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The Darcy-Weisbach equation states that the power-
Chapter 3, Problem 5SE(choose chapter or problem)
The Darcy–Weisbach equation states that the power-generating capacity in a hydroelectric system that is lost due to head loss is given by \(P=\eta \gamma Q H\), where \(\eta\) is the efficiency of the turbine, \(\gamma\) is the specific gravity of water, \(Q\) is the flow rate, and \(H\) is the head loss. Assume that \(\eta=0.85 \pm 0.02\), \(H=3.71\pm0.10\mathrm{\ m}\), \(Q=60\pm1\mathrm{\ m}^3/\mathrm{s}\), and \(\gamma=9800\mathrm{\ N}/\mathrm{m}^3\) with negligible uncertainty.
a. Estimate the power loss (the units will be in watts), and find the uncertainty in the estimate.
b. Find the relative uncertainty in the estimated power loss.
c. Which would provide the greatest reduction in the uncertainty in \(P\): reducing the uncertainty in \(\eta\) to 0.01, reducing the uncertainty in \(H\) to 0.05, or reducing the uncertainty in \(Q\) to 0.5?
Equation Transcription:
Text Transcription:
P = eta gamma QH
eta
gamma
Q
H
eta = 0.85 pm 0.02
H = 3.71 pm 0.10 m
Q = 60 pm 1 m^3 / s
gamma = 9800 N/m^3
P
Questions & Answers
QUESTION:
The Darcy–Weisbach equation states that the power-generating capacity in a hydroelectric system that is lost due to head loss is given by \(P=\eta \gamma Q H\), where \(\eta\) is the efficiency of the turbine, \(\gamma\) is the specific gravity of water, \(Q\) is the flow rate, and \(H\) is the head loss. Assume that \(\eta=0.85 \pm 0.02\), \(H=3.71\pm0.10\mathrm{\ m}\), \(Q=60\pm1\mathrm{\ m}^3/\mathrm{s}\), and \(\gamma=9800\mathrm{\ N}/\mathrm{m}^3\) with negligible uncertainty.
a. Estimate the power loss (the units will be in watts), and find the uncertainty in the estimate.
b. Find the relative uncertainty in the estimated power loss.
c. Which would provide the greatest reduction in the uncertainty in \(P\): reducing the uncertainty in \(\eta\) to 0.01, reducing the uncertainty in \(H\) to 0.05, or reducing the uncertainty in \(Q\) to 0.5?
Equation Transcription:
Text Transcription:
P = eta gamma QH
eta
gamma
Q
H
eta = 0.85 pm 0.02
H = 3.71 pm 0.10 m
Q = 60 pm 1 m^3 / s
gamma = 9800 N/m^3
P
ANSWER:Solution :
Step 1 of 4:
The Darcy- Weisbach equation states that the power - generating capacity in a hydroelectric system that is lost due to head lost is given by P = η γ QH . where is the efficiency of the turbine,
is the specific gravity of water ,
is the flow rate, H is the head loss. Here
, H= 3.71
,Q = 60
m/s , and
with negligible uncertainty.
We have to find
- The estimate of the power loss, and find the uncertainty in the estimate.
- The relative uncertainty in the estimated power loss.
- which would provide the greatest reduction in the uncertainty in P: reducing the uncertainty in η to 0.01, reducing the uncertainty in H to 0.05, or reducing the uncertainty in Q to 0.05.