The standard value for the density of water is \(\rho = 1000\;{\rm{kg/}}{{\rm{m}}^3}\).
The velocity relation between the ends of pipe by using discharge concept is calculated below:
\(\begin{array}{c}{A_1}{v_1} = {A_2}{v_2}\\\frac{\pi }{4}{\left( {{d_1}} \right)^2}{v_1} = \frac{\pi }{4}{\left( {{d_2}} \right)^2}{v_2}\\{v_2} = {\left( {\frac{{{d_1}}}{{{d_2}}}} \right)^2}{v_1}\end{array}\)
Here, \({v_1}\) and \({v_2}\) are the velocity of flow at two ends.
The velocity of flow at larger end is calculated below:
\(\begin{array}{l}{P_1} - {P_2} = \frac{1}{2}\rho \left( {{{\left( {{v_2}} \right)}^2} - {{\left( {{v_1}} \right)}^2}} \right)\\{P_1} - {P_2} = \frac{1}{2}\rho \left\{ {{{\left( {{{\left( {\frac{{{d_1}}}{{{d_2}}}} \right)}^2}{v_1}} \right)}^2} - {{\left( {{v_1}} \right)}^2}} \right\}\\{P_1} - {P_2} = \frac{1}{2}\rho \left\{ {\left( {{{\left( {\frac{{{d_1}}}{{{d_2}}}} \right)}^4}{{\left( {{v_1}} \right)}^2}} \right) - {{\left( {{v_1}} \right)}^2}} \right\}\\{P_1} - {P_2} = \frac{1}{2}\rho {\left( {{v_1}} \right)^2}\left( {{{\left( {\frac{{{d_1}}}{{{d_2}}}} \right)}^4} - 1} \right)\end{array}\)
On further simplifying the above relation.
\(\left( {{v_1}} \right) = \sqrt {\frac{{2\left( {{P_1} - {P_2}} \right)}}{{\rho \left( {{{\left( {\frac{{{d_1}}}{{{d_2}}}} \right)}^4} - 1} \right)}}} \)
Substitute the values in the above equation.
\(\begin{array}{c}{v_1} = \sqrt {\frac{{2\left( {33.5\;{\rm{kPa}} \times \frac{{{{10}^3}\;{\rm{Pa}}}}{{1\;{\rm{kPa}}}} - 22.6\;{\rm{kPa}} \times \frac{{{{10}^3}\;{\rm{Pa}}}}{{1\;{\rm{kPa}}}}} \right)}}{{\left( {1000\;{\rm{kg/}}{{\rm{m}}^3}} \right)\left( {{{\left( {\frac{{6\;{\rm{cm}}}}{{4.5\;{\rm{cm}}}}} \right)}^4} - 1} \right)}}} \\{v_1} = \sqrt {10.10\;{{\rm{m}}^2}{\rm{/}}{{\rm{s}}^2}} \\{v_1} = 3.17\;{\rm{m/s}}\end{array}\)
