---
title: "Dimensional analysis of forced convection"
subject: "Heat Transfer"
chapter: 3
key: "semester_5/heat_transfer:buckingham-pi-forced-convection"
source: https://www.bctengineer.com/q/semester_5/heat_transfer/buckingham-pi-forced-convection
site: BCT Engineering
---

# Dimensional analysis of forced convection

Heat Transfer · Chapter 3: Convection Heat Transfer

- Asked in: year not given, 8 marks
- Video: [HT EPISODE 64 FORCED CONVECTION DIMENSIONAL ANALYSIS](https://www.youtube.com/watch?v=RmjuDWKRYg4) — Professor kartik
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/heat_transfer/buckingham-pi-forced-convection
- Whole chapter: https://www.bctengineer.com/md/semester_5/heat_transfer/chapter-3

## Question

Using Buckingham's pi theorem, show that for forced convection inside a tube the Nusselt number is a function of the Reynolds and Prandtl numbers only. The heat transfer coefficient h depends on the tube diameter D, fluid velocity V, density rho, viscosity mu, thermal conductivity k and specific heat cp. Also state the physical meaning of each dimensionless group.

## Short answer

With variables h, D, V, rho, mu, k, cp, dimensional analysis using repeating variables D, V, mu, k yields three pi groups: Nu = hD/k, Re = rho V D/mu, Pr = cp mu/k. Thus Nu = f(Re, Pr). Nu represents dimensionless temperature gradient, Re inertia/viscous forces, Pr momentum/thermal diffusivity ratio.

## Answer

#### Variables and dimensions
Fundamental dimensions: mass M, length L, time T, temperature $\theta$ (heat is expressed in mechanical units, so there are 4).

| Variable | Symbol | Dimensions |
|---|---|---|
| Heat transfer coefficient | $h$ | $M T^{-3}\theta^{-1}$ |
| Diameter | $D$ | $L$ |
| Velocity | $V$ | $L T^{-1}$ |
| Density | $\rho$ | $M L^{-3}$ |
| Viscosity | $\mu$ | $M L^{-1}T^{-1}$ |
| Conductivity | $k$ | $M L T^{-3}\theta^{-1}$ |
| Specific heat | $c_p$ | $L^2T^{-2}\theta^{-1}$ |

$n = 7$ variables, $m = 4$ fundamental dimensions, so the number of $\pi$ groups is $n - m = 3$.

#### Repeating variables
Choose $D$, $V$, $\mu$, $k$ (together they contain $L, T, M, \theta$ and are independent). Each non-repeating variable ($h$, $\rho$, $c_p$) forms one group.

**Group 1 with $h$:** $\pi_1 = h\,D^aV^b\mu^ck^d$. Equating exponents to zero:
- $M$: $1 + c + d = 0$
- $L$: $a + b - c + d = 0$
- $T$: $-3 - b - c - 3d = 0$
- $\theta$: $-1 - d = 0 \Rightarrow d = -1$

Then $c = 0$, $b = 0$ and $a = 1$, so $\pi_1 = \dfrac{hD}{k} = Nu$.

**Group 2 with $\rho$:** the same procedure gives $\pi_2 = \dfrac{\rho V D}{\mu} = Re$.

**Group 3 with $c_p$:** $\pi_3 = \dfrac{c_p\mu}{k} = Pr$.

#### Result

$$
\pi_1 = f(\pi_2,\pi_3)\;\Rightarrow\; Nu = f(Re, Pr)
$$

Experiments give the form $Nu = C\,Re^m Pr^n$, for example Dittus-Boelter $Nu = 0.023\,Re^{0.8}Pr^{n}$.

#### Physical meaning

| Group | Meaning |
|---|---|
| Nusselt, $hD/k$ | Ratio of convective to conductive heat transfer in the fluid layer; dimensionless temperature gradient at wall |
| Reynolds, $\rho VD/\mu$ | Ratio of inertia to viscous forces; decides laminar or turbulent flow |
| Prandtl, $\mu c_p/k$ | Ratio of momentum diffusivity to thermal diffusivity; a fluid property |

## Check yourself

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

1. How many pi groups are formed and why?
   A. 4 = 7 variables - 3 dimensions
   B. 3 = 7 variables - 4 dimensions
   C. 7 = number of variables
   D. 2 = 4 - 2
   Answer: B. n - m = 7 - 4 = 3. (Key point: Number of groups)

2. What is required of repeating variables?
   A. They must include h
   B. They must be dimensionless
   C. Together contain all fundamental dimensions and are independent
   D. They must be the dependent variables
   Answer: C. D, V, mu, k cover M, L, T, theta. (Key point: Repeating variables)

3. The three groups are:
   A. Nu, Gr, Pr
   B. St, Re, Gr
   C. Nu, Re, Gr
   D. Nu, Re, Pr
   Answer: D. hD/k, rho V D/mu, cp mu/k. (Key point: Groups)

4. Prandtl number is the ratio of:
   A. Momentum diffusivity to thermal diffusivity
   B. Inertia to viscous force
   C. Convection to conduction
   D. Buoyancy to viscous force
   Answer: A. Pr = nu/alpha, a fluid property. (Key point: Meaning of Pr)

## 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
- [Dimensionless numbers in convection](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Asignificance-dimensionless-numbers-convection) — Heat Transfer, chapter 3
- [Newton's law of cooling and convection coefficient](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Anewtons-law-cooling-convection-coefficient) — 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
- [Water heating in a tube, Dittus-Boelter](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Adittus-boelter-water-heating-tube-length) — 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

## Related in other subjects

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- [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×
- [Insulating layer: minimum outside heat transfer coefficient](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Ainsulating-layer-min-outside-h-300c) — Fundamental of Thermodynamics & Heat Transfer, chapter 7, asked 2×
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