Question

A 1.0-cm-diameter pipe widens to 2.0 cm, then narrows to 0.5 cm. Liquid flows through the first segment at a speed of 9.0 m/s .

What is the speed in the second segment?

What is the speed in the third segment?

What is the volume flow rate through the pipe?

Answer 1

Diameter of the pipe in the first segment = D₁ = 1 cm = 0.01 m

Diameter of the pipe in the second segment = D₂ = 2 cm = 0.02 m

Diameter of the pipe in the third segment = D₃ = 0.5 cm = 0.005 m

Speed of flow in the first segment = V₁ = 9 m/s

Speed of flow in the second segment = V₂

Speed of flow in the third segment = V₃

Cross-sectional area of the pipe in the first segment = A₁ = D₁²/4

Cross-sectional area of the pipe in the second segment = A₂ = D₂²/4

Cross-sectional area of the pipe in the third segment = A₃ = D₃²/4

By continuity equation,

A₁V₁ = A₂V₂

(D₁²/4)V₁ = (D₂²/4)V₂

D₁²V₁ = D₂²V₂

(0.01)²(9) = (0.02)²V₂

V₂ = 2.25 m/s

A₁V₁ = A₃V₃

(D₁²/4)V₁ = (D₃²/4)V₃

D₁²V₁ = D₃²V₃

(0.01)²(9) = (0.005)²V₃

V₃ = 36 m/s

Volume flow rate through the pipe = Q

Q = A₁V₁

Q = (D₁²/4)V₁

Q = (0.01)²(9)/4

Q = 7.068 x 10^-4 m³/s

a) Speed in the second segment = 2.25 m/s

b) Speed in the third segment = 36 m/s

c) Volume flow rate through the pipe = 7.068 x 10^-4 m³/s