A pump delivers 0.06 m³/s against a total head of 32 m at 74% efficiency. What brake power is required?

Start with the thinking

• Water power first, then divide by efficiency.
• Dividing by efficiency increases the power — a common sign check.

Step-by-step solution

1. Water power — P_w = γQH = 9.81(0.06)(32)

2. Evaluate

   $$P_w = 18.8 kW$$

3. Brake power

   $$P = P_w/\eta = 18.8/0.74$$

4. Result

   $$P = 25.5 kW$$

Why the other options are there

• 13.9 kW (multiplied by efficiency)
• 18.8 kW (efficiency ignored)

Reference: FE Reference Handbook — Fluid Mechanics — Pump power

a positive-displacement blower feeding an aeration basin Given volumetric flow (Q) = 11.3000 m^3/s; pressure rise (\Delta p) = 59,900 Pa; efficiency (\eta) = 0.7500, determine the shaft power (W) in W.

Start with the thinking

• The governing relation printed in this handbook section is Blower / compressor fluid power.
• Everything except W is given, so isolate W 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 blower raises the pressure of an air stream across an aeration basin diffuser system.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$W = \dfrac{Q \Delta p}{\eta}$$

2. Step 2 — Rearrange symbolically for W:

   $$W = \dfrac{Q \Delta p}{\eta}$$

3. Step 3 — List the givens: volumetric flow (Q) = 11.3000 m^3/s, pressure rise (\Delta p) = 59,900 Pa, efficiency (\eta) = 0.7500.

4. Step 4 — Substitute the given values:

   $$W = \dfrac{11.3000 59900}{0.7500}$$

5. Step 5 — Evaluate:

   $$W = 902493\ \text{W}$$

6. Step 6 — Check: returning W = 902,493 W to

   $$W = \dfrac{Q \Delta p}{\eta}$$

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

Why the other options are there

• 1,804,987 — kept a factor of two that cancels in the correct rearrangement.
• 451,247 — dropped that same factor in the other direction.
• 992,743 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fans, Blowers, and Compressors

a centrifugal blower on a dust-collection duct Given shaft power (W) = 17,382 W; pressure rise (\Delta p) = 20,600 Pa; efficiency (\eta) = 0.8200, determine the volumetric flow (Q) in m^3/s.

Start with the thinking

• The governing relation printed in this handbook section is Blower / compressor fluid power.
• Everything except Q is given, so isolate Q 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 blower raises the pressure of an air stream across an aeration basin diffuser system.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$W = \dfrac{Q \Delta p}{\eta}$$

2. Step 2 — Rearrange symbolically for Q:

   $$Q = \dfrac{W \eta}{\Delta p}$$

3. Step 3 — List the givens: shaft power (W) = 17,382 W, pressure rise (\Delta p) = 20,600 Pa, efficiency (\eta) = 0.8200.

4. Step 4 — Substitute the given values:

   $$Q = \dfrac{17382 0.8200}{20600}$$

5. Step 5 — Evaluate:

   Q = 0.6919\ \text{m^3/s}

6. Step 6 — Check: returning Q = 0.6919 m^3/s to

   $$W = \dfrac{Q \Delta p}{\eta}$$

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

Why the other options are there

• 1.3838 — kept a factor of two that cancels in the correct rearrangement.
• 0.3460 — dropped that same factor in the other direction.
• 0.7611 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fans, Blowers, and Compressors

a compressor supplying pneumatic construction tools Given shaft power (W) = 63,819 W; volumetric flow (Q) = 3.2000 m^3/s; pressure rise (\Delta p) = 53,900 Pa, determine the efficiency (\eta).

Start with the thinking

• The governing relation printed in this handbook section is Blower / compressor fluid power.
• Everything except \eta is given, so isolate \eta 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 blower raises the pressure of an air stream across an aeration basin diffuser system.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$W = \dfrac{Q \Delta p}{\eta}$$

2. Step 2 — Rearrange symbolically for \eta:

   $$\eta = \dfrac{Q \Delta p}{W}$$

3. Step 3 — List the givens: shaft power (W) = 63,819 W, volumetric flow (Q) = 3.2000 m^3/s, pressure rise (\Delta p) = 53,900 Pa.

4. Step 4 — Substitute the given values:

   $$\eta = \dfrac{3.2000 53900}{63819}$$

5. Step 5 — Evaluate:

   $$\eta = 2.7026$$

6. Step 6 — Check: returning \eta = 2.7026 to

   $$W = \dfrac{Q \Delta p}{\eta}$$

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

Why the other options are there

• 5.4053 — kept a factor of two that cancels in the correct rearrangement.
• 1.3513 — dropped that same factor in the other direction.
• 2.9729 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fans, Blowers, and Compressors

a positive-displacement blower feeding an aeration basin Given volumetric flow (Q) = 10.2000 m^3/s; pressure rise (\Delta p) = 52,700 Pa; efficiency (\eta) = 0.5500, determine the shaft power (W) in W.

Start with the thinking

• The governing relation printed in this handbook section is Blower / compressor fluid power.
• Everything except W is given, so isolate W 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 blower raises the pressure of an air stream across an aeration basin diffuser system.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$W = \dfrac{Q \Delta p}{\eta}$$

2. Step 2 — Rearrange symbolically for W:

   $$W = \dfrac{Q \Delta p}{\eta}$$

3. Step 3 — List the givens: volumetric flow (Q) = 10.2000 m^3/s, pressure rise (\Delta p) = 52,700 Pa, efficiency (\eta) = 0.5500.

4. Step 4 — Substitute the given values:

   $$W = \dfrac{10.2000 52700}{0.5500}$$

5. Step 5 — Evaluate:

   $$W = 977345\ \text{W}$$

6. Step 6 — Check: returning W = 977,345 W to

   $$W = \dfrac{Q \Delta p}{\eta}$$

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

Why the other options are there

• 1,954,691 — kept a factor of two that cancels in the correct rearrangement.
• 488,673 — dropped that same factor in the other direction.
• 1,075,080 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fans, Blowers, and Compressors

a centrifugal blower on a dust-collection duct Given shaft power (W) = 85,678 W; pressure rise (\Delta p) = 59,700 Pa; efficiency (\eta) = 0.8400, determine the volumetric flow (Q) in m^3/s.

Start with the thinking

• The governing relation printed in this handbook section is Blower / compressor fluid power.
• Everything except Q is given, so isolate Q 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 blower raises the pressure of an air stream across an aeration basin diffuser system.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$W = \dfrac{Q \Delta p}{\eta}$$

2. Step 2 — Rearrange symbolically for Q:

   $$Q = \dfrac{W \eta}{\Delta p}$$

3. Step 3 — List the givens: shaft power (W) = 85,678 W, pressure rise (\Delta p) = 59,700 Pa, efficiency (\eta) = 0.8400.

4. Step 4 — Substitute the given values:

   $$Q = \dfrac{85678 0.8400}{59700}$$

5. Step 5 — Evaluate:

   Q = 1.2055\ \text{m^3/s}

6. Step 6 — Check: returning Q = 1.2055 m^3/s to

   $$W = \dfrac{Q \Delta p}{\eta}$$

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

Why the other options are there

• 2.4110 — kept a factor of two that cancels in the correct rearrangement.
• 0.6028 — dropped that same factor in the other direction.
• 1.3261 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fans, Blowers, and Compressors

a compressor supplying pneumatic construction tools Given shaft power (W) = 102,230 W; volumetric flow (Q) = 10.8000 m^3/s; pressure rise (\Delta p) = 41,200 Pa, determine the efficiency (\eta).

Start with the thinking

• The governing relation printed in this handbook section is Blower / compressor fluid power.
• Everything except \eta is given, so isolate \eta 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 blower raises the pressure of an air stream across an aeration basin diffuser system.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$W = \dfrac{Q \Delta p}{\eta}$$

2. Step 2 — Rearrange symbolically for \eta:

   $$\eta = \dfrac{Q \Delta p}{W}$$

3. Step 3 — List the givens: shaft power (W) = 102,230 W, volumetric flow (Q) = 10.8000 m^3/s, pressure rise (\Delta p) = 41,200 Pa.

4. Step 4 — Substitute the given values:

   $$\eta = \dfrac{10.8000 41200}{102230}$$

5. Step 5 — Evaluate:

   $$\eta = 4.3525$$

6. Step 6 — Check: returning \eta = 4.3525 to

   $$W = \dfrac{Q \Delta p}{\eta}$$

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

Why the other options are there

• 8.7051 — kept a factor of two that cancels in the correct rearrangement.
• 2.1763 — dropped that same factor in the other direction.
• 4.7878 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fans, Blowers, and Compressors

a positive-displacement blower feeding an aeration basin Given volumetric flow (Q) = 5.4000 m^3/s; pressure rise (\Delta p) = 56,600 Pa; efficiency (\eta) = 0.6800, determine the shaft power (W) in W.

Start with the thinking

• The governing relation printed in this handbook section is Blower / compressor fluid power.
• Everything except W is given, so isolate W 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 blower raises the pressure of an air stream across an aeration basin diffuser system.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$W = \dfrac{Q \Delta p}{\eta}$$

2. Step 2 — Rearrange symbolically for W:

   $$W = \dfrac{Q \Delta p}{\eta}$$

3. Step 3 — List the givens: volumetric flow (Q) = 5.4000 m^3/s, pressure rise (\Delta p) = 56,600 Pa, efficiency (\eta) = 0.6800.

4. Step 4 — Substitute the given values:

   $$W = \dfrac{5.4000 56600}{0.6800}$$

5. Step 5 — Evaluate:

   $$W = 449471\ \text{W}$$

6. Step 6 — Check: returning W = 449,471 W to

   $$W = \dfrac{Q \Delta p}{\eta}$$

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

Why the other options are there

• 898,941 — kept a factor of two that cancels in the correct rearrangement.
• 224,735 — dropped that same factor in the other direction.
• 494,418 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fans, Blowers, and Compressors

a centrifugal blower on a dust-collection duct Given shaft power (W) = 64,232 W; pressure rise (\Delta p) = 20,500 Pa; efficiency (\eta) = 0.7000, determine the volumetric flow (Q) in m^3/s.

Start with the thinking

• The governing relation printed in this handbook section is Blower / compressor fluid power.
• Everything except Q is given, so isolate Q 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 blower raises the pressure of an air stream across an aeration basin diffuser system.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$W = \dfrac{Q \Delta p}{\eta}$$

2. Step 2 — Rearrange symbolically for Q:

   $$Q = \dfrac{W \eta}{\Delta p}$$

3. Step 3 — List the givens: shaft power (W) = 64,232 W, pressure rise (\Delta p) = 20,500 Pa, efficiency (\eta) = 0.7000.

4. Step 4 — Substitute the given values:

   $$Q = \dfrac{64232 0.7000}{20500}$$

5. Step 5 — Evaluate:

   Q = 2.1933\ \text{m^3/s}

6. Step 6 — Check: returning Q = 2.1933 m^3/s to

   $$W = \dfrac{Q \Delta p}{\eta}$$

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

Why the other options are there

• 4.3866 — kept a factor of two that cancels in the correct rearrangement.
• 1.0966 — dropped that same factor in the other direction.
• 2.4126 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fans, Blowers, and Compressors

a compressor supplying pneumatic construction tools Given shaft power (W) = 10,187 W; volumetric flow (Q) = 3.2000 m^3/s; pressure rise (\Delta p) = 34,800 Pa, determine the efficiency (\eta).

Start with the thinking

• The governing relation printed in this handbook section is Blower / compressor fluid power.
• Everything except \eta is given, so isolate \eta 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 blower raises the pressure of an air stream across an aeration basin diffuser system.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$W = \dfrac{Q \Delta p}{\eta}$$

2. Step 2 — Rearrange symbolically for \eta:

   $$\eta = \dfrac{Q \Delta p}{W}$$

3. Step 3 — List the givens: shaft power (W) = 10,187 W, volumetric flow (Q) = 3.2000 m^3/s, pressure rise (\Delta p) = 34,800 Pa.

4. Step 4 — Substitute the given values:

   $$\eta = \dfrac{3.2000 34800}{10187}$$

5. Step 5 — Evaluate:

   $$\eta = 10.9316$$

6. Step 6 — Check: returning \eta = 10.9316 to

   $$W = \dfrac{Q \Delta p}{\eta}$$

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

Why the other options are there

• 21.8632 — kept a factor of two that cancels in the correct rearrangement.
• 5.4658 — dropped that same factor in the other direction.
• 12.0247 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fans, Blowers, and Compressors