P$5($optional$)$: An orifice plate flow meter, as shown in figure below, is situated in a horizontal pipe of $90$ mm diameter $($di$)$ in which water flows. The diameter of the sharp edged orifice $($do$)$ is $30$ mm $.$ The coefficient of discharge for the flow meter is $0\mathrm{.}65,$ and the coefficient of contraction is $0\mathrm{.}67.$ The static pressure difference across the orifice is $0\mathrm{.}3$ bar.

$(1)$ Apply Bernoulli's equation and the Continuity equation between positions $1$ and $2,$ and combine them together to derive an equation for the water velocity at position $1.$

$(2)$ Determine the equation for the ideal $($theoretical$)$ volumetric flow rate

$(3)$ Calculate the cross$-$sectional areas of the pipe and vena contracta.

$(4)$ Evaluate the value for the ideal $($theoretical$)$ volumetric flow rate of the water in the pipe.

$(5)$ Evaluate the value for actual $($real$)$ mass flow rate of the water in the pipe and find the coefficient of velocity for the orifice plate flow meter.