Reaeration By Fickian Diffusion

The user defines the reaeration rate at 20$\degree$C. The reaeration rate is adjusted to the local water temperature using the relationship:

$\kappa_2=\kappa_{2,20}*1.024^{(T_{water}-20)}$ 7:3.5.4

where $\kappa_2$ is the reaeration rate (day$^{-1}$ or hr$^{-1}$), $\kappa_{2,20}$ is the reaeration rate at 20$\degree$C (day$^{-1}$ or hr$^{-1}$), and $T_{water}$ is the average water temperature for the day or hour ($\degree$C).

Numerous methods have been developed to calculate the reaeration rate at 20$\degree$C, $\kappa_{2,20}$. A few of the methods are listed below. Brown and Barnwell (1987) provide additional methods.

Using field measurements, Churchill, Elmore and Buckingham (1962) derived the relationship:

$\kappa_{2,20}=5.03*v_c^{0.969}*depth^{-1.673}$ 7:3.5.5

where $\kappa_{2,20}$ is the reaeration rate at 20$\degree$C (day$^{-1}$), $v_c$ is the average stream velocity (m/s), and $depth$ is the average stream depth (m).

O’Connor and Dobbins (1958) incorporated stream turbulence characteristics into the equations they developed. For streams with low velocities and isotropic conditions,

$\kappa_{2,20} =294 * \frac{(D_m* v_c)^{0.5}}{depth^{1.5}}$ 7:3.5.6

where $\kappa_{2,20}$ is the reaeration rate at 20$\degree$C (day$^{-1}$), $D_m$ is the molecular diffusion coefficient (m$^2$/day), $v_c$ is the average stream velocity (m/s), and $depth$ is the average stream depth (m). For streams with high velocities and nonisotropic conditions,

$\kappa_{2,20}=2703*\frac{D_m^{0.5}*slp^{0.25}}{depth^{1.25}}$ 7:3.5.7

where $\kappa_{2,20}$ is the reaeration rate at 20$\degree$C (day$^{-1}$), $D_m$ is the molecular diffusion coefficient (m$^2$/day), $slp$ is the slope of the streambed (m/m), and $depth$ is the average stream depth (m). The molecular diffusion coefficient is calculated

$D_m=177*1.037^{\overline T_{water}-20}$ 7:3.5.8

where $D_m$ is the molecular diffusion coefficient (m$^2$/day), and $\overline T_{water}$ is the average water temperature ($\degree$C).

Owens et al. (1964) developed an equation to determine the reaeration rate for shallow, fast moving streams where the stream depth is 0.1 to 3.4 m and the velocity is 0.03 to 1.5 m/s.

$\kappa_{2,20}=5.34*\frac{v_c^{0.67}}{depth^{1.85}}$ 7:3.5.9

where $\kappa_{2,20}$ is the reaeration rate at 20$\degree$C (day$^{-1}$), $v_c$ is the average stream velocity (m/s), and $depth$ is the average stream depth (m).