Problem 8 - An Introduction to Materials Engineering and Science for Chemical and Materials Engineers

Question

Hydrogen dissolved in iron obeys Sievert's Law; that is, at equilibrium we have $$ \frac{1}{2} \mathrm{H}_{2}(g) \Longleftrightarrow[\mathrm{H}]_{\text {dissolved in iron }} $$ where \([\mathrm{H}]\) is the concentration of atomic hydrogen in the iron, in ppm by weight, in equilibrium with hydrogen gas at the pressure \(P_{\mathrm{H}_{2}} .\) At \(400^{\circ} \mathrm{C}\) and a hydrogen pressure of \(1.013 \times 10^{5} \mathrm{~Pa}(1 \mathrm{~atm})\), the solubility of hydrogen in iron is 3 ppm by weight. Calculate the flux of hydrogen through an iron tank of wall thickness \(0.001 \mathrm{~m}\) at \(400^{\circ} \mathrm{C}\). The density of iron \(400^{\circ} \mathrm{C}\) is \(7730 \mathrm{~kg} / \mathrm{m}^{3}\), and the diffusivity of hydrogen is \(10^{-8} \mathrm{~m}^{2} / \mathrm{s}\). Clearly state any assumptions you make.

Step-by-Step Solution

Answer

Flux of hydrogen is \\(2.319 \\ imes 10^{-4} ext{ kg/m}^2/s\\).

Step 1: Understand the problem setup

We are tasked to calculate the flux of hydrogen through an iron tank wall at a given temperature and pressure. The key parameters given are: the solubility of hydrogen in iron (3 ppm), the wall thickness (0.001 m), the density of iron (7730 kg/m³), and the diffusivity of hydrogen in iron (10⁻⁸ m²/s). We need to identify the driving force for diffusion based on Sievert's Law.

Step 2: Apply Sievert's Law for solubility

According to Sievert’s Law, the concentration of hydrogen in iron, \([ ext{H}]\), is proportional to the square root of the hydrogen gas pressure, \(P_{H_2}\). We are given the solubility of hydrogen as 3 ppm at 1 atm, and we assume any concentration difference drives diffusion. The given solubility is our concentration gradient from the internal to external surface, assuming the external hydrogen concentration is negligible.

Step 3: Use Fick's First Law for flux calculation

Fick's First Law states that the flux \(J\) is given by \(J = -Dabla C\), where \(D\) is the diffusivity and \(abla C\) is the concentration gradient. The concentration gradient \(abla C\) across the tank's wall is \(\frac{C_{1} - C_{2}}{\Delta x}\), where \(C_{1} = 3 ext{ ppm by weight} \ imes \ ext{density of iron}\) and \(\Delta x\) is \(0.001 ext{ m}\).

Step 4: Convert concentrations to SI units

Convert the concentration from ppm to a more consistent unit (kg/m³) for calculations. Since 1 ppm by weight corresponds to \(1 imes 10^{-6}\) kg of H per kg of Fe, for 3 ppm, \(3 imes 10^{-6} imes 7730 ext{ kg/m}^3 = 0.02319 ext{ kg/m}^3\). Assuming the concentration outside is negligible, \(C_{2} \approx 0\), and the gradient is simply \(\frac{0.02319}{0.001}\) kg/m³.

Step 5: Calculate the flux with Fick's Law

Plug in the values into Fick's Law equation: \(J = -Dabla C = -10^{-8} \ imes \frac{0.02319}{0.001} \ ext{ kg/m}^2/s\) resulting in \(J = -2.319 \ imes 10^{-4} ext{ kg/m}^2/s\). The negative sign indicates direction of primary to secondary side, consistent with outward diffusion.

Key Concepts

Hydrogen solubility in iron

Understanding hydrogen solubility in iron involves recognizing how hydrogen atoms interact with iron at certain conditions of temperature and pressure. According to Sievert's Law, the solubility of hydrogen in metals like iron is proportional to the square root of the pressure of the hydrogen gas with which it is in equilibrium. This means that at a given temperature, increasing the hydrogen pressure increases the amount of hydrogen dissolved in the iron.

However, this doesn't imply an endless increase, as there are limits to the solubility based on the material's characteristics and environmental conditions. In this exercise, it is specified that at 400°C and 1 atmosphere of pressure, the solubility of hydrogen in iron is 3 ppm by weight. This value serves as a crucial starting point for understanding how much hydrogen is actually present in the iron and is essential for further calculations related to diffusion.

Fick's First Law

Fick's First Law provides a way to calculate the flux of particles, such as hydrogen atoms, through a material. The law states that the flux, denoted as \(J\), is proportional to the concentration gradient across a certain distance. Mathematically, it is expressed as \(J = -Dabla C\), where \(D\) is the diffusivity coefficient, and \( abla C\) is the concentration gradient.

In simpler terms, this law implies that particles move from areas of high concentration to areas of low concentration, a process driven by differences in concentration across a material. The greater these differences, the larger the flux.

The negative sign in the equation indicates the direction of movement is from high to low concentration, aligning with the natural flow of particles in diffusion processes.

Diffusion and flux calculation

Diffusion in the context of this problem involves the movement of hydrogen atoms through the iron tank wall. To calculate how much hydrogen passes through a given area (flux), we apply Fick's First Law. In this exercise, the concentration of hydrogen inside the iron is calculated, assuming a negligible concentration outside (the external side).

The concentration gradient, \( abla C\), is determined by the solubility of hydrogen in iron, which is provided as 3 ppm. This is then converted to consistent units that can be used in the flux formula. The diffusivity, \(D\), is also provided as \(10^{-8} \text{ m}^2/ ext{s}\). Critical to accurate calculation is the wall thickness of 0.001 m, which defines the distance over which diffusion occurs.

Ultimately, plugging these values into the equation yields the flux, representing the rate at which hydrogen atoms move through the wall.

Concentration conversion

Converting concentration from ppm to relevant SI units is essential for accurate calculations in scientific applications. A concentration of 3 ppm by weight indicates 3 parts of hydrogen per million parts of iron by weight.

To use this in Fick’s Law, we convert ppm into kg/m³. Knowing that 1 ppm corresponds to \(1 \times 10^{-6}\) kg/kg, the conversion for 3 ppm in iron is calculated by multiplying 3 by \(1 \times 10^{-6}\) and subsequently multiplying by the density of iron, which is \(7730 \text{ kg/m}^3\) in this problem.

This conversion results in a concentration value in a unit that aligns with diffusivity units, allowing an accurate application of Fick’s Law to find the hydrogen flux.