Pumps, Turbines, and Flow Measurement

Energy Equation with Pump/Turbine

$\frac{P_1}{\gamma} + \frac{V_1^2}{2g} + z_1 + h_p = \frac{P_2}{\gamma} + \frac{V_2^2}{2g} + z_2 + h_t + h_L$

where:

• hp = head added by pump
• ht = head removed by turbine
• hL = total head losses

Pump Power

$P_{fluid} = \gamma Q h_p$

$P_{input} = \frac{\gamma Q h_p}{\eta}$

where:

• Pfluid = power added to the fluid (water horsepower)
• Pinput = power required from the motor (brake horsepower)
• η = pump efficiency (typically 60-85%)
• γ = specific weight
• Q = flow rate
• hp = pump head

Turbine power: $P_{output} = \gamma Q h_t \cdot \eta_t$

Pump Performance

Pump Curves

A pump performance curve shows:

• Head vs. Flow Rate — head decreases as flow increases
• Efficiency vs. Flow Rate — peaks at the Best Efficiency Point (BEP)
• Power vs. Flow Rate — generally increases with flow

System Curve

The system head requirement as a function of flow rate:

$h_{system} = h_{static} + h_L(Q)$

where static head is the elevation difference plus any pressure difference, and head loss increases with Q² (since hf ∝ V² ∝ Q²).

Operating point: Where the pump curve intersects the system curve.

Pumps in Series and Parallel

Cavitation

Cavitation occurs when the local static pressure drops below the fluid's vapor pressure, forming vapor bubbles that collapse violently.

Effects:

• Damage to pump impellers and turbine blades
• Noise and vibration
• Reduced performance
• Pitting and erosion

Net Positive Suction Head (NPSH)

$NPSH_A = \frac{P_{atm} - P_v}{\gamma} - z_s - h_{f,suction}$

where:

• Patm = atmospheric pressure
• Pv = vapor pressure of the fluid
• zs = suction lift (positive above pump)
• hf,suction = friction losses in suction line

Requirement: NPSHA ≥ NPSHR (available must exceed required)

Flow Measurement Devices

Discharge Coefficients (Cd)

• Venturi meter: Cd ≈ 0.95-0.99 (best accuracy)
• Flow nozzle: Cd ≈ 0.94-0.99
• Orifice plate: Cd ≈ 0.60-0.65 (most pressure loss)

Ideal Gas Law

$PV = nRT \quad \text{or} \quad P = \rho R_{specific} T$

where R = 8.314 J/(mol·K) is the universal gas constant.

Specific gas constant: Rspecific = R/M where M is the molar mass.

• For air: Rair = 287 J/(kg·K)

Compressibility factor (z): PV = znRT — accounts for real gas behavior. For an ideal gas, z = 1.