bed expansion during backwash of a rapid sand filter Given unexpanded bed depth (L) = 0.6800 m; backwash velocity (v_b) = 0.0185 m/s; particle settling velocity (v_s) = 0.0560 m/s, determine the expanded bed depth (L_e) in m.

Start with the thinking

• The governing relation printed in this handbook section is Bed Expansion.
• Everything except L_e is given, so isolate L_e 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.
• Bed expansion during filter backwashing relates the expanded bed depth to the backwash and particle settling velocities.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

2. Step 2 — Rearrange symbolically for L_e:

   $$L_{e} = L\left(\dfrac{v_b}{v_s}\right)^{0.22}$$

3. Step 3 — List the givens: unexpanded bed depth (L) = 0.6800 m, backwash velocity (v_b) = 0.0185 m/s, particle settling velocity (v_s) = 0.0560 m/s.

4. Step 4 — Substitute the given values:

   $$L_{e} = 0.6800\left(\dfrac{v_b}{v_s}\right)^{0.22}$$

5. Step 5 — Evaluate:

   $$L_{e} = 0.5329\ \text{m}$$

6. Step 6 — Check: returning L_e = 0.5329 m to

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

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

Why the other options are there

• 1.0659 — kept a factor of two that cancels in the correct rearrangement.
• 0.2665 — dropped that same factor in the other direction.
• 0.5862 — rounded an intermediate value before the final step.

Reference: FE Handbook — Bed Expansion (backwash)

bed expansion percentage calculated from backwash velocity Given backwash velocity (v_b) = 0.0075 m/s; particle settling velocity (v_s) = 0.1020 m/s; expanded bed depth (L_e) = 0.4700 m, determine the unexpanded bed depth (L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Bed Expansion.
• Everything except L is given, so isolate L 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.
• Bed expansion during filter backwashing relates the expanded bed depth to the backwash and particle settling velocities.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

2. Step 2 — Rearrange symbolically for L:

   $$L = \dfrac{L_e}{\left(v_b/v_s\right)^{0.22}}$$

3. Step 3 — List the givens: backwash velocity (v_b) = 0.0075 m/s, particle settling velocity (v_s) = 0.1020 m/s, expanded bed depth (L_e) = 0.4700 m.

4. Step 4 — Substitute the given values:

   $$L = \dfrac{L_e}{\left(v_b/v_s\right)^{0.22}}$$

5. Step 5 — Evaluate:

   $$L = 0.8346\ \text{m}$$

6. Step 6 — Check: returning L = 0.8346 m to

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

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

Why the other options are there

• 1.6692 — kept a factor of two that cancels in the correct rearrangement.
• 0.4173 — dropped that same factor in the other direction.
• 0.9181 — rounded an intermediate value before the final step.

Reference: FE Handbook — Bed Expansion (backwash)

filter bed expansion required to fluidize the media during backwash Given unexpanded bed depth (L) = 0.3300 m; particle settling velocity (v_s) = 0.0430 m/s; expanded bed depth (L_e) = 1.2000 m, determine the backwash velocity (v_b) in m/s.

Start with the thinking

• The governing relation printed in this handbook section is Bed Expansion.
• Everything except v_b is given, so isolate v_b 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.
• Bed expansion during filter backwashing relates the expanded bed depth to the backwash and particle settling velocities.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

2. Step 2 — Rearrange symbolically for v_b:

   $$v_{b} = v_s\left(\dfrac{L_e}{L}\right)^{1/0.22}$$

3. Step 3 — List the givens: unexpanded bed depth (L) = 0.3300 m, particle settling velocity (v_s) = 0.0430 m/s, expanded bed depth (L_e) = 1.2000 m.

4. Step 4 — Substitute the given values:

   $$v_{b} = v_s\left(\dfrac{L_e}{0.3300}\right)^{1/0.22}$$

5. Step 5 — Evaluate:

   $$v_{b} = 15.2039\ \text{m/s}$$

6. Step 6 — Check: returning v_b = 15.2039 m/s to

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

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

Why the other options are there

• 30.4079 — kept a factor of two that cancels in the correct rearrangement.
• 7.6020 — dropped that same factor in the other direction.
• 16.7243 — rounded an intermediate value before the final step.

Reference: FE Handbook — Bed Expansion (backwash)

bed expansion during backwash of a rapid sand filter Given unexpanded bed depth (L) = 0.3500 m; backwash velocity (v_b) = 0.0190 m/s; particle settling velocity (v_s) = 0.0560 m/s, determine the expanded bed depth (L_e) in m.

Start with the thinking

• The governing relation printed in this handbook section is Bed Expansion.
• Everything except L_e is given, so isolate L_e 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.
• Bed expansion during filter backwashing relates the expanded bed depth to the backwash and particle settling velocities.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

2. Step 2 — Rearrange symbolically for L_e:

   $$L_{e} = L\left(\dfrac{v_b}{v_s}\right)^{0.22}$$

3. Step 3 — List the givens: unexpanded bed depth (L) = 0.3500 m, backwash velocity (v_b) = 0.0190 m/s, particle settling velocity (v_s) = 0.0560 m/s.

4. Step 4 — Substitute the given values:

   $$L_{e} = 0.3500\left(\dfrac{v_b}{v_s}\right)^{0.22}$$

5. Step 5 — Evaluate:

   $$L_{e} = 0.2759\ \text{m}$$

6. Step 6 — Check: returning L_e = 0.2759 m to

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

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

Why the other options are there

• 0.5519 — kept a factor of two that cancels in the correct rearrangement.
• 0.1380 — dropped that same factor in the other direction.
• 0.3035 — rounded an intermediate value before the final step.

Reference: FE Handbook — Bed Expansion (backwash)

bed expansion percentage calculated from backwash velocity Given backwash velocity (v_b) = 0.0110 m/s; particle settling velocity (v_s) = 0.1480 m/s; expanded bed depth (L_e) = 1.1500 m, determine the unexpanded bed depth (L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Bed Expansion.
• Everything except L is given, so isolate L 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.
• Bed expansion during filter backwashing relates the expanded bed depth to the backwash and particle settling velocities.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

2. Step 2 — Rearrange symbolically for L:

   $$L = \dfrac{L_e}{\left(v_b/v_s\right)^{0.22}}$$

3. Step 3 — List the givens: backwash velocity (v_b) = 0.0110 m/s, particle settling velocity (v_s) = 0.1480 m/s, expanded bed depth (L_e) = 1.1500 m.

4. Step 4 — Substitute the given values:

   $$L = \dfrac{L_e}{\left(v_b/v_s\right)^{0.22}}$$

5. Step 5 — Evaluate:

   $$L = 2.0373\ \text{m}$$

6. Step 6 — Check: returning L = 2.0373 m to

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

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

Why the other options are there

• 4.0745 — kept a factor of two that cancels in the correct rearrangement.
• 1.0186 — dropped that same factor in the other direction.
• 2.2410 — rounded an intermediate value before the final step.

Reference: FE Handbook — Bed Expansion (backwash)

filter bed expansion required to fluidize the media during backwash Given unexpanded bed depth (L) = 0.5900 m; particle settling velocity (v_s) = 0.1230 m/s; expanded bed depth (L_e) = 1.1700 m, determine the backwash velocity (v_b) in m/s.

Start with the thinking

• The governing relation printed in this handbook section is Bed Expansion.
• Everything except v_b is given, so isolate v_b 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.
• Bed expansion during filter backwashing relates the expanded bed depth to the backwash and particle settling velocities.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

2. Step 2 — Rearrange symbolically for v_b:

   $$v_{b} = v_s\left(\dfrac{L_e}{L}\right)^{1/0.22}$$

3. Step 3 — List the givens: unexpanded bed depth (L) = 0.5900 m, particle settling velocity (v_s) = 0.1230 m/s, expanded bed depth (L_e) = 1.1700 m.

4. Step 4 — Substitute the given values:

   $$v_{b} = v_s\left(\dfrac{L_e}{0.5900}\right)^{1/0.22}$$

5. Step 5 — Evaluate:

   $$v_{b} = 2.7633\ \text{m/s}$$

6. Step 6 — Check: returning v_b = 2.7633 m/s to

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

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

Why the other options are there

• 5.5265 — kept a factor of two that cancels in the correct rearrangement.
• 1.3816 — dropped that same factor in the other direction.
• 3.0396 — rounded an intermediate value before the final step.

Reference: FE Handbook — Bed Expansion (backwash)

bed expansion during backwash of a rapid sand filter Given unexpanded bed depth (L) = 0.6800 m; backwash velocity (v_b) = 0.0090 m/s; particle settling velocity (v_s) = 0.0410 m/s, determine the expanded bed depth (L_e) in m.

Start with the thinking

• The governing relation printed in this handbook section is Bed Expansion.
• Everything except L_e is given, so isolate L_e 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.
• Bed expansion during filter backwashing relates the expanded bed depth to the backwash and particle settling velocities.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

2. Step 2 — Rearrange symbolically for L_e:

   $$L_{e} = L\left(\dfrac{v_b}{v_s}\right)^{0.22}$$

3. Step 3 — List the givens: unexpanded bed depth (L) = 0.6800 m, backwash velocity (v_b) = 0.0090 m/s, particle settling velocity (v_s) = 0.0410 m/s.

4. Step 4 — Substitute the given values:

   $$L_{e} = 0.6800\left(\dfrac{v_b}{v_s}\right)^{0.22}$$

5. Step 5 — Evaluate:

   $$L_{e} = 0.4871\ \text{m}$$

6. Step 6 — Check: returning L_e = 0.4871 m to

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

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

Why the other options are there

• 0.9742 — kept a factor of two that cancels in the correct rearrangement.
• 0.2436 — dropped that same factor in the other direction.
• 0.5358 — rounded an intermediate value before the final step.

Reference: FE Handbook — Bed Expansion (backwash)

bed expansion percentage calculated from backwash velocity Given backwash velocity (v_b) = 0.0165 m/s; particle settling velocity (v_s) = 0.0330 m/s; expanded bed depth (L_e) = 1.5800 m, determine the unexpanded bed depth (L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Bed Expansion.
• Everything except L is given, so isolate L 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.
• Bed expansion during filter backwashing relates the expanded bed depth to the backwash and particle settling velocities.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

2. Step 2 — Rearrange symbolically for L:

   $$L = \dfrac{L_e}{\left(v_b/v_s\right)^{0.22}}$$

3. Step 3 — List the givens: backwash velocity (v_b) = 0.0165 m/s, particle settling velocity (v_s) = 0.0330 m/s, expanded bed depth (L_e) = 1.5800 m.

4. Step 4 — Substitute the given values:

   $$L = \dfrac{L_e}{\left(v_b/v_s\right)^{0.22}}$$

5. Step 5 — Evaluate:

   $$L = 1.8403\ \text{m}$$

6. Step 6 — Check: returning L = 1.8403 m to

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

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

Why the other options are there

• 3.6806 — kept a factor of two that cancels in the correct rearrangement.
• 0.9201 — dropped that same factor in the other direction.
• 2.0243 — rounded an intermediate value before the final step.

Reference: FE Handbook — Bed Expansion (backwash)

filter bed expansion required to fluidize the media during backwash Given unexpanded bed depth (L) = 1.4000 m; particle settling velocity (v_s) = 0.1040 m/s; expanded bed depth (L_e) = 0.8800 m, determine the backwash velocity (v_b) in m/s.

Start with the thinking

• The governing relation printed in this handbook section is Bed Expansion.
• Everything except v_b is given, so isolate v_b 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.
• Bed expansion during filter backwashing relates the expanded bed depth to the backwash and particle settling velocities.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

2. Step 2 — Rearrange symbolically for v_b:

   $$v_{b} = v_s\left(\dfrac{L_e}{L}\right)^{1/0.22}$$

3. Step 3 — List the givens: unexpanded bed depth (L) = 1.4000 m, particle settling velocity (v_s) = 0.1040 m/s, expanded bed depth (L_e) = 0.8800 m.

4. Step 4 — Substitute the given values:

   $$v_{b} = v_s\left(\dfrac{L_e}{1.4000}\right)^{1/0.22}$$

5. Step 5 — Evaluate:

   $$v_{b} = 0.0126\ \text{m/s}$$

6. Step 6 — Check: returning v_b = 0.0126 m/s to

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

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

Why the other options are there

• 0.0252 — kept a factor of two that cancels in the correct rearrangement.
• 0.0063 — dropped that same factor in the other direction.
• 0.0139 — rounded an intermediate value before the final step.

Reference: FE Handbook — Bed Expansion (backwash)

bed expansion during backwash of a rapid sand filter Given unexpanded bed depth (L) = 1.4000 m; backwash velocity (v_b) = 0.0055 m/s; particle settling velocity (v_s) = 0.0610 m/s, determine the expanded bed depth (L_e) in m.

Start with the thinking

• The governing relation printed in this handbook section is Bed Expansion.
• Everything except L_e is given, so isolate L_e 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.
• Bed expansion during filter backwashing relates the expanded bed depth to the backwash and particle settling velocities.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

2. Step 2 — Rearrange symbolically for L_e:

   $$L_{e} = L\left(\dfrac{v_b}{v_s}\right)^{0.22}$$

3. Step 3 — List the givens: unexpanded bed depth (L) = 1.4000 m, backwash velocity (v_b) = 0.0055 m/s, particle settling velocity (v_s) = 0.0610 m/s.

4. Step 4 — Substitute the given values:

   $$L_{e} = 1.4000\left(\dfrac{v_b}{v_s}\right)^{0.22}$$

5. Step 5 — Evaluate:

   $$L_{e} = 0.8246\ \text{m}$$

6. Step 6 — Check: returning L_e = 0.8246 m to

   $$\dfrac{L_e}{L} = \left(\dfrac{v_b}{v_s}\right)^{0.22}$$

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

Why the other options are there

• 1.6492 — kept a factor of two that cancels in the correct rearrangement.
• 0.4123 — dropped that same factor in the other direction.
• 0.9070 — rounded an intermediate value before the final step.

Reference: FE Handbook — Bed Expansion (backwash)