B1-3. Conduction, Convection and Radiation

1. Thermal Conduction

Conduction is the primary method of thermal energy transfer in solids.

Thermal energy is transferred via atomic vibrations and, in metals, delocalised free electrons. The rate at which this heat flows depends on the material and dimensions.

Rate of Thermal Conduction:

$$\dfrac{\Delta Q}{\Delta t} = k A \dfrac{\Delta T}{\Delta x}$$

$\dfrac{\Delta Q}{\Delta t}$: Rate of heat transfer (Power, in W)
$k$: Thermal conductivity of the material (W m⁻¹ K⁻¹)
$A$: Cross-sectional area (m²)
$\dfrac{\Delta T}{\Delta x}$: Temperature gradient (K m⁻¹)

Example 1: Heat Loss Through a Window

Problem: A glass window has a width of $2.0$ m, a height of $1.5$ m, and a thickness of $5.0$ mm. The temperature inside the room is $20^\circ\text{C}$ and the temperature outside is $5^\circ\text{C}$. Given that the thermal conductivity of glass is $0.80$ W m⁻¹ K⁻¹, calculate the rate of heat transfer through the window.

Solution:

Step 1: Calculate the Area ($A$) and Temperature Difference ($\Delta T$).
$A = 2.0 \times 1.5 = 3.0$ m²
$\Delta T = 20 - 5 = 15$ K (Remember, a change in Celsius equals a change in Kelvin).
Step 2: Convert thickness ($\Delta x$) to SI units.
$\Delta x = 5.0$ mm $= 0.005$ m.
Step 3: Apply the Conduction formula.
$$\dfrac{\Delta Q}{\Delta t} = k A \dfrac{\Delta T}{\Delta x}$$
$$\dfrac{\Delta Q}{\Delta t} = (0.80) \times (3.0) \times \dfrac{15}{0.005}$$
$$\dfrac{\Delta Q}{\Delta t} = 2.4 \times 3000 = \mathbf{7200 \text{ W}}$$

2. Thermal Convection

Convection is the primary method of heat transfer in fluids (liquids and gases). Heating a fluid causes it to expand. This decreases its density ($\rho = \dfrac{m}{V}$), causing the hotter fluid to rise while cooler, denser fluid sinks, creating a continuous convection current.

Example 2: Observing Convection

One method of observing a convection current in the lab is by heating a beaker of water containing purple potassium permanganate crystals. Heat is initially transferred through the glass wall of the beaker by conduction. The water directly above the flame is heated, expands, becomes less dense, and rises. This causes the dissolved purple crystal to flow upwards. As the water reaches the top and cools, it becomes denser and falls back down, visibly tracing the convection current.

3. Thermal Radiation

Thermal radiation transfers energy via electromagnetic waves (primarily infrared) and does not require a medium—it can travel through a vacuum.

Stefan-Boltzmann Law:

This relates the total power emitted by a body to its temperature, surface area, and how perfectly it acts like a black body (emissivity).

$$P = e \sigma A T^4$$

$P$: Total Power emitted (W)
$e$: Emissivity (ranges from 0 to 1, where 1 is a perfect black body)
$\sigma$: Stefan-Boltzmann constant ($5.67 \times 10^{-8}$ W m⁻² K⁻⁴)
$A$: Surface area (m²)
$T$: Absolute surface temperature (K)

Example 3: Power Emitted by a Hot Sphere

Problem: A solid iron sphere of radius $0.10$ m is heated to a temperature of $500$ K in a vacuum. The emissivity of the iron surface is $0.80$. Calculate the total thermal power radiated by the sphere.