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).

Reaeration occurs by diffusion of oxygen from the atmosphere into the stream and by the mixing of water and air that occurs during turbulent flow.

An adequate dissolved oxygen concentration is a basic requirement for a healthy aquatic ecosystem. Dissolved oxygen concentrations in streams are a function of atmospheric reareation, photosynthesis, plant and animal respiration, benthic (sediment) demand, biochemical oxygen demand, nitrification, salinity, and temperature. The change in dissolved oxygen concentration on a given day is calculated:

$\Delta Ox_{str}=(\kappa_2*(Ox_{sat}-Ox_{str})+(\alpha_3* \mu _a-\alpha_4*\rho_a)*algae-\kappa_1*cbod-\frac{\kappa_4}{1000*depth}-\alpha_5*\beta_{N,1}*NH4_{str}-\alpha_6*\beta_{N,2}*NO2_{str})*TT$

7:3.5.1

where $\Delta Ox_{str}$ is the change in dissolved oxygen concentration (mg O$_2$/L), $\kappa_2$ is the reaeration rate for Fickian diffusion (day$^{-1}$ or hr$^{-1}$), $Ox_{sat}$ is the saturation oxygen concentration (mg O$_2$/L), $Ox_{str}$ is the dissolved oxygen concentration in the stream (mg O$_2$/L), $\alpha_3$ is the rate of oxygen production per unit of algal photosynthesis (mg O$_2$/mg alg), $\mu _a$ is the local specific growth rate of algae (day$^{-1}$ or hr$^{-1}$), $\alpha _4$ is the rate of oxygen uptake per unit of algae respired (mg O$_2$/mg alg), $\rho_a$ is the local respiration or death rate of algae (day$^{-1}$ or hr$^{-1}$), $algae$ is the algal biomass concentration at the beginning of the day (mg alg/L), $\kappa_1$ is the CBOD deoxygenation rate (day$^{-1}$ or hr$^{-1}$), $cbod$ is the carbonaceous biological oxygen demand concentration (mg CBOD/L), $\kappa_4$ is the sediment oxygen demand rate (mg O$_2$/(m$^2$.day) or mg O$_2$/(m$^2$.hr)), $depth$ is the depth of water in the channel (m), $\alpha_5$ is the rate of oxygen uptake per unit NH$^+_4$ oxidation (mg O$_2$/mg N), $\beta_{N,1}$ is the rate constant for biological oxidation of ammonia nitrogen (day$^{-1}$ or hr$^{-1}$), $NH4_{str}$ is the ammonium concentration at the beginning of the day (mg N/L), $\alpha_6$ is the rate of oxygen uptake per unit $NO_2^-$ oxidation (mg O$_2$/mg N), $\beta_{N,2}$ is the rate constant for biological oxidation of nitrite to nitrate (day$^{-1}$ or hr$^{-1}$), $NO2_{str}$ is the nitrite concentration at the beginning of the day (mg N/L) and $TT$ is the flow travel time in the reach segment (day or hr). The user defines the rate of oxygen production per unit algal photosynthesis, the rate of oxygen uptake per unit algal respiration, the rate of oxygen uptake per unit NH$_4^+$ oxidation and rate of oxygen uptake per unit $NO_2^-$ oxidation. Section 7:3.1.2.1 describes the calculation of the local growth rate of algae while equation 7:3.1.17 describes the calculation of the local respiration rate of algae. The rate constant for biological oxidation of NH$^+_4$ is calculated with equation 7:3.2.5 while the rate constant for $NO_2^-$ oxidation is determined with equation 7:3.2.9. The CBOD deoxygenation rate is calculated using equation 7:3.4.2. The calculation of depth and travel time are reviewed in Chapter 7:1.

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

$\kappa_4=\kappa_{4,20}*1.060^{(T_{water}-20)}$ 7:3.5.2

where $\kappa_4$ is the sediment oxygen demand rate (mg O$_2$/(m$^2$.day) or mg O$_2$/(m$^2$.hr)), $\kappa_{4,20}$ is the sediment oxygen demand rate at 20$\degree$C (mg O$_2$/(m$^2$.day) or mg O$_2$/(m$^2$.hr)), and $T_{water}$ is the average water temperature for the day or hour ($\degree$C).

The amount of oxygen that can be dissolved in water is a function of temperature, concentration of dissolved solids, and atmospheric pressure. An equation developed by APHA (1985) is used to calculate the saturation concentration of dissolved oxygen:

$Ox_{sat}=exp[-139.34410+\frac{1.575701*10^5}{T_{wat,K}}-\frac{6.642308*10^7}{T_{wat,K}^2}+\frac{1.243800*10^{10}}{T_{wat,K}^3}-\frac{8.621949*10^{11}}{T_{wat,K}^4}]$ 7:3.5.3

where $Ox_{sat}$ is the equilibrium saturation oxygen concentration at 1.00 atm (mg O$_2$/L), and $T_{wat,K}$ is the water temperature in Kelvin (273.15+$\degree$C).

Reareation will occur when water falls over a dam, weir, or other structure in the stream. To simulate this form of reaeration, a “structure” command line is added in the watershed configuration file (.fig) at every point along the stream where flow over a structure occurs.

The amount of reaeration that occurs is a function of the oxygen deficit above the structure and a reaeration coefficient:

$\Delta Ox_{str}=D_a-D_b=D_a(1-\frac{1}{rea})$ 7:3.5.10

where $\Delta Ox_{str}$ is the change in dissolved oxygen concentration (mg O$_2$/L), $D_a$ is the oxygen deficit above the structure (mg O$_2$/L), $D_b$ is the oxygen deficit below the structure (mg O$_2$/L), and $rea$ is the reaeration coefficient.

The oxygen deficit above the structure, $D_a$, is calculated:

$D_a=Ox_{sat}-Ox_{str}$ 7:3.5.11

where $Ox_{sat}$ is the equilibrium saturation oxygen concentration (mg O$_2$/L), and $Ox_{str}$ is the dissolved oxygen concentration in the stream (mg O$_2$/L).

Butts and Evans (1983) documents the following relationship that can be used to estimate the reaeration coefficient:

$rea=1+0.38*coef_a*coef_b*h_{fall}*(1-0.11*h_{fall})*(1+0.046*\overline T_{water})$ 7:3.5.12

where $rea$ is the reaeration coefficient, $coef_a$ is an empirical water quality factor, $coef_b$ is an empirical dam aeration coefficient, $h_{fall}$ is the height through which water falls (m), and $\overline T_{water}$ is the average water temperature ($\degree$C).

The empirical water quality factor is assigned a value based on the condition of the stream:

$coef_a$ = 1.80 in clean water

$coef_a$ = 1.60 in slightly polluted water

$coef_a$ = 1.00 in moderately polluted water

$coef_a$ = 0.65 in grossly polluted water

The empirical dam aeration coefficient is assigned a value based on the type of structure:

$coef_b$ = 0.70 to 0.90 for flat broad crested weir

$coef_b$ = 1.05 for sharp crested weir with straight slope face

$coef_b$ = 0.80 for sharp crested weir with vertical face

$coef_b$ = 0.05 for sluice gates with submerged discharge

Table 7:3-5: SWAT+ input variables used in in-stream oxygen calculations.