---
title: "Natural convection from a vertical plate"
subject: "Heat Transfer"
chapter: 3
key: "semester_5/heat_transfer:natural-convection-vertical-plate-heat-loss"
source: https://www.bctengineer.com/q/semester_5/heat_transfer/natural-convection-vertical-plate-heat-loss
site: BCT Engineering
---

# Natural convection from a vertical plate

Heat Transfer · Chapter 3: Convection Heat Transfer

- Asked in: year not given, 8 marks
- Video: [Free Convection over a Vertical Plate](https://www.youtube.com/watch?v=kNT88CcpLyM) — LearnChemE
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/heat_transfer/natural-convection-vertical-plate-heat-loss
- Whole chapter: https://www.bctengineer.com/md/semester_5/heat_transfer/chapter-3

## Question

A vertical plate 0.5 m high and 0.8 m wide is maintained at 90 °C in still air at 30 °C. Using properties at the film temperature of 60 °C (nu = 18.97 x 10^-6 m^2/s, k = 0.02896 W/m K, Pr = 0.7202, beta = 1/Tf), calculate (a) the Grashof and Rayleigh numbers, (b) the average Nusselt number and heat transfer coefficient, and (c) the heat lost from one side of the plate. Use Nu = 0.59 Ra^(1/4) for 10^4 < Ra < 10^9. Also explain the physical meaning of the Grashof number.

## Short answer

Grashof number Gr = 6.14×10^8 (ratio of buoyancy to viscous force), Rayleigh number Ra = 4.42×10^8; using Nu = 0.59 Ra^1/4 gives Nu = 85.5, average h = 4.95 W/m^2·K, heat loss from one side = 119 W.

## Answer

#### Grashof number: physical meaning
$Gr = \dfrac{g\beta\,\Delta T\,L^3}{\nu^2}$ is the ratio of buoyancy force to viscous force acting on the fluid. In natural convection it plays the role that the Reynolds number plays in forced convection. A higher $Gr$ means stronger buoyancy-driven motion and higher $h$.

#### (a) Grashof and Rayleigh numbers
$T_f = 60$ °C $= 333.15$ K, so $\beta = 1/333.15 = 3.0\times10^{-3}\ \text{K}^{-1}$.

$$
\begin{aligned}
Gr_L &= \frac{9.81\times(1/333.15)\times(90-30)\times(0.5)^3}{(18.97\times10^{-6})^2} = 6.14\times10^{8}\\
Ra_L &= Gr_L\,Pr = 6.14\times10^8\times0.7202 = 4.42\times10^{8}
\end{aligned}
$$

$Ra < 10^9$, so the flow is laminar and the given correlation is valid.

#### (b) Nusselt number and $h$

$$
Nu = 0.59\,Ra^{1/4} = 0.59\,(4.42\times10^8)^{0.25} = 0.59\times145.0 = 85.5
$$

$$
\bar h = \frac{Nu\,k}{L} = \frac{85.5\times0.02896}{0.5} = 4.95\ \text{W/m}^2\text{K}
$$

#### (c) Heat loss

$$
Q = \bar h A\,\Delta T = 4.95\times(0.5\times0.8)\times60 = 119\ \text{W}
$$

**Answer:** $Gr = 6.14\times10^8$, $Ra = 4.42\times10^8$; $Nu = 85.5$; $\bar h = 4.95\ \text{W/m}^2$K; $Q \approx 119$ W.

## Check yourself

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

1. Grashof number is the ratio of:
   A. Inertia to viscous force
   B. Buoyancy to viscous force
   C. Convection to conduction
   D. Momentum to thermal diffusivity
   Answer: B. Gr is the natural-convection analogue of Re. (Key point: Grashof meaning)

2. What is beta for air as an ideal gas?
   A. 1/Tf with Tf in Celsius
   B. Tf
   C. 1/Tf with Tf in kelvin
   D. 273/Tf
   Answer: C. beta = 1/333.15 K^-1. (Key point: Beta)

3. Ra is:
   A. Gr / Pr
   B. Re x Pr
   C. Gr x Re
   D. Gr x Pr = 4.42 x 10^8
   Answer: D. 0.7202 x 6.14e8 = 4.42e8, below 1e9 so laminar. (Key point: Rayleigh number)

4. Heat loss from one side is about:
   A. 119 W
   B. 238 W
   C. 4.95 W
   D. 1500 W
   Answer: A. Q = 4.95 x 0.4 x 60 = 119 W. (Key point: Heat loss)

## Related in Heat Transfer

- [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
- [Condensation of steam on a vertical plate](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Afilm-condensation-vertical-plate-numerical) — Heat Transfer, chapter 5
- [Natural and forced convection compared](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Afree-versus-forced-convection) — Heat Transfer, chapter 3
- [Heat loss from a cylinder in cross flow](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Across-flow-air-cylinder-heat-loss) — Heat Transfer, chapter 3

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