Problem 79
The roof of a house is \(15 \mathrm{~m} \times 8 \mathrm{~m}\) and is made of a 30 -cm-thick concrete layer. The interior of the house is maintained at \(25^{\circ} \mathrm{C}\) and 50 percent relative humidity and the local atmospheric pressure is \(100 \mathrm{kPa}\). Determine the amount of water vapor that will migrate through the roof in \(24 \mathrm{~h}\) if the average outside conditions during that period are \(3^{\circ} \mathrm{C}\) and 30 percent relative humidity. The permeability of concrete to water vapor is $24.7 \times 10^{-12} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m} \cdot \mathrm{Pa}$.
Problem 82
The diffusion of water vapor through plaster boards and its condensation in the wall insulation in cold weather are of concern since they reduce the effectiveness of insulation. Consider a house that is maintained at \(20^{\circ} \mathrm{C}\) and 60 percent relative humidity at a location where the atmospheric pressure is \(97 \mathrm{kPa}\). The inside of the walls is finished with \(9.5\)-mm-thick gypsum wallboard. Taking the vapor pressure at the outer side of the wallboard to be zero, determine the maximum amount of water vapor that will diffuse through a \(3-\mathrm{m} \times 8-\mathrm{m}\) section of a wall during a 24 - \(h\) period. The permeance of the \(9.5-\mathrm{mm}\)-thick gypsum wallboard to water vapor is $2.86 \times 10^{-9} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{Pa}$.
Problem 83
Reconsider Prob. 14-82. In order to reduce the migration of water vapor through the wall, it is proposed to use a \(0.051-\mathrm{mm}\)-thick polyethylene film with a permeance of $9.1 \times 10^{-12} \mathrm{~kg} / \mathrm{s}^{2} \mathrm{~m}^{2}$.Pa. Determine the amount of water vapor that will diffuse through the wall in this case during a \(24-\mathrm{h}\) period. Answer: \(26.4 \mathrm{~g}\)
Problem 84
In transient mass diffusion analysis, can we treat the diffusion of a solid into another solid of finite thickness (such as the diffusion of carbon into an ordinary steel component) as a diffusion process in a semi-infinite medium? Explain.
Problem 85
When the density of a species \(A\) in a semi-infinite medium is known at the beginning and at the surface, explain how you would determine the concentration of the species \(A\) at a specified location and time.
Problem 86
Define the penetration depth for mass transfer, and explain how it can be determined at a specified time when the diffusion coefficient is known.
Problem 87
A thick wall made of natural rubber is exposed to pure oxygen gas on one side of its surface. Both the wall and oxygen gas are isothermal at $25^{\circ} \mathrm{C}$, and the oxygen concentration at the wall surface is constant. Determine the time required for the oxygen concentration at \(x=5 \mathrm{~mm}\) to reach 5 percent of its concentration at the wall surface.
Problem 89
Consider a piece of steel undergoing a decarburization process at $925^{\circ} \mathrm{C}\(. The mass diffusivity of carbon in steel at \)925^{\circ} \mathrm{C}\( is \)1 \times 10^{-7} \mathrm{~cm}^{2} / \mathrm{s}$. Determine the depth below the surface of the steel at which the concentration of carbon is reduced to 40 percent from its initial value as a result of the decarburization process for \((a)\) an hour and \((b) 10\) hours. Assume the concentration of carbon at the surface is zero throughout the decarburization process.
Problem 92
A long nickel bar with a diameter of \(5 \mathrm{~cm}\) has been stored in a hydrogen-rich environment at \(358 \mathrm{~K}\) and \(300 \mathrm{kPa}\) for a long time, and thus it contains hydrogen gas throughout uniformly. Now the bar is taken into a well-ventilated area so that the hydrogen concentration at the outer surface remains at almost zero at all times. Determine how long it will take for the hydrogen concentration at the center of the bar to drop by half. The diffusion coefficient of hydrogen in the nickel bar at the room temperature of \(298 \mathrm{~K}\) can be taken to be \(D_{A B}=\) $1.2 \times 10^{-12} \mathrm{~m}^{2} / \mathrm{s}\(. Answer: \)3.3$ years
Problem 97
A steel part whose initial carbon content is \(0.10\) percent by mass is to be case-hardened in a furnace at \(1150 \mathrm{~K}\) by exposing it to a carburizing gas. The diffusion coefficient of carbon in steel is strongly temperature dependent, and at the furnace temperature it is given to be $D_{A B}=7.2 \times 10^{-12} \mathrm{~m}^{2} / \mathrm{s}$. Also, the mass fraction of carbon at the exposed surface of the steel part is maintained at \(0.011\) by the carbon-rich environment in the furnace. If the hardening process is to continue until the mass fraction of carbon at a depth of \(0.6 \mathrm{~mm}\) is raised to \(0.32\) percent, determine how long the part should be held in the furnace.
