# Fin effectiveness: Definition, Equation [with Pdf]

## What is fin effectiveness in heat exchanger?

Fin effectiveness is the ratio of the rate of heat transfer by use of fin to the rate of heat transfer without fin.

Therefore,

\epsilon=\frac{Q_{\text{Fin}}}{Q_{\text{Without fin}}}

The value of effectiveness indicates the increase in the rate of convective heat transfer from the body after the addition of fins, therefore the value of effectiveness should be greater than 1.

## Fin effectiveness equation for different fin conditions:

A) For infinitely long fin:-

The rate of heat transfer for the infinitely long fin is given by,

Q_{\text{Fin}}=\sqrt{[PhKA]}.(t_{o}-t_{a})

Where,
A = Cross-setional area of fin
h = Convective heat transfer coefficient
t_{0} = Fin base temperature
t_{a} = Surrounding temperature
K = Thermal conductivity of the fin material
P = Cross-section perimeter

The rate of heat transfer without fin is given by,

Q_{\text{Without fin}}=hA.(t_{o}-t_{a})

Now the effectiveness of infinitely long fin is given by,

\epsilon = \frac{Q_{\text{Fin}}}{Q_{\text{Without fin}}} = \frac{\sqrt{[PhKA]}.(t_{o}-t_{a})}{hA.(t_{o}-t_{a})}

= \frac{\sqrt{[PhKA]}}{hA}

\epsilon=\sqrt{\frac{PK}{hA}}

This is the fin effectiveness equation for infinitely long fin.

From above equation, we can say that,

i) The effectiveness of the heat exchanger increases with an increase in the K/h ratio.
ii) It also increases with an increase in the P/A ratio.
iii) The high thermal conductivity of material gives higher effectiveness.

B) For the fin with insulated tip:-

The rate of heat transfer for the fin with insulated tip is given by,

Q_{\text{Actual}}=\sqrt{[PhKA]}.(t_{o}-t_{a}).tanh(ml)

The rate of heat transfer without fin is given by,

Q_{\text{Without fin}}=hA.(t_{o}-t_{a})

Therefore effectiveness is given by,

\epsilon = \frac{Q_{\text{Fin}}}{Q_{\text{Without fin}}} = \frac{\sqrt{[PhKA]}.(t_{o}-t_{a}).tanh(ml)}{hA.(t_{o}-t_{a})}

= \frac{\sqrt{[PhKA]}\times tanh(ml)}{hA}

\epsilon=\sqrt{\frac{PK}{hA}}.tanh(ml)

This is the fin effectiveness equation for fin with insulated tip. 