The efficiency of a pin fin with insulated tip is

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(T₀ - 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 \(m=\sqrt \frac{hP}{kA_c}\)