This question was previously asked in

UPPSC AE Mechanical 2019 Official Paper II (Held on 13 Dec 2020)

Option 2 : \(\frac{{\tanh mL}}{{mL}}\)

__ Explanation__:

**Fin efficiency**:

The efficiency of a fin is defined as the ratio of the actual heat transferred by the fin to the maximum heat transferable by fin, if the entire fin area were at base temperature, for an infinitely long fin.

\(\eta_{fin}=\frac{(\dot Q_{fin})_{actual}}{(\dot Q_{fin})_{maximum}}\)

where \((̇ Q_{fin})_{maximum}\) = hA_{fin}(T_{0} - T_{∞});

where A_{fin} = surface area of the fin = Perimeter × Length.

**Case I**:

**Insulated tip**:

\((\dot Q_{fin})_{actual}=\sqrt{hPkA_c}(T_o-T_{\infty})\tan h(mL)\)

\(\eta_{fin}=\frac{(\dot Q_{fin})_{actual}}{(\dot Q_{fin})_{maximum}}\)

\(\eta_{fin}=\frac{\sqrt{hPkA_c}(T_o-T_{\infty})\tan h(mL)}{h(pL)(T_o-T_{\infty})}=\frac{\tanh(mL)}{mL}\)

where \(m=\sqrt \frac{hP}{kA_c}\)

**Case II**:

**Infinitely long fin**:

\((\dot Q_{fin})_{actual}=\sqrt{hPkA_c}(T_o-T_{\infty})\)

\(\eta_{fin}=\frac{(\dot Q_{fin})_{actual}}{(\dot Q_{fin})_{maximum}}\)

\(\eta_{fin}=\frac{\sqrt{hPkA_c}(T_o-T_{\infty})}{h(pL)(T_o-T_{\infty})}=\frac{1}{mL}\)

where \(m=\sqrt \frac{hP}{kA_c}\)