Abstract
Well-mixed zone models are often employed to compute indoor air quality and occupant exposures. While effective, a potential downside to assuming instantaneous, perfect mixing is underpredicting exposures to high intermittent concentrations within a room. When such cases are of concern, more spatially resolved models, like computational-fluid dynamics methods, are used for some or all of the zones. But, these models have higher computational costs and require more input information. A preferred compromise would be to continue with a multi-zone modeling approach for all rooms, but with a better assessment of the spatial variability within a room. To do so, we present a quantitative method for estimating a room’s spatiotemporal variability, based on influential room parameters. Our proposed method disaggregates variability into the variability in a room’s average concentration, and the spatial variability within the room relative to that average. This enables a detailed assessment of how variability in particular room parameters impacts the uncertain occupant exposures. To demonstrate the utility of this method, we simulate contaminant dispersion for a variety of possible source locations. We compute breathing-zone exposure during the releasing (source is active) and decaying (source is removed) periods. Using CFD methods, we found after a 30 minutes release the average standard deviation in the spatial distribution of exposure was approximately 28% of the source average exposure, whereas variability in the different average exposures was lower, only 10% of the total average. We also find that although uncertainty in the source location leads to variability in the average magnitude of transient exposure, it does not have a particularly large influence on the spatial distribution during the decaying period, or on the average contaminant removal rate. By systematically characterizing a room’s average concentration, its variability, and the spatial variability within the room important insights can be gained as to how much uncertainty is introduced into occupant exposure predictions by assuming a uniform in-room contaminant concentration. We discuss how the results of these characterizations can improve our understanding of the uncertainty in occupant exposures relative to well-mixed models.
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Abbreviations
- C :
-
concentration (conc.) [kg/m3]
- \({\cal C}\) :
-
massless concentration [m−3]
- E :
-
exposure (exp.) [s·kg/m3]
- \({\cal E}\) :
-
massless exposure [s/m3]
- t :
-
time/time from start of release [s]
- t avg :
-
averaging time [s]
- t d :
-
time from end of release [s]
- t mix :
-
mixing time [s]
- t rel :
-
releasing time [s]
- \({\cal T}\) :
-
temporary time-integration variable
- \({\bar X}\) :
-
arithmetic mean of a set X
- \({\tilde X}\) :
-
variable/uncertain value of X
- D:
-
decaying period
- fit:
-
parameter from curve fit
- i :
-
measuring location index
- j :
-
source location index
- R:
-
releasing period
- WM:
-
well-mixed (WM) model
- *:
-
mean-normalized quantity
- G :
-
contaminant generation rate [kg/s]
- V :
-
room volume [m3]
- Q :
-
volume flow rate [m3/s]
- C in :
-
supply air conc. [kg/m3]
- C(t):
-
WM room conc. [kg/m3]
- C 0,WM :
-
WM initial conc. [kg/m3]
- C D,WM :
-
decaying period WM conc. [kg/m3]
- C R,WM :
-
releasing period WM conc. [kg/m3]
- C SS,WM :
-
WM steady-state conc. [kg/m3]
- E R,WM :
-
releasing period WM exp. [kg/m3]
- E D,WM :
-
decaying period WM exp. [kg/m3]
- τ WM :
-
WM room time constant [s]
- s X :
-
sample standard deviation in set X
- \(s_X^2\) :
-
sample standard variance in set X
- δ X :
-
general measure of variability in X
- δ * X :
-
normalized variability in X
- \({{\cal C}_{ij}} \equiv {{\cal C}_j}\left( {t,{{\vec x}_i}} \right)\) :
-
conc. of source-j at point \({{\vec x}_i}\)
- \({{\bar {\cal C}}_j}\) :
-
source-averaged conc. of source-j
- \({\overline{\overline {\cal C}} _j}\) :
-
total-average concentration
- \({{\cal E}_{ij}} \equiv {{\cal E}_j}\left( {t,{{\vec x}_i}} \right)\) :
-
exp. to source-j at point \({{\vec x}_i}\)
- \({{\bar {\cal E}}_j}\) :
-
source-averaged exp. to source-j
- \({\overline{\overline {\cal E}} _j}\) :
-
total-average exposure
- \({{\cal C}_{0,{\rm{fit}}}}\) :
-
decaying period initial fit conc.
- \({{\cal C}_{0,{\rm{CFD}}}}\) :
-
decaying period CFD initial conc.
- \({{\cal C}_{{\rm{D}},{\rm{fit}}}}\left( {{t_{\rm{d}}}} \right)\) :
-
decaying period fitted avg. conc.
- \({{\cal C}_{{\rm{R}},{\rm{fit}}}}\left( t \right)\) :
-
releasing period fitted avg. conc.
- \({{\cal C}_{{\rm{SS}},{\rm{fit}}}}\) :
-
releasing period scaling term
- \({{\cal C}_{{\rm{SS}},{\rm{fi}}{{\rm{t}}^ * }}}\) :
-
steady-state conc.
- \({{\cal C}_{{\rm{I,D}},{\rm{fit}}}}\) :
-
decaying period mixing-stage offset
- \({{\cal C}_{{\rm{I,R}},{\rm{fit}}}}\) :
-
releasing period mixing-stage offset
- τ R,fit :
-
releasing period time constant
- τ D,fit :
-
decaying period time constant
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Acknowledgements
This research was supported in parts by the U.S. Defense Threat Reduction Agency and performed under U.S. Department of Energy Contract No. DE-AC02-05CH11231. We would like to thank Dr. Ruoyou You for providing us with the experimental data used to validate our model, and would also like to thank ANSYS Inc. who provided us with the software licenses used for this study.
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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by John Castellini. The first draft of the manuscript was written by John Castellini and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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Castellini, J.E., Faulkner, C.A., Zuo, W. et al. Quantifying spatiotemporal variability in occupant exposure to an indoor airborne contaminant with an uncertain source location. Build. Simul. 16, 889–913 (2023). https://doi.org/10.1007/s12273-022-0971-3
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DOI: https://doi.org/10.1007/s12273-022-0971-3