non-steady state continuous flow completely mixed reactor start-up Given initial tank concentration (C_0) = 46.0000 mg/L; influent concentration (C_in) = 35.5000 mg/L; flow rate (Q) = 1,680 m^3/day; tank volume (V) = 16,620 m^3; time (t) = 7.7000 day, determine the tank concentration at time t (C) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Non-steady State Continuous Flow.
• Everything except C is given, so isolate C 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.
• Non-steady state continuous flow in a completely mixed reactor tracks how tank concentration approaches the influent concentration over time.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

2. Step 2 — Rearrange symbolically for C:

   $$C = C_0 e^{-\frac{Q}{V}t}+C_{in}\left(1-e^{-\frac{Q}{V}t}\right)$$

3. Step 3 — List the givens: initial tank concentration (C_0) = 46.0000 mg/L, influent concentration (C_in) = 35.5000 mg/L, flow rate (Q) = 1,680 m^3/day, tank volume (V) = 16,620 m^3, time (t) = 7.7000 day.

4. Step 4 — Substitute the given values:

   $$C = C_0 e^{-\frac{1680}{16620}7.7000}+35.5000\lef7.7000(1-e^{-\frac{1680}{16620}7.7000}\righ7.7000)$$

5. Step 5 — Evaluate:

   $$C = 40.3213\ \text{mg/L}$$

6. Step 6 — Check: returning C = 40.3213 mg/L to

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

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

Why the other options are there

• 80.6425 — kept a factor of two that cancels in the correct rearrangement.
• 20.1606 — dropped that same factor in the other direction.
• 44.3534 — rounded an intermediate value before the final step.

Reference: FE Handbook — Non-steady State Continuous Flow (CMFR)

non-steady state continuous flow tank concentration approaching equilibrium Given initial tank concentration (C_0) = 46.0000 mg/L; flow rate (Q) = 3,440 m^3/day; tank volume (V) = 12,910 m^3; time (t) = 6.9000 day; tank concentration at time t (C) = 11.7000 mg/L, determine the influent concentration (C_in) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Non-steady State Continuous Flow.
• Everything except C_in is given, so isolate C_in 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.
• Non-steady state continuous flow in a completely mixed reactor tracks how tank concentration approaches the influent concentration over time.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

2. Step 2 — Rearrange symbolically for C_in:

   $$C_{in} = \dfrac{C-C_0 e^{-\frac{Q}{V}t}}{1-e^{-\frac{Q}{V}t}}$$

3. Step 3 — List the givens: initial tank concentration (C_0) = 46.0000 mg/L, flow rate (Q) = 3,440 m^3/day, tank volume (V) = 12,910 m^3, time (t) = 6.9000 day, tank concentration at time t (C) = 11.7000 mg/L.

4. Step 4 — Substitute the given values:

   $$C_{in} = \dfrac{11.7000-C_0 e^{-\frac{3440}{12910}6.9000}}{1-e^{-\frac{3440}{12910}6.9000}}$$

5. Step 5 — Evaluate:

   $$C_{in} = 5.2131\ \text{mg/L}$$

6. Step 6 — Check: returning C_in = 5.2131 mg/L to

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

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

Why the other options are there

• 10.4262 — kept a factor of two that cancels in the correct rearrangement.
• 2.6065 — dropped that same factor in the other direction.
• 5.7344 — rounded an intermediate value before the final step.

Reference: FE Handbook — Non-steady State Continuous Flow (CMFR)

non-steady state continuous flow reactor with changing influent concentration Given initial tank concentration (C_0) = 45.0000 mg/L; influent concentration (C_in) = 95.0000 mg/L; flow rate (Q) = 4,720 m^3/day; tank volume (V) = 18,470 m^3; tank concentration at time t (C) = 60.3000 mg/L, determine the time (t) in day.

Start with the thinking

• The governing relation printed in this handbook section is Non-steady State Continuous Flow.
• Everything except t is given, so isolate t 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.
• Non-steady state continuous flow in a completely mixed reactor tracks how tank concentration approaches the influent concentration over time.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

2. Step 2 — Rearrange symbolically for t:

   $$t = -\dfrac{V}{Q}\ln\left(\dfrac{C-C_{in}}{C_0-C_{in}}\right)$$

3. Step 3 — List the givens: initial tank concentration (C_0) = 45.0000 mg/L, influent concentration (C_in) = 95.0000 mg/L, flow rate (Q) = 4,720 m^3/day, tank volume (V) = 18,470 m^3, tank concentration at time t (C) = 60.3000 mg/L.

4. Step 4 — Substitute the given values:

   $$t = -\dfrac{18470}{4720}\ln\left(\dfrac{60.3000-95.0000}{C_0-95.0000}\right)$$

5. Step 5 — Evaluate:

   $$t = 1.4294\ \text{day}$$

6. Step 6 — Check: returning t = 1.4294 day to

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

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

Why the other options are there

• 2.8588 — kept a factor of two that cancels in the correct rearrangement.
• 0.7147 — dropped that same factor in the other direction.
• 1.5723 — rounded an intermediate value before the final step.

Reference: FE Handbook — Non-steady State Continuous Flow (CMFR)

non-steady state continuous flow completely mixed reactor start-up Given initial tank concentration (C_0) = 36.5000 mg/L; influent concentration (C_in) = 50.5000 mg/L; flow rate (Q) = 1,620 m^3/day; tank volume (V) = 18,270 m^3; time (t) = 2.5000 day, determine the tank concentration at time t (C) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Non-steady State Continuous Flow.
• Everything except C is given, so isolate C 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.
• Non-steady state continuous flow in a completely mixed reactor tracks how tank concentration approaches the influent concentration over time.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

2. Step 2 — Rearrange symbolically for C:

   $$C = C_0 e^{-\frac{Q}{V}t}+C_{in}\left(1-e^{-\frac{Q}{V}t}\right)$$

3. Step 3 — List the givens: initial tank concentration (C_0) = 36.5000 mg/L, influent concentration (C_in) = 50.5000 mg/L, flow rate (Q) = 1,620 m^3/day, tank volume (V) = 18,270 m^3, time (t) = 2.5000 day.

4. Step 4 — Substitute the given values:

   $$C = C_0 e^{-\frac{1620}{18270}2.5000}+50.5000\lef2.5000(1-e^{-\frac{1620}{18270}2.5000}\righ2.5000)$$

5. Step 5 — Evaluate:

   $$C = 39.2835\ \text{mg/L}$$

6. Step 6 — Check: returning C = 39.2835 mg/L to

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

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

Why the other options are there

• 78.5671 — kept a factor of two that cancels in the correct rearrangement.
• 19.6418 — dropped that same factor in the other direction.
• 43.2119 — rounded an intermediate value before the final step.

Reference: FE Handbook — Non-steady State Continuous Flow (CMFR)

non-steady state continuous flow tank concentration approaching equilibrium Given initial tank concentration (C_0) = 45.5000 mg/L; flow rate (Q) = 2,690 m^3/day; tank volume (V) = 18,220 m^3; time (t) = 1.7000 day; tank concentration at time t (C) = 82.4000 mg/L, determine the influent concentration (C_in) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Non-steady State Continuous Flow.
• Everything except C_in is given, so isolate C_in 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.
• Non-steady state continuous flow in a completely mixed reactor tracks how tank concentration approaches the influent concentration over time.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

2. Step 2 — Rearrange symbolically for C_in:

   $$C_{in} = \dfrac{C-C_0 e^{-\frac{Q}{V}t}}{1-e^{-\frac{Q}{V}t}}$$

3. Step 3 — List the givens: initial tank concentration (C_0) = 45.5000 mg/L, flow rate (Q) = 2,690 m^3/day, tank volume (V) = 18,220 m^3, time (t) = 1.7000 day, tank concentration at time t (C) = 82.4000 mg/L.

4. Step 4 — Substitute the given values:

   $$C_{in} = \dfrac{82.4000-C_0 e^{-\frac{2690}{18220}1.7000}}{1-e^{-\frac{2690}{18220}1.7000}}$$

5. Step 5 — Evaluate:

   $$C_{in} = 211.7\ \text{mg/L}$$

6. Step 6 — Check: returning C_in = 211.7 mg/L to

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

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

Why the other options are there

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

Reference: FE Handbook — Non-steady State Continuous Flow (CMFR)

non-steady state continuous flow reactor with changing influent concentration Given initial tank concentration (C_0) = 39.5000 mg/L; influent concentration (C_in) = 20.5000 mg/L; flow rate (Q) = 120.0 m^3/day; tank volume (V) = 13,240 m^3; tank concentration at time t (C) = 37.2000 mg/L, determine the time (t) in day.

Start with the thinking

• The governing relation printed in this handbook section is Non-steady State Continuous Flow.
• Everything except t is given, so isolate t 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.
• Non-steady state continuous flow in a completely mixed reactor tracks how tank concentration approaches the influent concentration over time.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

2. Step 2 — Rearrange symbolically for t:

   $$t = -\dfrac{V}{Q}\ln\left(\dfrac{C-C_{in}}{C_0-C_{in}}\right)$$

3. Step 3 — List the givens: initial tank concentration (C_0) = 39.5000 mg/L, influent concentration (C_in) = 20.5000 mg/L, flow rate (Q) = 120.0 m^3/day, tank volume (V) = 13,240 m^3, tank concentration at time t (C) = 37.2000 mg/L.

4. Step 4 — Substitute the given values:

   $$t = -\dfrac{13240}{120.0}\ln\left(\dfrac{37.2000-20.5000}{C_0-20.5000}\right)$$

5. Step 5 — Evaluate:

   $$t = 14.2363\ \text{day}$$

6. Step 6 — Check: returning t = 14.2363 day to

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

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

Why the other options are there

• 28.4727 — kept a factor of two that cancels in the correct rearrangement.
• 7.1182 — dropped that same factor in the other direction.
• 15.6600 — rounded an intermediate value before the final step.

Reference: FE Handbook — Non-steady State Continuous Flow (CMFR)

non-steady state continuous flow completely mixed reactor start-up Given initial tank concentration (C_0) = 30.5000 mg/L; influent concentration (C_in) = 58.0000 mg/L; flow rate (Q) = 2,420 m^3/day; tank volume (V) = 14,970 m^3; time (t) = 2.4000 day, determine the tank concentration at time t (C) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Non-steady State Continuous Flow.
• Everything except C is given, so isolate C 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.
• Non-steady state continuous flow in a completely mixed reactor tracks how tank concentration approaches the influent concentration over time.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

2. Step 2 — Rearrange symbolically for C:

   $$C = C_0 e^{-\frac{Q}{V}t}+C_{in}\left(1-e^{-\frac{Q}{V}t}\right)$$

3. Step 3 — List the givens: initial tank concentration (C_0) = 30.5000 mg/L, influent concentration (C_in) = 58.0000 mg/L, flow rate (Q) = 2,420 m^3/day, tank volume (V) = 14,970 m^3, time (t) = 2.4000 day.

4. Step 4 — Substitute the given values:

   $$C = C_0 e^{-\frac{2420}{14970}2.4000}+58.0000\lef2.4000(1-e^{-\frac{2420}{14970}2.4000}\righ2.4000)$$

5. Step 5 — Evaluate:

   $$C = 39.3432\ \text{mg/L}$$

6. Step 6 — Check: returning C = 39.3432 mg/L to

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

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

Why the other options are there

• 78.6864 — kept a factor of two that cancels in the correct rearrangement.
• 19.6716 — dropped that same factor in the other direction.
• 43.2775 — rounded an intermediate value before the final step.

Reference: FE Handbook — Non-steady State Continuous Flow (CMFR)

non-steady state continuous flow tank concentration approaching equilibrium Given initial tank concentration (C_0) = 23.5000 mg/L; flow rate (Q) = 2,280 m^3/day; tank volume (V) = 19,490 m^3; time (t) = 5.5000 day; tank concentration at time t (C) = 68.8000 mg/L, determine the influent concentration (C_in) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Non-steady State Continuous Flow.
• Everything except C_in is given, so isolate C_in 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.
• Non-steady state continuous flow in a completely mixed reactor tracks how tank concentration approaches the influent concentration over time.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

2. Step 2 — Rearrange symbolically for C_in:

   $$C_{in} = \dfrac{C-C_0 e^{-\frac{Q}{V}t}}{1-e^{-\frac{Q}{V}t}}$$

3. Step 3 — List the givens: initial tank concentration (C_0) = 23.5000 mg/L, flow rate (Q) = 2,280 m^3/day, tank volume (V) = 19,490 m^3, time (t) = 5.5000 day, tank concentration at time t (C) = 68.8000 mg/L.

4. Step 4 — Substitute the given values:

   $$C_{in} = \dfrac{68.8000-C_0 e^{-\frac{2280}{19490}5.5000}}{1-e^{-\frac{2280}{19490}5.5000}}$$

5. Step 5 — Evaluate:

   $$C_{in} = 119.0\ \text{mg/L}$$

6. Step 6 — Check: returning C_in = 119.0 mg/L to

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

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

Why the other options are there

• 237.9 — kept a factor of two that cancels in the correct rearrangement.
• 59.4844 — dropped that same factor in the other direction.
• 130.9 — rounded an intermediate value before the final step.

Reference: FE Handbook — Non-steady State Continuous Flow (CMFR)

non-steady state continuous flow reactor with changing influent concentration Given initial tank concentration (C_0) = 47.0000 mg/L; influent concentration (C_in) = 88.5000 mg/L; flow rate (Q) = 1,600 m^3/day; tank volume (V) = 9,340 m^3; tank concentration at time t (C) = 19.9000 mg/L, determine the time (t) in day.

Start with the thinking

• The governing relation printed in this handbook section is Non-steady State Continuous Flow.
• Everything except t is given, so isolate t 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.
• Non-steady state continuous flow in a completely mixed reactor tracks how tank concentration approaches the influent concentration over time.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

2. Step 2 — Rearrange symbolically for t:

   $$t = -\dfrac{V}{Q}\ln\left(\dfrac{C-C_{in}}{C_0-C_{in}}\right)$$

3. Step 3 — List the givens: initial tank concentration (C_0) = 47.0000 mg/L, influent concentration (C_in) = 88.5000 mg/L, flow rate (Q) = 1,600 m^3/day, tank volume (V) = 9,340 m^3, tank concentration at time t (C) = 19.9000 mg/L.

4. Step 4 — Substitute the given values:

   $$t = -\dfrac{9340}{1600}\ln\left(\dfrac{19.9000-88.5000}{C_0-88.5000}\right)$$

5. Step 5 — Evaluate:

   $$t = -2.9339\ \text{day}$$

6. Step 6 — Check: returning t = -2.9339 day to

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

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

Why the other options are there

• -5.8678 — kept a factor of two that cancels in the correct rearrangement.
• -1.4670 — dropped that same factor in the other direction.
• -3.2273 — rounded an intermediate value before the final step.

Reference: FE Handbook — Non-steady State Continuous Flow (CMFR)

non-steady state continuous flow completely mixed reactor start-up Given initial tank concentration (C_0) = 27.0000 mg/L; influent concentration (C_in) = 6.0000 mg/L; flow rate (Q) = 4,930 m^3/day; tank volume (V) = 10,310 m^3; time (t) = 2.3000 day, determine the tank concentration at time t (C) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Non-steady State Continuous Flow.
• Everything except C is given, so isolate C 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.
• Non-steady state continuous flow in a completely mixed reactor tracks how tank concentration approaches the influent concentration over time.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

2. Step 2 — Rearrange symbolically for C:

   $$C = C_0 e^{-\frac{Q}{V}t}+C_{in}\left(1-e^{-\frac{Q}{V}t}\right)$$

3. Step 3 — List the givens: initial tank concentration (C_0) = 27.0000 mg/L, influent concentration (C_in) = 6.0000 mg/L, flow rate (Q) = 4,930 m^3/day, tank volume (V) = 10,310 m^3, time (t) = 2.3000 day.

4. Step 4 — Substitute the given values:

   $$C = C_0 e^{-\frac{4930}{10310}2.3000}+6.0000\lef2.3000(1-e^{-\frac{4930}{10310}2.3000}\righ2.3000)$$

5. Step 5 — Evaluate:

   $$C = 12.9916\ \text{mg/L}$$

6. Step 6 — Check: returning C = 12.9916 mg/L to

   $$C = C_0 e^{-\frac{Q}{V} t} + C_{in}\left(1 - e^{-\frac{Q}{V} t}\right)$$

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

Why the other options are there

• 25.9833 — kept a factor of two that cancels in the correct rearrangement.
• 6.4958 — dropped that same factor in the other direction.
• 14.2908 — rounded an intermediate value before the final step.

Reference: FE Handbook — Non-steady State Continuous Flow (CMFR)