Rose equation headloss through a clean rapid sand filter bed Given drag coefficient (C_D) = 2.2000; filter media depth (L) = 0.7900 m; filtration velocity (v_f) = 0.0038 m/s; shape factor (phi) = 0.8700; gravitational acceleration (g) = 9.8100 m/s^2; grain diameter (d) = 0.0019 m; porosity (eps) = 0.4040, determine the headloss through clean bed (h_L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Rose Equation.
• Everything except h_L is given, so isolate h_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.
• The Rose equation calculates head loss through a clean granular filter bed based on grain size, depth, and porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

2. Step 2 — Rearrange symbolically for h_L:

   $$h_{L} = \dfrac{1.067C_DLv_f^2}{\phi g d \,\epsilon^4}$$

3. Step 3 — List the givens: drag coefficient (C_D) = 2.2000, filter media depth (L) = 0.7900 m, filtration velocity (v_f) = 0.0038 m/s, shape factor (phi) = 0.8700, gravitational acceleration (g) = 9.8100 m/s^2, grain diameter (d) = 0.0019 m, porosity (eps) = 0.4040.

4. Step 4 — Substitute the given values:

   $$h_{L} = \dfrac{1.067C_DLv_f^2}{0.8700 9.8100 0.0019 \,\epsilon^4}$$

5. Step 5 — Evaluate:

   $$h_{L} = 0.0620\ \text{m}$$

6. Step 6 — Check: returning h_L = 0.0620 m to

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

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

Why the other options are there

• 0.1240 — kept a factor of two that cancels in the correct rearrangement.
• 0.0310 — dropped that same factor in the other direction.
• 0.0682 — rounded an intermediate value before the final step.

Reference: FE Handbook — Rose Equation (filter headloss)

Rose equation used to estimate clean-bed head loss in filtration Given drag coefficient (C_D) = 1.6500; filtration velocity (v_f) = 0.0027 m/s; shape factor (phi) = 0.7800; gravitational acceleration (g) = 9.8100 m/s^2; grain diameter (d) = 0.0009 m; porosity (eps) = 0.3990; headloss through clean bed (h_L) = 0.6650 m, determine the filter media depth (L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Rose Equation.
• 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.
• The Rose equation calculates head loss through a clean granular filter bed based on grain size, depth, and porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

2. Step 2 — Rearrange symbolically for L:

   $$L = \dfrac{h_L \phi g d \,\epsilon^4}{1.067C_Dv_f^2}$$

3. Step 3 — List the givens: drag coefficient (C_D) = 1.6500, filtration velocity (v_f) = 0.0027 m/s, shape factor (phi) = 0.7800, gravitational acceleration (g) = 9.8100 m/s^2, grain diameter (d) = 0.0009 m, porosity (eps) = 0.3990, headloss through clean bed (h_L) = 0.6650 m.

4. Step 4 — Substitute the given values:

   $$L = \dfrac{h_L 0.7800 9.8100 0.0009 \,\epsilon^4}{1.067C_Dv_f^2}$$

5. Step 5 — Evaluate:

   $$L = 8.7422\ \text{m}$$

6. Step 6 — Check: returning L = 8.7422 m to

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

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

Why the other options are there

• 17.4844 — kept a factor of two that cancels in the correct rearrangement.
• 4.3711 — dropped that same factor in the other direction.
• 9.6164 — rounded an intermediate value before the final step.

Reference: FE Handbook — Rose Equation (filter headloss)

Rose equation filter headloss calculation for uniform media Given drag coefficient (C_D) = 1.6500; filter media depth (L) = 1.4600 m; shape factor (phi) = 0.7500; gravitational acceleration (g) = 9.8100 m/s^2; grain diameter (d) = 0.0013 m; porosity (eps) = 0.3910; headloss through clean bed (h_L) = 0.7850 m, determine the filtration velocity (v_f) in m/s.

Start with the thinking

• The governing relation printed in this handbook section is Rose Equation.
• Everything except v_f is given, so isolate v_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.
• The Rose equation calculates head loss through a clean granular filter bed based on grain size, depth, and porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

2. Step 2 — Rearrange symbolically for v_f:

   $$v_{f} = \sqrt{\dfrac{h_L \phi g d\,\epsilon^4}{1.067C_DL}}$$

3. Step 3 — List the givens: drag coefficient (C_D) = 1.6500, filter media depth (L) = 1.4600 m, shape factor (phi) = 0.7500, gravitational acceleration (g) = 9.8100 m/s^2, grain diameter (d) = 0.0013 m, porosity (eps) = 0.3910, headloss through clean bed (h_L) = 0.7850 m.

4. Step 4 — Substitute the given values:

   $$v_{f} = \sqrt{\dfrac{h_1.4600 0.7500 9.8100 0.0013\,\epsilon^4}{1.067C_D1.4600}}$$

5. Step 5 — Evaluate:

   $$v_{f} = 0.0083\ \text{m/s}$$

6. Step 6 — Check: returning v_f = 0.0083 m/s to

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

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

Why the other options are there

• 0.0165 — kept a factor of two that cancels in the correct rearrangement.
• 0.0041 — dropped that same factor in the other direction.
• 0.0091 — rounded an intermediate value before the final step.

Reference: FE Handbook — Rose Equation (filter headloss)

Rose equation headloss through a clean rapid sand filter bed Given drag coefficient (C_D) = 0.5800; filter media depth (L) = 1.2300 m; filtration velocity (v_f) = 0.0043 m/s; shape factor (phi) = 0.9500; gravitational acceleration (g) = 9.8100 m/s^2; grain diameter (d) = 0.0005 m; porosity (eps) = 0.4080, determine the headloss through clean bed (h_L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Rose Equation.
• Everything except h_L is given, so isolate h_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.
• The Rose equation calculates head loss through a clean granular filter bed based on grain size, depth, and porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

2. Step 2 — Rearrange symbolically for h_L:

   $$h_{L} = \dfrac{1.067C_DLv_f^2}{\phi g d \,\epsilon^4}$$

3. Step 3 — List the givens: drag coefficient (C_D) = 0.5800, filter media depth (L) = 1.2300 m, filtration velocity (v_f) = 0.0043 m/s, shape factor (phi) = 0.9500, gravitational acceleration (g) = 9.8100 m/s^2, grain diameter (d) = 0.0005 m, porosity (eps) = 0.4080.

4. Step 4 — Substitute the given values:

   $$h_{L} = \dfrac{1.067C_DLv_f^2}{0.9500 9.8100 0.0005 \,\epsilon^4}$$

5. Step 5 — Evaluate:

   $$h_{L} = 0.1090\ \text{m}$$

6. Step 6 — Check: returning h_L = 0.1090 m to

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

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

Why the other options are there

• 0.2180 — kept a factor of two that cancels in the correct rearrangement.
• 0.0545 — dropped that same factor in the other direction.
• 0.1199 — rounded an intermediate value before the final step.

Reference: FE Handbook — Rose Equation (filter headloss)

Rose equation used to estimate clean-bed head loss in filtration Given drag coefficient (C_D) = 2.8600; filtration velocity (v_f) = 0.0040 m/s; shape factor (phi) = 0.8600; gravitational acceleration (g) = 9.8100 m/s^2; grain diameter (d) = 0.0020 m; porosity (eps) = 0.4470; headloss through clean bed (h_L) = 0.8400 m, determine the filter media depth (L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Rose Equation.
• 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.
• The Rose equation calculates head loss through a clean granular filter bed based on grain size, depth, and porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

2. Step 2 — Rearrange symbolically for L:

   $$L = \dfrac{h_L \phi g d \,\epsilon^4}{1.067C_Dv_f^2}$$

3. Step 3 — List the givens: drag coefficient (C_D) = 2.8600, filtration velocity (v_f) = 0.0040 m/s, shape factor (phi) = 0.8600, gravitational acceleration (g) = 9.8100 m/s^2, grain diameter (d) = 0.0020 m, porosity (eps) = 0.4470, headloss through clean bed (h_L) = 0.8400 m.

4. Step 4 — Substitute the given values:

   $$L = \dfrac{h_L 0.8600 9.8100 0.0020 \,\epsilon^4}{1.067C_Dv_f^2}$$

5. Step 5 — Evaluate:

   $$L = 11.3575\ \text{m}$$

6. Step 6 — Check: returning L = 11.3575 m to

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

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

Why the other options are there

• 22.7150 — kept a factor of two that cancels in the correct rearrangement.
• 5.6787 — dropped that same factor in the other direction.
• 12.4932 — rounded an intermediate value before the final step.

Reference: FE Handbook — Rose Equation (filter headloss)

Rose equation filter headloss calculation for uniform media Given drag coefficient (C_D) = 8.9700; filter media depth (L) = 0.9500 m; shape factor (phi) = 0.8200; gravitational acceleration (g) = 9.8100 m/s^2; grain diameter (d) = 0.0018 m; porosity (eps) = 0.4300; headloss through clean bed (h_L) = 1.6550 m, determine the filtration velocity (v_f) in m/s.

Start with the thinking

• The governing relation printed in this handbook section is Rose Equation.
• Everything except v_f is given, so isolate v_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.
• The Rose equation calculates head loss through a clean granular filter bed based on grain size, depth, and porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

2. Step 2 — Rearrange symbolically for v_f:

   $$v_{f} = \sqrt{\dfrac{h_L \phi g d\,\epsilon^4}{1.067C_DL}}$$

3. Step 3 — List the givens: drag coefficient (C_D) = 8.9700, filter media depth (L) = 0.9500 m, shape factor (phi) = 0.8200, gravitational acceleration (g) = 9.8100 m/s^2, grain diameter (d) = 0.0018 m, porosity (eps) = 0.4300, headloss through clean bed (h_L) = 1.6550 m.

4. Step 4 — Substitute the given values:

   $$v_{f} = \sqrt{\dfrac{h_0.9500 0.8200 9.8100 0.0018\,\epsilon^4}{1.067C_D0.9500}}$$

5. Step 5 — Evaluate:

   $$v_{f} = 0.0095\ \text{m/s}$$

6. Step 6 — Check: returning v_f = 0.0095 m/s to

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

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

Why the other options are there

• 0.0189 — kept a factor of two that cancels in the correct rearrangement.
• 0.0047 — dropped that same factor in the other direction.
• 0.0104 — rounded an intermediate value before the final step.

Reference: FE Handbook — Rose Equation (filter headloss)

Rose equation headloss through a clean rapid sand filter bed Given drag coefficient (C_D) = 2.1400; filter media depth (L) = 0.7800 m; filtration velocity (v_f) = 0.0045 m/s; shape factor (phi) = 0.8700; gravitational acceleration (g) = 9.8100 m/s^2; grain diameter (d) = 0.0004 m; porosity (eps) = 0.3700, determine the headloss through clean bed (h_L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Rose Equation.
• Everything except h_L is given, so isolate h_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.
• The Rose equation calculates head loss through a clean granular filter bed based on grain size, depth, and porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

2. Step 2 — Rearrange symbolically for h_L:

   $$h_{L} = \dfrac{1.067C_DLv_f^2}{\phi g d \,\epsilon^4}$$

3. Step 3 — List the givens: drag coefficient (C_D) = 2.1400, filter media depth (L) = 0.7800 m, filtration velocity (v_f) = 0.0045 m/s, shape factor (phi) = 0.8700, gravitational acceleration (g) = 9.8100 m/s^2, grain diameter (d) = 0.0004 m, porosity (eps) = 0.3700.

4. Step 4 — Substitute the given values:

   $$h_{L} = \dfrac{1.067C_DLv_f^2}{0.8700 9.8100 0.0004 \,\epsilon^4}$$

5. Step 5 — Evaluate:

   $$h_{L} = 0.5369\ \text{m}$$

6. Step 6 — Check: returning h_L = 0.5369 m to

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

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

Why the other options are there

• 1.0737 — kept a factor of two that cancels in the correct rearrangement.
• 0.2684 — dropped that same factor in the other direction.
• 0.5905 — rounded an intermediate value before the final step.

Reference: FE Handbook — Rose Equation (filter headloss)

Rose equation used to estimate clean-bed head loss in filtration Given drag coefficient (C_D) = 6.9800; filtration velocity (v_f) = 0.0040 m/s; shape factor (phi) = 0.8500; gravitational acceleration (g) = 9.8100 m/s^2; grain diameter (d) = 0.0004 m; porosity (eps) = 0.4540; headloss through clean bed (h_L) = 1.2580 m, determine the filter media depth (L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Rose Equation.
• 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.
• The Rose equation calculates head loss through a clean granular filter bed based on grain size, depth, and porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

2. Step 2 — Rearrange symbolically for L:

   $$L = \dfrac{h_L \phi g d \,\epsilon^4}{1.067C_Dv_f^2}$$

3. Step 3 — List the givens: drag coefficient (C_D) = 6.9800, filtration velocity (v_f) = 0.0040 m/s, shape factor (phi) = 0.8500, gravitational acceleration (g) = 9.8100 m/s^2, grain diameter (d) = 0.0004 m, porosity (eps) = 0.4540, headloss through clean bed (h_L) = 1.2580 m.

4. Step 4 — Substitute the given values:

   $$L = \dfrac{h_L 0.8500 9.8100 0.0004 \,\epsilon^4}{1.067C_Dv_f^2}$$

5. Step 5 — Evaluate:

   $$L = 1.4959\ \text{m}$$

6. Step 6 — Check: returning L = 1.4959 m to

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

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

Why the other options are there

• 2.9919 — kept a factor of two that cancels in the correct rearrangement.
• 0.7480 — dropped that same factor in the other direction.
• 1.6455 — rounded an intermediate value before the final step.

Reference: FE Handbook — Rose Equation (filter headloss)

Rose equation filter headloss calculation for uniform media Given drag coefficient (C_D) = 9.5600; filter media depth (L) = 0.7300 m; shape factor (phi) = 0.7400; gravitational acceleration (g) = 9.8100 m/s^2; grain diameter (d) = 0.0013 m; porosity (eps) = 0.4860; headloss through clean bed (h_L) = 0.3790 m, determine the filtration velocity (v_f) in m/s.

Start with the thinking

• The governing relation printed in this handbook section is Rose Equation.
• Everything except v_f is given, so isolate v_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.
• The Rose equation calculates head loss through a clean granular filter bed based on grain size, depth, and porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

2. Step 2 — Rearrange symbolically for v_f:

   $$v_{f} = \sqrt{\dfrac{h_L \phi g d\,\epsilon^4}{1.067C_DL}}$$

3. Step 3 — List the givens: drag coefficient (C_D) = 9.5600, filter media depth (L) = 0.7300 m, shape factor (phi) = 0.7400, gravitational acceleration (g) = 9.8100 m/s^2, grain diameter (d) = 0.0013 m, porosity (eps) = 0.4860, headloss through clean bed (h_L) = 0.3790 m.

4. Step 4 — Substitute the given values:

   $$v_{f} = \sqrt{\dfrac{h_0.7300 0.7400 9.8100 0.0013\,\epsilon^4}{1.067C_D0.7300}}$$

5. Step 5 — Evaluate:

   $$v_{f} = 0.0051\ \text{m/s}$$

6. Step 6 — Check: returning v_f = 0.0051 m/s to

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

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

Why the other options are there

• 0.0102 — kept a factor of two that cancels in the correct rearrangement.
• 0.0025 — dropped that same factor in the other direction.
• 0.0056 — rounded an intermediate value before the final step.

Reference: FE Handbook — Rose Equation (filter headloss)

Rose equation headloss through a clean rapid sand filter bed Given drag coefficient (C_D) = 8.2500; filter media depth (L) = 0.8000 m; filtration velocity (v_f) = 0.0028 m/s; shape factor (phi) = 0.9400; gravitational acceleration (g) = 9.8100 m/s^2; grain diameter (d) = 0.0011 m; porosity (eps) = 0.4810, determine the headloss through clean bed (h_L) in m.

Start with the thinking

• The governing relation printed in this handbook section is Rose Equation.
• Everything except h_L is given, so isolate h_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.
• The Rose equation calculates head loss through a clean granular filter bed based on grain size, depth, and porosity.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

2. Step 2 — Rearrange symbolically for h_L:

   $$h_{L} = \dfrac{1.067C_DLv_f^2}{\phi g d \,\epsilon^4}$$

3. Step 3 — List the givens: drag coefficient (C_D) = 8.2500, filter media depth (L) = 0.8000 m, filtration velocity (v_f) = 0.0028 m/s, shape factor (phi) = 0.9400, gravitational acceleration (g) = 9.8100 m/s^2, grain diameter (d) = 0.0011 m, porosity (eps) = 0.4810.

4. Step 4 — Substitute the given values:

   $$h_{L} = \dfrac{1.067C_DLv_f^2}{0.9400 9.8100 0.0011 \,\epsilon^4}$$

5. Step 5 — Evaluate:

   $$h_{L} = 0.0981\ \text{m}$$

6. Step 6 — Check: returning h_L = 0.0981 m to

   $$h_L = \dfrac{1.067 C_D L v_f^2}{\phi g d} \left(\dfrac{1}{\epsilon^4}\right)$$

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

Why the other options are there

• 0.1962 — kept a factor of two that cancels in the correct rearrangement.
• 0.0491 — dropped that same factor in the other direction.
• 0.1079 — rounded an intermediate value before the final step.

Reference: FE Handbook — Rose Equation (filter headloss)