---
title: "Fourier law and thermal conductivity"
subject: "Heat Transfer"
chapter: 2
key: "semester_5/heat_transfer:fourier-law-thermal-conductivity"
source: https://www.bctengineer.com/q/semester_5/heat_transfer/fourier-law-thermal-conductivity
site: BCT Engineering
---

# Fourier law and thermal conductivity

Heat Transfer · Chapter 2: Conduction Heat Transfer

- Asked in: year not given, 5 marks
- Video: [Thermal Conductivity  ( fourier law of heat conduction )](https://www.youtube.com/watch?v=8I76NK3V1r0) — Techno Education Academy 
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/heat_transfer/fourier-law-thermal-conductivity
- Whole chapter: https://www.bctengineer.com/md/semester_5/heat_transfer/chapter-2

## Question

State Fourier's law of heat conduction. Define thermal conductivity and explain how it varies with temperature for metals, gases and insulating materials. Give typical values for copper, steel, water, air and glass wool.

## Short answer

Fourier's law: Q = -kA dT/dx. Thermal conductivity k is heat flow per unit area per unit temperature gradient. k of metals decreases with temperature; for gases and insulators it increases. Typical values: Cu 385 W/m·K, steel 45, water 0.6, air 0.026, glass wool 0.04.

## Answer

**Fourier's law:** the rate of heat conduction in a direction is proportional to the area normal to that direction and to the temperature gradient in it:

$$
Q_x = -kA\frac{dT}{dx}\qquad q'' = -k\frac{dT}{dx}
$$

The negative sign makes $Q$ positive in the direction of decreasing temperature (second law of thermodynamics).

#### Thermal conductivity
It is the heat flow per unit time through a unit area of a material of unit thickness when the temperature difference across it is 1 K: $k = \dfrac{Q\,L}{A\,\Delta T}$, in W/m K. It is a property of the material and measures how well it conducts heat.

#### Variation with temperature
- **Pure metals:** $k$ is high (free electrons) and decreases slightly as temperature rises, because lattice vibration scatters the electrons. Alloys and impurities reduce $k$ strongly; for alloys $k$ may rise with temperature.
- **Gases:** $k$ rises with temperature (about as $\sqrt{T}$), since molecular speed and collision energy exchange increase. Pressure has little effect.
- **Liquids:** $k$ generally falls with temperature (water is the exception, rising to about 130 °C).
- **Insulating solids:** $k$ increases with temperature and with moisture or density, because pore conduction and radiation across the voids increase.

#### Typical values (about room temperature)
| Material | $k$ (W/m K) |
|---|---|
| Copper | 385 |
| Mild steel | 45 |
| Water | 0.6 |
| Air | 0.026 |
| Glass wool | 0.04 |

## Check yourself

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

1. Fourier's law states that conduction rate is proportional to:
   A. Area and temperature itself
   B. Thickness and temperature difference squared
   C. Area normal to flow and temperature gradient
   D. Fluid velocity and area
   Answer: C. Q = -kA dT/dx. (Key point: Fourier's law)

2. How does k change with temperature for pure metals and for gases?
   A. Metals increase; gases decrease
   B. Both increase
   C. Both decrease
   D. Metals decrease slightly; gases increase
   Answer: D. In metals lattice vibration scatters electrons; in gases molecular speed rises with T. (Key point: k of metals and gases)

3. Why does k of an insulating solid rise with moisture and temperature?
   A. Pore conduction and radiation across voids increase
   B. Free electrons increase
   C. Lattice vibration scatters more
   D. Pressure falls
   Answer: A. Moisture and temperature increase heat transfer through the pores. (Key point: Insulating solids)

## Related in Heat Transfer

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- [Diffusion of water vapour through stagnant air](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Aevaporation-water-test-tube-stagnant-air) — Heat Transfer, chapter 7
- [Composite furnace wall heat loss](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Acomposite-furnace-wall-interface-temperatures) — Heat Transfer, chapter 2
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- [Conduction through hollow sphere](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Ahollow-sphere-heat-flow) — Heat Transfer, chapter 2

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