Part A A liquid of density flows at speed v0 through a horizontal pipe that expa

Question

Part A

A liquid of density flows at speed v0 through a horizontal pipe that expands smoothly from diameter d0 to a larger diameter d1. The pressure in the narrower section is p0. Find an expression for the pressure p1 in the wider section. Express your answer in terms of the variables p0, , v0, d0 and d1.

Part B

A pressure gauge reads 50 kPa as water flows at 13.0 m/s through a 16.8-cm-diameter horizontal pipe. What is the reading of a pressure gauge after the pipe has expanded to 23.0 cm in diameter? Express your answer with the appropriate units.

Explanation / Answer

A) By Bernoulli’s equation for horizontal pipe,

P0 + 1/2v0^2 = P1 + 1/2v1^2

But v0A0= v1A1

v0(d0/2)^2 = v1(d1/2)^2

v0(d0/2)^2 = v1(d1/2)^2

v1 = v0(d0/d1)^2

Thus,

P0 + 1/2v0^2 = P1 + 1/2[v0(d0/d1)^2]^2

P0 + 1/2v0^2 = P1 + 1/2v0^2(d0/d1)^4

P1 = P0 + 1/2v0^2 - 1/2v0^2(d0/d1)^4 = P0 + 1/2v0^2[ 1- (d0/d1)^4]

B) Plugging values in above equation,

P1 = 50*10^3 + ½*1000*13.0^2[ 1- (0.168/0.230)^4] = 110.4 kPa

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