A $300m,1^{\text{'}\text{'}}$ schedule $40$ commercial steel pipe is used to trarsport propane at ${105}^{**}C.$ The pressures upstrean and dowrstream are $5\mathrm{.}52$MPa and $4\mathrm{.}14$MPa respect vely. At such high pressures gropane does not behwe ideally, therefore the followirg PV data is used;

$\backslash$table$[[P(Mpa),,V^{**}{10}^3(\frac{m^2}{k}g),$h $($at ${105}^{**}C$ in cP$)],[4\mathrm{.}14,9\mathrm{.}46,0\mathrm{.}0172,],[4\mathrm{.}48,6\mathrm{.}19,0\mathrm{.}0192,],[4\mathrm{.}65,5\mathrm{.}19,0\mathrm{.}0200,],[4\mathrm{.}83,4\mathrm{.}39,0\mathrm{.}0280,],[5\mathrm{.}00,3\mathrm{.}75,0\mathrm{.}0356,],[5\mathrm{.}17,3\mathrm{.}26,0\mathrm{.}0369,],[5\mathrm{.}52,3\mathrm{.}53,0\mathrm{.}0380,]]$

a$)$ Calculate the Mass flow rate $(k\frac{g}{s}$

Hints

a$)$ Use the $F$ balance in thes form ${\displaystyle \underset{1}{\overset{2}{}}}\frac{dF}{F}+G^{2}ln(\frac{v_1}{v_2})+\frac{2r^2{c}^2}{s_4}$

b$)$ Use the PV data to evaluate$/$estimate ${\displaystyle \underset{1}{\overset{2}{}}}\frac{dr}{V}$; integrate graphically or numerically.

c$)$ Estimate I $=0\mathrm{.}005$ fi$.$why??