## The aggregated dead zone model

There is, however, an alternative approach that is based on the direct analysis of concentration curves. This makes use of the type of DBM transfer function methodology that has already been described in Section 14.2.6 above). The resulting model has been called the aggregated dead zone (ADZ) model (Wallis et al., 1989; Young and Wallis, 1993). It is easiest to apply when the flow is approximately steady and the pollutant or tracer is already well mixed, but does not require that discharge be constant downstream. It then treats the advection of pollution as a pure time delay and the dispersion as the result of mixing in one or more linear stores representing the aggregated effect of dead zone dispersion. If only a single store is needed, then there are only two parameters, the advective time delay, t, and the mean residence time, T. The sum of these two times is the mean residence time in the reach. The ratio

T + t is called the dispersive fraction. The dispersive fraction can also be interpreted under the steady flow assumption as the ratio of an effective mixing volume to the total volume of water participating in the flow in the reach. It is worth noting that this may not be the total volume in the reach if there are real 'dead zones' in the reach that do not interact with either the flow or the pollutant.

These parameters are easily determined for a reach of river where well-mixed concentration curves are available at the entry to and exit from a river reach. Fig. 14.20 shows how well the ADZ model can fit a concentration curve observed during a tracer experiment. For a number of reaches, such curves have been available for a variety of discharges so that relationships between the advective time delay and the dispersive fraction can be developed (Fig. 14.21). Interestingly, in many (but not all) reaches studied with this model, the dispersive fraction has proven to be a near constant (Fig. 14.21b), while the slope of the log discharge v. log advective travel time relationship appears to be rather similar for many (but not all) reaches. This implies a scaling relationship between the effective mixing volume and the total volume in the reach as the total volume increases with discharge. There is no physical reason why this should be the case, it is a purely empirical result. It means, however, that it is relatively simple to set up a model of pollutant transport in a reach.

Thus, in studying the 1994 pollution incident in the River Eden, for example, there was some uncertainty about the timing of the release of the ammonium sulphate that caused the fish kill, so a series of tracer experiments was carried out in the river at a

River Brock

River Brock

Fig. 14.21 Example of change in the aggregated dead zone model parameters with discharge for a reach of the River Brock, south of Lancaster, UK derived from tracer experiments: (a) advective time delay; and (b) dispersive fraction.

Fig. 14.21 Example of change in the aggregated dead zone model parameters with discharge for a reach of the River Brock, south of Lancaster, UK derived from tracer experiments: (a) advective time delay; and (b) dispersive fraction.

similar discharge (4.1 m3 s-1) to that on the day of the pollution incident (3.1 m3 s-1). The difference in discharge would, however, make a significant difference to the travel times and dispersion. Thus, relationships of the form of those shown in Fig. 14.21 were used to estimate the parameters of the ADZ model at the lower discharge, which could then be used to predict the transport of the pollutant and resolve the timing of the input (Fig. 14.22).

The ADZ model can also be applied to the initial mixing problem from an estimate of the point input of tracer, but higher order DBM models are then generally required. Lees et al. (2000), by matching the solutions of the ADE and ADZ models under steady flow conditions have provided relationships that better match the long tails of real concentration curves.

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