---
title: "Fin heat dissipation, efficiency and effectiveness"
subject: "Heat Transfer"
chapter: 6
key: "semester_5/heat_transfer:fin-equation-tip-conditions-efficiency"
source: https://www.bctengineer.com/q/semester_5/heat_transfer/fin-equation-tip-conditions-efficiency
site: BCT Engineering
---

# Fin heat dissipation, efficiency and effectiveness

Heat Transfer · Chapter 6: Applications of Heat Transfer

- Asked in: year not given, 8 marks
- Video: [Heat Transfer - Chapter 3 - Extended Surfaces (Fins)](https://www.youtube.com/watch?v=SUjhwuKtU14) — Kody Powell
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/heat_transfer/fin-equation-tip-conditions-efficiency
- Whole chapter: https://www.bctengineer.com/md/semester_5/heat_transfer/chapter-6

## Question

What is a fin and why is it used? Name common types of fins. Derive the temperature distribution and heat dissipation for a straight fin of uniform cross-section with an insulated tip. Define fin efficiency and fin effectiveness and state when a fin is worthwhile.

## Answer

#### Fins
A fin is an extended surface that increases the area for convection, so that more heat leaves a hot surface. Heat flow $Q = hA\Delta T$ can be increased by raising $h$, $\Delta T$ or $A$; fins increase $A$ when $h$ is low (gas side). Examples: engine cylinder fins, radiator tubes, heat sinks.

**Types:** straight (rectangular, triangular, parabolic profile), pin (cylindrical or tapered), annular (circumferential) fins on tubes.

#### Derivation (uniform fin, insulated tip)
Assumptions: steady one-dimensional conduction along the fin, constant $k$ and $h$, uniform fin temperature across thickness. Let $A_c$ be the cross-section, $P$ the perimeter, $T_b$ the base temperature and $\theta = T - T_\infty$.

Energy balance on a slice $dx$: heat conducted in = heat conducted out + heat convected:

$$
-kA_c\frac{dT}{dx} = -kA_c\left(\frac{dT}{dx}+\frac{d^2T}{dx^2}dx\right)+hP\,dx\,(T-T_\infty)
$$

$$
\frac{d^2\theta}{dx^2}-m^2\theta = 0,\qquad m=\sqrt{\frac{hP}{kA_c}}
$$

Solution: $\theta = C_1e^{mx}+C_2e^{-mx}$. Boundary conditions: $\theta = \theta_b$ at $x = 0$; $d\theta/dx = 0$ at $x = L$ (insulated tip). This gives

$$
\frac{\theta}{\theta_b} = \frac{\cosh m(L-x)}{\cosh mL}
$$

Heat dissipated equals conduction at the base:

$$
Q_{fin} = -kA_c\frac{d\theta}{dx}\Big|_{x=0} = \sqrt{hPkA_c}\;\theta_b\tanh(mL)
$$

Other tip conditions: infinitely long fin $Q = \sqrt{hPkA_c}\,\theta_b$; convecting tip, use corrected length $L_c = L + A_c/P$ in the above formula.

#### Efficiency and effectiveness
$$
\eta_{fin}=\frac{Q_{fin}}{Q_{max}}=\frac{Q_{fin}}{hPL\,\theta_b}=\frac{\tanh mL}{mL}
\qquad
\varepsilon_{fin}=\frac{Q_{fin}}{hA_c\theta_b}
$$

Efficiency is the actual heat loss over the heat that would be lost if the entire fin were at the base temperature. Effectiveness is the heat with the fin over the heat without the fin.

A fin is worthwhile if $\varepsilon_{fin} > 2$ (preferably much greater). This needs high $k$ (copper, aluminium), thin closely spaced fins, and low $h$ on the fin side (gas, free convection). Fins are useless when $h$ is high, e.g. boiling or condensation.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why are fins used, and where are they most useful?
   A. To raise the temperature difference, on the liquid side
   B. To lower the thermal conductivity, on the gas side
   C. To increase the surface area, mainly where h is low (gas side)
   D. To increase h, mainly during boiling
   Answer: C. Q = hA(dT) can be raised by raising A; fins do this and pay off when h is low. (Key point: Purpose of fins)

2. The energy balance on a fin slice gives which governing equation?
   A. d2(theta)/dx2 + m^2 theta = 0
   B. d(theta)/dx = m theta
   C. d2(theta)/dx2 - m^2 theta = 0, with m = sqrt(hP/(k Ac))
   D. d2(theta)/dx2 = 0
   Answer: C. Conduction net into the slice balances convection from its surface, producing the exponential-type equation. (Key point: Fin equation)

3. For the insulated-tip fin, the heat dissipated is
   A. sqrt(hPkAc) theta_b tanh(mL)
   B. sqrt(hPkAc) theta_b
   C. hPL theta_b
   D. sqrt(hPkAc) theta_b cosh(mL)
   Answer: A. Differentiating theta/theta_b = cosh m(L-x)/cosh mL at x = 0 gives this expression. (Key point: Insulated-tip result)

4. A fin is worthwhile when
   A. effectiveness is below 1
   B. effectiveness is greater than 2
   C. efficiency is 0
   D. h on the fin side is very high, as in boiling
   Answer: B. An effectiveness above 2 means the fin at least doubles the heat transfer; it needs high k and low h. (Key point: Fin worthwhile)

## Related in Heat Transfer

- [Heat dissipation from a pin fin](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Apin-fin-aluminium-heat-dissipation) — Heat Transfer, chapter 6
- [Effectiveness-NTU analysis of a heat exchanger](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Aeffectiveness-ntu-counterflow-oil-water) — Heat Transfer, chapter 6
- [Heat loss from insulated steam pipe](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Ainsulated-steam-pipe-heat-loss) — Heat Transfer, chapter 2

## Related in other subjects

- [Earth cross-section: composition, density, temperature](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Aearth-crosssection-composition-density-temperature) — Engineering Geology I, chapter 2
- [Cantilever tip deflection, temperature gradient](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Abeam-temperature-gradient-cantilever) — Theory of Structures I, chapter 3
- [Heat equation, insulated ends, u(x,0)=lx-x²](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aheat-eq-insulated-ends-lx-x2) — Applied Mathematics, chapter 3, asked 2×
- [Heat equation, insulated ends, u(x,0)=x](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aheat-eq-insulated-ends-u-x) — Applied Mathematics, chapter 3