A 7-hp (shaft) pump is used to raise water to an elevation of 15 m. If the mechanical efficiency of the pump is 82 percent, determine the maximum volume flow rate of water.

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A pump with a specified shaft power and efficiency is used to raise water to a higher elevation. The maximum flow rate of water is to be determined.

Assumptions 1 The flow is steady and incompressible. 2 The elevation difference between the reservoirs is constant. 3 We assume the flow in the pipes to be frictionless since the maximum flow rate is to be determined,

Properties We take the density of water to be [latex] ho=1000 \mathrm{~kg} / \mathrm{m}^3[/latex].

Analysis The useful pumping power (the part converted to mechanical energy of water) is

[latex]\displaystyle \dot{W}_{ ext {pump,u }}=\eta_{ ext {pump }} \dot{W}_{ ext {pump, shaft }}=(0.82)(7 \mathrm{~hp})=5.74 \mathrm{~hp}[/latex]

The elevation of water and thus its potential energy changes during pumping, but it experiences no changes in its velocity and pressure. Therefore, the change in the total mechanical energy of water is equal to the change in its potential energy, which is [latex] g z[/latex] per unit mass, and [latex] \dot{m} g z[/latex] for a given mass flow rate. That is,

[latex]\displaystyle \Delta \dot{E}_{\mathrm{mech}}=\dot{m} \Delta e_{\mathrm{mech}}=\dot{m} \Delta p e=\dot{m} g \Delta z= ho \dot{\boldsymbol{V}} g \Delta z[/latex]

Noting that [latex] \Delta \dot{E}_{ ext {mech }}=\dot{W}_{ ext {pump }, \mathrm{u}}[/latex], the volume flow rate of water is determined to be

[latex]\displaystyle \dot{\boldsymbol{V}}=\frac{\dot{W}_{ ext {pump,u }}}{ ho g z_2}=\frac{5.74 \mathrm{~hp}}{\left(1000 \mathrm{~kg} / \mathrm{m}^3 ight)\left(9.81 \mathrm{~m} / \mathrm{s}^2 ight)(15 \mathrm{~m})}\left(\frac{745.7 \mathrm{~W}}{1 \mathrm{~hp}} ight)\left(\frac{1 \mathrm{~N} \cdot \mathrm{m} / \mathrm{s}}{1 \mathrm{~W}} ight)\left(\frac{1 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}^2}{1 \mathrm{~N}} ight)=\mathbf{0 . 0 2 9 1} \mathbf{~m^3/s}[/latex]

Discussion This is the maximum flow rate since the frictional effects are ignored. In an actual system, the flow rate of water will be less because of friction in pipes.

Question 2.75: A 7-hp (shaft) pump is used to raise water to an elevation o......

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