material balance around a wastewater treatment unit process Given mass flow in (In) = 352.0 kg/day; mass flow out (Out) = 1,633 kg/day; generation rate (Gen) = 366.0 kg/day, determine the accumulation rate (Acc) in kg/day.

Start with the thinking

• The governing relation printed in this handbook section is Material Balance.
• Everything except Acc is given, so isolate Acc 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.
• A material balance around a treatment process accounts for accumulation, inflow, outflow, and generation of a constituent.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Accumulation = In - Out + Generation$$

2. Step 2 — Rearrange symbolically for Acc:

   $$Acc = In-Out+Gen$$

3. Step 3 — List the givens: mass flow in (In) = 352.0 kg/day, mass flow out (Out) = 1,633 kg/day, generation rate (Gen) = 366.0 kg/day.

4. Step 4 — Substitute the given values:

   $$Acc = 352.0-1633+366.0$$

5. Step 5 — Evaluate:

   $$Acc = -915.0\ \text{kg/day}$$

6. Step 6 — Check: returning Acc = -915.0 kg/day to

   $$Accumulation = In - Out + Generation$$

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

Why the other options are there

• -1,830 — kept a factor of two that cancels in the correct rearrangement.
• -457.5 — dropped that same factor in the other direction.
• -1,007 — rounded an intermediate value before the final step.

Reference: FE Handbook — Material Balance

material balance for a pollutant across a mixing basin Given mass flow out (Out) = 390.0 kg/day; generation rate (Gen) = 476.0 kg/day; accumulation rate (Acc) = 243.0 kg/day, determine the mass flow in (In) in kg/day.

Start with the thinking

• The governing relation printed in this handbook section is Material Balance.
• Everything except In is given, so isolate 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.
• A material balance around a treatment process accounts for accumulation, inflow, outflow, and generation of a constituent.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Accumulation = In - Out + Generation$$

2. Step 2 — Rearrange symbolically for In:

   $$In = Acc+Out-Gen$$

3. Step 3 — List the givens: mass flow out (Out) = 390.0 kg/day, generation rate (Gen) = 476.0 kg/day, accumulation rate (Acc) = 243.0 kg/day.

4. Step 4 — Substitute the given values:

   $$In = 243.0+390.0-476.0$$

5. Step 5 — Evaluate:

   $$In = 157.0\ \text{kg/day}$$

6. Step 6 — Check: returning In = 157.0 kg/day to

   $$Accumulation = In - Out + Generation$$

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

Why the other options are there

• 314.0 — kept a factor of two that cancels in the correct rearrangement.
• 78.5000 — dropped that same factor in the other direction.
• 172.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Material Balance

steady state material balance on a treatment reactor Given mass flow in (In) = 3,485 kg/day; generation rate (Gen) = 220.0 kg/day; accumulation rate (Acc) = 1,267 kg/day, determine the mass flow out (Out) in kg/day.

Start with the thinking

• The governing relation printed in this handbook section is Material Balance.
• Everything except Out is given, so isolate Out 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.
• A material balance around a treatment process accounts for accumulation, inflow, outflow, and generation of a constituent.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Accumulation = In - Out + Generation$$

2. Step 2 — Rearrange symbolically for Out:

   $$Out = In-Acc+Gen$$

3. Step 3 — List the givens: mass flow in (In) = 3,485 kg/day, generation rate (Gen) = 220.0 kg/day, accumulation rate (Acc) = 1,267 kg/day.

4. Step 4 — Substitute the given values:

   $$Out = 3485-1267+220.0$$

5. Step 5 — Evaluate:

   $$Out = 2438\ \text{kg/day}$$

6. Step 6 — Check: returning Out = 2,438 kg/day to

   $$Accumulation = In - Out + Generation$$

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

Why the other options are there

• 4,876 — kept a factor of two that cancels in the correct rearrangement.
• 1,219 — dropped that same factor in the other direction.
• 2,682 — rounded an intermediate value before the final step.

Reference: FE Handbook — Material Balance

material balance around a wastewater treatment unit process Given mass flow in (In) = 3,065 kg/day; mass flow out (Out) = 3,323 kg/day; generation rate (Gen) = 489.0 kg/day, determine the accumulation rate (Acc) in kg/day.

Start with the thinking

• The governing relation printed in this handbook section is Material Balance.
• Everything except Acc is given, so isolate Acc 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.
• A material balance around a treatment process accounts for accumulation, inflow, outflow, and generation of a constituent.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Accumulation = In - Out + Generation$$

2. Step 2 — Rearrange symbolically for Acc:

   $$Acc = In-Out+Gen$$

3. Step 3 — List the givens: mass flow in (In) = 3,065 kg/day, mass flow out (Out) = 3,323 kg/day, generation rate (Gen) = 489.0 kg/day.

4. Step 4 — Substitute the given values:

   $$Acc = 3065-3323+489.0$$

5. Step 5 — Evaluate:

   $$Acc = 231.0\ \text{kg/day}$$

6. Step 6 — Check: returning Acc = 231.0 kg/day to

   $$Accumulation = In - Out + Generation$$

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

Why the other options are there

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

Reference: FE Handbook — Material Balance

material balance for a pollutant across a mixing basin Given mass flow out (Out) = 2,486 kg/day; generation rate (Gen) = 395.0 kg/day; accumulation rate (Acc) = -1,069 kg/day, determine the mass flow in (In) in kg/day.

Start with the thinking

• The governing relation printed in this handbook section is Material Balance.
• Everything except In is given, so isolate 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.
• A material balance around a treatment process accounts for accumulation, inflow, outflow, and generation of a constituent.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Accumulation = In - Out + Generation$$

2. Step 2 — Rearrange symbolically for In:

   $$In = Acc+Out-Gen$$

3. Step 3 — List the givens: mass flow out (Out) = 2,486 kg/day, generation rate (Gen) = 395.0 kg/day, accumulation rate (Acc) = -1,069 kg/day.

4. Step 4 — Substitute the given values:

   $$In = -1069+2486-395.0$$

5. Step 5 — Evaluate:

   $$In = 1022\ \text{kg/day}$$

6. Step 6 — Check: returning In = 1,022 kg/day to

   $$Accumulation = In - Out + Generation$$

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

Why the other options are there

• 2,044 — kept a factor of two that cancels in the correct rearrangement.
• 511.0 — dropped that same factor in the other direction.
• 1,124 — rounded an intermediate value before the final step.

Reference: FE Handbook — Material Balance

steady state material balance on a treatment reactor Given mass flow in (In) = 1,257 kg/day; generation rate (Gen) = 356.0 kg/day; accumulation rate (Acc) = -1,632 kg/day, determine the mass flow out (Out) in kg/day.

Start with the thinking

• The governing relation printed in this handbook section is Material Balance.
• Everything except Out is given, so isolate Out 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.
• A material balance around a treatment process accounts for accumulation, inflow, outflow, and generation of a constituent.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Accumulation = In - Out + Generation$$

2. Step 2 — Rearrange symbolically for Out:

   $$Out = In-Acc+Gen$$

3. Step 3 — List the givens: mass flow in (In) = 1,257 kg/day, generation rate (Gen) = 356.0 kg/day, accumulation rate (Acc) = -1,632 kg/day.

4. Step 4 — Substitute the given values:

   $$Out = 1257--1632+356.0$$

5. Step 5 — Evaluate:

   $$Out = 3245\ \text{kg/day}$$

6. Step 6 — Check: returning Out = 3,245 kg/day to

   $$Accumulation = In - Out + Generation$$

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

Why the other options are there

• 6,490 — kept a factor of two that cancels in the correct rearrangement.
• 1,623 — dropped that same factor in the other direction.
• 3,570 — rounded an intermediate value before the final step.

Reference: FE Handbook — Material Balance

material balance around a wastewater treatment unit process Given mass flow in (In) = 1,566 kg/day; mass flow out (Out) = 3,384 kg/day; generation rate (Gen) = 453.0 kg/day, determine the accumulation rate (Acc) in kg/day.

Start with the thinking

• The governing relation printed in this handbook section is Material Balance.
• Everything except Acc is given, so isolate Acc 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.
• A material balance around a treatment process accounts for accumulation, inflow, outflow, and generation of a constituent.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Accumulation = In - Out + Generation$$

2. Step 2 — Rearrange symbolically for Acc:

   $$Acc = In-Out+Gen$$

3. Step 3 — List the givens: mass flow in (In) = 1,566 kg/day, mass flow out (Out) = 3,384 kg/day, generation rate (Gen) = 453.0 kg/day.

4. Step 4 — Substitute the given values:

   $$Acc = 1566-3384+453.0$$

5. Step 5 — Evaluate:

   $$Acc = -1365\ \text{kg/day}$$

6. Step 6 — Check: returning Acc = -1,365 kg/day to

   $$Accumulation = In - Out + Generation$$

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

Why the other options are there

• -2,730 — kept a factor of two that cancels in the correct rearrangement.
• -682.5 — dropped that same factor in the other direction.
• -1,502 — rounded an intermediate value before the final step.

Reference: FE Handbook — Material Balance

material balance for a pollutant across a mixing basin Given mass flow out (Out) = 1,144 kg/day; generation rate (Gen) = 303.0 kg/day; accumulation rate (Acc) = 1,935 kg/day, determine the mass flow in (In) in kg/day.

Start with the thinking

• The governing relation printed in this handbook section is Material Balance.
• Everything except In is given, so isolate 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.
• A material balance around a treatment process accounts for accumulation, inflow, outflow, and generation of a constituent.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Accumulation = In - Out + Generation$$

2. Step 2 — Rearrange symbolically for In:

   $$In = Acc+Out-Gen$$

3. Step 3 — List the givens: mass flow out (Out) = 1,144 kg/day, generation rate (Gen) = 303.0 kg/day, accumulation rate (Acc) = 1,935 kg/day.

4. Step 4 — Substitute the given values:

   $$In = 1935+1144-303.0$$

5. Step 5 — Evaluate:

   $$In = 2776\ \text{kg/day}$$

6. Step 6 — Check: returning In = 2,776 kg/day to

   $$Accumulation = In - Out + Generation$$

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

Why the other options are there

• 5,552 — kept a factor of two that cancels in the correct rearrangement.
• 1,388 — dropped that same factor in the other direction.
• 3,054 — rounded an intermediate value before the final step.

Reference: FE Handbook — Material Balance

steady state material balance on a treatment reactor Given mass flow in (In) = 2,182 kg/day; generation rate (Gen) = 98.0000 kg/day; accumulation rate (Acc) = -1,363 kg/day, determine the mass flow out (Out) in kg/day.

Start with the thinking

• The governing relation printed in this handbook section is Material Balance.
• Everything except Out is given, so isolate Out 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.
• A material balance around a treatment process accounts for accumulation, inflow, outflow, and generation of a constituent.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Accumulation = In - Out + Generation$$

2. Step 2 — Rearrange symbolically for Out:

   $$Out = In-Acc+Gen$$

3. Step 3 — List the givens: mass flow in (In) = 2,182 kg/day, generation rate (Gen) = 98.0000 kg/day, accumulation rate (Acc) = -1,363 kg/day.

4. Step 4 — Substitute the given values:

   $$Out = 2182--1363+98.0000$$

5. Step 5 — Evaluate:

   $$Out = 3643\ \text{kg/day}$$

6. Step 6 — Check: returning Out = 3,643 kg/day to

   $$Accumulation = In - Out + Generation$$

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

Why the other options are there

• 7,286 — kept a factor of two that cancels in the correct rearrangement.
• 1,822 — dropped that same factor in the other direction.
• 4,007 — rounded an intermediate value before the final step.

Reference: FE Handbook — Material Balance

material balance around a wastewater treatment unit process Given mass flow in (In) = 4,430 kg/day; mass flow out (Out) = 707.0 kg/day; generation rate (Gen) = 7.0000 kg/day, determine the accumulation rate (Acc) in kg/day.

Start with the thinking

• The governing relation printed in this handbook section is Material Balance.
• Everything except Acc is given, so isolate Acc 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.
• A material balance around a treatment process accounts for accumulation, inflow, outflow, and generation of a constituent.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Accumulation = In - Out + Generation$$

2. Step 2 — Rearrange symbolically for Acc:

   $$Acc = In-Out+Gen$$

3. Step 3 — List the givens: mass flow in (In) = 4,430 kg/day, mass flow out (Out) = 707.0 kg/day, generation rate (Gen) = 7.0000 kg/day.

4. Step 4 — Substitute the given values:

   $$Acc = 4430-707.0+7.0000$$

5. Step 5 — Evaluate:

   $$Acc = 3730\ \text{kg/day}$$

6. Step 6 — Check: returning Acc = 3,730 kg/day to

   $$Accumulation = In - Out + Generation$$

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

Why the other options are there

• 7,460 — kept a factor of two that cancels in the correct rearrangement.
• 1,865 — dropped that same factor in the other direction.
• 4,103 — rounded an intermediate value before the final step.

Reference: FE Handbook — Material Balance