a. Starting from the appropriately simplified differential forms of Fick’s flux equation and the general differential equation for mass transfer relevant to the physical system of interest, develop the final, analytical, integrated equation to determine the total rate of drug release (WA) from the capsule under conditions where the saturated concentration of A within the liquid core of the capsule remains constant....
b. What is the maximum possible rate of drug release from the capsule, in units of gmole A per hour, when cAo ≈0?
Get solution
24.2 Consider the “drug patch” shown in the figure on the next page. The drug patch consists of a pure drug source mounted on top of a gel diffusion barrier. The gel diffusion barrier has a thickness of 2.0 mm. The gel diffusion barrier is in direct contact with the skin. The cumulative drug release vs. time profile for a 3.0-cm×3.0-cm square patch at 20°C is also shown below. Other experiments showed that the drug was immediately taken up into the body after exiting the patch. The drug is only slightly soluble in the gel material. The maximum solubility of the drug in the gel diffusion barrier is 0.50 μmole/ cm3, and the solubility of the drug in the gel diffusion barrier is not affected by temperature.
a. From the data shown below, estimate the effective diffusion coefficient of the drug in the diffusion barrier.......
b. When used on the body, heat transfer raises the temperature of the drug patch to about 35°C. What is the new drug delivery rate (WA) at this temperature in units of μmole/day? For purposes of this analysis, assume that the gel-like diffusion barrier material approximates the properties of liquid water.
Get solution
24.3 Consider the microscale apparatus shown in the figure on the next page. This apparatus is designed to deliver a small, steady stream of methanol (MeOH) vapor to a separate device that reforms the methanol vapor into hydrogen gas needed for a miniature fuel cell. In the present system, liquid methanol is vaporized at a constant temperature. The methanol vapor passes through a tube, and then through a porous ceramic membrane. A steady flow of O2 gas over the membrane keeps the partial pressure of methanol vapor in the exit gas constant. Small amounts of liquid methanol are constantly added to the base of the apparatus to keep the liquid methanol level steady. The apparatus is heated to maintain a constant temperature of 20 C, and the total system pressure is kept constant at 1.0 atm. Let A = MeOH vapor and B = O2 gas. Consider the tube System 1 .(z = 0 to z . L1. the porous ceramic membrane (to z . L1 to z = L2) System 2.
a. The porous ceramic membrane in System 2 consists of a parallel array of cylindrical pores. Each pore has a uniform diameter of 5.0 microns (μm). Estimate the effective diffusion coefficient for MeOH vapor within the porous ceramic membrane....
b. Based upon the nomenclature provided in the figure below, state boundary conditions for Systems 1 and 2 for MeOH in algebraic form.
c. State relevant assumptions for the mass-transfer process for Systems 1 and 2. Based on these assumptions, develop the final, integrated mathematical expression for predicting the total MeOH transfer rate WA through Systems 1 and 2. In this analysis, include the appropriate simplification of the general differential equation for mass transfer and Fick’s flux equation. Leave all boundary conditions in algebraic form. Hint: To help link System 1 to System 2, consider the dilute UMD assumption.
d. Using the dilute UMD assumption, estimate the total transfer rate of MeOH vapor from the apparatus in units of μ mole/h.
Get solution
24.4 Consider the biosensor device shown in the figure in the next column. The biosensor is designed to measure the concentration of solute A in the well-mixed liquid phase. At the base of the device is an electrode of surface area 2.0 cm2. The electrode is coated with an enzyme that catalyzes the reaction A → 2D. When solute A reacts to product D, product D is detected by the electrode, enabling for direct measurement of the flux of product D, which at steady state can be used to determine the concentration of A in the bulk liquid. The rate of reaction of A at the enzyme surface is rapid relative to the rate of diffusion of A down to the surface. Directly above the enzyme-coated electrode is a gel layer of 0.30 cm thickness that serves as a diffusion barrier for solute A and protects the enzyme. The gel layer is designed to make the flux of A down to the enzyme-coated surface diffusion limited. The effective diffusion coefficient of solute A in this gel layer is DAe = 4.0 × 10–7 cm2 / s at 20°C. Above the gel layer is a well-mixed liquid containing a constant concentration of solute A, CˊAo. The solubility of solute A in the liquid differs from the solubility of A in the gel layer. Specifically, the equilibrium solubility of A in the liquid layer (CˊAo) is related to the solubility of A in the gel layer (cA) by CˊA . K cA, with equilibrium partitioning constant K = 0.8 cm3 gel/cm3 liquid. The process is considered very dilute, and the total molar concentration of the gel layer is unknown. The concentration of product D in the wellmixed liquid is very small so that cDo ≈ 0. At 20°C, the electrode measures that the formation of product D is equal to 3.6 × 10–5 mmole D/h. What is the concentration of solute A in the bulk well-mixed liquid phase, Cˊ Ao, in units of mmole/cm3?...
Get solution
24.5 A porous “water vapor barrier” is placed over the tissue implant shown in the figure in the next column. The purpose of the porous “water vapor barrier” is to allow O2 gas direct access to the tissue while minimizing the diffusion-limited rate of evaporation of water from the tissue. Both the vapor barrier and the tissue implant possess “slab” geometry. The process is slightly pressurized and operates at 37 °C and 1.2 atm total system pressure (P). The O2 gas stream contains water vapor at 20% relative humidity at 37°C. The vapor barrier material is a random microporous polymer with mean pore size of 50 nm (1 × 107 nm = 1.0 cm) and void fraction (ε) of 0.40. The tissue approximates the properties of liquid water. At 37 C, the vapor pressure of liquid water is 47 mm Hg (1.0 atm = 760 mm Hg), and the Henry’s law constant (H) for the dissolution of O2 gas in water is 800 L atm/gmole. Let A = H2O; B = O2.
a. Making use of the Fuller–Schettler–Giddings correlation in the calculations, what is the effective diffusion coefficient (DAe) of H2O vapor in the randomly microporous water vapor barrier?...
b. What is the thickness of the vapor barrier (L) required to limit the rate of water evaporation from the tissue to 0.180 g H2O/cm2 day? State all assumptions for your analysis.
c. What is the concentration of dissolved oxygen in the tissue (CBL* , gmole O2/L tissue) at the interface between the tissue and the porous vapor barrier (z = 0)?
Get solution
24.6 An open well contains water contaminated with volatile benzene at the bottom of the well, with dimensions shown in the figure on page 525. The concentration of dissolved benzene in the water is 156 g/m3, and remains constant. The system is isothermal at 25°C. We are interested in determining the emission of benzene, a carcinogen, into the atmosphere from the well.
a. Define the source, sink, and system boundary for all of the species undergoing mass transfer. State three reasonable assumptions that describe the mass-transfer process.Potentially useful data: The molecular diffusion coefficient of benzene in dissolved water is 1.1×10–5 cm2 = s at 25°C, and the molecular diffusion coefficient of benzene vapor in air is 0.093 cm2/s at 1.0 atm and 25°C. The Henry’s law constant for benzene partitioning into water is H = 4.84×10–3 m3 atm/mole at 25°C. The vapor pressure of liquid water is 0.0317 bar (0.031 atm) at 25°C, and the density is 1000 kg/m3 at 25°C. The vapor pressure of benzene is 0.13 atm at 25°C. The humidity of the air flowing over the well hole is 40% of relative saturation at 25°C. The molecular weight of benzene is 78 g/mole....
b. State reasonable boundary conditions, and specify their numerical values with units, for all of the species undergoing mass transfer.
c. What are the maximum emission rates (in mole/day) of benzene and water vapor from the well? What is cumulative benzene emission (in grams) over a period of 30 days? Is it significant?
Get solution
24.7 A spherical ball of solid, nonporous naphthalene, a “mothball,” is suspended in still air. The naphthalene ball slowly sublimes, releasing the naphthalene vapor into the surrounding air by molecular diffusion-limited process. Estimate the time required to reduce the diameter from 2.0 to 0.50 cm when the surrounding air is at 347 K and 1.0 atm. Naphthalene has a molecular weight of 128 g/mol, a solid density of 1.145 g/cm3, and a diffusivity in air of 8.19 × 10–6 m2 /s, and exerts a vapor pressure of 5.0 Torr (666 Pa) at 347 K.
Get solution
24.8 The mass transfer device shown in the figure at the top of the next column is used to carry out the controlled release of a vapor-phase pheromone drug used in pest control. The solid drug sublimes at a vapor pressure P*A within the gas space of the reservoir. A polymer layer of thickness L = 0.15 cm covers the drug reservoir. The drug vapor (species A) absorbs into a polymer diffusion layer by a linear relationship pA = S · CˊA, where Cˊ A is the concentration of the pheromone drug dissolved in the polymer (gmole species A/cm3 polymer), pA is the partial pressure of the drug vapor (atm), and S is the partitioning constant for the drug between the vapor phase and the polymer phase (cm3-atm/mol). The pheromone is highly soluble in the polymer. The drug then diffuses through the polymer layer with diffusion coefficient DAe, and then exits to the surroundings as a vapor. Air flow over the top surface of the polymer layer generates a “fluid boundary layer.” The flux of the drug vapor across this boundary layer is given by...where kG is the gas-phase mass-transfer coefficient (gmole/ cm2 s atm). Generally, kG increases as the air flow rate over the surface increases. At steady state, the flux of drug (species A) through the polymer layer equals the flux through the boundary layer.
a. Develop a mathematical model, in final integrated form, for the drug vapor flux NA. The final model can only contain the following terms: NA, DAe, PA* , pA∞, L, S, kG. State all assumptions for analysis....
b. Determine the maximum possible drug vapor flux associated with the mass-transfer device, in units of μmole/cm2 s (1 μmole = 1.0 × 10–6 mole), under conditions where pA∞ ≈0, 30°C, and 1.0 atm total system pressure. The diffusion coefficient of drug vapor through polymer, DAe, is 1.0 × 10–6 cm2/s. The Henry’s law constant for absorption (dissolution) of drug vapor into polymer, S, is 0.80 cm3 atm/gmole. The “mass-transfer coefficient” for boundary layer, kG, is 1.0 × 10–5 mol/cm2 s atm. The vapor pressure of pheromone drug at 30°C is 1.1 atm.
Get solution
24.9 “Microvia” are microscopic passages between two thin films on a microelectronic device. Often, microvia are filled with a conductive metal to make a microscopic conductor for the flow of electrons between the two thin films. In one particular process, tungsten is deposited onto the base of the microvia by the following chemical vapor deposition reaction of tungsten hexaflouride (WF6) vapor:...As the tungsten metal forms, it fills the microvia (2.0 mm depth, 0.5 mm diameter), as shown in the figure below. The tungsten metal does not coat the side walls of the microvia; it only grows upward from the base of the microvia where the tungsten was initially seeded. The reactants are significantly diluted in inert helium (He) gas to lower the deposition rate. The temperature is 700 K, the total system pressure is 75 Pa, and the composition of WF6 in the bulk gas space over the microvia is 0.001 mole%. Assume that the tungsten deposition rate is limited by molecular diffusion. The molecular weight of tungsten (W) is 184 g/mole, the molecular weight of fluorine is 19 g/mole, and the density of solid tungsten is 19.4 g/cm3.
a. Develop a pseudo-steady-state (PSS) molecular diffusion mass transfer model to predict the depth of tungsten metal within the microvia as a function of time....
b. Estimate time required to completely fill the microvia, assuming Knudsen diffusion for WF6 vapor at the low total system pressure of 75 Pa.
Get solution
24.10 The data provided in Figure 26.7 are based on the diffusion of O2 into SiO2 formed from the oxidation of (100) crystalline silicon at 1000°C. Estimate the diffusion coefficient of O2 in SiO2 formed from the oxidation of (111) crystalline silicon at 1000°C, using the data in the table below, provided by Hess (1990).*The maximum solubility of O2 in the SiO2 is 9.6 10–8 mole O2/cm3 solid at 1000°C and 1.0 atm O2 gas partial pressure....*D.W. Hess, Chem. Eng. Education, 24, 34 (1990).Figure 26.7 Silicon dioxide(SiO2) film thickness vs. time at 1000°C....
Get solution
24.11 A flat surface containing many parallel pores is clogged with “coke” from a manufacturing process, as shown in the figure below. Pure oxygen gas (O2) at high temperature is used to oxidize the coke, which is mainly solid carbon, to carbon dioxide (CO2) gas. This process will remove the solid carbon clogging the pores, and hence clean the surface. A large excess of O2 is in the bulk gas over the surface, and so it may be assumed that bulk gas composition is always 100% O2. It may also be assumed that the oxidation reaction is very rapid relative to the rate of diffusion, so that the production of CO2 is limited by mass transfer, and the O2 concentration at the carbon surface is essentially zero. The pores are cylindrical, with diameter of 1.0 mm and depth of 5 mm. The oxidation process is carried out at 2.0 atm total system pressure and 600 C. The density of solid carbon is 2.25 g/cm3. Let A = O2, and B = CO2.
a. At some time after the oxidation process, the cleaned depth of the pore is 3.0 mm (0.3 cm) from the mouth of the pore. What is the total emissions rate of CO2 gas (WB) at this point in the process?...
b. How long will the oxidation process take to reach this cleaned depth of 0.3 cm from the mouth of the pore?
Get solution
24.12 A “biofilm” (component B) coats the surface of a nonporous inert sphere. The diameter of the nonporous inert core is 4.0 mm, and the overall diameter of the spherical biofilm particle is 8.0 mm. The spherical biofilm particle is suspended within water containing a known, constant, dilute concentration of solute A (cA∞). Within the biofilm, a homogeneous, first-order reaction ... D takes place.
a. Define the system, the source and sink for mass transfer of reactant A. Consider that the process is dilute with respect to species A and D. Propose three additional reasonable assumptions for this process.
b. Develop the differential material balance for the process in terms of concentration profile cA. State all boundary conditions necessary to completely specify this differential equation.
Get solution
24.13 Consider a spherical gel bead containing a biocatalyst uniformly distributed within the gel. Within the gel bead, a homogeneous, first-order reaction ... D is promoted by the biocatalyst. The gel bead is suspended within water containing a known, constant, dilute concentration of solute A (cA∞).
a. Define the system, and identify the source and the sink for the mass-transfer process with respect to reactant A. List three reasonable assumptions for this process. Then, using the “shell balance” approach, develop the differential material balance model for the process in terms of concentration profile cA. State all boundary conditions necessary to completely specify this differential equation.
b. The analytical solution for the concentration profile is given by...What is the total consumption rate of solute A by one single bead in units of mmol A per hour? The bead is 6.0 mm in diameter. The diffusion coefficient of solute A within the gel is 2 · 10–6 cm2/s, k1 is 0.019 s–1, and cA∞ is 0.02 μmole/cm3. Hint: Differentiate the relationship for cA(r) with respect to r, then estimate the flux NA at r = R.
Get solution
24.14 A biofilm reactor with a well-mixed liquid phase shown below will be used to treat wastewater contaminated with trichloroethylene (TCE) at a concentration 0.25 mg/L (1.9 mmole/m3, MTCE = 131.4 g/gmole). If the available surface area of the biofilm in the reactor is 800 m2, and the volumetric flow rate of wastewater into the reactor is 100 m3/h, what is the desired outlet concentration of TCE? The temperature of the process is constant at 20°C. In a well-mixed, continuous-flow reactor at steady state, the concentration of the solute of the liquid phase of inside the reactor is assumed equal to the concentration of the solute in the liquid that exits the reactor. It may also be assumed that the TCE degradation in the biofilm proceeds by homogeneous first-order reaction kinetics. The biofilm is of δ = 100 μm thickness.Potentially useful data* : kTCE = 4.31 s–1 (first-order rate constant for TCE in biofilm); DTCE-biofilm= 9.03 × 10–10 m2 / s (diffusion coefficient TCE in biofilm).*J.P. Arcangeli, E. Arvin, Environ. Sci. Technol., 31, 3044 (1997).
Get solution
24.15 Carbon dioxide (CO2) from waste sources is a sustainable feedstock for chemicals production if new technologies can be developed to carry out the reduction of CO2, the most oxidized form of carbon, to a more reactive molecule. Recently, solar-energy driven reactor concepts have emerged that harness the sun’s energy to sustainably drive the thermo-catalytic reduction of CO2 to reactive CO over a Ceria catalyst at high temperatures. A highly simplified version of this concept is provided in the figure in the next column, which is operated at 800°C and 1.0 atm. The diameter of the cylindrical reactor is 10.0 cm, the thickness of the porous catalyst lining the base of the reactor is 1.0 cm, and the gas space above the porous catalyst layer can be considered well mixed. Pure CO2 is fed into the reactor and diffuses into the porous catalyst layer, which drives the reaction CO2(g) → CO(g) + 1/2O2(g), which is approximated as first order with rate constant of k = 6.0 s–1 at 800°C. The effective diffusion coefficient of CO2 in the gas mixture within the porous catalyst is 0.40 cm2/s at 1.0 atm and 800°C.
a. It is desired to achieve 5.0 mole% CO in the exiting outlet gas. What is the molar flow rate exiting the reactor (n2), in units of gmole/min? What is the inlet molar flow rate of CO2?...
b. What is the mole fraction of CO2 on the back side of the catalyst layer?
Get solution
24.16 Recall from example 3, Chapter 25, that the differential model for the radial concentration profile of dissolved oxygen within one cylindrical engineered tissue bundle (Figure 25.8) is...with boundary conditions...Often, KA is very small relative to cA so that the homogeneous reaction term approaches a zero-order process that is not dependent on concentration:...So that...where m is the metabolic respiration rate of the tissue. In the present process, m = 0.25 mole / m3 · h at 25 °C, and R1 and R2 are equal to 0.25 cm and 0.75 cm, respectively. Pure O2 gas at 1.0 atm flows through the tube of length 15 cm. The masstransfer resistance due to the thin-walled tube is neglected, and the Henry’s law constant for dissolution of O2 in the tissue is 0.78 atm m3/mole at 25°C. The diffusion coefficient of oxygen in water is 2.1 10–5 cm2/s at 25°C, which approximates the diffusivity of oxygen dissolved in the tissue.
a. Develop a model, in final integrated form, to predict the concentration profile cA(r), and then plot out the concentration profile. Note that for diffusion with zero-order homogeneous chemical reaction, there will be a critical radius, Rc, where the dissolved oxygen concentration goes to zero. Therefore, if Rc<R2, then cA(r) = 0 from r = Rc to r=R2example 3......
b. Using this model and the process input parameter detailed above, determine Rc. Then, plot of the concentration profile from cA(r) from r = R1 to r = Rc.
c. Develop a model, in final algebraic form, to predict WA, total oxygen transfer rate through one tube. From the process input parameters given in the problem statement, calculate WA.
Get solution
24.17 In the distillation of a benzene/toluene mixture, a vapor richer in the more volatile component benzene is produced from the benzene/toluene liquid solution. Benzene is transferred from the liquid to the vapor phase, and the less volatile toluene is transferred in the opposite direction, as shown in the figure below. At the system temperature and pressure, the latent heats of vaporization of benzene (A) and toluene (B) are 30 and 33 kJ/gmole, respectively. Both components are diffusing through a gas film of thickness δ. Develop the final, integrated form of Fick’s flux equation to predict the steady-state mass transfer of benzene through the gas film. The equation must include terms for the bulk gas-phase mole fraction of benzene, the gas-phase mole fraction of benzene in equilibrium with the liquid solution, the diffusion coefficient of benzene in toluene, the diffusion path δ, and the total molar gas concentration, and the latent heats of vaporization for benzene (ΔHv,A) and toluene (ΔHv,B). Assume that distillation is an adiabatic process....
Get solution
