---
title: "Dimensionless numbers in convection"
subject: "Heat Transfer"
chapter: 3
key: "semester_5/heat_transfer:significance-dimensionless-numbers-convection"
source: https://www.bctengineer.com/q/semester_5/heat_transfer/significance-dimensionless-numbers-convection
site: BCT Engineering
---

# Dimensionless numbers in convection

Heat Transfer · Chapter 3: Convection Heat Transfer

- Asked in: year not given, 5 marks
- Video: [Convective heat transfer - Dimensionless numbers](https://www.youtube.com/watch?v=Cm_Ggi9_FtY) — R. Paul Singh
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/heat_transfer/significance-dimensionless-numbers-convection
- Whole chapter: https://www.bctengineer.com/md/semester_5/heat_transfer/chapter-3

## Question

Define Reynolds, Prandtl, Nusselt, Grashof and Stanton numbers. State the physical significance of each and the relation between them in forced and natural convection.

## Short answer

Re = VL/ν (inertia/viscous), Pr = ν/α (momentum/thermal diffusivity), Nu = hL/k (convection/conduction), Gr = gβΔT L^3/ν^2 (buoyancy/viscous), St = h/(ρVc_p) (heat transferred/thermal capacity). Forced convection: Nu = f(Re,Pr); natural: Nu = f(Gr,Pr) with Ra = Gr Pr.

## Answer

Dimensionless numbers let results from one experiment be applied to similar situations of different size, fluid or velocity.

| Number | Definition | Physical significance |
|---|---|---|
| Reynolds, $Re$ | $\dfrac{\rho V L}{\mu} = \dfrac{VL}{\nu}$ | Inertia force / viscous force. Decides laminar or turbulent flow |
| Prandtl, $Pr$ | $\dfrac{\mu c_p}{k} = \dfrac{\nu}{\alpha}$ | Momentum diffusivity / thermal diffusivity. Fluid property; compares thickness of velocity and thermal layers |
| Nusselt, $Nu$ | $\dfrac{hL}{k}$ | Convection / conduction in the fluid layer of thickness $L$; dimensionless heat transfer coefficient |
| Grashof, $Gr$ | $\dfrac{g\beta\Delta T L^3}{\nu^2}$ | Buoyancy force / viscous force in natural convection |
| Stanton, $St$ | $\dfrac{h}{\rho V c_p} = \dfrac{Nu}{Re\,Pr}$ | Heat transferred to the fluid / thermal capacity of the flowing fluid |

#### Relations
- **Forced convection:** $Nu = f(Re, Pr)$, e.g. $Nu = 0.023\,Re^{0.8}Pr^{0.4}$ for turbulent flow in tubes.
- **Natural convection:** $Nu = f(Gr, Pr)$, usually written $Nu = C\,(Gr\,Pr)^m$, where the product $Gr\,Pr$ is the Rayleigh number $Ra$.
- **Mixed convection:** $Gr/Re^2$ near 1 shows both effects are important.
- $St\,Pr^{2/3} = f/2$ is the Colburn analogy linking heat transfer with skin friction.

## Check yourself

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

1. Nusselt number represents:
   A. Inertia to viscous force
   B. Buoyancy to viscous force
   C. Heat transferred to thermal capacity
   D. Convection to conduction in the fluid layer
   Answer: D. Nu = hL/k. (Key point: Nusselt)

2. Stanton number St equals:
   A. Nu/(Re Pr)
   B. Nu Re Pr
   C. Re Pr/Nu
   D. Gr/Re^2
   Answer: A. St = h/(rho V cp) = Nu/(Re Pr). (Key point: Stanton)

3. Natural convection correlations are usually written:
   A. Nu = C Re^m Pr^n
   B. Nu = C (Gr Pr)^m
   C. Nu = C St^m
   D. Nu = C Gr/Re
   Answer: B. Gr Pr is the Rayleigh number. (Key point: Natural convection relation)

4. When are both natural and forced convection important?
   A. Gr/Re^2 much less than 1
   B. Re = 0
   C. Gr/Re^2 near 1
   D. Pr = 0
   Answer: C. Gr/Re^2 near 1 means buoyancy and forced flow are comparable. (Key point: Mixed convection)

## Related in Heat Transfer

- [Natural and forced convection compared](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Afree-versus-forced-convection) — Heat Transfer, chapter 3
- [Dimensional analysis of forced convection](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Abuckingham-pi-forced-convection) — Heat Transfer, chapter 3
- [Natural convection from a vertical plate](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Anatural-convection-vertical-plate-heat-loss) — Heat Transfer, chapter 3
- [Laminar forced convection over a heated flat plate](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Alaminar-flow-over-heated-plate-air) — Heat Transfer, chapter 3

## Related in other subjects

- [Dimensionless parameters and their significance](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Adimensionless-parameters-turbomachines) — Turbo Machines, chapter 1
- [Dimensionless numbers and types of similarity](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Adimensionless-numbers-similitude-types) — Fluid Mechanics, chapter 5
- [Forced response depends on input, natural does not](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aforced-vs-natural-response-example) — Electrical Circuit Theory, chapter 3
- [Natural vs forced commutation](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Anatural-vs-forced-commutation) — Power Electronics, chapter 1