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# The Darcy-Weisbach equation states that the power- ISBN: 9780073401331 38

## Solution for problem 5SE Chapter 3

Statistics for Engineers and Scientists | 4th Edition

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Problem 5SE

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 = η γ QH. where η is the efficiency of the turbine, γ is the specific gravity of water, Q is the flow rate, and H is the head loss. Assume that η = 0.85 ± 0.02, H = 3.71 ± 0.10 m, Q = 60± 1 m3/s, and γ= 9800 N/m3 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 η to 0.01, reducing the uncertainty in H to 0.05, or reducing the uncertainty in Q to 0.5?

Step-by-Step 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.

Step 2 of 4:

Here the Darcy- Weisbach is given as

P= Since , =0.02 (uncertainty) , (uncertainty), H= 3.71 , So

P= =  (0.85 3.71)

=  1.854 watt

To find the uncertainty in the power loss we have to find the uncertainty of P due to each factor.  = =  ( 9800 =  218148 = =  ( 0.85 )

=  30904.3 = =  ( 0.85 )

=  499800

So the uncertainty due to all factors is = =     73188.9 [in watts]

=    .07318  watt

So the  estimate of the power loss in watts is P = ( 1.854 watts. And the uncertainty in the power loss is 0.073.

Step 3 of 4

Step 4 of 4

##### ISBN: 9780073401331

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