exposure scenario, and we are often asked to provide quick answers regarding probable concentrations. In these and other scenarios, modeling can provide estimates of potential exposures. This article shows how a relatively simple “well mixed room” (WMR) model can be used to predict current and future exposures. To illustrate, let’s look at a hypothetical scenario involving welding operations.
We have been asked to evaluate the health risk associated with a newly installed welding operation at a distant repair facility. It has come to the attention of the shop manager that the welding rod manufacturer recommends using the recently revised ACGIH TLV for manganese of 0.02 mg/m3 (respirable) rather than the archaic OSHA limit. The repairman/welder uses a respirator, but now there is concern regarding the manganese exposures for nearby employees. Can we
y predict likely manganese levels prior to visiting the site and collecting personal exposure measurements? After several phone calls, we know the following:
Room and general ventilation.
The shop’s inside dimensions are roughly 12 m by 9 m, with a ceiling height of just over 5 m, giving us a volume (ignoring objects within the shop) of 540 m3. The bay doors are generally open in warm weather, but usually the shop relies upon mechanical ventilation, with fresh air and return ducts mounted high on opposing walls. The mechanical ventilation was designed to provide a ventilation rate (Q) of around 54 m3/min (6 air changes per hour).
A work table placed roughly in the middle of the room has been dedicated to welding repair. A typical shift involves around 10 welding repairs, with a range of 1 to 20 per shift. Each repair takes roughly 10 to 20 minutes and requires 2 to 5 minutes of arc time, with 3 minutes being typical. This is followed by several minutes of non-welding activities and preparation for the next repair. Shielded metal arc welding (SMAW) is used, and the most common welding rod used is a 1/8-inch E7018. (For simplicity, let’s ignore voltage and amperage settings.) The repairman recalls that he can detect a metallic “welding” smell during these tasks, but after 30 to 60 minutes the smell is no longer noticeable. This suggests that nearly all of the welding fumes have been removed via Q from the shop within a fairly short period.
Agent and generation rate.
Manganese is considered the agent of concern when welding in an open, well-ventilated space using mild and medium steel welding consumables (rods and wires) and base metals. A quick search reveals a recent article in the
Annals of Occupational Hygiene
that determined the fume generation rate for 3/16-inch E7018 rods in a controlled, laboratory setting. From the data presented, we calculate a generation rate (G) of 11.5 mg/min for manganese fume (which may overestimate the actual generation rate for the smaller-diameter rod used in the shop).
Local controls.
No local ventilation—installed or portable—is being utilized. To protect nearby workers from viewing the high intensity arc, the table is surrounded by opaque welding curtains (that is, screens). The repairman feels that curtains impede air flow and create a somewhat restricted space.
Personal protection.
The repairman uses a low-profile half-mask with an assigned protection factor (APF) of 10, but according to the manufacturer, the filters are expected to remove 99.97 percent of all airborne particles.
At this point, while we would prefer to measure actual exposures (and will eventually), we have sufficient information for some back-of-the-envelope calculations to predict current bystander (far-field) and welder (near-field) manganese fume levels, as well as future levels should the subject of a portable fume exhaust system come up (and we will ensure that it does).

What is a (nearly) worst-case far-field concentration?
We can use the standard one-box well-mixed room (WMR), constant emission (CE) model, as described in AIHA’s
Mathematical Models for Estimating Occupational Exposure to Chemicals
, to predict the steady-state, general room (that is, far-field) concentration. A welding process is a well-known CE source—one that emits at a fairly constant rate—since fumes are emitted only during the “arc time.” (Welding fumes are particulates, but because of their small particle size, they tend to act like vapors and follow the air streams.) Associated with each WMR model are two sets of equations: one or two “steady-state” equations, and one or two “transient” equations. The steady-state equations are easy to use, whereas the transient equations require a spreadsheet or software. In this scenario we will use the steady-state equation to estimate a worst-case concentration. Here we assume that the welder is welding constantly, without significant breaks; that sufficient time has elapsed for the room concentration to increase and then level out at the steady-state concentration; and the airflow in the room allows for fairly rapid, complete mixing throughout the room. Given these assumptions, the steady-state concentration is predicted to be:
where G = generation rate (mg/min) and Q = room ventilation rate (m3/min).
A steady-state concentration is reached whenever the mass released into the room exactly balances the mass removed from the room via the general ventilation rate Q (and any other source of ventilation). For a CE scenario it also represents the maximum “average room” concentration that can be achieved, and in this case it is more than ten times the OEL, as is shown in Figure 1. The concentration curve, produced using the standard one-box, WMR “transient” model, shows that the steady-state concentration will be reached after about an hour. We conclude that nearly uninterrupted welding is highly likely to produce unacceptable far-field levels in this shop, upwards of ten times the OEL (but perhaps higher or lower, depending upon room location and the actual degree of mixing).
Figure 1.
Modeling predicts that far-field, steady-state exposure to manganese fumes will be ten times greater than the OEL within one hour.
Figure 2.
Modeling a single welding task predicts far-field TWA exposures well below the PEL.
What are probable high, medium, and low far-field concentrations?
If the predicted worst-case concentration is less than the OEL, we could predict with reasonable accuracy that the majority of actual exposures are likely to also be less than the OEL. (Of course, sampling would be required to verify this prediction.) Unfortunately, the predicted worst-case levels are considerably greater than the OEL. We now need to predict more realistic exposure levels that take into account the frequency and duration of welding. Figure 2 shows the predicted far-field concentration curve for a low exposure day—that is, a single welding repair task—but using the reported maximum of 5 minutes of arc time per task. The figure shows, starting 30 minutes into the shift, a five-minute rise in concentration, corresponding to the welding arc time, followed by a period of concentration decay that extends to the end of the shift. (The curve was calculated using the transient concentration rise and decay equations for the WMR one-box model in AIHA’s
Mathematical Models
textbook, and can be reproduced using the free IHMod spreadsheet available from the
web page
of AIHA's Exposure Assessment Strategies Committee.) The far-field, full-shift TWA is predicted to be 0.002 mg/m3. The room concentration an hour after welding ceased approaches zero (that is, nearly all of the fume has been removed from the room via the general ventilation system). Figure 3 shows the predicted far-field curve for a medium exposure day—ten welding repair tasks—again assuming five minutes of arc time per task. The far-field TWA is predicted to slightly exceed the OEL. Figure 4 shows a probable high exposure day, with twenty welding repairs. The predicted far-field TWA is more than twice the OEL. The TWA in each figure was calculated by integrating (that is, measuring) the area under each concentration curve. (A sampling device does exactly the same thing when it gives us the average level across the sampling period.) Yes, we applied calculus integration techniques and computer algorithms to calculate the TWA in each figure using custom software. But given one reasonable assumption, we can calculate a full- or partial-shift TWA with just a hand calculator.
Figure 3.
Modeling ten welding tasks predicts a far-field TWA slightly above the PEL.
Figure 4.
The far-field TWA is expected to be more than twice the OEL on a high-exposure workday.
Two papers in the January issue of JOEH propose slight modifications to the standard one-box steady-state equation used earlier, and the standard two-box (near-field and far-field) steady-state equations. (I coauthored these papers with Gary H. Ganser.) If we assume that the initial and final concentrations are nearly always zero or near zero, it is possible to calculate the full-shift (or even partial-shift) TWA, for nearly any number of tasks, using the modified one-box steady-state equation. For example, using Equation 4 from the first JOEH paper we can calculate an identical “high” far-field concentration:
where γ = fraction of time that welding occurs, n = number of repairs, tg = fume generation time per repair (mg/min), and T = total time (min). Similarly, the shift TWAs in Figures 2 and 3 can be calculated by simply changing n to 1 and 10: 0.002 and 0.022 mg/m3.
We now have estimates of low, medium, and high far-field, general room manganese levels that are just as accurate as those determined using a calculus integration approach. The medium and high estimates suggest the possibility that actual far-field levels routinely approach and exceed the OEL.
What if a portable LEV unit is purchased?
The two JOEH papers also propose several new one-box and two-box steady-state and transient models. One model, called “Model 108 - 1Box.CE.LevR.Gv.SS,” was designed for scenarios where a portable local control (with filtered return to the work space) is used to capture a contaminant. The Model 108 steady-state equation (below) requires that we know, or can approximate, the exhaust rate of the portable unit and its filtration efficiency. We also need an estimate of its operational collection efficiency. We quickly find, via the internet, a candidate unit with an advertised exhaust rate of 20 m3/min and a filtration efficiency of 99.99 percent (for all particles). Let’s assume that the welder will be careful enough so that the moveable exhaust hood captures, on average, 75 percent of the freshly generated fume. The predicted “high” far-field exposure is:
where εL = fraction of welding fume that is immediately captured by the LEV unit, εL.F = portable LEV filtration efficiency, and QL=LEV exhaust flowrate (m3/min). By changing n to 1 and 10, we calculate probable low and medium far-field TWA exposures of 0.0004 and 0.004 mg/m3. It appears that a portable LEV system will be adequate for controlling far-field concentrations to less than half of the OEL, provided it is used properly.
We could also use the two-box steady-state models from the second JOEH paper to calculate the probable near-field concentrations for the welder (for comparison to the “maximum use concentration” for the respirator), with and without local controls, but I will leave this for you to do.

So far, the modeling illustrated in this article has been deterministic. A single value was assigned to each variable, so only a single concentration could be calculated, but the variables can be changed to generate estimates of the probable low, medium, and high concentrations. What would be more useful is an estimate of the overall exposure profile. For this we must apply probabilistic modeling methods, where one or more of the variables are themselves modeled using a statistical distribution.
Figure 5.
A Monte Carlo simulation predicts an exposure profile with a 95th percentile that is greater than the OEL.
Figure 6.
The predicted exposure profile with use of a portable LEV.
For example, Figure 5 shows a histogram of 10,000 artificial full-shift TWAs, created using a Monte Carlo simulation. This predicted exposure profile has a 95th percentile that exceeds the OEL by nearly 50 percent, so we probably have a Category 4 exposure scenario for employees in the far field. Figure 6 shows the predicted exposure profile whenever a portable LEV is used. The predicted 95th percentile is less than half of the OEL, suggesting a Category 2 exposure profile. (The details of these simulations are beyond the scope of this article, but AIHA’s Exposure Assessment Strategies Committee offers a PDC on applying probabilistic methods to modeling.)
Our conclusion, using either deterministic or probabilistic modeling, is that current far-field levels are likely to frequently exceed the manganese OEL, and that the use of LEV will probably control the majority of far-field exposures to less than half of the OEL.
The WMR models require estimates of the relevant physical and chemical determinants of exposure for a specific work scenario, and use equations that place the various determinants in their proper mathematical relationship. In principle, WMR models correctly calculate the overall average far-field and near-field levels provided that reality conforms to the model’s assumptions.
But rooms are seldom “well mixed”; there are concentration hot spots, dead spots, and gradients throughout. Furthermore, workloads vary, the generation rate varies, and each worker moves throughout the room (sometimes leaving it) in a unique pattern each day, accumulating exposure daily in different ways. In addition, there may be unknown sources of the contaminant, as well as unrecognized contaminant-loss mechanisms, due to chemical reactions, adsorption to surfaces, or settling (for particulates). Last, our parameter estimates may be biased. For example, our estimate of the manganese fume generation rate was based on published data reported for a thicker welding rod. However, the empirical evidence indicates that we can expect a predicted concentration to be usually within half to two times the true concentration, provided that an appropriate model is selected and reasonable values are assigned to the model’s variables.
A welding fume scenario was chosen because it is a classic constant emissions source, but also because welding generates thermal plumes that can complicate the interpretation of the modeling results. The simple WMR models do not take into account your knowledge of the process, materials, work practices, contaminant loss mechanisms, nearby sources, and work flow, to name a few of many possible considerations. You may have to mentally adjust the predicted concentrations to reflect your professional judgment and monitoring experience.
Take a one-box or two-box model out for a drive on your next survey. Compare the predictions to your measurements. You may find that they can be reasonably predictive and you can amaze your boss and colleagues with your newfound prognosticating abilities. With a little study and practice, you could easily be the eight-hundred-pound “exposure science” gorilla in the room.
is a consultant with Exposure Assessment Solutions, Inc. in Morgantown, W.Va. He can be reached at
Send feedback to

Articles published in the last twenty years on the app
lication of one- and two-box WMR models are too numerous to list here. Most appeared in the JOEH and the
Annals of Occupational Hygiene
(now called the
Annals of Work Exposure and Health
AIHA’s Exposure Assessment Strategies Committee (EASC) sponsors PDCs on modeling and modeling using Monte Carlo simulation. (Unfortunately, neither will be presented at AIHce 2017.) IHMod, an Excel spreadsheet designed to be used with AIHA’s
Mathematical Models
textbook, can be downloaded from the EASC
web page
. The IHMod user’s guide (
) is also available.
AIHA: Mathematical Models for Estimating Occupational Exposure to Chemicals
(2009). Annals of Occupational Hygiene: “Profiling Mild Steel Welding Processes to Reduce Fume Emissions and Costs in the Workplace” (May 2014).
Journal of Occupational and Environmental Hygiene
: “Models for Nearly Every Occasion: Part I - One Box Models” and “Models for Nearly Every Occasion: Part II - Two Box Models” (January 2017).
The Synergist
: “
Patterns of Exposure: The Power and Utility of Mathematical Models
” (January 2017).
Given one reasonable assumption, we can calculate a full- or partial-shift TWA with just a hand calculator.