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Chapter 14: Problem 189

Air flows in a 4-cm-diameter wet pipe at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) with an average velocity of \(4 \mathrm{~m} / \mathrm{s}\) in order to dry the surface. The Nusselt number in this case can be determined from \(\mathrm{Nu}=0.023 \mathrm{Re}^{0.8} \mathrm{Pr}^{0.4}\) where \(\mathrm{Re}=10,550\) and \(\operatorname{Pr}=0.731\). Also, the diffusion coefficient of water vapor in air is \(2.42 \times\) \(10^{-5} \mathrm{~m}^{2} / \mathrm{s}\). Using the analogy between heat and mass transfer, the mass transfer coefficient inside the pipe for fully developed flow becomes (a) \(0.0918 \mathrm{~m} / \mathrm{s}\) (b) \(0.0408 \mathrm{~m} / \mathrm{s}\) (c) \(0.0366 \mathrm{~m} / \mathrm{s}\) (d) \(0.0203 \mathrm{~m} / \mathrm{s}\) (e) \(0.0022 \mathrm{~m} / \mathrm{s}\)

Short Answer

Expert verified
Answer: The mass transfer coefficient inside the pipe for fully developed flow is approximately 0.0366 m/s.

Step by step solution

01

1: Calculate Nusselt number

To calculate the Nusselt number, use the given formula Nu = 0.023 Re^0.8 Pr^0.4, and substitute the values of Re=10,550 and Pr=0.731: Nu = 0.023 * (10,550)^0.8 * (0.731)^0.4
02

2: Calculate Sherwood number

As the given problem is based on an analogy between heat and mass transfer, the Sherwood number would be equal to the Nusselt number: Sh = Nu
03

3: Compute the mass transfer coefficient

Now we will find the mass transfer coefficient, km, by using the equation: \(k_{\mathrm{m}}=\frac{\mathrm{Sh} \cdot \mathrm{D_{wv}}}{\mathrm{d}}\) We need to substitute the given values: Sh (which is equal to Nu), \(D_{wv} = 2.42 \times 10^{-5} \, \mathrm{m^2} \!/\! \mathrm{s}\), and d = 4 cm (0.04 m): \(k_{\mathrm{m}}=\frac{\mathrm{Nu} \cdot 2.42 \times 10^{-5}\mathrm{~m}^{2} /\mathrm{s}}{0.04\mathrm{~m}}\)
04

4: Evaluate and compare

Evaluate the expression for \(k_{\mathrm{m}}\) and compare the result with the given options: \(k_{\mathrm{m}} \approx 0.0366 \, \mathrm{m} \!/\! \mathrm{s}\) This value matches option (c). Therefore, the mass transfer coefficient inside the pipe for fully developed flow is \(0.0366 \mathrm{~m} / \mathrm{s}\).

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Most popular questions from this chapter

Air at \(40^{\circ} \mathrm{C}\) and 1 atm flows over a \(5-\mathrm{m}\)-long wet plate with an average velocity of \(2.5 \mathrm{~m} / \mathrm{s}\) in order to dry the surface. Using the analogy between heat and mass transfer, determine the mass transfer coefficient on the plate.

Consider steady one-dimensional mass diffusion through a wall. Mark these statements as being True or False. (a) Other things being equal, the higher the density of the wall, the higher the rate of mass transfer. (b) Other things being equal, doubling the thickness of the wall will double the rate of mass transfer. (c) Other things being equal, the higher the temperature, the higher the rate of mass transfer. (d) Other things being equal, doubling the mass fraction of the diffusing species at the high concentration side will double the rate of mass transfer.

What is the relation \((f / 2) \mathrm{Re}=\mathrm{Nu}=\mathrm{Sh}\) known as? Under what conditions is it valid? What is the practical importance of it? \(\mathrm{St}_{\text {mass }} \mathrm{Sc}^{2 / 3}\) and what are the names of the variables in it? Under what conditions is it valid? What is the importance of it in engineering?

Consider a carbonated drink in a bottle at \(37^{\circ} \mathrm{C}\) and \(130 \mathrm{kPa}\). Assuming the gas space above the liquid consists of a saturated mixture of \(\mathrm{CO}_{2}\) and water vapor and treating the drink as water, determine \((a)\) the mole fraction of the water vapor in the \(\mathrm{CO}_{2}\) gas and \((b)\) the mass of dissolved \(\mathrm{CO}_{2}\) in a 200-ml drink.

Exposure to high concentration of gaseous ammonia can cause lung damage. The acceptable shortterm ammonia exposure level set by the Occupational Safety and Health Administration (OSHA) is 35 ppm for 15 minutes. Consider a vessel filled with gaseous ammonia at \(30 \mathrm{~mol} / \mathrm{L}\), and a 10 -cm- diameter circular plastic plug with a thickness of \(2 \mathrm{~mm}\) is used to contain the ammonia inside the vessel. The ventilation system is capable of keeping the room safe with fresh air, provided that the rate of ammonia being released is below \(0.2 \mathrm{mg} / \mathrm{s}\). If the diffusion coefficient of ammonia through the plug is \(1.3 \times 10^{-10} \mathrm{~m}^{2} / \mathrm{s}\), determine whether or not the plug can safely contain the ammonia inside the vessel.
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