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Qwestrum Engineering360 · Mechanical Engineering · Fluid Machinery

Specific Speed and Similarity

Specific speed characterises a machine's type: N_s = N√Q/H^(3/4) for pumps and N√P/H^(5/4) for turbines. Geometrically similar machines share N_s and follow the affinity laws, 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.

  • Specific speed selects turbine/pump type (Pelton, Francis, Kaplan)
  • Geometrically similar machines: same N_s, same efficiency curves
  • Cavitationparameterσ=HaHCavitation parameter \sigma = \frac{H_{a}}{H}

Topic details

Introduction

Specific speed and dimensional similarity let engineers select machine type and scale model results to prototypes. Modi & Seth derive specific speed from dimensional analysis and present the affinity (similarity) laws for homologous machines.

Scope in B.Tech and GATE syllabus

Specific speed is a single number that groups speed, flow (or power), and head; low N_s indicates high-head machines (Pelton, radial pumps), high N_s indicates low-head high-flow machines (Kaplan, axial pumps). This guides turbine/pump selection before detailed design.

Why this topic matters in practice

The affinity laws relate discharge, head, and power of the same machine at different speeds (Q ∝ N, H ∝ N², P ∝ N³) and of geometrically similar machines of different size. Model testing uses unit quantities (unit speed, discharge, power) to predict prototype performance — a favourite numerical.

Key relations & formulas

Ns=NQ/H(34)N_{s} = N\sqrt{Q}/H^(\frac{3}{4})
(pump, SI: N rpm, Q m³/s, H m)
N_{s}_turbine = N\sqrt{P}/H^(\frac{5}{4})
(turbine specific speed)
Q2Q1=(N2N1)(D2D1)3\frac{Q_{2}}{Q_{1}} = (\frac{N_{2}}{N_{1}})(\frac{D_{2}}{D_{1}})^{3}
(affinity laws, same pump)

Formulas (Indian textbook notation)

  • H2H1=(N2N1)2(D2D1)2\frac{H_{2}}{H_{1}} = (\frac{N_{2}}{N_{1}})^{2}(\frac{D_{2}}{D_{1}})^{2}

Notation and sign conventions

Relation 1 —
N_{s} = N\sqrt{Q}/H^
Ns=NQ/H(34)N_{s} = N\sqrt{Q}/H^(\frac{3}{4})
(pump, SI: N rpm, Q m³/s, H m)
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 —
N_{s}_turbine = N\sqrt{P}/H^
N_{s}_turbine = N\sqrt{P}/H^(\frac{5}{4})
(turbine specific speed)
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 —
Q2Q1=\frac{Q_{2}}{Q_{1}} =
Q2Q1=(N2N1)(D2D1)3\frac{Q_{2}}{Q_{1}} = (\frac{N_{2}}{N_{1}})(\frac{D_{2}}{D_{1}})^{3}
(affinity laws, same pump)
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 —
H2H1=\frac{H_{2}}{H_{1}} =

Formulas (Indian textbook notation)

  • H2H1=(N2N1)2(D2D1)2\frac{H_{2}}{H_{1}} = (\frac{N_{2}}{N_{1}})^{2}(\frac{D_{2}}{D_{1}})^{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.

Fundamentals and definitions

Dimensional analysis of turbomachines yields dimensionless groups; specific speed collapses them into one design number. For pumps N_s = N√Q/H^(3/4); for turbines N_s = N√P/H^(5/4), with consistent units stated in the problem.

Governing relations in practice

A given specific speed corresponds to a particular impeller/runner geometry, so N_s selects the machine type. Two machines that are geometrically similar have equal specific speed and superimposable non-dimensional performance curves (homologous series).

Design and analysis considerations

The affinity laws for one machine at varying speed are Q₂/Q₁ = N₂/N₁, H₂/H₁ = (N₂/N₁)², P₂/P₁ = (N₂/N₁)³. For similar machines of different diameter, D replaces or multiplies N accordingly.

Advanced theory and extensions

Model testing scales small-model data to full-size prototypes using unit quantities: unit speed N_u = N/√H, unit discharge Q_u = Q/√H, unit power P_u = P/H^(3/2). These allow prototype performance and cavitation limits to be predicted economically — the practical purpose of similarity.

Assumptions and validity limits

State assumptions explicitly before using any relation for specific speed and similarity — 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 specific speed and similarity.
4. Use equation 1:
N_{s} = N\sqrt{Q}/H^
.
5. Use equation 2:
N_{s}_turbine = N\sqrt{P}/H^
.
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

Specific Speed and Similarity 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 specific speed and similarity with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use specific speed and similarity?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Using the pump specific-speed formula (with Q) for a turbine (which uses P)
• Applying affinity laws across geometrically dissimilar machines
• Wrong exponents in the affinity laws (H ∝ N², P ∝ N³)
• Inconsistent units in specific speed, which is unit-dependent

Quick revision checklist

Before attempting specific speed and similarity problems, confirm you can:
1. Specific speed selects turbine/pump type (Pelton, Francis, Kaplan)
2. Geometrically similar machines: same N_s, same efficiency curves
3.
Cavitationparameterσ=HaHCavitation parameter \sigma = \frac{H_{a}}{H}
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.

Specific speed of a pump

Problem

A pump runs at N = 1450 rpm, delivering Q = 0.25 m³/s at head H = 25 m. Find its specific speed.

Solution

N_s = N√Q/H^(3/4) = 1450 × √0.25/25^0.75 = 1450 × 0.5/11.18 = 725/11.18 = 64.8.

Conceptual check — Specific Speed and Similarity

Problem

In a Fluid Machinery semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of specific speed and similarity." 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 Specific Speed and Similarity, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    Specific speed characterises a machine's type: N_s = N√Q/H^(3/4) for pumps and N√P/H^(5/4) for turbines. Geometrically similar machines share N_s and follow the affinity laws, per Modi & Seth.
  2. 2
    State the relation N_s = N√Q/H^ and name each symbol.

    Model answer

    The governing relation is N_{s} = N\sqrt{Q}/H^. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation N_s_turbine = N√P/H^ and name each symbol.

    Model answer

    The governing relation is N_{s}_turbine = N\sqrt{P}/H^. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation Q₂/Q₁ = and name each symbol.

    Model answer

    The governing relation is Q2Q1=\frac{Q_{2}}{Q_{1}} =. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation H₂/H₁ = and name each symbol.

    Model answer

    The governing relation is H2H1=\frac{H_{2}}{H_{1}} =. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: Specific speed selects turbine/pump type (Pelton, Francis, Kaplan)

    Model answer

    Specific speed selects turbine/pump type (Pelton, Francis, Kaplan) — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Geometrically similar machines: same N_s, same efficiency curves

    Model answer

    Geometrically similar machines: same N_s, same efficiency curves — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: Cavitation parameter σ = H_a/H

    Model answer

    Cavitationparameterσ=HaHCavitation parameter \sigma = \frac{H_{a}}{H} — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Using the pump specific-speed formula (with Q) for a turbine (which uses P)?

    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: Applying affinity laws across geometrically dissimilar 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: Wrong exponents in the affinity laws (H ∝ N², P ∝ N³)?

    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: Inconsistent units in specific speed, which is unit-dependent?

    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
    Modi & Seth Ch. 20 — unit speed, unit discharge, unit power for model testing.
  • 2
    Avoid: Using the pump specific-speed formula (with Q) for a turbine (which uses P)
  • 3
    Avoid: Applying affinity laws across geometrically dissimilar machines
  • 4
    Avoid: Wrong exponents in the affinity laws (H ∝ N², P ∝ N³)

📖 Standard books (India)

  • Fluid Mechanics & Hydraulic Machines — Modi & Seth

    Read: Syllabus unit

    Fluid statics, dynamics, pipes, and turbomachinery

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