flow through a packed bed of granular filter media Given dynamic viscosity (mu) = 0.0006 Pa*s; bed porosity (epsilon) = 0.3100; superficial velocity (V) = 0.0360 m/s; gravity (g) = 9.7400 m/s^2; fluid density (rho) = 820.0 kg/m^3; particle diameter (D_p) = 0.0028 m; bed depth (L) = 1.2000 m, determine the head loss (h_f) in m.

Start with the thinking

• The governing relation printed in this handbook section is Flow through a packed bed (Ergun-type head loss).
• Everything except h_f is given, so isolate h_f 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.
• Flow through a packed bed filter or reactor obeys a Darcy/Ergun-type head loss relation with the bed porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

2. Step 2 — Rearrange the relation so that h_f stands alone on the left-hand side.

3. Step 3 — List the givens: dynamic viscosity (mu) = 0.0006 Pa*s, bed porosity (epsilon) = 0.3100, superficial velocity (V) = 0.0360 m/s, gravity (g) = 9.7400 m/s^2, fluid density (rho) = 820.0 kg/m^3, particle diameter (D_p) = 0.0028 m, bed depth (L) = 1.2000 m.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$h_{f} = 0.9923\ \text{m}$$

6. Step 6 — Check: returning h_f = 0.9923 m to

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

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

Why the other options are there

• 1.9846 — kept a factor of two that cancels in the correct rearrangement.
• 0.4962 — dropped that same factor in the other direction.
• 1.0915 — rounded an intermediate value before the final step.

Reference: FE Handbook — Flow Through a Packed Bed

flow through a packed bed reactor with spherical particles Given dynamic viscosity (mu) = 0.0017 Pa*s; bed porosity (epsilon) = 0.3100; gravity (g) = 9.8300 m/s^2; fluid density (rho) = 1,030 kg/m^3; particle diameter (D_p) = 0.0068 m; bed depth (L) = 0.4000 m; head loss (h_f) = 3.3580 m, determine the superficial velocity (V) in m/s.

Start with the thinking

• The governing relation printed in this handbook section is Flow through a packed bed (Ergun-type head loss).
• Everything except V is given, so isolate V 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.
• Flow through a packed bed filter or reactor obeys a Darcy/Ergun-type head loss relation with the bed porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

2. Step 2 — Rearrange the relation so that V stands alone on the left-hand side.

3. Step 3 — List the givens: dynamic viscosity (mu) = 0.0017 Pa*s, bed porosity (epsilon) = 0.3100, gravity (g) = 9.8300 m/s^2, fluid density (rho) = 1,030 kg/m^3, particle diameter (D_p) = 0.0068 m, bed depth (L) = 0.4000 m, head loss (h_f) = 3.3580 m.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$V = 0.9644\ \text{m/s}$$

6. Step 6 — Check: returning V = 0.9644 m/s to

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

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

Why the other options are there

• 1.9289 — kept a factor of two that cancels in the correct rearrangement.
• 0.4822 — dropped that same factor in the other direction.
• 1.0609 — rounded an intermediate value before the final step.

Reference: FE Handbook — Flow Through a Packed Bed

flow through a packed bed used in a water treatment column Given dynamic viscosity (mu) = 0.0008 Pa*s; bed porosity (epsilon) = 0.5200; superficial velocity (V) = 0.0480 m/s; gravity (g) = 9.7900 m/s^2; fluid density (rho) = 1,050 kg/m^3; particle diameter (D_p) = 0.0040 m; head loss (h_f) = 1.6330 m, determine the bed depth (L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Flow through a packed bed (Ergun-type head loss).
• 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.
• Flow through a packed bed filter or reactor obeys a Darcy/Ergun-type head loss relation with the bed porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

2. Step 2 — Rearrange the relation so that L stands alone on the left-hand side.

3. Step 3 — List the givens: dynamic viscosity (mu) = 0.0008 Pa*s, bed porosity (epsilon) = 0.5200, superficial velocity (V) = 0.0480 m/s, gravity (g) = 9.7900 m/s^2, fluid density (rho) = 1,050 kg/m^3, particle diameter (D_p) = 0.0040 m, head loss (h_f) = 1.6330 m.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$L = 28.4566\ \text{m}$$

6. Step 6 — Check: returning L = 28.4566 m to

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

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

Why the other options are there

• 56.9132 — kept a factor of two that cancels in the correct rearrangement.
• 14.2283 — dropped that same factor in the other direction.
• 31.3023 — rounded an intermediate value before the final step.

Reference: FE Handbook — Flow Through a Packed Bed

flow through a packed bed of granular filter media Given dynamic viscosity (mu) = 0.0016 Pa*s; bed porosity (epsilon) = 0.3100; superficial velocity (V) = 0.0250 m/s; gravity (g) = 9.8200 m/s^2; fluid density (rho) = 870.0 kg/m^3; particle diameter (D_p) = 0.0086 m; bed depth (L) = 2.7500 m, determine the head loss (h_f) in m.

Start with the thinking

• The governing relation printed in this handbook section is Flow through a packed bed (Ergun-type head loss).
• Everything except h_f is given, so isolate h_f 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.
• Flow through a packed bed filter or reactor obeys a Darcy/Ergun-type head loss relation with the bed porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

2. Step 2 — Rearrange the relation so that h_f stands alone on the left-hand side.

3. Step 3 — List the givens: dynamic viscosity (mu) = 0.0016 Pa*s, bed porosity (epsilon) = 0.3100, superficial velocity (V) = 0.0250 m/s, gravity (g) = 9.8200 m/s^2, fluid density (rho) = 870.0 kg/m^3, particle diameter (D_p) = 0.0086 m, bed depth (L) = 2.7500 m.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$h_{f} = 0.4173\ \text{m}$$

6. Step 6 — Check: returning h_f = 0.4173 m to

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

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

Why the other options are there

• 0.8346 — kept a factor of two that cancels in the correct rearrangement.
• 0.2087 — dropped that same factor in the other direction.
• 0.4591 — rounded an intermediate value before the final step.

Reference: FE Handbook — Flow Through a Packed Bed

flow through a packed bed reactor with spherical particles Given dynamic viscosity (mu) = 0.0015 Pa*s; bed porosity (epsilon) = 0.5300; gravity (g) = 9.8400 m/s^2; fluid density (rho) = 1,070 kg/m^3; particle diameter (D_p) = 0.0090 m; bed depth (L) = 1.2500 m; head loss (h_f) = 2.1130 m, determine the superficial velocity (V) in m/s.

Start with the thinking

• The governing relation printed in this handbook section is Flow through a packed bed (Ergun-type head loss).
• Everything except V is given, so isolate V 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.
• Flow through a packed bed filter or reactor obeys a Darcy/Ergun-type head loss relation with the bed porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

2. Step 2 — Rearrange the relation so that V stands alone on the left-hand side.

3. Step 3 — List the givens: dynamic viscosity (mu) = 0.0015 Pa*s, bed porosity (epsilon) = 0.5300, gravity (g) = 9.8400 m/s^2, fluid density (rho) = 1,070 kg/m^3, particle diameter (D_p) = 0.0090 m, bed depth (L) = 1.2500 m, head loss (h_f) = 2.1130 m.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$V = 4.3182\ \text{m/s}$$

6. Step 6 — Check: returning V = 4.3182 m/s to

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

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

Why the other options are there

• 8.6364 — kept a factor of two that cancels in the correct rearrangement.
• 2.1591 — dropped that same factor in the other direction.
• 4.7500 — rounded an intermediate value before the final step.

Reference: FE Handbook — Flow Through a Packed Bed

flow through a packed bed used in a water treatment column Given dynamic viscosity (mu) = 0.0019 Pa*s; bed porosity (epsilon) = 0.4500; superficial velocity (V) = 0.0010 m/s; gravity (g) = 9.8700 m/s^2; fluid density (rho) = 870.0 kg/m^3; particle diameter (D_p) = 0.0022 m; head loss (h_f) = 1.0040 m, determine the bed depth (L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Flow through a packed bed (Ergun-type head loss).
• 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.
• Flow through a packed bed filter or reactor obeys a Darcy/Ergun-type head loss relation with the bed porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

2. Step 2 — Rearrange the relation so that L stands alone on the left-hand side.

3. Step 3 — List the givens: dynamic viscosity (mu) = 0.0019 Pa*s, bed porosity (epsilon) = 0.4500, superficial velocity (V) = 0.0010 m/s, gravity (g) = 9.8700 m/s^2, fluid density (rho) = 870.0 kg/m^3, particle diameter (D_p) = 0.0022 m, head loss (h_f) = 1.0040 m.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$L = 44.1045\ \text{m}$$

6. Step 6 — Check: returning L = 44.1045 m to

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

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

Why the other options are there

• 88.2090 — kept a factor of two that cancels in the correct rearrangement.
• 22.0522 — dropped that same factor in the other direction.
• 48.5149 — rounded an intermediate value before the final step.

Reference: FE Handbook — Flow Through a Packed Bed

flow through a packed bed of granular filter media Given dynamic viscosity (mu) = 0.0007 Pa*s; bed porosity (epsilon) = 0.5600; superficial velocity (V) = 0.0090 m/s; gravity (g) = 9.7500 m/s^2; fluid density (rho) = 1,110 kg/m^3; particle diameter (D_p) = 0.0092 m; bed depth (L) = 1.5000 m, determine the head loss (h_f) in m.

Start with the thinking

• The governing relation printed in this handbook section is Flow through a packed bed (Ergun-type head loss).
• Everything except h_f is given, so isolate h_f 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.
• Flow through a packed bed filter or reactor obeys a Darcy/Ergun-type head loss relation with the bed porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

2. Step 2 — Rearrange the relation so that h_f stands alone on the left-hand side.

3. Step 3 — List the givens: dynamic viscosity (mu) = 0.0007 Pa*s, bed porosity (epsilon) = 0.5600, superficial velocity (V) = 0.0090 m/s, gravity (g) = 9.7500 m/s^2, fluid density (rho) = 1,110 kg/m^3, particle diameter (D_p) = 0.0092 m, bed depth (L) = 1.5000 m.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$h_{f} = 0.0017\ \text{m}$$

6. Step 6 — Check: returning h_f = 0.0017 m to

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

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

Why the other options are there

• 0.0034 — kept a factor of two that cancels in the correct rearrangement.
• 0.0009 — dropped that same factor in the other direction.
• 0.0019 — rounded an intermediate value before the final step.

Reference: FE Handbook — Flow Through a Packed Bed

flow through a packed bed reactor with spherical particles Given dynamic viscosity (mu) = 0.0014 Pa*s; bed porosity (epsilon) = 0.5400; gravity (g) = 9.8300 m/s^2; fluid density (rho) = 860.0 kg/m^3; particle diameter (D_p) = 0.0066 m; bed depth (L) = 1.4500 m; head loss (h_f) = 4.5030 m, determine the superficial velocity (V) in m/s.

Start with the thinking

• The governing relation printed in this handbook section is Flow through a packed bed (Ergun-type head loss).
• Everything except V is given, so isolate V 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.
• Flow through a packed bed filter or reactor obeys a Darcy/Ergun-type head loss relation with the bed porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

2. Step 2 — Rearrange the relation so that V stands alone on the left-hand side.

3. Step 3 — List the givens: dynamic viscosity (mu) = 0.0014 Pa*s, bed porosity (epsilon) = 0.5400, gravity (g) = 9.8300 m/s^2, fluid density (rho) = 860.0 kg/m^3, particle diameter (D_p) = 0.0066 m, bed depth (L) = 1.4500 m, head loss (h_f) = 4.5030 m.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$V = 4.0525\ \text{m/s}$$

6. Step 6 — Check: returning V = 4.0525 m/s to

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

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

Why the other options are there

• 8.1049 — kept a factor of two that cancels in the correct rearrangement.
• 2.0262 — dropped that same factor in the other direction.
• 4.4577 — rounded an intermediate value before the final step.

Reference: FE Handbook — Flow Through a Packed Bed

flow through a packed bed used in a water treatment column Given dynamic viscosity (mu) = 0.0016 Pa*s; bed porosity (epsilon) = 0.3200; superficial velocity (V) = 0.0330 m/s; gravity (g) = 9.8900 m/s^2; fluid density (rho) = 1,090 kg/m^3; particle diameter (D_p) = 0.0050 m; head loss (h_f) = 1.0040 m, determine the bed depth (L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Flow through a packed bed (Ergun-type head loss).
• 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.
• Flow through a packed bed filter or reactor obeys a Darcy/Ergun-type head loss relation with the bed porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

2. Step 2 — Rearrange the relation so that L stands alone on the left-hand side.

3. Step 3 — List the givens: dynamic viscosity (mu) = 0.0016 Pa*s, bed porosity (epsilon) = 0.3200, superficial velocity (V) = 0.0330 m/s, gravity (g) = 9.8900 m/s^2, fluid density (rho) = 1,090 kg/m^3, particle diameter (D_p) = 0.0050 m, head loss (h_f) = 1.0040 m.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$L = 2.4210\ \text{m}$$

6. Step 6 — Check: returning L = 2.4210 m to

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

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

Why the other options are there

• 4.8421 — kept a factor of two that cancels in the correct rearrangement.
• 1.2105 — dropped that same factor in the other direction.
• 2.6632 — rounded an intermediate value before the final step.

Reference: FE Handbook — Flow Through a Packed Bed

flow through a packed bed of granular filter media Given dynamic viscosity (mu) = 0.0008 Pa*s; bed porosity (epsilon) = 0.4100; superficial velocity (V) = 0.0090 m/s; gravity (g) = 9.7300 m/s^2; fluid density (rho) = 870.0 kg/m^3; particle diameter (D_p) = 0.0082 m; bed depth (L) = 2.7500 m, determine the head loss (h_f) in m.

Start with the thinking

• The governing relation printed in this handbook section is Flow through a packed bed (Ergun-type head loss).
• Everything except h_f is given, so isolate h_f 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.
• Flow through a packed bed filter or reactor obeys a Darcy/Ergun-type head loss relation with the bed porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

2. Step 2 — Rearrange the relation so that h_f stands alone on the left-hand side.

3. Step 3 — List the givens: dynamic viscosity (mu) = 0.0008 Pa*s, bed porosity (epsilon) = 0.4100, superficial velocity (V) = 0.0090 m/s, gravity (g) = 9.7300 m/s^2, fluid density (rho) = 870.0 kg/m^3, particle diameter (D_p) = 0.0082 m, bed depth (L) = 2.7500 m.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$h_{f} = 0.0264\ \text{m}$$

6. Step 6 — Check: returning h_f = 0.0264 m to

   $$\dfrac{h_f}{L} = \dfrac{150 \mu (1-\epsilon)^2 V}{\epsilon^3 g \rho D_p^2}$$

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

Why the other options are there

• 0.0527 — kept a factor of two that cancels in the correct rearrangement.
• 0.0132 — dropped that same factor in the other direction.
• 0.0290 — rounded an intermediate value before the final step.

Reference: FE Handbook — Flow Through a Packed Bed