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Engineering Calculation Report
Date: 2026-06-04 20:32
Annular (Circular) Fin Efficiency Calculator
Compute fin efficiency Ξ·, effectiveness Ξ΅, and heat transfer rate for annular fins using exact Bessel function solutions.
Calculation Domain Inputs
Specify the annular fin geometry and thermal boundary conditions:
- Inner Radius ($r_1$): Tube or pipe outer radius (fin base).
- Outer Radius ($r_2$): Tip of the fin (actual outer edge).
- Fin Thickness ($t$): Profile height (uniform). Used to compute corrected length $r_{2c} = r_2 + t/2$.
- Thermal Conductivity ($k$): Fin material heat conduction capacity.
- Convection Coefficient ($h$): Heat transfer coefficient of surrounding fluid.
- Temperatures: Base $T_0$ and ambient $T_\infty$.
Fin Configuration
Annular Fin Efficiency (Incropera Table 3.5):
$$\eta_f = \frac{2r_1}{m(r_{2c}^2 - r_1^2)} \cdot \frac{K_1(mr_1)I_1(mr_{2c}) - I_1(mr_1)K_1(mr_{2c})}{I_0(mr_1)K_1(mr_{2c}) + K_0(mr_1)I_1(mr_{2c})}$$ $$m = \sqrt{\frac{2h}{kt}}, \quad r_{2c} = r_2 + \frac{t}{2}$$ β’ $I_n, K_n$ = Modified Bessel functions of order $n$
β’ $\eta_f = Q_{fin} / Q_{max}$ = Fin efficiency
β’ $\varepsilon = Q_{fin} / (h A_{base} \theta_b)$ = Effectiveness
$$\eta_f = \frac{2r_1}{m(r_{2c}^2 - r_1^2)} \cdot \frac{K_1(mr_1)I_1(mr_{2c}) - I_1(mr_1)K_1(mr_{2c})}{I_0(mr_1)K_1(mr_{2c}) + K_0(mr_1)I_1(mr_{2c})}$$ $$m = \sqrt{\frac{2h}{kt}}, \quad r_{2c} = r_2 + \frac{t}{2}$$ β’ $I_n, K_n$ = Modified Bessel functions of order $n$
β’ $\eta_f = Q_{fin} / Q_{max}$ = Fin efficiency
β’ $\varepsilon = Q_{fin} / (h A_{base} \theta_b)$ = Effectiveness
Results & Visualization
Results and visualizations will be displayed here upon completion of the computation.
Calculation Methodology
Mathematical Model & Theory
Annular fins are commonly found on heat exchangers, motor housings, and pipe systems. The temperature distribution satisfies the modified Bessel equation due to the cylindrical geometry:
$$\frac{d^2\theta}{dr^2} + \frac{1}{r}\frac{d\theta}{dr} - m^2\theta = 0, \quad m = \sqrt{\frac{2h}{kt}}$$
$$\theta(r) = C_1 I_0(mr) + C_2 K_0(mr)$$
Fin Efficiency (Table 3.5, Incropera):
$$\eta_f = \frac{2r_1}{m(r_{2c}^2 - r_1^2)} \cdot \frac{K_1(mr_1)I_1(mr_{2c}) - I_1(mr_1)K_1(mr_{2c})}{I_0(mr_1)K_1(mr_{2c}) + K_0(mr_1)I_1(mr_{2c})}$$
$$r_{2c} = r_2 + \frac{t}{2} \quad \text{(corrected tip radius)}$$
Variable Definitions & Units:
- $r_1$: Inner (base) radius [m]
- $r_2$: Outer (tip) radius [m]
- $r_{2c} = r_2 + t/2$: Corrected outer radius (accounts for tip convection) [m]
- $t$: Fin thickness [m]
- $k$: Thermal conductivity [W/mΒ·K]
- $h$: Convection coefficient [W/mΒ²Β·K]
- $I_n, K_n$: Modified Bessel functions of the first and second kind, order $n$
- $Q_{fin} = \eta_f \cdot h \cdot A_{fin} \cdot (T_0 - T_\infty)$
Assumptions & Boundary Conditions:
- Steady-state, 1D conduction in the radial direction.
- Uniform fin thickness $t$ (constant cross-section profile).
- Constant thermal conductivity $k$.
- Uniform convection coefficient $h$ over all surfaces.
- Adiabatic at corrected tip $r_{2c}$ (models convective tip via length correction).
- Valid for $Bi_{fin} = ht/2k < 0.1$.
Academic References:
- Incropera, F.P. et al. (2011). Fundamentals of Heat and Mass Transfer, 7th ed., Table 3.5.
- Γengel, Y.A. (2015). Heat and Mass Transfer: Fundamentals and Applications, Table 3-3.
- Abramowitz & Stegun (1972). Handbook of Mathematical Functions, Β§9.8.
Worked Engineering Example
Problem Statement:
An aluminum annular fin ($k = 200$ W/mΒ·K) is mounted on a tube of outer radius $r_1 = 25$ mm. The fin extends to $r_2 = 64$ mm with a uniform thickness $t = 2$ mm. The base temperature is $T_0 = 100$Β°C, ambient is $T_\infty = 20$Β°C, and the convection coefficient is $h = 25$ W/mΒ²Β·K. Find the fin efficiency and heat transfer rate.
Step-by-step Solution:
1. Corrected tip radius and fin parameter:
$$r_{2c} = 0.064 + 0.001 = 0.065 \text{ m}$$ $$m = \sqrt{\frac{2 \times 25}{200 \times 0.002}} = 11.18 \text{ m}^{-1}$$ $$m r_1 = 0.2795, \quad m r_{2c} = 0.7267$$ 2. Fin efficiency (Bessel functions):
$$\eta_f \approx 90.4\%$$ 3. Corrected fin surface area:
$$A_{fin} = 2\pi(r_{2c}^2 - r_1^2) = 2\pi(0.065^2 - 0.025^2) = 0.02262 \text{ m}^2$$ 4. Heat transfer rate:
$$Q_{fin} = \eta_f \cdot h \cdot A_{fin} \cdot \Delta T = 0.904 \times 25 \times 0.02262 \times 80 = 40.9 \text{ W}$$
Final Result:
Fin efficiency: 90.4%. Heat dissipated: 40.9 W.
An aluminum annular fin ($k = 200$ W/mΒ·K) is mounted on a tube of outer radius $r_1 = 25$ mm. The fin extends to $r_2 = 64$ mm with a uniform thickness $t = 2$ mm. The base temperature is $T_0 = 100$Β°C, ambient is $T_\infty = 20$Β°C, and the convection coefficient is $h = 25$ W/mΒ²Β·K. Find the fin efficiency and heat transfer rate.
Step-by-step Solution:
1. Corrected tip radius and fin parameter:
$$r_{2c} = 0.064 + 0.001 = 0.065 \text{ m}$$ $$m = \sqrt{\frac{2 \times 25}{200 \times 0.002}} = 11.18 \text{ m}^{-1}$$ $$m r_1 = 0.2795, \quad m r_{2c} = 0.7267$$ 2. Fin efficiency (Bessel functions):
$$\eta_f \approx 90.4\%$$ 3. Corrected fin surface area:
$$A_{fin} = 2\pi(r_{2c}^2 - r_1^2) = 2\pi(0.065^2 - 0.025^2) = 0.02262 \text{ m}^2$$ 4. Heat transfer rate:
$$Q_{fin} = \eta_f \cdot h \cdot A_{fin} \cdot \Delta T = 0.904 \times 25 \times 0.02262 \times 80 = 40.9 \text{ W}$$
Final Result:
Fin efficiency: 90.4%. Heat dissipated: 40.9 W.