$(20pts)$ A volumetric flow rate $Q$ of isopentane of density ${}_1$ flows through a pipe of crosssectional area $A,$ and then passes through the Venturi "meter" shown in the following figure, whose "throat" cross$-$sectional area is $a.$ A manometer containing mercury $({}_M)$ is connected between an upstream point $($station $1)$ and the throat $($station $2),$ and registers a difference in mercury levels of $h.$

$,q,q,$

$Q=C_{D}A\sqrt[\phantom{2}]{\frac{2({}_m-{}_i)gh}{{}_i(\frac{A^2}{a^2}-1)}}$

$($b$)(10pts$ If the diameters of the pipe and the throat of the Venturi are $6$ in$.$ and $3$ in$.,$ respectivelv. what flow rate $($gDm$)$ of iso$-$ventane $({}_i=38\mathrm{.}75l\frac{b_m}{f}{t}^3)$ would register a $h$

of $20$ in$.$ on a mercury $(s=13\mathrm{.}57)$ manometer? What is the corresponding pressure drop in psi? Assume a discharge coefficient of $C_D=0\mathrm{.}98.[1053gpm$ and $9\mathrm{.}35$ psi $]$

Note: Those parts of the manometer not occupied by the mercury are filled with isopentane since there is free communication with the pipe via the pressure tappings. You may neglect friction in this problem. Manometers are discussed in Lecture Notes $1$ and on p$.93$ of the textbook.