In this section, some representative empirical equations of the reaeration rate are given. The classical context for such equations are analyses in rivers, describing for example the oxygen dynamics in a river downstream of a continuous discharge of liquid organic waste. As the organic waste is being advected downstream, it degrades. This biodegradation process consumes oxygen representing a sink of DO in the river. This oxygen consumption of the organic waste is in its initial stage of predominantly carbonaceous degradation usually given as a first-order process:

in which L is the biological oxygen demand (BOD) in units of mass of concentration and K1 is the first-order decay coefficient. Typical values of K1 range from 0.05 to 0.4 day-1. In a flowing river with mean velocity U, the time change is represented by travel time t — x/U, where x is the distance downstream from the discharge point; thus

The reduced DO bulk concentration C resulting from this process drives a simultaneous reaeration effect as a source effect at the free interface of the water body.

The overall mass balance for this sink/source process is given as dC Kl .

dx H

in which K2 — KL/H is the 'reaeration coefficient', a volumetric coefficient (in units of time-1) that averages the surface transfer velocity KL over the water depth H.

The solution of eqns [5] and [6] is given by the classical Streeter-Phelps equation for oxygen sag in a river:

K1Lo

-exp

"U

Do exp

in which D represents the DO deficit, D = Cs — C, and Lo and Do are the initial values of L and D at the discharge location, respectively. This equation shows clearly the interplay between sink (K1) and source (K2) effects on DO concentration.

Reliable predictions of the reaeration coefficient K2, or the transfer velocity KL, are crucial for the modeling of water quality in aquatic environments. In the last few decades, many such relations have been proposed. The approaches can be generally classified into two categories. (1) In conceptual models, a mechanism for the interaction between the molecular diffusion and turbulent flows near the interface is postulated. The main problem of such models is to develop a reliable relation between the variables of the near-surface turbulence and the global stream parameters. (2) In empirical models, regression techniques are employed in order to acquire the best relation between the observed reaeration rate K2 in field or laboratory experiments as a function of some global stream parameters, such as the mean velocity, slope, and/or water depth. The main drawback of these models is the limited applicability of locally developed equations to other streams' conditions. Implementation of one equation to other stream conditions beyond the range of parameters considered in the original equation may lead to enormous errors in the estimation of K2.

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