Heat Transfer

Overview

Heat can be transferred from one place to another by three mechanisms: conduction, convection, and radiation. Each operates by different physical principles.

Conduction

Transfer of heat through direct molecular contact, without bulk motion of the material.

Fourier's Law

$$P = \frac{Q}{t} = \frac{kA(T_H - T_C)}{L}$$

Where:

• $P$ = power (rate of heat flow) in Watts
• $k$ = thermal conductivity (W/m·K)
• $A$ = cross-sectional area
• $T_H - T_C$ = temperature difference
• $L$ = length/thickness

Thermal Conductivity Values

Material	$k$ (W/m·K)
Silver	429
Copper	401
Aluminum	237
Steel	50
Glass	0.8
Water	0.6
Wood	0.15
Fiberglass	0.04
Air	0.026

Thermal Resistance

$$R = \frac{L}{kA}$$

For heat flow:

$$P = \frac{\Delta T}{R}$$

Series Combination

$$R_{\text{total}} = R_1 + R_2 + R_3 + \cdots$$

Parallel Combination

$$\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots$$

Convection

Heat transfer by bulk motion of a fluid (liquid or gas).

Newton's Law of Cooling

$$P = hA(T_{\text{surface}} - T_{\text{fluid}})$$

Where $h$ = convection coefficient (depends on fluid properties and flow)

Types of Convection

Natural (Free) Convection:

• Driven by buoyancy (density differences)
• Hot fluid rises, cold fluid sinks
• Example: warm air rising from a heater

Forced Convection:

• Driven by external means (fan, pump)
• More efficient heat transfer
• Example: car radiator with fan

Factors Affecting Convection

• Fluid velocity
• Fluid properties (viscosity, thermal conductivity)
• Surface geometry
• Temperature difference

Radiation

Heat transfer through electromagnetic waves. No medium required.

Stefan-Boltzmann Law

$$P = \varepsilon\sigma AT^4$$

Where:

• $\varepsilon$ = emissivity (0 to 1)
• $\sigma$ = Stefan-Boltzmann constant = $5.67 \times 10^{-8}$ W/m²·K⁴
• $A$ = surface area
• $T$ = absolute temperature (Kelvin)

Net Radiation Exchange

$$P_{\text{net}} = \varepsilon\sigma A(T^4 - T_{\text{env}}^4)$$

Emissivity

• Perfect blackbody: $\varepsilon = 1$
• Perfect reflector: $\varepsilon = 0$
• Most real surfaces: $0.1 < \varepsilon < 0.95$

Surface	$\varepsilon$
Blackbody	1.0
Human skin	0.97
Water	0.96
White paint	0.90
Polished aluminum	0.05

Wien's Displacement Law

Peak wavelength of radiation:

$$\lambda_{\max} = \frac{b}{T}$$

Where $b = 2.898 \times 10^{-3}$ m·K

Examples

Example 1: Window Heat Loss

A glass window ($k = 0.8$ W/m·K) is 1 m × 2 m and 5 mm thick. Inside: 20°C, Outside: 0°C.

$$P = \frac{kA\Delta T}{L} = \frac{0.8 \times 2 \times 20}{0.005} = 6400 \text{ W} = 6.4 \text{ kW}$$

Example 2: Composite Wall

A wall has 2 cm wood ($k = 0.15$) and 5 cm fiberglass ($k = 0.04$). Area = 10 m², $\Delta T = 30$°C.

$$R_{\text{wood}} = \frac{0.02}{0.15 \times 10} = 0.0133 \text{ K/W}$$ $$R_{\text{fiber}} = \frac{0.05}{0.04 \times 10} = 0.125 \text{ K/W}$$ $$R_{\text{total}} = 0.138 \text{ K/W}$$ $$P = \frac{30}{0.138} = 217 \text{ W}$$

Example 3: Radiation from Hot Object

A steel ball ($\varepsilon = 0.8$, radius = 5 cm) at 500°C in 25°C room.

$$A = 4\pi r^2 = 4\pi(0.05)^2 = 0.0314 \text{ m}^2$$ $$P = \varepsilon\sigma A(T^4 - T_{\text{env}}^4)$$ $$P = 0.8 \times 5.67 \times 10^{-8} \times 0.0314 \times [(773)^4 - (298)^4]$$ $$P = 0.8 \times 5.67 \times 10^{-8} \times 0.0314 \times (3.57 \times 10^{11} - 7.9 \times 10^9)$$ $$P \approx 497 \text{ W}$$

Example 4: Human Body Radiation

Body ($\varepsilon = 0.97$, $A = 1.8$ m²) at 33°C in 20°C room.

$$P = 0.97 \times 5.67 \times 10^{-8} \times 1.8 \times [(306)^4 - (293)^4]$$ $$P = 0.97 \times 5.67 \times 10^{-8} \times 1.8 \times (8.76 \times 10^9 - 7.37 \times 10^9)$$ $$P \approx 138 \text{ W}$$

Example 5: Peak Radiation Wavelength

Find peak wavelength for Sun ($T \approx 5800$ K) and human body ($T \approx 310$ K).

Sun:

$$\lambda_{\max} = \frac{2.898 \times 10^{-3}}{5800} = 5.0 \times 10^{-7} \text{ m} = 500 \text{ nm (visible light)}$$

Human body:

$$\lambda_{\max} = \frac{2.898 \times 10^{-3}}{310} = 9.3 \times 10^{-6} \text{ m} = 9.3 \text{ μm (infrared)}$$