1. Conduction free study notes

1. What is Conduction?

Conduction is a mode of heat transfer in which thermal energy is transferred from a region of higher temperature to a region of lower temperature without any macroscopic movement of the material as a whole. It occurs due to molecular interactions and the movement of free electrons, primarily in solids.

Conduction is the dominant mode of heat transfer in solids, less significant in liquids, and least effective in gases.

Examples:

Heat flow through a metal rod
Heating of a spoon placed in hot tea

2. Physical Mechanism of Conduction

The mechanism of heat conduction depends on the state of matter:

a) In Solids

Heat transfer occurs due to:
- Lattice vibrations (phonons) in non-metallic solids
- Free electrons in metallic solids
Metals are good conductors because free electrons transport energy rapidly.

b) In Liquids

Heat conduction occurs due to molecular collisions.
Since molecules are loosely packed, conduction is weaker compared to solids.

c) In Gases

Heat transfer occurs due to collisions between gas molecules.
Large intermolecular spacing results in poor conduction.

3. Temperature Gradient

The temperature gradient is the rate of change of temperature with respect to distance in a particular direction. Heat conduction takes place only when there is a temperature gradient, i.e., change of temperature with distance. Temperature gradient is essential to Fourier’s law.

Mathematically we can represent temperature gradient as following

$\text{Temperature gradient} = \frac{dT}{dx}$ Where:

$x$ = distance in the direction of heat flow
T = temperature

Note:

Heat always flows in the direction of negative temperature gradient
Higher the gradient, higher the rate of heat transfer

4. Fourier’s Law of Heat Conduction

The fundamental law governing conduction was proposed by Joseph Fourier.

Fourier’s Law states that “The rate of heat transfer by conduction is directly proportional to the area normal to the direction of heat flow and the temperature gradient”.

Mathematically, $Q = -kA \frac{dT}{dx}$

Where:

$Q$ = Rate of heat transfer (W)
$k$ = Thermal conductivity of the material (W/m·K)
$A$ = Area perpendicular to heat flow (m²)
$\frac{dT}{dx}$ = Temperature gradient
Negative sign indicates heat flows from high to low temperature

5. Thermal Conductivity (k)

Thermal conductivity (k) is defined as the amount of heat transferred per unit time through a unit area of a material of unit thickness when a unit temperature gradient exists. Thermal conductivity is a material property that indicates its ability to conduct heat. It is different for different materials.

Unit of Thermal Conductivity

$k = \frac{W}{m \cdot K}$ Watt per meter per Kelvin (W/m·K)

Characteristics

Higher $k$ → Better conductor
Lower $k$ → Better insulator

Typical Values

Material	Thermal Conductivity (W/m·K)
Copper	~385
Aluminum	~205
Steel	~45
Brick	~0.6
Glass Wool	~0.04

6. One-Dimensional Steady-State

In many engineering problems, heat flow is assumed to be:

One-dimensional
Steady-state (temperature does not change with time)
No internal heat generation

For a plane wall of thickness $L$ : $Q = \frac{kA(T_1 – T_2)}{L}$

Where:

$T_1$ and $T_2$ are surface temperatures

7. Thermal Resistance Concept

Conduction heat transfer can be analyzed using the thermal resistance analogy, similar to electrical resistance. $R_{cond} = \frac{L}{kA}$ $Q = \frac{(T_1 – T_2)}{R_{cond}}$

This concept is useful in analyzing composite walls and multi-layer insulation systems.

8. Conduction Through Different Geometries

a) Plane Wall

$Q = \frac{kA(T_1 – T_2)}{L}$

b) Hollow Cylinder

$Q = \frac{2\pi k L (T_1 – T_2)}{\ln(r_2/r_1)}$

c) Hollow Sphere

$Q = \frac{4\pi k (T_1 – T_2)}{\left(\frac{1}{r_1} – \frac{1}{r_2}\right)}$

9. Factors Affecting Heat Conduction

Temperature difference
Thermal conductivity of material
Cross-sectional area
Length or thickness of the material
Nature and structure of the material

10. Engineering Applications of Conduction

Heat flow through boiler walls and furnace linings
Design of insulation materials
Heat transfer in engine components
Cooling of electrical and electronic devices
Heat loss calculation in buildings and pipelines