Unit 12: Rate of Heat Flow
Conduction
Conduction is the transfer of thermal energy through a material without any bulk movement of the substance. Energy is passed from particle to particle via collisions and lattice vibrations, making it the dominant mode in solids.
Fourier’s Law of Heat Conduction
The rate of heat transfer Q/t through a slab of uniform cross‑sectional area A and thickness dx is proportional to the temperature gradient dT/dx:
Q/t = -k A (dT/dx)
where:
- k = thermal conductivity of the material (W·m⁻¹·K⁻¹)
- A = cross‑sectional area perpendicular to heat flow (m²)
- dT/dx = temperature gradient along the direction of heat flow (K·m⁻¹)
- The negative sign indicates heat flows from higher to lower temperature.
Thermal Conductivity (k)
Thermal conductivity is an intrinsic property that quantifies a material’s ability to conduct heat. Metals exhibit high k due to free electron transport, whereas insulators have low k because energy transfer relies on slower phonon interactions.
Measurement Techniques
- Searle’s Bar Method: A metal bar is heated at one end and cooled at the other; steady‑state temperature distribution is measured to calculate
kusing Fourier’s law. - Lee’s Disc Method: Suitable for poor conductors; a disc of the material is placed between a heated metal plate and a conducting bottom plate. The rate of cooling of the disc yields
k.
Examples of Conductors and Insulators
| Good Conductors (high k) | Poor Conductors / Insulators (low k) |
|---|---|
| Copper (Cu) – ~400 W·m⁻¹·K⁻¹ | Wood – ~0.12 W·m⁻¹·K⁻¹ |
| Aluminium (Al) – ~235 W·m⁻¹·K⁻¹ | Rubber – ~0.16 W·m⁻¹·K⁻¹ |
| Silver (Ag) – ~429 W·m⁻¹·K⁻¹ | Glass – ~0.8–1.0 W·m⁻¹·K⁻¹ |
| Iron (Fe) – ~80 W·m⁻¹·K⁻¹ | Air (still) – ~0.024 W·m⁻¹·K⁻¹ |
Convection
Convection involves the transfer of heat by the macroscopic movement of fluid (liquid or gas). The fluid carries thermal energy from one region to another, enhancing the overall heat transfer rate compared to pure conduction.
Types of Convection
- Natural (Free) Convection: Driven by buoyancy forces arising from density differences caused by temperature gradients. Example: warm air rising near a heater.
- Forced Convection: Induced by an external agent such as a fan, pump, or wind. Example: air forced over a car radiator.
Governing Concept (Newton’s Law of Cooling)
For many engineering applications, the convective heat flux q'' is expressed as:
q'' = h (T_s - T_∞)
where:
- h = convective heat transfer coefficient (W·m⁻²·K⁻¹)
- T_s = surface temperature (K)
- T_∞ = bulk fluid temperature far from the surface (K)
Examples
- Sea Breeze: During the day, land heats faster than water; warm air over land rises, drawing cooler air from the sea—a natural convection cycle.
- Central Heating System: Hot water pumped through radiators transfers heat to room air via forced convection (fan‑assisted) and natural convection currents.
- Radiator in a Car: Engine coolant transfers heat to the radiator tubes; air forced by the fan removes heat from the fins, cooling the engine.
Radiation
Radiation is the transfer of energy by electromagnetic waves. Unlike conduction and convection, it does not require a material medium and can occur through a vacuum.
Emission and Absorption
All bodies with a temperature above absolute zero (0 K) emit thermal radiation. Simultaneously, they absorb incident radiation. The balance between emission and absorption determines the net radiative exchange.
Perfect (Ideal) Radiator
An ideal radiator, also called a black body, absorbs all incident radiation regardless of wavelength or direction and emits radiation with the maximum possible intensity for its temperature.
Black‑Body Radiation
A black body is both a perfect absorber and a perfect emitter. Its radiative properties depend solely on temperature, not on the material composition or shape.
Emissivity (e)
Emissivity quantifies how closely a real surface approximates a black body:
e = (E_actual) / (E_blackbody), 0 ≤ e ≤ 1
where E_actual is the emissive power of the surface and E_blackbody is that of an ideal black body at the same temperature.
Wien’s Displacement Law
The wavelength at which the emission spectrum of a black body peaks (λ_max) is inversely proportional to its absolute temperature:
λ_max T = b, b ≈ 2.898 × 10⁻³ m·K
Thus, hotter objects emit peak radiation at shorter wavelengths (e.g., the Sun peaks in the visible range, while a room‑temperature object peaks in the infrared).
Stefan‑Boltzmann Law
The total power radiated per unit area from a black body is proportional to the fourth power of its absolute temperature. For a real surface, emissivity modifies this relationship.
Radiative Power Emitted
P = e σ A T⁴
where:
- P = net radiative power emitted (W)
- e = emissivity of the surface (dimensionless)
- σ = Stefan‑Boltzmann constant =
5.67 × 10⁻⁸ W·m⁻²·K⁻⁴ - A = surface area emitting radiation (m²)
- T = absolute temperature of the surface (K)
Net Radiative Exchange Between Two Surfaces
When a surface at temperature T exchanges radiation with surroundings at temperature T_s, the net power is:
P_net = e σ A (T⁴ - T_s⁴)
If T > T_s, the surface loses heat; if T < T_s, it gains heat.
Practical Example
Consider a metal plate of area 0.5 m² with emissivity e = 0.8 at T = 400 K in an environment at T_s = 300 K:
P_net = 0.8 × (5.67×10⁻⁸) × 0.5 × (400⁴ - 300⁴)≈ 0.8 × 5.67×10⁻⁸ × 0.5 × (2.56×10¹⁰ - 8.1×10⁹)≈ 0.8 × 5.67×10⁻⁸ × 0.5 × 1.75×10¹⁰≈ 0.8 × 5.67×10⁻⁸ × 8.75×10⁹≈ 0.8 × 496.125≈ 397 W
Thus, the plate radiates roughly 400 W to its surroundings.
Summary of Key Equations
| Concept | Formula | Variables |
|---|---|---|
| Fourier’s Law (Conduction) | Q/t = -k A (dT/dx) | k = thermal conductivity, A = area, dT/dx = temp. gradient |
| Newton’s Law of Cooling (Convection) | q'' = h (T_s - T_∞) | h = convective coefficient, T_s = surface temp., T_∞ = fluid temp. |
| Wien’s Displacement Law | λ_max T = b | b = 2.898×10⁻³ m·K |
| Stefan‑Boltzmann Law | P = e σ A T⁴ | e = emissivity, σ = 5.67×10⁻⁸ W·m⁻²·K⁻⁴, A = area, T = temp. |
| Net Radiation | P_net = e σ A (T⁴ - T_s⁴) | T_s = surroundings temp. |