# 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).