Final answer:
Using the Darcy-Weisbach equation, the static pressure at the outlet of the pipe is calculated to be approximately 65.85234 kPa. However, since no answer option closely matches this result, it is likely that there has been an error in the provided options or in the calculation. The closest given answer (c. 4.2 kPa) does not accurately reflect the calculated result.
Step-by-step explanation:
To find the static pressure at the outlet of the steel pipe, the Darcy-Weisbach equation can be used, which is derived from the Bernoulli's equation and takes into account head loss due to friction in pipes. The pressure drop through the pipe can be calculated as:
ΔP = f (L/D) (ρv^2 / 2)
Where:
- f is the friction factor
- L is the length of the pipe
- D is the diameter of the pipe
- ρ is the density of water
- v is the mean velocity of the fluid flow
Given values:
- f = 0.0259
- L = 20 m
- D = 0.025 m
- Flow rate (Q) = 4.5 m^3/h
- Static pressure at inlet (P_inlet) = 70 kPa = 70,000 Pa
First, we convert the volume flow rate to m^3/s:
Q = 4.5 m^3/h = 4.5 / 3600 m^3/s
Then, we calculate the velocity (v) using Q = vA, where A is the cross-sectional area of the pipe, A = π(D/2)^2:
v = Q / A = Q / (π(D/2)^2)
The density (ρ) of water at approximately 10°C is about 1000 kg/m^3. Now we calculate the velocity (v):
v = (4.5/3600) / (π(0.025/2)^2) = 2.563 m/s
Next, we calculate the pressure drop (ΔP):
ΔP = 0.0259 * (20 / 0.025) * (1000 * 2.563^2 / 2) Pa
ΔP = 4147.66 Pa
The static pressure at the outlet (P_outlet) is then P_inlet - ΔP:
P_outlet = 70000 Pa - 4147.66 Pa = 65852.34 Pa
Converting back to kPa:
P_outlet = 65.85234 kPa
The closest answer from the options provided is: