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Chapter 14: Problem 52

A thin plastic membrane separates hydrogen from air. The molar concentrations of hydrogen in the membrane at the inner and outer surfaces are determined to be \(0.045\) and \(0.002 \mathrm{kmol} / \mathrm{m}^{3}\), respectively. The binary diffusion coefficient of hydrogen in plastic at the operation temperature is \(5.3 \times 10^{-10} \mathrm{~m}^{2} / \mathrm{s}\). Determine the mass flow rate of hydrogen by diffusion through the membrane under steady conditions if the thickness of the membrane is (a) \(2 \mathrm{~mm}\) and (b) \(0.5 \mathrm{~mm}\).

Short Answer

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Question: Calculate the mass flow rate of hydrogen through a thin plastic membrane under steady conditions for two different thicknesses, (a) 2 mm and (b) 0.5 mm, given the molar concentrations of hydrogen at the inner and outer surfaces are 0.045 kmol/m³ and 0.002 kmol/m³, and the binary diffusion coefficient of hydrogen in the plastic is 5.3 x 10⁻¹⁰ m²/s. Answer: To calculate the mass flow rate of hydrogen through the membrane, follow these steps: 1. Calculate the concentration gradient: ΔC = C₁ - C₂ 2. Apply Fick's law of diffusion to find the molar flux: J_A = -D_AB * (ΔC / L) 3. Calculate the molar flux for each membrane thickness. 4. Calculate the mass flow rate by multiplying the molar flux by the molar mass of hydrogen (M_H = 2 kg/kmol): ṁ = J_A * M_H 5. Solve for the mass flow rate for each membrane thickness. After following these steps, you will find the mass flow rate of hydrogen for both thicknesses.

Step by step solution

01

Identify the given variables

We are given the following information: - Molar concentrations of hydrogen at the inner and outer surfaces: \(C_1 = 0.045 \mathrm{kmol/m^3}\) and \(C_2 = 0.002 \mathrm{kmol/m^3}\) - Binary diffusion coefficient of hydrogen in plastic: \(D_{AB} = 5.3 \times 10^{-10} \mathrm{m^2/s}\) - Two membrane thicknesses: (a) \(L_1 = 2 \mathrm{mm} = 2 \times 10^{-3} \mathrm{m}\), (b) \(L_2 = 0.5 \mathrm{mm} = 0.5 \times 10^{-3} \mathrm{m}\)
02

Calculate the concentration gradient

The concentration gradient, \(\Delta C\), is the difference in concentration across the membrane and can be calculated as: \(\Delta C = C_1 - C_2\)
03

Apply Fick's law of diffusion

For a given membrane thickness, \(L\), the molar flux, \(J_A\), can be determined using Fick's law: \(J_A = -D_{AB} \frac{\Delta C}{L}\) where \(D_{AB}\) is the binary diffusion coefficient.
04

Calculate the molar flux for each membrane thickness

Using the information given in the problem, calculate the molar flux for each membrane thickness. For (a) \(L_1 = 2 \times 10^{-3} \mathrm{m}\) and (b) \(L_2 = 0.5 \times 10^{-3} \mathrm{m}\), we obtain: \(J_{A1} = -D_{AB} \frac{\Delta C}{L_1}\) \(J_{A2} = -D_{AB} \frac{\Delta C}{L_2}\)
05

Calculate the mass flow rate

To find the mass flow rate, \(\dot{m}\), of hydrogen, we multiply the molar flux by the molar mass of hydrogen, \(M_H = 2 \mathrm{kg/kmol}\): \(\dot{m}_{1} = J_{A1} \cdot M_H\) \(\dot{m}_{2} = J_{A2} \cdot M_H\)
06

Solve for the mass flow rate

By substituting the given values and calculated values, we can now solve for the mass flow rate for each membrane thickness: \(\dot{m}_{1} = J_{A1} \cdot M_H = \left(-D_{AB} \frac{\Delta C}{L_1}\right) \cdot M_H\) \(\dot{m}_{2} = J_{A2} \cdot M_H = \left(-D_{AB} \frac{\Delta C}{L_2}\right) \cdot M_H\) After solving for the mass flow rates, we will have the mass flow rate of hydrogen through the membrane under steady conditions for both thicknesses.

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Most popular questions from this chapter

A steel part whose initial carbon content is \(0.12\) percent by mass is to be case-hardened in a furnace at \(1150 \mathrm{~K}\) by exposing it to a carburizing gas. The diffusion coefficient of carbon in steel is strongly temperature dependent, and at the furnace temperature it is given to be \(D_{A B}=7.2 \times 10^{-12} \mathrm{~m}^{2} / \mathrm{s}\). Also, the mass fraction of carbon at the exposed surface of the steel part is maintained at \(0.011\) by the carbon-rich environment in the furnace. If the hardening process is to continue until the mass fraction of carbon at a depth of \(0.7 \mathrm{~mm}\) is raised to \(0.32\) percent, determine how long the part should be held in the furnace.

Carbon at \(1273 \mathrm{~K}\) is contained in a \(7-\mathrm{cm}\)-innerdiameter cylinder made of iron whose thickness is \(1.2 \mathrm{~mm}\). The concentration of carbon in the iron at the inner surface is \(0.5 \mathrm{~kg} / \mathrm{m}^{3}\) and the concentration of carbon in the iron at the outer surface is negligible. The diffusion coefficient of carbon through iron is \(3 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\). The mass flow rate of carbon by diffusion through the cylinder shell per unit length of the cylinder is (a) \(2.8 \times 10^{-9} \mathrm{~kg} / \mathrm{s}\) (b) \(5.4 \times 10^{-9} \mathrm{~kg} / \mathrm{s}\) (c) \(8.8 \times 10^{-9} \mathrm{~kg} / \mathrm{s}\) (d) \(1.6 \times 10^{-8} \mathrm{~kg} / \mathrm{s}\) (e) \(5.2 \times 10^{-8} \mathrm{~kg} / \mathrm{s}\) 14-185 The surface of an iron component is to be hardened by carbon. The diffusion coefficient of carbon in iron at \(1000^{\circ} \mathrm{C}\) is given to be \(3 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\). If the penetration depth of carbon in iron is desired to be \(1.0 \mathrm{~mm}\), the hardening process must take at least (a) \(1.10 \mathrm{~h}\) (b) \(1.47 \mathrm{~h}\) (c) \(1.86 \mathrm{~h}\) (d) \(2.50 \mathrm{~h}\) (e) \(2.95 \mathrm{~h}\)

Consider a 20 -cm-thick brick wall of a house. The indoor conditions are \(25^{\circ} \mathrm{C}\) and 50 percent relative humidity while the outside conditions are \(50^{\circ} \mathrm{C}\) and 50 percent relative humidity. Assuming that there is no condensation or freezing within the wall, determine the amount of moisture flowing through a unit surface area of the wall during a \(24-\mathrm{h}\) period.

Benzene-free air at \(25^{\circ} \mathrm{C}\) and \(101.3 \mathrm{kPa}\) enters a 5 -cm-diameter tube at an average velocity of \(5 \mathrm{~m} / \mathrm{s}\). The inner surface of the \(6-m\)-long tube is coated with a thin film of pure benzene at \(25^{\circ} \mathrm{C}\). The vapor pressure of benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) at \(25^{\circ} \mathrm{C}\) is \(13 \mathrm{kPa}\), and the solubility of air in benezene is assumed to be negligible. Calculate \((a)\) the average mass transfer coefficient in \(\mathrm{m} / \mathrm{s},(b)\) the molar concentration of benzene in the outlet air, and \((c)\) the evaporation rate of benzene in \(\mathrm{kg} / \mathrm{h}\).

A 2-mm-thick 5-L vessel made of nickel is used to store hydrogen gas at \(358 \mathrm{~K}\) and \(300 \mathrm{kPa}\). If the total inner surface area of the vessel is \(1600 \mathrm{~cm}^{2}\), determine the rate of gas loss from the nickel vessel via mass diffusion. Also, determine the fraction of the hydrogen lost by mass diffusion after one year of storage.
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