Extended Surfaces and Fins

Fins extend surface area to enhance convection. For a straight fin of uniform cross-section, with. Tip conditions give standard formulas; efficiency (RC Sachdeva / Incropera).

What is a fin (extended surface)? Why are fins used?

A fin is a protrusion from a surface that increases area for convection (or radiation). Used when is low (gases) to raise heat transfer for a given base-to-fluid.

Write the governing equation for a straight fin of uniform cross-section (steady, 1-D).

where and. perimeter, cross-section.

Define fin efficiency.

— actual heat transfer divided by heat transfer if the entire fin were at base temperature.

Define fin effectiveness. When is a fin justified?

— ratio to heat transfer from the base area if unfinned. Typically require for the fin to be worthwhile.

Give heat transfer for a fin with insulated tip (longitudinal, uniform ).

with (approx. neglecting tip area).

What is an infinitely long fin? Write.

Tip temperature approaches ( large).. Temperature.

How does thermal conductivity of the fin material affect performance?

Higher lowers and raises and toward the ideal. Poor conductors need short, stubby fins; metals (Al, Cu) are preferred.

Explain the corrected fin length for a convective tip.

Approximate a convective tip by an insulated-tip fin of length (rod: or ). Then use formulas.

Effectiveness of an infinitely long fin equals:

since and is the bare-base rate.

True/False: Adding a fin always increases heat transfer from a surface.

False — if is very low or very high, may be; the fin can insulate rather than help.

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Study notes

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Qwestrum Engineering360 · Mechanical Engineering · Heat & Mass Transfer

Extended Surfaces & Fins

Fins extend surface area to enhance convection. For a straight fin of uniform cross-section,

\dfrac{d^2\theta}{dx^2}-m^2\theta=0

with

m=\sqrt{hP/(kA_c)}

. Tip conditions give standard

q_{\mathrm{fin}}

formulas; efficiency

\eta_f=q_{\mathrm{fin}}/q_{\max}

(RC Sachdeva / Incropera).

Exam tip: never replace LMTD with an arithmetic mean of end ΔT unless ΔT_a ≈ ΔT_b; keep U and A in consistent SI units.

Key formulas & points

Skim these first — then read the full notes below.

Fin parameter: $m=\sqrt{\dfrac{hP}{kA_c}}$ , $\theta=T-T_\infty$ .
ODE: $\dfrac{d^2\theta}{dx^2}-m^2\theta=0$ (steady, 1-D, const $h,k$ ).
Infinite fin: $\theta/\theta_b=e^{-mx}$ , $q=\sqrt{hPkA_c}\,\theta_b$ .
Adiabatic tip: $q=\sqrt{hPkA_c}\,\theta_b\tanh(mL)$ .
Convective tip: use corrected length $L_{c} = L+\frac{A_{c}}{P}$ (approx.) with adiabatic formula.
Efficiency: $\eta_f=q_{\mathrm{fin}}/(h A_f\theta_b)$ ; effectiveness $\varepsilon_f=q_{\mathrm{fin}}/(h A_c\theta_b)$ .
Fins justified when $\varepsilon_f>2$ typically; high $k$ , adequate $\frac{hP}{(kA_{c})}$ .

Topic details

Definition and physical meaning

An extended surface (fin) is a protrusion from a hot (or cold) wall that increases the area exposed to convection, raising total heat transfer when the wall area alone is insufficient (electronics cooling, IC engines, heat exchangers).

Excess temperature:

\theta(x)=T(x)-T_\infty

Fin parameter:

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

where

P

= perimeter exposed to convection,

A_{c}

= cross-sectional area for conduction along the fin.

Physical parameters

Symbol	Meaning	SI unit
$T$ , $T_\infty$ , $T_{b}$	Local / ambient / base temperature	$\mathrm{K}$
$\theta$ , $\theta_b$	Excess temperature	$\mathrm{K}$
$h$	Convective coefficient	$\mathrm{W/(m^2\cdot K)}$
$k$	Fin conductivity	$\mathrm{W/(m\cdot K)}$
$A_{c}$	Cross-sectional area	$\mathrm{m^2}$
$P$	Perimeter	$\mathrm{m}$
$L$	Fin length	$\mathrm{m}$
$m$	Fin parameter	$\mathrm{1/m}$
$q_{\mathrm{fin}}$	Heat transfer through fin	$\mathrm{W}$
$\eta_f$	Fin efficiency	—
$\varepsilon_f$	Fin effectiveness	—

Fig — Straight fin of uniform cross-section

Schematic diagram for study — aligned with standard B.Tech / GATE syllabus.

Core assumptions (state these in exams)

1. Steady state; no heat generation in the fin.
2. One-dimensional conduction along fin length

x

(temperature uniform on each cross-section).
3. Constant

k

h

, and

A_{c}

(uniform cross-section straight fin).
4. Negligible radiation (or lumped into effective

h

).
5. Uniform

T_\infty

and base temperature

T_{b}

prescribed.
6. Contact resistance at the base neglected unless given.

Longitudinal thin fins and pin fins of constant section use the same ODE; annular and triangular fins need modified

A_{c}(x)

or charts.

Derivation summary — fin equation

Energy balance on a slice

dx

: conduction in − conduction out − convection from surface = 0 (steady):

q_x-q_{x+dx}-h(P\,dx)\theta=0

With Fourier

q_x=-kA_c d\theta/dx

and

dq_{x} = (\frac{dq_{x}}{dx})dx

\frac{d}{dx}\left(kA_c\frac{d\theta}{dx}\right)-hP\theta=0

For constant

kA_{c}

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

General solution:

\theta(x)=C_1\sinh(mx)+C_2\cosh(mx)

(or equivalent exponentials). Constants from BCs at base and tip.

Standard tip cases and heat rates

Case 1 — infinitely long fin (

\theta\to 0

x\to\infty

\frac{\theta}{\theta_b}=e^{-mx},\qquad q_{\mathrm{fin}}=\sqrt{hPkA_c}\,\theta_b

Case 2 — adiabatic tip (

d\theta/dx|_{x=L}=0

\frac{\theta}{\theta_b}=\frac{\cosh\big[m(L-x)\big]}{\cosh(mL)}

q_{\mathrm{fin}}=\sqrt{hPkA_c}\,\theta_b\tanh(mL)

Case 3 — convective tip (exact hyperbolic form with

h

at tip). Engineering approximation: replace

L

by corrected length

L_c=L+\frac{A_c}{P}

and use the adiabatic-tip formula with

L_{c}

(very common in GATE/Sachdeva).

Case 4 — prescribed tip temperature

T(L) = T_{L}

: use both end BCs on the general solution.

Efficiency, effectiveness, and when to use fins

Fin efficiency (actual heat / heat if entire fin at

T_{b}

\eta_f=\frac{q_{\mathrm{fin}}}{h A_f\theta_b}

For adiabatic tip,

A_{f} = PL

, and

\eta_f=\tanh(mL)/(mL)

Fin effectiveness (heat with fin / heat through base area

A_{c}

without fin):

\varepsilon_f=\frac{q_{\mathrm{fin}}}{h A_c\theta_b}

For infinite fin:

\varepsilon_f=\sqrt{kP/(hA_c)}=k m A_c/(h A_c)\cdots=\sqrt{kP/(h A_c)}

Design guidance
- Prefer high

k

(Al, Cu) and geometries with large

\frac{P}{A_{c}}

(thin fins).
- Fins help most when

h

is low (gas cooling); less benefit in boiling/high-

h

liquids.
- Rule of thumb: use fins if

\varepsilon_f\gtrsim 2

Step-by-step problem approach

1. Identify fin type (pin, rectangular longitudinal) and tip condition.
2. Compute

A_{c}

P

, then

m=\sqrt{hP/(kA_c)}

.
3.

\theta_b=T_b-T_\infty

.
4. Select

q

formula (infinite /

\tanh(mL)

L_{c}

).
5. If asked,

\eta_f=q/(h A_f\theta_b)

\varepsilon_f=q/(h A_c\theta_b)

.
6. Units: mm → m for

A_{c}

P

L

;

h

in W/(m²·K),

k

in W/(m·K).
7. Check

mL

: if

mL\gtrsim 3

, infinite-fin formula is often adequate.

Common mistakes in exams

• Using diameter instead of radius when computing pin

A_c=\pi d^2/4

and

P=\pi d

.
• Forgetting corrected length for convective tip.
• Mixing

\eta_f

and

\varepsilon_f

definitions.
• Using °C absolute incorrectly — only

\Delta T=\theta

matters here.
• Applying uniform-

A_{c}

formulas to triangular/annular fins without modification.
• Inconsistent mm/m in

m

(which has units 1/m).

Calculator

Fin parameter m

m = \sqrt{\frac{h P}{k A_c}}

hW/m²·KPerimeter PmConductivity kW/m·KArea A_cm²

Result

7.07111/m

m = √(hP/(kA_c)) = √((25×0.04)/(200×1.000e-4)) = 7.071 m⁻¹

Worked examples

Try the problem first — open the solution when you are ready to check.

Adiabatic-tip straight fin heat rate

Problem

An aluminium fin (

k=200\,\mathrm{W/(m\cdot K)}

) has

L=50\,\mathrm{mm}

, rectangular section

t=2\,\mathrm{mm}

w=40\,\mathrm{mm}

(use

A_{c} = t w

P\approx 2(w+t)

T_b=100^\circ\mathrm{C}

T_\infty=25^\circ\mathrm{C}

h=25\,\mathrm{W/(m^2\cdot K)}

. Assume adiabatic tip. Find

q_{\mathrm{fin}}

Solution