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Ultra-pure hydrogen is required in applications ranging from the manufacturing of semiconductors to powering fuel cells. The crystalline structure of palladium allows only the transfer of atomic hydrogen (H) through its thickness, and therefore palladium membranes are used to filter hydrogen from contaminated streams containing mixtures of hydrogen and other gases. Hydrogen molecules \(\left(\mathrm{H}_{2}\right)\) are first adsorbed onto the palladium's surface and are then dissociated into atoms \((\mathrm{H})\), which subsequently diffuse through the metal. The H atoms recombine on the opposite side of the membrane, forming pure \(\mathrm{H}_{2}\). The surface concentration of H takes the form \(C_{\mathrm{H}}=K_{s} p_{\mathrm{H}_{2}}^{05}\), where \(K_{s} \approx\) \(1.4 \mathrm{kmol} / \mathrm{m}^{3} \cdot \mathrm{bar}^{0.5}\) is known as Sieverts constant. Consider an industrial hydrogen purifier consisting of an array of palladium tubes with one tube end connected to a collector plenum and the other end closed. The tube bank is inserted into a shell. Impure \(\mathrm{H}_{2}\) at \(T=\) \(600 \mathrm{~K}, p=15\) bars, \(x_{\mathrm{H}_{2}}=0.85\) is introduced into the shell while pure \(\mathrm{H}_{2}\) at \(p=6\) bars, \(T=600 \mathrm{~K}\) is extracted through the tubes. Determine the production rate of pure hydrogen \((\mathrm{kg} / \mathrm{h})\) for \(N=100\) tubes which are of inside diameter \(D_{i}=1.6 \mathrm{~mm}\), wall thickness \(t=75 \mu \mathrm{m}\), and length \(L=80 \mathrm{~mm}\). The mass diffusivity of hydrogen \((\mathrm{H})\) in palladium at \(600 \mathrm{~K}\) is approximately \(D_{\mathrm{AB}}=7 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\).

Step by step solution

01

Calculate the concentration gradient of atomic hydrogen on the surfaces of the tubes

First, we need to calculate the surface concentration of atomic hydrogen on both sides of the tube. We can use the Sieverts constant relationship provided: \(C_{H} = K_s p_{H_2}^{0.5}\) where \(C_H\) is the surface concentration of atomic hydrogen, \(K_s\) is Sieverts constant, and \(p_{H_2}\) is the partial pressure of hydrogen. On the impure side of the palladium tubes, the partial pressure of hydrogen is given by \(p_{H_2, impure} = p x_{H_2} = 15\textrm{ bars} \cdot 0.85 = 12.75\textrm{ bars}\), so the surface concentration of atomic hydrogen on the impure side can be calculated as follows: \(C_{H, impure} = K_s p_{H_2, impure}^{0.5} \approx 1.4 \frac{\mathrm{kmol}}{\mathrm{m}^3\cdot\mathrm{bar}^{0.5}} (12.75\textrm{ bars})^{0.5}\) On the pure side of the palladium tubes, the partial pressure of hydrogen is given by \(p_{H_2, pure} = 6\textrm{ bars}\), so the surface concentration of atomic hydrogen on the pure side can be calculated as follows: \(C_{H, pure} = K_s p_{H_2, pure}^{0.5} \approx 1.4 \frac{\mathrm{kmol}}{\mathrm{m}^3\cdot\mathrm{bar}^{0.5}} (6\textrm{ bars})^{0.5}\)

02

Calculate the mass transfer flux using Fick's law

The mass transfer flux, \(j_m\), can be calculated using Fick's law with the given mass diffusivity, \(D_{AB}\), concentration gradient, and tube wall thickness, \(t\): \(j_m = -D_{AB} \frac{C_{H, impure} - C_{H, pure}}{t}\) To find the mass transfer flux, substitute the values for \(D_{AB}\), \(C_{H, impure}\), \(C_{H, pure}\), and \(t\).

03

Compute the total mass transfer rate

To determine the total mass transfer rate, we need to find the transfer rate per palladium tube and then multiply by the number of tubes, \(N\). First, we need the inside surface area of one tube, which can be found using: \(A_{inside} = \pi D_i L\) Then, the mass transfer rate per tube can be obtained by multiplying the mass transfer flux by the inside surface area: \(m_{H_2, tube} = j_m A_{inside}\) Finally, by multiplying the mass transfer rate per tube by the total number of tubes, we can calculate the total mass transfer rate: \(m_{H_2, total} = m_{H_2, tube} N\)

04

Convert mass transfer rate to the required units

The mass transfer rate calculated previously is in units of kmol/s. To convert it to kg/h, we can multiply by the molar mass of hydrogen and convert the time unit: \(m_{H_2, kg/h} = m_{H_2, total} \times 2 \frac{\textrm{kg}}{\textrm{kmol}} \times \frac{3600\textrm{ s}}{1\textrm{ h}}\) Now we have the production rate of pure hydrogen in kg/h.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mass Transfer in Palladium

Palladium, a precious metal with unique properties, plays a vital role in purifying hydrogen due to its ability to selectively allow hydrogen atoms to diffuse through it. This phenomenon of selective permeation is driven by a process called mass transfer, which is the movement of atoms from a region of higher concentration to a region of lower concentration.

When impure hydrogen gas comes in contact with a palladium membrane, hydrogen molecules (\text{\footnotesize{\(\text{\text{\footnotesize{\)\begin{math}\text{\text{\footnotesize{\(\begin{math}\text{{H}}_2\text{\text{\footnotesize{\)\begin{math}\text{\text{\footnotesize{\(\begin{math}\text{{H}}_2\text{\text{\footnotesize{\)\begin{math}\text{{H}}_2\text{\text{\footnotesize{\(\begin{math}\text{{H}}_2\text{\text{\footnotesize{\)\begin{math}\text{{H}}_2\text{\text{\footnotesize{\(\begin{math}\text{{H}}_2\text{\text{\footnotesize{\)\begin{math}\begin{math}$}}}}}}}}}}}}}}}}}}}}