---
title: "Dimensionless parameters and their significance"
subject: "Turbo Machines"
chapter: 1
key: "semester_7/turbo_machines:dimensionless-parameters-turbomachines"
source: https://www.bctengineer.com/q/semester_7/turbo_machines/dimensionless-parameters-turbomachines
site: BCT Engineering
---

# Dimensionless parameters and their significance

Turbo Machines · Chapter 1: Introduction

- Asked in: year not given, 6 marks
- Video: [Significance of π Terms in Turbomachines | Dimensionless Parameters | VTU](https://www.youtube.com/watch?v=yWOCfyjSq10) — Thermalprof
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/turbo_machines/dimensionless-parameters-turbomachines
- Whole chapter: https://www.bctengineer.com/md/semester_7/turbo_machines/chapter-1

## Question

Explain the dimensionless parameters used for turbomachines (flow coefficient, head or pressure coefficient, power coefficient, Reynolds number, Mach number and specific speed) and state the physical significance of each.

## Short answer

Flow coefficient φ = Q/(ND³) relates meridional velocity to blade speed. Head coefficient ψ = gH/(N²D²) indicates loading. Power coefficient λ = P/(ρN³D⁵). Reynolds number Re = ρND²/μ governs viscous effects. Mach number Ma = ND/a indicates compressibility. Specific speed N_s = N√Q/(gH)^(3/4) determines machine type (low N_s radial, high N_s axial).

## Answer

Dimensional analysis of a turbomachine (variables: $D$, $N$, $\rho$, $\mu$, $Q$, $gH$, $P$) gives dimensionless groups that allow geometrically similar machines to be compared and model test results to be scaled up.

| Parameter | Expression | Physical significance |
|---|---|---|
| Flow coefficient | $\phi = \dfrac{Q}{N D^3}$ | Proportional to the ratio of axial (meridional) velocity to blade speed. Fixes the shape of the velocity triangles |
| Head (pressure) coefficient | $\psi = \dfrac{gH}{N^2 D^2}$ | Ratio of head (energy per unit mass) to the kinetic energy based on blade speed. Measures the loading of the machine |
| Power coefficient | $\lambda = \dfrac{P}{\rho N^3 D^5}$ | Power per unit of $\rho N^3 D^5$; with $\phi$ and $\psi$ it gives the efficiency $\eta = \phi\psi/\lambda$ |
| Reynolds number | $Re = \dfrac{\rho N D^2}{\mu}$ | Ratio of inertia to viscous forces; controls friction loss. Its effect is small at high $Re$ |
| Mach number | $Ma = \dfrac{ND}{a}$ | Blade speed to speed of sound; important for compressors and gas turbines, where compressibility and shock losses appear |
| Specific speed | $N_s = \dfrac{N\sqrt{Q}}{(gH)^{3/4}}$ | Speed of a geometrically similar machine delivering unit flow at unit head. Decides the type: low $N_s$ radial, high $N_s$ axial |

#### Use in practice
1. **Similarity:** two machines are dynamically similar if $\phi$ and $\psi$ (and $Re$, $Ma$) are equal, and their efficiencies are then equal.
2. **Scaling laws:** for the same fluid and constant $\phi$: $Q \propto N D^3$, $H \propto N^2 D^2$, $P \propto N^3 D^5$.
3. **Selection of type:** a high $N_s$ means large flow and low head, which needs an axial machine; a low $N_s$ means small flow and high head, which needs a radial one.
4. **Performance prediction:** a single non-dimensional curve of $\psi$ and $\eta$ against $\phi$ represents a whole family of machines.

For compressible flow, the mass flow parameter $\dot m \sqrt{T_{01}}/p_{01}$ and the speed parameter $N/\sqrt{T_{01}}$ are used in place of $\phi$ and $N$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the flow coefficient phi = Q/(N D^3) physically represent?
   A. Ratio of inertia to viscous forces
   B. Ratio of meridional (axial) velocity to blade speed, fixing the velocity triangle shape
   C. Ratio of blade speed to sound speed
   D. Power per unit of rho N^3 D^5
   Answer: B. phi is proportional to flow velocity over blade speed, so it fixes the velocity triangles. (Key point: Flow coefficient)

2. A machine with a high specific speed is typically:
   A. Radial, for small flow and high head
   B. A positive displacement machine
   C. A reciprocating pump
   D. Axial, for large flow and low head
   Answer: D. High Ns means large flow and low head, which needs an axial machine; low Ns means radial. (Key point: Specific speed)

3. When are two geometrically similar machines dynamically similar?
   A. When phi and psi (and Re, Ma) are equal
   B. When their diameters are equal
   C. When their speeds are equal
   D. When their power is equal
   Answer: A. Equal flow and head coefficients (plus Re and Ma) give equal efficiencies and allow scaling. (Key point: Similarity)

4. For the same fluid and constant phi, how does power scale?
   A. P proportional to N D
   B. P proportional to N^2 D^2
   C. P proportional to N^3 D^5
   D. P proportional to N D^3
   Answer: C. From the power coefficient P/(rho N^3 D^5) constant, P is proportional to N^3 D^5. (Key point: Scaling laws)

5. Why is the Mach number important mainly for compressors and gas turbines?
   A. Because viscous friction dominates
   B. Because compressibility and shock losses appear at high blade speed relative to sound speed
   C. Because they have no blades
   D. Because the Reynolds number is zero
   Answer: B. Ma = ND/a compares blade speed to sound speed; compressible machines suffer shock loss when it gets high. (Key point: Mach number)

## Related in Turbo Machines

- [Stagnation properties and Mach number relations](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Astagnation-temperature-pressure-relations) — Turbo Machines, chapter 4
- [Rocket engine thrust and specific impulse calculation](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Arocket-thrust-specific-impulse-numerical) — Turbo Machines, chapter 5
- [Brayton cycle efficiency and optimum pressure ratio](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Abrayton-cycle-efficiency-optimum-pressure-ratio) — Turbo Machines, chapter 3

## Related in other subjects

- [Drag coefficient of cylinder vs Reynolds number](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Acylinder-drag-coefficient-reynolds-variation) — Fluid Mechanics, chapter 9, asked 2×
- [Reynolds number and flow regime in pipes](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Areynolds-number-flow-regime) — Applied Computational Fluid Dynamics, chapter 1
- [Speed of sound and Mach number](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Amach-number-speed-of-sound-regimes) — Fluid Mechanics, chapter 9
- [Head, specific speed and part load in turbine choice](https://www.bctengineer.com/question.md?key=semester_8%2Fpower_plant_design%3Aturbine-head-speed-part-load) — Power Plant Design, chapter 3