The separately uploaded figure shows a chemical processing system, where liquid reagents $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ react in the elevated tank, and produce a very corrosive product $\backslash ($ C $\backslash ).$ The reaction requires equal volume ratio of $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ meaning half of the tank should be filled by each of these. The volume of the product C will be exactly sum of the volumes of the reagent, meaning the product $\backslash ($ C $\backslash )$ will fill the tank after the reaction if $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ each occupies half of the tank.

As Cis very corrosive, the tank is pre$-$fabricated with fixed vertical height $5$ m and fixed $5$ m by $4$ m cross$-$sectional dimensions. Also, it has two inlets of $2\mathrm{.}5$ cm diameter at the top for $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash ),$ while it drains from the bottom via an outlet of $2\mathrm{.}5$ cm diameter. The product C cannot be pumped out, because no pump can survive handling it$.$ This means that the only way C can be taken out of the tank by draining under gravity from its elevated location. Only one kind of properly treated pipe of diameter $2\mathrm{.}5$ cm can be used for this purpose. The product has to be delivered in two separate locations as shown in the figure.

Reagents $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ are pumped into the system by two separate pumps. However, one needs to ensure velocity of liquid $\backslash ($ A $\backslash )$ does not exceed $\backslash (25\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash$cdots $\backslash )$ otherwise it would be unstable. Also, take the diameter of inlet pipes to be $2\mathrm{.}5$ cm $.$

Part $1,$ you have to design the optimum elevation and the piping network so that the system becomes most profitable. For each batch, the company spends $\backslash (\backslash$$ $502\backslash )$ in labor, transport and materials, but gets $\backslash (\backslash$$ $645\backslash )$ as revenue. The time required for a batch is summation of the time to fill and time to drain, as instantaneous reaction does not require any settling time as long as A and B are fed in same rates. The operation cost also involves expenses for running two pumps pumping reagents A and B in the inlet, where the industrial rate for energy is $50$ cents per kilo$-$watt$-$hour $($note the unit$).$ Fortunately, there is no investment cost, as the company got zero interest lgan$--$ so you do not have to worry about it$.$

Try to design the elevation for maximum profit. If you have the tank elevation too high, the energy cost will reduce your profit. On the other hand, too low elevation would drain the tank slowly causing loss of revenue. Also, indicate the piping lay$-$out while computing the optimum elevation. At the same time, give an estimate of the annual profit assuming the tank will be continuously fed.

Use the following values for your calculations. The densities of A$,$ B and C are $\backslash (900\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{\backslash$wedge$\}3,1400\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{\backslash$wedge$\}3\backslash )$ and $\backslash (1100\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{\backslash$wedge$\}3\backslash )($those who are concerned about mass conservations in the reaction should note a gas also escapes during the reaction$).$ Take friction factor of all pipes to be $0\mathrm{.}0125$ for turbulent flows. There are no minor losses. The efficiency of the pumps is $0\mathrm{.}8.$

Part $2$

The only class of pump ideal for the inlet flow is identified. This class of pumps has the following data obtained by testing a pump of $20$ cm diameter rotating with $50$ rpm driving water.

Q $($in m$^3/$s$)\backslash (\backslash$quad $0\mathrm{.}00010\mathrm{.}000150\mathrm{.}00020\mathrm{.}000250\mathrm{.}00030\mathrm{.}000350\mathrm{.}00040\mathrm{.}000450\mathrm{.}0005\backslash )$

Delta h $($in meter$)2\mathrm{.}235\mathrm{.}947\mathrm{.}448\mathrm{.}338\mathrm{.}788\mathrm{.}537\mathrm{.}725\mathrm{.}582\mathrm{.}77$

Power input $($in kW$)3\mathrm{.}4312\mathrm{.}3819\mathrm{.}5826\mathrm{.}4933\mathrm{.}0237\mathrm{.}2740\mathrm{.}1134\mathrm{.}6420\mathrm{.}69$

Design the diameter of the two pumps and their operating angular speed for best possible performance for your system. The schematic of the processing plant