Calculate the entropy generation rate contributed by a rectangular plate fin whose dimension parallel to the flow (and to the base wall) is L. The boundary layer that coats the length L is turbulent:...

Calculate the entropy generation rate contributed by a rectangular plate fin whose dimension parallel to the flow (and to the base wall) is L.

The boundary layer that coats the length L is turbulent: It is known that the fin surface shear stress 𝜏_w
and the fin stream heat transfer coefficient h are functions of the longitudinal position x measured along the boundary layer:

such that x = 0 represents the leading edge of the plate fin and x = L, the trailing edge. The free-stream velocity and temperature are U_∞
and T_∞, respectively. If q′′ (constant) is the local heat flux from the fin surface to the turbulent boundary layer, then in this design the heat transfer rate to the entire boundary layer (q′ ) is considered fixed: q′ = q′′L. By minimizing the total entropy generation rate Ṡ ′
_gen
of the L-wide fin, determine the optimal L that should be placed in contact with the flow.