Solve Fluid Flow Pump System: Volume Flowrate & Power

[RazerM]

I am currently doing revision for an upcoming exam and can't work out how this particular question is solved. Attached is an image of the system.

Homework Statement

A pump draws water from a large tank and delivers it to atmosphere through a 50 mm diameter nozzle at a level 10 m above the free water surface in the tank. The pressure rise across the pump (the pipes entering and leaving the pump have the same diameter) is measured as 136 $\textrm{kN}/\textrm{m}^{2}$.
Calculate the volume flowrate through the system and the power in the discharge jet. Ignore energy losses in the pump, pipeline and nozzle.

Answers given; $q=0.0171\,\textrm{m}^{3}/\textrm{s}$
$W=645\,\textrm{W}$

Homework Equations

$$\begin{align} A&=\frac{\pi}{4}D^{2}\\ q&=Av\\ \textrm{Power} &= \rho q\cdot W_{\textrm{in}} \end{align}$$

The Attempt at a Solution

I am choosing two points to apply bernoulli's equation; the reservoir surface and the point where fluid leaves the nozzle.
$$\begin{align*}P_{1}&=0\,(\textrm{Gauge Pressure})\\ P_{2}&=136\textrm{kN}/\textrm{m}^{2}\\ v_{1}&=0\\ v_{2}&=? \end{align*}$$

Now the equation from what I understand:
$$\begin{align*} \frac{P_{1}}{\rho}+\frac{v_{1}^{2}}{2}+gZ_{1}+W_{\textrm{in}}&=\frac{P_{2}}{\rho}+\frac{v_{2}^{2}}{2}+gZ_{2}+\textrm{losses}\\ W_{\textrm{in}} &= \frac{P_{2}}{\rho}+\frac{v_{2}^{2}}{2}+gZ_{2} \end{align*}$$

Using the given answers for q gives the incorrect Power from equation (3) which means I am misunderstanding where to take values from in the system or how to implement the pump pressure change into the equations.

Any help is appreciated.

 

[Valkarie]

It looks like you are forgetting to take into account the pressure drop across the nozzle. Bernoulli's equation applies between two points, so at the reservoir surface (Point 1) and at the nozzle exit (Point 2). The pressure drop across the nozzle is given by the equation:P2 = P1 - (ρ*V2^2)/2where V2 is the velocity of the water leaving the nozzle. Using this equation, you can calculate the velocity of the water leaving the nozzle, and then use that velocity to calculate the flow rate and power in the discharge jet.