1) A state that shall remain nameless has poor water infrastructure (a-hem- lifo

Question

1) A state that shall remain nameless has poor water infrastructure (a-hem- lifornia). A 0.12m radius pipe is broken. The pipe runs under a freeway that is 100m wide. The break is 10m in length and is located 70m from the entrance to the pipe (so there is 20m of pipe after the break before the water exits). The break has caused both a leak in the system and has hollowed out a cavity under the freeway. a) Water enters the pipe at a rate of 0.5m3/s. What is the speed of the water at the entrance to the pipe? b) At the far side of the freeway the water leaves the pipe with a speed of 5.5 m/s. How much water is lost per second due to the leak? c) Under the freeway, the flowing water has hollowed out a cavity. To determine the size of the cavity the UNLV-trained engineer injects some dye at the entrance to the pipe and measures the time that it arrives at the far end to be 30 seconds. What is the speed of the flow through the broken part of the pipe? d) To estimate the maximum cavity cross sectional area, assume that the flow rate matches the input flow rate throughout the break (so the leak occurs at the end of the cavity rather than the beginning). What is the effective cross sectional area of the cavity? to a middle

Explanation / Answer

1. a. radius of pipe, r = 0.12 m

volume flow rate, inlet, qi = 0.5 m^3/s

hence

speed of ewater at inlet = vi

vi*pi*r^2 = qi

hence

vi = 11.05242 m/s

b. water leaves pipe at ve = 5.5 m/s

exit flow rate qe ve*pi*r^2 = 0.248814 m^3/s

hence wasted water = qw

qw = qi - qe = 0.251185 m^3/s

c. speed of flow through broken part of the pipe = v

then

Av = 0.25118 m^3/s ( A is effective area of cross section of the cavity)

now, t = 30 s

assumiing speed of flow in the broken part be v

t = 70/vi + (10)/v + 20/ve = 30

hence

v = 0.499246592212 m/s

d. cavity cross section area = A

Av = 0.251185 m^3/s

A = 0.503128 m^2

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