depth of sorption zone in a granular activated carbon column Given carbon bed depth (D) = 4.2000 m; time to total exhaustion (t_t) = 177.0 day; time to breakthrough (t_b) = 82.0000 day, determine the sorption zone depth (D_s) in m.

Start with the thinking

• The governing relation printed in this handbook section is Depth of Sorption Zone.
• Everything except D_s is given, so isolate D_s 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.
• Depth of sorption zone in a carbon adsorption column is derived from breakthrough and exhaustion times of the bed.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

2. Step 2 — Rearrange symbolically for D_s:

   $$D_{s} = D\left(\dfrac{t_t-t_b}{t_t-0.5t_b}\right)$$

3. Step 3 — List the givens: carbon bed depth (D) = 4.2000 m, time to total exhaustion (t_t) = 177.0 day, time to breakthrough (t_b) = 82.0000 day.

4. Step 4 — Substitute the given values:

   $$D_{s} = 4.2000\left(\dfrac{t_t-t_b}{t_t-0.5t_b}\right)$$

5. Step 5 — Evaluate:

   $$D_{s} = 2.9338\ \text{m}$$

6. Step 6 — Check: returning D_s = 2.9338 m to

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

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

Why the other options are there

• 5.8676 — kept a factor of two that cancels in the correct rearrangement.
• 1.4669 — dropped that same factor in the other direction.
• 3.2272 — rounded an intermediate value before the final step.

Reference: FE Handbook — Depth of Sorption Zone

depth of sorption zone calculated from breakthrough curve data Given time to total exhaustion (t_t) = 42.0000 day; time to breakthrough (t_b) = 112.0 day; sorption zone depth (D_s) = 1.8100 m, determine the carbon bed depth (D) in m.

Start with the thinking

• The governing relation printed in this handbook section is Depth of Sorption Zone.
• 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.
• Depth of sorption zone in a carbon adsorption column is derived from breakthrough and exhaustion times of the bed.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

2. Step 2 — Rearrange symbolically for D:

   $$D = D_s\left(\dfrac{t_t-0.5t_b}{t_t-t_b}\right)$$

3. Step 3 — List the givens: time to total exhaustion (t_t) = 42.0000 day, time to breakthrough (t_b) = 112.0 day, sorption zone depth (D_s) = 1.8100 m.

4. Step 4 — Substitute the given values:

   $$D = D_s\left(\dfrac{t_t-0.5t_b}{t_t-t_b}\right)$$

5. Step 5 — Evaluate:

   $$D = 0.3620\ \text{m}$$

6. Step 6 — Check: returning D = 0.3620 m to

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

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

Why the other options are there

• 0.7240 — kept a factor of two that cancels in the correct rearrangement.
• 0.1810 — dropped that same factor in the other direction.
• 0.3982 — rounded an intermediate value before the final step.

Reference: FE Handbook — Depth of Sorption Zone

sorption zone depth used to design carbon bed replacement schedule Given carbon bed depth (D) = 2.4500 m; time to total exhaustion (t_t) = 144.0 day; time to breakthrough (t_b) = 117.0 day, determine the sorption zone depth (D_s) in m.

Start with the thinking

• The governing relation printed in this handbook section is Depth of Sorption Zone.
• Everything except D_s is given, so isolate D_s 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.
• Depth of sorption zone in a carbon adsorption column is derived from breakthrough and exhaustion times of the bed.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

2. Step 2 — Rearrange symbolically for D_s:

   $$D_{s} = D\left(\dfrac{t_t-t_b}{t_t-0.5t_b}\right)$$

3. Step 3 — List the givens: carbon bed depth (D) = 2.4500 m, time to total exhaustion (t_t) = 144.0 day, time to breakthrough (t_b) = 117.0 day.

4. Step 4 — Substitute the given values:

   $$D_{s} = 2.4500\left(\dfrac{t_t-t_b}{t_t-0.5t_b}\right)$$

5. Step 5 — Evaluate:

   $$D_{s} = 0.7737\ \text{m}$$

6. Step 6 — Check: returning D_s = 0.7737 m to

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

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

Why the other options are there

• 1.5474 — kept a factor of two that cancels in the correct rearrangement.
• 0.3868 — dropped that same factor in the other direction.
• 0.8511 — rounded an intermediate value before the final step.

Reference: FE Handbook — Depth of Sorption Zone

depth of sorption zone in a granular activated carbon column Given time to total exhaustion (t_t) = 66.0000 day; time to breakthrough (t_b) = 41.0000 day; sorption zone depth (D_s) = 2.7700 m, determine the carbon bed depth (D) in m.

Start with the thinking

• The governing relation printed in this handbook section is Depth of Sorption Zone.
• 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.
• Depth of sorption zone in a carbon adsorption column is derived from breakthrough and exhaustion times of the bed.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

2. Step 2 — Rearrange symbolically for D:

   $$D = D_s\left(\dfrac{t_t-0.5t_b}{t_t-t_b}\right)$$

3. Step 3 — List the givens: time to total exhaustion (t_t) = 66.0000 day, time to breakthrough (t_b) = 41.0000 day, sorption zone depth (D_s) = 2.7700 m.

4. Step 4 — Substitute the given values:

   $$D = D_s\left(\dfrac{t_t-0.5t_b}{t_t-t_b}\right)$$

5. Step 5 — Evaluate:

   $$D = 5.0414\ \text{m}$$

6. Step 6 — Check: returning D = 5.0414 m to

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

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

Why the other options are there

• 10.0828 — kept a factor of two that cancels in the correct rearrangement.
• 2.5207 — dropped that same factor in the other direction.
• 5.5455 — rounded an intermediate value before the final step.

Reference: FE Handbook — Depth of Sorption Zone

depth of sorption zone calculated from breakthrough curve data Given carbon bed depth (D) = 4.4000 m; time to total exhaustion (t_t) = 185.0 day; time to breakthrough (t_b) = 95.0000 day, determine the sorption zone depth (D_s) in m.

Start with the thinking

• The governing relation printed in this handbook section is Depth of Sorption Zone.
• Everything except D_s is given, so isolate D_s 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.
• Depth of sorption zone in a carbon adsorption column is derived from breakthrough and exhaustion times of the bed.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

2. Step 2 — Rearrange symbolically for D_s:

   $$D_{s} = D\left(\dfrac{t_t-t_b}{t_t-0.5t_b}\right)$$

3. Step 3 — List the givens: carbon bed depth (D) = 4.4000 m, time to total exhaustion (t_t) = 185.0 day, time to breakthrough (t_b) = 95.0000 day.

4. Step 4 — Substitute the given values:

   $$D_{s} = 4.4000\left(\dfrac{t_t-t_b}{t_t-0.5t_b}\right)$$

5. Step 5 — Evaluate:

   $$D_{s} = 2.8800\ \text{m}$$

6. Step 6 — Check: returning D_s = 2.8800 m to

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

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

Why the other options are there

• 5.7600 — kept a factor of two that cancels in the correct rearrangement.
• 1.4400 — dropped that same factor in the other direction.
• 3.1680 — rounded an intermediate value before the final step.

Reference: FE Handbook — Depth of Sorption Zone

sorption zone depth used to design carbon bed replacement schedule Given time to total exhaustion (t_t) = 132.0 day; time to breakthrough (t_b) = 40.0000 day; sorption zone depth (D_s) = 2.0200 m, determine the carbon bed depth (D) in m.

Start with the thinking

• The governing relation printed in this handbook section is Depth of Sorption Zone.
• 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.
• Depth of sorption zone in a carbon adsorption column is derived from breakthrough and exhaustion times of the bed.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

2. Step 2 — Rearrange symbolically for D:

   $$D = D_s\left(\dfrac{t_t-0.5t_b}{t_t-t_b}\right)$$

3. Step 3 — List the givens: time to total exhaustion (t_t) = 132.0 day, time to breakthrough (t_b) = 40.0000 day, sorption zone depth (D_s) = 2.0200 m.

4. Step 4 — Substitute the given values:

   $$D = D_s\left(\dfrac{t_t-0.5t_b}{t_t-t_b}\right)$$

5. Step 5 — Evaluate:

   $$D = 2.4591\ \text{m}$$

6. Step 6 — Check: returning D = 2.4591 m to

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

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

Why the other options are there

• 4.9183 — kept a factor of two that cancels in the correct rearrangement.
• 1.2296 — dropped that same factor in the other direction.
• 2.7050 — rounded an intermediate value before the final step.

Reference: FE Handbook — Depth of Sorption Zone

depth of sorption zone in a granular activated carbon column Given carbon bed depth (D) = 1.7000 m; time to total exhaustion (t_t) = 188.0 day; time to breakthrough (t_b) = 64.0000 day, determine the sorption zone depth (D_s) in m.

Start with the thinking

• The governing relation printed in this handbook section is Depth of Sorption Zone.
• Everything except D_s is given, so isolate D_s 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.
• Depth of sorption zone in a carbon adsorption column is derived from breakthrough and exhaustion times of the bed.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

2. Step 2 — Rearrange symbolically for D_s:

   $$D_{s} = D\left(\dfrac{t_t-t_b}{t_t-0.5t_b}\right)$$

3. Step 3 — List the givens: carbon bed depth (D) = 1.7000 m, time to total exhaustion (t_t) = 188.0 day, time to breakthrough (t_b) = 64.0000 day.

4. Step 4 — Substitute the given values:

   $$D_{s} = 1.7000\left(\dfrac{t_t-t_b}{t_t-0.5t_b}\right)$$

5. Step 5 — Evaluate:

   $$D_{s} = 1.3513\ \text{m}$$

6. Step 6 — Check: returning D_s = 1.3513 m to

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

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

Why the other options are there

• 2.7026 — kept a factor of two that cancels in the correct rearrangement.
• 0.6756 — dropped that same factor in the other direction.
• 1.4864 — rounded an intermediate value before the final step.

Reference: FE Handbook — Depth of Sorption Zone

depth of sorption zone calculated from breakthrough curve data Given time to total exhaustion (t_t) = 153.0 day; time to breakthrough (t_b) = 9.0000 day; sorption zone depth (D_s) = 0.9700 m, determine the carbon bed depth (D) in m.

Start with the thinking

• The governing relation printed in this handbook section is Depth of Sorption Zone.
• 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.
• Depth of sorption zone in a carbon adsorption column is derived from breakthrough and exhaustion times of the bed.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

2. Step 2 — Rearrange symbolically for D:

   $$D = D_s\left(\dfrac{t_t-0.5t_b}{t_t-t_b}\right)$$

3. Step 3 — List the givens: time to total exhaustion (t_t) = 153.0 day, time to breakthrough (t_b) = 9.0000 day, sorption zone depth (D_s) = 0.9700 m.

4. Step 4 — Substitute the given values:

   $$D = D_s\left(\dfrac{t_t-0.5t_b}{t_t-t_b}\right)$$

5. Step 5 — Evaluate:

   $$D = 1.0003\ \text{m}$$

6. Step 6 — Check: returning D = 1.0003 m to

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

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

Why the other options are there

• 2.0006 — kept a factor of two that cancels in the correct rearrangement.
• 0.5002 — dropped that same factor in the other direction.
• 1.1003 — rounded an intermediate value before the final step.

Reference: FE Handbook — Depth of Sorption Zone

sorption zone depth used to design carbon bed replacement schedule Given carbon bed depth (D) = 2.2000 m; time to total exhaustion (t_t) = 74.0000 day; time to breakthrough (t_b) = 7.0000 day, determine the sorption zone depth (D_s) in m.

Start with the thinking

• The governing relation printed in this handbook section is Depth of Sorption Zone.
• Everything except D_s is given, so isolate D_s 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.
• Depth of sorption zone in a carbon adsorption column is derived from breakthrough and exhaustion times of the bed.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

2. Step 2 — Rearrange symbolically for D_s:

   $$D_{s} = D\left(\dfrac{t_t-t_b}{t_t-0.5t_b}\right)$$

3. Step 3 — List the givens: carbon bed depth (D) = 2.2000 m, time to total exhaustion (t_t) = 74.0000 day, time to breakthrough (t_b) = 7.0000 day.

4. Step 4 — Substitute the given values:

   $$D_{s} = 2.2000\left(\dfrac{t_t-t_b}{t_t-0.5t_b}\right)$$

5. Step 5 — Evaluate:

   $$D_{s} = 2.0908\ \text{m}$$

6. Step 6 — Check: returning D_s = 2.0908 m to

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

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

Why the other options are there

• 4.1816 — kept a factor of two that cancels in the correct rearrangement.
• 1.0454 — dropped that same factor in the other direction.
• 2.2999 — rounded an intermediate value before the final step.

Reference: FE Handbook — Depth of Sorption Zone

depth of sorption zone in a granular activated carbon column Given time to total exhaustion (t_t) = 26.0000 day; time to breakthrough (t_b) = 8.0000 day; sorption zone depth (D_s) = 2.7000 m, determine the carbon bed depth (D) in m.

Start with the thinking

• The governing relation printed in this handbook section is Depth of Sorption Zone.
• 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.
• Depth of sorption zone in a carbon adsorption column is derived from breakthrough and exhaustion times of the bed.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

2. Step 2 — Rearrange symbolically for D:

   $$D = D_s\left(\dfrac{t_t-0.5t_b}{t_t-t_b}\right)$$

3. Step 3 — List the givens: time to total exhaustion (t_t) = 26.0000 day, time to breakthrough (t_b) = 8.0000 day, sorption zone depth (D_s) = 2.7000 m.

4. Step 4 — Substitute the given values:

   $$D = D_s\left(\dfrac{t_t-0.5t_b}{t_t-t_b}\right)$$

5. Step 5 — Evaluate:

   $$D = 3.3000\ \text{m}$$

6. Step 6 — Check: returning D = 3.3000 m to

   $$D_s = D\left(\dfrac{t_t - t_b}{t_t - 0.5 t_b}\right)$$

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

Why the other options are there

• 6.6000 — kept a factor of two that cancels in the correct rearrangement.
• 1.6500 — dropped that same factor in the other direction.
• 3.6300 — rounded an intermediate value before the final step.

Reference: FE Handbook — Depth of Sorption Zone