Hydraulic Machinery

Learning Objectives

  • Distinguish positive-displacement and rotodynamic machines and select suitable pump or turbine types.
  • Calculate pump head, water power, shaft power, motor input, turbine output, and efficiencies with consistent units.
  • Construct pump and system curves and determine the actual operating point.
  • Evaluate best-efficiency-point operation, series and parallel pump combinations, and variable-speed affinity laws.
  • Calculate NPSH available, compare it with NPSH required, and recognize cavitation, air-entry, priming, and suction limitations.
  • Apply angular momentum and Euler's turbomachinery equation conceptually to pumps and turbines.
  • Use specific speed and unit quantities correctly while recognizing unit-system dependence.
  • Check machine selection over the complete operating envelope rather than at a single design point.

Hydraulic machines exchange energy between a fluid and a rotating or reciprocating mechanical system. Pumps add head to water; turbines extract head as shaft power. Real performance is governed by machine curves, the connected system, efficiency, cavitation margin, speed, controls, and off-design operation.

Pump

A machine that transfers mechanical shaft energy to a fluid, increasing its pressure, velocity, elevation, or a combination of these quantities.

Turbine

A machine that extracts hydraulic energy from flowing water and converts it to rotating shaft power.

Major Machine Classes

  • Positive-displacement pumps: Trap and displace a nearly fixed volume per cycle. Examples include piston, diaphragm, gear, screw, and progressive-cavity pumps. They can generate high pressure and require relief protection against blocked discharge.
  • Rotodynamic pumps: Continuously transfer angular momentum through an impeller. Centrifugal, mixed-flow, and axial-flow pumps are the principal types.
  • Impulse turbines: Convert pressure head to jet velocity before the runner; the runner operates near atmospheric pressure. Pelton is the common high-head example.
  • Reaction turbines: Pressure changes through the runner and casing. Francis and Kaplan machines are common for medium- and low-head installations.

Pump Head from the Energy Equation

The head added by a pump between suction section 1 and discharge section 2 is

Pump Total Head

Mechanical head increase across a pump using section-average quantities.

Hp=(p2γ+α2V222g+z2)−(p1γ+α1V122g+z1)H_p = \left( \frac{p_2}{\gamma} +\alpha_2\frac{V_2^2}{2g} +z_2 \right) - \left( \frac{p_1}{\gamma} +\alpha_1\frac{V_1^2}{2g} +z_1 \right)

Variables

SymbolDescriptionUnit
HpH_pHead transferred from pump to fluidm
α1,α2\alpha_1,\alpha_2Kinetic-energy correction coefficients-

Static Head and Total Dynamic Head Are Not the Same

Static head comes from elevation and pressure boundary differences at zero flow. Total dynamic head at the operating flow also includes velocity-head changes and all major and minor losses. Motor sizing from static head alone can be seriously inadequate.

Water Power

Useful hydraulic power delivered to the fluid.

Pw=ρgQHp=γQHpP_w=\rho gQH_p=\gamma QH_p

If γ\gamma is entered in kN/m³, QQ in m³/s, and HpH_p in m, the result is directly in kW.

Variables

SymbolDescriptionUnit
PwP_wWater or hydraulic powerW or kW
QQDischargem3/sm^3/s

Pump Shaft and Motor Power

Accounts for pump and motor efficiencies.

Pshaft=PwηpP_{shaft}=\frac{P_w}{\eta_p}Pelectric=PshaftηmP_{electric}=\frac{P_{shaft}}{\eta_m}

Variables

SymbolDescriptionUnit
ηp\eta_pPump efficiency-
ηm\eta_mMotor efficiency-

Efficiency Components

Pump overall efficiency combines hydraulic losses in the impeller and casing, volumetric leakage, disk friction, bearing and seal loss, and mechanical transmission loss. Efficiency depends on discharge, head, speed, impeller diameter, viscosity, wear, and clearances.

Pump Characteristic Curve

A manufacturer-tested relation between head and discharge at a specified speed, impeller diameter, fluid, and test condition. Performance data normally include head, efficiency, shaft power, and NPSH required.

System Curve

The head required by the connected system as discharge changes. A common steady form is

Hsys=Hstatic+KQ2H_{sys}=H_{static}+KQ^2

for losses approximately proportional to velocity squared.

Operating Point

A pump does not independently choose its discharge. The steady operating point occurs where

Hpump(Q)=Hsys(Q)H_{pump}(Q)=H_{sys}(Q)

Changing a valve, pipe roughness, tank level, parallel pump state, speed, or demand moves the intersection.

Interactive Pump–System Curves

The simulation calculates the intersection of an illustrative pump curve and system curve, then reports flow, head, efficiency, water power, and brake power.

Best Efficiency Point, BEP

The flow and head at which a pump operates at maximum hydraulic efficiency for a given speed and impeller. Near BEP, internal incidence, recirculation, radial thrust, vibration, and hydraulic loss are generally minimized.

Allowable Operating Range

Continuous operation far below or above BEP can cause suction or discharge recirculation, vibration, heat rise, seal and bearing damage, unstable head, cavitation, excessive power, or motor overload. Selection should use the manufacturer's preferred and allowable operating ranges, not only a single efficiency value.

Pump Power Curve and Motor Sizing

The required shaft power can rise or fall with discharge depending on pump type and impeller geometry. The motor must cover the maximum expected absorbed power over the allowed operating range, including density, viscosity, speed, and impeller tolerances, while accounting for service factor and starting method.

Pumps in Series

For pumps carrying the same discharge in series, heads add:

Hseries(Q)=∑iHi(Q)H_{series}(Q)=\sum_i H_i(Q)

Series operation is used when additional head is required. Check intermediate casing pressure, minimum-flow limits, controls, and whether both pumps can operate stably over the combined curve.

Pumps in Parallel

Parallel pumps operate between common suction and discharge headers, so their individual discharges add at a common head:

Qparallel(H)=∑iQi(H)Q_{parallel}(H)=\sum_i Q_i(H)

The total flow increase is less than the arithmetic sum of separate free-delivery flows because the system head rises as total flow increases.

Parallel Pumps Must Share Stably

Mismatched curves, unequal suction conditions, check-valve behavior, or operation near a flat/unstable curve can cause one pump to carry most flow or experience reverse flow. Parallel operation requires compatible curves and coordinated controls.

Pump Affinity Laws

Approximate scaling for geometrically similar operation with the same fluid.

For constant impeller diameter,

Q2Q1=N2N1\frac{Q_2}{Q_1}=\frac{N_2}{N_1}H2H1=(N2N1)2\frac{H_2}{H_1}=\left(\frac{N_2}{N_1}\right)^2P2P1=(N2N1)3\frac{P_2}{P_1}=\left(\frac{N_2}{N_1}\right)^3

Variables

SymbolDescriptionUnit
NNRotational speedrpm or rad/s

Impeller-Diameter Affinity Relations

Approximate trimming relations at constant speed for modest diameter changes.

Q2Q1≈D2D1,H2H1≈(D2D1)2,P2P1≈(D2D1)3\frac{Q_2}{Q_1}\approx\frac{D_2}{D_1}, \qquad \frac{H_2}{H_1}\approx\left(\frac{D_2}{D_1}\right)^2, \qquad \frac{P_2}{P_1}\approx\left(\frac{D_2}{D_1}\right)^3

Affinity Laws Are Similarity Approximations

Efficiency, Reynolds number, clearances, impeller exit geometry, motor cooling, NPSH required, and system controls do not remain perfectly similar over large speed changes or aggressive trimming. Use manufacturer-corrected curves for final selection.

Variable-Speed Control

Reducing speed can lower both head and friction demand efficiently in variable-flow systems. However, static head weakens cubic energy savings, and minimum speed may be limited by motor cooling, pump stability, check valves, tank levels, and service pressure requirements.

Vapor Pressure

The absolute pressure at which a liquid can coexist with its vapor at a specified temperature. Vapor pressure increases with temperature.

NPSH Available, NPSHANPSH_A

The absolute stagnation head at the pump suction above vapor-pressure head, evaluated at the pump suction reference:

NPSH Available at Suction Flange

Calculates suction pressure margin above vapor pressure.

NPSHA=ps,absγ+αsVs22g−pvγNPSH_A = \frac{p_{s,abs}}{\gamma} +\alpha_s\frac{V_s^2}{2g} -\frac{p_v}{\gamma}

Variables

SymbolDescriptionUnit
ps,absp_{s,abs}Absolute pressure at pump suction referencePa
pvp_vLiquid vapor pressure at operating temperaturePa

NPSH Available from an Open Suction Reservoir

Common reservoir-to-pump expression with reservoir velocity neglected.

NPSHA=patmγ+zsurface−zpump−hL,suction−pvγNPSH_A = \frac{p_{atm}}{\gamma} +z_{surface}-z_{pump} -h_{L,suction} -\frac{p_v}{\gamma}

NPSH Required, NPSHRNPSH_R

A manufacturer-tested pump characteristic indicating the suction head margin associated with a stated cavitation-performance criterion, commonly a specified percentage head drop. It is not a universal no-cavitation boundary.

Provide Margin Above $NPSH_R$

Reliable design requires NPSHANPSH_A greater than NPSHRNPSH_R by a project-appropriate margin. Margin depends on pump energy level, speed, service criticality, fluid properties, inlet distortion, dissolved gas, transients, uncertainty, and the selected industry standard.

Factors That Reduce NPSHANPSH_A

  • Higher liquid temperature and vapor pressure.
  • Greater suction lift or lower source level.
  • High-elevation sites with lower atmospheric pressure.
  • Small, long, rough, or obstructed suction piping.
  • Dirty strainers, partially closed valves, and entrance losses.
  • Excessive flow or acceleration during transients.
  • Air pockets, vortices, and poor sump submergence.

Cavitation

Formation and collapse of vapor cavities in low-pressure regions. In pumps it can cause noise, vibration, pitting, loss of head and efficiency, unstable operation, and shortened bearing or seal life.

Air Entrainment Is Not the Same as Cavitation

Entrained or ingested air can create symptoms similar to cavitation but has different causes and remedies. Check sump vortices, leaking suction joints, inadequate submergence, gas release, and priming as well as NPSH.

Priming and Suction Layout

Most centrifugal pumps cannot evacuate air from a dry suction line without a priming system. Suction piping should minimize high points and air pockets, use smooth entrances and long-radius fittings, provide adequate straight approach, avoid unnecessary throttling, and maintain sufficient sump submergence.

Euler Turbomachinery Equation

Angular momentum gives the ideal specific shaft work transferred between runner and fluid:

ws=u2Vθ2−u1Vθ1w_s=u_2V_{\theta2}-u_1V_{\theta1}

and ideal Euler head is

HE=u2Vθ2−u1Vθ1gH_E=\frac{u_2V_{\theta2}-u_1V_{\theta1}}{g}

where u=rωu=r\omega is blade speed and VθV_\theta is the tangential component of absolute velocity. Sign and inlet/outlet convention determine whether the machine acts as a pump or turbine.

Turbine Power

The available hydraulic power at net head HnH_n is ρgQHn\rho gQH_n. Actual shaft output is

Pshaft=ηtρgQHnP_{shaft}=\eta_t\rho gQH_n

Generator output additionally includes generator and transmission efficiency.

Turbine Selection by Head and Flow

  • Pelton: High head, relatively low discharge; impulse jets.
  • Francis: Medium head and discharge; mixed-flow reaction runner.
  • Kaplan/propeller: Low head, high discharge; axial-flow reaction runner.
  • Crossflow and small hydro machines: Useful over selected low-to-medium head ranges and variable flows.

Final selection also considers cavitation setting, sediment, fish passage, runaway speed, part-load efficiency, governing, civil layout, and maintenance.

Specific Speed

An index relating rotational speed to flow and head at the best-efficiency condition. A common pump form is

Ns=NQH3/4N_s=\frac{N\sqrt{Q}}{H^{3/4}}

and a common turbine power form is

Ns=NPH5/4N_s=\frac{N\sqrt{P}}{H^{5/4}}

Specific-Speed Values Depend on Definition and Units

The conventional numerical specific speed is not dimensionless unless explicitly normalized. Values differ between SI and US customary definitions and between pump, turbine-flow, and turbine-power forms. Always state the formula, units, speed basis, and whether head is per stage.

Dimensionless Specific Speed

For comparison independent of unit system, dimensionless groups can be formed, such as

Ωs=ωQ(gH)3/4\Omega_s=\frac{\omega\sqrt{Q}}{(gH)^{3/4}}

for pumps. Dimensionless form is preferred in similarity analysis, while manufacturer literature often uses conventional unit-dependent indices.

Machine Selection Checks

  • Required duty range and system curves for all tank levels and valve states.
  • BEP proximity and allowable operating range.
  • Water, shaft, and electrical power over the entire curve.
  • NPSH margin, suction specific speed, sump approach, and transient minimum pressure.
  • Motor starting, variable-speed range, harmonics, and minimum cooling speed.
  • Series/parallel staging and check-valve behavior.
  • Minimum continuous flow, recirculation, bypass, and thermal limits.
  • Materials, corrosion, solids, viscosity, temperature, and maintenance access.
  • Surge, pump trip, power failure, and emergency operating scenarios.

Pump Selection Workflow

  1. Develop system curves for minimum, normal, and maximum static head and resistance conditions.
  2. Select candidate pumps whose curves intersect the required duty range.
  3. Check BEP proximity and manufacturer preferred operating region.
  4. Calculate shaft and motor power at every credible operating point.
  5. Calculate worst-case NPSHANPSH_A and compare with NPSHRNPSH_R plus required margin.
  6. Check suction layout, priming, sump submergence, and air/vortex risk.
  7. Evaluate variable speed, parallel/series staging, controls, minimum flow, and standby duty.
  8. Perform transient analysis for trip, start, valve movement, and check-valve closure.
  9. Confirm materials, maintenance, lifecycle energy, and redundancy before final procurement.
Key Takeaways
  • Pump operating flow is set by the intersection of pump and system curves, not by the pump nameplate alone.
  • Water power is ρgQH\rho gQH; pump and motor efficiencies increase required input power, while turbine efficiency reduces available output.
  • BEP and allowable operating range govern reliability, vibration, and efficiency.
  • Affinity laws are useful similarity approximations but do not replace corrected manufacturer curves.
  • NPSHANPSH_A uses absolute pressure and actual vapor pressure; it must exceed NPSHRNPSH_R by an appropriate margin.
  • Series pumps add head, parallel pumps add flow at common head, and both arrangements require stable curve matching and controls.
  • Specific speed must be accompanied by its definition and unit convention.
  • Final machinery design must include off-design operation, suction layout, controls, transient pressure, and lifecycle performance.
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