Heat Transfer — Conduction, Convection, Radiation 🔥

Heat Transfer — Conduction, Convection, Radiation 🔥

Three scenes ☕: a metal spoon in hot tea, whose handle slowly heats up (conduction). Warm steam rising from the pot (convection). The Sun’s warmth reaching your face from 150 million kilometres away, crossing a total vacuum (radiation) ☀️. Those three are the only ways heat travels in nature — and each one has a formula you can compute with. Let’s do it precisely.

The core idea in one paragraph 📌

Conduction (\( Q/t = kA\Delta T/L \)): heat moves from hot to cold via collisions between neighboring particles, without matter moving. Convection (\( Q/t = hA\Delta T \)): heat moves by bulk fluid motion — warm rises, cold sinks. Radiation (\( P = e\sigma A T^4 \)): electromagnetic waves — needs no medium, works even in vacuum. Conductors (metals) have high \( k \); insulators (fibreglass, still air) have low \( k \). Radiation scales as \( T^4 \), so hot objects radiate far more than warm ones. A thermos defeats all three (vacuum, mirrored walls, insulating stopper) to keep drinks hot or cold for hours.

Three key formulas 📐

\[ \boxed{\text{Conduction:} \quad \frac{Q}{t} = k\, A\, \frac{\Delta T}{L}} \]

Fourier’s law — \( k \): thermal conductivity (W/(m·K)), \( A \): area, \( L \): thickness, \( \Delta T \): temperature difference across.

\[ \boxed{\text{Convection:} \quad \frac{Q}{t} = h\, A\, \Delta T} \]

Newton’s law of cooling — \( h \): convective heat-transfer coefficient, depending on flow regime (natural vs forced).

\[ \boxed{\text{Radiation:} \quad P = e\, \sigma\, A\, T^4 \qquad P_\text{net} = e\sigma A (T^4 – T_\text{env}^4)} \]

Stefan–Boltzmann law — \( \sigma = 5.67\times 10^{-8}\,\text{W/(m}^2\text{K}^4) \), \( e \): emissivity (\( 0 \le e \le 1 \); ideal blackbody \( e=1 \)).

Thermal conductivity of common materials 📊

Material \( k \) (W/(m·K)) Note
Diamond 2000+ Highest of natural materials
Silver 429
Copper 401 Heat pipes, wiring
Aluminium 237 Pans, radiators
Brass 109
Iron/steel 80
Glass 0.8 ~500× less than copper
Liquid water 0.6
Concrete 1.7
Wood 0.15 That’s why pan handles are wooden 🪵
Fibreglass 0.04 Building insulation
Still air 0.024 ~20,000× less than copper — best cheap insulator
Vacuum 0 ← what thermos flasks exploit

Method 1: Conduction 🥄

Molecules stay put; only kinetic energy passes hand-to-hand from hot to cold. In metals, on top of atomic vibrations, free electrons also carry heat — that’s why metallic \( k \) is orders of magnitude larger than non-metals’.

Example 1: A single-pane window \( A = 2\,\text{m}^2 \), thickness \( L = 4\,\text{mm} = 0.004\,\text{m} \), room \( 20° \)C, outside \( 0° \)C. Heat lost per second?

\[ \frac{Q}{t} = \frac{k A \Delta T}{L} = \frac{0.8 \times 2 \times 20}{0.004} = 8000\ \text{W} \]

8 kilowatts from a single window! That’s why double-glazed windows exist — an air layer (\( k=0.024 \)) between the panes cuts conduction ~30×.

Method 2: Convection 🌊

Heated fluid expands ⇒ lower density ⇒ rises; cold fluid sinks ⇒ convective cell. Two kinds:

Example 2: A room radiator with \( A=1\,\text{m}^2 \), surface \( 60° \)C, air \( 20° \)C, \( h = 8\,\text{W/(m}^2\text{K)} \) (natural convection):

\[ \frac{Q}{t} = hA\Delta T = 8 \times 1 \times 40 = 320\ \text{W} \]

Turn on a fan (\( h=25 \)) and the power jumps to \( 1000 \) W — three times more. That’s why air conditioners have fans.

Method 3: Radiation ☀️

Every body with \( T > 0 \) K emits electromagnetic waves — the wavelength range depends on temperature:

Example 3: A nude human, \( A = 1.8\,\text{m}^2 \), skin \( T_s = 33° \)C = \( 306 \) K, room \( T_e = 20° \)C = \( 293 \) K, \( e = 0.98 \) for skin:

\[ P_\text{net} = e\sigma A(T_s^4 – T_e^4) = 0.98 \times 5.67\times 10^{-8} \times 1.8 \times (306^4 – 293^4) \]
\[ \approx 143\ \text{W} \]

143 W just from radiation! That’s why a nude body chills fast in a cool room — and why clothing is dramatically warming.

Beautiful application: the thermos flask ☕

A thermos keeps drinks hot or cold for hours because it blocks all three:

Result: only a few watts of heat leak out ⇒ tea stays hot 8 hours.

Why are clear nights colder? 🌌

The ground absorbs solar radiation all day and re-emits it as infrared at night. Clouds reflect that IR back down (like a blanket). A clear sky = no blanket ⇒ heat escapes freely to space ⇒ ground cools further. That’s why winter deserts get bitterly cold at night despite blazing days.

The greenhouse effect — same physics, planetary scale 🌍

Earth’s atmosphere is transparent to the Sun’s visible light (short wavelengths) but partly absorbs and re-emits Earth’s outgoing infrared (long wavelengths) via CO₂, H₂O, and methane. That atmospheric “blanket” keeps Earth about 33°C warmer than it would be without one — the difference between a living planet and an ice ball. Adding CO₂ thickens the blanket ⇒ global warming.

Python analysis 🐍

1) Heat loss through three window types

def conduction(k, A, L, dT):
    return k * A * dT / L    # W

windows = [
    ("Single pane (4-mm glass)",     0.8,   2, 0.004),
    ("Double pane (12-mm air gap)",  0.024, 2, 0.012),
    ("Triple pane + argon",          0.017, 2, 0.020),
]
dT = 20
for name, k, A, L in windows:
    P = conduction(k, A, L, dT)
    print(f"{name:32s} → {P:>7.1f} W")
# Standard double-glazing is ~80× better than single pane

2) Stefan–Boltzmann — radiated power vs temperature

import numpy as np, matplotlib.pyplot as plt

sigma = 5.67e-8
e, A = 0.9, 1.0
T = np.linspace(200, 2000, 300)     # K
P = e * sigma * A * T**4

plt.plot(T, P/1000)
plt.xlabel("Temperature (K)"); plt.ylabel("Radiated power (kW/m²)")
plt.title("Stefan-Boltzmann law — P ∝ T⁴")
plt.grid(alpha=0.3); plt.yscale("log")
plt.axvline(310, color="red", ls=":", label="Human body")
plt.axvline(5778, color="orange", ls=":", label="Sun surface")
plt.legend(); plt.show()

# Doubling T → 16× radiation
for T in [300, 600, 1200]:
    print(f"T = {T} K → P = {e*sigma*A*T**4:.1f} W/m²")

3) Tea cooling: open cup vs lidded vs thermos — Newton’s law

import numpy as np, matplotlib.pyplot as plt

def newton_cooling(T0, T_env, tau_min, t_min):
    """T(t) = T_env + (T0 - T_env) * exp(-t/tau)"""
    return T_env + (T0 - T_env) * np.exp(-t_min / tau_min)

t = np.linspace(0, 480, 200)   # 8 hours
T_open  = newton_cooling(85, 22, tau_min=30,  t_min=t)   # open cup
T_lid   = newton_cooling(85, 22, tau_min=90,  t_min=t)   # cup with lid
T_flask = newton_cooling(85, 22, tau_min=600, t_min=t)   # thermos

plt.plot(t/60, T_open,  label="Open cup",     color="red")
plt.plot(t/60, T_lid,   label="Lidded cup")
plt.plot(t/60, T_flask, label="Thermos",      color="steelblue")
plt.axhline(60, color="gray", ls="--", label="Min drinkable")
plt.xlabel("Time (h)"); plt.ylabel("Tea temperature (°C)")
plt.title("Tea cooling under three conditions")
plt.legend(); plt.grid(alpha=0.3); plt.show()

for name, T in [("open", T_open), ("lid", T_lid), ("thermos", T_flask)]:
    i = np.searchsorted(-T, -60)   # first time below 60°C
    if i < len(T): print(f"{name}: below 60°C after {t[i]/60:.1f} h")
    else:          print(f"{name}: still above 60°C even after 8 h 🎯")

Take-home summary 🎁

Three routes: conduction (direct contact, \( Q/t=kA\Delta T/L \)), convection (fluid motion, \( Q/t=hA\Delta T \)), radiation (EM waves, works even in vacuum, \( P=e\sigma A T^4 \)). Metals conduct; air and wood insulate. Radiation scales as \( T^4 \) ⇒ explosive growth with temperature. A thermos defeats all three routes and keeps tea hot for 8 hours. The greenhouse effect is the same radiation physics on a planetary scale. Clear night = no cloud blanket = cold. Radiators sit low because warm air rises; AC units sit high because cold air sinks 🎯.

“Nice to know” box: How can flimsy aluminium foil block heat? 🪞

Wrap food in kitchen foil? That 15-µm sheet does two opposite jobs: (1) it conducts heat well (k ≈ 237) so it spreads temperature fast; (2) but its surface is highly polished ⇒ emissivity \( e \approx 0.03 \) ⇒ it reflects ~97% of incoming radiation. That’s why firefighters’ emergency blankets and spacecraft shielding use it — a “thermal mirror.” One step further: NASA’s rescue blankets use Mylar coated with aluminium (\( e \approx 0.05 \)) and keep astronauts warm in lunar shade (3 K) and cool in lunar sun (\( 120° \)C). The whole system is a clever game with \( e \) ✨.

Test yourself 📝

References and further exploration 📚

Articles and reference

Videos (YouTube)

External simulators

On this site 🔗

The last section of this chapter: gas laws 🎈 — the relationships among pressure, volume, and temperature, from Boyle to the ideal-gas equation. See you there! 👋

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