A multiple$-$effect evaporator is to be used for evaporating a feed of $5\%$ solids to a product of $20\%$ solids with a feed rate of $50,000$ lbM$/$hr$.$ Each sq ft of surface will evaporate $25$ IbM$/$hr under the operating conditions. A $5,000-$sq ft evaporator costs $$25,000.$ If N is the number of effects, the investment costs may be considered proportional to N$,$ and the pertinent annual direct costs will vary as $$48,000/$N$0\mathrm{.}9.$ What is the most economical number of effects to use if $5-$yr life is assumed and interest is neglected?