Rate of Heat Flow

SYLLABUS

Conduction: Thermal conductivity and measurement, Convection, Radiation: Ideal radiator, Black-body radiation, Stefan-Boltzmann law.

What is This Chapter About?

Touch a metal rod with one end in a fire and within seconds your hand feels hot. Hold a piece of wood in the same fire and the end you hold stays cool for much longer. Place your hand near (but not touching) a glowing electric heater and you feel warmth even across an air gap. These three experiences illustrate the three fundamentally different mechanisms by which heat energy travels from one place to another — conduction, convection, and radiation.

Chapter 12 of NEB Grade 11 Physics studies how heat flows — quantitatively — through and between bodies. It introduces the precise mathematical laws governing each mode of heat transfer and connects them to real-world applications ranging from building insulation to the temperature of the Sun. Carrying 5 teaching hours, this chapter has both strong theoretical content and rich practical relevance.

Key Concepts You Will Learn

Conduction

Thermal conduction is the transfer of heat through a material by direct molecular interaction, without any bulk movement of the material itself. When one end of a solid is heated, the molecules there vibrate more energetically and transfer energy to neighbouring molecules through collisions, gradually propagating the heat along the material.

The rate of heat flow by conduction through a slab of material is given by Fourier’s Law:

[latex]\frac{dQ}{dt} = -kA\frac{dT}{dx}[/latex]

where k is the coefficient of thermal conductivity of the material, A is the cross-sectional area, and [latex]\frac{dT}{dx}[/latex] is the temperature gradient — the rate of change of temperature with distance along the material. The negative sign indicates that heat flows in the direction of decreasing temperature.

The coefficient of thermal conductivity k characterises how well a material conducts heat. Metals have high k values (copper: ~400 W m⁻¹ K⁻¹) because free electrons carry thermal energy efficiently. Non-metals and insulators have low k values (glass wool: ~0.04 W m⁻¹ K⁻¹), which is why they are used as building insulation.

Searle’s Method is the standard experimental technique for measuring k for good conductors. A long metal bar is heated at one end by steam and cooled at the other end by flowing water. Thermometers at known positions along the bar measure the temperature gradient, and the flow rate and temperature rise of the cooling water measure the rate of heat transfer. Combining these measurements gives k precisely.

Convection

Thermal convection is the transfer of heat by the bulk movement of a fluid (liquid or gas). When a fluid is heated from below, it expands, becomes less dense, and rises — carrying thermal energy with it. Cooler, denser fluid sinks to replace it, creating a convection current that continuously circulates and distributes heat throughout the fluid.

Convection is the dominant mode of heat transfer in fluids and is responsible for ocean currents, atmospheric circulation, sea breezes, the boiling of water, and the convective zone of the Sun. It cannot occur in solids because the molecules cannot move from place to place.

Radiation

Thermal radiation is the transfer of heat by electromagnetic waves — primarily infrared radiation — that requires no medium and can travel through a vacuum. All objects at any temperature above absolute zero emit thermal radiation. The hotter the object, the more radiation it emits and the shorter the peak wavelength of that radiation (which is why very hot objects glow red, then orange, then white as temperature increases).

Absorption, Reflection, and Transmission

When radiation falls on a surface, it can be absorbed, reflected, or transmitted. The three coefficients satisfy: [latex]a + r + t = 1[/latex], where a is the absorptivity, r is the reflectivity, and t is the transmissivity. A perfect absorber (a = 1) is called a black body — it absorbs all radiation falling on it and reflects none. In practice, a small hole in a cavity acts as an ideal black body.

A perfect black body is also the most efficient emitter of radiation at every wavelength. The emissivity (e) of a surface measures how well it emits radiation compared to a black body at the same temperature — for a black body, e = 1; for all real surfaces, e < 1.

Stefan-Boltzmann Law

The total power radiated per unit area by a black body is given by the Stefan-Boltzmann Law:

[latex]P = \sigma T^4[/latex]

where [latex]\sigma = 5.67 \times 10^{-8}[/latex] W m⁻² K⁻⁴ is the Stefan-Boltzmann constant and T is the absolute temperature in kelvin. The fourth-power dependence on temperature is dramatic — doubling the absolute temperature increases the radiated power by a factor of 16. For a real body with emissivity e and surface area A, the net power radiated is:

[latex]P = e\sigma A(T^4 – T_0^4)[/latex]

where [latex]T_0[/latex] is the temperature of the surroundings.

Estimating the Temperature of the Sun

An elegant application of Stefan’s Law allows us to estimate the apparent surface temperature of the Sun. By measuring the total power received per unit area at the Earth’s surface (the solar constant, approximately 1360 W/m²) and knowing the distance from the Earth to the Sun and the Sun’s radius, we can calculate the total power radiated by the Sun and then use Stefan’s Law in reverse to find the temperature of the solar surface — approximately 5778 K. This beautiful calculation connects laboratory physics to astronomy.

Why This Chapter Matters for NEB Exams

Rate of Heat Flow is examined consistently in NEB Grade 11 Physics. Fourier’s Law of conduction — stating it, applying it, and solving numerical problems — is a standard question. Stefan-Boltzmann Law problems (calculating radiated power or temperature given radiated power) appear regularly. The distinction between the three modes of heat transfer, the definitions of absorptivity, reflectivity, and emissivity, and the principle of Searle’s method are all common short-answer topics. Estimating the Sun’s temperature using Stefan’s Law is a classic application problem.

Important Topics to Focus On

• Three modes of heat transfer — conduction, convection, radiation — with examples
• Fourier’s Law: [latex]\frac{dQ}{dt} = -kA\frac{dT}{dx}[/latex] — definition of thermal conductivity
• Temperature gradient and its role in conduction
• Searle’s method for measuring thermal conductivity — principle and procedure
• Coefficients of absorption, reflection, and transmission: [latex]a + r + t = 1[/latex]
• Ideal radiator (black body) — definition, emissivity = 1
• Stefan-Boltzmann Law: [latex]P = e\sigma A(T^4 – T_0^4)[/latex]
• Emissive power and emissivity of real surfaces
• Estimating the apparent temperature of the Sun using Stefan’s Law
• Numerical problems on thermal conductivity and Stefan’s Law

Connection to Other Chapters

Rate of Heat Flow builds on Chapter 9 (Heat and Temperature) — heat flows because of temperature differences — and Chapter 11 (Quantity of Heat) — the heat transferred is quantified using specific heat capacity and latent heat. The concept of radiation connects to Chapter 13 (Ideal Gas) through blackbody radiation and molecular energy, and to Grade 12 Chapter 21 (Photons), where the quantum nature of radiation is explored. Stefan’s Law connects forward to Grade 12 thermodynamics through the concept of entropy and disorder. The variation of density with temperature studied in Chapter 10 (Thermal Expansion) drives the convection currents described here.