Modeling Dissolved Oxygen
Oxygen Balance in Rivers and Streams
Selecting a Metric for DO Fluxes
Typically use oxygen demand (OD):
- measure of the concentration of oxidizable materials in a water sample
- metric for organic waste contamination
- reflects how oxygen will be depleted in a given segment
But there are several different processes affecting total OD!
Biochemical Oxygen Demand (BOD)
Oxygen used by microbes during aerobic decomposition of organic materials: \[\text{Organic Matter} + \text{O}_2 \rightarrow \text{CO}_2 + \text{H}_2\text{O} + \text{NO}_3 + \text{SO}_2 + \text{Residuals}\]
Broadly speaking, we care about two types of BOD: Carbonaceous BOD and Nitrogenous BOD.
Carbonaceous BOD (CBOD)
Oxygen consumed during microbial decomposition of carbon compounds, e.g.: \[\text{C}_a\text{H}_b\text{O}_c + d\text{O}_2 \rightarrow e\text{H}_2\text{O} + f\text{CO}_2\]
Nitrogenous BOD (NBOD)
Oxygen consumed during microbial decomposition of nitrogen compounds: \[2\text{NH}_2^+ + 4\text{O}_2 \rightarrow 2\text{H}_2\text{O} + 4\text{H}^+ + 2\text{NO}_3^-\]
BOD and Time
Moreover, BOD is differentiated based on time frame, e.g.:
- BOD5: oxygen demand over 5 days
- BOD20: oxygen demand over 20 days
DO Modeling Needs
Need a model that will predict DO as a function of CBOD, NBOD.
Use a fate and transport modeling approach: how are relevant quantities moved downstream?
Note: Can’t assume homogeneous processes.
Modeling DO
So what do we do?
Start by assuming steady-state waste in each section (or box…).
Modeling DO
We’ll track the mass balance in terms of rates (not absolute mass).
What happens to an element of water?
Steady-State Waste, DO Mass Balance
Let \(U\) be the river velocity (km/d), \(x\) the distance downstream from a waste release site in km, and \(C(x)\) the DO concentration at \(x\) in mg/L.
\[\begin{align}
U \frac{dC}{dx} &= \text{Change in DO} \\[0.5em]
&= \text{Reaeration} + \text{Photosynthesis} - \text{Respiration} \\[0.5em]
& \qquad - \text{Benthal Uptake} - \text{CBOD} - \text{NBOD}
\end{align}\]
BOD Oxygen Uptake
Assume deoxygenation from waste decomposition is first-order (rate \(k\)):
\[\begin{aligned}
\frac{dM}{dt} &= -kM \\\\
\Rightarrow M &= M_0 \exp(-kt)
\end{aligned}\]
But our equations are formulated in terms of distance:
\[M = M_0 \exp(-kx/U)\]
BOD Oxygen Uptake
For biochemical organics, if \(k_c\) is the deoxygenation rate (d\(^{-1}\));
\[B(x) = B_0 \exp(-k_c x / U);\]
For nitrification, if \(k_n\) is the deoxygenation rate (d\(^{-1}\)):
\[N(x) = N_0 \exp(-k_n x / U).\]
Steady-State Waste, DO Mass Balance
So the corresponding oxygen uptake rates are \[k_c B(x) = k_c B_0 \exp(-k_c x / U)\] and
\[k_n N(x) = k_n N_0 \exp(-k_n x / U).\]
Steady-State Waste, DO Mass Balance
Other processes:
Reaeration, assume a simple linear model based on difference from saturation level \(C_s\): \(k_a (C_s - C)\)
Assume measured, constant values for photosynthesis (\(P_s\)), respiration (\(R\)), benthal uptake (\(S_B\))
Steady-State Waste, DO Mass Balance
Putting it all together:
\[\begin{aligned}
U \frac{dC}{dx} &= k_a (C_s - C) + P - R - S_B \\\\
&\quad - k_cB_0\exp\left(\frac{-k_cx}{U}\right) - k_n N_0\exp\left(\frac{-k_nx}{U}\right)
\end{aligned}\]
