Two heat exchangers in a circulating water loop transfer heat from a fluid

condensing at $175$F to a boiling fluid at $68$F$,$ as shown in Figure P$6\mathrm{.}14.$ The

overall heat transfer coefficients of the evaporator and condenser are UE $=$

$5\mathrm{.}3$ Btu$/$hft$2$

F and UC $=7$ Btu$/$hft$2$

F$,$ respectively. The first cost of the heat

exchangers is $$21\mathrm{.}50/$ft$2,$ and the present worth of the lifetime pumping costs

is $11\mathrm{.}75[$$h$/$lbm$]$mw $.$ This system must accomplish a heat transfer rate of

$220,000$ Btu$/$h between the condensing and boiling fluids. Using a numerical

optimization algorithm, determine the heat exchanger areas, A$1$ and A$2,$ that

minimize the total present worth of costs.