Pumps, Turbines, and Flow Measurement

Energy Equation with a Pump or Turbine

Adding machines to the energy equation:

p₁/γ + V₁²/2g + z₁ + h_p = p₂/γ + V₂²/2g + z₂ + h_t + h_L

where h_p is head added by a pump, h_t is head removed by a turbine, and h_L is total head loss. Solve for h_p when sizing a pump to move fluid from point 1 to point 2 against elevation, pressure, velocity, and friction.

Pump Power

The power delivered to the fluid (water power) and the power the motor must supply (brake power) differ by efficiency:

P_fluid = γ Q h_p P_input = γ Q h_p / η

with η ≈ 0.60–0.85 typical. Turbine output is P_output = γ Q h_t · η_t (efficiency multiplies, because the turbine gives power).

Worked example — pump input power

A pump delivers Q = 0.05 m³/s against h_p = 30 m at η = 75%:

1. Fluid power = γQh_p = 9,810 × 0.05 × 30 = 14,715 W.
2. Input power = 14,715 / 0.75 = 19,620 W ≈ 19.6 kW.

The motor must supply more than the fluid receives because of internal losses — divide by η, never multiply.

Pump and System Curves

A pump performance curve plots head vs. flow: head falls as flow rises. Efficiency peaks at the Best Efficiency Point (BEP).

The system curve is the head the piping demands at each flow:

h_system = h_static + h_L(Q)

Static head is the elevation (plus pressure) the pump must overcome regardless of flow; the loss term grows with Q² since h_f ∝ V² ∝ Q². The operating point is the intersection of the pump and system curves — that is the actual flow and head the installed pump will deliver.

Pumps in Series and Parallel

Because the operating point shifts along the system curve, two parallel pumps do not give exactly double the flow — the higher flow raises system losses, moving the intersection to a higher head and slightly less than 2× flow.

Cavitation and NPSH

Cavitation happens when the local static pressure falls below the fluid's vapor pressure (p_v), flashing vapor bubbles that then collapse violently against the impeller. Effects: pitting and erosion, noise and vibration, and lost performance. It is most likely on the suction side where pressure is lowest.

The guard against cavitation is Net Positive Suction Head (NPSH):

NPSH_A = (p_atm − p_v)/γ − z_s − h_{f,suction}

where z_s is the suction lift (positive when the pump is above the source). The design requirement is:

NPSH_available ≥ NPSH_required

To raise NPSH_A: lower the pump (reduce z_s), shorten/widen the suction line (reduce h_f), or cool the fluid (lower p_v). Pumping hot or volatile liquids, or lifting from a deep source, makes cavitation more likely.

Flow Measurement Devices

Discharge coefficients

• Venturi meter: C_d ≈ 0.95–0.99 (smooth, best accuracy, low permanent loss)
• Flow nozzle: C_d ≈ 0.94–0.99
• Orifice plate: C_d ≈ 0.60–0.65 (cheap but high permanent pressure loss)

The venturi recovers most of its pressure downstream; the orifice plate does not — the engineering trade-off is cost vs. energy loss.

Ideal Gas Law (Compressible-Flow Foundation)

Many FE fluids questions touch gas behavior:

PV = nRT or P = ρ R_specific T

with the universal gas constant R = 8.314 J/(mol·K). The specific gas constant is R_specific = R/M (M = molar mass); for air R_air = 287 J/(kg·K). Real gases add a compressibility factor z (PV = znRT); for an ideal gas z = 1.

Worked example — air density from the ideal gas law

Air at 300 K and 100 kPa: ρ = P/(R_air·T) = 100,000/(287 × 300) = 100,000/86,100 = 1.16 kg/m³. This is the density used in continuity and Reynolds-number calculations for airflow.

Hydraulic Radius (Carry-Over from Open Channels)

The hydraulic radius R_h = A/P (flow area ÷ wetted perimeter) appears in Manning's equation and non-circular duct friction.

Worked example — half-full pipe

For a half-full circular pipe of diameter D: A = πD²/8 and wetted perimeter P = πD/2, so R_h = (πD²/8)/(πD/2) = D/4. (A full circular pipe also gives R_h = D/4.)

Common Traps

• Divide by efficiency for a pump's input power; multiply by efficiency for a turbine's output.
• Parallel ≠ double flow. The system curve fixes the real operating point.
• Cavitation is a suction-side, vapor-pressure phenomenon — not an over-pressure or temperature-freezing issue.
• Orifice C_d ≈ 0.6, venturi C_d ≈ 0.98 — don't interchange them.

Quick reference

• P_input = γQh_p/η; P_turbine = γQh_t·η.
• Operating point = pump curve ∩ system curve; series adds head, parallel adds flow.
• Cavitation when p < p_v; require NPSH_A ≥ NPSH_R.
• Venturi C_d ≈ 0.98 (low loss); orifice C_d ≈ 0.6 (high loss).