Thin-walled aluminum tubes of diameter D = 10 mm are used in the condenser of an air conditioner. Under normal operating conditions, a convection coefficient of h_i = 5000 W/m² ∙ K is associated with condensation on the inner surface of the tubes, while a coefficient of ho = 100 W/m² ∙ K is maintained by airflow over the tubes.

(a) What is the overall heat transfer coefficient if the tubes are un-finned?

(b) What is the overall heat transfer coefficient based on the inner surface U_i if aluminum annular fins of thickness t = 1.5 mm. outer diameter D_o = 20 mm, and pitch S = 3.5 mm are added to the outer surface? Base your calculations on a 1-m-long section of tube. Subject to the requirements that t > 1 mm and (S – t) > 1.5 mm. explore the effect of variations in t and S on U_i, what combination of t and S would yield the best heat transfer performance?