Vertical mixing, continuous model

From these semiquantitative considerations we conclude that competition between, on one hand, processes which eliminate concentration gradients like vertical mixing, and on the other hand, processes which produce concentration gradients like in-situ reactions and boundary fluxes, can be highly variable in time. Sometimes vertical mixing is intensive enough to warrant the box-model approach, other times we would need a model that allows us to describe spatially continuous concentration profiles. Sometimes we need a tool which could handle both situations. Such a model will be discussed next. 

Continuous crystallizers frequently employ draft tubes to enforce mixing up and down the vertical axis. Even with draft tubes, however, the solids will not be uniformly distributed in general. Figure 8.3 shows solids distribution in a laboratory model of a draft tube crystallizer (Green and Robertson 1993). Notice that there are variations with vertical position inside the crystallizer. In addition, the average concentration inside the draft tube is different from that outside because the net particle velocity and therefore particle hold-up is different where the particle settling velocity is opposite the net flow, i.e., in up flow than where they settle in the net direction of the flow, i.e., down flow. This is discussed in more detail in Mullin (1994), Jones and Mullin (1973), and the experimental modeling section below. 

The smallest spatial scale at which outdoor air pollution is usually modeled corresponds to the air volume affected by chemical emissions from a point source, such as a smokestack (Fig. 4.27). Chemicals are carried downwind by advection, while turbulent transport (typically modeled as Fickian transport) causes the chemical concentrations to become more diluted as the plume mixes laterally and vertically. Typically, smokestacks produce emissions continuously and thus steady-state analysis is often appropriate. At some distance downwind, the plume disperses sufficiently to reach the groimd the point at which this occurs, and the concentrations of the chemicals at this point and elsewhere, can be estimated from solutions to the advection-dispersion-reaction equation (Eq. 1.5). 

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