Section 7.2: Heat Transfer

Heat transfer is the movement of thermal energy from a hotter body to a cooler one. It occurs via three main mechanisms: conduction, convection, and radiation.

Conduction: Transfer of heat through direct contact of particles.  
        Equation: 
        
        \[
        Q = \frac{k A \Delta T t}{d}
        \]  
        where \( k \) = thermal conductivity (W/m·K), \( A \) = area (m²), \( \Delta T \) = temperature difference (K), \( t \) = time (s), \( d \) = thickness (m)
        

        Convection: Transfer of heat through a fluid (liquid or gas) motion.  
        Depends on fluid properties and temperature difference.
      

Radiation: Transfer of heat through electromagnetic waves, can occur in vacuum.  
        Stefan-Boltzmann law: 
        
        \[
        P = \sigma \epsilon A T^4
        \]  
        where \( \sigma \) = 5.67×10⁻⁸ W/m²K⁴, \( \epsilon \) = emissivity, \( A \) = surface area, \( T \) = absolute temperature
        

Example 1: Heat Conduction

A metal rod of length 0.5 m and cross-sectional area 0.01 m² has a temperature difference of 80 K. If the rod's thermal conductivity is 200 W/m·K, find the heat conducted in 10 s.

\[ Q = \frac{k A \Delta T t}{d} = \frac{200 \cdot 0.01 \cdot 80 \cdot 10}{0.5} = 3200 \text{ J} \]
Heat conducted is 3200 J.

Example 2: Radiation

A blackbody of area 0.5 m² at 600 K radiates energy. Find the power emitted. (\( \epsilon = 1 \))

\[ P = \sigma \epsilon A T^4 = 5.67\times10^{-8} \cdot 1 \cdot 0.5 \cdot 600^4 \approx 3672 \text{ W} \]
Power emitted is approximately 3672 W.

Practice Problems

A wall 0.2 m thick has thermal conductivity 1.5 W/m·K. Temperature difference is 30 K. Area is 5 m². Find heat conducted in 1 hour.
A pot of water is heated on a stove. Identify the primary mode of heat transfer to the water.
A 1 m² black plate at 500 K radiates energy into surroundings at 300 K. Compute net power radiated. (\( \epsilon = 0.9 \))
Air is heated by a radiator. Explain the mechanism of heat transfer involved.
A metal rod of length 1 m, cross-section 0.02 m², \( k = 150 \) W/m·K, \( \Delta T = 50 \) K. Calculate heat conducted in 5 minutes.
A thin metal sheet is exposed to sunlight. Identify the mechanism of heat transfer causing it to warm.
Compare the efficiency of heat transfer via conduction, convection, and radiation in a house wall.
Calculate the energy radiated by a 0.3 m² surface at 400 K over 10 s. (\( \epsilon = 0.8 \))
Explain why convection is faster in fluids than in solids.
A rod of thickness 0.05 m has thermal conductivity 100 W/m·K. Temperature difference is 60 K. Find heat conducted in 2 minutes over an area of 0.02 m².