Question

The outflow pipe of a water (1000 ?? ?3 ) tower is 20 m above the ground. The outflow pipe leaves the tower and drop down into the ground. At 5 m below the ground, the pipe then runs horizontally for some distance. At some point the pipe comes back up out of the ground and runs horizontally at 1 m above it. The diameter of the pipe as it leaves the tower is 50 cm. The diameter of the pipe while it is underground is 10 cm. The diameter of the pipe that is 1 m above the ground is 2 cm. The pressure in the pipe while it is underground is 506500 ??. The pressure in the pipe while it is 1m above the ground is 101300 ??.

a) Find the velocity of the water in the underground portion of the pipe. ??????: 1.05 ? ?

b) Find the velocity of the water in the pipe that is 1 ? above the ground. ??????: 26.3 ? ?

c) Find the velocity of the water in the pipe as it leaves the water tower. ??????: 0.04 ? ? d) Find the pressure of the fluid as it leaves the water tower. ??????: 262054 ?�

Answer 1

h1 = +20 m above ground ; diameter of pipe at top = d1= 0.5 m; Radius = r1 =0.25 m P1=? ; V1 = ?

h2= +1 m above ground ; dia of pipe= d2= 0.02 m; Radius =r2 =0.01m; Pressure =P2 = 101300 Pa; V2=?

h3= -5m below ground; dia pipe = d3 = 0.1 m Radius = r3 = 0.05m Pressure=P3 = 506500 Pa ; V3 = ?

Applying Bernoulli's Equation:

P1 + 0.5 x 1000 Kg/m^3 x v1^2 + 1000 Kg/m^3 X 9.8 m/s^2 X 20 m =

101300 Pa + 0.5 x 1000 Kg/m^3 X (v2)^2 + 1000 Kg/m^3 x 9.8 m/s^2 x 1 m =

506500 Pa + 0.5 x 1000 Kg/m^3 X (v3)^2 + 1000 Kg/m^3 x 9.8 m/s^2 x (-5) m

V3 x a3 = V2 x a2 , where a = area of the pipe ( this because in steady state volume of water flowing across any given cross section per second is same)

V3 = V2 x a2 / a3 = V2 x [ (0.01 m)^2/ (0.05 m)^2 ] = 0.04 V2 V3 = 0.04 x V2

101300 Pa + 500 V2^2 + 9800 = 506500 Pa + 500 V3^2 - 49000

500 V2^2 - 500 V3^2 = 506500 - 101300 - (49000+9800)

500 ( V2^2 - 0.04^2 x v2^2 ) = 346400

v2^2 ( 1- 0.04^2) = 692.8

V2 = 26.3 m/s

V3 = 0.04 x V2 = 0.04 x 26.3 m/s = 1.05 m/s V3 = 1.05 m/s

V1 x a1 = V2 x a2  

V1 = V2 ( a2/a1) = 26.3 m/s x (0.01m^2 /0.25 m^2 ) = 0.04 m/s V1 = 0.04 m/s

P1 + 0.5 x 1000 Kg/m^3 x (0.04m/s)^2 + 1000 Kg/m^3 X 9.8 m/s^2 X 20 m =

506500 Pa + 0.5 x 1000 Kg/m^3 X (1.05m/s)^2 + 1000 Kg/m^3 x 9.8 m/s^2 x (-5) m

P1 = 0.5 x 1000 (1.05^2 - 0.04^2 ) + 1000 x 9.8 ( -20-5) + 506500

P1 = 262050 Pa