PROBLEMS

13.2-1.

Diffusion Through Liquids and a Membrane. A membrane process is being designed to recover solute A from a dilute solution where c1 = 2.0 × 10-2 kg mol A/m3 by dialysis through a membrane to a solution where c2 = 0.3 × 10-2. The membrane thickness is 1.59 × 10-5 m, the distribution coefficient K' = 0.75, DAB = 3.5 × 10-11 m2/s in the membrane, the mass-transfer coefficient in the dilute solution is kc1 = 3.5 × 10-5 m/s, and kc2 = 2.1 × 10-5.

  1. Calculate the individual resistances, total resistance, and total percent resistance of the two films.

  2. Calculate the flux at steady state and the total area in m2 for a transfer of 0.01 kg mol solute/h.

  3. Increasing the velocity of both liquid phases flowing past the surface of the membrane will increase the mass-transfer coefficients, which are approximately proportional to v0.6, where v is velocity. If the velocities are doubled, calculate the total percent resistance of the two films and the percent increase in flux.

A1: Ans. (a) Total resistance = 6.823 × 105 s/m, 11.17% resistance; (b) NA = 2.492 × 10-8 kg mol A/s · m2, area = 111.5 m2
13.2-2.

Suitability of a Membrane for Hemodialysis. Experiments are being conducted to determine the suitability of a cellophane membrane 0.029 mm thick for use in an artificial-kidney device. In an experiment at 37°C using NaCl as the diffusing solute, the membrane separates two components containing stirred aqueous solutions of NaCl, where c1 = 1.0 × 10-4 g mol/cm3 (100 g mol/m3) and c2 = 5.0 × 10-7. The mass-transfer coefficients on either side of the membrane have been estimated as kc1 = kc2 = 5.24 × 10-5 m/s. Experimental data obtained gave a flux NA = 8.11 × 10-4 g mol NaCl/s · m2 at pseudo-steady-state conditions.

  1. Calculate the permeability pM in m/s and DABK' in m2/s.

  2. Calculate the percent resistance to diffusion in the liquid films.

13.3-1.

Gas-Permeation Membrane for Oxygenation. To determine the suitability of silicone rubber for its use as a membrane for a heart-lung machine to oxygenate blood, an experimental value of the permeability at 30°C of oxygen was obtained, where PM = 6.50 × 10-7 cm3 O2 (STP)/(s · cm2 · cm Hg/mm).

  1. Predict the maximum flux of O2 in kg mol/s · m2 with an O2 pressure of 700 mm Hg on one side of the membrane and an equivalent pressure in the blood film side of 50 mm. The membrane is 0.165 mm thick. Since the gas film is pure oxygen, the gas film resistance is zero. Neglect the blood film resistance in this case.

  2. Assuming a maximum requirement for an adult of 300 cm3 O2 (STP) per minute, calculate the membrane surface area required in m2. (Note: The actual area needed should be considerably larger since the blood film resistance, which must be determined by experiment, can be appreciable.)

A3: Ans. (b) 1.953 m2
13.4-1.

Derivation of Equation for Permeate Concentration. Derive Eq. (13.4-11) for Case 2 for complete mixing. Note that xo from Eq. (13.4-9) must first be substituted into Eq. (13.4-6) before multiplying out the equation and solving for yp.

13.4-2.

Use of Complete-Mixing Model for Membrane Design. A membrane having a thickness of 2 × 10-3 cm, permeability = 400 × 10-10 cm3 (STP) · cm/(s · cm2 · cm Hg), and = 10 is to be used to separate a gas mixture of A and B. The feed flow rate is Lf = 2 × 103 cm3 (STP)/s and its composition is xf = 0.413. The feed-side pressure is 80 cm Hg and the permeate-side pressure is 20 cm Hg. The reject composition is to be xo = 0.30. Using the complete-mixing model, calculate the permeate composition, fraction of feed permeated, and membrane area.

A5: Ans. yp = 0.678
13.4-3.

Design Using Complete-Mixing Model. A gaseous feed stream having a composition xf = 0.50 and a flow rate of 2 × 103 cm3 (STP)/s is to be separated in a membrane unit. The feed-side pressure is 40 cm Hg and the permeate is 10 cm Hg. The membrane has a thickness of 1.5 × 10-3 cm, permeability = 40 × 10-10 cm3 (STP) cm/(s · cm2 · cm Hg), and = 10. The fraction of feed permeated is 0.529.

  1. Use the complete-mixing model to calculate the permeate composition, reject composition, and membrane area.

  2. Calculate the minimum reject concentration.

  3. If the feed composition is increased to xf = 0.60, what is this new minimum reject concentration?

A6: Ans. (a) Am = 5.153 × 107 cm2 (c) xoM = 0.2478
13.4-4.

Effect of Permeabilities on Minimum Reject Concentration. For the conditions of Problem 13.4-2, xf = 0.413, = 10, pl = 20 cm Hg, ph = 80 cm Hg, and xo = 0.30. Calculate the minimum reject concentration for the following cases:

  1. Calculate xoM for the given conditions.

  2. Calculate the effect on xoM if the permeability of B increases so that decreases to 5.

  3. Calculate the limiting value of xoM when is lowered to its minimum value of 1.0. Make a plot of xoM versus for these three cases.

13.4-5.

Minimum Reject Concentration and Pressure Effect. For Example 13.4-2 for separation of air, do as follows:

  1. Calculate the minimum reject concentration.

  2. If the pressure on the feed side is reduced by one-half, calculate the effect on xoM.

A8: Ans. (b) xoM = 0.0624
13.5-1.

Separation of Multicomponent Gas Mixtures. Using the same feed composition and flow rate, pressures, and membrane as in Example 13.5-1, do the following, using the complete-mixing model:

  1. Calculate the permeate composition, reject composition, and membrane area for a fraction permeated of 0.50 instead of 0.25.

  2. Repeat part (a) but for θ = 0.90.

  3. Make a plot of permeate composition ypA versus θ and also of area Am versus θ using the calculated values for θ = 0.25, 0.50, and 0.90.

13.5-2.

Separation of Helium from Natural Gas. A typical composition of a natural gas (S1) is 0.5% He (A), 17.0% N2 (B), 76.5% CH4 (C), and 6.0% higher hydrocarbons (D). The membrane proposed to separate helium has a thickness of 2.54 × 10-3 cm, and the permeabilities are = 60 × 10-10 cm3 (STP) · cm/(s · cm2 · cm Hg), = 3.0 × 10-10, and = 1.5 × 10-10. It is assumed that the higher hydrocarbons are essentially nonpermeable ( ≅ 0). The feed flow rate is 2.0 × 105 cm3 (STP)/s. The feed pressure ph = 500 cm Hg and the permeate pressure pl = 20 cm Hg.

  1. For a fraction permeated of 0.2, calculate the permeate composition, reject composition, and membrane area using the complete mixing model.

  2. Use the permeate from part (a) as feed to a completely mixed second stage. The pressure ph = 500 cm Hg and pl = 20 cm. For a fraction permeated of 0.20, calculate the permeate composition and membrane area.

13.6-1.

Design Using Cross-Flow Model for Membrane. Use the same conditions for the separation of an air stream as given in Example 13.6-1. The given values are xf = 0.209, = 10, ph = 190 cm Hg, pl = 19 cm Hg, Lf = 1 × 106 cm3 (STP)/s, = 500 × 10-10 cm3 (STP) · cm/(s · cm2 · cm Hg), and t = 2.54 × 10-3 cm. Do as follows using the cross-flow model:

  1. Calculate yp, xo, and Am for θ = 0.40.

  2. Calculate yp and xo for θ = 0.

A11: Ans. (a) yp = 0.452, xo = 0.0303, Am = 6.94 × 108 cm2 (S6); (b) yp = 0.655, xo = 0.209
13.7-1.

Equations for Cocurrent and Countercurrent Flow Models. Derive the equations for the following cases:

  1. For cocurrent flow, show the detailed steps for deriving Eqs. (13.7-35) and (13.7-36).

  2. For countercurrent flow, show the detailed steps in obtaining Eq. (13.7-22) from (13.7-4) and (13.7-5). Also, show the detailed steps to derive Eq. (13.7-24).

13.7-2.

Design Using Countercurrent-Flow Model for Membrane. Use the same conditions as given in Example 13.6-1 for the separation of an air stream. The given values are xf = 0.209, = 10, ph = 190 cm Hg, pl = 19 cm Hg, Lf = 1 × 106 cm3 (STP)/s, = 500 × 10-10 cm3 (STP) · cm/(s · cm2 · cm Hg), and t = 2.54 × 10-3 cm. Using the countercurrent-flow model, calculate yp, xo, and Am for θ = 0.40. (Note that this problem involves a trial-and-error procedure along with the numerical solution of two differential equations.)

13.8-1.

Effect of Pressure Drop on Asymmetric Membrane Calculations. Using the same conditions as in Example 13.8-1, do as follows:

  1. Use the shortcut method to calculate the overall pressure drop Δph using Eq. (13.8-16) over the entire length. Use typical data for air given in Section 13.8C to calculate NT, fiber length z2 - z1, and so forth. Assume T = 25° + 273 K and ph = 1034 kPa, Ts = 273 K, ps = 101.325 kPa.

  2. With the above values of ph, use the shortcut method in a second iteration to recalculate Am. (Be sure to correct for Pho at xo and ).

  3. Calculate the pressure drop if the number of tubes NT in part (a) is reduced by 50%.

A14: Ans. (a) NT = 1.267 × 105, (z2z1) = 1.413 m, Δph = 20.74 kPa; (b) Am = 54.10 m2; (c) Δph = 82.96 kPa
13.8-2.

Spreadsheet Calculation for Asymmetric Membrane. Using the conditions for Example 13.8-1, write the detailed spreadsheet and calculate the results. Compare these results with those in Table 13.8-2.

13.8-3.

Comparison of Experimental and Predicted Spreadsheet Results in a Pilot-Unit Asymmetric Membrane. A pilot-size membrane used to separate air in order to obtain nitrogen has the following dimensions with the feed inside the tubes: total NT = 3.8 × 104 fibers, ID = 95 μm, OD = 135 μm, length of fibers = 19 in. An experimental run gave the following results. Feed rate Lf = 1.086 m3/h, Lo = 0.654 m3/h, Vp = 0.450 m3/h, xo = 0.067 mole fraction oxygen, yp = 0.413, ph = 703.81 kPa, pl = 98.10 kPa, and T = 25°C. The permeances determined previously are (P'O2/t) = 1.50 × 10-10 m3 (STP)/(s · m2 · Pa), (P'N2/t) = 2.47 × 10-11. Flow is counter current. Do as follows:

  1. Predict the pressure drop in the tubes using the shortcut method.

  2. Using a spreadsheet, predict the performance of this run and compare these results with the experimental values. Note that this is trial and error, since the area is fixed but the value of xo is not. First assume a value of xo. Then divide the difference between xf = 0.209 and xo into nine equal parts and perform the spreadsheet calculation. If the calculated and actual areas do not agree, assume another value of xo, and so forth. Neglect pressure drop.

A16: Ans. (a) Δph = 4.455 kPa; (b) xo = 0.079, Vp = 0.442 m3/hr, yp = 0.398
13.9-1.

Osmotic Pressure of Salt and Sugar Solutions. Calculate the osmotic pressure of the following solutions at 25°C and compare with the experimental values:

  1. Solution of 0.50 g mol NaCl/kg H2O. (See Table 13.9-1 for the experimental value.)

  2. Solution of 1.0 g sucrose/kg H2O. (Experimental value = 0.0714 atm.)

  3. Solution of 1.0 g MgCl2/kg H2O. (Experimental value = 0.660 atm.)

A17: Ans. (a) π = 24.39 atm; (b) π = 0.0713 atm; (c) π = 0.768 atm
13.9-2.

Determination of Permeability Constants for Reverse Osmosis. A cellulose-acetate membrane with an area of 4.0 × 10-3 m2 is used at 25°C to determine the permeability constants for reverse osmosis of a feed salt solution containing 12.0 kg NaCl/m3 (ρ = 1005.5 kg/m3). The product solution has a concentration of 0.468 kg NaCl/m3 (ρ = 997.3 kg/m3). The measured product flow rate is 3.84 × 10-8 m3/s and the pressure difference used is 56.0 atm. Calculate the permeability constants and the solute rejection R.

A18: Ans. Aw = 2.013 × 10-4 kg solvent/s · m2 · atm, R = 0.961
13.9-3.

Performance of a Laboratory Reverse-Osmosis Unit. A feed solution at 25°C contains 3500 mg NaCl/L (ρ = 999.5 kg/m3). The permeability constant Aw = 3.50 × 10-4 kg solvent/s · m2 · atm and As = 2.50 × 10-7 m/s. Using a ΔP = 35.50 atm, calculate the fluxes, solute rejection R, and product solution concentration in mg NaCl/L. Repeat, but using a feed solution of 3500 mg BaCl2/L. Use the same value of Aw, but As = 1.00 × 10-7 m/s (A1).

13.10-1.

Effect of Pressure on Performance of Reverse-Osmosis Unit. Using the same conditions and permeability constants as in Example 13.10-1, calculate the fluxes, solute rejection R, and product concentration c2 for ΔP pressures of 17.20, 27.20, and 37.20 atm. (Note: The values for 27.20 atm have already been calculated.) Plot the fluxes, R, and c2 versus the pressure.

13.10-2.

Effect of Concentration Polarization on Reverse Osmosis. Repeat Example 13.10-1 but use a concentration polarization of 1.5. (Note: The flux equations and the solute rejection R should be calculated using this new value of c1.)

A21: Ans. Nw = 1.171 × 10-2 kg solvent/s · m2, c2 = 0.1361 kg NaCl/m3
13.10-3.

Performance of a Complete-Mixing Model for Reverse Osmosis. Use the same feed conditions and pressures given in Example 13.10-1. Assume that the cut or fraction recovered of the solvent water will be 0.10 instead of the very low water recovery assumed in Example 13.10-1. Hence, the concentrations of the entering feed solution and the exit feed will not be the same. The flow rate q2 of the permeate water solution is 100 gal/h. Calculate c1 and c2 in kg NaCl/m3 and the membrane area in m2.

A22: Ans. c1 = 2.767 kg/m3, c2 = 0.0973 kg/m3, area = 8.68 m2
13.11-1.

Flux for Ultrafiltration. A solution containing 0.9 wt % protein is to undergo ultrafiltration using a pressure difference of 5 psi. The membrane permeability is Aw = 1.37 × 10-2 kg/s · m2 · atm. Assuming no effects of polarization, predict the flux in kg/s · m2 and in units of gal/ft2 · day, which are often used in industry.

A23: Ans. 9.88 gal/ft2 · day
13.11-2.

Time for Ultrafiltration Using Recirculation. It is desired to use ultrafiltration for 800 kg of a solution containing 0.05 wt % of a protein to obtain a solution of 1.10 wt %. The feed is recirculated past the membrane with a surface area of 9.90 m2. The permeability of the membrane is Aw = 2.50 × 10-2 kg/s · m2 · atm. Neglecting the effects of concentration polarization, if any, calculate the final amount of solution and the time to achieve this, using a pressure difference of 0.50 atm.

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