Consider a fluid flow through a pipe from left to right. The figure above shows

Question

Consider a fluid flow through a pipe from left to right. The figure above shows a cross sectional view of the pipe; the pipe is cylindrical and consists of two sections with different radii. In the wider section, the radius is R1 , the fluid flow speed is V1 , and the gauge pressure of the fluid is P1. In the narrower section, the radius, the fluid flow speed, and the gauge pressure of the fluid are R2 , V2 , and P2 . The fluid is incompressible.

(a) If R1 = 4.6 cm, R2 = 1.5 cm, and the fluid flow speed in the wider section is V1 = 15.2 cm/s, what is the flow rate at the narrower section of the pipe?
cm3/s

(b) If the fluid is water with a mass density of 1000 kg/m3, what is P1 - P2 , the pressure difference between the wider and the narrower section of the pipe?
atm.

(c) For the given pipe dimensions of R1 = 4.6 cm, R2 = 1.5 cm, in order to have the fluid flow speed in the narrower section of the pipe V2 = 228.3 cm/s, what should be the fluid flow speed in the wider section V1?
cm/s

Explanation / Answer

Rate of volume of fluid flowing = pi * r^2 * v

In a single pipe, rate of volume flow should be the same at any cross section.

So, rate of flow in wider section = rate of flow in narrower section.

Rate of flow in wider section = pi* 4.6^2 * 15.2 = 1030.38 cm^3 /s

(b) Flow rate in narrower section = 1010.44 cm^3 /s

So, pi* 1.5^2 * v2 = 1030.38

v2 = 142.95 m/s

From Bernoulli's equation,

P/d + v^2 /2 + gh = constant

where P is pressure

d is mass density of fluid

v is velocity

h is height

At narrow and wide portions, height from ground is the same. So, h is same

P1 /d + v1^2 /2 + gh = P2/d + v2^2/2 + gh

(P1-P2)/d = (v2^2 - v1^2)/2

P1 - P2 = 1000* (1.4295 m/s ^2 - 0.152 m/s ^2)

P1 - P2 = 2020 Pa = 0.02 atm

(c) From the same formula in (a),

pi* r1^2 * v1 = pi* r2^2 * v2

4.6^2 * v1 = 1.5^2 * 228.3

v1 = 24.28 cm/s

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.