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Qwestrum Engineering360 · Mechanical Engineering · Fluid Machinery

Pelton and Francis Turbines

Turbine work per unit mass follows Euler's equation W = u(V_w1 − V_w2)/g; the Pelton is an impulse machine for high heads and the Francis a reaction machine for medium heads. Bucket/runner speed u = πDN/60, per Modi & Seth.
Exam tip: keep density/pressure in SI (Pa, kg/m³); convert head carefully before Bernoulli or Darcy–Weisbach.

Key formulas & points

Skim these first — then read the full notes below.

  • Pelton: impulse turbine, high head low flow
  • Francis: reaction turbine, medium head, radial flow
  • DegreeofreactionR=(h1−h2)(h1−h3)≈0.5forFrancisDegree of reaction R = \frac{(h_{1} - h_{2})}{(h_{1} - h_{3})} \approx 0.5 for Francis

Topic details

Introduction

Hydraulic turbines are a core fluid-machinery topic, examined through velocity triangles and efficiency calculations. Modi & Seth classify turbines by head and specific speed: Pelton (impulse, high head), Francis (reaction, medium head), and Kaplan (reaction, low head, high flow).

Scope in B.Tech and GATE syllabus

For the Pelton wheel, the jet strikes buckets and momentum change gives the driving force; maximum efficiency is at bucket speed u ≈ V/2. For the Francis turbine, water flows through the runner with both pressure and velocity change (reaction), and the degree of reaction is around 0.5.

Why this topic matters in practice

Velocity triangles at inlet and outlet give the whirl components feeding Euler's turbine equation. Drawing these triangles correctly, with blade and flow angles, is the marks-critical skill.

Key relations & formulas

u=πDN60u = \frac{\pi DN}{60}
(bucket velocity, optimum u ≈ V/2)
ηhyd=ρgQH⋅ηmech\eta_{hyd} = \rho gQH\cdot \eta_{mech}
(power P = η·ρgQH)
ψ=H(u2/2g)ψ = \frac{H}{(u^{2}/2g)}
(head coefficient)
ϕ=Q(A⋅2gH)\phi = \frac{Q}{(A\cdot \sqrt{2gH})}
(flow coefficient)

Notation and sign conventions

Relation 1 —
u=πDN60u = \frac{\pi DN}{60}
u=πDN60u = \frac{\pi DN}{60}
(bucket velocity, optimum u ≈ V/2)
Write this relation with symbols exactly as in Fluid Mechanics & Hydraulic Machines — Modi & Seth before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
ηhyd=ρgQH⋅ηmech\eta_{hyd} = \rho gQH\cdot \eta_{mech}
ηhyd=ρgQH⋅ηmech\eta_{hyd} = \rho gQH\cdot \eta_{mech}
(power P = η·ρgQH)
Write this relation with symbols exactly as in Fluid Mechanics & Hydraulic Machines — Modi & Seth before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
ψ=H/ψ = H/
ψ=H(u2/2g)ψ = \frac{H}{(u^{2}/2g)}
(head coefficient)
Write this relation with symbols exactly as in Fluid Mechanics & Hydraulic Machines — Modi & Seth before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
ϕ=Q/\phi = Q/
ϕ=Q(A⋅2gH)\phi = \frac{Q}{(A\cdot \sqrt{2gH})}
(flow coefficient)
Write this relation with symbols exactly as in Fluid Mechanics & Hydraulic Machines — Modi & Seth before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

Euler's turbine equation W = (V_w1·u₁ − V_w2·u₂)/g gives the work per unit weight, where V_w are the whirl (tangential) velocity components and u the blade speeds. Maximising V_w1 and minimising V_w2 (ideally zero) maximises work.

Governing relations in practice

The Pelton wheel is an impulse turbine: the entire head is converted to jet velocity V = C_v√(2gH) in the nozzle, then the buckets extract momentum at atmospheric pressure. Optimum bucket speed is u = V/2, and the deflection angle (~165°) sets the force.

Design and analysis considerations

The Francis turbine is a reaction machine: water enters radially under pressure through guide vanes, and both pressure and kinetic energy change through the runner. The degree of reaction R ≈ 0.5 means roughly half the energy conversion is by pressure drop in the runner.

Advanced theory and extensions

Overall efficiency η = shaft power/(ρgQH) combines hydraulic, mechanical, and volumetric efficiencies. Specific speed selects the type: low for Pelton, medium for Francis, high for Kaplan. Velocity triangles at inlet and outlet are the computational backbone.

Assumptions and validity limits

State assumptions explicitly before using any relation for pelton and francis turbines — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Fluid Machinery viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Fluid Machinery papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to pelton and francis turbines.
4. Use equation 1:
u=πDN60u = \frac{\pi DN}{60}
.
5. Use equation 2:
ηhyd=ρgQH⋅ηmech\eta_{hyd} = \rho gQH\cdot \eta_{mech}
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Pelton and Francis Turbines appears in hydropower, water supply, and process plants. In Indian mechanical curricula this topic is tested because it connects theory to turbines, pumps, and fluid power devices.
GATE and semester exams often combine pelton and francis turbines with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use pelton and francis turbines?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Setting outlet whirl V_w2 arbitrarily instead of using the outlet velocity triangle
• Confusing impulse (Pelton, atmospheric runner) with reaction (Francis, pressure drop) machines
• Using jet velocity where blade velocity u = πDN/60 is required
• Forgetting the nozzle coefficient C_v when finding Pelton jet velocity

Quick revision checklist

Before attempting pelton and francis turbines problems, confirm you can:
1. Pelton: impulse turbine, high head low flow
2. Francis: reaction turbine, medium head, radial flow
3.
DegreeofreactionR=(h1−h2)(h1−h3)≈0.5forFrancisDegree of reaction R = \frac{(h_{1} - h_{2})}{(h_{1} - h_{3})} \approx 0.5 for Francis
Revise the solved examples in Fluid Mechanics & Hydraulic Machines — Modi & Seth and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Pelton bucket speed

Problem

A Pelton wheel of diameter D = 1.5 m runs at N = 300 rpm. Find the bucket (peripheral) velocity.

Solution

u = πDN/60 = π × 1.5 × 300/60 = π × 7.5 = 23.56 m/s.

Conceptual check — Pelton and Francis Turbines

Problem

In a Fluid Machinery semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of pelton and francis turbines." What should a complete answer include?

Practice questions

Most-asked interview and GATE questions for this topic — expand any item for a model answer.

  1. 1
    What is Pelton and Francis Turbines, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    Turbine work per unit mass follows Euler's equation W = u(V_w1 − V_w2)/g; the Pelton is an impulse machine for high heads and the Francis a reaction machine for medium heads. Bucket/runner speed u = πDN/60, per Modi & Seth.
  2. 2
    State the relation u = πDN/60 and name each symbol.

    Model answer

    The governing relation is u=πDN60u = \frac{\pi DN}{60}. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation η_hyd = ρgQH·η_mech and name each symbol.

    Model answer

    The governing relation is ηhyd=ρgQH⋅ηmech\eta_{hyd} = \rho gQH\cdot \eta_{mech}. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation ψ = H/ and name each symbol.

    Model answer

    The governing relation is ψ=H/ψ = H/. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation φ = Q/ and name each symbol.

    Model answer

    The governing relation is ϕ=Q/\phi = Q/. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: Pelton: impulse turbine, high head low flow

    Model answer

    Pelton: impulse turbine, high head low flow — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Francis: reaction turbine, medium head, radial flow

    Model answer

    Francis: reaction turbine, medium head, radial flow — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: Degree of reaction R = (h₁ − h₂)/(h₁ − h₃) ≈ 0.5 for Francis

    Model answer

    DegreeofreactionR=(h1−h2)(h1−h3)≈0.5forFrancisDegree of reaction R = \frac{(h_{1} - h_{2})}{(h_{1} - h_{3})} \approx 0.5 for Francis — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Setting outlet whirl V_w2 arbitrarily instead of using the outlet velocity triangle?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  10. 10
    How would you correct this error in a viva: Confusing impulse (Pelton, atmospheric runner) with reaction (Francis, pressure drop) machines?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  11. 11
    How would you correct this error in a viva: Using jet velocity where blade velocity u = πDN/60 is required?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  12. 12
    How would you correct this error in a viva: Forgetting the nozzle coefficient C_v when finding Pelton jet velocity?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.

Exams & GATE

  • 1
    Velocity triangles at inlet/outlet — Euler turbine equation W = u(V_w1 − V_w2).
  • 2
    Avoid: Setting outlet whirl V_w2 arbitrarily instead of using the outlet velocity triangle
  • 3
    Avoid: Confusing impulse (Pelton, atmospheric runner) with reaction (Francis, pressure drop) machines
  • 4
    Avoid: Using jet velocity where blade velocity u = πDN/60 is required

📖 Standard books (India)

  • Fluid Mechanics & Hydraulic Machines — Modi & Seth

    Read: Syllabus unit

    Fluid statics, dynamics, pipes, and turbomachinery

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