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Qwestrum Engineering360 · Mechanical Engineering · Applied Thermodynamics

Compressors and Nozzles

Polytropic compressor work is w = n/(n−1)·RT₁[(P₂/P₁)^((n−1)/n) − 1]; multistage compression with intercooling minimises it. In a nozzle flow chokes at the throat when Mach number M = 1, with a = √(γRT), per P.K. Nag.
Exam tip: lock the sign convention (Q into system, W by system in P.K. Nag) before substituting; use absolute temperature for ideal-gas and efficiency ratios.

Key formulas & points

Skim these first — then read the full notes below.

  • Multistage compression with intercooling reduces total work
  • Choked flow in nozzle when M = 1 at throat
  • Stagnationproperties:h0=h+V22Stagnation properties: h_{0} = h + \frac{V^{2}}{2}

Topic details

Introduction

Compressors and nozzles apply the steady-flow energy equation to work-absorbing and accelerating devices. P.K. Nag derives reciprocating-compressor work with and without clearance, and multistage compression with intercooling for minimum work.

Scope in B.Tech and GATE syllabus

Isentropic efficiency compares ideal to actual work; for compressors it is w_isentropic/w_actual (<1). Volumetric efficiency and clearance ratio affect reciprocating-compressor capacity.

Why this topic matters in practice

Nozzle flow introduces compressible-flow ideas: stagnation properties, critical pressure ratio, and choking at Mach 1 in the throat. A converging nozzle can only reach sonic velocity; supersonic flow needs a converging-diverging (de Laval) nozzle. Identifying whether the flow is choked is the decisive step in nozzle numericals.

Key relations & formulas

Wcomp=∫vdP≈n(n−1)⋅R⋅T1[(P2P1)((n−1)n)−1]W_{comp} = \int v dP \approx \frac{n}{(n-1)}\cdot R\cdot T_{1}[(\frac{P_{2}}{P_{1}})^(\frac{(n-1)}{n}) - 1]
(polytropic)
ηis=WisWactual\eta_{is} = \frac{W_{is}}{W_{actual}}
(isentropic efficiency)
m˙=ρAV=AVaṁ = \rho AV = \frac{AV}{a}
(continuity, a = sonic velocity)
a=γRTa = \sqrt{\gamma RT}
(speed of sound)

Notation and sign conventions

Relation 1 —
Wcomp=∫vdP≈n/W_{comp} = \int v dP \approx n/
Wcomp=∫vdP≈n(n−1)⋅R⋅T1[(P2P1)((n−1)n)−1]W_{comp} = \int v dP \approx \frac{n}{(n-1)}\cdot R\cdot T_{1}[(\frac{P_{2}}{P_{1}})^(\frac{(n-1)}{n}) - 1]
(polytropic)
Write this relation with symbols exactly as in Engineering Thermodynamics — P.K. Nag before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
ηis=WisWactual\eta_{is} = \frac{W_{is}}{W_{actual}}
ηis=WisWactual\eta_{is} = \frac{W_{is}}{W_{actual}}
(isentropic efficiency)
Write this relation with symbols exactly as in Engineering Thermodynamics — P.K. Nag before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
m˙=ρAV=AVaṁ = \rho AV = \frac{AV}{a}
m˙=ρAV=AVaṁ = \rho AV = \frac{AV}{a}
(continuity, a = sonic velocity)
Write this relation with symbols exactly as in Engineering Thermodynamics — P.K. Nag before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
a=√a = √
a=γRTa = \sqrt{\gamma RT}
(speed of sound)
Write this relation with symbols exactly as in Engineering Thermodynamics — P.K. Nag before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

Compressor work equals ∫v dP; for a polytropic process w = n/(n−1)·RT₁[(P₂/P₁)^((n−1)/n) − 1]. Cooling during compression (lower n) reduces work, which is why intercooling is used.

Governing relations in practice

For multistage compression with perfect intercooling, work is minimised when each stage has the same pressure ratio (P₂/P₁)^(1/N); this also equalises stage work and discharge temperatures. Isentropic efficiency η_is = w_ideal/w_actual accounts for real irreversibilities.

Design and analysis considerations

Nozzles convert enthalpy to kinetic energy: h₀ = h + V²/2 (stagnation). The exit velocity V = √(2(h₁ − h₂)). As back-pressure falls, mass flow rises until the throat reaches Mach 1 (choked), after which flow is fixed.

Advanced theory and extensions

The speed of sound a = √(γRT) sets the Mach number M = V/a. The critical pressure ratio P*/P₀ = (2/(γ+1))^(γ/(γ−1)) marks choking. A converging-diverging nozzle is required to accelerate beyond sonic. Determining choked vs unchoked flow governs the whole calculation.

Assumptions and validity limits

State assumptions explicitly before using any relation for compressors and nozzles — 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 Applied Thermodynamics 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 Applied Thermodynamics 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 compressors and nozzles.
4. Use equation 1:
Wcomp=∫vdP≈n/W_{comp} = \int v dP \approx n/
.
5. Use equation 2:
ηis=WisWactual\eta_{is} = \frac{W_{is}}{W_{actual}}
.
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

Compressors and Nozzles appears in IC engines, gas turbines, and compressors. In Indian mechanical curricula this topic is tested because it connects theory to air-standard and vapour power cycles.
GATE and semester exams often combine compressors and nozzles with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use compressors and nozzles?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Using isentropic (γ) exponent where polytropic (n) applies, or vice versa
• Forgetting that a converging nozzle cannot exceed Mach 1 (choking limit)
• Ignoring intercooling's effect and computing single-stage work for a multistage machine
• Evaluating speed of sound with the wrong temperature (must be local static T)

Quick revision checklist

Before attempting compressors and nozzles problems, confirm you can:
1. Multistage compression with intercooling reduces total work
2. Choked flow in nozzle when M = 1 at throat
3.
Stagnationproperties:h0=h+V22Stagnation properties: h_{0} = h + \frac{V^{2}}{2}
Revise the solved examples in Engineering Thermodynamics — P.K. Nag 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.

Speed of sound in air

Problem

Find the speed of sound in air at T = 300 K (γ = 1.4, R = 287 J/kgK).

Solution

a = √(γRT) = √(1.4 × 287 × 300) = √120540 = 347.2 m/s.

Conceptual check — Compressors and Nozzles

Problem

In a Applied Thermodynamics semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of compressors and nozzles." 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 Compressors and Nozzles, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    Polytropic compressor work is w = n/(n−1)·RT₁[(P₂/P₁)^((n−1)/n) − 1]; multistage compression with intercooling minimises it. In a nozzle flow chokes at the throat when Mach number M = 1, with a = √(γRT), per P.K. Nag.
  2. 2
    State the relation W_comp = ∫v dP ≈ n/ and name each symbol.

    Model answer

    The governing relation is Wcomp=∫vdP≈n/W_{comp} = \int v dP \approx n/. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation η_is = W_is/W_actual and name each symbol.

    Model answer

    The governing relation is ηis=WisWactual\eta_{is} = \frac{W_{is}}{W_{actual}}. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation ṁ = ρAV = AV/a and name each symbol.

    Model answer

    The governing relation is m˙=ρAV=AVaṁ = \rho AV = \frac{AV}{a}. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation a = √ and name each symbol.

    Model answer

    The governing relation is a=√a = √. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: Multistage compression with intercooling reduces total work

    Model answer

    Multistage compression with intercooling reduces total work — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Choked flow in nozzle when M = 1 at throat

    Model answer

    Choked flow in nozzle when M = 1 at throat — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: Stagnation properties: h₀ = h + V²/2

    Model answer

    Stagnationproperties:h0=h+V22Stagnation properties: h_{0} = h + \frac{V^{2}}{2} — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Using isentropic (γ) exponent where polytropic (n) applies, or vice versa?

    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: Forgetting that a converging nozzle cannot exceed Mach 1 (choking limit)?

    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: Ignoring intercooling's effect and computing single-stage work for a multistage machine?

    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: Evaluating speed of sound with the wrong temperature (must be local static T)?

    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
    P.K. Nag Ch. 19–20 — isentropic efficiency always < 1 for real devices.
  • 2
    Avoid: Using isentropic (γ) exponent where polytropic (n) applies, or vice versa
  • 3
    Avoid: Forgetting that a converging nozzle cannot exceed Mach 1 (choking limit)
  • 4
    Avoid: Ignoring intercooling's effect and computing single-stage work for a multistage machine

📖 Standard books (India)

  • Engineering Thermodynamics — P.K. Nag

    Read: Syllabus unit

    The standard thermodynamics text in most Indian universities

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