$($a$)$ The energy loss per unit mass due to an expansion in a circular channel can be written as

$E_x=\frac{(u_1-u_2{)}^2}{2}$

with $u_1$ and $u_2$ the average velocities in the smaller and larger section of the pipe, respectively.

$($i$)$ Draw a fully labelled diagram of the pipe and include a control volume enabling a momentum balance for the calculation of $E_x.$

$($ii$)$ By applying the momentum balance and the modified Bernoulli's equation, derive the above equation for $E_x.$

$($b$)$ Consider the pipe in Part $($a$).$ If the pipe diameters are $d_1=0\mathrm{.}1m$ and $d_2=0\mathrm{.}2m,$ and the mass flow rate is $9\mathrm{.}5kg{s}^{-1},$ calculate the expansion head loss. Assume the liquid density to be $1000kg{m}^{-3}.$

$($c$)$ Consider the pipe in Part $($a$).$ Draw a control volume enabling the calculation of the force required to keep the pipe stationary and provide an expression for the force.