Monday, December 30, 2013

The Eddy Diffusion Near Field Model is Now Useable

I am going back to a “nuts and bolts” piece on inhalation exposure modeling this week.   The subject is a near field model that has been around for many years but was not very useful until recent work has promised to make it so.

The model is the Eddy Diffusivity Model.   The basic model is presented below:

It may not look like it but the math is pretty straightforward especially if you let IH MOD do the work (see previous blog MODELING MATH MADE EASY OR AT LEAST EASIER to learn about IH MOD Excel Spreadsheet Models and documentation).    Conceptually, the model is pretty simple.   If you have a small source its vapors will radiate out as a sphere if it is suspended in air,  They will radiate as a hemisphere if it’s on a flat surface, as a quarter-sphere if it is on a floor surface along a wall and as a 1/8 sphere if it’s in the corner.  The above equation is for a sphere, the 4 become a 2 for a hemisphere, a 1 for a quarter-sphere and 0.5 for a 1/8 sphere.  What is cool about it is that the concentration is a continuously decreasing gradient as you go away from the point source.   That is, as the distance from the source (r) increases then C decreases.   It does NOT need or use the well-mixed assumption of the 1 zone or 2 zone models.

Seems like it would be the ideal model for such sources but there was one major problem.   All the parameters in the model are relatively easy or straightforward to estimate or measure except D.  Indeed, the predictions of this model are highly depended on D as defined above.   D is dependent on how the air moves about randomly in the indoor environment and it has historically proven itself to be very difficult to measure.   As a result we have had to use a very wide range of estimates for D and as such the utility of this model was quite limited.

Enter some sharp researchers from Stanford University and their work on estimating D from parameters in the room that are much easier to measure; namely, ventilation rate expressed as air changer per hour (ACH) and the dimensions of the room.  They published their work in the journal of Environmental Science and Technology (ES&T) which has a very good reputation.    This part of that paper boils down to the following simple regression relationship:

D  = L2  (0.60 (ACH) + 0.25)/60    (units:  m2/min)
                L = cube root of the room volume (m)
               ACH  = mixing air changes per hour in the room volume (hr-1)

The R2  regression fit for this sub-model is 0.70 which means that 70% of the relationship between D and the room volume and ventilation and D is explained or predicted by the model and about 30% is unexplained or random noise.  In my experience, given the other uncertainties involved, this is pretty good.  This algorithm is applicable over an ACH range of 0.1 to 2.0.    Dr. Kai-Chung Cheng was first author on this paper and it is my understanding that he is pursuing additional work to sharpen up this relationship and to add to its applicability.   Dr. Rachael Jones (University of Illinois at Chicago, School or Public Health) is also a brilliant modeler and a very active researcher in this area.  I understand that she is also planning research to deepen our quantitative understanding of these relationships.   In the mean time I have put the above algorithm to estimate D into a spreadsheet which I will happily send to whoever asks me for it at

I plan to use it whenever I use the 2 box model (see previous blog: THE MOST VERSATILE AND WELL-TESTED INHALATION MODEL) to compare the results and try and learn something about what these different models are telling us. 

The reference for the Stanford paper is:

Kai-Chung Cheng, Viviana Acevedo-Bolton, Ruo-Ting Jiang, Neil E. Klepeis, Wayne R. Ott, Oliver B. Fringer, and Lynn: Modeling Exposure Close to Air Pollution Sources in Naturally Ventilated Residences: Association of Turbulent Diffusion Coefficient with Air Change Rate M. Hildemann, | Environ. Sci. Technol. 2011, 45, 4016–4022

I am quite sure that Dr. Cheng will be happy to send you a pdf copy if you write to him at: KaiChung Cheng


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