Question

What is the difference between the pressure head at the end of a 150m long pipe of diameter 1m coming from the bottom of a reservoir with a water surface 40m above a receiving reservoir delivering 10m3s-1; and water coming through an identical route in an open rectangular channel of width 1m with the same delivery. Assume that the Darcey Weisbach friction factor is 0.0019 and that the Manning n for the channel is 0.013.

Answer 1

`\frac {p_2- p_1}{\rho g} = 31.06 m`

Step-by-step explanation:

from bernoulli's theorem we have

`(p_1)/(\rho g) + (v_1^(2))/(2g) +z_1 = (p_2)/(\rho g) + (v_2^(2))/(2g) +z_2  + h_f`

we need to find pressure head difference i.e.

`\frac {p_2- p_1}{\rho g} = (z_1 - z_2) - h_f`

where h_f id head loss

`h_f = (flv^(2))/(D 2g)`

velocity v =
`(1)/(n) * R^(2/3) S^(2/3)`

`S = (\delta h)/(L) = (40)/(150) = 0.267`

hydraulic mean radius R =
`(A)/(P) = (hw)/(2h+w)`

`R = (40*1)/(2*40+1) = 0.493 m`

so velocity is =
`(1)/(0.013) * 0.493^(2/3) 0.267^(1/2)`

v = 24.80 m/s

head loss

`h_f = (0.0019*150*24.80^(2))/(1* 2*9.81)`

`h_f  =8.93 m`

pressure difference is

`\frac {p_2- p_1}{\rho g} = 40 - 8.93 = 31.06 m`

`\frac {p_2- p_1}{\rho g} = 31.06 m`