"Heat Transfer and Solar Energy"

About

Heat always flows from a hotter body to a colder body. But HOW does it travel? This chapter explores the three modes of heat transfer — conduction (through solids), convection (through fluids), and radiation (through electromagnetic waves, even in vacuum). You will also learn about solar energy, the ultimate source of most energy on Earth.

Key Concepts

12.1 Modes of Heat Transfer

12.2 Conduction

Heat transfer through a material without the movement of particles as a whole. Particles vibrate and pass energy to neighbouring particles.

Rate of heat flow (Fourier's law): $\frac{Q}{t}=\frac{KA(T_h-T_c)}{d}$

Where:

• $K$ = thermal conductivity (SI: J⋅s⁻¹⋅m⁻¹⋅°C⁻¹ or W⋅m⁻¹⋅K⁻¹)
• $A$ = cross-sectional area
• $d$ = thickness
• $T_h-T_c$ = temperature difference

Definition of $K$: When $A=1$ m², $d=1$ m, $\Delta T=1$°C, and $t=1$ s, then $Q=K$ joules. $K$ is the heat conducted per second across a unit cube with unit temperature difference.

Good conductors: Metals (copper, silver, aluminium — high $K$) Poor conductors (insulators): Wood, wool, glass, air — low $K$

Why wool keeps us warm: Wool traps a large amount of air between its fibres. Air is a poor conductor, so body heat does not escape easily. The trapped air layer insulates the body.

12.3 Convection

Heat transfer through the actual movement of fluid particles. Warmer (lighter) fluid rises, cooler (denser) fluid sinks — setting up convection currents.

Sea breeze (daytime):

• Land heats up faster than ocean (lower specific heat)
• Air over land becomes hot and rises → low pressure
• Cool air from ocean moves toward land → sea breeze

Land breeze (nighttime):

• Land cools faster than ocean
• Air over ocean is warmer and rises
• Cool air from land moves toward sea

12.4 Radiation

Heat transfer through electromagnetic waves — no medium required. All bodies above 0 K emit thermal radiation.

Stefan-Boltzmann law: $E=\sigma T^4$

Where $\sigma =5.67\times 10^{-8}$ W⋅m⁻²⋅K⁻⁴ (Stefan-Boltzmann constant).

For a body that is not a perfect black body: $E=ϵ\sigma T^4$ ($ϵ$ = emissivity, 0 < $ϵ$ ≤ 1).

Wien's displacement law: ${\lambda }_{m}T=b$

Where $b=2.884\times 10^{-3}$ m⋅K (Wien's constant).

• ${\lambda }_m$ is the wavelength at which maximum radiation is emitted
• Hotter bodies emit at shorter wavelengths
• At 300 K (room temp): ${\lambda }_m\approx 9.6$ µm (infrared)

Colour and radiation absorption:

• Dark colours absorb more radiation → feel warmer in sunlight
• Light colours reflect most radiation → stay cooler
• That's why light-coloured clothing is preferred in summer

12.5 Solar Energy

The Sun radiates energy at the rate of about $3.8\times 10^{26}$ W. The solar constant (energy received per unit area per second at Earth's outer atmosphere) is approximately 1.36 kW/m².

Applications of solar energy:

• Solar cookers and water heaters
• Solar cells (photovoltaic panels)
• Solar furnaces
• Passive solar heating in buildings

INTEXT QUESTIONS 12.1

Q1. Distinguish between conduction and convection.

Ans:

Q2. Verify that the units of K are J s⁻¹ m⁻¹ °C⁻¹.

Ans: From Fourier's law: $K=\frac{Q\cdot d}{A\cdot t\cdot (T_h-T_c)}$

Units: $\frac{\text{J}\cdot \text{m}}{{\text{m}}^2\cdot \text{s}\cdot \text{°C}}=\frac{\text{J}}{\text{s}\cdot \text{m}\cdot \text{°C}}={\text{J⋅s}}^{-1}{\text{⋅m}}^{-1}{\text{⋅°C}}^{-1}$

Q3. Explain why do humans wrap themselves in woollens in winter season?

Ans: Wool is a very good insulator because it traps a large amount of air between its fibres. Air is a poor conductor of heat, so body heat does not escape easily. The trapped air layer around the body prevents heat loss and shields from cold external temperature. Wool also feels warm and provides comfortable covering.

Q4. A cubical slab of surface area 1 m², thickness 1 m, and made of a material of thermal conductivity K. The opposite faces of the slab are maintained at 1°C temperature difference. Compute the energy transferred across the surface in one second and hence give a numerical definition of K.

Ans: $A=1$ m², $d=1$ m, $T_h-T_c=1$°C, $t=1$ s

$\frac{Q}{t}=\frac{KA(T_h-T_c)}{d}⟹Q=K\cdot 1\cdot 1\cdot 1=K\text{ J}$

$K$ is numerically equal to the heat conducted per second across a unit cube of the material when its opposite faces are maintained at a unit temperature difference.

Q5. During the summer, the land mass gets very hot. But the air over the ocean does not get as hot. This results in the onset of sea breezes. Explain.

Ans: Land heats up faster than ocean because land has lower specific heat capacity. Hot air over land rises → low pressure over land. Cooler, denser air over ocean creates high pressure. Cool air from the sea moves toward the land to balance the pressure — this is the sea breeze. This daytime phenomenon provides natural cooling in coastal areas.

INTEXT QUESTIONS 12.2

Q1. At what wavelength does a cavity radiator at 300 K emit most radiation?

Ans: Using Wien's law: ${\lambda }_{m}T=2.884\times 10^{-3}$ m⋅K

${\lambda }_m=\frac{2.884\times 10^{-3}}{300}=9.613\times 10^{-6}\text{ m}=9.6\text{ µm}$

Q2. Why do we wear light colour clothing during summer?

Ans: Light colours reflect most of the sunlight (including heat), rather than absorbing it. Dark-coloured clothes absorb more solar energy, making the fabric and body warmer. Light colours keep the fabric cooler and help maintain normal body temperature. Lightweight, breathable materials further assist air circulation and sweat evaporation.

Terminal Exercise

1. Distinguish between the three modes of heat transfer. Give two examples of each.

2. State Fourier's law of heat conduction. Define coefficient of thermal conductivity.

3. A metal rod of length 2 m and cross-sectional area 0.01 m² has its ends maintained at 100°C and 0°C. If $K=400$ W⋅m⁻¹⋅K⁻¹, calculate the rate of heat flow.

4. Explain the formation of (a) sea breeze, (b) land breeze using convection.

5. State Stefan-Boltzmann law. A black body at 727°C radiates energy at the rate of $E$. At what temperature will it radiate at rate $16E$?

6. State Wien's displacement law. The surface temperature of the Sun is about 6000 K. At what wavelength does the Sun emit maximum radiation?

7. Explain why: (a) a thermos flask has a vacuum between double walls, (b) cooking utensils have wooden/plastic handles, (c) the filament of a bulb appears white-hot while the connecting wires remain cool.

8. Define emissivity and absorptivity. State Kirchhoff's law of radiation.

9. Explain the principle and working of a solar water heater.

10. What is the solar constant? If the Sun's surface temperature is 5800 K and its radius is $7\times 10^8$ m, estimate the solar energy received per second by Earth. (Earth-Sun distance = $1.5\times 10^{11}$ m)

11. A room is heated by a 1 kW electric heater. The room has a glass window of area 2 m² and thickness 5 mm. If the inside temperature is 20°C and outside is 5°C, find the thermal conductivity of glass if half the heat is lost through the window.

12. Why does a piece of metal feel colder than a piece of wood at the same temperature?

Worked Examples

Example 1: Conduction

Problem: An iron rod of length 1 m and cross-section 0.001 m² has one end in boiling water (100°C) and the other in ice (0°C). $K_{\text{iron}}=80$ W⋅m⁻¹⋅K⁻¹. Find the heat conducted per minute.

Solution: $\frac{Q}{t}=\frac{KA(T_h-T_c)}{d}=\frac{80\times 0.001\times 100}{1}=8\text{ W}$ $Q_{\text{per min}}=8\times 60=480\text{ J}$

Example 2: Wien's Law

Problem: A star emits maximum radiation at 300 nm. Find its surface temperature.

Solution: $T=\frac{b}{{\lambda }_m}=\frac{2.884\times 10^{-3}}{300\times 10^{-9}}=\frac{2.884\times 10^{-3}}{3\times 10^{-7}}\approx 9613\text{ K}$

Example 3: Stefan-Boltzmann

Problem: A black body at 500 K radiates 100 W. At what temperature will it radiate 400 W?

Solution: $E\propto T^4$, so $\frac{E_2}{E_1}={\left(\frac{T_2}{T_1}\right)}^4$

$\frac{400}{100}={\left(\frac{T_2}{500}\right)}^4⟹4={\left(\frac{T_2}{500}\right)}^4⟹T_2=500\times 4^{1/4}\approx 707\text{ K}$

Common Mistakes

1. Thinking conduction occurs in liquids/gases: Conduction mainly occurs in solids. Liquids and gases transfer heat primarily by convection.
2. Forgetting that radiation does not need a medium: The Sun's heat reaches Earth through the vacuum of space.
3. Mixing up Wien's law and Stefan-Boltzmann: Wien's law gives peak wavelength (colour); Stefan-Boltzmann gives total energy radiated.
4. Using Celsius instead of Kelvin in radiation laws: $T$ must be in kelvin for $T^4$ and Wien's law.
5. Confusing good conductors with good radiators: A good absorber is also a good radiator (Kirchhoff's law).

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