Thermal Energy Transfer

Thermal Energy Transfer

Info: Board Coverage AQA Paper 2 | Edexcel CP9/CB6 | OCR (A) Paper 2 | CIE P3

Energy moves from hot to cold by three mechanisms: conduction, convection and radiation. This page consolidates the transfer mechanisms, U-values, and the latent-heat calculations that sit alongside the specific heat capacity work in Thermal Properties.

1. Conduction

Conduction is the transfer of energy through a material without any bulk movement of the material itself. In metals, the dominant mechanism is free (delocalised) electrons colliding with the lattice; in non-metals, energy passes through lattice vibrations (phonons). This is why metals conduct heat far better than insulators: the same free electrons that carry current carry thermal energy.

The Conduction Equation (Fourier’s Law)

The rate of energy flow through a slab of material is

where $k$ is the thermal conductivity ($\mathrm{W\,m^{-1}\,K^{-1}}$), $A$ the cross-sectional area, $d$ the thickness, and $\Delta\theta$ the temperature difference across the slab.

Worked example. A window pane has area $2.0\ \mathrm{m^2}$, thickness $4.0\ \mathrm{mm}$ and $k = 0.80\ \mathrm{W\,m^{-1}K^{-1}}$. The inside is at $20\ ^\circ\mathrm{C}$, the outside at $5.0\ ^\circ\mathrm{C}$.

In reality the glass surfaces are not at room temperature; the temperature drops mostly across boundary air layers. That is exactly why double glazing works: a trapped argon gap with low $k$ adds a large effective thickness.

2. Convection

Convection requires a fluid. The heated fluid expands, becomes less dense, and rises; cooler fluid replaces it, forming a convection current. Energy is then carried by the bulk motion of the fluid itself.

• Natural convection: driven by density differences alone (radiator warming a room).
• Forced convection: driven by an external pump or fan (a fan heater, or coolant pumped through a nuclear reactor — see Nuclear Power and Safety).

The rate of transfer is commonly modelled as

with $h$ a heat transfer coefficient. You are rarely asked to calculate with $h$ at A-level, but you are expected to explain designs that reduce convection: trapping air in foam or fibreglass works because air is a poor conductor and the fibres prevent the air from circulating.

3. Radiation

All bodies emit electromagnetic radiation. The power radiated by a body at Kelvin temperature $T$ is given by the Stefan—Boltzmann law in the form used at A-level:

where $e$ is the emissivity ($0 \le e \le 1$; $e = 1$ for a perfect black body) and $\sigma = 5.67\times10^{-8}\ \mathrm{W\,m^{-2}K^{-4}}$ is the Stefan—Boltzmann constant.

For an object in surroundings at temperature $T_s$, the net exchange is

Key qualitative points examiners award marks for:

• Emission depends on $T^4$: doubling the Kelvin temperature multiplies the radiated power by 16. Using the Celsius temperature here is one of the most common errors in the paper.
• Matte black surfaces are the best absorbers and the best emitters; shiny silver surfaces are the worst at both.
• The Earth’s radiative equilibrium temperature follows from balancing absorbed solar flux against $e\sigma T^4$ — a standard application question.

4. U-Values

For a building element, the U-value bundles conduction through all layers into one figure:

A double-glazed unit might have $U \approx 1.5\ \mathrm{W\,m^{-2}K^{-1}}$ against $U \approx 5.0$ for single glazing. Lower U-value means better insulation. Exam questions typically give several layers in series and ask you to compare energy loss before and after an upgrade, then comment on cost savings from a stated energy price.

5. Specific Latent Heat and Phase Changes

During a phase change the temperature of a pure substance is constant even though energy is still being supplied: the energy goes into breaking intermolecular bonds, not into kinetic energy. The energy required is

with $L_f$ the specific latent heat of fusion (solid $\leftrightarrow$ liquid) and $L_v$ the specific latent heat of vaporisation (liquid $\leftrightarrow$ gas). For water, $L_v \approx 2.26\times10^{6}\ \mathrm{J\,kg^{-1}}$ — roughly seven times $L_f$, which is why scalding steam burns are so severe: the steam releases $mL_v$ plus the cooling water releases $m c \Delta\theta$ as it condenses and cools.

Worked example. An electric kettle delivers $2.5\ \mathrm{kW}$ to $0.30\ \mathrm{kg}$ of boiling water. How long to boil it dry? ($L_v = 2.26\times10^{6}\ \mathrm{J\,kg^{-1}}$)

Mixing Problems (Calorimetry)

Multi-stage problems chain latent heat with specific heat capacity. A standard ice-to-steam calculation has five stages: warm the ice to $0\ ^\circ\mathrm{C}$, melt it, heat the water to $100\ ^\circ\mathrm{C}$, boil it, superheat the steam. Set total energy supplied equal to the sum over stages and solve for the unknown. Always compute and compare stage-by-stage when the question asks “does the ice all melt?” — answer that sub-question explicitly first.

6. Common Pitfalls

• Celsius vs Kelvin: $T^4$ radiation laws and the ideal gas law need Kelvin. Specific heat questions only need differences, where Celsius is fine.
• Area vs radius: radiation questions give a diameter. $A = 4\pi r^2$ needs $r = d/2$.
• Latent heat has no $\Delta\theta$: writing $Q = mL\Delta\theta$ in a phase-change stage is an automatic mark loss.
• Direction of energy flow: in net-radiation questions, quote the sign convention you are using (net loss from the object vs net gain by the surroundings).
• Conduction in a steady state: the rate is the same through every layer in series; the temperature differences distribute in proportion to the thermal resistances.

7. Summary

Phase changes: $Q = mL$ at constant temperature. Mixing problems: sum the stages.

Cross-References

• Thermal Properties — specific heat capacity, gas laws and kinetic theory
• Thermodynamics — first law, $p$-$V$ work and the Carnot cycle
• Nuclear Power and Safety — forced-convection coolant loops in reactors