Question

The 3.0 cm-diameter water line in the figure splits into two 1.0 cm-diameter pipes. All pipes are circular and at thesame elevation. At point A, the water speed is 2.0 m/s and the gauge pressure is 50 kPa.What is the gauge pressure at point B?

Answer 1

Prepare:

Let be the gauge pressure at point A.

Let be the gauge pressure at point B.

Given that, all pipes are circular, and they are at the same elevation.

Bernoulli’s equation relating pressure p, speed v, and height y at any two points along a streamline in an ideal fluid can be expressed as

Here, the two points are in same elevation, so

Therefore, the above equation becomes

Therefore, the gauge pressure at point B can be expressed as

. . . . . . (1)

Here,

Speed of the fluid at point A

Speed of the fluid at point B

Density of the fluid

Figure:

The water pipe splits into two small pipes as shown in below figure.

Solve:

Given data

The diameter of the big pipe,

The radius of the big pipe,

The diameter of each small pipe,

The radius of the small pipe,

The speed of the water at point A,

The gauge pressure at point A,

The density of the water,

The volume flow rate of water through point A can be expressed as

Here, the cross-sectional area of the pipe,

Therefore, the above equation becomes

. . . . . . (2)

Similarly, the volume flow rate of water through each small pipe can be expressed as

. . . . . . (3)

According to given figurer, the volume flow rate of water at point A is equals the sum of the volume flow rate through one small pipe and the volume flow rate through the other small pipe.

Substituting the equations (2) and (3) in above relation,

. . . . . . (4)

Using equation (4), the speed of the water at point B is

Using equation (1), the gauge pressure at point B can be calculated as