Chapter 1: Q39E (page 390)

Water is flowing in a pipe with a circular cross section but with varying cross-sectional area, and at all points the water completely fills the pipe. (a) At one point in the pipe the radius is 0.150 m. What is the speed of the water at this point if water is flowing into this pipe at a steady rate of 1.20 m³/s? (b) At a second point in the pipe the water speed is 3.80 m/s. What is the radius of the pipe at this point?

Short Answer

Expert verified

Speed of water = 17 m/s

Radius of pipe = 0.32 m

Step by step solution

Identification of given data

Radius of circle = 0.150 m
Flow rate of flow tube = 1.20 $m^{3} / s$

Significance of continuity equation

The continuity equation states that no matter where you are along the length of a pipe, the cross-sectional area and fluid velocity will always be the same, regardless of where you are.

Calculating speed of water

$\begin{aligned} A_{1} V_{1} = A_{2} V_{2} \\ 1.20 \frac{m^{3}}{s} = A_{2} V_{2} \\ \frac{1.20 m^{3}}{s} = (π \times r^{2}) V_{2} \end{aligned}$

Putting value in equation we get,

$V_{2} = \frac{1.20 m^{3} / s}{(π \times {(0.150 m)}^{2})} = 17 m / s$

Calculating flow rate of flow tube

$\begin{aligned} A_{1} V_{1} = A_{2} V_{2} \\ \frac{1.20 m^{3}}{s} = A_{2} (\frac{3.80 m}{s}) \\ \frac{1.20 m^{3} / s}{3.80 m / s} = A_{2} = π \times r^{2} \\ r = 0.32 m \end{aligned}$

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Short Answer

Step by step solution

Identification of given data

Significance of continuity equation

Calculating speed of water

Calculating flow rate of flow tube

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