Provide the expression for fin efficiency for a planar fin with convection.

• A. η_fin = tanh(m L) /(m L) where m = sqrt(h P /(k A))
• B. η_fin = tan(m L) /(m L)
• C. η_fin = sinh(m L) /(m L)
• D. η_fin = cosh(m L) /(m L)

Answer: (A)

The key idea is how a planar fin conducts heat to a surrounding fluid when the tip loses heat by convection. Solve the fin equation for a constant-cross-section fin: the dimensionless temperature theta = (T − T∞)/(Tb − T∞) satisfies d^2 theta/dx^2 − m^2 theta = 0 with m^2 = h P /(k A). The general solution is a combination of hyperbolic functions, and applying the base temperature condition theta(0) = 1 along with the convective boundary condition at the tip gives the fin heat transfer Q_fin = h P (Tb − T∞) [tanh(mL)/(mL)]. Fin efficiency is defined as the ratio of this fin heat transfer to the heat transfer from the same base area if it were at Tb, which yields eta_fin = tanh(mL)/(mL). The parameter m is m = sqrt(h P /(k A)). This is why the correct expression uses tanh(mL) divided by (mL). Other forms do not satisfy the boundary conditions for a fin with convective tip, so they don’t represent the fin efficiency in this scenario.