Through a pipe 17.0 cm in diameter,water is pumped from the Colorado River up to

Question

Through a pipe 17.0 cm in diameter,water is pumped from the Colorado River up to Grand Canyon Village,located on the rim of the canyon. The river is at an elevation of564 m and the village is at an elevation of 2093 m. (Note: Assume that the free-fallacceleration and the density of air are constant over this range ofelevations. (a) What is the minimum pressure at which thewater must be pumped if it is to arrive at the village?
1 MPa

(b) If 4500 m3 are pumped per day, what is the speed ofthe water in the pipe?
2 m/s

(c) What additional pressure is necessary to deliver the flow?
3 kPa (a) What is the minimum pressure at which thewater must be pumped if it is to arrive at the village?
1 MPa

(b) If 4500 m3 are pumped per day, what is the speed ofthe water in the pipe?
2 m/s

(c) What additional pressure is necessary to deliver the flow?
3 kPa

Explanation / Answer

   Given that Elevation of the riverh1 = 564m       Elevation of the village h2 = 2093m     Now the height of the village from theriver is h = ( 2093m - 564m )                                                                           = 1529m     diameter of the pipe is D =17.0cm    radius of the pipe R is 0.0085m 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 ) ( 1529m )                    = ................... Mpa b) volume of water pumped per day is V =4500m3                     then volume rate of flow V /t     =   4500m3 / 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                                                                         = ( 0.0085m )2                                                                    = 0.02669m2                            then 4500m3 / 86400s   = (0.02669m2 ) V                                  then the speed of water in the pipe is V =1.95 m/s c) According to Bernoulli's theorem additionalpressure is necessary to deliver this flow is                       P = 1/2 V2                             = 1/2 ( 1000kg /m3 ) ( 1.95 m/s )                             = 1902 pa.                              

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