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.

The fluid is water with a mass density of 1000 kg/m3. If R1 = 4.4 cm, V2 = 115.3 cm/s, and the water flow rate in the wider section of the pipe is 815 cm3/s, what is the flow speed at the wider section of the pipe V1?

For such a water flow, what is the pressure difference between the wider and the narrower section of the pipe?

For the water flow through this pipe, if you change the flow speed V1 the pressure difference P1 - P2 will change. At what value of V1 the pressure difference P1 - P2 will be 0.0364 atm?

Explanation / Answer

1. use the equation of continuity as A1V1 = A2 V2

where A1 is area of r1 and A2 is the area of r2

olume flow rate dV/dt = A2* V2

where A is area and v is velocity

V2 = 815e -6/(3.14 * 0.044* 0.044)

V2 = 13 cm/s

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apply Pressure Difference P2-P1 = 0.5 rho (V2^2-V1^2)

P2-P1 = 0.5* 1000* (1.15^2 - 0.13^2)

dP = P2-P1 = 652.8 Pa

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1atm = 1.013 e 4 pa

0.5 rho (V2^2- V1^2) = 0.0364* 1.013 e 4

V2^2 - V1^2 = 368.2* 2/1000 = 0.7362

V2^2 = 1.152^2 + 0.736^2

V2 = 1.367 m.s or 136.7 cm/s

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