Question

Increasing this To this Increases the capacity

pipe diameter diameter by a factor of
(inches) (inches)
4 6 2.25
4 8 4.00
6 8 1.78
6 10 2.78
6 12 4.00
8 10 1.56
8 12 2.25
8 15 3.52
10 12 1.44
10 15 2.25
12 15 1.56

A 12 inch pipeline is flowing full of water and is necked down to a four inch pipeline, does the flow velocity of the water in the 4 inch line increase or decrease?
a. Increases
b. Decreases
c. Flow is not impacted

Answer 1

The flow velocity of water in a four inch pipeline increases when a 12 inch pipeline is necked down to it, due to the principle of conservation of mass for incompressible fluids, which requires that the product of cross-sectional area and flow velocity be constant.

Step-by-step explanation:

When a 12 inch pipeline is flowing full of water and is necked down to a four inch pipeline, the flow velocity of the water in the 4 inch line increases. This occurs due to the principle of conservation of mass, also known as the continuity equation, which in the context of incompressible fluids like water states that the product of the cross-sectional area of the pipe and the velocity of the fluid flowing through it must remain constant. Since the area of the cross section decreases when the diameter of the pipe decreases, the velocity must increase to keep the flow rate constant. Therefore, in accordance with the continuity equation, v1A1 = v2A2, where 'v' is velocity and 'A' is cross-sectional area, a reduction in the diameter of a pipe leads to an increase in flow velocity. As the area A is proportional to the square of the diameter (A = πd^2/4), when the diameter decreases, the area decreases by the square of the factor of diameter reduction, causing the velocity to increase by the inverse square of that factor, keeping the product of area and velocity constant.