12 Water is pumped from the Colorado River to supply Grand Canyon Village, locat
ID: 1272312 • Letter: 1
Question
12
Water is pumped from the Colorado River to supply Grand Canyon Village, located on the rim of the canyon. The river is at an elevation of 564 m and the village is at an elevation of 2094 m. Imagine that the water is pumped through a pipe 10.0 cm in diameter, driven by a single pump at the bottom end(a What is the minimum pressure at which the water must be pumped if it is to arrive at the village?(bIf 4000 m3 are pumped per day, what is the speed of the water in the pipe? Note: Assume the free-fall acceleration and the density of air are constant over this range of elevations. The pressures you calculate are too high for an ordinary pipe. The water is actually lifted in stages by several pumps through shorter pipes.
Explanation / Answer
Given that Elevation of the riverh1 = 564m
Elevation of the village h2 = 2094m
Now the height of the village from theriver is h = ( 2094m - 564m )
= 1530m
diameter of the pipe is D =10.0cm
radius of the pipe R is 0.005m
a) The minimum pressure at which the water must bepumped if it is to arrive at the village is
P = P0 + ?gh
? is density of water = 1000kg /m3 , P0 = 1 atm , g = 9.8m/s2
then p = 1 atm + (1000kg /m3 ) (9.8m/s2 ) ( 1530m )
= .14.99 Mpa
b) volume of water pumped per day is V =4000m3
then volume rate of flow V /t = 4000m3 / 86400s
we know that the volume rate of flow V /t = a* v
where a is area of cross section of the pipe is ? R2
= ? ( 0.005m )2
= 0.07854m2
then 4000m3 / 86400s = (0.007854m2 ) V
then the speed of water in the pipe is V =5.89 m/s
c) According to Bernoulli's theorem additionalpressure is necessary to deliver this flow is
?P = 1/2 ? V2
= 1/2 ( 1000kg /m3 ) ( 5.89 m/s )
= 2945 pa.
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