Hydraulic Machinery

Hydraulic Machinery

Principles of pumps and turbines, power calculations, characteristic curves, and cavitation.

Concept Overview

Hydraulic machines convert fluid energy into mechanical energy (turbines) or mechanical energy into fluid energy (pumps).

Pumps

Pumps increase the pressure head of a fluid. The most common types are centrifugal (radial flow) and axial flow pumps.

Pump Power

Hydraulic pump power is analyzed in two ways:

Water Power ( $P_w$ ): The power delivered to the fluid.
Brake Power ( $P_b$ ): The power supplied to the pump shaft by the motor.

Water Power Formula

Calculates the useful fluid power delivered to the water by the pump.

P_w = \gamma Q H

Variables

Symbol	Description	Unit
		$k N / m^{3}$
$Q$	Discharge	$m^{3} / s$
$H$	Total dynamic head (TDH)	m

Brake Power Formula

Calculates the input power required at the pump shaft from the motor, accounting for pump efficiency.

P_b = \frac{P_w}{\eta}

Variables

Symbol	Description	Unit
$P_{w}$	Water power	kW
	Pump efficiency	decimal

Total Dynamic Head (TDH)

The total head against which the pump operates, representing the net energy increase per unit weight of fluid.

Total Dynamic Head (TDH)

Calculates the total dynamic head by summing static heads, friction losses, and velocity head differences between discharge and suction.

H = h_s + h_d + h_f + \frac{V_d^2}{2g} - \frac{V_s^2}{2g}

Variables

Symbol	Description	Unit
$h_{s}$	Static suction lift	m
$h_{d}$	Static discharge head	m
$h_{f}$	Friction losses in suction and discharge pipes	m
$V_{d}$	Velocity in discharge pipe	m/s
$V_{s}$	Velocity in suction pipe	m/s
$g$	Acceleration due to gravity	$m / s^{2}$

Turbines

Turbines extract energy from fluid flow to generate mechanical power (e.g., hydroelectric dams).

Turbine Power

The output power of a turbine represents the mechanical power extracted from the fluid. The turbine efficiency ( $\eta$ ) accounts for hydraulic, volumetric, and mechanical losses.

Turbine Output Power

Calculates the mechanical power output generated by a hydraulic turbine from the available fluid power.

P = \gamma Q H \eta

Variables

Symbol	Description	Unit
$P$	Output power	W
	Specific weight of fluid	$N / m^{3}$
$Q$	Discharge	$m^{3} / s$
$H$	Net head	m
	Turbine efficiency	decimal

Specific Speed

Specific speed is a dimensionless parameter used to classify and select the most efficient type of pump or turbine for a given application.

Pump Specific Speed ( $N_{s}$ )

Pump specific speed is defined as the speed in RPM at which a geometrically similar pump would deliver 1 unit of flow rate at 1 unit of head. It depends only on the design shape of the impeller, not its physical size or actual operating speed.

Impeller selection based on $N_s$ :

Low $N_s$ : Centrifugal (Radial flow). High head, low flow.
Medium $N_s$ : Mixed flow.
High $N_s$ : Axial flow (Propeller). Low head, very high flow.

Pump Specific Speed

Calculates the specific speed of a pump to classify the impeller type based on operational parameters at the Best Efficiency Point (BEP).

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

Variables

Symbol	Description	Unit
$N$	Pump rotational speed	RPM
$Q$	Discharge at Best Efficiency Point (BEP)	$m^{3} / s$
$H$	Head per stage at BEP	m

Turbine Specific Speed ( $N_{s}$ )

For turbines, specific speed is defined based on power output ( $P$ ) rather than discharge ( $Q$ ), since generating power is the primary goal.

Turbine selection based on $N_s$ :

Low $N_s$ : Pelton Wheel (Impulse turbine). Very high head ( $> 200\text{ m}$ ), low flow rate.
Medium $N_s$ : Francis Turbine (Reaction turbine). Medium head ( $40\text{ m} - 400\text{ m}$ ), medium flow.
High $N_s$ : Kaplan Turbine (Reaction turbine, axial flow). Low head ( $< 40\text{ m}$ ), high flow.

Turbine Specific Speed

Calculates the specific speed of a turbine to classify and select the optimal turbine type based on power output and head.

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

Variables

Symbol	Description	Unit
$N$	Rotational speed	RPM
$P$	Power output at design point	kW
$H$	Net head	m

Characteristic Curves

Performance curves plot Head ( $H$ ), Power ( $P$ ), and Efficiency ( $\eta$ ) against Discharge ( $Q$ ) for a constant speed ( $N$ ).

Head-Discharge Curve (H-Q): Typically drops as $Q$ increases.
Efficiency Curve: Increases to a peak (Best Efficiency Point - BEP) and then drops.
Power Curve: Increases with $Q$ .

Affinity Laws

Affinity laws are scaling rules to predict pump or turbine performance under different operating conditions.

Pump Affinity Laws

When testing a single pump (where impeller diameter $D$ is constant) at different rotational speeds, performance parameters scale predictably. These scaling laws assume that hydraulic efficiency remains constant between different speeds, which is highly accurate for moderate changes in rotational speed.

The scaling relationships are:

Discharge ( $Q$ ): Varies directly with the rotational speed.
Head ( $H$ ): Varies with the square of the rotational speed.
Power ( $P$ ): Varies with the cube of the rotational speed.

Affinity Law for Discharge

Scales the volumetric flow rate of a pump linearly with rotational speed.

\frac{Q_1}{Q_2} = \frac{N_1}{N_2}

Variables

Symbol	Description	Unit
$Q_{1}$	Initial discharge	$m^{3} / s$
$Q_{2}$	Final discharge	$m^{3} / s$
$N_{1}$	Initial speed	RPM
$N_{2}$	Final speed	RPM

Affinity Law for Head

Scales the developed head of a pump quadratically with rotational speed.

\frac{H_1}{H_2} = \left(\frac{N_1}{N_2}\right)^2

Variables

Symbol	Description	Unit
$H_{1}$	Initial head	m
$H_{2}$	Final head	m
$N_{1}$	Initial speed	RPM
$N_{2}$	Final speed	RPM

Affinity Law for Power

Scales the required shaft power of a pump cubically with rotational speed.

\frac{P_1}{P_2} = \left(\frac{N_1}{N_2}\right)^3

Variables

Symbol	Description	Unit
$P_{1}$	Initial shaft power	kW
$P_{2}$	Final shaft power	kW
$N_{1}$	Initial speed	RPM
$N_{2}$	Final speed	RPM

Cavitation and NPSH

Cavitation occurs when the absolute pressure at the pump inlet drops below the vapor pressure ( $P_v$ ) of the liquid. Bubbles form and collapse violently, causing damage, pitting, vibration, and loss of efficiency.

Net Positive Suction Head (NPSH)

The absolute head at the pump inlet above the vapor pressure head of the liquid, serving as the margin of safety against cavitation.

Cavitation Prevention Criterion

The governing condition that must be met to ensure cavitation does not occur within the pump.

NPSH_{\text{available}} > NPSH_{\text{required}}

Variables

Symbol	Description	Unit
	Net Positive Suction Head available from the system installation	m
	Net Positive Suction Head required by the pump manufacturer	m

NPSH Available ( $N P S H_{A}$ )

Calculates the Net Positive Suction Head available in a static suction lift installation.

NPSH_A = \frac{P_{\text{atm}}}{\gamma} - h_s - h_{fs} - \frac{P_v}{\gamma}

Variables

Symbol	Description	Unit
	Atmospheric pressure	kPa
	Specific weight of the fluid	$k N / m^{3}$
$h_{s}$	Static suction lift (vertical distance from liquid level to impeller centerline)	m
$h_{fs}$	Friction head loss in the suction piping	m
$P_{v}$	Vapor pressure of the liquid at operating temperature	kPa

Key Takeaways

Pumps add energy to a fluid system, increasing its total dynamic head (TDH).
Affinity Laws predict performance changes when altering pump speed: Flow scales linearly ( $Q \propto N$ ), Head scales quadratically ( $H \propto N^2$ ), and Power scales cubically ( $P \propto N^3$ ).
Water Power ( $P_w$ ) is the useful energy gained by the fluid, while Brake Power ( $P_b$ ) is the total energy supplied by the motor. Brake power is always greater due to efficiency losses ( $\eta < 1$ ).
Turbines extract energy from a moving fluid to convert it into mechanical work, where efficiency ( $\eta$ ) is in the numerator since output power is less than input fluid power.
Specific Speed ( $N_s$ ) is a design index used to select the optimal machinery type: centrifugal/impulse (low $N_s$ ), mixed/Francis (medium $N_s$ ), and axial/Kaplan (high $N_s$ ).
Cavitation is avoided when the available Net Positive Suction Head ( $NPSH_A$ ) exceeds the required head ( $NPSH_R$ ).

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Concept Overview

Pumps

Pump Power

Water Power Formula

Brake Power Formula

Total Dynamic Head (TDH)

Total Dynamic Head (TDH)

Turbines

Turbine Power

Turbine Output Power

Specific Speed

Pump Specific Speed (NsN_sNs​)

Pump Specific Speed

Turbine Specific Speed (NsN_sNs​)

Turbine Specific Speed

Characteristic Curves

Affinity Laws

Pump Affinity Laws

Affinity Law for Discharge

Affinity Law for Head

Affinity Law for Power

Cavitation and NPSH

Net Positive Suction Head (NPSH)

Cavitation Prevention Criterion

NPSH Available (NPSHANPSH_ANPSHA​)

Pump Specific Speed ( $N_{s}$ )

Turbine Specific Speed ( $N_{s}$ )

NPSH Available ( $N P S H_{A}$ )