Heat Transfer from Extended Surface (Fins)

Introduction

• Heat conduction through solid medium.
• Heat removal from solid surface to surrounding by convection.
• Newton's Law of cooling: Rate of heat transfer by convection is proportional to:
  • Convection heat transfer coefficient (H)
  • Heat transfer area (A)
  • Temperature difference (T_S - T_∞)

Enhancing Heat Transfer

• Increase the convection heat transfer coefficient (H)
  • Not straightforward due to dependency on fluid velocity, flow motion, and surface conditions.
  • Can use fans or pumps (expensive).
• Increase heat transfer area (A) using fins.
  • More economical.

Examples of Fins

• Refrigerator back: Coils with attached fins to increase surface area.
• Heat sinks on CPUs/GPUs: Enhance heat removal from electronic devices.
• Car radiators: Fins attached to channels for better heat transfer between coolant and surrounding air.
• Fans and motors: Combined with fins for forced convection.
• Heat pipes: Combined with plate fins for efficient heat transfer.

Analysis of Heat Transfer across Fins

• Setup: Solid surface with temperature T_S, and cylindrical fin with one-dimensional heat conduction.
• Energy Balance Equation (Steady State, 1D, No Heat Generation)
  • Heat conduction: Q_x, Q_{x + Δx}
  • Heat convection from surface
  • Governing equation derived using limits and derivative definitions.
• Excess Temperature (θ) Defined as (T_x – T_∞)
  • Solution involves second-order linear homogeneous differential equation with constant coefficients.

Boundary Conditions

• At the fin base: T_B = T_S
• Types at fin tip:
  1. Convective heat transfer: Hθ = K dθ/dx
  2. Adiabatic condition: dθ/dx = 0
  3. Prescribed temperature: θ = θ_L
  4. Infinitely long fin: θ = 0 (temperature equals T_∞ at the tip)

Solutions and Temperature Profiles

• Different boundary conditions result in different temperature profiles and heat transfer rates.
  • Convective: Decreasing temperature profile from base to tip, significant heat transfer at base.
  • Adiabatic: Slope at fin tip = 0, decreasing temperature but no heat loss at tip.
  • Prescribed Temperature: Specific temperature at the tip, varying temperature gradient.
  • Infinitely Long Fin: Temperature reaches T_∞, minimal heat transfer at tip.

Performance Metrics of Fins

• Fin Efficiency
  • Actual heat transfer / Ideal heat transfer
  • Ideal case: Uniform temperature distribution.
  • Efficiency < 1.
• Fin Effectiveness
  • Actual heat transfer / Heat transfer without fin.
  • Effectiveness > 1.

Optimal Fin Shape and Efficiency

• Higher performance with flat or slender fins (higher p/AC ratio).
• More effective for low H value (typically in gas).

Efficiency of Common Fin Shapes

• Use tables provided in textbooks for calculating efficiency of different fin shapes with uniform cross-sections.
  • e.g., Table 3.5 for different fin shapes.
  • Applies to fins with uniform cross-sections (Table 3.4).