# 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:

&#x20;   $$\kappa_2=\kappa_{2,20}*1.024^{(T_{water}-20)}$$                                                                               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).

&#x20;              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.

&#x20; Using field measurements, Churchill, Elmore and Buckingham (1962) derived the relationship:

&#x20;           $$\kappa_{2,20}=5.03*v_c^{0.969}*depth^{-1.673}$$                                                        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).

&#x20;          O’Connor and Dobbins (1958) incorporated stream turbulence characteristics into the equations they developed. For streams with low velocities and isotropic conditions,

&#x20;           $$\kappa_{2,20} =294 * \frac{(D_m* v_c)^{0.5}}{depth^{1.5}}$$                                                                       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,  &#x20;

&#x20;         $$\kappa_{2,20}=2703*\frac{D_m^{0.5}*slp^{0.25}}{depth^{1.25}}$$                                                                   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

&#x20;             $$D_m=177*1.037^{\overline T_{water}-20}$$                                                               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).

&#x20;      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.

&#x20;               $$\kappa_{2,20}=5.34*\frac{v_c^{0.67}}{depth^{1.85}}$$                                                                    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).
