Problem 53
During cold weather periods, vapor in a room diffuses through the dry wall and condenses in the adjoining insulation. This process decreases the thermal resistance and degrades the insulation. Consider a condition at which the vapor pressure in the air at \(25^{\circ} \mathrm{C}\) inside a room is $3 \mathrm{kPa}$, and the vapor pressure in the insulation is negligible. The \(3-\mathrm{m}\)-high and 10 -m-wide dry wall is \(12 \mathrm{~mm}\) thick with a solubility of water vapor in the wall material of approximately $0.007 \mathrm{kmol} / \mathrm{m}^{3}$. bar, and the diffusion coefficient of water vapor in the wall is \(0.2 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\). Determine the mass diffusion rate of water vapor through the wall.
Problem 54
Consider one-dimensional mass diffusion of species \(A\) through a plane wall. Does the species \(A\) content of the wall change during steady mass diffusion? How about during transient mass diffusion?
Problem 55
Write down the relations for steady one-dimensional heat conduction and mass diffusion through a plane wall, and identify the quantities in the two equations that correspond to each other.
Problem 56
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.
Problem 57
A thin plastic membrane separates hydrogen from air. The molar concentrations of hydrogen in the membrane at the inner and outer surfaces are determined to be \(0.045\) and \(0.002 \mathrm{kmol} / \mathrm{m}^{3}\), respectively. The binary diffusion coefficient of hydrogen in plastic at the operation temperature is \(5.3 \times\) \(10^{-10} \mathrm{~m}^{2} / \mathrm{s}\). Determine the mass flow rate of hydrogen by diffusion through the membrane under steady conditions if the thickness of the membrane is (a) \(2 \mathrm{~mm}\) and $(b) 0.5 \mathrm{~mm}$.
Problem 58
Exposure to high concentrations of gaseous short-term ammonia exposure level set by the Occupational Safety and Health Administration (OSHA) is $35 \mathrm{ppm}\( for \)15 \mathrm{~min}$. Consider a vessel filled with gaseous ammonia at \(30 \mathrm{~mol} / \mathrm{L}\), and a \(10-\mathrm{cm}\)-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.
Problem 64
Helium gas is stored at \(293 \mathrm{~K}\) in a 3-m-outer-diameter spherical container made of 3 -cm-thick Pyrex. The molar concentration of helium in the Pyrex is \(0.00069 \mathrm{kmol} / \mathrm{m}^{3}\) at the inner surface and negligible at the outer surface. Determine the mass flow rate of helium by diffusion through the Pyrex container.
Problem 65
Hydrogen can cause fire hazards, and hydrogen gas leaking into surrounding air can lead to spontaneous ignition with extremely hot flames. Even at very low leakage rate, hydrogen can sustain combustion causing extended fire damages. Hydrogen gas is lighter than air, so if a leakage occurs it accumulates under roofs and forms explosive hazards. To prevent such hazards, buildings containing source of hydrogen must have adequate ventilation system and hydrogen sensors. Consider a metal spherical vessel, with an inner diameter of \(5 \mathrm{~m}\) and a thickness of \(3 \mathrm{~mm}\), containing hydrogen gas at \(2000 \mathrm{kPa}\). The vessel is situated in a room with atmospheric air at 1 atm. The ventilation system for the room is capable of keeping the air fresh, provided that the rate of hydrogen leakage is below $5 \mu \mathrm{g} / \mathrm{s}$. If the diffusion coefficient and solubility of hydrogen gas in the metal vessel are \(1.5 \times\) \(10^{-12} \mathrm{~m}^{2} / \mathrm{s}\) and \(0.005 \mathrm{kmol} / \mathrm{m}^{3}\)-bar, respectively, determine whether or not the vessel is safely containing the hydrogen gas.
Problem 66
Helium gas is stored at \(293 \mathrm{~K}\) and \(500 \mathrm{kPa}\) in a \(1-\mathrm{cm}-\) thick, 2-m-inner-diameter spherical tank made of fused silica \(\left(\mathrm{SiO}_{2}\right)\). The area where the container is located is well ventilated. Determine \((a)\) the mass flow rate of helium by diffusion through the tank and \((b)\) the pressure drop in the tank in one week as a result of the loss of helium gas.
Problem 67
Pure \(\mathrm{N}_{2}\) gas at \(1 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C}\) is flowing through a 10 -m-long, 3-cm-inner diameter pipe made of 2 -mm-thick rubber. Determine the rate at which \(\mathrm{N}_{2}\) leaks out of the pipe if the medium surrounding the pipe is \((a)\) a vacuum and \((b)\) atmospheric air at \(1 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C}\) with 21 percent \(\mathrm{O}_{2}\) and 79 percent \(\mathrm{N}_{2}\). Answers: (ca) $2.28 \times 10^{-10} \mathrm{kmol} / \mathrm{s}\(, (b) \)4.78 \times\( \)10^{-11} \mathrm{kmol} / \mathrm{s}$
