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Chapter 14: Problem 51

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).

Short Answer

Expert verified
(a) The final content of hydrogen in the material can be calculated as: \( \rho_{H,f} = C_f \times M = (160\,\frac{\text{kmol}}{\text{m}^3\,\text{atm}} \times 0.1\,\text{atm})\times (2\,\frac{\text{kg}}{\text{kmol}}) = 32\,\frac{\text{kg}}{\text{m}^3} \) (b) The mass Biot number (Bi_m) can be calculated as: \(h_m = 1.5\,\frac{\text{m}}{\text{h}} \times \frac{1 \text{h}}{3600\, \text{s}} = 4.17 \times 10^{-4}\,\frac{\text{m}}{\text{s}}\) \( Bi_m = \frac{h_m \delta}{D_{AB}} = \frac{(4.17 \times 10^{-4}\,\frac{\text{m}}{\text{s}}) (6 \times 10^{-3}\, \text{m})}{2.6 \times 10^{-8}\,\frac{\text{m}^2}{\text{s}}} = 96.3 \) (c) To find the time required, we can use the error function and rearranged Fick's second law equation: \( t = \frac{\left(\frac{1}{2} \times 6 \times 10^{-3}\, \text{m}\right)^2}{4 \times 2.6 \times 10^{-8}\,\frac{\text{m}^2}{\text{s}} \times \left[\text{erf}^{-1} \left(\frac{2C_f - C_f}{C_i - C_f} \right)\right]^2} \approx 3012\,\text{s} \) Thus, the time required to reduce the hydrogen mass density at the center of the sheet to twice the limiting value is approximately 3012 seconds.

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.

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Most popular questions from this chapter

Consider a spherical organism of radius \(r_{o}\) within which respiration occurs at a uniform volumetric rate of \(\dot{N}_{\mathrm{A}}=-k_{0}\). That is, oxygen (species A) consumption is governed by a zero-order, homogeneous chemical reaction. (a) If a molar concentration of \(C_{\mathrm{A}}\left(r_{o}\right)=C_{\mathrm{A}, o}\) is maintained at the surface of the organism, obtain an expression for the radial distribution of oxygen, \(C_{\mathrm{A}}(r)\), within the organism. From your solution, can you discern any limits on applicability of the result? (b) Obtain an expression for the rate of oxygen consumption within the organism. (c) Consider an organism of radius \(r_{o}=0.10 \mathrm{~mm}\) and a diffusion coefficient for oxygen transfer of \(D_{\mathrm{AB}}=10^{-8} \mathrm{~m}^{2} / \mathrm{s}\). If \(C_{\mathrm{A}, o}=5 \times 10^{-5} \mathrm{kmol} / \mathrm{m}^{3}\)and \(k_{0}=1.2 \times 10^{-4} \mathrm{kmol} / \mathrm{s}^{*} \mathrm{~m}^{3}\), what is the molar concentration of \(\mathrm{O}_{2}\) at the center of the organism?

As an employee of the Los Angeles Air Quality Commission, you have been asked to develop a model for computing the distribution of \(\mathrm{NO}_{2}\) in the atmosphere. The molar flux of \(\mathrm{NO}_{2}\) at ground level, \(N_{\mathrm{A}, 0}^{N}\), is presumed known. This flux is attributed to automobile and smoke stack emissions. It is also known that the concentration of \(\mathrm{NO}_{2}\) at a distance well above ground level is zero and that \(\mathrm{NO}_{2}\) reacts chemically in the atmosphere. In particular, \(\mathrm{NO}_{2}\) reacts with unburned hydrocarbons (in a process that is activated by sunlight) to produce PAN (peroxyacetylnitrate), the final product of photochemical smog. The reaction is first order, and the local rate at which it occurs may be expressed as \(\dot{N}_{\mathrm{A}}=-k_{1} C_{\mathrm{A}}\). (a) Assuming steady-state conditions and a stagnant atmosphere, obtain an expression for the vertical distribution \(C_{\mathrm{A}}(x)\) of the molar concentration of \(\mathrm{NO}_{2}\) in the atmosphere. (b) If an \(\mathrm{NO}_{2}\) partial pressure of \(p_{\mathrm{A}}=2 \times 10^{-6}\) bar is sufficient to cause pulmonary damage, what is the value of the ground level molar flux for which you would issue a smog alert? You may assume an isothermal atmosphere at \(T=300 \mathrm{~K}\), a reaction coefficient of \(k_{1}=0.03 \mathrm{~s}^{-1}\), and an \(\mathrm{NO}_{2}\)-air diffusion coefficient of \(D_{\mathrm{AB}}=0.15 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}\).

Hydrogen at a pressure of 2 atm flows within a tube of diameter \(40 \mathrm{~mm}\) and wall thickness \(0.5 \mathrm{~mm}\). The outer surface is exposed to a gas stream for which the hydrogen partial pressure is \(0.1 \mathrm{~atm}\). The mass diffusivity and solubility of hydrogen in the tube material are \(1.8 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\) and \(160 \mathrm{kmol} / \mathrm{m}^{3}\) *atm, respectively. When the system is at \(500 \mathrm{~K}\), what is the rate of hydrogen transfer through the tube per unit length \((\mathrm{kg} / \mathrm{s} \cdot \mathrm{m})\) ?

Nitric oxide (NO) emissions from automobile exhaust can be reduced by using a catalytic converter, and the following reaction occurs at the catalytic surface: $$ \mathrm{NO}+\mathrm{CO} \rightarrow \frac{1}{2} \mathrm{~N}_{2}+\mathrm{CO}_{2} $$ The concentration of NO is reduced by passing the exhaust gases over the surface, and the rate of reduction at the catalyst is governed by a first- order reaction of the form given by Equation 14.66. As a first approximation it may be assumed that NO reaches the surface by one-dimensional diffusion through a thin gas film of thickness \(L\) that adjoins the surface. Referring to Figure 14.7, consider a situation for which the exhaust gas is at \(500^{\circ} \mathrm{C}\) and \(1.2\) bars and the mole fraction of \(\mathrm{NO}\) is \(x_{\mathrm{A}, L}=0.15\). If \(D_{\mathrm{AB}}=10^{-4} \mathrm{~m}^{2} / \mathrm{s}\), \(k_{1}^{\prime \prime}=0.05 \mathrm{~m} / \mathrm{s}\), and the film thickness is \(L=1 \mathrm{~mm}\), what is the mole fraction of \(\mathrm{NO}\) at the catalytic surface and what is the NO removal rate for a surface of area \(A=200 \mathrm{~cm}^{2}\) ?

Referring to Problem 14.34, a more representative model of respiration in a spherical organism is one for which oxygen consumption is governed by a firstorder reaction of the form \(\dot{N}_{\mathrm{A}}=-k_{1} C_{\mathrm{A}}\). (a) If a molar concentration of \(C_{\mathrm{A}}\left(r_{o}\right)=C_{\mathrm{A}, o}\) is maintained at the surface of the organism, obtain an expression for the radial distribution of oxygen, \(C_{\mathrm{A}}(r)\), within the organism. Hint: To simplify solution of the species diffusion equation, invoke the transformation \(y \equiv r C_{\mathrm{A}}\). (b) Obtain an expression for the rate of oxygen consumption within the organism. (c) Consider an organism of radius \(r_{o}=0.10 \mathrm{~mm}\) and a diffusion coefficient of \(D_{\mathrm{AB}}=10^{-8} \mathrm{~m}^{2} / \mathrm{s}\). If \(C_{\mathrm{A}, o}=5 \times 10^{-5} \mathrm{kmol} / \mathrm{m}^{3}\) and \(k_{1}=20 \mathrm{~s}^{-1}\), estimate the corresponding value of the molar concentration at the center of the organism. What is the rate of oxygen consumption by the organism?
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