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Hydrogen gas is used in a process to manufacture a sheet material of \(6-\mathrm{mm}\) thickness. At the end of the process, \(\mathrm{H}_{2}\) remains in solution in the material with a uniform concentration of \(320 \mathrm{kmol} / \mathrm{m}^{3}\). To remove \(\mathrm{H}_{2}\) from the material, both surfaces of the sheet are exposed to an airstream at \(500 \mathrm{~K}\) and a total pressure of \(3 \mathrm{~atm}\). Due to contamination, the hydrogen partial pressure is \(0.1 \mathrm{~atm}\) in the airstream, which provides a convection mass transfer coefficient of \(1.5 \mathrm{~m} / \mathrm{h}\). The mass diffusivity and solubility of hydrogen (A) in the sheet material (B) are \(D_{\mathrm{AB}}=2.6 \times 10^{-8} \mathrm{~m}^{2} / \mathrm{s}\) and \(S_{\mathrm{AB}}=160 \mathrm{kmol} / \mathrm{m}^{3} \cdot\) atm, respectively. (a) If the sheet material is left exposed to the airstream for a long time, determine the final content of hydrogen in the material \(\left(\mathrm{kg} / \mathrm{m}^{3}\right)\). (b) Identify and evaluate the parameter that can be used to determine whether the transient mass diffusion process in the sheet can be assumed to be characterized by a uniform concentration at any time during the process. Hint: This situation is analogous to that used to determine the validity of the lumped-capacitance method for a transient heat transfer analysis. (c) Determine the time required to reduce the hydrogen mass density at the center of the sheet to twice the limiting value calculated in part (a).

Step by step solution

01

(a) Find the final content of hydrogen in the material

To find the final content of hydrogen in the material, we'll need to use the solubility property of hydrogen in the sheet material (B), given as \(S_{AB}=160\,\text{kmol}/\text{m}^3\,\text{atm}\). The hydrogen partial pressure at the air interface is given as 0.1 atm. Calculate the final concentration of hydrogen, \( C_{f} = S_{AB} \times p_H = 160\,\frac{\text{kmol}}{\text{m}^3\,\text{atm}} \times 0.1\,\text{atm} \) Now, convert the final concentration of hydrogen to mass density using the molar mass of hydrogen, M (2 kg/kmol). \( \rho_{H,f} = C_f \times M = (160\,\frac{\text{kmol}}{\text{m}^3\,\text{atm}} \times 0.1\,\text{atm})\times (2\,\frac{\text{kg}}{\text{kmol}}) \)

02

(b) Identify and evaluate the parameter for uniform concentration

The hint suggests that there is an analogy between mass transfer and heat transfer problems. In transient heat transfer analysis, the Biot number (Bi) is used to determine the validity of the lumped-capacitance method. Similarly, for mass transfer, we can define a mass Biot number (Bi_m) as follows: \( Bi_m = \frac{h_m \delta}{D_{AB}}\) where \(h_m\) is the mass transfer coefficient (1.5 m/h), \(\delta\) is the sheet thickness (6 mm), and \(D_{AB}\) is the mass diffusivity (2.6 脳 10鈦烩伕 m虏/s). First, convert the mass transfer coefficient to the same unit as mass diffusivity (m/s), \( h_m = 1.5\,\frac{\text{m}}{\text{h}} \times \frac{1 \text{h}}{3600\, \text{s}}\) Now, calculate the mass Biot number, \( Bi_m = \frac{h_m \delta}{D_{AB}}\)

03

(c) Time required to reduce hydrogen mass density to twice the limiting value

We have to determine the time required to reduce the hydrogen mass density at the center of the sheet to twice the final content of hydrogen. We can use Fick's second law of diffusion, which involves the error function (erf): \( \frac{C - C_f}{C_i - C_f} = \text{erf}\left(\frac{x}{2\sqrt{D_{AB}t}}\right) \) Rearranging the equation to solve for time, we get \( t = \frac{x^2}{4D_{AB}\left[\text{erf}^{-1} \left(\frac{C - C_f}{C_i - C_f} \right)\right]^2} \) To find the time required for the hydrogen mass density to be twice the limiting value, we need to plug in x as half the sheet thickness, Ci as the initial concentration (320 kmol/m鲁), and C as twice the final concentration: \( t = \frac{\left(\frac{1}{2} \times 6 \times 10^{-3}\, \text{m}\right)^2}{4 \times 2.6 \times 10^{-8}\,\frac{\text{m}^2}{\text{s}} \times \left[\text{erf}^{-1} \left(\frac{2C_f - C_f}{C_i - C_f} \right)\right]^2} \) Calculate the time t.

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Key Concepts

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

Hydrogen Diffusion

Hydrogen diffusion is a fundamental concept in mass transfer processes. It involves the movement of hydrogen molecules through a material due to a concentration gradient. Diffusion is driven by the random motion of molecules, moving from an area of higher concentration to one of lower concentration.

There are several factors affecting hydrogen diffusion:

• Concentration gradient: A higher difference in concentration increases the rate of diffusion.
• Temperature: Higher temperatures increase molecular motion, enhancing diffusion.
• Material properties: The nature of the sheet material affects diffusion. Porous or less dense materials may allow faster diffusion.
• Diffusivity: Defined as the extent to which a molecule spreads out over time within a given medium, diffusivity (\( D_{AB} \) in this case) is crucial. For hydrogen, the molar mass is small, meaning it can diffuse quickly.

The diffusion of hydrogen can be described using Fick's laws, particularly when we're interested in situations where concentration varies with both time and position.

Solubility

Solubility is the measure of how much of a particular solute can dissolve in a solvent at a given temperature and pressure. In the context of hydrogen diffusion into a material, solubility indicates the maximum amount of hydrogen that can be dissolved in the sheet material.

The solubility of hydrogen in the sheet material is characterized by the parameter \( S_{AB} \), which represents the amount of hydrogen (\( \text{kmol/m}^3 \)) that can be dissolved under a unit partial pressure of hydrogen (\( \text{atm} \)).

• It is essential for predicting how much hydrogen will remain in the material when subjected to certain conditions.
• In the problem, solubility helps us calculate the final concentration of hydrogen by multiplying it with the hydrogen partial pressure in the air.

Therefore, understanding solubility helps determine equilibrium conditions where the rate of hydrogen entering the material equals the rate leaving it.

Biot Number

The Biot number is a dimensionless parameter that describes the ratio of internal resistance to diffusive transport (within a material) compared to external convective forces. It is commonly used in heat and mass transfer to assess the effectiveness of a process.

Defined as \( Bi_m = \frac{h_m \delta}{D_{AB}} \), where:

• \( h_m \) is the convective mass transfer coefficient indicating the ease of mass transfer over the surface.
• \( \delta \) is the characteristic length, usually the thickness of the material.
• \( D_{AB} \) is the diffusivity, showing how well the substance moves within the medium.

A small Biot number suggests that internal transport within the material occurs faster than external transfer, implying a uniform concentration within the material. Conversely, a high Biot number indicates dominating external mass transfer resistance, meaning possible concentration gradients within the material.

Fick's Second Law

Fick's Second Law of Diffusion describes how diffusion causes the concentration of a substance to change over time. While Fick's First Law relates to steady-state diffusion, Fick's Second Law considers dynamic situations where concentration varies with time.This law is represented by the equation:\[\frac{\partial C}{\partial t} = D_{AB} abla^2 C\]where:

• \( C \) represents concentration.
• \( t \) denotes time.
• \( D_{AB} \) is the mass diffusivity.
• \( abla^2 \) is the Laplacian operator, which can include second derivatives of \( x, y, z \) coordinates.

This law is powerful for predicting how the concentration of hydrogen within the material will decay over time due to diffusion. It is applied specifically when calculating the time required to achieve a particular concentration within the material, as seen in transient mass transfer problems. Mathematical tools, such as the error function, are often used to solve these equations efficiently.