Question

You are working as a summer intern for the Illinois Department of Natural Resources (DNR) and are assigned to some initial work on water resources project. The department will be overseeing the construction of a dam to create a large fresh water lake that will be approximately 14 meters deep. A horizontal pipe 1.2 meters long and 3 cm in diameter will pass through the dam at a depth of 7 meters to allow for release of the water in emergencies and for sampling. In normal situations, a plug will secure the pipe opening.

1)What will be the total force on the left side of the plug?FLEFT =___________ N2)What is the total force on the right side of the plug?FRIGHT =__________ N3)What is the gauge pressure on the plug?Pgauge =___________ Pa4)If the plug were to be removed, how much water would flow out in 3 hours??VOL =_____________ m3

Answer 1

1) F = 71.6 N , 2) F = 120.2 N , 3) P_m= 68.600 Pa, 4) V = 2.4210-5 m³

Explanation:

This is a problem of fluid mechanics, to find the force we must use its definition

P = F / A

F = P A

The area of ​​the circular pipe is

A = π r² = π d²/4

The pressure is given by the expression

P = P_atm + ρ g h

1) the force on the outer side is

P = P_atm

we substitute in the expression of force

F = P_atm π d² / 4

let's calculate

F = 1,013 10⁵ π 0.03²/4

F = 7.16 10¹ N

F = 71.6 N

2) the force on the inner side

the pressure

P = P_atm + ρ g h

P = 1,015 10⁵ + 1,000 9.8 7

P = 1,701 10⁵ Pa

F = 1,701 10⁵ π 0.03² / 4

F = 1,202 10²

F = 120.2 N

3) manometric pressure

Pm = ρ g h

P m = 1000 9.8 7

P_m= 68.600 Pa

4) In this part they ask for the volume that comes out in time t= 3 h

to calculate this volume we can use the flow ratio

Q = A v

V t = A v

V = A v / t

sent is the velocity of the water that comes out, to calculate it we use the Bernoulli equation

we will use index 1 for the lake surface and ionice 2 apa the position of the plug

P₁ + ρ g v₁² + ρ g y₁ = P₂ + ρ g v₂² + ρ g y₂

As the lake has much more capacity than the pipeline, the velocity of the surface of the lake is peeling, in this case we approximate it steel

(P₁-P₂) + ρ G (y₁ -y₂) = ρ g v₂²

1000 9.8 v₂² = ρ g h + 1000 9.8 (7-0)

9800 v₂² = 1000 9.8 7 + 68600

v₂ = √ (137200)

v₂ = 370.4 m / s

t = 3 h (3600s / h) = 10800 s

we substitute in the volume equation

V = π d²/4 370.4 / 10800

V = 2.4210-5 m³