Elevated Storage Tank
Consider a pump delivers water from a lake up to an elevated storage tank through a 10 in pipe diameter that is 2,000 ft long. If the Hazen-Williams roughness coefficient is 100 and the pump’s discharge rate is 500 gpm, what is the friction loss? Ignore minor losses and assume turbulent flow.
Expand Hint
Hazen-Williams Equation:
$$$Q=k_1CAR_{H}^{0.63}S^{0.54}$$$
where
$$Q$$
is the discharge rate,
$$k_1$$
is 0.849 for SI units or 1.318 for USCS,
$$C$$
is the Hazen-Williams roughness coefficient,
$$A$$
is the cross sectional flow area,
$$R_H$$
is the hydraulic radius, and
$$S$$
is the energy grade line slope.
Hint 2
$$$S=\frac{h_f}{L}$$$
where
$$h_f$$
is the friction loss, and
$$L$$
is the total length.
Hazen-Williams Equation:
$$$Q=k_1CAR_{H}^{0.63}S^{0.54}$$$
where
$$Q$$
is the discharge rate,
$$k_1$$
is 0.849 for SI units or 1.318 for USCS,
$$C$$
is the Hazen-Williams roughness coefficient,
$$A$$
is the cross sectional flow area,
$$R_H$$
is the hydraulic radius, and
$$S$$
is the slope of energy grade line.
Remember, the hydraulic radius is:
$$$R_H=\frac{cross \: sectional\:area}{wetted\:perimeter}=\frac{D_H}{4}$$$
where
$$D_H$$
is the hydraulic diameter.
Since the energy grade line slope is:
$$$S=\frac{h_f}{L}$$$
where
$$h_f$$
is the friction loss, and
$$L$$
is the total length. The friction loss is:
$$$\frac{Q}{k_1CA(D_{H}/4)^{0.63}}=(\frac{h_f}{L})^{0.54}$$$
$$$h_f=L[\frac{Q}{k_1CA(D_{H}/4)^{0.63}}]^{\frac{1}{0.54}}$$$
$$$=(2,000ft)[\frac{\frac{500gal}{min}\cdot \frac{0.134ft^3}{1gal}\cdot \frac{1min}{60sec}}{(1.318)(100)(\frac{\pi (10/12ft)^2}{4})(\frac{(10/12ft)}{4})^{0.63}}]^{1.85}$$$
$$$=(2,000ft)[\frac{1.117ft^3/s}{(131.8)(0.5451ft^2)(0.372ft)}]^{1.85}=(2,000ft)[\frac{1.117ft^3/s}{26.73ft^3}]^{1.85}$$$
$$$=(2,000ft)(0.04179)^{1.85}=(2,000ft)(0.00281)=5.6\:ft$$$
Keep in mind that the non-applicable units don’t cancel out completely when using the Hazen-Williams equation. It’s important to intentionally alter the units on the final answer. You can always verify the answer via an online calculator.
5.6 ft
Time Analysis
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