Wastewater Modeling Example

Lecture 04

August 30, 2023

Review and Questions

Mathematical Models

A mathematical model of a system is a set of equations which maps inputs (forcings, decisions) to outputs (outcomes, metrics).

Conceptual Model of an Environmental System

Mathematical Models

Mathematical models of systems can be:

  • Deterministic vs. Stochastic
  • Descriptive vs. Prescriptive

Questions?

Poll Everywhere QR Code

Text: VSRIKRISH to 22333

URL: https://pollev.com/vsrikrish

See Results

Wastewater Modeling Example

Wastewater Treatment Example

Diagram of CRUD release from two factories

Two factories are discharging a chemical, chlororadiated ureadicarboxyl (CRUD), into the Riley River.

Wastewater Treatment Example

Diagram of CRUD release from two factories

Environmental authorities have sampled water from the river and CRUD concentrations exceed the legal standard (1 mg/L).

Wastewater Treatment Example

Diagram of CRUD release from two factories

We want to design a CRUD removal plan to get back in compliance.

Wastewater Treatment Example

Diagram of CRUD release from two factories

  • Where do we start?
  • What do we need to know?

CRUD Decay Rate

CRUD decays in the river with first-order decay coefficient \(k=0.45 \ \text{d}^{-1}\).

How Much To Remove?

Consider the problem at each point of release.

  1. What is the inflow? What is the outflow for a given \(E_i\)?
  2. How does this impact the concentration downstream?

\[\begin{align*} \text{(recall that } 1 \ \text{mg/L} &= 1 \ \text{g/m}^3 \\ &= 10^{-3} \ \text{kg/m}^3\text{)} \end{align*}\]

Mass-Balance at Release 1

Factory 1 releases \(1000 \ \text{kg/d}\) without treatment.

What is the outflow for a given treatment fraction \(E_1\)?

Mass-Balance at Release 1

Factory 1 releases \(1000 \ \text{kg/d}\) without treatment.

What is the outflow for a given \(E_1\)?

Total CRUD after factory 1 release: \(\color{blue}100 + \color{red} 1000(1-E_1) \color{black} \ \text{kg/d}\)

Mass-Balance at Release 2

Is this the inflow at Release 2?

CRUD First-Order Mass Decay

Given first-order decay rate of \(0.45 \ \text{d}^{-1}\): \[\frac{dM}{dt} = -0.45 M \Rightarrow \frac{dM}{M} = -0.45 dt\]

\[\int_{M(0)}^{M(T)} \frac{dM}{M} = -0.45 \int_0^T dt\]

\[\ln\left(\frac{M(T)}{M(0)}\right) = -0.45 T\]

First-Order Mass Decay

So, after \(t\) days: \[M(t) = M(0) \exp\left(-0.45 t\right)\]

Is this what we need?

No! We need \(M(x)\), where \(x\) is some distance downstream.

Decay in Terms of Distance

Since the velocity of the river is \(25 \ \text{km/d}\):

\[M(x) = M_0 \exp\left(-\frac{0.45 x}{25}\right), \quad x \leq 10 \ \text{km}.\]

Simplifying and plugging in \(x = 10\) and \(M_0\), the inflow of CRUD at the factory 2 release is:

\[M(10) = (1100 - 1000E_1) \exp(-0.18) \ \text{kg/d}.\]

Mass-Balance at Release 2

This means that after factory 2 releases CRUD, the mass is:

\[\color{red}(1100 - 1000E_1) \exp(-0.18) \color{black}+ \color{blue}1200(1 - E_2) \ \color{black}\text{kg/d}.\]

CRUD Mass Downstream

We can use this as an initial condition for \(M(x), x > 10\):

\[M(x) = M_1 \exp\left(-\frac{0.45x}{25}\right), \quad x > 10,\] where \[M_1 = (1100 - 1000E_1) \exp(-0.18) + 1200(1 - E_2).\]

CRUD Concentration Downstream

Therefore the concentration of CRUD \(C(x)\) (in \(\text{mg/L}\)) at any point \(x \ \text{km}\) downstream of the first release is:

\[ C(x) = \begin{cases} M_0\exp\left(-0.45 x/25\right) / 600, & x \leq 10 \ \text{km} \\ M_1 \exp\left(-0.45x/25\right) / 660, & x > 10 \ \text{km} \\ \end{cases} \] where \[ \begin{align*} M_0 &= 1100 - 1000E_1,\\ M_1 &= (1100 - 1000E_1) \exp(-0.18) + 1200(1 - E_2). \end{align*} \]

What Does This Mean?

Looking at these equations:

The downstream concentration (past Release 2) depends on both \(E_1\) and \(E_2\).

This means whether the regulatory limit is achieved depends on both factories!

We don’t know what needed levels of \(E_2\) are without knowing \(E_1\).

Systems Management Requires the Full Picture

One option: We could leave \(E_1\) up to the owners of Factory 1 and require \(E_2\) be set based on this level.

  • Is this fair?
  • What other information would we need to make this decision?

Key Takeaways

Key Takeaways

  • Systems models can combine different types of processes (e.g. mass balance and reactions).
  • Outcomes in one part of a system depend on others.
  • Often need to consider decisions in different parts of the system together.
  • Key questions about fairness with multiple outcomes or decisions.

Upcoming Schedule

Next Classes

Friday: Lake Eutrophication Example

Wednesday: Box Models and Simulation