🍢 Pin Fin & Spine Calculator

Calculate efficiency, effectiveness, and heat dissipation for cylindrical, conical, parabolic, and rectangular pin fins or spines under multiple tip conditions.

⚡ Fortran 90 Engine Double Precision (IEEE 754) ✓ ISO / ASME Validated

Heat Transfer

📊 Solver Telemetry ● ACTIVE

👁️ Views 195

⚡ Solves 153

💾 Downloads 365 📦 Fortran Code 3.4 KB

📅 Released Jun 2026

⏱️ Latency < 1 ms

Configurator

Geometry

Fin shape profile:

Diameter (d) [mm]:

Thickness (t) [mm]:

Length (L) [mm]:

Thermal Parameters

Material Preset:

Thermal Conductivity ($k$) [W/m·K]:

Convection Coefficient ($h$) [W/m²·K]:

Temperatures & Boundary Conditions

Base Temperature ($T_b$) [°C]:

Ambient Temperature ($T_\infty$) [°C]:

Tip Condition:

Tip Convection ($h_{tip}$) [W/m²·K]:

Tip Temperature ($T_{tip}$) [°C]:

Results & Plots

Results and visualizations will be displayed here upon completion of the computation.

Calculation Methodology

Mathematical Model & Theory

Pin fins (or spines) are critical for electronics cooling, motor housings, and heat sinks. The steady state 1D heat conduction along a uniform fin is described by:

$$\frac{d^2\theta}{dx^2} - m^2\theta = 0, \quad m = \sqrt{\frac{hP}{kA_c}}$$ $$\theta(x) = T(x) - T_\infty$$

For tapering circular profiles like Conical Spines and Parabolic Spines, cross-sectional area changes, transforming the heat equation into modified Bessel forms with exact efficiencies:

$$\text{Conical Spine: } \eta_f = \frac{2}{mL} \frac{I_2(2mL)}{I_1(2mL)}$$ $$\text{Parabolic Spine: } \eta_f = \frac{1.5}{mL} \frac{I_1(4/3 mL)}{I_0(4/3 mL)}$$

Academic References:

Incropera, F.P. et al. (2011). Fundamentals of Heat and Mass Transfer, 7th ed., Table 3.6.
Kraus, A.D. and Bar-Cohen, A. (1995). Design and Analysis of Heat Sinks. John Wiley & Sons.

Convective Length Corrections

For uniform fins (cylindrical and rectangular) with convective tip boundary conditions, the standard length correction $L_c$ simplifies the model to an adiabatic boundary equation:

$$\text{Cylindrical corrected length: } L_c = L + \frac{d}{4}$$ $$\text{Rectangular corrected length: } L_c = L + \frac{t}{2}$$

This calculator implements the **exact analytical convective boundary condition** without relying on the length approximation for Cylindrical and Rectangular profiles to guarantee thermodynamic precision.