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26.2 A thin (1.0-mm-thick) coat of fresh paint has just been sprayed over a 1.5-m by 1.5-m square steel body part, which approximates a flat surface. The paint contains a volatile solvent that initially constitutes 30 wt% of the wet paint. The initial density of the wet paint is 1.5 g/cm3. The freshly painted part is introduced into a drying chamber. Air is blown into the rectangular drying chamber at a volumetric flow rate of 60 m3/min, as shown in the figure below, which has dimensions L = 1.5 m, H = 1.0 m, W = 1.5 m). The temperature of the air stream and the steel body part are both maintained at 27°C, and the total system pressure is 1.0 atm. The molecular weight of the solvent is 78 g/ gmole, the vapor pressure exerted by the solvent at 27°C is 105mm Hg, and the molecular diffusion coefficient of solvent vapor in air at 27°C and 1.0 atm is 0.097 cm2/s....
a. What is the Schmidt number and the average Sherwood number (ShL) for the mass-transfer process?
b. What is the estimated solvent evaporation rate from the surface of the whole body part in units of g/min? It may be assumed that convection mass-transfer limits the evaporation rate, and that the concentration of solvent vapor in the bulk gas is finite, but can be approximated as cA;∞ ≈ 0.
c. Using the results from part (a), how long will it take for the paint to completely dry?
d. What are the hydrodynamic (δ) and concentration boundary-layer (δc) thicknesses at x = L = 1.5 m? How does this compare to H, the height of the drying chamber?
e. What is the new required air flow rate (m3/min) if the desired solvent mass-transfer rate is 150 g/min?
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26.3 A horizontal chemical vapor deposition (CVD) reactor similar to the configuration shown in Example 3, Figure 28.6 will be used for growth of gallium arsenide (GaAs) thin films. In this process, arsine vapor, trimethylgallium vapor, and H2 gas are fed into the reactor. Inside the reactor, the silicon wafer rests on a heated plate called a susceptor. The reactant gases flow parallel to the surface of the wafer and deposit a GaAs thin film according to the simplified CVD reactions...If the reactant gas is considerably diluted in H2 gas, then the mass transfer of each species in the H2 carrier gas can be treated separately. These surface reactions are considered to be very rapid, and so the mass transfer of the gaseous reactants to the surface of the wafer limits the rate of GaAs thin film formation. In the present process, a 15 cm × 15 cm square silicon wafer is positioned at the leading edge of the susceptor plate. The process temperature is 800 K, and the total system pressure 101.3 kPa (1.0 atm). The feed gas delivered to the reactor results in a bulk linear velocity of 100 cm/s. The composition of arsine and trimethylgallium in the feed gas are both 0.10 mole%, which is very dilute. You may assume that the amount of arsine and trimethylgallium delivered with the feed gas is much higher than the amount of arsine and trimethylgallium consumed by the reactions, so that the concentration of these reactants in the bulk gas phase is essentially constant down the length of the reactor. You may also assume that the surface-reaction rates are instantaneous relative to the rates of mass transfer, so that the gas-phase concentrations of both arsine vapor and trimethylgallium vapor at the surface of the wafer are essentially zero. The binary gas phase diffusion coefficient of trimethylgallium in H2 is 1.55 cm2/s at 800K and 1.0 atm.
a. What are the average mass-transfer rates for arsine and trimethylgallium over the whole wafer?.........
b. Based on the ratio of the arsine and trimethylgallium mass-transfer rates, what is the composition of the GaAs composite thin film—e.g., the molar composition of gallium (Ga) and arsenic (As) in the solid? How could the feed-gas composition be adjusted so that the molar ratio of Ga to As within the solid thin film is 1:1?
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26.4 Boundary-layer analysis for fluid flow over a flat plate predicts the following relationships between the local Sherwood (Shx), Reynolds (Re), and Schmidt (Sc) numbers:...with the transition beginning at Rex = 2.0 × 105. Determine what percentage of the mass transfer occurs within the laminar zone of the flow over the flat plate if the Reynolds number at the end of the plate is ReL = 3.0 × 106.
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26.5 In using the von Kármán approximate method for analyzing the turbulent boundary layer over a flat plate, the following velocity and concentration profiles were assumed:...and...The four constants—α, β, η, and ξ—are determined by the appropriate boundary conditions at the surface and at the outer edge of the hydrodynamic and concentration layers.
a. Determine α, β, η, and ξ, and provide the resulting equations for velocity and concentration profiles.
b. Upon the application of the von Kármán momentum integral equation, the thickness of the turbulent boundary layer is given by...Use this relationship, and the solution to von Kármán concentration integral equation for Sc = 1.0, to obtain the following equation for the local mass-transfer coefficient:...
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26.6 A well-mixed open pond contains wastewater that is contaminated with a dilute concentration of dissolved methylene chloride. The pond is rectangular with dimensions of 500 m by 100 m, as shown in the figure (above right). Air at 27°C and 1.0 atm blows parallel to the surface of the pond with a bulk velocity of 7.5 m/s. At 20°C and 1.0 atm, for the gas phase (A = methylene chloride, B = air), the diffusion coefficient (DAB) is 0.085 cm2/s, and kinematic viscosity (vB) is 0.15 cm2/s. At 27°C, for the liquid phase, (A = methylene chloride, B = liquid water), the diffusion coefficient (DAB) is 1.07 × 10−5 cm2/s, and the kinematic viscosity (vB) is 0.010 cm2/s.
a. At what position across the pond is the air flow no longer laminar? Would it be reasonable to assume that the mean gas film mass-transfer coefficient for methylene chloride in air is dominated by turbulent flow mass transfer?...
b. As part of an engineering analysis to predict the emissions rate of methylene chloride (species A) from the pond, determine the average gas film mass-transfer coefficient associated with the mass-transfer methylene chloride from the liquid surface to the bulk air stream.
c. Compare the Schmidt number for methylene chloride in the gas phase vs. the liquid phase, and explain why the values are different.
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26.7 Gasoline from an under-storage storage tank leaked down onto an impermeable clay barrier and collected into a liquid pool. A simplified picture of the situation is provided in the figure below. Directly over this underground pool of liquid gasoline (n-octane, species A) is a layer of gravel of 1.0 m thickness and width of 10.0 m. The volatile n-octane vapors diffuse through the highly porous gravel layer, and then through a gas boundary layer formed by flow of air over the top surface of the gravel bed, and finally out to the bulk atmosphere where the n-octane is diluted to below detectable levels. There is no adsorption of n-octane vapor onto the porous gravel layer, and n-octane vapor concentration is dilute. Assume that the mass-transfer process is allowed to achieve a steady state. The temperature of the system is constant at 15°C, and the total system pressure is 1.0 atm. At this temperature, liquid n-octane exerts a vapor pressure of 1039 Pa. The void spaces in porous layer create a void fraction (ε) of 0.40, and but the pore size is large enough that Knudsen diffusion can be neglected.
a. What is the average mole fraction of n-octane vapor at the top surface of the rock layer (yAs = cAs/C) if the air velocity is very low, only 2.0 cm/s? What is the average flux of n-octane vapor emitted to the atmosphere?...
b. What would be the average mole fraction of n-octane vapor at the top surface of the rock layer if the air velocity is 50.0 cm/s? What is the average flux of n-octane vapor?
c. The Biot number associated with a mass-transfer process involving diffusion and convection in series is defined as...where L refers to the path length for molecular diffusion within the porous gravel layer and DAe refers to the diffusion coefficient of species A within this porous medium, which is not the same as the molecular diffusion coefficient as octane vapor in air. Determine the Biot number for parts (a) and (b), and then assess the relative importance of convective mass transfer in determining the n-octane vapor emissions rate.
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26.8 Consider the process shown in the figure (next page). A bulk gas stream containing 0.10 mole% of carbon monoxide (CO) gas, 2.0mole% O2 gas, and 97.9 mole% of CO2 gas flows over a flat catalytic surface of length 0.50 m at a bulk velocity of 40 m/s at 1.0atm and 600 K. Heat-transfer processes maintain the gas stream and catalytic surface at 600 K. At this temperature, the catalytic surface promotes the oxidation reaction CO(g) + 1=2O2.(g) → CO2(g). Let A = CO, B = O2, C = CO2. The gas-phase diffusion coefficients at 1.0 atm and 300K are DAB= 0.213 cm2/s, DAC = 0.155 cm2/s, DBC = 0.166 cm2/s.
a. What are the Schmidt numbers for CO and O2 mass transfer? What species (CO, O2, CO2) is considered the carrier gas?...
b. For CO mass transfer, what is the average convective mass-transfer coefficient (kc) over the 0.50mlength of the catalytic surface, and the local mass transfer coefficient (kc;x) at the far edge of the catalytic surface (x = L = 0.50 m)?
c. Using boundary-layer theory, scale kc for CO mass transfer to kc for O2 transfer.
d. At 600 K, the surface reaction constant for the first-order oxidation reaction with respect to CO concentration is ks = 1.5 cm/s. What is the average molar flux of CO to the catalytic surface, assuming that the composition of CO in the bulk gas is maintained at 0.10 mole%?
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26.9 A small droplet of liquid detergent, falling through air in a spray drying tower, has its diameter reduced as water evaporates from the surface. If it is assumed that the temperature of the liquid within the drop remains at 290K and the dry air is at 310 K, determine the concentration of water vapor in the surrounding bulk air stream within the drying tower. The total system pressure is 1.0 atm, and the film average gas temperature is 300 K.Potentially useful data: kinematic viscosity of air at 300 K, vair = 1.57 × 10−5 m2/s; thermal diffusivity of air at 300 K, α = 2.22 × 10−5m2/s; gas-phase diffusion coefficient of water vapor in air at 300 K, DA-air = 2.63 × 10−5 m2/s; density of air at 300 K, ρG = 1.18 kg/m3; heat capacity of air at 300 K, Cp,air = 1006 J/kg · K; latent heat of vaporization of water at 290 K, ∆Hv,A = 2.46 kJ/g H2O; vapor pressure of water at 290 K, PA = 1.94 × 103 Pa.
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26.10 In a spray column, a liquid is sprayed into a gas stream, and mass is transferred between the liquid and gas phases. The formation of liquid drops from the spray nozzle is considered to be a function of the nozzle diameter, gravitational acceleration, surface tension of the liquid against the gas, liquid density, liquid viscosity, velocity, and the viscosity and density of the surrounding gas medium. Arrange these variables into dimensionless groups. Should any other variables have been included?
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26.11 A falling liquid film within a gas–liquid contactor of 1.50 m length is in contact with 100% carbon dioxide gas at 1.0 atm and 25°C. The wetted surface area is 0.50 m2, and the liquid film thickness is 2.0 mm, which is thin enough to prevent ripples or waves in the falling liquid film. The liquid delivered to the contactor does not initially contain any dissolved CO2. At 25°C, the Henry’s law constant for the dissolution of CO2 gas in water is 29.5 m3 × atm/kgmole, and the molecular diffusion coefficient for CO2 in liquid water is 2.0 × 10−5 cm2/s. What is the average molar flux of CO2 into the film? What is the estimated bulk concentration of dissolved CO2 in the liquid exiting the process?
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