13.1 Water quality

Achievable FE Civil

13. Environmental engineering

Our FE Civil course is currently in development and is a work-in-progress.

Water quality

6 min read

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This chapter covers the following:

Biochemical oxygen demand (BOD)
BOD exertion
Kinetic temperature corrections
Stream modeling - Streeter Phelps
Oxygen saturation
Dilution purification of wastewater streams

Biochemical oxygen demand (BOD)

Biochemical oxygen demand (BOD) measures how much dissolved oxygen microorganisms need to biologically oxidize biodegradable organic matter under aerobic conditions. In practice, BOD is most commonly reported as the 5-day test at 20°C, written as $BOD_{5}$ .

Because BOD reflects the amount of biodegradable organic pollution present, it’s widely used in:

wastewater treatment design
regulatory compliance
evaluating impacts on receiving waters

Basic BOD testing/sampling equation

$BOD (mg/L) = \frac{D _{1} - D _{2}}{P}$

When the dilution water is seeded:

$BOD (mg/L) = \frac{( D _{1} - D _{2} ) - ( B _{1} - B _{2} ) f}{P}$

Where:

$D_{1}$ = dissolved oxygen of diluted sample immediately after preparation (mg/L)
$D_{2}$ = dissolved oxygen of diluted sample after 5-day incubation at 20°C (mg/L)
$B_{1}$ = dissolved oxygen of seed control before incubation (mg/L)
$B_{2}$ = dissolved oxygen of seed control after incubation (mg/L)
$f$ = fraction of seeded dilution water volume in sample to volume of seeded dilution water in seed control
$P$ = fraction of wastewater sample volume to total combined volume

$BO D_{t}$ :

$BOD_{t}$ is the BOD exerted after $t$ days of incubation, based on the difference between the blank and the sample, scaled by the dilution factor.

$BOD_{t} = (D O_{b, t} - D O_{s, t}) \times dilution factor$

Where:

$D O_{b, t}$ = dissolved oxygen concentration in blank after $t$ days in incubation (mg/L)
$D O_{s, t}$ = dissolved oxygen concentration in sample after $t$ days in incubation (mg/L)

Dilution factor:

$Dilution factor = \frac{volume of bottle or diluted sample}{volume of undiluted sample}$

Volume of seed, undiluted (mL):

$= \frac{allowable depletion \times volume of bottle}{estimated BOD of sample}$

BOD exertion

BOD exertion describes how oxygen demand is applied over time as microorganisms decompose organic matter. The demand isn’t used up instantly; it increases gradually as biochemical reactions proceed. Under ideal conditions, this behavior is modeled with first-order kinetics.

$BOD_{t} = L_{0} (1 - e^{- k t})$

Where:

$BOD_{t}$ = amount of BOD exerted at time $t$ (mg/L)
$k$ = BOD decay rate constant, base $e$ (day⁻¹)
$k = 2.30 K$
$K$ = rate constant, base 10 (day⁻¹)
$L_{0}$ = ultimate BOD (mg/L)
$t$ = time (days)

Kinetic temperature corrections

Reaction rates in water quality processes depend strongly on temperature. To model conditions at temperatures other than the standard reference of 20°C, you adjust the rate constant using a temperature correction.

$k_{T} = k_{20} θ^{T - 20}$

Where:

$T$ = temperature of interest (°C)
$k_{T}$ = BOD rate constant at the temperature of interest (day⁻¹)
$k_{20}$ = BOD rate constant determined at 20°C (day⁻¹)
$θ$ = temperature coefficient
Reaeration: $θ = 1.024$

Stream modeling - Streeter Phelps

The Streeter-Phelps model describes dissolved oxygen (DO) changes in a stream after an organic waste discharge. It combines two competing processes:

oxygen depletion from BOD exertion (deoxygenation)
oxygen replenishment from the atmosphere (reaeration)

The model predicts the DO deficit downstream, including the critical point where DO reaches its minimum before recovering toward saturation.

A two-part diagram showing BOD oxygen consumption over time and the resulting dissolved oxygen sag curve (Streeter-Phelps model) with initial deficit, critical point, and recovery to saturation downstream of discharge. — Stream modeling with Streeter Phelps

$D = \frac{k _{d} L _{a}}{k _{r} - k _{d}} [e^{- k_{d} t} - e^{- k_{r} t}] + D_{a} e^{- k_{r} t}$

$t_{c} = \frac{1}{k _{r} - k _{d}} ln (\frac{k _{r}}{k _{d}} (1 - D_{a} \frac{k _{r} - k _{d}}{k _{d} L _{a}}))$

$D O = D O_{sat} - D$

Where:

$D$ = dissolved oxygen deficit (mg/L)
$D O$ = dissolved oxygen concentration (mg/L)
$D_{a}$ = initial dissolved oxygen deficit in mixing zone (mg/L)
$D O_{sat}$ = saturated dissolved oxygen concentration (mg/L)
$k_{d}$ = deoxygenation rate constant, base $e$ (day⁻¹)
$k_{r}$ = reaeration rate constant, base $e$ (day⁻¹)
$L_{a}$ = initial ultimate BOD in mixing zone (mg/L)
$t$ = time (days)
$t_{c}$ = time at which minimum dissolved oxygen occurs (days)

Oxygen saturation

Oxygen saturation is the maximum dissolved oxygen concentration water can hold when it’s in equilibrium with the atmosphere. It depends mainly on:

temperature
atmospheric pressure
salinity

Saturation is a key reference value because it sets the upper limit for DO and helps define the oxygen deficit used in stream modeling.

$D O_{sat} = K_{H} \cdot P_{O_{2}}$

Where:

$K_{H}$ = Henry’s Law constant (moles/L·atm)
$P_{O_{2}}$ = partial pressure of oxygen (atm)

$D = D O_{stream} - D O$

Where:

$D$ = oxygen deficit (mg/L)
$D O_{stream}$ = saturation concentration of dissolved oxygen at the temperature of the stream after mixing (mg/L)
$D O$ = actual concentration of dissolved oxygen in stream (mg/L)

Dilution purification of wastewater streams

Dilution and natural purification control what happens to wastewater after it enters a receiving water.

Dilution lowers pollutant concentrations through mixing with cleaner water.
Purification reduces pollutants through processes such as biodegradation, sedimentation, and reaeration.

Together, these processes determine a stream’s assimilative capacity and help guide discharge permit limits.

If untreated or partially treated sewage is instantly mixed upon discharge into a large water body, the resulting parameter levels (e.g., temperature, DO, BOD, suspended solids) can be estimated using a weighted average. If the mixed conditions meet water quality standards, pretreatment may not be required.

Consider a wastewater flow rate $Q_{w}$ with ultimate BOD $L_{w}$ and dissolved oxygen $D O_{w}$ mixing with a river with flow rate $Q_{r}$ , ultimate BOD $L_{r}$ , and dissolved oxygen $D O_{r}$ .

The initial (immediately after mixing) ultimate BOD of the river-wastewater mix is:

$L_{o} = \frac{Q _{w} L _{w} + Q _{r} L _{r}}{Q _{w} + Q _{r}}$

Initial dissolved oxygen immediately after mixing $D O_{o}$ is given by:

$D O_{o} = \frac{Q _{w} D O _{w} + Q _{r} D O _{r}}{Q _{w} + Q _{r}}$

Temperature immediately after mixing $T_{o}$ is given by:

$T_{o} = \frac{Q _{w} T _{w} + Q _{r} T _{r}}{Q _{w} + Q _{r}}$

Initial oxygen deficit after mixing $D_{o}$ is given by:

$D_{o} = D O_{s a t} - \frac{Q _{w} D O _{w} + Q _{r} D O _{r}}{Q _{w} + Q _{r}}$

Biochemical oxygen demand (BOD)

Measures oxygen needed by microbes to oxidize organic matter
Key for wastewater treatment design, compliance, and impact assessment
Main formula: $BOD = \frac{D _{1} - D _{2}}{P}$
- Adjust for seeded dilution: $BOD = \frac{( D _{1} - D _{2} ) - ( B _{1} - B _{2} ) f}{P}$
$BOD_{t}$ after $t$ days: $(D O_{b, t} - D O_{s, t}) \times dilution factor$

BOD exertion

Describes time-dependent oxygen demand as organics decompose
Modeled by first-order kinetics: $BOD_{t} = L_{0} (1 - e^{- k t})$
- $L_{0}$ = ultimate BOD, $k$ = decay rate constant
$k = 2.30 K$ (base 10 to base $e$ conversion)

Kinetic temperature corrections

Reaction rates increase with temperature
Adjust rate constant: $k_{T} = k_{20} θ^{T - 20}$
- $θ = 1.024$ for reaeration
- $k_{20}$ = rate at 20°C, $T$ = temperature of interest

Stream modeling - Streeter Phelps

Predicts DO changes after waste discharge
- Deoxygenation (BOD exertion) vs. reaeration (atmospheric O $_{2}$ )
DO deficit: $D = \frac{k _{d} L _{a}}{k _{r} - k _{d}} [e^{- k_{d} t} - e^{- k_{r} t}] + D_{a} e^{- k_{r} t}$
Critical point (minimum DO): $t_{c} = \frac{1}{k _{r} - k _{d}} ln (\frac{k _{r}}{k _{d}} (1 - D_{a} \frac{k _{r} - k _{d}}{k _{d} L _{a}}))$
$D O = D O_{sat} - D$

Oxygen saturation

Maximum DO water can hold at equilibrium with atmosphere
- Depends on temperature, pressure, salinity
$D O_{sat} = K_{H} \cdot P_{O_{2}}$
Oxygen deficit: $D = D O_{stream} - D O$

Dilution purification of wastewater streams

Dilution: lowers pollutant concentration by mixing
Purification: pollutant reduction via biodegradation, sedimentation, reaeration
Initial mixed values (after discharge):
- Ultimate BOD: $L_{o} = \frac{Q _{w} L _{w} + Q _{r} L _{r}}{Q _{w} + Q _{r}}$
- DO: $D O_{o} = \frac{Q _{w} D O _{w} + Q _{r} D O _{r}}{Q _{w} + Q _{r}}$
- Temperature: $T_{o} = \frac{Q _{w} T _{w} + Q _{r} T _{r}}{Q _{w} + Q _{r}}$
- Oxygen deficit: $D_{o} = D O_{s a t} - \frac{Q _{w} D O _{w} + Q _{r} D O _{r}}{Q _{w} + Q _{r}}$

Next | 13.2 Water treatment and distribution

Water quality

This chapter covers the following:

Biochemical oxygen demand (BOD)
BOD exertion
Kinetic temperature corrections
Stream modeling - Streeter Phelps
Oxygen saturation
Dilution purification of wastewater streams

Biochemical oxygen demand (BOD)

Because BOD reflects the amount of biodegradable organic pollution present, it’s widely used in:

wastewater treatment design
regulatory compliance
evaluating impacts on receiving waters

Basic BOD testing/sampling equation

$BOD (mg/L) = \frac{D _{1} - D _{2}}{P}$

When the dilution water is seeded:

$BOD (mg/L) = \frac{( D _{1} - D _{2} ) - ( B _{1} - B _{2} ) f}{P}$

Where:

$D_{1}$ = dissolved oxygen of diluted sample immediately after preparation (mg/L)
$D_{2}$ = dissolved oxygen of diluted sample after 5-day incubation at 20°C (mg/L)
$B_{1}$ = dissolved oxygen of seed control before incubation (mg/L)
$B_{2}$ = dissolved oxygen of seed control after incubation (mg/L)
$f$ = fraction of seeded dilution water volume in sample to volume of seeded dilution water in seed control
$P$ = fraction of wastewater sample volume to total combined volume

$BO D_{t}$ :

$BOD_{t}$ is the BOD exerted after $t$ days of incubation, based on the difference between the blank and the sample, scaled by the dilution factor.

$BOD_{t} = (D O_{b, t} - D O_{s, t}) \times dilution factor$

Where:

$D O_{b, t}$ = dissolved oxygen concentration in blank after $t$ days in incubation (mg/L)
$D O_{s, t}$ = dissolved oxygen concentration in sample after $t$ days in incubation (mg/L)

Dilution factor:

$Dilution factor = \frac{volume of bottle or diluted sample}{volume of undiluted sample}$

Volume of seed, undiluted (mL):

$= \frac{allowable depletion \times volume of bottle}{estimated BOD of sample}$

BOD exertion

$BOD_{t} = L_{0} (1 - e^{- k t})$

Where:

$BOD_{t}$ = amount of BOD exerted at time $t$ (mg/L)
$k$ = BOD decay rate constant, base $e$ (day⁻¹)
$k = 2.30 K$
$K$ = rate constant, base 10 (day⁻¹)
$L_{0}$ = ultimate BOD (mg/L)
$t$ = time (days)

Kinetic temperature corrections

$k_{T} = k_{20} θ^{T - 20}$

Where:

$T$ = temperature of interest (°C)
$k_{T}$ = BOD rate constant at the temperature of interest (day⁻¹)
$k_{20}$ = BOD rate constant determined at 20°C (day⁻¹)
$θ$ = temperature coefficient
Reaeration: $θ = 1.024$

Stream modeling - Streeter Phelps

The Streeter-Phelps model describes dissolved oxygen (DO) changes in a stream after an organic waste discharge. It combines two competing processes:

oxygen depletion from BOD exertion (deoxygenation)
oxygen replenishment from the atmosphere (reaeration)

The model predicts the DO deficit downstream, including the critical point where DO reaches its minimum before recovering toward saturation.

$D = \frac{k _{d} L _{a}}{k _{r} - k _{d}} [e^{- k_{d} t} - e^{- k_{r} t}] + D_{a} e^{- k_{r} t}$

$t_{c} = \frac{1}{k _{r} - k _{d}} ln (\frac{k _{r}}{k _{d}} (1 - D_{a} \frac{k _{r} - k _{d}}{k _{d} L _{a}}))$

$D O = D O_{sat} - D$

Where:

$D$ = dissolved oxygen deficit (mg/L)
$D O$ = dissolved oxygen concentration (mg/L)
$D_{a}$ = initial dissolved oxygen deficit in mixing zone (mg/L)
$D O_{sat}$ = saturated dissolved oxygen concentration (mg/L)
$k_{d}$ = deoxygenation rate constant, base $e$ (day⁻¹)
$k_{r}$ = reaeration rate constant, base $e$ (day⁻¹)
$L_{a}$ = initial ultimate BOD in mixing zone (mg/L)
$t$ = time (days)
$t_{c}$ = time at which minimum dissolved oxygen occurs (days)

Oxygen saturation

Oxygen saturation is the maximum dissolved oxygen concentration water can hold when it’s in equilibrium with the atmosphere. It depends mainly on:

temperature
atmospheric pressure
salinity

Saturation is a key reference value because it sets the upper limit for DO and helps define the oxygen deficit used in stream modeling.

$D O_{sat} = K_{H} \cdot P_{O_{2}}$

Where:

$K_{H}$ = Henry’s Law constant (moles/L·atm)
$P_{O_{2}}$ = partial pressure of oxygen (atm)

$D = D O_{stream} - D O$

Where:

$D$ = oxygen deficit (mg/L)
$D O_{stream}$ = saturation concentration of dissolved oxygen at the temperature of the stream after mixing (mg/L)
$D O$ = actual concentration of dissolved oxygen in stream (mg/L)

Dilution purification of wastewater streams

Dilution and natural purification control what happens to wastewater after it enters a receiving water.

Dilution lowers pollutant concentrations through mixing with cleaner water.
Purification reduces pollutants through processes such as biodegradation, sedimentation, and reaeration.

Together, these processes determine a stream’s assimilative capacity and help guide discharge permit limits.

Consider a wastewater flow rate $Q_{w}$ with ultimate BOD $L_{w}$ and dissolved oxygen $D O_{w}$ mixing with a river with flow rate $Q_{r}$ , ultimate BOD $L_{r}$ , and dissolved oxygen $D O_{r}$ .

The initial (immediately after mixing) ultimate BOD of the river-wastewater mix is:

$L_{o} = \frac{Q _{w} L _{w} + Q _{r} L _{r}}{Q _{w} + Q _{r}}$

Initial dissolved oxygen immediately after mixing $D O_{o}$ is given by:

$D O_{o} = \frac{Q _{w} D O _{w} + Q _{r} D O _{r}}{Q _{w} + Q _{r}}$

Temperature immediately after mixing $T_{o}$ is given by:

$T_{o} = \frac{Q _{w} T _{w} + Q _{r} T _{r}}{Q _{w} + Q _{r}}$

Initial oxygen deficit after mixing $D_{o}$ is given by:

$D_{o} = D O_{s a t} - \frac{Q _{w} D O _{w} + Q _{r} D O _{r}}{Q _{w} + Q _{r}}$

Achievable FE Civil

13. Environmental engineering

Our FE Civil course is currently in development and is a work-in-progress.

Water quality

6 min read

Font

Discuss

Feedback

This chapter covers the following:

Biochemical oxygen demand (BOD)
BOD exertion
Kinetic temperature corrections
Stream modeling - Streeter Phelps
Oxygen saturation
Dilution purification of wastewater streams

Biochemical oxygen demand (BOD)

Because BOD reflects the amount of biodegradable organic pollution present, it’s widely used in:

wastewater treatment design
regulatory compliance
evaluating impacts on receiving waters

Basic BOD testing/sampling equation

$BOD (mg/L) = \frac{D _{1} - D _{2}}{P}$

When the dilution water is seeded:

$BOD (mg/L) = \frac{( D _{1} - D _{2} ) - ( B _{1} - B _{2} ) f}{P}$

Where:

$D_{1}$ = dissolved oxygen of diluted sample immediately after preparation (mg/L)
$D_{2}$ = dissolved oxygen of diluted sample after 5-day incubation at 20°C (mg/L)
$B_{1}$ = dissolved oxygen of seed control before incubation (mg/L)
$B_{2}$ = dissolved oxygen of seed control after incubation (mg/L)
$f$ = fraction of seeded dilution water volume in sample to volume of seeded dilution water in seed control
$P$ = fraction of wastewater sample volume to total combined volume

$BO D_{t}$ :

$BOD_{t}$ is the BOD exerted after $t$ days of incubation, based on the difference between the blank and the sample, scaled by the dilution factor.

$BOD_{t} = (D O_{b, t} - D O_{s, t}) \times dilution factor$

Where:

$D O_{b, t}$ = dissolved oxygen concentration in blank after $t$ days in incubation (mg/L)
$D O_{s, t}$ = dissolved oxygen concentration in sample after $t$ days in incubation (mg/L)

Dilution factor:

$Dilution factor = \frac{volume of bottle or diluted sample}{volume of undiluted sample}$

Volume of seed, undiluted (mL):

$= \frac{allowable depletion \times volume of bottle}{estimated BOD of sample}$

BOD exertion

$BOD_{t} = L_{0} (1 - e^{- k t})$

Where:

$BOD_{t}$ = amount of BOD exerted at time $t$ (mg/L)
$k$ = BOD decay rate constant, base $e$ (day⁻¹)
$k = 2.30 K$
$K$ = rate constant, base 10 (day⁻¹)
$L_{0}$ = ultimate BOD (mg/L)
$t$ = time (days)

Kinetic temperature corrections

$k_{T} = k_{20} θ^{T - 20}$

Where:

$T$ = temperature of interest (°C)
$k_{T}$ = BOD rate constant at the temperature of interest (day⁻¹)
$k_{20}$ = BOD rate constant determined at 20°C (day⁻¹)
$θ$ = temperature coefficient
Reaeration: $θ = 1.024$

Stream modeling - Streeter Phelps

The Streeter-Phelps model describes dissolved oxygen (DO) changes in a stream after an organic waste discharge. It combines two competing processes:

oxygen depletion from BOD exertion (deoxygenation)
oxygen replenishment from the atmosphere (reaeration)

The model predicts the DO deficit downstream, including the critical point where DO reaches its minimum before recovering toward saturation.

$D = \frac{k _{d} L _{a}}{k _{r} - k _{d}} [e^{- k_{d} t} - e^{- k_{r} t}] + D_{a} e^{- k_{r} t}$

$t_{c} = \frac{1}{k _{r} - k _{d}} ln (\frac{k _{r}}{k _{d}} (1 - D_{a} \frac{k _{r} - k _{d}}{k _{d} L _{a}}))$

$D O = D O_{sat} - D$

Where:

$D$ = dissolved oxygen deficit (mg/L)
$D O$ = dissolved oxygen concentration (mg/L)
$D_{a}$ = initial dissolved oxygen deficit in mixing zone (mg/L)
$D O_{sat}$ = saturated dissolved oxygen concentration (mg/L)
$k_{d}$ = deoxygenation rate constant, base $e$ (day⁻¹)
$k_{r}$ = reaeration rate constant, base $e$ (day⁻¹)
$L_{a}$ = initial ultimate BOD in mixing zone (mg/L)
$t$ = time (days)
$t_{c}$ = time at which minimum dissolved oxygen occurs (days)

Oxygen saturation

Oxygen saturation is the maximum dissolved oxygen concentration water can hold when it’s in equilibrium with the atmosphere. It depends mainly on:

temperature
atmospheric pressure
salinity

Saturation is a key reference value because it sets the upper limit for DO and helps define the oxygen deficit used in stream modeling.

$D O_{sat} = K_{H} \cdot P_{O_{2}}$

Where:

$K_{H}$ = Henry’s Law constant (moles/L·atm)
$P_{O_{2}}$ = partial pressure of oxygen (atm)

$D = D O_{stream} - D O$

Where:

$D$ = oxygen deficit (mg/L)
$D O_{stream}$ = saturation concentration of dissolved oxygen at the temperature of the stream after mixing (mg/L)
$D O$ = actual concentration of dissolved oxygen in stream (mg/L)

Dilution purification of wastewater streams

Dilution and natural purification control what happens to wastewater after it enters a receiving water.

Dilution lowers pollutant concentrations through mixing with cleaner water.
Purification reduces pollutants through processes such as biodegradation, sedimentation, and reaeration.

Together, these processes determine a stream’s assimilative capacity and help guide discharge permit limits.

Consider a wastewater flow rate $Q_{w}$ with ultimate BOD $L_{w}$ and dissolved oxygen $D O_{w}$ mixing with a river with flow rate $Q_{r}$ , ultimate BOD $L_{r}$ , and dissolved oxygen $D O_{r}$ .

The initial (immediately after mixing) ultimate BOD of the river-wastewater mix is:

$L_{o} = \frac{Q _{w} L _{w} + Q _{r} L _{r}}{Q _{w} + Q _{r}}$

Initial dissolved oxygen immediately after mixing $D O_{o}$ is given by:

$D O_{o} = \frac{Q _{w} D O _{w} + Q _{r} D O _{r}}{Q _{w} + Q _{r}}$

Temperature immediately after mixing $T_{o}$ is given by:

$T_{o} = \frac{Q _{w} T _{w} + Q _{r} T _{r}}{Q _{w} + Q _{r}}$

Initial oxygen deficit after mixing $D_{o}$ is given by:

$D_{o} = D O_{s a t} - \frac{Q _{w} D O _{w} + Q _{r} D O _{r}}{Q _{w} + Q _{r}}$

Biochemical oxygen demand (BOD)

Measures oxygen needed by microbes to oxidize organic matter
Key for wastewater treatment design, compliance, and impact assessment
Main formula: $BOD = \frac{D _{1} - D _{2}}{P}$
- Adjust for seeded dilution: $BOD = \frac{( D _{1} - D _{2} ) - ( B _{1} - B _{2} ) f}{P}$
$BOD_{t}$ after $t$ days: $(D O_{b, t} - D O_{s, t}) \times dilution factor$

BOD exertion

Describes time-dependent oxygen demand as organics decompose
Modeled by first-order kinetics: $BOD_{t} = L_{0} (1 - e^{- k t})$
- $L_{0}$ = ultimate BOD, $k$ = decay rate constant
$k = 2.30 K$ (base 10 to base $e$ conversion)

Kinetic temperature corrections

Reaction rates increase with temperature
Adjust rate constant: $k_{T} = k_{20} θ^{T - 20}$
- $θ = 1.024$ for reaeration
- $k_{20}$ = rate at 20°C, $T$ = temperature of interest

Stream modeling - Streeter Phelps

Predicts DO changes after waste discharge
- Deoxygenation (BOD exertion) vs. reaeration (atmospheric O $_{2}$ )
DO deficit: $D = \frac{k _{d} L _{a}}{k _{r} - k _{d}} [e^{- k_{d} t} - e^{- k_{r} t}] + D_{a} e^{- k_{r} t}$
Critical point (minimum DO): $t_{c} = \frac{1}{k _{r} - k _{d}} ln (\frac{k _{r}}{k _{d}} (1 - D_{a} \frac{k _{r} - k _{d}}{k _{d} L _{a}}))$
$D O = D O_{sat} - D$

Oxygen saturation

Maximum DO water can hold at equilibrium with atmosphere
- Depends on temperature, pressure, salinity
$D O_{sat} = K_{H} \cdot P_{O_{2}}$
Oxygen deficit: $D = D O_{stream} - D O$

Dilution purification of wastewater streams

Dilution: lowers pollutant concentration by mixing
Purification: pollutant reduction via biodegradation, sedimentation, reaeration
Initial mixed values (after discharge):
- Ultimate BOD: $L_{o} = \frac{Q _{w} L _{w} + Q _{r} L _{r}}{Q _{w} + Q _{r}}$
- DO: $D O_{o} = \frac{Q _{w} D O _{w} + Q _{r} D O _{r}}{Q _{w} + Q _{r}}$
- Temperature: $T_{o} = \frac{Q _{w} T _{w} + Q _{r} T _{r}}{Q _{w} + Q _{r}}$
- Oxygen deficit: $D_{o} = D O_{s a t} - \frac{Q _{w} D O _{w} + Q _{r} D O _{r}}{Q _{w} + Q _{r}}$

Next | 13.2 Water treatment and distribution

Water quality

This chapter covers the following:

Biochemical oxygen demand (BOD)
BOD exertion
Kinetic temperature corrections
Stream modeling - Streeter Phelps
Oxygen saturation
Dilution purification of wastewater streams

Biochemical oxygen demand (BOD)

Because BOD reflects the amount of biodegradable organic pollution present, it’s widely used in:

wastewater treatment design
regulatory compliance
evaluating impacts on receiving waters

Basic BOD testing/sampling equation

$BOD (mg/L) = \frac{D _{1} - D _{2}}{P}$

When the dilution water is seeded:

$BOD (mg/L) = \frac{( D _{1} - D _{2} ) - ( B _{1} - B _{2} ) f}{P}$

Where:

$D_{1}$ = dissolved oxygen of diluted sample immediately after preparation (mg/L)
$D_{2}$ = dissolved oxygen of diluted sample after 5-day incubation at 20°C (mg/L)
$B_{1}$ = dissolved oxygen of seed control before incubation (mg/L)
$B_{2}$ = dissolved oxygen of seed control after incubation (mg/L)
$f$ = fraction of seeded dilution water volume in sample to volume of seeded dilution water in seed control
$P$ = fraction of wastewater sample volume to total combined volume

$BO D_{t}$ :

$BOD_{t}$ is the BOD exerted after $t$ days of incubation, based on the difference between the blank and the sample, scaled by the dilution factor.

$BOD_{t} = (D O_{b, t} - D O_{s, t}) \times dilution factor$

Where:

$D O_{b, t}$ = dissolved oxygen concentration in blank after $t$ days in incubation (mg/L)
$D O_{s, t}$ = dissolved oxygen concentration in sample after $t$ days in incubation (mg/L)

Dilution factor:

$Dilution factor = \frac{volume of bottle or diluted sample}{volume of undiluted sample}$

Volume of seed, undiluted (mL):

$= \frac{allowable depletion \times volume of bottle}{estimated BOD of sample}$

BOD exertion

$BOD_{t} = L_{0} (1 - e^{- k t})$

Where:

$BOD_{t}$ = amount of BOD exerted at time $t$ (mg/L)
$k$ = BOD decay rate constant, base $e$ (day⁻¹)
$k = 2.30 K$
$K$ = rate constant, base 10 (day⁻¹)
$L_{0}$ = ultimate BOD (mg/L)
$t$ = time (days)

Kinetic temperature corrections

$k_{T} = k_{20} θ^{T - 20}$

Where:

$T$ = temperature of interest (°C)
$k_{T}$ = BOD rate constant at the temperature of interest (day⁻¹)
$k_{20}$ = BOD rate constant determined at 20°C (day⁻¹)
$θ$ = temperature coefficient
Reaeration: $θ = 1.024$

Stream modeling - Streeter Phelps

The Streeter-Phelps model describes dissolved oxygen (DO) changes in a stream after an organic waste discharge. It combines two competing processes:

oxygen depletion from BOD exertion (deoxygenation)
oxygen replenishment from the atmosphere (reaeration)

The model predicts the DO deficit downstream, including the critical point where DO reaches its minimum before recovering toward saturation.

$D = \frac{k _{d} L _{a}}{k _{r} - k _{d}} [e^{- k_{d} t} - e^{- k_{r} t}] + D_{a} e^{- k_{r} t}$

$t_{c} = \frac{1}{k _{r} - k _{d}} ln (\frac{k _{r}}{k _{d}} (1 - D_{a} \frac{k _{r} - k _{d}}{k _{d} L _{a}}))$

$D O = D O_{sat} - D$

Where:

$D$ = dissolved oxygen deficit (mg/L)
$D O$ = dissolved oxygen concentration (mg/L)
$D_{a}$ = initial dissolved oxygen deficit in mixing zone (mg/L)
$D O_{sat}$ = saturated dissolved oxygen concentration (mg/L)
$k_{d}$ = deoxygenation rate constant, base $e$ (day⁻¹)
$k_{r}$ = reaeration rate constant, base $e$ (day⁻¹)
$L_{a}$ = initial ultimate BOD in mixing zone (mg/L)
$t$ = time (days)
$t_{c}$ = time at which minimum dissolved oxygen occurs (days)

Oxygen saturation

Oxygen saturation is the maximum dissolved oxygen concentration water can hold when it’s in equilibrium with the atmosphere. It depends mainly on:

temperature
atmospheric pressure
salinity

Saturation is a key reference value because it sets the upper limit for DO and helps define the oxygen deficit used in stream modeling.

$D O_{sat} = K_{H} \cdot P_{O_{2}}$

Where:

$K_{H}$ = Henry’s Law constant (moles/L·atm)
$P_{O_{2}}$ = partial pressure of oxygen (atm)

$D = D O_{stream} - D O$

Where:

$D$ = oxygen deficit (mg/L)
$D O_{stream}$ = saturation concentration of dissolved oxygen at the temperature of the stream after mixing (mg/L)
$D O$ = actual concentration of dissolved oxygen in stream (mg/L)

Dilution purification of wastewater streams

Dilution and natural purification control what happens to wastewater after it enters a receiving water.

Dilution lowers pollutant concentrations through mixing with cleaner water.
Purification reduces pollutants through processes such as biodegradation, sedimentation, and reaeration.

Together, these processes determine a stream’s assimilative capacity and help guide discharge permit limits.

Consider a wastewater flow rate $Q_{w}$ with ultimate BOD $L_{w}$ and dissolved oxygen $D O_{w}$ mixing with a river with flow rate $Q_{r}$ , ultimate BOD $L_{r}$ , and dissolved oxygen $D O_{r}$ .

The initial (immediately after mixing) ultimate BOD of the river-wastewater mix is:

$L_{o} = \frac{Q _{w} L _{w} + Q _{r} L _{r}}{Q _{w} + Q _{r}}$

Initial dissolved oxygen immediately after mixing $D O_{o}$ is given by:

$D O_{o} = \frac{Q _{w} D O _{w} + Q _{r} D O _{r}}{Q _{w} + Q _{r}}$

Temperature immediately after mixing $T_{o}$ is given by:

$T_{o} = \frac{Q _{w} T _{w} + Q _{r} T _{r}}{Q _{w} + Q _{r}}$

Initial oxygen deficit after mixing $D_{o}$ is given by:

$D_{o} = D O_{s a t} - \frac{Q _{w} D O _{w} + Q _{r} D O _{r}}{Q _{w} + Q _{r}}$

Water quality

Biochemical oxygen demand (BOD)

Basic BOD testing/sampling equation

When the dilution water is seeded:

BODt​:

BOD exertion

Kinetic temperature corrections

Stream modeling - Streeter Phelps

Oxygen saturation

Dilution purification of wastewater streams

Water quality

Biochemical oxygen demand (BOD)

Basic BOD testing/sampling equation

When the dilution water is seeded:

BODt​:

BOD exertion

Kinetic temperature corrections

Stream modeling - Streeter Phelps

Oxygen saturation

Dilution purification of wastewater streams

Water quality

Biochemical oxygen demand (BOD)

Basic BOD testing/sampling equation

When the dilution water is seeded:

BODt​:

BOD exertion

Kinetic temperature corrections

Stream modeling - Streeter Phelps

Oxygen saturation

Dilution purification of wastewater streams

Water quality

Biochemical oxygen demand (BOD)

Basic BOD testing/sampling equation

When the dilution water is seeded:

BODt​:

BOD exertion

Kinetic temperature corrections

Stream modeling - Streeter Phelps

Oxygen saturation

Dilution purification of wastewater streams

$BO D_{t}$ :

$BO D_{t}$ :

$BO D_{t}$ :

$BO D_{t}$ :