Question

The solubility of hydrogen gas in steel in terms of its mass fraction is given as \(w_{\mathrm{H}_{2}}=2.09 \times 10^{-4} \exp (-3950 / T) P_{\mathrm{H}_{2}}^{0.5}\) where \(P_{\mathrm{H}_{2}}\) is the partial pressure of hydrogen in bars and \(T\) is the temperature in \(\mathrm{K}\). If natural gas is transported in a 1-cm-thick, 3-m-internal-diameter steel pipe at \(500 \mathrm{kPa}\) pressure and the mole fraction of hydrogen in the natural gas is 8 percent, determine the highest rate of hydrogen loss through a 100 -m-long section of the pipe at steady conditions at a temperature of \(293 \mathrm{~K}\) if the pipe is exposed to air. Take the diffusivity of hydrogen in steel to be \(2.9 \times 10^{-13} \mathrm{~m}^{2} / \mathrm{s}\).

Short answer

The highest rate of hydrogen loss through a 100-m-long section of the pipe at steady conditions is approximately 5.754 x 10^-9 mol/s.

Step-by-step solution

Find the mass fraction of hydrogen in steel

To find the mass fraction of hydrogen in steel, we will use the given equation: \(w_{\mathrm{H_{2}}} = 2.09 \times 10^{-4} \exp (-3950 / T) P_{\mathrm{H_{2}}}^{0.5}\) where: \(w_{\mathrm{H_{2}}}\) is the mass fraction of hydrogen in steel \(T\) is the temperature in K (293 K) \(P_{\mathrm{H_{2}}}\) is the partial pressure of hydrogen in bars First, we need to find the partial pressure of hydrogen in the natural gas. The total pressure is \(500 \mathrm{kPa}\), and the mole fraction of hydrogen is 8%, so we can calculate the partial pressure of hydrogen as follows: \(P_{\mathrm{H_{2}}} = 0.08 \times 500 \mathrm{kPa} = 40 \mathrm{kPa}\) We need to convert this pressure to bars: \(P_{\mathrm{H_{2}}} = 40 \mathrm{kPa} \times (\frac{1 \mathrm{bar}}{100 \mathrm{kPa}}) = 0.4 \mathrm{bars}\) Now we can find the mass fraction of hydrogen in steel: \(w_{\mathrm{H_{2}}} = 2.09 \times 10^{-4} \exp (-3950 / 293) (0.4)^{0.5} = 4.166 \times 10^{-7}\)

Convert mass fraction to concentration in moles

To convert the mass fraction to a concentration in terms of moles, we need the molar mass of hydrogen (\(M_{\mathrm{H_{2}}}\) = 2 g/mol): \(\mathrm{Concentration}_{\mathrm{H_{2}}} = \frac{w_{\mathrm{H_{2}}}}{M_{\mathrm{H_{2}}}} = \frac{4.166 \times 10^{-7}}{2 \times 10^{-3} \mathrm{~kg/mol}}\) \(\mathrm{Concentration}_{\mathrm{H_{2}}} = 2.083 \times 10^{-4} \mathrm{~mol/kg}\)

Find the concentration difference

Assuming the concentration of hydrogen on the outside of the pipe is zero, the concentration difference is: \(\Delta \mathrm{Concentration}_{\mathrm{H_{2}}} = 2.083 \times 10^{-4} \mathrm{~mol/kg}\)

Calculate the highest rate of hydrogen loss using Fick's Law

Fick's Law states that the mass transfer rate through a wall (mol/s) is the flux multiplied by the wall area. The flux is given as the product of the diffusivity and the concentration gradient (concentration difference divided by the pipe wall thickness): \(\mathrm{Mass~Transfer~Rate}_{\mathrm{H_{2}}} = \mathrm{Diffusivity}_{\mathrm{H_{2}}} \times \frac{\Delta \mathrm{Concentration}_{\mathrm{H_{2}}}}{\mathrm{Wall~Thickness}} \times \mathrm{Wall~Area}\) We know the diffusivity of hydrogen in steel is \(2.9 \times 10^{-13} \mathrm{m^2/s}\), the wall thickness is 0.01 m, and the pipe is 3 m in diameter and 100 m long. We can find the wall area as follows: \(\mathrm{Wall~Area} = 2 \times \pi \times \frac{3}{2} \mathrm{~m} \times 100 \mathrm{~m} = 300\pi \mathrm{~m^2}\) Now, we can calculate the mass transfer rate: \(\mathrm{Mass~Transfer~Rate}_{\mathrm{H_{2}}} = 2.9 \times 10^{-13} \mathrm{~m^2/s} \times \frac{2.083 \times 10^{-4} \mathrm{~mol/kg}}{0.01 \mathrm{~m}} \times 300\pi \mathrm{~m^2}\) \(\mathrm{Mass~Transfer~Rate}_{\mathrm{H_{2}}} = 5.754 \times 10^{-9} \mathrm{~mol/s}\) The highest rate of hydrogen loss through a 100-m-long section of the pipe at steady conditions is \(5.754 \times 10^{-9} \mathrm{~mol/s}\).

Key concepts

Mass Fraction

Understanding the mass fraction of a component in a mixture is crucial for determining its concentration and its potential impact on the overall properties of the mixture. In our exercise, the mass fraction of hydrogen in steel, denoted as \(w_{\mathrm{H_{2}}}\), represents the proportion of hydrogen by mass relative to the total mass of the steel. The equation provided connects this fraction to the temperature (T) and the partial pressure of hydrogen (\(P_{\mathrm{H_{2}}}\)). The mass fraction is essential when calculating the rate of hydrogen permeation through steel, as it establishes a baseline for how much hydrogen is initially present within the steel matrix.

Moreover, understanding mass fraction is pivotal when assessing material durability, as certain concentrations of hydrogen can lead to phenomena like hydrogen embrittlement, affecting a material's mechanical properties.

Fick's Law of Diffusion

Fick's Law of diffusion is a principle that predicts the movement of particles from areas of higher concentration to areas of lower concentration. In the context of our problem, Fick's law helps us to determine the rate at which hydrogen atoms will migrate through the steel pipe wall. This rate of migration, or diffusivity, is a function of the concentration gradient and the physical characteristics of the material – in this case, steel.

The formula we use is an application of Fick's first law, which relates the diffusive flux to the concentration gradient. It is the mathematical expression that quantifies the amount of substance that will transfer per unit area per unit time, depending on the concentration difference across the steel wall and its diffusivity coefficient. Understanding Fick's law is not only crucial for solving this problem but is also fundamental in material science and engineering disciplines, dealing with mass transfer in various applications.

Partial Pressure

Partial pressure is a concept from gas laws that refers to the pressure a single gas component in a mixture would exert if it occupied the entire volume alone at the same temperature. In our exercise, hydrogen's partial pressure is a fraction of the total pressure exerted by the natural gas within the pipe. The mole fraction of hydrogen in the gas mix, along with the total pressure, determines its partial pressure, which in turn influences the mass fraction of hydrogen dissolved in the steel, according to the provided solubility formula.

This correlation is essential because it links the chemical potential of hydrogen (driven by partial pressure) to its solubility in the metal, directly affecting the concentration gradient that is established for diffusion. An understanding of partial pressure is necessary when working in fields like chemistry, materials science, and environmental engineering.

Molecular Concentration

The molecular concentration provides a measure of the amount of a substance within a given volume or mass of material. We observe in the exercise that converting the mass fraction of hydrogen to a concentration in moles per kilogram of steel requires the molar mass of hydrogen. This conversion is a crucial step as it allows us to apply Fick's law, which uses concentration (in moles per volume) as a factor to determine the mass transfer rate.

Being able to accurately describe the molecular concentration plays a fundamental role in predicting the behavior of substances in various environments. For instance, when engineers are designing pipelines, they must understand how different concentrations of gases can affect the integrity and performance of the materials used in construction. Grasping the concept of molecular concentration is valuable across various scientific fields, including chemistry, physics, and environmental studies.