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Chapter 8: Problem 29

Water at \(10^{\circ} \mathrm{C}\) flows through a smooth 60 -mm-diameter pipe with an average velocity of \(8 \mathrm{m} / \mathrm{s}\). Would a layer of rust of height \(0.005 \mathrm{mm}\) on the pipe wall protrude through the viscous sublayer? Justify your answer with appropriate calculations.

Short Answer

Expert verified
No, a layer of rust of height \(0.005 \mathrm{mm}\) on the pipe wall would not protrude through the viscous sublayer, as the thickness of the viscous sublayer is \(0.82 \mathrm{mm}\), which is greater than the height of the rust.

Step by step solution

01

Identify the given variables

The given variables are: temperature \(T = 10^{\circ} \mathrm{C}\), diameter of the pipe \(d = 60 \mathrm{mm}\), the velocity of the fluid \(u = 8 \mathrm{m/s}\), and the height of the layer of rust \(h = 0.005 \mathrm{mm}\).
02

Compute the Reynolds number

The Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces. It is calculated by the equation \(Re = \frac{ud}{\nu}\), where \(\nu\) is the kinematic viscosity of water at \(10^{\circ} \mathrm{C}\), which is \(1.31 \times 10^{-6} \mathrm{m^{2} / s}\). Then \(Re = \frac{(8 \mathrm{m/s})(0.06 \mathrm{m})}{1.31 \times 10^{-6} \mathrm{m^{2} / s}} = 3.67 \times 10^{5}\).
03

Compute the thickness of the viscous sublayer

The thickness of the viscous sublayer \(\delta\) can be calculated by the equation \(\delta = \frac{5\nu}{u} = \frac{5(1.31 \times 10^{-6} \mathrm{m^{2} / s})}{8 \mathrm{m/s}} = 0.82 \mathrm{mm}\).
04

Compare the height of the rust with the thickness of the viscous sublayer

The thickness of the viscous sublayer is \(0.82 \mathrm{mm}\), while the height of the rust is \(0.005 \mathrm{mm}\). Since the height of the rust is less than the thickness of the viscous sublayer, the rust would not protrude through the viscous sublayer.

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