$3.(34$ points$)$ Milk is concentrated in a double effect, feed forward, evaporator at a feed rate of $1000$ lb$/$hr$.$ We wish to concentrate the milk from an initial concentration of $87\%$ water, $13\%$ solids to a final concentration of $74\%$ water, $26\%$ solids, i$.$e$.$ double the solids concentration. Dry saturated steam is available to the first effect at $5\mathrm{.}3$ psig; the pressure in the vapor space of the second effect is $5$ psia. The temperature in the first effect is $190$oF$,$ the maximum allowable. The overall heat transfer coefficients are $150$ and $140$ BTU$/$hr$-$ft$2-$oF$,$ in the first and second effects, respectively. Ignore BPE $($any dissolved fat or protein are large molecules and won$$t have a BPE$),$ heat losses, heats of dilution, and any subcooling of the condensate. Calculate:

a$.$ The necessary feed temperature $(79$oF$)$

b$.$ The steam usage rate $(363$ lb$/$hr$)$

c$.$ The economies per effect and overall $(0\mathrm{.}67$; $1\mathrm{.}07$; $1\mathrm{.}38)$

d$.$ Heat transfer area required per effect assuming two identical effects $(61$ ft$2)$

Since both feed and product solutions are comprised of $74\%$ or more H$2$O$,$ you can use Cp of water for both, Cp $=1$ BTU$/$lb$.$oF