Multi-box models

Abstract

The boundary between the numerical advection—diffusion methods of volume elements and the multi-box models tends to be blurred. In the latter, the region to be studied is divided into cells not necessarily identical in area and height. No diffusion between the boxes is assumed; the concentration in each cell or box is considered uniform. This is perhaps the most important difference from previously outlined models where a (variable) diffusion constant had to be used. As Scriven and Fisher (1975) pointed out, in the ideal box model infinite diffusivity inside the individual box is assumed. The mass-conservation equation is solved for each of the boxes. The uniform concentration within a particular box at any time is a function of the box volume, of the rate at which material is being imported, of the emission rate, of the concentration within the box in the preceding time increment and of the residual fractions of these three terms in describing the amount of material remaining in the box. For box n, the functional relationship is, as expressed by Reiquam (1969, 1970, 1971),

$$$$

(6.1)

where x_n,t is the concentration, V_n.t the volume, q_n.t the rate at which pollutants are advected into box n, Q _n.t the emission rate within box n, r_n.t the residual of x _n,t-1 remaining, r_n.t the residual of Q_n.t remaining, p _n.t the residual of remaining, all at the end of time increment, and x _n.t-1 the concentration in box n at time t −1. The residual fractions are simple geometrical relationships between the resultant wind vector and the box dimensions.