Hydraulic Turbines Classifications: Francis, Pelton & Kaplan Selection (2026)

1. Fundamental Principles of Hydraulic Energy Conversion

Hydropower generation relies on converting the potential and kinetic energy of water into mechanical shaft torque. The primary machine executing this energy conversion is the hydraulic turbine. Water stored in elevated reservoirs flows through pressurized penstocks, developing substantial hydrodynamic pressure and velocity at the runner interface.

The design, operational envelope, and mechanical configuration of these systems depend directly on hydraulic turbines classifications. Selecting an inappropriate runner geometry causes severe flow separation, cavitation pitting, structural vibration, and precipitous efficiency loss. Practicing hydro-mechanical engineers must evaluate net head $H_{net}$, volumetric discharge $Q$, rotational speed $N$, and specific speed $N_s$ to match the hydraulic profile of a catchment basin with optimal runner kinematics.

Understanding hydraulic turbines classifications establishes the mathematical basis for determining whether energy extraction occurs at constant atmospheric pressure through pure kinetic impulse or across continuous pressure gradients within sealed reaction passages.

2. Primary Hydraulic Turbines Classifications Framework

Hydraulic engineers organize turbines into distinct taxonomy groups based on three primary engineering criteria: the mode of action on runner vanes, the direction of fluid path across the rotor, and the specific speed index.

2.1 Action Mechanism: Impulse vs Reaction Machines

In an impulse machine, water potential energy converts entirely into kinetic energy inside stationary converging nozzles before contacting the runner. The high-velocity jet discharges into an open, unpressurized casing. Water strikes curved buckets at atmospheric pressure. The fluid undergoes direction reversal, imparting momentum to the rotor purely through dynamic impulse forces.

In a reaction machine, the runner remains fully submerged within a pressurized spiral casing. Fluid entering the runner carries both pressure energy and kinetic energy. As water traverses the curved runner passages, static pressure drops continuously from inlet to outlet. This static pressure drop accelerates fluid relative to the blade passages, generating reactive aerodynamic lift forces that drive shaft rotation.

2.2 Fluid Flow Direction Geometry

The kinematic trajectory of water particles relative to the axis of rotation establishes four spatial flow classifications:

1. Tangential Flow: Fluid strikes the runner periphery tangentially along the pitch circle diameter (Pelton wheel).
2. Radial Inflow: Fluid enters radially inward through guide vanes toward the center axis (classical Fourneyron and early Francis designs).
3. Mixed Flow (Radial-to-Axial): Fluid enters radially inward through stationary stay vanes and wicket gates, curving smoothly through a 90-degree bend to discharge parallel to the shaft axis (modern Francis runner).
4. Axial Flow: Fluid enters and discharges parallel to the rotational shaft axis across propeller-type blades (Kaplan and bulb tubular units).

2.3 Head and Discharge Operational Regimes

Site hydrology dictates turbine morphology. High-head mountainous catchments with modest river discharge require impulse wheels to handle immense hydrostatic pressures. Broad lowland river basins characterized by high volumetric flow rates and small elevation drops necessitate high-capacity axial reaction units.

Head Class      | Net Head Envelope ($H_{net}$) | Preferred Turbine Technology
----------------+-------------------------------+-----------------------------------------
Ultra-High Head | $H > 800\\text{ m}$            | Multi-jet Pelton Wheels
High Head       | $300\\text{ m} < H \\le 800\\text{ m}$ | Pelton / High-Head Francis Runners
Medium Head     | $50\\text{ m} < H \\le 300\\text{ m}$  | Francis Reaction Turbines
Low Head        | $15\\text{ m} < H \\le 50\\text{ m}$   | Kaplan / Deriaz / Propeller Units
Ultra-Low Head  | $H \\le 15\\text{ m}$          | Bulb, Pit, and S-Type Tubular Turbines

3. Kinematics and Governing Physics: Euler’s Turbomachine Equation

Angular momentum conservation governs all rotary turbomachinery. Applying Reynolds Transport Theorem to a control volume encompassing the rotating runner establishes the relationship between fluid velocities and mechanical shaft work.

3.1 Derivation of Euler Head and Dynamic Momentum Exchange

Consider a fluid stream of mass flow rate $\dot{m} = \rho Q$ entering the runner at radius $r_1$ with absolute velocity $\mathbf{v}_1$ and exiting at radius $r_2$ with absolute velocity $\mathbf{v}_2$. The net hydrodynamic torque $\mathbf{T}$ exerted on the runner equals the net rate of angular momentum efflux:

$$T = \rho Q \left( r_1 v_{w1} – r_2 v_{w2} \right)$$

Where $v_{w1}$ and $v_{w2}$ represent the tangential (whirl) components of absolute velocities $\mathbf{v}_1$ and $\mathbf{v}_2$. Multiplying torque by angular velocity $\omega = 2\pi N / 60$ gives power extracted by the rotor:

$$P_{rotor} = T \omega = \rho Q \left( \omega r_1 v_{w1} – \omega r_2 v_{w2} \right) = \rho Q \left( u_1 v_{w1} – u_2 v_{w2} \right)$$

Here, $u_1 = \omega r_1$ and $u_2 = \omega r_2$ denote runner peripheral tangential speeds at inlet and outlet. Dividing power by weight flow rate $\gamma Q = \rho g Q$ yields the theoretical Euler head $H_e$:

$$H_e = \frac{u_1 v_{w1} – u_2 v_{w2}}{g}$$

To maximize hydraulic efficiency $\eta_h = H_e / H_{net}$, the discharge whirl velocity $v_{w2}$ must be reduced toward zero ($v_{w2} \to 0$), ensuring fluid exits the runner with purely axial or radial velocity.

3.2 Inlet and Outlet Velocity Triangle Analysis

Vector decomposition at runner boundaries splits absolute velocity $\mathbf{v}$ into blade relative velocity $\mathbf{v}_r$ and peripheral speed $\mathbf{u}$:

$$\mathbf{v} = \mathbf{u} + \mathbf{v}_r$$

From trigonometric decomposition of velocity triangles at inlet (index 1) and outlet (index 2):

$$v_{r1}^2 = v_1^2 + u_1^2 – 2 u_1 v_1 \cos\alpha_1 = v_1^2 + u_1^2 – 2 u_1 v_{w1}$$

Rearranging for $u_1 v_{w1}$:

$$u_1 v_{w1} = \frac{1}{2} \left( v_1^2 + u_1^2 – v_{r1}^2 \right)$$

Similarly for the exit triangle:

$$u_2 v_{w2} = \frac{1}{2} \left( v_2^2 + u_2^2 – v_{r2}^2 \right)$$

Substituting these relations into Euler’s equation yields the fundamental three-component expansion of turbomachinery head:

$$H_e = \underbrace{\frac{v_1^2 – v_2^2}{2g}}_{\text{Dynamic Head Change}} + \underbrace{\frac{u_1^2 – u_2^2}{2g}}_{\text{Centrifugal Head Field}} + \underbrace{\frac{v_{r2}^2 – v_{r1}^2}{2g}}_{\text{Relative Diffusive Head}}$$

3.3 Degree of Reaction Mathematical Definition

The degree of reaction $R$ quantifies the proportion of static pressure drop occurring within the runner relative to total energy transferred:

$$R = \frac{\Delta p_{runner} / \gamma}{H_e} = \frac{\frac{p_1 – p_2}{\gamma}}{H_e} = \frac{\left( u_1^2 – u_2^2 \right) + \left( v_{r2}^2 – v_{r1}^2 \right)}{2 g H_e} = 1 – \frac{v_1^2 – v_2^2}{2 g H_e}$$

• Pure Impulse Machine (Pelton): $p_1 = p_2 = p_{atm}$, $u_1 = u_2$, $v_{r1} = v_{r2}$ (neglecting boundary friction). Thus, $R = 0$.

• Francis Reaction Machine: $0.30 \le R \le 0.70$. Energy conversion splits between static pressure reduction and absolute velocity deceleration.

• Kaplan Axial Machine: $R \ge 0.75$. Pressure drop dominates the energy transfer mechanism.

4. Detailed Engineering Analysis by Turbine Family

4.1 High-Head Impulse Systems: The Pelton Wheel

Lester Pelton engineered the modern impulse wheel featuring double-hemispherical ellipsoidal buckets divided by a central splitter ridge. Water discharges from spear-regulated converging nozzles at jet velocity:

$$v_1 = C_v \sqrt{2 g H_{net}}$$

Where $C_v \approx 0.97 – 0.99$ denotes the nozzle coefficient of discharge. The optimal peripheral speed ratio $\phi = u_1 / \sqrt{2 g H_{net}}$ for peak theoretical efficiency is:

$$\phi_{opt} = \frac{C_v \cos\beta_2}{2} \approx 0.45 – 0.48$$

Buckets turn fluid through an angle $180^\circ – \beta_2 \approx 165^\circ – 170^\circ$, avoiding mechanical interference with trailing buckets while ejecting water laterally with negligible residual kinetic energy.

4.2 Medium-Head Radial-Axial Reaction: The Francis Runner

James B. Francis developed the inward-flow reaction runner. Modern Francis units utilize a convergent volute scroll case maintaining uniform velocity distribution around the periphery. Stationary stay vanes absorb internal hydraulic hoop loads, while adjustable wicket gates (guide vanes) modulate flow angle $\alpha_1$ and discharge volume $Q$.

Water leaves the wicket gates at absolute angle $\alpha_1 \approx 12^\circ – 28^\circ$, accelerates over the 3D-curved suction and pressure surfaces of runner blades, and discharges into a divergent draft tube at outlet angle $\beta_2 \approx 15^\circ – 22^\circ$.

4.3 Low-Head Axial Reaction: Kaplan and Propeller Turbines

Viktor Kaplan modified the propeller turbine by introducing servomotor-driven rotatable blades synchronized with wicket gate openings (double regulation). This conjugate control maintains tangent flow entry across broad operational ranges, suppressing flow separation during part-load conditions.

Axial flow velocity through runner annular area $A = \frac{\pi}{4}(D_e^2 – D_h^2)$ remains constant:

$$v_f = \frac{Q}{\frac{\pi}{4}(D_e^2 – D_h^2)}$$

Where $D_e$ denotes outer tip diameter and $D_h$ represents runner hub diameter. Hub-to-tip diameter ratios vary from $D_h / D_e \approx 0.30$ (ultra-low head) to $0.55$ (medium-head Kaplan).

5. Cavitation Phenomena and Thoma’s Sigma Analysis

Cavitation represents the most destructive operational hazard in reaction turbomachinery. Local static pressure falling below water saturation vapor pressure $p_v(T)$ causes micro-bubbles to nucleate, expand, and implode violently against solid boundaries.

5.1 Vapor Bubble Collapse Dynamics and Material Erosion

Implosion of vapor cavities near metal surfaces generates asymmetric micro-jets striking boundary walls at velocities exceeding $1,000\text{ m/s}$. Local shock pressures reach $1.5\text{ to }2.0\text{ GPa}$, stripping protective oxide layers, inducing work hardening, and pitting high-grade stainless steel runners (such as 13Cr-4Ni martensitic alloys).

5.2 Thoma Cavitation Parameter and Setting Elevation

Dieter Thoma established the non-dimensional cavitation index $\sigma$:

$$\sigma = \frac{H_{atm} – H_v – Z_s}{H_{net}}$$

Where:

• $H_{atm} = \frac{p_{atm}}{\gamma}$ is atmospheric pressure head ($\approx 10.33\text{ m}$ at sea level).

• $H_v = \frac{p_v}{\gamma}$ is saturated vapor pressure head ($\approx 0.24\text{ m}$ at $20^\circ\text{ C}$).

• $Z_s$ is vertical distance from downstream tailrace water level to runner centerline (positive if above tailrace, negative if submerged).

• $H_{net}$ is net available head.

To prevent cavitation inception, the available plant cavitation factor $\sigma_{plant}$ must strictly exceed critical cavitation factor $\sigma_c$:

$$\sigma_{plant} \ge 1.15 \cdot \sigma_c$$

The critical factor scales with specific speed according to empirical relations:

$$\sigma_c \approx 0.0432 \left( \frac{N_{st}}{100} \right)^2 \quad \text{(for Francis runners)}$$

$$\sigma_c \approx 0.28 + \frac{1}{7.5} \left( \frac{N_{st}}{100} \right)^3 \quad \text{(for Kaplan runners)}$$

6. Draft Tube Pressure Recovery and Energy Efficiency

Impulse turbines discharge freely into air, but reaction turbines require an airtight divergent conduit connecting the runner exit to the tailrace channel.

The draft tube performs two vital hydraulic functions:
1. Recovers kinetic energy exiting the runner ($v_2^2 / 2g$) by converting dynamic head into static pressure head through continuous flow deceleration.
2. Allows the turbine runner to be positioned above tailrace flood level without sacrificing net suction head.

Applying Bernoulli’s equation between runner exit (2) and draft tube discharge (3):

$$\frac{p_2}{\gamma} + \frac{v_2^2}{2g} + Z_s = \frac{p_3}{\gamma} + \frac{v_3^2}{2g} + 0 + h_{f,dt}$$

Setting $p_3 = p_{atm}$ at tailrace surface gives runner exit static pressure:

$$\frac{p_2}{\gamma} = \frac{p_{atm}}{\gamma} – Z_s – \left[ \eta_{dt} \frac{v_2^2}{2g} – \frac{v_3^2}{2g} \right]$$

Where $\eta_{dt} = 1 – \frac{h_{f,dt}}{v_2^2 / 2g}$ denotes draft tube diffuser efficiency (typically $0.80 – 0.88$ for elbow-type diffusers).

7. Comprehensive Worked Engineering Calculation: Francis Runner Sizing

7.1 Operating Conditions and Dimensional Parameters

A proposed hydroelectric station requires a medium-head Francis turbine installation with the following verified site parameters:

• Net Available Head: $H_{net} = 125.0\text{ m}$

• Turbine Shaft Power: $P_{shaft} = 48.0\text{ MW} = 48{,}000\text{ kW}$

• Rotational Speed: $N = 375\text{ rpm}$ ($\omega = 39.27\text{ rad/s}$)

• Hydraulic Efficiency: $\eta_h = 0.940$ ($94.0\%$)

• Overall Plant Efficiency: $\eta_o = 0.910$ ($91.0\%$)

• Flow Velocity Ratio: $\psi = \frac{v_{f1}}{\sqrt{2 g H_{net}}} = 0.220$

• Peripheral Speed Ratio: $\phi = \frac{u_1}{\sqrt{2 g H_{net}}} = 0.720$

• Runner Breadth Ratio: $n_b = \frac{B_1}{D_1} = 0.120$

• Discharge Whirl: Purely radial ($v_{w2} = 0$, $\alpha_2 = 90^\circ$)

• Atmospheric Head: $H_{atm} = 10.10\text{ m}$; Vapor Head: $H_v = 0.25\text{ m}$

Step 1: Determine design volumetric discharge $Q$:

$$P_{shaft} = \eta_o \gamma Q H_{net} \implies Q = \frac{48{,}000 \times 10^3}{0.910 \times 9810 \times 125.0} = 43.08\text{ m}^3/\text{ s}$$

Step 2: Calculate runner inlet peripheral speed $u_1$ and flow velocity $v_{f1}$:

$$v_{ideal} = \sqrt{2 g H_{net}} = \sqrt{2 \times 9.81 \times 125.0} = 49.52\text{ m/s}$$

$$u_1 = \phi \sqrt{2 g H_{net}} = 0.720 \times 49.52 = 35.65\text{ m/s}$$

$$v_{f1} = \psi \sqrt{2 g H_{net}} = 0.220 \times 49.52 = 10.89\text{ m/s}$$

Step 3: Calculate runner outer diameter $D_1$ and inlet blade height $B_1$:

$$u_1 = \frac{\pi D_1 N}{60} \implies D_1 = \frac{60 \times 35.65}{\pi \times 375} = 1.816\text{ m}$$

$$B_1 = 0.120 \times D_1 = 0.120 \times 1.816 = 0.218\text{ m} = 218\text{ mm}$$

Check flow continuity through inlet area ($A_1 = \pi D_1 B_1 k_t$, assuming blade blockage thickness factor $k_t = 0.95$):

$$Q_{calc} = 0.95 \times \pi \times 1.816 \times 0.218 \times 10.89 = 12.86\text{ m}^3/\text{ s}$$

For full 3D Francis runner geometry, setting flow passage depth establishes flow matching across all intermediate streamlines.

7.2 Velocity Components and Vane Angles Determination

Step 4: Compute inlet whirl velocity $v_{w1}$ using hydraulic efficiency formulation:

$$\eta_h = \frac{u_1 v_{w1}}{g H_{net}} \implies v_{w1} = \frac{\eta_h g H_{net}}{u_1} = \frac{0.940 \times 9.81 \times 125.0}{35.65} = 32.36\text{ m/s}$$

Step 5: Calculate guide vane (wicket gate) flow angle $\alpha_1$:

$$\tan\alpha_1 = \frac{v_{f1}}{v_{w1}} = \frac{10.89}{32.36} = 0.3365 \implies \alpha_1 = 18.60^\circ$$

Step 6: Calculate runner blade inlet angle $\beta_1$:

$$\tan\beta_1 = \frac{v_{f1}}{v_{w1} – u_1} = \frac{10.89}{|32.36 – 35.65|} = \frac{10.89}{-3.29} = -3.310 \implies \beta_1 = 106.8^\circ$$

The inlet angle $\beta_1 > 90^\circ$ confirms a standard backward-curved high-efficiency Francis vane profile.

7.3 Cavitation Limit and Setting Level Verification

Step 7: Calculate specific speed in metric units ($N_{st}$):

$$N_{st} = \frac{N \sqrt{P_{kW}}}{H_{net}^{5/4}} = \frac{375 \times \sqrt{48{,}000}}{(125.0)^{1.25}} = \frac{375 \times 219.09}{417.82} = 196.6\text{ rpm}$$

Step 8: Estimate critical Thoma cavitation factor $\sigma_c$:

$$\sigma_c = 0.0432 \left( \frac{N_{st}}{100} \right)^2 = 0.0432 \times (1.966)^2 = 0.167$$

Step 9: Calculate maximum allowable setting level above tailrace ($Z_{s,max}$), applying safety factor $SF = 1.15$:

$$\sigma_{plant} = 1.15 \times \sigma_c = 1.15 \times 0.167 = 0.192$$

$$Z_{s,max} = H_{atm} – H_v – \sigma_{plant} H_{net} = 10.10 – 0.25 – (0.192 \times 125.0) = 9.85 – 24.00 = -14.15\text{ m}$$

The calculation demonstrates that the runner centerline must be set at least $14.15\text{ m}$ below minimum downstream tailrace level ($Z_s = -14.15\text{ m}$) to suppress cavitation inception during full load.

8. Specific Speed Selection Criteria and Sizing Envelopes

The specific speed $N_s$ serves as the universal index for matching runner geometry to catchment hydrology:

$$N_s = \frac{N \sqrt{P}}{\rho^{1/2} (g H_{net})^{5/4}} \quad \text{(Dimensionless Form)}$$

In conventional metric engineering practice:

$$N_{st} = \frac{N [\text{ rpm}] \sqrt{P [\text{ kW}]}}{H_{net} [\text{ m}]^{5/4}}$$

     100 % ┌─────────────────────────────────────────────────────────────┐
           │                     KAPLAN (Double Regulated)               │
           │  PELTON            /─────────────────────────\              │
      80 % │  /─────────\      /                           \             │
Efficiency │ /           \    /                             \   FRANCIS  │
           │/             \  /                               \ /───────\ │
      60 % │               \/                                 \         ││
           │                                                   \        ││
      40 % └─────────────────────────────────────────────────────────────┘
           0%             20%         40%         60%         80%       100%
                                Turbine Rated Discharge (Q / Q_rated)

9. Synthesis and Hydrodynamic Equilibrium Wrap-Up

Mastering hydraulic turbines classifications allows hydro engineers to transform raw fluid momentum into stable megawatt capacity across complex geographic terrains. Balancing velocity triangles, dynamic runner kinematics, draft tube diffusion, and cavitation limits guarantees that hydraulic machinery operates with high reliability throughout multi-decade operating lifecycles.

References & Standards Cited

1. IEC 60193: Hydraulic Turbines, Storage Pumps and Pump-Turbines – Model Acceptance Tests. International Electrotechnical Commission, Geneva, Switzerland.
2. IEC 60041: Field Acceptance Tests to Determine the Hydraulic Performance of Hydraulic Turbines, Storage Pumps and Pump-Turbines.
3. USBR (1980): Selecting Hydraulic Reaction Turbines. Engineering Monograph No. 20, United States Bureau of Reclamation, Denver, CO.
4. ASCE Committee on Hydropower Development (1995): Civil Engineering Guidelines for Planning and Designing Hydroelectric Developments. American Society of Civil Engineers, Reston, VA.
5. Dixon, S.L., and Hall, C.A. (2014): Fluid Mechanics and Thermodynamics of Turbomachinery. 7th Edition, Butterworth-Heinemann, Oxford, UK.
6. Brennen, C.E. (1995): Cavitation and Multiphase Flow. Oxford University Press, New York.

Frequently Asked Questions (FAQ)

Why does a Pelton turbine operate without a draft tube while Francis units require one?

A Pelton turbine is an impulse machine operating at atmospheric pressure. Water discharges into unpressurized air without needing static pressure recovery. In contrast, Francis reaction runners discharge fluid at sub-atmospheric pressure. A sealed, divergent draft tube is mandatory to decelerate exiting flow, recover residual kinetic energy, and prevent tailrace flooding from breaking internal suction.

How does specific speed dictate runner profile geometry?

Low specific speed machines ($N_{st} 450$) handle large discharge volumes under low heads, requiring wide, unobstructed axial flow passages characteristic of multi-blade Kaplan propellers.

What physical mechanism causes cavitation erosion on Francis runner trailing edges?

When local absolute pressure drops below the saturated vapor pressure of water ($p < p_v$), micro-vapor cavities nucleate within high-velocity vortices. As these cavities sweep into downstream regions of recovering pressure, they implode violently. These implosions generate localized liquid micro-jets striking blade surfaces at supersonic velocities with contact stresses exceeding $1.5text{ GPa}$, causing metal fatigue and pitting.

What is the fundamental difference between fixed propeller and Kaplan turbines?

Propeller turbines have fixed runner blades and adjustable wicket gates (single regulation), resulting in narrow peak efficiency curves that drop sharply at off-design discharge. Kaplan turbines feature synchronized rotation of both wicket gates and runner blades (double regulation), maintaining tangential flow entry and high efficiency ($>90%$) across a broad operating envelope from $30%$ to $100%$ rated discharge.

Why is runway speed a critical parameter in hydroelectric station structural design?

Runway speed represents the maximum rotational speed reached if generator electrical load drops instantly while governor valves fail open. Runway speed can reach $1.8text{ to }2.2times$ rated operational speed in Francis units and up to $2.8times$ in Kaplan turbines. Centrifugal stresses scale with $omega^2$, meaning runner disks, rotor poles, and bearings must withstand up to $8times$ normal operational hoop tension without plastic rupture.