- 1.1 What Is Mass Transfer?
- 1.2 Preliminaries: Continuum and Concentration
- 1.3 Flux Vector
- 1.4 Concentration Jump at Interface
- 1.5 Application Examples
- 1.6 Basic Methodology of Model Development
- 1.7 Conservation Principle
- 1.8 Differential Models
- 1.9 Macroscopic Scale
- 1.10 Mesoscopic or Cross-Section Averaged Models
- 1.11 Compartmental Models
- Summary
- Review Questions
- Problems
Problems
1.1 Mass fraction to mole fractions. Show that mass fractions can be converted to mole fractions by the use of the following equation:
Derive an expression for dy_{i} as a function of dω_{i} values. Do this for a binary mixture. Expression for multicomponent mixture becomes rather unwieldy!
1.2 Mole fraction to mass fractions. Show that mole fractions can be converted to mass fractions by the use of the following equation:
Derive an expression for dω_{i} as a function of dy_{i} values for a binary mixture.
1.3 Average molecular weight. At a point in a methane reforming furnace, we have a gas of the composition CH_{4} = 10%, H_{2} = 15%, CO = 15%, and H_{2}O = 10% by moles. Find the mass fractions and the average molecular weight of the mixture. Find the density of the gas.
1.4 Average molecular weight variations. Two bulbs are separated by a 20-cm-long capillary tube. One bulb contains hydrogen and the other bulb contains nitrogen. The mole fraction profile varies in a linear manner along the length of the capillary. Calculate the mass fraction profile and show that the variation is not linear. Also calculate the average molecular weight as a function of the length along the capillary.
1.5 Mass fraction gradient. For a diffusion process across a stagnant film, the mole fraction gradient of the diffusing species was found to be constant. What is the mass fraction gradient? Is it also linear? The mixture is benzene–air.
1.6 Total concentration in a liquid mixture. Find the total molar concentration and species concentrations of 10% ethyl alcohol by mass in water at room temperature.
1.7 Effect of coordinate rotation on flux components. In Figure 1.1, the flux vector is 2e_{x} + e_{y}. Now consider a coordinate system that is rotated by an angle θ. Find this angle such that the flux component N_{Ay} is zero. What is the value of N_{Ax} in this coordinate system?
1.8 Flux vector in cylindrical coordinates. Define flux vector in terms of its components in cylindrical coordinates. Sketch the planes over which the components act. Show the relations between these components and the components in Cartesian coordinates.
1.9 Flux vector in spherical coordinates. Define the flux vector in terms of its components in spherical coordinates. Sketch the planes over which the components act. Show the relations between these components and the components in Cartesian coordinates.
1.10 Different forms of the Henry’s law constant. Express the Henry’s law constants reported in Table 1.2 as H_{i,pc} and H_{i,cp}.
1.11 Henry’s law constants: Unit conversions. Henry’s law constants for O_{2} and CO_{2} are reported as 760.2 L · atm/mol and 29.41 L · atm/mol. Which form of Henry’s law is being used? Convert these constants to values for the other forms of Henry’s law shown in the text.
1.12 Solubility of CO_{2}. The Henry’s law constant values for CO_{2} are shown below as a function of temperature.
Temperature K |
280 |
300 |
320 |
H. bar |
960 |
1730 |
2650 |
Fit an equation of the type
ln H = A + B/T
What is the physical significance of the parameter B? Find the solubility of pure CO_{2} in water at these temperatures.
1.13 Vapor pressure calculations: The Antoine equation. The Antoine constants for water are A = 8.07131, B = 1730.63, C = 233.426 in units of mm Hg for pressure and degrees Celsius for temperature. Convert these constants to a form where pressure is in Pa and temperature is in kelvins. Also rearrange the Antoine equation to a form where temperature can be calculated explicitly. This represents the boiling point at that pressure. What is the boiling point of water at Denver, Colorado (the “Mile High City”)?
1.14 Vant Hoff relation. Given the Antoine constants for a species, can you calculate the heat of vaporization of that species? Find this value for water from the data given in Problem 1.13.
1.15 Concentration jump. A solid rock of NaCl is in contact with water. Calculate the concentration of NaCl on the water side and salt side of the interface in mol/m^{3}.
1.16 A well-mixed reactor: Mass balance calculation. Consider a well-stirred reactor where a reaction A → B is taking place. The volumetric flow rate is 1 m^{3}/s and the reactor volume is 0.3 m^{3}. The inlet concentration of A is 1000 mol/m^{3} and the exit concentration is 200 mol/m^{3}. What is the rate of reaction of A in the system? If the reaction is first order and the contents are well mixed, what is the value of the rate constant?
1.17 Mass transfer coefficient calculation. A naphthalene ball (M_{A} = 128 g/mol and ρ_{A} = 1145 kg/m^{3}) is suspended in a flowing stream of air at 347 K and 1 atm pressure. The vapor pressure of naphthalene is 666 Pa for the given temperature. The diameter of the ball was found to change from 2.1 cm to 1.9 cm over a time interval of 1 hour. Estimate the mass transfer coefficient from the solid to the flowing gas.
1.18 A model for VOC loss from a holding tank. Wastewater containing a VOC at a concentration of 10 mol/m^{3} enters an open tank at a volumetric flow rate of 0.2 m^{3}/min and exits at the same rate. The tank has a diameter of 4 m and the depth of liquid in the tank is 1 m. The concentration of VOC in the exit stream and the rate of release of VOC is requested by the EPA. Use the conservation law to set a up a model. State further assumptions you may need to complete the model. List the parameters needed to solve the problem.
1.19 Averaging. Velocity profile in laminar flow is
v_{z}(r) = v_{max}[1 – (r/R)^{2}]
What does v_{max} represent? How is it related to the pressure drop? Find the average velocity.
The concentration distribution at a given axial position for a solid dissolving from the wall is given by Equation 1.24 Find the cross-sectional average concentration and compare the value with the cup mixing concentration calculated in Example 1.2.
1.20 Cup mixing versus cross-sectional average. The variation of scaled concentration in a laminar flow tubular reactor was measured and fitted to the following equation at a specified axial position:
c_{A} = 0.5[1 – (r/R)^{2} + (r/R)^{4}/2]
Calculate the center, wall, cup mixing, and cross-sectional average concentrations.
1.21 Turbulent flow velocity profile. Velocity profile in turbulent flow is commonly modeled by the 1/7-th law:
v_{z}(r) = v_{c}[1 – (r/R)]^{1/7}
What does v_{c} represent? Find the average velocity. Compare the average velocity and the center line velocity.