2.1 Calculation of Average Velocities and Fluxes: Imagine a cubic volume which is 1 cm on each side and is being convected at 1 cm/s in the horizontal direction (z-direction). Assume that 2 mol of A (molecular weight = 2), 3 mol of B (molecular weight = 3) and 4 mol of C (molecular weight = 4) are present in the cubic volume at steady-state, and also an unequimolar counter-diffusion is superimposed on the flow. Molecules of A diffuse in the +z direction at a rate of 2 mol/s;
molecules of B diffuse in the opposite direction at a rate of 1 mol/s. Find mass average velocity and molar average velocity. Also, calculate the fluxes relative to the mass average velocity and relative to molar average velocity.
[Ans: Mass average velocity = 1.034 cm/s, Molar average velocity = 1.11 cm/s; Molar fluxes relative to mass average velocity of: A = 1.931, B = -1.1034, C = -0.1379 mol/cm2$s; Mass fluxes relative to mass average velocity of: A = 3.8621, B = -3.3103, C = -0.5517 g/cm2$s;
Molar fluxes relative to molar average velocity of: A = 1.7778, B = - 1.3333, C = -0.4444 mol/cm2$s; Mass fluxes relative to molar average velocity of: A = 3.5556, B = -4.0, C = -1.7778 g/cm2$s.
2.2 Molecular Diffusion of Oxygen Through a Stagnant Layer of Methane: Oxygen is diffusing through a stagnant layer of methane 5 mm thick. The temperature is 20°C and the pressure is 1 atm.
2.3 Diffusion of Ammonia in Stagnant Air : Ammonia is diffusing through a stagnant air film of 0.25 mm thick. The total pressure is 2 atm and the temperature 50°C. Calculate the rate of diffusion of ammonia in kg/hr through 1 square metre surface if the concentrations of ammonia on the two sides of the film are 10% and 2% by volume, respectively
Diffusivity of ammonia in air at 0°C and 1 atm pressure is 0.198 cm2/s
[Ans: 20.04 kg/hr$m2]
2.4 Diffusion of Oxygen in Stagnant Nitrogen: Oxygen is diffusing in a mixture of oxygen-nitrogen at 1 atm pressure and at a temperature of 25°C; concentrations of oxygen at planes 2 mm apart are 10 and 20 volume percent, respectively. Nitrogen is non-diffusing. Calculate the flux of oxygen.
Given: diffusivity of oxygen in nitrogen is 1.89 # 10-3 m2/s
Ans: 4.55 # 10-3 mol/m2$s.
2.5 Diffusional Flux of Oxygen in Stagnant Carbon Monoxide: Oxygen (A) is diffusing through carbon monoxide (B) under steady-state conditions with carbon monoxide non-diffusing. The total pressure is 1 # 105 N/m2 and the temperature is 0°C. The partial pressures of oxygen at two planes 2.0 mm apart are 13000 and 6500 N/m2, respectively. The diffusivity for the mixture is 1.87 # 10-5 m2/s. Calculate the amount of oxygen diffused in one hour in kmol through each square metre of the two planes. [Ans: 0.107 kmol/hr$m2]
2.6 Rate of Evaporation of Water Through Air Film: A layer of water 20 mm thick is maintained at a constant temperature of 25°C in contact with dry air at 30°C and 1 atm pressure. Assuming evaporation to take place by molecular diffusion through an air film of 0.5 mm thick, calculate the time necessary for evaporation of 50% of the water.
The diffusivity of water vapour in air may be taken as 0.258 cm2/s. The vapour pressure of water at 20°C is 17.6 mm Hg. [Ans: 3.04 hr]
2 . 7 Diffusion of Oxygen in Stagnant Nitrogen at Elevated Pressure: The operation as stated in Numerical Problem 2.4 takes place at 10 atm pressure keeping other conditions the same.
Calculate the rate of diffusion of oxygen in g/cm2$hr, taking the value of diffusivity of oxygen in nitrogen 0.181 cm2/s [Ans: 5.02 g/cm2$hr]
2.8 Effect of Elevated Pressure on the Rate of Molecular Diffusion of Ammonia in Air : Ammonia is diffusing through a stagnant air film 2 mm thick at a temperature of 27°C and a pressure of 1 atm.
Estimate the effect on the rate of diffusion of raising the total pressure to 10 atm for the following cases:
(i) the concentration of ammonia is 10% (by volume) on one side of the film and zero on the other side;
(ii) the partial pressure of ammonia is 0.10 atm on one side of the film and zero on the other side.
The diffusivity of ammonia in air at 0°C and 1 atm pressure is 0.198 cm2/s
[Ans: (i) Rate of diffusion remains unchanged, (ii) Rate of diffusion becomes 4.65 # 10-7 gmol/cm2$s]
2.9 Equimolal Counter-Diffusion of Two Gases: Gases A and B are diffusing through each other under conditions of equal molal counter-diffusion. The concentrations of gas A, 4 mm apart, are 10%
and 2% by volume. The total pressure is 1 atm and the temperature is 20°C. The diffusivity of the two gases under these conditions is 0.15 cm2/s Determine the rate of diffusion. [Ans: 1.248 gmol/cm2$s]
2.10 Equimolal Counter-Diffusion of Hydrogen and Oxygen at Elevated Pressure: In a gas mixture of hydrogen and oxygen, steady-state equimolar counter-diffusion is occurring at a total pressure of 100 kPa and temperature of 20°C. If the partial pressure of oxygen at two planes 0.01 m apart, and perpendicular to the direction of diffusion are 15 kPa and 5 kPa, respectively and the mass diffusion flux of oxygen in the mixture is 1.6 # 10-5 kmol/m2$s, calculate the molecular diffusivity for the system. [Ans. 38.98 # 10-6 m2/s]
2.11 Diffusion of a Gas Through a Mixture of Gases: Hydrogen is diffusing through a stagnant mixture of gases containing 40% nitrogen, 30% ammonia and 30% carbon dioxide by volume. The temperature is 25°C and total pressure is 1 atm. The partial pressure of hydrogen at two planes 6 mm apart in the direction of diffusion are 100 mm Hg and 40 mm Hg, respectively. Determine the rate of diffusion of hydrogen.
Given: At 25°C and 1 atm, the diffusivities of hydrogen with respect to the constituents of the gas mixture are as under:
DH2–N2 = 0.773 cm2/s; DH2–NH3 = 0.783 cm2/s; DH2–CO2 = 0.646 cm2/s
[Ans: 4.41 # 10-6 gmol/s]
2.12 Diffusion Through Variable Area: A conical duct 50 cm long having internal diameters of 15 cm and 7.5 cm at the larger and smaller ends respectively, connects two very large flasks A and B.
Flask A contains a uniform mixture of 75 mole% ammonia and 25 mole% hydrogen. Flask B contains 20 mole% ammonia and 80 mole% hydrogen. The temperature is throughout 25°C and the total pressure is 1 atm. Under these conditions, the diffusivity of NH3–H2 is 7.83 # 10-5 cm2/s.
Assuming complete absence of convection and that ammonia diffuses in the direction of decreasing diameter, determine the rate of transfer of ammonia between the two flasks. [Ans: 3.11 # 10-5 gmol/s]
2.13 Calculation of the Rate of Combustion of Spherical Charcoal Particle: Spheres of charcoal, 25 mm in diameter are being burnt in a stream of air at 1 atm pressure. Assuming the controlling resistance is equivalent to that for molecular diffusion of oxygen through a stagnant air film 2.0 mm thick and neglecting the effect of back diffusion of the products of combustion, calculate the theoretical combustion rate in kg/hr$m2.
The partial pressures of oxygen are 0.21 atm in the air stream and zero at the carbon-gas interface.
The bulk temperature of the gas is 1550 K. The diffusivity of O2 in air at 0°C and 1 atm pressure is 0.18 cm2/s [Ans: 9.76 kg/hr$m2]
2.14 Estimation of Loss of Hydrogen from a Spherical Container: Hydrogen at 10 bar pressure and at 300 K temperature is stored in a spherical steel tank of 100 mm diameter and 2 mm thickness.
The concentration of hydrogen in the tank is 1.5 kmol/m3 and the diffusion coefficient of hydrogen in steel is 0.3 # 10-12 m2/s. Calculate the rate of loss of hydrogen from the tank.
[Ans: 7.35 # 10-12 kmol/s or 14.7 # 10-12 kg/s]
2.15 Effect of Temperature on Diffusivity of Gas System: A diffusivity value of 0.151 cm2/s has been reported for CO2-air system at 293 K and 1 atm pressure. Using Chapman-Enskog equation, estimate the diffusivity for the system at 1500 K.
[Ans: 2.49 cm2/s]
2.16 Estimation of Diffusivities in Gas Mixtures using Chapman-Enskog Equation: Calculate the diffusivities of the following pairs of gases using Chapman-Enskog equation:
(i) Ammonia in air at standard atmospheric pressure and 0°C.
(ii) Equimolar mixture of CO2 and N2 at 288.2 K and 40 atm pressure.
(iii) Methane-ethane system at 40°C and 1 atm pressure.
[Ans. (i) 1.84 # 10-5 m2/s = 0.184 cm2/s, (ii) 7.5 # 10-5 cm2/s = 7.5 # 10-9 m2/s, (iii) 1.56 # 10-5 m2/s.]
2.17 Diffusivity of Acetone in Air: A narrow test tube 3 mm id, 50 mm long was filled with acetone to
a height of 39 mm from the bottom and maintained at a constant temperature of 20°C in a gentle current of air. The total pressure was atmospheric. After 5 hr, the liquid level was 29.5 mm from the bottom. The barometric pressure was 750 mm Hg.
Calculate the diffusivity of acetone in air under the conditions of the experiment.
Given: Density of liquid acetone at 20°C = 0.79 g/cm3, Vapour pressure of acetone at 20°C = 180 mm Hg.
[Ans: 0.10 cm2/s]
2.18 Rate of Diffusion of Water in Dry Air in a Well: Calculate the rate of diffusion of water vapour from a thin layer of water at the bottom of a well 6 m in height to dry air flowing over the top of the well. Assume the entire system is at 298 K and 1 atm pressure. If the well diameter is 3 m, find out the total weight of water diffused per second from the surface of the water in the well.
The diffusion coefficient of water vapour in dry air at the operating conditions is 0.256 # 10-4 m2/s. The partial pressure of water vapour at 298 K is 0.0323 # 104 kg/m2. [Ans: 7.0 # 10-4 g/s]
2.19 Calculation of the Amount of n-octane evaporated in nitrogen Atmosphere: Liquid n-octane evaporates and diffuses upward through a long tube initially filled with nitrogen. As the liquid evaporates, it pushes the gas upward. How many grams of liquid n-octane will evaporate into nitrogen in 24.5 hr in a system kept at 20°C and 1 atm pressure if the area of the liquid surface is 1.29 cm2? The vapour pressure of n-octane at 20°C is 10.45 mm Hg. [Ans: 6.71 mg]
2.20 Rate of Evaporation of Chloropicrin in Air: Find the rate of evaporation (in g/hr) of chloropicrin (CCl3NO2) kept in a long tube into air (considering it a pure substance) at 25°C.
Given: Total pressure = 770 mm Hg, Diffusivity = 0.088 cm2/s, Vapour pressure = 23.81 mm Hg, Distance from liquid level to the top of the tube = 11.14 cm, Density of chloropicrin = 1.65 g/cm3, and surface area of liquid exposed for evaporation = 2.29 cm2. [Ans: 0.014 g/hr]
2.21 Rate of Absorption of Oxygen in Pyrogallate Solution: A test tube 2.0 cm in diameter and 15 cm tall is partly filled with a solution of alkaline pyrogallate. The depth of the empty space above the solution is 5 cm. The temperature is 25°C and the total pressure is 1 atm. Air may be assumed to contain 21% O2 and 79% N2. The diffusivity of O2 in N2 at the given condition is 0.21 cm2/s.
Calculate the rate of absorption of O2 from air in the solution in kg/s at steady-state if air flows gently over the open end of the test tube. [Ans: 71.38 # 10-9 kg/s]
2.22 Rate of Diffusion of Acetic Acid Through a Stagnant Film: Acetic acid is diffusing across a 1.5 mm thick stagnant film of aqueous solution at 20°C. The concentrations of acetic acid on the two sides of the film are 11% and 5% acid by weight respectively. Calculate the rate of diffusion of acetic acid.
Given: Densities of 11% and 5% acetic acid solution at 20°C are 1100 and 1010 kg/m3 respectively.
Diffusivity of acetic acid in aqueous solution at 20°C is 1 # 10–5 cm2/s. [Ans: 0.812 # 10-6 kmol/m2$s]
2.23 Prediction of Self-Diffusivity in Liquid Mercury: The diffusivity of Hg203 in normal liquid Hg has been measured along with viscosity and volume per unit mass. Compare the experimentally
measured DAA with the values calculated using Eq. (2.60).
T, K DAA, cm2/sn, cpVA, cm3/g 275.7 1.52 # 10-5 1.68 0.0736 289.6 1.68 # 10-5 1.56 0.0737 364.2 2.57 # 10-5 1.27 0.0748
2.24 Estimation of Diffusivity of Liquid: Calculate the diffusion coefficient for hydrogen sulphide in water at 40°C. [Ans: 2.7 # 10-8 m2/s]
2.25 Estimation of Liquid Diffusivity Using Wilke-Chang Equation: Estimate the diffusion coefficient for the following situations using Wilke-Chang equation:
(i) Acetic acid in dilute aqueous solution at 12.5°C. The density of acetic acid at its normal boiling point is 0.937 g/cm3.
(ii) A dilute solution of methanol in water at 100°C while the diffusivity for the same solution at 15°C is 1.28 # 10-5 cm2/s.
(iii) Bromonaphthalene in dilute ethyl alcohol solution at 20°C. Atomic volume: C = 14.8, H = 3.7, Br = 27.0. For naphthalene ring subtract 30. Viscosity of dilute alcohol solution at 20°C = 1.192 cp; z = 1.5.
[Ans: (i) Experimental value: 0.91 ± 0.04 # 10-5 cm2/s, (ii) 6.7 # 10-5 cm2/s, (iii) 7.0 # 10-6 cm2/s]
2.26 Diffusion through a Solid Slab: A slab of clay 40 mm thick with the four thin edges sealed, is being dried from two flat faces by exposure to dry air. The initial moisture is 18%. Drying takes place by internal diffusion of liquid water followed by evaporation at the surface. The surface moisture content is 2.5%. The average moisture content has fallen to 9.75% in 6 hr.
Assuming diffusivity to be independent of moisture content and uniform in all directions, calculate (i) the diffusivity of clay in cm2/s, (ii) the time required to reduce the moisture content to 6%. [Ans:
(i) 4.42 # 10-5 cm2/s, (ii) 14.075 hr]
2.27 Diffusion Through a Solid Sphere: Spheres of porous clay 12 mm in diameter were thoroughly impregnated with an aqueous solution of sodium chloride of concentration 0.15 g/cm3. When exposed to a running supply of fresh water at 18°C, the spheres lost 60% of their salt content in 2.25 hr. At 18 °C, the average diffusivity of NaCl in water is 1.36 # 10-9 m2/s. Assume Di/a2 = 0.1175F.
Estimate the time required for removal of 80% of the dissolved solute if the spheres were impregnated with an aqueous solution of ethyl alcohol of concentration 0.17 g/cm3 when exposed to a running supply of water containing 0.01 g/cm3 of ethanol. The average solubility of ethanol in water at 18°C is 0.53 # 10-9 m2/s.
[Ans: 17.18 hr]
2.28 Calculation of Pressure in Knudsen Diffusion: A stainless -steel tubing is of 1.6 # 10-3 m inside diameter and 4 m long. One end of the tube is evacuated. Calculate the pressure in the other end in order the Knudsen number to be 10. The gas inside is of molecular weight 92, viscosity 6.5 #
10-4 kg/m$s., and temperature 300 K.
[Ans: 8.43 # 10-4 atm]
2.29 Calculation of Knudsen Diffusion Coefficient and Flux: Hydrogen gas is kept in a container at 200°C and 1 atm pressure. The mouth of the container is closed with a porous plug of 5 # 10-3 m long with average pore size of 100 Å. Calculate the Knudsen diffusion coefficient and the steady flux through the plug.
[Ans: 0.0746 cm2/s, 3.84 # 10-6 gmol/cm2$s]