This question aims to find the **length** of the pin fin of a uniform fabricated **aluminum alloy** and its **effectiveness** in accounting for tip convection.

The question is based on the concepts of **convection heat transfer.** **Convection heat transfer** is the movement of heat from one medium to another due to **fluid motion**. We can calculate the heat transfer using the **thermal conductivity** of the metal, its **efficiency,** and **heat transfer coefficient.**

## Expert Answer

The information is given in the problem to find the **length $L$** of the fin; its **effectiveness $\varepsilon_f$** is given as follows:

\[ \text{Thermal Conductivity, $k$}\ =\ 160\ W/mK \]

\[ \text{Diameter, $D$}\ =\ 4 mm \]

\[ \text{Fin Efficiency , $\eta_f$}\ =\ 0.65 \]

\[ \text{Heat Transfer Coefficient, $h$}\ =\ 220\ W/m^2K \]

a) To find the **length $L$** of the **fin,** we will use the **efficiency** formula given as:

\[ \eta_f = \dfrac{ \tanh mL_c} {m L_c} \]

**$m$** is the **effective mass** of the **fin.** We can find the value for **$m$** by using this formula:

\[ m = \sqrt{ \dfrac{4 h} {D k}} \]

Substituting the values, we get:

\[ m = \sqrt{ \dfrac{4 \times 220} {4 \times 10^{-3} \times 160}} \]

By solving, we get:

\[ m = 37.08\ m^ {-3} \]

Putting this value of **effective mass $m$** in the formula for **efficiency,** we get:

\[ 0.65 = \dfrac{ \tanh (37.08 \times L_c)} {37.08\ L_c} \]

Solving for $L_c$, we get:

\[ L_c = 36.2\ mm \]

**$L_c$** is the **convection length** of the fin. To find the **length $L$** of the fin, we can use the following formula:

\[ L = L_c\ -\ \dfrac {D} {4} \]

\[ L = 36.2\ -\ \dfrac {4} {4} \]

\[ L = 35.2\ mm \]

b) The formula gives the **fin effectiveness $\varepsilon_f$:**

\[ \varepsilon_f = \dfrac{ \tanh (m L_c)} {\sqrt {\dfrac {D h} {4 k}}} \]

Putting the value in the above equation, we get:

\[ \varepsilon_f = \dfrac {\tanh (37.08 \times 0.0362)}{\sqrt{ \dfrac{0.004 \times 220} {4 \times 160}}} \]

By solving this equation we get the value of **effectiveness** of the **fin $\varepsilon_f$:**

\[ \varepsilon_f = 23.52 \]

## Numerical Result

The **length $L$** of the fin is calculated to be:

\[ L = 35.2\ mm \]

The **effectiveness** of the **fin $\varepsilon_f$** is calculated to be:

\[ \varepsilon_f = 23.52 \]

## Example

The **diameter** of an **aluminum alloy** is **$3mm$** and its **convection length $L_c=25.6mm$.** Find the length $L$.

\[ \text{Diameter, $D$}\ =\ 3\ mm \]

\[ \text{Convection Length, $L_c$}\ =\ 25.6\ mm \]

Using the formula for finding length $L$, we get:

\[ L\ =\ L_c\ -\ \dfrac {D} {4} \]

\[ L\ =\ 25.6\ -\ \dfrac {3} {4} \]

\[ L\ =\ 24.85\ mm \]

The **length $L$** is calculated to be **$24.85mm$.**