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Appendix C A Simple Box Mode! with Recirculation One can formulate a simple mode] to follow the evolution of pollutant concentrations based on looking at the air mass above an area of interest as a well-mixed box. in this case, the time varying volume of the box, V(t), is given as the area over which the model is being applied, multiplied by the effective height into which the pollutants are mixed (i.e., the mixing height, H(t)), noting that it can vary with time. V(t) = L ~v7I(t) where L is the length of the box (in the direction of the wind) and His the width of the box perpendicular to the wind, as shown in Figure C-1 . Applying the principle of conservation of mass, one then gets dLWHc dt = Min - MOut + E where Min is the mass flux in (mass/time) due to the wind carrying the contaminant (CO) in from outside and Moot is the mass flux out; c is the time dependent CO concentration (mass/volume) in the box, and E the time dependent emission rate from all sources in the box. The mass flux in is given by the wind velocity, U. multiplied by the area of the side of the box (OH) and background CO concentration Cb, Min = UW~6t9Cb. 194

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Appendix C 195 / UCb ~ H(t) ~ EU c(t) T W Uc FIGURE C-1 Diagram of a simple box model win a box of width W. length L, and time dependent height H(t) and wind speed U. E is the total mass rate of pollutant emissions within the box, which is assumed to be well mixed, with CO concentration c throughout; cb is the background concentration. If the background concentration is small it can be neglected. As noted below, the assumption that the mass flow into the box depends only on the background concentration may be wrong if there is CO that leaves the box but is recirculated. The mass flux out is given by Mout= UVVH(t)c where the assumption that the box is well mixed leads to having the con- taminant concentration leaving the box the same as that within it. Using the above two expressions for Min and Moot leads to the classical box model formulation for the evolution of pollutant concentration c, TIC UCb Uc E 1 dH~ At L :L 1,WH(~t) H(t~a7tl dH >0 dt . The last term causes the concentration to decrease if the mixing height increases with time, because the CO mixes in an expanding volume. If the mixing height decreases with time, the change in height has no effect on the concentration in the box (the last term is zero).

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196 Appendix C if the emission rate is much greater than the flux in due to the wind (E >> Mind, the first term can be dropped. This equation can account for the time variation in both the mixing height and emission rate, but assumes that the length and width of the box are fixed. Further, it assumes that the pol- lutant concentrations do not vary spatially within the box, which implies that the emissions do not vary significantly spatially. This limits the size of the mode] application area, and means that it should not be used to esti- mate hot spot concentrations. Also, it assumes that none of the contami- nant that leaves the box returns, except after being added to the background levels. This assumption can be wrong. In that case, the mode! should be modified to account for the fraction of contaminant that leaves the box and is recirculated back into it. This can be done by adding a recirculation coefficient ~ that is the Faction of contaminant that returns to the box after being advected out. Physically, or must be between O and 1. The resulting mode! becomes TIC UCb U(~-a~c E c do Ucb E ~U(~-~) = + _ _ =+ _ + C'. At L L LWH(t) H(t) dt dH>o L LWH(t) L H(t) If or is zero, the original box model is obtained. As cc approaches one, virtually all of the air and contaminant leaving the box re-enters, allowing the contaminant concentration to build up dramatically.