You must design and specify equipment for transporting $100 %$ acetic acid (density $=1000 \mathrm{~kg} / \mathrm{m}^{3}$, $\mu=1 \mathrm{mPa}$ s), at a rate of $11.3 \mathrm{~m}^{3} / \mathrm{h}$, from a large vessel at ground level into a storage tank that is $6 \mathrm{~m}$ above the vessel. The line includes $185 \mathrm{~m}$ of pipe and eight flanged elbows. It is necessary to use stainless steel for the system, for which the pipe is hydraulically smooth, and you must determine the most economical diameter to use. You have $38.1 \mathrm{~mm}$ (1.5 in.) and $50.2 \mathrm{~mm}$ (2 in.) nominal Sch. 40 pipe available. The cost may be determined from the following approximate formulas:

Pump cost: $\quad$ Cost $(\$)=3.1\left(\mathrm{~m}^{3} / \mathrm{s}\right)^{0.3}(\mathrm{~m} \text { of head })^{0.25}$

Motor cost: $\quad \operatorname{Cost}(\$)=75(\mathrm{kWh})^{0.85}$

Pipe cost: $\quad$ Cost $(\$) / \mathrm{ft}=2.5$ (Nom. Dia., in. $)^{3 / 2}$

$90^{\circ}$ elbow: $\quad$ Cost $(\$)=5$ (nom. Dia., in. $)^{3 / 2}$

Power cost: $\quad=0.03 \$ / \mathrm{kWh}$

(a) Calculate the total pump head required for each size pipe, in $\mathrm{ft}$ of head.

(b) Calculate the motor hp required for each size pipe, assuming $80 %$ efficiency (motors available only in multiples of $1 / 4 \mathrm{hp}$ ).

(c) Calculate the total capital cost for pipe, pump, motor, and fittings for each size pipe.

(d) Assuming that the useful life of the installation is 5 years, calculate the total operating cost over this period for each size pipe.

(e) Which size pipe results in the lowest total cost over the 5-year period?