For installation and operation of a pipeline for an incompressible fluid, the total cost (in dollars per year) can be represented as follows:

\[C=C_{1} D^{1.5} L+C_{2} m \Delta p / ho\]

where

$C_{1}=$ the installed cost of the pipe per foot of length computed on an annual basis $\left(C_{1} D^{1.5}\right.$ is expressed in dollars per year per foot length, $C_{2}$ is based on $\$ 0.05

/ \mathrm{kWh}, 365$ days $/$ year and 60 percent pump efficiency).

\[\begin{aligned}D & =\text { diameter }(\text { to be optimized) } \\L & =\text { pipeline length }=100 \text { miles } \\m & =\text { mass flow rate }=200,000 \mathrm{lb} / \mathrm{h} \\\Delta p & =\left(2 ho v^{2} L / D g_{c}\right) f=\text { pressure drop, psi } \\ho & =\text { density }=60 \mathrm{lb} / \mathrm{ft} 3 \\v & =\text { velocity }=(4 m) /\left(ho \pi D^{2}\right) \\f & =\text { friction factor }=\left(0.046 \mu^{0.2}\right) /\left(D^{0.2} v^{0.2} ho^{0.2}\right) \\\mu & =\text { viscosity }=1 \mathrm{cP}\end{aligned}\]

a. Find general expressions for $D^{\text {opt }}, v^{\text {opt }}$, and $C^{\text {opt }}$

b. For $C_{1}=0.3$ ( $D$ expressed in inches for installed cost), calculate $D^{\text {opt }}$ and $v^{\text {opt }}$ for the following pairs of values of $\mu$ and $ho ; \mu=0.2,10 \mathrm{cP}, ho=50,80 \mathrm{lb} / \mathrm{ft}^{3}$