Question #80117

A water pump draws incompressible fluid (density $\rho = 1000 \text{ kg/m}^3$) from a large stationary reservoir at the origin $(0,0,0)$ and delivers it to an open tank at position vector $\vec{r} = (3\hat{i} + 4\hat{j} + 12\hat{k}) \text{ m}$. The delivery pipe is straight and consists of two segments. Segment 1 has length $L_1 = 5 \text{ m}$ and diameter $D = 20 \text{ cm}$. Segment 2 has length $L_2 = 8 \text{ m}$ and diameter $d = 10 \text{ cm}$. The pump maintains a constant volume flow rate $Q = 0.02\pi \text{ m}^3/\text{s}$.

Assume the flow in Segment 1 is ideal with negligible viscous dissipation. In Segment 2, the constriction induces severe turbulence, resulting in a head loss given by the Darcy-Weisbach equation $h_L = f \frac{L_2}{d} \frac{v_2^2}{2g}$, where the friction factor is $f = 0.05$ and $v_2$ is the flow velocity in Segment 2.

Assuming the fluid is discharged at atmospheric pressure with velocity $v_2$, and taking $g = 10 \text{ m/s}^2$ in the $-\hat{k}$ direction, what is the mechanical power output required from the pump?