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Chapter 8: Problem 42

Water at \(15^{\circ} \mathrm{C}\) is flowing through a 5 -cm-diameter smooth tube with a length of \(200 \mathrm{~m}\). Determine the Darcy friction factor and pressure loss associated with the tube for (a) mass flow rate of \(0.02 \mathrm{~kg} / \mathrm{s}\) and \((b)\) mass flow rate of \(0.5 \mathrm{~kg} / \mathrm{s}\).

Short Answer

Expert verified
Question: Calculate the Darcy friction factor and the pressure loss associated with the tube for two different mass flow rates: (a) 0.02 kg/s and (b) 0.5 kg/s.

Step by step solution

01

Calculate average water velocity

Calculate the average water velocity (v) in the tube using the mass flow rate (ṁ) and the cross-sectional area (A) of the tube. The formula to calculate average water velocity is: \(v = \frac{ṁ}{\rho A}\) where ρ is the water density at \(15^{\circ} \mathrm{C}\), which is about \(1000 \mathrm{~kg/m^3}\), and A is the cross-sectional area of the tube calculated as: \(A = \pi \frac{D^2}{4}\) where D is the diameter of the tube.
02

Calculate the Reynolds number

Calculate the Reynolds number (Re) using the average water velocity (v), the tube diameter (D), and the kinematic viscosity (ν) of the water at \(15^{\circ} \mathrm{C}\). The formula for Reynolds number is: \(Re = \frac{vD}{\nu}\) The kinematic viscosity of water at \(15^{\circ} \mathrm{C}\) is about \(1.14 \times 10^{-6} \mathrm{~m^2/s}\).
03

Determine the Darcy friction factor

Determine the Darcy friction factor (f) associated with the tube using the Reynolds number (Re). For a smooth tube, the Darcy friction factor can be obtained using the Blasius equation: \(f = 0.079 \times Re^{-1/4}\)
04

Calculate the pressure loss

Calculate the pressure loss (ΔP) in the tube using the Darcy friction factor (f), the tube length (L), the average water velocity (v), and the tube diameter (D). The pressure loss formula is: \(ΔP = f \frac{L}{D} \frac{1}{2} \rho v^2\) Now, let's find the Darcy friction factor and pressure loss for both mass flow rates. (a) Mass flow rate (\(ṁ = 0.02 \mathrm{~kg/s}\)) 1. Calculate average water velocity 2. Calculate Reynolds number 3. Determine the Darcy friction factor 4. Calculate the pressure loss (b) Mass flow rate (\(ṁ = 0.5 \mathrm{~kg/s}\)) 1. Calculate average water velocity 2. Calculate Reynolds number 3. Determine the Darcy friction factor 4. Calculate the pressure loss

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Most popular questions from this chapter

Air enters a 25-cm-diameter 12 -m-long underwater duct at \(50^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) at a mean velocity of \(7 \mathrm{~m} / \mathrm{s}\), and is cooled by the water outside. If the average heat transfer coefficient is \(85 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and the tube temperature is nearly equal to the water temperature of \(10^{\circ} \mathrm{C}\), determine the exit temperature of air and the rate of heat transfer. Evaluate air properties at a bulk mean temperature of \(30^{\circ} \mathrm{C}\). Is this a good assumption?

In a manufacturing plant that produces cosmetic products, glycerin is being heated by flowing through a \(25-\mathrm{mm}\)-diameter and \(10-\mathrm{m}\)-long tube. With a mass flow rate of \(0.5 \mathrm{~kg} / \mathrm{s}\), the flow of glycerin enters the tube at \(25^{\circ} \mathrm{C}\). The tube surface is maintained at a constant surface temperature of \(140^{\circ} \mathrm{C}\). Determine the outlet mean temperature and the total rate of heat transfer for the tube. Evaluate the properties for glycerin at \(30^{\circ} \mathrm{C}\).

What does the logarithmic mean temperature difference represent for flow in a tube whose surface temperature is constant? Why do we use the logarithmic mean temperature instead of the arithmetic mean temperature?

In the fully developed region of flow in a circular tube, will the velocity profile change in the flow direction? How about the temperature profile?

Combustion gases passing through a 3-cm-internaldiameter circular tube are used to vaporize waste water at atmospheric pressure. Hot gases enter the tube at \(115 \mathrm{kPa}\) and \(250^{\circ} \mathrm{C}\) at a mean velocity of \(5 \mathrm{~m} / \mathrm{s}\), and leave at \(150^{\circ} \mathrm{C}\). If the average heat transfer coefficient is \(120 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and the inner surface temperature of the tube is \(110^{\circ} \mathrm{C}\), determine \((a)\) the tube length and (b) the rate of evaporation of water.
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