06 Dissolved Oxygen

.title[Fate and Transport Modeling] .subtitle[BEE 4750/5750] .subtitle[Environmental Systems Analysis, Fall 2022] <hr> .author[Vivek Srikrishnan] .date[September 14, 2022]

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# Outline

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1. Questions?
2. Fate and Transport Modeling
3. Example: Dissolved Oxygen

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# Poll

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URL: [https://pollev.com/vsrikrish](https://pollev.com/vsrikrish)

Text: **VSRIKRISH** to 22333, then message]

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***Any questions?***

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# Last Class

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* Monte Carlo Confidence Intervals
  * Risk

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# Fate and Transport Modeling

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* How do nutrients and other quantities move through environmental mediums?
  * The mass balances can be the result of more complex processes, not just decay/dissipation, requiring simulation one "chunk" of the domain at a time.

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# Example: Dissolved Oxygen

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.left-column[Dissolved oxygen (DO) is the free, non-compound oxygen present in water or other liquids.

Freshwater can only hold small amounts, and thie capacity is regulated by temperature.]

.right-column[.center[![Dissolved Oxygen by temperature](https://www.fondriest.com/environmental-measurements/wp-content/uploads/2013/11/dissolvedoxygen_river-levels.jpg)

.cite[Source: [fondriest.com](https://www.fondriest.com/environmental-measurements/parameters/water-quality/dissolved-oxygen/)]]]

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# Dissolved Oxygen

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.center[![Dissolved Oxygen and Temperature Plot](https://d9-wret.s3.us-west-2.amazonaws.com/assets/palladium/production/s3fs-public/thumbnails/image/do_temp_0.png) .cite[Source: [usgs.gov](https://www.usgs.gov/special-topics/water-science-school/science/dissolved-oxygen-and-water)]]

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# Dissolved Oxygen and Life

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**Hypoxia** occurs when DO levels are $< 2$ mg/L. ]

.right-column[.center[![Minimum DO requirements for freshwater fish](https://www.fondriest.com/environmental-measurements/wp-content/uploads/2013/11/dissolvedoxygen_levels-fresh.jpg)] .center[.cite[Source: [fondriest.com](https://www.fondriest.com/environmental-measurements/parameters/water-quality/dissolved-oxygen/)]] ]

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# Factors Influencing DO

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* Temperature, Pressure, Depth
  * Salinity
  * Mixing
  * Plant and Microbial Life
  * Organic Matter

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# Impact of Paris On DO In The Seine, 1874

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.center[![Dissolved Oxygen Downstream of Paris, 1874](https://www.researchgate.net/publication/325721176/figure/fig10/AS:962439298445322@1606474818380/Longitudinal-profiles-x-axis-in-km-downstream-of-Paris-center-of-dissolved-oxygen_W640.jpg)

.cite[Source: Dmitrieva, T., et al. (2018). [https://doi.org/10.1007/s12685-018-0216-7](https://doi.org/10.1007/s12685-018-0216-7)]]

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# DO Regulatory Standards

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**Objective**: Keep DO *above* the regulatory standard.

In NY:

* DO levels may not fall below 3 mg/L
  * DO may not be below 4.8 mg/L for an extended period

.cite[Source: [Westlaw](https://govt.westlaw.com/nycrr/Document/I4ed90412cd1711dda432a117e6e0f345?viewType=FullText&originationContext=documenttoc&transitionType=CategoryPageItem&contextData=(sc.Default)&bhcp=1)]

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# Oxygen Balance in Rivers and Streams

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# Selecting a Metric for DO Fluxes

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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!

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# Types of Oxygen Demand

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* **Biochemical Oxygen Demand (BOD)**: Oxygen used by microbes during aerobic decomposition of organic materials:

$$
\text{Organic Matter} + \text{O}_2 \rightarrow \text{CO}_2 +  \text{H}_2\text{O} + \text{NO}_3 + \text{SO}_2 + \text{Residuals}
$$

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* **Carbonaceous BOD (CBOD)**: Oxygen consumed during microbial decomposition of carbon compounds, *e.g.*:

$$
\text{C}_a\text{H}_b\text{O}_c + d\text{O}_2 \rightarrow e\text{H}_2\text{O} + f\text{CO}_2
$$

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* **Nitrogenous BOD (NBOD)**: Oxygen consumed during microbial decomposition of nitrogen compounds:

$$
2\text{NH}_2^+ + 4\text{O}_2 \rightarrow 2\text{H}_2\text{O} + 4\text{H}^+ + 2\text{NO}_3^-
$$

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Moreover, BOD is differentiated based on time frame, *e.g.*:

* BOD5: oxygen demand over 5 days
 * BOD20: oxygen demand over 20 days

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# Modeling DO

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* Need a model that will predict DO as a function of CBOD, NBOD
  * But now, use a *fate and transport* modeling approach.
  * Can't assume homogeneous processes.

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# Modeling DO

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Start by assuming *steady-state* waste in each section.]

.right-column[.center[![Processes influencing oxygen balance in moving freshwater](figures/do-processes.svg)]]

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# Steady-State Waste, DO Mass Balance

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$$
\text{Flow in} - \text{Flow out} = U \frac{dC}{dx},
$$

where $C(x)$ is DO (mg/l) $x$ km downstream, $U$ (km/d) is the water velocity.

Additional notation:

* CBOD (mg/l): $B(x)$
  * NBOD (mg/l): $N(x)$

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# Steady-State Waste, DO Mass Balance

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Assume decomposition of wastes is first-order:

$$
\begin{aligned}
\frac{dM}{dt} &= -kM \\\\
\Rightarrow M &= M_0 \exp(-kt)
\end{aligned}
$$

where $M$ is the mass (kg) of waste, $M_0$ is the mass at $t=0, x=0$.

Since $U$ is the velocity:

$$
M = M_0 \exp(-kx/U)
$$

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# Steady-State Waste, DO Mass Balance

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This means:

* For biochemical organics,

$$
B(x) = B_0 \exp(-k_c x / U);
$$

* For nitrification,

$$
N(x) = N_0 \exp(-k_n x / U).
$$

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# Steady-State Waste, DO Mass Balance

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Other processes:

* Reaeration, assume a simple linear model: $k_a (C_s - C)$
  * Assume measured, constant values for photosynthesis ($P_s$), respiration ($R$), benthal uptake ($S_B$)

DO uptake from waste decomposition:

* CBOD: $k_cB = k_cB_0 \exp(-k_cx/U)$
  * NBOD: $k_nN = k_nN_0 \exp(-k_nx/U)$

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# Steady-State Waste, DO Mass Balance

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Putting it all together:

$$
\begin{aligned}
U \frac{dC}{dx} &= k_a (C_s - C) + P - R - S_B \\\\
&\quad - k_cB_0\exp\left(\frac{-k_cx}{U}\right) - k_n N_0\exp\left(\frac{-k_nx}{U}\right)
\end{aligned}
$$

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# Steady-State Mass Balance Solution

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$$
\begin{aligned}
C(x) &= C_s(1 - \alpha_1) + C_0 \alpha_1 - B_0 \alpha_2 - N_0 \alpha_3 \\\\
&\quad + \left(\frac{P-R-S_B}{k_a}\right) (1-\alpha_1),
\end{aligned}
$$

$$
\begin{aligned}
\alpha_1 &= \exp\left(-\frac{k_a x}{U}\right)\\\\
\alpha_2 &= \left(\frac{k_c}{k_a-k_c}\right)\left[\exp\left(\frac{-k_c x}{U}\right) - \exp\left(\frac{-k_ax}{U}\right)\right]\\\\
\alpha_3 &= \left(\frac{k_n}{k_a-k_n}\right)\left[\exp\left(\frac{-k_n x}{U}\right) - \exp\left(\frac{-k_ax}{U}\right)\right]
\end{aligned}
$$

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# Steady-State Mass Balance Solution

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$$
\begin{aligned}
C(x) &= C_s(1 - \alpha_1) + C_0 \alpha_1 - B_0 \alpha_2 - N_0 \alpha_3 \\\\
&\quad + \left(\frac{P-R-S_B}{k_a}\right) (1-\alpha_1),
\end{aligned}
$$

**Note**: Usually, these models ignore $P$, $R$, and $S_B$.

Why do you think that might be?

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# Steady-State Mass Balance Solution

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$$
C(x) = C_s(1 - \alpha_1) + C_0 \alpha_1 - B_0 \alpha_2 - N_0 \alpha_3
$$

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# Multiple Discharges

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**Note**: Need to calculate new concentrations at each discharge due to volume changes!

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# Multiple Discharges

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**Question**: Is it fair to ask every discharge to comply with the regulation? Why or why not?

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# Example DO Questions:

* What treatment % (CBOD/NBOD reduction) is needed to meet standard?
  * What is the probability of noncompliance?

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# Simulation Modeling Framework

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# Next Class

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* Prescriptive Modeling
  * Introduction to Optimization