“…The results indicated that the heat transfer coefficient of the inner tube was larger than that of the outer tube and the heat transfer rate was the highest when heat transfer rates of both sides of the annuli were equal. As to heat exchanger optimization, some researchers chose the sum of investment and optional costs, which were related to heat transfer area and fluid head loss respectively, as objective function (Fesanghary et al, 2009;Ponce-Ortega et al, 2009;Ravagnani and Caballero, 2007), some researchers chose the cost of heat transfer surface area or capital investment as objective function (Costa and Queiroz, 2008;Rao et al, 1991), and some others chose the sum of entropy generation of stream as objective function (Bejan et al, 1996;Johannessen et al, 2002;Zhang et al, 2010). Pontryagin's Maximum Principle (PMP) is proposed by Soviet mathematician Pontryagin (Pontryagin, 1987) and it has been widely used in the field of optimal control (Peng, 1990;Wang and Chen, 1999;Van Damme et al, 2014;Ozparpucu et al, 2014).…”