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23.2 The moisture in hot, humid, stagnant air surrounding a cold-water pipeline continually diffuses to the cold surface where it condenses. The condensed water forms a liquid film around the pipe, and then continuously drops off the pipe to the ground below. At a distance of 10 cm from the surface of the pipe, the moisture content of the air is constant. Close to the pipe, the moisture content of the air approaches the vapor pressure of water evaluated at the temperature of the pipe.
a. Draw a picture of the physical system, select the coordinate system that best describes the transfer process, and state at least five reasonable assumptions of the mass-transfer aspects of the process.
b. What is the simplified form of the general differential equation for mass transfer in terms of the flux of water vapor, NA?
c. What is the simplified differential form of Fick’s flux equation for water vapor, NA?
d. What is the simplified form of the general differential equation for mass transfer in terms of the concentration of water vapor, cA?
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23.3 A device has been proposed that will serve as a “blood oxygenator” for a heart–lung bypass machine, as shown in the figure below. In this process, blood containing no dissolved oxygen (O2, species A) enters the top of the chamber and then falls vertically down as a liquid film of uniform thickness, along a surface designed to appropriately wet blood. Contacting the liquid surface is a 100% O2 gas phase. Oxygen is soluble in blood, where the equilibrium solubility cA is function of the partial pressure of oxygen gas. In analyzing the mass transport of dissolved oxygen into the falling film, you may assume the following: (1) the process has a constant source of O2 (gas) and a constant sink (falling liquid film), and so is at steady state; (2) the process is dilute with respect to dissolved oxygen dissolved the fluid; (3) the falling liquid film has a flat velocity profile with velocity vmax; (4) the gas space always contains 100% oxygen; (5) the width of the liquid film,W, is much larger than the length of the liquid film, L.
a. Simplify the general differential equation for O2 transfer, leaving the differential equation in terms of the fluxes. If your analysis suggests more than one dimension for flux, provide a simplified flux equation for each coordinate of interest....
b. Provide one simplified differential equation in terms of the oxygen concentration cA.
c. Provide boundary conditions associated with the oxygen mass-transfer process.
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23.4 Consider the process shown in the figure below, where carbon monoxide (CO) gas is being oxidized to carbon dioxide (CO2). This process is similar to how the catalytic converter in your car works to clean up CO in the exhaust. The inlet gas, which is 1.0 mole% CO diluted in O2, is fed into a rectangular chamber with a nonporous catalyst layer lining the top and bottom walls. The catalyst surface drives the reaction CO(g) + 1/2O2(g) → CO2(g), which is extremely rapid at the temperature of operation so that the gas-phase concentration of CO at the catalyst surface is essentially zero. For this system, you may assume that the gas velocity profile is flat—i.e., vx(y) = v, and the gas velocity is relatively slow.
a. Develop the final form of the general differential equation for mass transfer in terms of the concentration profile for CO, cA (species A). State all relevant assumptions, and also describe the source and sink for CO mass transfer....
b. State all necessary boundary conditions needed to specify the system.
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23.5 Consider the drug treatment system shown in the figure below. A hemispherical cluster of unhealthy cells is surrounded by a larger hemisphere of stagnant dead tissue (species B), which is turn surrounded by a flowing fluid. The bulk, wellmixed fluid contains a drug compound (species A) of constant but dilute bulk concentration cAo. Drug A is also soluble in the unhealthy tissue but does not preferentially partition into it relative to the fluid. The drug (species A) enters the dead tissue and targets the unhealthy cells. At the unhealthy cell boundary (r = R1), the consumption of drug A is so fast that the flux of A to the unhealthy cells is diffusion limited. All metabolites of drug A produced by the unhealthy cells stay within the unhealthy cells. However, drug A can also degrade to inert metabolite D by a first-order reaction dependent on cA—i.e., ...—that occurs only within the stagnant dead tissue.
a. State all reasonable assumptions and conditions that appropriately describe the system for mass transfer....
b. Develop the differential form of Fick’s flux equation for drug A within the multicomponent system without the “dilute system” assumption. Then, simplify this equation for a dilute solution. State all additional assumptions as necessary.
c. Appropriately simplify the general differential equation for mass transfer for drug A. Specify the final differential equation in two ways: in terms of NA, and in terms of concentration cA.
d. Specify the boundary conditions for both components A and D.
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23.6 A common process for increasing the moisture content of air is to bubble it through a column of water. The air bubbles are assumed to be spheres having a radius of 1.0 mm, and are in thermal equilibrium with the surrounding water at 298 K. The vapor pressure of water is 0.03 atm at 298 K, and the total pressure of the gas inside the air bubble is 1.0 atm.
a. Draw a picture of the physical system, and state at least five reasonable assumptions for the mass-transfer aspects of the water evaporation process. What coordinate system should be used?
b. What are the simplified differential forms of Fick’s flux equation for water vapor (species A), and the general differential equation for mass transfer in terms of concentration cA?
c. Propose reasonable boundary and initial conditions.
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23.7 Consider the diffusion of solute A into the single cylindrical pore shown in the figure below. The end of the pore at z = L is sealed. The pore space is initially filled with inert fluid B. As solute A diffuses into the quiescent fluid space inside the pore space, it adsorbs onto the inner walls of the pore. The “adsorption isotherm” of solute A onto the solid surface of the pore is described by the Langmuir equation, given by...where qA is the amount adsorbed on the surface (moles A/cm2 surface area), cA is the local concentration of solute A right above the surface (moles A/cm3), and K is the equilibrium constant (moles/cm3), and qA,max is the maximum amount of solute A, which can be adsorbed on the surface (moles A/cm2 surface area). At high concentrations where cA >> K; qA ≈ qA;max, and at low concentrations where K >> cA, the adsorption isothermbecomes linear, so that......
a. Think of a specific physical system—i.e., propose specific materials for solute A, the fluid B, and the solid surface from an outside literature reference. What does the plot of the Langmuir isotherm (qA vs. cA) look like for this specific physical system? Develop an algebraic expression that describes the maximum amount of solute A, which can be adsorbed within a single pore.
b. You may now consider that the concentration profiles of solute A is only in the axial direction, not in the radial direction. You may also assume that the process is dilute with respect to solute A, the linear adsorption isotherm is valid, and the rate processes of adsorption are extremely fast. Using the “shell balance” approach, develop the differential forms of the general differential equation for mass transfer and Fick’s flux equation, taking into account the adsorption of solute A onto the surface of pore in the differential mass balance. Then combine the simplified forms of the general differential equation of mass transfer and Fick’s flux equation to arrive at a single differential equation for the transfer of solute A within the pore in terms of concentration cA. State all assumptions and boundary/initial conditions as part of the analysis.
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