aerobic digestion tank volume for waste activated sludge stabilization Given influent sludge flow (Q_i) = 243.0 m^3/day; influent volatile solids (X_i) = 3,380 mg/L; BOD fraction factor (F) = 0.7200; influent BOD5 (S_i) = 2,840 mg/L; digester volatile solids (X_d) = 15,500 mg/L; reaction rate constant (K_d) = 0.1770 1/day; volatile fraction (P_v) = 0.7300; digester detention time (theta_c) = 39.5000 day, determine the digester volume (V_d) in m^3.

Start with the thinking

• The governing relation printed in this handbook section is Aerobic Digestion.
• Everything except V_d is given, so isolate V_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.
• Aerobic digestion sizes the digester volume needed to stabilize waste activated sludge volatile solids.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V_d = \dfrac{Q_i (X_i + F S_i)}{X_d (K_d P_v + 1/\theta_c)}$$

2. Step 2 — Rearrange symbolically for V_d:

   $$V_{d} = \dfrac{Q_i(X_i+FS_i)}{X_d\left(K_dP_v+\dfrac{1}{\theta_c}\right)}$$

3. Step 3 — List the givens: influent sludge flow (Q_i) = 243.0 m^3/day, influent volatile solids (X_i) = 3,380 mg/L, BOD fraction factor (F) = 0.7200, influent BOD5 (S_i) = 2,840 mg/L, digester volatile solids (X_d) = 15,500 mg/L, reaction rate constant (K_d) = 0.1770 1/day, volatile fraction (P_v) = 0.7300, digester detention time (theta_c) = 39.5000 day.

4. Step 4 — Substitute the given values:

   $$V_{d} = \dfrac{Q_i(X_i+FS_i)}{X_d\left(K_dP_v+\dfrac{1}{\theta_c}\right)}$$

5. Step 5 — Evaluate:

   V_{d} = 550.4\ \text{m^3}

6. Step 6 — Check: returning V_d = 550.4 m^3 to

   $$V_d = \dfrac{Q_i (X_i + F S_i)}{X_d (K_d P_v + 1/\theta_c)}$$

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

Why the other options are there

• 1,101 — kept a factor of two that cancels in the correct rearrangement.
• 275.2 — dropped that same factor in the other direction.
• 605.4 — rounded an intermediate value before the final step.

Reference: FE Handbook — Aerobic Digestion

anaerobic digestion of sludge for biogas recovery Given influent BOD ultimate (S_0) = 11,720 mg/L; effluent soluble BOD (S) = 60.0000 mg/L; flow rate (Q) = 570.0 m^3/day, determine the methane gas production (V_g) in m^3/day.

Start with the thinking

• The governing relation printed in this handbook section is Anaerobic Digestion.
• Everything except V_g is given, so isolate V_g 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.
• Anaerobic digestion estimates methane gas production from the BOD stabilized in an anaerobic sludge digester.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V_g = 5.62\left(S_0 - S\right) Q \times 10^{-3}$$

2. Step 2 — Rearrange symbolically for V_g:

   $$V_{g} = 5.62(S_0-S)Q\times 10^{-3}$$

3. Step 3 — List the givens: influent BOD ultimate (S_0) = 11,720 mg/L, effluent soluble BOD (S) = 60.0000 mg/L, flow rate (Q) = 570.0 m^3/day.

4. Step 4 — Substitute the given values:

   $$V_{g} = 5.62(S_0-60.0000)570.0\times 10^{-3}$$

5. Step 5 — Evaluate:

   V_{g} = 37352\ \text{m^3/day}

6. Step 6 — Check: returning V_g = 37,352 m^3/day to

   $$V_g = 5.62\left(S_0 - S\right) Q \times 10^{-3}$$

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

Why the other options are there

• 74,703 — kept a factor of two that cancels in the correct rearrangement.
• 18,676 — dropped that same factor in the other direction.
• 41,087 — rounded an intermediate value before the final step.

Reference: FE Handbook — Anaerobic Digestion Gas Production

aerobic digestion design of a sludge digester Given influent volatile solids (X_i) = 5,420 mg/L; BOD fraction factor (F) = 0.5800; influent BOD5 (S_i) = 2,340 mg/L; digester volatile solids (X_d) = 13,460 mg/L; reaction rate constant (K_d) = 0.0870 1/day; volatile fraction (P_v) = 0.6500; digester detention time (theta_c) = 27.0000 day; digester volume (V_d) = 8,763 m^3, determine the influent sludge flow (Q_i) in m^3/day.

Start with the thinking

• The governing relation printed in this handbook section is Aerobic Digestion.
• Everything except Q_i is given, so isolate Q_i 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.
• Aerobic digestion sizes the digester volume needed to stabilize waste activated sludge volatile solids.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V_d = \dfrac{Q_i (X_i + F S_i)}{X_d (K_d P_v + 1/\theta_c)}$$

2. Step 2 — Rearrange symbolically for Q_i:

   $$Q_{i} = \dfrac{V_d X_d\left(K_dP_v+\dfrac{1}{\theta_c}\right)}{X_i+FS_i}$$

3. Step 3 — List the givens: influent volatile solids (X_i) = 5,420 mg/L, BOD fraction factor (F) = 0.5800, influent BOD5 (S_i) = 2,340 mg/L, digester volatile solids (X_d) = 13,460 mg/L, reaction rate constant (K_d) = 0.0870 1/day, volatile fraction (P_v) = 0.6500, digester detention time (theta_c) = 27.0000 day, digester volume (V_d) = 8,763 m^3.

4. Step 4 — Substitute the given values:

   $$Q_{i} = \dfrac{V_d X_d\left(K_dP_v+\dfrac{1}{\theta_c}\right)}{X_i+FS_i}$$

5. Step 5 — Evaluate:

   Q_{i} = 1629\ \text{m^3/day}

6. Step 6 — Check: returning Q_i = 1,629 m^3/day to

   $$V_d = \dfrac{Q_i (X_i + F S_i)}{X_d (K_d P_v + 1/\theta_c)}$$

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

Why the other options are there

• 3,258 — kept a factor of two that cancels in the correct rearrangement.
• 814.4 — dropped that same factor in the other direction.
• 1,792 — rounded an intermediate value before the final step.

Reference: FE Handbook — Aerobic Digestion

anaerobic digestion gas production estimate from BOD stabilization Given influent BOD ultimate (S_0) = 9,660 mg/L; effluent soluble BOD (S) = 320.0 mg/L; methane gas production (V_g) = 4,590 m^3/day, determine the flow rate (Q) in m^3/day.

Start with the thinking

• The governing relation printed in this handbook section is Anaerobic Digestion.
• Everything except Q is given, so isolate Q 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.
• Anaerobic digestion estimates methane gas production from the BOD stabilized in an anaerobic sludge digester.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V_g = 5.62\left(S_0 - S\right) Q \times 10^{-3}$$

2. Step 2 — Rearrange symbolically for Q:

   $$Q = \dfrac{V_g}{5.62(S_0-S)\times 10^{-3}}$$

3. Step 3 — List the givens: influent BOD ultimate (S_0) = 9,660 mg/L, effluent soluble BOD (S) = 320.0 mg/L, methane gas production (V_g) = 4,590 m^3/day.

4. Step 4 — Substitute the given values:

   $$Q = \dfrac{V_g}{5.62(S_0-320.0)\times 10^{-3}}$$

5. Step 5 — Evaluate:

   Q = 87.4420\ \text{m^3/day}

6. Step 6 — Check: returning Q = 87.4420 m^3/day to

   $$V_g = 5.62\left(S_0 - S\right) Q \times 10^{-3}$$

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

Why the other options are there

• 174.9 — kept a factor of two that cancels in the correct rearrangement.
• 43.7210 — dropped that same factor in the other direction.
• 96.1862 — rounded an intermediate value before the final step.

Reference: FE Handbook — Anaerobic Digestion Gas Production

aerobic digestion volatile solids reduction sizing calculation Given influent sludge flow (Q_i) = 230.0 m^3/day; influent volatile solids (X_i) = 11,870 mg/L; BOD fraction factor (F) = 0.6900; influent BOD5 (S_i) = 1,540 mg/L; digester volatile solids (X_d) = 9,460 mg/L; reaction rate constant (K_d) = 0.0880 1/day; volatile fraction (P_v) = 0.8000; digester detention time (theta_c) = 20.5000 day, determine the digester volume (V_d) in m^3.

Start with the thinking

• The governing relation printed in this handbook section is Aerobic Digestion.
• Everything except V_d is given, so isolate V_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.
• Aerobic digestion sizes the digester volume needed to stabilize waste activated sludge volatile solids.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V_d = \dfrac{Q_i (X_i + F S_i)}{X_d (K_d P_v + 1/\theta_c)}$$

2. Step 2 — Rearrange symbolically for V_d:

   $$V_{d} = \dfrac{Q_i(X_i+FS_i)}{X_d\left(K_dP_v+\dfrac{1}{\theta_c}\right)}$$

3. Step 3 — List the givens: influent sludge flow (Q_i) = 230.0 m^3/day, influent volatile solids (X_i) = 11,870 mg/L, BOD fraction factor (F) = 0.6900, influent BOD5 (S_i) = 1,540 mg/L, digester volatile solids (X_d) = 9,460 mg/L, reaction rate constant (K_d) = 0.0880 1/day, volatile fraction (P_v) = 0.8000, digester detention time (theta_c) = 20.5000 day.

4. Step 4 — Substitute the given values:

   $$V_{d} = \dfrac{Q_i(X_i+FS_i)}{X_d\left(K_dP_v+\dfrac{1}{\theta_c}\right)}$$

5. Step 5 — Evaluate:

   V_{d} = 2638\ \text{m^3}

6. Step 6 — Check: returning V_d = 2,638 m^3 to

   $$V_d = \dfrac{Q_i (X_i + F S_i)}{X_d (K_d P_v + 1/\theta_c)}$$

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

Why the other options are there

• 5,277 — kept a factor of two that cancels in the correct rearrangement.
• 1,319 — dropped that same factor in the other direction.
• 2,902 — rounded an intermediate value before the final step.

Reference: FE Handbook — Aerobic Digestion

anaerobic digestion methane gas production for a sludge digester Given effluent soluble BOD (S) = 250.0 mg/L; flow rate (Q) = 4,840 m^3/day; methane gas production (V_g) = 7,201 m^3/day, determine the influent BOD ultimate (S_0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Anaerobic Digestion.
• Everything except S_0 is given, so isolate S_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.
• Anaerobic digestion estimates methane gas production from the BOD stabilized in an anaerobic sludge digester.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V_g = 5.62\left(S_0 - S\right) Q \times 10^{-3}$$

2. Step 2 — Rearrange symbolically for S_0:

   $$S_{0} = \dfrac{V_g}{5.62\,Q\times 10^{-3}}+S$$

3. Step 3 — List the givens: effluent soluble BOD (S) = 250.0 mg/L, flow rate (Q) = 4,840 m^3/day, methane gas production (V_g) = 7,201 m^3/day.

4. Step 4 — Substitute the given values:

   $$S_{0} = \dfrac{V_g}{5.62\,4840\times 10^{-3}}+250.0$$

5. Step 5 — Evaluate:

   $$S_{0} = 514.7\ \text{mg/L}$$

6. Step 6 — Check: returning S_0 = 514.7 mg/L to

   $$V_g = 5.62\left(S_0 - S\right) Q \times 10^{-3}$$

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

Why the other options are there

• 1,029 — kept a factor of two that cancels in the correct rearrangement.
• 257.4 — dropped that same factor in the other direction.
• 566.2 — rounded an intermediate value before the final step.

Reference: FE Handbook — Anaerobic Digestion Gas Production

aerobic digestion tank volume for waste activated sludge stabilization Given influent volatile solids (X_i) = 17,370 mg/L; BOD fraction factor (F) = 0.7800; influent BOD5 (S_i) = 4,730 mg/L; digester volatile solids (X_d) = 8,140 mg/L; reaction rate constant (K_d) = 0.1460 1/day; volatile fraction (P_v) = 0.7300; digester detention time (theta_c) = 32.5000 day; digester volume (V_d) = 9,961 m^3, determine the influent sludge flow (Q_i) in m^3/day.

Start with the thinking

• The governing relation printed in this handbook section is Aerobic Digestion.
• Everything except Q_i is given, so isolate Q_i 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.
• Aerobic digestion sizes the digester volume needed to stabilize waste activated sludge volatile solids.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V_d = \dfrac{Q_i (X_i + F S_i)}{X_d (K_d P_v + 1/\theta_c)}$$

2. Step 2 — Rearrange symbolically for Q_i:

   $$Q_{i} = \dfrac{V_d X_d\left(K_dP_v+\dfrac{1}{\theta_c}\right)}{X_i+FS_i}$$

3. Step 3 — List the givens: influent volatile solids (X_i) = 17,370 mg/L, BOD fraction factor (F) = 0.7800, influent BOD5 (S_i) = 4,730 mg/L, digester volatile solids (X_d) = 8,140 mg/L, reaction rate constant (K_d) = 0.1460 1/day, volatile fraction (P_v) = 0.7300, digester detention time (theta_c) = 32.5000 day, digester volume (V_d) = 9,961 m^3.

4. Step 4 — Substitute the given values:

   $$Q_{i} = \dfrac{V_d X_d\left(K_dP_v+\dfrac{1}{\theta_c}\right)}{X_i+FS_i}$$

5. Step 5 — Evaluate:

   Q_{i} = 528.8\ \text{m^3/day}

6. Step 6 — Check: returning Q_i = 528.8 m^3/day to

   $$V_d = \dfrac{Q_i (X_i + F S_i)}{X_d (K_d P_v + 1/\theta_c)}$$

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

Why the other options are there

• 1,058 — kept a factor of two that cancels in the correct rearrangement.
• 264.4 — dropped that same factor in the other direction.
• 581.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Aerobic Digestion

anaerobic digestion of sludge for biogas recovery Given influent BOD ultimate (S_0) = 17,690 mg/L; effluent soluble BOD (S) = 465.0 mg/L; flow rate (Q) = 1,790 m^3/day, determine the methane gas production (V_g) in m^3/day.

Start with the thinking

• The governing relation printed in this handbook section is Anaerobic Digestion.
• Everything except V_g is given, so isolate V_g 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.
• Anaerobic digestion estimates methane gas production from the BOD stabilized in an anaerobic sludge digester.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V_g = 5.62\left(S_0 - S\right) Q \times 10^{-3}$$

2. Step 2 — Rearrange symbolically for V_g:

   $$V_{g} = 5.62(S_0-S)Q\times 10^{-3}$$

3. Step 3 — List the givens: influent BOD ultimate (S_0) = 17,690 mg/L, effluent soluble BOD (S) = 465.0 mg/L, flow rate (Q) = 1,790 m^3/day.

4. Step 4 — Substitute the given values:

   $$V_{g} = 5.62(S_0-465.0)1790\times 10^{-3}$$

5. Step 5 — Evaluate:

   V_{g} = 173280\ \text{m^3/day}

6. Step 6 — Check: returning V_g = 173,280 m^3/day to

   $$V_g = 5.62\left(S_0 - S\right) Q \times 10^{-3}$$

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

Why the other options are there

• 346,560 — kept a factor of two that cancels in the correct rearrangement.
• 86,640 — dropped that same factor in the other direction.
• 190,608 — rounded an intermediate value before the final step.

Reference: FE Handbook — Anaerobic Digestion Gas Production

aerobic digestion design of a sludge digester Given influent sludge flow (Q_i) = 70.0000 m^3/day; influent volatile solids (X_i) = 6,840 mg/L; BOD fraction factor (F) = 0.6600; influent BOD5 (S_i) = 2,810 mg/L; digester volatile solids (X_d) = 15,100 mg/L; reaction rate constant (K_d) = 0.0570 1/day; volatile fraction (P_v) = 0.7100; digester detention time (theta_c) = 27.0000 day, determine the digester volume (V_d) in m^3.

Start with the thinking

• The governing relation printed in this handbook section is Aerobic Digestion.
• Everything except V_d is given, so isolate V_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.
• Aerobic digestion sizes the digester volume needed to stabilize waste activated sludge volatile solids.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V_d = \dfrac{Q_i (X_i + F S_i)}{X_d (K_d P_v + 1/\theta_c)}$$

2. Step 2 — Rearrange symbolically for V_d:

   $$V_{d} = \dfrac{Q_i(X_i+FS_i)}{X_d\left(K_dP_v+\dfrac{1}{\theta_c}\right)}$$

3. Step 3 — List the givens: influent sludge flow (Q_i) = 70.0000 m^3/day, influent volatile solids (X_i) = 6,840 mg/L, BOD fraction factor (F) = 0.6600, influent BOD5 (S_i) = 2,810 mg/L, digester volatile solids (X_d) = 15,100 mg/L, reaction rate constant (K_d) = 0.0570 1/day, volatile fraction (P_v) = 0.7100, digester detention time (theta_c) = 27.0000 day.

4. Step 4 — Substitute the given values:

   $$V_{d} = \dfrac{Q_i(X_i+FS_i)}{X_d\left(K_dP_v+\dfrac{1}{\theta_c}\right)}$$

5. Step 5 — Evaluate:

   V_{d} = 520.0\ \text{m^3}

6. Step 6 — Check: returning V_d = 520.0 m^3 to

   $$V_d = \dfrac{Q_i (X_i + F S_i)}{X_d (K_d P_v + 1/\theta_c)}$$

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

Why the other options are there

• 1,040 — kept a factor of two that cancels in the correct rearrangement.
• 260.0 — dropped that same factor in the other direction.
• 572.0 — rounded an intermediate value before the final step.

Reference: FE Handbook — Aerobic Digestion

anaerobic digestion gas production estimate from BOD stabilization Given influent BOD ultimate (S_0) = 3,390 mg/L; effluent soluble BOD (S) = 425.0 mg/L; methane gas production (V_g) = 6,994 m^3/day, determine the flow rate (Q) in m^3/day.

Start with the thinking

• The governing relation printed in this handbook section is Anaerobic Digestion.
• Everything except Q is given, so isolate Q 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.
• Anaerobic digestion estimates methane gas production from the BOD stabilized in an anaerobic sludge digester.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V_g = 5.62\left(S_0 - S\right) Q \times 10^{-3}$$

2. Step 2 — Rearrange symbolically for Q:

   $$Q = \dfrac{V_g}{5.62(S_0-S)\times 10^{-3}}$$

3. Step 3 — List the givens: influent BOD ultimate (S_0) = 3,390 mg/L, effluent soluble BOD (S) = 425.0 mg/L, methane gas production (V_g) = 6,994 m^3/day.

4. Step 4 — Substitute the given values:

   $$Q = \dfrac{V_g}{5.62(S_0-425.0)\times 10^{-3}}$$

5. Step 5 — Evaluate:

   Q = 419.7\ \text{m^3/day}

6. Step 6 — Check: returning Q = 419.7 m^3/day to

   $$V_g = 5.62\left(S_0 - S\right) Q \times 10^{-3}$$

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

Why the other options are there

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

Reference: FE Handbook — Anaerobic Digestion Gas Production