The conditions at two different positions along a pipeline (at points 1 and 2) are related by the Bernoulli equation (see Problem 11). For flow in a pipe,

\[e_{f}=\left(\frac{4 f L}{D}\right)\left(\frac{V^{2}}{2}\right)\]

where

$D$ is the pipe diameter

$L$ is the pipe length between points 1 and 2

If the flow is laminar $\left(N_{R e}<2000\right)$, the value of $\alpha=2$ and $f=16 / N_{R e}$, but for turbulent flow in a smooth pipe, $\alpha=1$ and $f=0.0791 / N_{R e}^{1 / 4}$. The work done by a pump on the fluid $(-w)$ is related to the power delivered to the fluid $(H P)$ and the mass flow rate of the fluid $(\dot{m})$ by $H P=-w \dot{m}$. Consider water $(ho=1 \mathrm{~g} / \mathrm{cc}, \mu=1 \mathrm{cP}$ ) being pumped at a rate of $150 \mathrm{gpm}$ through a $2000 \mathrm{ft}$ long $3 \mathrm{in}$. diameter pipe. The water is transported from a reservoir $(z=0)$ at atmospheric pressure to a condenser at the top of a column that is at an elevation of $30 \mathrm{ft}$ and a pressure of $5 \mathrm{psig}$. Determine

(a) The value of the Reynolds number in the pipe

(b) The value of the friction factor in the pipe (assuming that it is smooth)

(c) The power that the pump must deliver to the water, in horsepower (hp)

Problem 11,

When the energy balance on the fluid in a stream tube is written in the following form, it is known as the Bernoulli equation:

\[ \frac{P_{2}-P_{1}}{ho}+g\left(z_{2}-z_{1}\right)+\frac{\alpha}{2}\left(V_{2}^{2}-V_{1}^{2}\right)+e_{f}+w=0, \]

where

$w$ is the work done on a unit mass of fluid

$e_{f}$ is the energy per unit mass dissipated by friction in the fluid, including all thermal energy effects due to heat transfer or internal generation

$\alpha$ is equal to either 1 or 2 for turbulent or laminar flow, respectively

If $P_{1}=25 \mathrm{psig}, P_{2}=10 \mathrm{psig}, z_{1}=5 \mathrm{~m}, z_{2}=8 \mathrm{~m}, V_{1}=20 \mathrm{ft} / \mathrm{s}, V_{2}=5 \mathrm{ft} / \mathrm{s}, ho=62.4 \mathrm{lb}_{\mathrm{m}} / \mathrm{ft}^{3}, \alpha=1$, and $w=0$, calculate the value of $e_{f}$ in each of the following systems of units:

(a) $\mathrm{SI}$

(b) mks engineering (e.g., metric engineering)

(c) English engineering

(d) English scientific (with $\mathrm{M}$ as a fundamental dimension)

(e) English thermal units (e.g., Btu)

(f) Metric thermal units (e.g., calories)