Extended Surfaces

Extended surfaces (fins) enhance heat transfer by increasing the surface area available for convection. Common applications include air-cooled heat exchangers, electronic cooling, HVAC coils, and radiators.

Fundamental Principles

Fin Equation

The general differential equation governing fin heat transfer:

$$\frac{d^2T}{dx^2} - m^2(T - T_\infty) = 0$$

Where:

• m² = hP/(kAc) for uniform cross-section fins
• P = fin perimeter (m)
• Ac = cross-sectional area (m²)
• k = thermal conductivity (W/m·K)
• h = convection coefficient (W/m²·K)

Fin Parameter

The fin parameter m determines fin behavior:

$$m = \sqrt{\frac{hP}{kA_c}}$$

Physical significance:

• High m: steep temperature gradient, ineffective fin
• Low m: gradual temperature gradient, effective fin
• Units: m⁻¹

Fin Efficiency

Fin efficiency compares actual heat transfer to ideal heat transfer if the entire fin were at base temperature.

$$\eta_f = \frac{Q_{actual}}{Q_{ideal}} = \frac{Q_{actual}}{hA_{fin}(T_b - T_\infty)}$$

Rectangular Fin Efficiency

For a straight rectangular fin (constant cross-section):

$$\eta_f = \frac{\tanh(mL)}{mL}$$

Where L = fin length from base to tip.

Corrected length (accounts for tip heat loss):

$$L_c = L + \frac{t}{2}$$

For thin fins where t « L, use:

$$\eta_f = \frac{\tanh(mL_c)}{mL_c}$$

Circular Fin Efficiency

For annular fins (constant thickness):

$$\eta_f = \frac{2r_1}{r_2^2 - r_1^2} \cdot \frac{K_1(mr_1)I_1(mr_2) - I_1(mr_1)K_1(mr_2)}{I_0(mr_1)K_1(mr_2) + K_0(mr_1)I_1(mr_2)}$$

Where:

• r₁ = tube outer radius (m)
• r₂ = fin outer radius (m)
• I, K = modified Bessel functions

Simplified approximation for r₂/r₁ < 2:

$$\eta_f \approx \frac{\tanh(mL_e)}{mL_e}$$

Where equivalent length Le = r₂ - r₁.

Triangular and Parabolic Fins

Triangular fin (thickness varies linearly to zero at tip):

$$\eta_f = \frac{1}{mL} \cdot \frac{I_1(2mL)}{I_0(2mL)}$$

Parabolic fin (optimized profile):

$$\eta_f = \frac{2}{1 + \sqrt{1 + (2mL)^2}}$$

Fin Efficiency Table

Fin Effectiveness

Fin effectiveness compares heat transfer with fin to heat transfer without fin.

$$\varepsilon_f = \frac{Q_{with,fin}}{Q_{without,fin}} = \frac{Q_{fin}}{hA_b(T_b - T_\infty)}$$

For a straight fin with insulated tip:

$$\varepsilon_f = \sqrt{\frac{kP}{hA_c}} \cdot \tanh(mL)$$

Effectiveness Criteria

Design guideline: Use high thermal conductivity materials (aluminum, copper) for maximum effectiveness.

Rectangular Fin Analysis

Straight Rectangular Fin

Geometry:

• Width: W (m)
• Thickness: t (m)
• Length: L (m)
• Ac = W × t
• P = 2(W + t) ≈ 2W for thin fins

Heat transfer rate:

$$Q = \sqrt{hPkA_c} \cdot (T_b - T_\infty) \cdot \tanh(mL)$$

Design Parameters

Circular Fin Analysis

Annular Fin on Tube

Configuration:

• Tube outer diameter: D₀
• Fin outer diameter: D_f
• Fin thickness: t
• Fin spacing: s

Heat dissipation per fin:

$$Q = 2\pi r_1 kt(T_b - T_\infty) \cdot m \cdot \eta_f \cdot \phi$$

Where φ is a dimensionless parameter from Bessel function solution.

Circular Fin Optimization

Optimal fin spacing (natural convection):

$$s_{opt} = 2.714 \cdot L \cdot \left(\frac{Ra_L}{4}\right)^{-0.25}$$

Where Ra_L is Rayleigh number based on fin height.

Heat Sink Design

Overall Surface Efficiency

For a finned surface with N fins:

$$\eta_o = 1 - \frac{N \cdot A_{fin}}{A_{total}}(1 - \eta_f)$$

Where:

• A_total = total heat transfer area (base + fins)
• A_fin = surface area of single fin

Total heat transfer rate:

$$Q_{total} = h \cdot A_{total} \cdot \eta_o \cdot (T_b - T_\infty)$$

Heat Sink Performance Parameters

Thermal Resistance

The total thermal resistance from junction to ambient:

$$R_{total} = R_{junction-case} + R_{contact} + R_{sink}$$

Heat sink resistance:

$$R_{sink} = \frac{1}{h \cdot A_{total} \cdot \eta_o}$$

HVAC Coil Fin Analysis

Plate Fin Coils

Typical specifications:

• Fin spacing: 8-20 fins per inch (FPI)
• Fin thickness: 0.1-0.15 mm aluminum
• Tube diameter: 7-10 mm
• Row depth: 2-8 rows

Fin efficiency for plate fins on round tubes:

$$\eta_f = \frac{\tanh(m \cdot r_{eq} \cdot \phi)}{m \cdot r_{eq} \cdot \phi}$$

Where:

• r_eq = equivalent radius = 0.5 × (fin spacing)
• φ = correction factor from charts (typically 0.8-1.0)

Coil Surface Effectiveness

Air-side heat transfer:

$$Q = h_o \cdot A_o \cdot \eta_o \cdot LMTD$$

Where:

• h₀ = outside convection coefficient (W/m²·K)
• A₀ = total outside surface area (m²)
• LMTD = log mean temperature difference

Fin Spacing Selection

Higher FPI increases surface area but decreases air-side heat transfer coefficient due to increased pressure drop and reduced flow area.

Fin Parameter Equations

Dimensionless Groups

Biot number for fins:

$$Bi_f = \frac{hL_c}{k}$$

Represents ratio of conduction resistance to convection resistance.

Fin length parameter:

$$mL = L\sqrt{\frac{hP}{kA_c}}$$

Critical fin length (beyond which additional length adds negligible heat transfer):

$$L_{crit} = \frac{2.3}{m}$$

For mL > 2.3, tanh(mL) ≈ 1.0, and additional length is wasteful.

Optimization Strategies

Mass Optimization

For a given heat duty Q and material volume V:

Rectangular fin optimum thickness:

$$t_{opt} = \sqrt{\frac{2k}{h}}$$

Optimum fin spacing (vertical plate array, natural convection):

$$s_{opt} = 2.714 \cdot \left(\frac{L}{Ra_s^{0.25}}\right)$$

Cost-Performance Optimization

Performance index:

$$PI = \frac{Q}{Cost} = \frac{\eta_f \cdot A_{fin} \cdot h \cdot \Delta T}{\rho \cdot V \cdot C_{material}}$$

Where:

• ρ = material density (kg/m³)
• V = fin volume (m³)
• C_material = material cost ($/kg)

Material Selection Criteria

The parameter √(k/ρC) represents thermal performance per unit cost.

Design Procedure

1. Calculate required heat transfer rate Q from thermal load
2. Estimate convection coefficient h based on fluid, velocity, geometry
3. Select fin material based on temperature, corrosion, cost
4. Calculate fin parameter m = √(hP/kAc)
5. Determine fin efficiency η_f from appropriate equation
6. Calculate overall surface efficiency η_o
7. Size fin array to achieve Q = hA_total·η_o·ΔT
8. Check constraints: spacing, pressure drop, manufacturability
9. Optimize if needed for cost, weight, or volume

Performance Degradation

Fouling Effects

Fouling reduces effective fin efficiency:

$$\eta_{f,fouled} = \eta_f \cdot \frac{h_{clean}}{h_{fouled}}$$

Typical fouling factors for HVAC coils: 0.85-0.95 after 1 year of operation.

Contact Resistance

For bonded or press-fit fins, contact resistance R_c reduces performance:

$$\eta_{effective} = \eta_f \cdot \frac{1}{1 + \frac{R_c \cdot k \cdot t}{A_c}}$$

Thermal greases or brazing eliminate contact resistance.