Conceptual Models

As has been shown above, the most severe obstacle in understanding the actual mechanism of the reaeration processes (or any transfer of gases with low solubility) lies in the limited thickness of the aqueous boundary layer (Figure 1). Performing measurements to elucidate the transport characteristics within this thin layer is extremely difficult even though novel modern measurement methods will surely bring further insights.

The predominant approach to formulating theoretical models of gas transfer has been based on conceptual descriptions of the eddy motions, pioneered by Higbie's 'penetration model'. This model assumed that the turbulence in the bulk region of the fluid would bring up fresh packages of liquid to the surface, where gas transfer takes place for a certain renewal time T (Figure 1). Danckwerts generalized Higbie's 'penetration model' by allowing the constant renewal time T to follow an exponential probability distribution of surface renewal rate r, so that

where 1/r may be thought as the mean time between surface renewal events.

The renewal models have shown that KL depends on the square root of the molecular diffusivity D. The hydrodynamics affecting the gas-transfer process, however, are hidden in the r term that must be determined experimentally for individual turbulence conditions. Some researchers tried to explicitly relate r with measurable turbulent parameters (the hydrodynamic behavior). Fortescue and Pearson assumed that the largest turbulent eddies dominate the gas-transfer process so that r can be approximated by «'/and the so-called 'large eddy model' can be deduced, Kl = (D-u /L)1/2, with «' the root mean square turbulent fluctuation, and L the turbulent integral length scale. On the other hand, Lamont and Scott and also Banerjee et al. suggested that small eddies of Kolmogorov scale are the dominant mechanism controlling the transfer process so that r«(e/v)1/2 with e the turbulent energy dissipation rate near the interface and v the kinematic viscosity. This yields the so-called 'small eddy model' KL = yD(e/v)1=4. The range of large and small eddy models can also be written as

where Sc is the Schmidt number (Sc = v/D, approximately 500 for oxygen), c is a constant, and b has a value of —0.5 and —0.25 for the large and small eddy model, respectively. Up to now, there is still no general agreement on the power dependence of the Reynolds number. Theofanus interpreted the differences in the exponent as being dependent on the range of the turbulence intensity level involved and proposed a two-regime model in which the large eddy model is relevant at low turbulent Reynolds numbers ReT < 500 and the small eddy model at high ReT > 500. Another approach to unify the two regimes uses an expression relating KL to the surface divergence of the turbulent eddies.

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