Answer:
The power required to drive the pump is reduced in [tex]\frac{1}{8}[/tex].
Explanation:
From Turbomachinery, we find that power required to drive the pump ([tex]\dot W[/tex]) is directly proportional to the cube of the speed of rotation of the impeller ([tex]\dot n[/tex]). That is:
[tex]\dot W \propto \dot n^{3}[/tex]
[tex]\dot W = k\cdot \dot n ^{3}[/tex] (Eq. 1)
Where [tex]k[/tex] is the proportionality constant.
And now we eliminate the proportionality constant by constructing the following relationship:
[tex]\frac{\dot W_{B}}{\dot W_{A}} = \left(\frac{\dot n_{B}}{\dot n_{A}}\right)^{3}[/tex] (Eq. 2)
If we know that [tex]\frac{\dot n_{B}}{\dot n_{A}} = \frac{1}{2}[/tex], then:
[tex]\frac{\dot W_{B}}{\dot W_{A}} = \frac{1}{8}[/tex]
The power required to drive the pump is reduced in [tex]\frac{1}{8}[/tex].