Water of density 1000 kg m\(^{-3}\) flows in a horizontal pipe of 10 cm diameter at an average velocity of 0.5 m s\(^{-1}\). The following plot shows the pressure measured at various distances from the pipe entrance.
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For fully developed flow, use the Darcy-Weisbach equation to find friction factor using pressure drop and pipe dimensions.
To determine the Fanning friction factor in the pipe when the flow is fully developed, we can use the Darcy-Weisbach equation:
\[
\Delta P = f \cdot \frac{L}{D} \cdot \frac{\rho V^2}{2}
\]
where:
\(\Delta P\) is the pressure drop (Pa),
\(f\) is the Fanning friction factor,
\(L\) is the length of the pipe,
\(D\) is the diameter of the pipe,
\(\rho\) is the fluid density (1000 kg/m³), and
\(V\) is the velocity of the fluid (0.5 m/s).
From the plot, we can see the pressure at the pipe entrance (\(P_1 = 1205 \, \text{Pa}\)) and at a distance of 6 meters (\(P_2 = 1000 \, \text{Pa}\)). The pressure drop \(\Delta P = P_1 - P_2 = 1205 - 1000 = 205 \, \text{Pa}\).
We know that the flow is fully developed after a certain distance (here 6 meters), so we will use the pressure drop over this length \(L = 6 \, \text{m}\).
Substituting the values into the equation:
\[
205 = f \cdot \frac{6}{0.1} \cdot \frac{1000 \times 0.5^2}{2}
\]
Solving for \(f\):
\[
f = \frac{205 \cdot 0.1}{6 \cdot 250} = 0.0074
\]
Thus, the Fanning friction factor is 0.0074.
