$($Need help with question a$,$d$,$e$,$f$).$

Figure Q$1$a$.$In an industrial plant, $2$ litre plastic bottles are filled with orange juice. The filling process is achieved by draining the juice from a storage tank into the bottle through a straight circular pipe connected to the bottom of the tank $($see Figure Q$1$a$).$ The juice density and dynamic viscosity are $1250kg{m}^{-3}$ and $1\mathrm{.}53{10}^{-3}$Pas, respectively. The height, $H,$ of the storage tank is $9\mathrm{.}0m.$ The tank is open to the atmosphere, and it has a circular cross section with a diameter $D=1\mathrm{.}5m.$ The level of juice in the tank at a time, $t,$ is denoted as $z(t),$ and, at the time $t=0,$ the juice level, $z(0),$ is equal to two third of the tank height. The pipe diameter and length are $d=1cm$ and $L=80cm,$ respectively.

$($a$)$ Calculate how many juice bottles can be filled during the process. Assume that the filling process ends when the juice level reaches the bottom of the tank.$($Answer$=5300)$

$($b$)$ Demonstrate that the juice average velocity at the outlet of the pipe, $u_2,$ can be expressed as

$u_2=\sqrt[\phantom{2}]{2g(z(t)+L)}$

Assume a turbulent flow and neglect the effects of friction.

$($c$)$ Demonstrate that the time, $t,$ taken for the level of juice in the tank to fall from $z(0)$ to $z(t)$ is given by the following expression:

$t=(\frac{D}{d}{)}^2\sqrt[\phantom{2}]{\frac{2}{g}}(\sqrt[\phantom{2}]{z(0)+L}-\sqrt[\phantom{2}]{z(t)+L})$

$($d$)$ Calculate how long it takes for the level of juice in the tank to reach the bottom of the tank $-$ namely, how long it takes to fill the number of bottles calculated at point $($a$).(4\mathrm{.}8$hr$)$

$($e$)$ Show that the flow at the outlet of the pipe is turbulent at any time during the process.

$($f$)$ Assume that the storage tank is pressurised. By denoting as $P$ the tank pressure, determine the new expressions for the juice average velocity at the outlet of the pipe, $u_2,$ and the time, $t,$ taken for the level of juice in the tank to fall from $z(0)$ to $z(t).$