Hydraulic Power (P_hyd)  

Hydraulic Power (also called water horsepower or liquid horsepower) is the theoretical power required to move a fluid from one point to another. It represents the energy added to the liquid to overcome resistance due to elevation (head), friction, and velocity.

                                                                                 P_hyd = (ρ × g × Q × H) / 1000

Where:
ρ = Fluid density (kg/m³)
g = Acceleration due to gravity (9.81 m/s²)
Q = Flow rate (m³/s)
H = Total developed head (m)

Division by 1000 converts watts to kilowatts

For water at standard conditions, ρ ≈ 1000 kg/m³, the formula simplifies to:

                                                                                P_hyd = 9.81 × Q × H (in kW)

Shaft Power (P_sh):

Shaft Power is the actual power delivered to the pump shaft by the motor. Since all pumps have some efficiency loss, the shaft power is greater than the hydraulic power.

                                                                                P_sh = P_hyd / η

Where:
η = Pump efficiency (decimal)
Example: If the pump efficiency is 70%, then η = 0.7

Pump Efficiency ($\eta$)

Pump efficiency is the ratio of hydraulic power output to shaft power input. It accounts for losses due to friction, leakage, and other mechanical inefficiencies. Typical centrifugal pumps have efficiencies ranging from 50% to 90%, depending on design and operating conditions.

Specific Gravity (SG)

Specific gravity is the ratio of the fluid density to the density of water at 4°C (1000 kg/m³). It affects power calculations because heavier fluids require more power to pump. For fluids other than water, multiply the hydraulic power by the specific gravity to adjust for density differences.

• For fluids heavier than water, more power is needed.
• For lighter fluids, less power is needed.

Modified Hydraulic Power Formula with Specific Gravity:

                                                                                P_hyd = 9.81 × Q × H × SG (in kW)

Where:
SG = Specific Gravity = ρ_fluid / ρ_water
For water, SG = 1.0
For other fluids like sulfuric acid (SG ≈ 1.84), use the actual value accordingly

How is Pump Power Calculated – Sample Calculation Using Water

Let’s calculate the hydraulic and shaft power for a pump handling water with the following parameters:

• Fluid: Water (specific gravity, SG = 1)
• Flow rate, Q = 400 m³/h
• Density of water, ρ = 1000 kg/m³
• Pump head, H = 30 m
• Pump efficiency, η = 70% or 0.7
• Motor efficiency, η_motor = 90% or 0.9
• Operating time: 24 hours/day

Step 1: Calculate Hydraulic Power

Hydraulic power is the power imparted to the fluid, calculated by:

                                                                            P_hyd = (Q × ρ × g × H) / (3.6 × 10⁶)

Where:
Q = 400 m³/h
ρ = 1000 kg/m³
g = 9.81 m/s²
H = 30 m
3.6 × 10⁶ is used to convert the result to kilowatts (kW)

                                                          Calculate numerator: 400 × 1000 × 9.81 × 30 = 117,720,000

                                                          Now calculate hydraulic power: P_hyd = 117,720,000 / 3,600,000 = 32.7 kW

Step 2: Calculate Shaft Power

Shaft power accounts for pump efficiency:

                                                          P_sh = P_hyd / η = 32.7 / 0.7 = 46.71 kW

Step 3: Calculate Electrical Power Input to Motor

Accounting for motor efficiency:

                                                          P_electric = P_sh / η_motor = 46.71 / 0.9 = 51.9 kW

Step 4: Calculate Daily Energy Consumption

Operating 24 hours a day:

                                                          E_daily = P_electric × t = 51.9 × 24 = 1245.6 kWh/day

Step 5: Calculate Operating Cost (Optional)

Assuming electricity cost of $0.12 per kWh:

                                                         Cost_daily = E_daily × cost per kWh = 1245.6 × 0.12 = 149.47 USD/day

Summary