9.1 Heat Transfer

9.1 Heat Transfer

1. Conduction, Convection, Radiation

1. Conduction:

   • Heat transfer through a stationary medium by molecular activity.

   • Requires physical contact between objects or within an object.

   • Governed by Fourier's Law: $q = -kA \frac{dT}{dx}$

     • $q$ = heat transfer rate (W)

     • $k$ = thermal conductivity (W/m·K)

     • $A$ = cross-sectional area

     • $dT/dx$ = temperature gradient

2. Convection:

   • Heat transfer between a surface and a moving fluid.

   • Two mechanisms: conduction at surface + fluid motion.

   • Governed by Newton's Law of Cooling: $q = hA(T_s - T_\infty)$

     • $h$ = convective heat transfer coefficient (W/m²·K)

     • $T_s$ = surface temperature

     • $T_\infty$ = fluid temperature

3. Radiation:

   • Heat transfer by electromagnetic waves.

   • Requires no medium (works in vacuum).

   • All objects above absolute zero emit thermal radiation.

2. Heat Transfer through Walls, Tubes and Spheres

1. Plane Wall (Slab):

   • One-dimensional steady-state conduction.

   • Thermal resistance: $R_{cond} = \frac{L}{kA}$

   • Heat transfer: $q = \frac{T_1 - T_2}{R_{cond}}$

   • Composite wall: Series resistances: $q = \frac{T_1 - T_4}{\sum R_i}$

2. Cylindrical Tube (Pipe):

   • Radial conduction through tube wall.

   • Thermal resistance: $R_{cyl} = \frac{\ln(r_2/r_1)}{2\pi kL}$

   • Heat transfer per unit length: $\frac{q}{L} = \frac{2\pi k(T_1 - T_2)}{\ln(r_2/r_1)}$

3. Spherical Shell:

   • Radial conduction through sphere wall.

   • Thermal resistance: $R_{sph} = \frac{1}{4\pi k}\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$

   • Heat transfer: $q = \frac{4\pi k(T_1 - T_2)}{(1/r_1) - (1/r_2)}$

3. Stefan–Boltzmann Law

1. Blackbody Radiation:

   • Ideal emitter and absorber of radiation.

   • Emits maximum possible radiation at given temperature.

2. Stefan-Boltzmann Law:

   • Total emissive power of blackbody: $E_b = \sigma T^4$

     • $E_b$ = emissive power (W/m²)

     • $\sigma$ = Stefan-Boltzmann constant = $5.67 \times 10^{-8}$ W/m²·K⁴

     • $T$ = absolute temperature (K)

3. Real Surfaces:

   • Emissivity ($\epsilon$): Ratio of actual emission to blackbody emission.

   • $E = \epsilon \sigma T^4$ (where $0 \le \epsilon \le 1$)

   • Absorptivity ($\alpha$): Fraction of incident radiation absorbed.

4. Radiation Exchange:

   • Net radiation between two surfaces depends on:

     • Temperatures $T_1$ and $T_2$

     • Surface properties (emissivity, absorptivity)

     • View factor (geometric configuration)

4. Overall Heat Transfer Coefficient

1. Definition (U):

   • Combined effect of all resistances in a heat transfer system.

   • Relates total heat transfer to temperature difference: $q = UA\Delta T$

   • Units: W/m²·K

2. For Plane Wall with Convection:

   • Total resistance: $R_{total} = \frac{1}{h_1A} + \frac{L}{kA} + \frac{1}{h_2A}$

   • Overall U: $\frac{1}{UA} = R_{total}$ or $U = \frac{1}{\frac{1}{h_1} + \frac{L}{k} + \frac{1}{h_2}}$

3. For Cylindrical System (Pipe):

   • Based on either inner or outer area.

   • For outer area $A_o$: $\frac{1}{U_oA_o} = \frac{1}{h_iA_i} + \frac{\ln(r_o/r_i)}{2\pi kL} + \frac{1}{h_oA_o}$

4. Applications:

   • Heat exchangers design and analysis.

   • Building insulation calculations.

   • Equipment sizing for heating/cooling systems.

5. Limiting Resistance:

   • Smallest h-value often controls overall U.

   • Improving largest resistance has minimal effect.