Streeter Phelps oxygen sag curve downstream of a wastewater outfall Given deoxygenation rate (k_d) = 0.3100 1/day; reaeration rate (k_r) = 0.8900 1/day; initial ultimate BOD (L_0) = 30.0000 mg/L; initial oxygen deficit (D_0) = 2.7000 mg/L; time of travel (t) = 2.2000 day, determine the dissolved oxygen deficit (D) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Streeter Phelps.
• Everything except D is given, so isolate D symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The Streeter Phelps oxygen sag equation models dissolved oxygen deficit downstream of a wastewater discharge in a stream.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

2. Step 2 — Rearrange symbolically for D:

   $$D = \dfrac{k_d L_0}{k_r-k_d}\left(e^{-k_d t}-e^{-k_r t}\right)+D_0 e^{-k_r t}$$

3. Step 3 — List the givens: deoxygenation rate (k_d) = 0.3100 1/day, reaeration rate (k_r) = 0.8900 1/day, initial ultimate BOD (L_0) = 30.0000 mg/L, initial oxygen deficit (D_0) = 2.7000 mg/L, time of travel (t) = 2.2000 day.

4. Step 4 — Substitute the given values:

   $$D = \dfrac{k_d L_0}{k_r-k_d}\lef2.2000(e^{-k_d 2.2000}-e^{-k_r 2.2000}\righ2.2000)+D_0 e^{-k_r 2.2000}$$

5. Step 5 — Evaluate:

   $$D = 6.2251\ \text{mg/L}$$

6. Step 6 — Check: returning D = 6.2251 mg/L to

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 12.4502 — kept a factor of two that cancels in the correct rearrangement.
• 3.1125 — dropped that same factor in the other direction.
• 6.8476 — rounded an intermediate value before the final step.

Reference: FE Handbook — Streeter Phelps

Streeter Phelps model computing dissolved oxygen deficit in a river Given deoxygenation rate (k_d) = 0.3200 1/day; reaeration rate (k_r) = 0.9100 1/day; initial oxygen deficit (D_0) = 4.0000 mg/L; time of travel (t) = 0.5000 day; dissolved oxygen deficit (D) = 10.4200 mg/L, determine the initial ultimate BOD (L_0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Streeter Phelps.
• Everything except L_0 is given, so isolate L_0 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The Streeter Phelps oxygen sag equation models dissolved oxygen deficit downstream of a wastewater discharge in a stream.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

2. Step 2 — Rearrange symbolically for L_0:

   $$L_{0} = \dfrac{\left(D-D_0 e^{-k_r t}\right)(k_r-k_d)}{k_d\left(e^{-k_d t}-e^{-k_r t}\right)}$$

3. Step 3 — List the givens: deoxygenation rate (k_d) = 0.3200 1/day, reaeration rate (k_r) = 0.9100 1/day, initial oxygen deficit (D_0) = 4.0000 mg/L, time of travel (t) = 0.5000 day, dissolved oxygen deficit (D) = 10.4200 mg/L.

4. Step 4 — Substitute the given values:

   $$L_{0} = \dfrac{\lef0.5000(10.4200-D_0 e^{-k_r 0.5000}\righ0.5000)(k_r-k_d)}{k_d\lef0.5000(e^{-k_d 0.5000}-e^{-k_r 0.5000}\righ0.5000)}$$

5. Step 5 — Evaluate:

   $$L_{0} = 66.7575\ \text{mg/L}$$

6. Step 6 — Check: returning L_0 = 66.7575 mg/L to

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 133.5 — kept a factor of two that cancels in the correct rearrangement.
• 33.3787 — dropped that same factor in the other direction.
• 73.4332 — rounded an intermediate value before the final step.

Reference: FE Handbook — Streeter Phelps

Streeter Phelps equation predicting the critical DO sag point Given deoxygenation rate (k_d) = 0.4700 1/day; reaeration rate (k_r) = 1.0400 1/day; initial ultimate BOD (L_0) = 24.0000 mg/L; time of travel (t) = 0.7000 day; dissolved oxygen deficit (D) = 4.3900 mg/L, determine the initial oxygen deficit (D_0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Streeter Phelps.
• Everything except D_0 is given, so isolate D_0 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The Streeter Phelps oxygen sag equation models dissolved oxygen deficit downstream of a wastewater discharge in a stream.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

2. Step 2 — Rearrange symbolically for D_0:

   $$D_{0} = \dfrac{D-\dfrac{k_d L_0}{k_r-k_d}\left(e^{-k_d t}-e^{-k_r t}\right)}{e^{-k_r t}}$$

3. Step 3 — List the givens: deoxygenation rate (k_d) = 0.4700 1/day, reaeration rate (k_r) = 1.0400 1/day, initial ultimate BOD (L_0) = 24.0000 mg/L, time of travel (t) = 0.7000 day, dissolved oxygen deficit (D) = 4.3900 mg/L.

4. Step 4 — Substitute the given values:

   $$D_{0} = \dfrac{4.3900-\dfrac{k_d L_0}{k_r-k_d}\lef0.7000(e^{-k_d 0.7000}-e^{-k_r 0.7000}\righ0.7000)}{e^{-k_r 0.7000}}$$

5. Step 5 — Evaluate:

   $$D_{0} = -0.6120\ \text{mg/L}$$

6. Step 6 — Check: returning D_0 = -0.6120 mg/L to

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -1.2241 — kept a factor of two that cancels in the correct rearrangement.
• -0.3060 — dropped that same factor in the other direction.
• -0.6732 — rounded an intermediate value before the final step.

Reference: FE Handbook — Streeter Phelps

Streeter Phelps oxygen sag curve downstream of a wastewater outfall Given deoxygenation rate (k_d) = 0.2500 1/day; reaeration rate (k_r) = 0.7800 1/day; initial ultimate BOD (L_0) = 44.5000 mg/L; initial oxygen deficit (D_0) = 4.6000 mg/L; time of travel (t) = 1.6000 day, determine the dissolved oxygen deficit (D) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Streeter Phelps.
• Everything except D is given, so isolate D symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The Streeter Phelps oxygen sag equation models dissolved oxygen deficit downstream of a wastewater discharge in a stream.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

2. Step 2 — Rearrange symbolically for D:

   $$D = \dfrac{k_d L_0}{k_r-k_d}\left(e^{-k_d t}-e^{-k_r t}\right)+D_0 e^{-k_r t}$$

3. Step 3 — List the givens: deoxygenation rate (k_d) = 0.2500 1/day, reaeration rate (k_r) = 0.7800 1/day, initial ultimate BOD (L_0) = 44.5000 mg/L, initial oxygen deficit (D_0) = 4.6000 mg/L, time of travel (t) = 1.6000 day.

4. Step 4 — Substitute the given values:

   $$D = \dfrac{k_d L_0}{k_r-k_d}\lef1.6000(e^{-k_d 1.6000}-e^{-k_r 1.6000}\righ1.6000)+D_0 e^{-k_r 1.6000}$$

5. Step 5 — Evaluate:

   $$D = 9.3650\ \text{mg/L}$$

6. Step 6 — Check: returning D = 9.3650 mg/L to

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 18.7300 — kept a factor of two that cancels in the correct rearrangement.
• 4.6825 — dropped that same factor in the other direction.
• 10.3015 — rounded an intermediate value before the final step.

Reference: FE Handbook — Streeter Phelps

Streeter Phelps model computing dissolved oxygen deficit in a river Given deoxygenation rate (k_d) = 0.4100 1/day; reaeration rate (k_r) = 0.9000 1/day; initial oxygen deficit (D_0) = 2.1000 mg/L; time of travel (t) = 2.3000 day; dissolved oxygen deficit (D) = 4.4400 mg/L, determine the initial ultimate BOD (L_0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Streeter Phelps.
• Everything except L_0 is given, so isolate L_0 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The Streeter Phelps oxygen sag equation models dissolved oxygen deficit downstream of a wastewater discharge in a stream.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

2. Step 2 — Rearrange symbolically for L_0:

   $$L_{0} = \dfrac{\left(D-D_0 e^{-k_r t}\right)(k_r-k_d)}{k_d\left(e^{-k_d t}-e^{-k_r t}\right)}$$

3. Step 3 — List the givens: deoxygenation rate (k_d) = 0.4100 1/day, reaeration rate (k_r) = 0.9000 1/day, initial oxygen deficit (D_0) = 2.1000 mg/L, time of travel (t) = 2.3000 day, dissolved oxygen deficit (D) = 4.4400 mg/L.

4. Step 4 — Substitute the given values:

   $$L_{0} = \dfrac{\lef2.3000(4.4400-D_0 e^{-k_r 2.3000}\righ2.3000)(k_r-k_d)}{k_d\lef2.3000(e^{-k_d 2.3000}-e^{-k_r 2.3000}\righ2.3000)}$$

5. Step 5 — Evaluate:

   $$L_{0} = 18.9524\ \text{mg/L}$$

6. Step 6 — Check: returning L_0 = 18.9524 mg/L to

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 37.9049 — kept a factor of two that cancels in the correct rearrangement.
• 9.4762 — dropped that same factor in the other direction.
• 20.8477 — rounded an intermediate value before the final step.

Reference: FE Handbook — Streeter Phelps

Streeter Phelps equation predicting the critical DO sag point Given deoxygenation rate (k_d) = 0.3000 1/day; reaeration rate (k_r) = 0.7300 1/day; initial ultimate BOD (L_0) = 27.5000 mg/L; time of travel (t) = 2.4000 day; dissolved oxygen deficit (D) = 3.8700 mg/L, determine the initial oxygen deficit (D_0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Streeter Phelps.
• Everything except D_0 is given, so isolate D_0 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The Streeter Phelps oxygen sag equation models dissolved oxygen deficit downstream of a wastewater discharge in a stream.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

2. Step 2 — Rearrange symbolically for D_0:

   $$D_{0} = \dfrac{D-\dfrac{k_d L_0}{k_r-k_d}\left(e^{-k_d t}-e^{-k_r t}\right)}{e^{-k_r t}}$$

3. Step 3 — List the givens: deoxygenation rate (k_d) = 0.3000 1/day, reaeration rate (k_r) = 0.7300 1/day, initial ultimate BOD (L_0) = 27.5000 mg/L, time of travel (t) = 2.4000 day, dissolved oxygen deficit (D) = 3.8700 mg/L.

4. Step 4 — Substitute the given values:

   $$D_{0} = \dfrac{3.8700-\dfrac{k_d L_0}{k_r-k_d}\lef2.4000(e^{-k_d 2.4000}-e^{-k_r 2.4000}\righ2.4000)}{e^{-k_r 2.4000}}$$

5. Step 5 — Evaluate:

   $$D_{0} = -12.3480\ \text{mg/L}$$

6. Step 6 — Check: returning D_0 = -12.3480 mg/L to

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -24.6961 — kept a factor of two that cancels in the correct rearrangement.
• -6.1740 — dropped that same factor in the other direction.
• -13.5828 — rounded an intermediate value before the final step.

Reference: FE Handbook — Streeter Phelps

Streeter Phelps oxygen sag curve downstream of a wastewater outfall Given deoxygenation rate (k_d) = 0.2600 1/day; reaeration rate (k_r) = 0.7300 1/day; initial ultimate BOD (L_0) = 49.5000 mg/L; initial oxygen deficit (D_0) = 4.2000 mg/L; time of travel (t) = 2.3000 day, determine the dissolved oxygen deficit (D) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Streeter Phelps.
• Everything except D is given, so isolate D symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The Streeter Phelps oxygen sag equation models dissolved oxygen deficit downstream of a wastewater discharge in a stream.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

2. Step 2 — Rearrange symbolically for D:

   $$D = \dfrac{k_d L_0}{k_r-k_d}\left(e^{-k_d t}-e^{-k_r t}\right)+D_0 e^{-k_r t}$$

3. Step 3 — List the givens: deoxygenation rate (k_d) = 0.2600 1/day, reaeration rate (k_r) = 0.7300 1/day, initial ultimate BOD (L_0) = 49.5000 mg/L, initial oxygen deficit (D_0) = 4.2000 mg/L, time of travel (t) = 2.3000 day.

4. Step 4 — Substitute the given values:

   $$D = \dfrac{k_d L_0}{k_r-k_d}\lef2.3000(e^{-k_d 2.3000}-e^{-k_r 2.3000}\righ2.3000)+D_0 e^{-k_r 2.3000}$$

5. Step 5 — Evaluate:

   $$D = 10.7332\ \text{mg/L}$$

6. Step 6 — Check: returning D = 10.7332 mg/L to

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 21.4663 — kept a factor of two that cancels in the correct rearrangement.
• 5.3666 — dropped that same factor in the other direction.
• 11.8065 — rounded an intermediate value before the final step.

Reference: FE Handbook — Streeter Phelps

Streeter Phelps model computing dissolved oxygen deficit in a river Given deoxygenation rate (k_d) = 0.2300 1/day; reaeration rate (k_r) = 0.6800 1/day; initial oxygen deficit (D_0) = 3.2000 mg/L; time of travel (t) = 1.7000 day; dissolved oxygen deficit (D) = 5.7600 mg/L, determine the initial ultimate BOD (L_0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Streeter Phelps.
• Everything except L_0 is given, so isolate L_0 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The Streeter Phelps oxygen sag equation models dissolved oxygen deficit downstream of a wastewater discharge in a stream.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

2. Step 2 — Rearrange symbolically for L_0:

   $$L_{0} = \dfrac{\left(D-D_0 e^{-k_r t}\right)(k_r-k_d)}{k_d\left(e^{-k_d t}-e^{-k_r t}\right)}$$

3. Step 3 — List the givens: deoxygenation rate (k_d) = 0.2300 1/day, reaeration rate (k_r) = 0.6800 1/day, initial oxygen deficit (D_0) = 3.2000 mg/L, time of travel (t) = 1.7000 day, dissolved oxygen deficit (D) = 5.7600 mg/L.

4. Step 4 — Substitute the given values:

   $$L_{0} = \dfrac{\lef1.7000(5.7600-D_0 e^{-k_r 1.7000}\righ1.7000)(k_r-k_d)}{k_d\lef1.7000(e^{-k_d 1.7000}-e^{-k_r 1.7000}\righ1.7000)}$$

5. Step 5 — Evaluate:

   $$L_{0} = 25.7136\ \text{mg/L}$$

6. Step 6 — Check: returning L_0 = 25.7136 mg/L to

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 51.4272 — kept a factor of two that cancels in the correct rearrangement.
• 12.8568 — dropped that same factor in the other direction.
• 28.2850 — rounded an intermediate value before the final step.

Reference: FE Handbook — Streeter Phelps

Streeter Phelps equation predicting the critical DO sag point Given deoxygenation rate (k_d) = 0.1700 1/day; reaeration rate (k_r) = 0.8400 1/day; initial ultimate BOD (L_0) = 15.0000 mg/L; time of travel (t) = 1.1000 day; dissolved oxygen deficit (D) = 1.7500 mg/L, determine the initial oxygen deficit (D_0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Streeter Phelps.
• Everything except D_0 is given, so isolate D_0 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The Streeter Phelps oxygen sag equation models dissolved oxygen deficit downstream of a wastewater discharge in a stream.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

2. Step 2 — Rearrange symbolically for D_0:

   $$D_{0} = \dfrac{D-\dfrac{k_d L_0}{k_r-k_d}\left(e^{-k_d t}-e^{-k_r t}\right)}{e^{-k_r t}}$$

3. Step 3 — List the givens: deoxygenation rate (k_d) = 0.1700 1/day, reaeration rate (k_r) = 0.8400 1/day, initial ultimate BOD (L_0) = 15.0000 mg/L, time of travel (t) = 1.1000 day, dissolved oxygen deficit (D) = 1.7500 mg/L.

4. Step 4 — Substitute the given values:

   $$D_{0} = \dfrac{1.7500-\dfrac{k_d L_0}{k_r-k_d}\lef1.1000(e^{-k_d 1.1000}-e^{-k_r 1.1000}\righ1.1000)}{e^{-k_r 1.1000}}$$

5. Step 5 — Evaluate:

   $$D_{0} = 0.2617\ \text{mg/L}$$

6. Step 6 — Check: returning D_0 = 0.2617 mg/L to

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.5233 — kept a factor of two that cancels in the correct rearrangement.
• 0.1308 — dropped that same factor in the other direction.
• 0.2878 — rounded an intermediate value before the final step.

Reference: FE Handbook — Streeter Phelps

Streeter Phelps oxygen sag curve downstream of a wastewater outfall Given deoxygenation rate (k_d) = 0.5000 1/day; reaeration rate (k_r) = 0.8700 1/day; initial ultimate BOD (L_0) = 46.0000 mg/L; initial oxygen deficit (D_0) = 4.6000 mg/L; time of travel (t) = 1.3000 day, determine the dissolved oxygen deficit (D) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Streeter Phelps.
• Everything except D is given, so isolate D symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The Streeter Phelps oxygen sag equation models dissolved oxygen deficit downstream of a wastewater discharge in a stream.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

2. Step 2 — Rearrange symbolically for D:

   $$D = \dfrac{k_d L_0}{k_r-k_d}\left(e^{-k_d t}-e^{-k_r t}\right)+D_0 e^{-k_r t}$$

3. Step 3 — List the givens: deoxygenation rate (k_d) = 0.5000 1/day, reaeration rate (k_r) = 0.8700 1/day, initial ultimate BOD (L_0) = 46.0000 mg/L, initial oxygen deficit (D_0) = 4.6000 mg/L, time of travel (t) = 1.3000 day.

4. Step 4 — Substitute the given values:

   $$D = \dfrac{k_d L_0}{k_r-k_d}\lef1.3000(e^{-k_d 1.3000}-e^{-k_r 1.3000}\righ1.3000)+D_0 e^{-k_r 1.3000}$$

5. Step 5 — Evaluate:

   $$D = 13.8756\ \text{mg/L}$$

6. Step 6 — Check: returning D = 13.8756 mg/L to

   $$D = \dfrac{k_d L_0}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_0 e^{-k_r t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 27.7512 — kept a factor of two that cancels in the correct rearrangement.
• 6.9378 — dropped that same factor in the other direction.
• 15.2631 — rounded an intermediate value before the final step.

Reference: FE Handbook — Streeter Phelps