Problem 6 - An Introduction to Materials Engineering and Science for Chemical and Materials Engineers

Question

A plate of iron is exposed to a carburizing (carbon-rich) atmosphere on one side and a decarburizing (carbon-deficient) atmosphere on the other side at \(700^{\circ} \mathrm{C}\). If a condition of steady state is achieved, calculate the diffusion flux of carbon through the plate if the concentration of carbon at positions of 5 and \(10 \mathrm{~mm}\) beneath the carburizing surface are \(1.2\) and \(0.8 \mathrm{~kg} / \mathrm{m}^{3}\), respectively. Assume a diffusion coefficient of \(3 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\) at this temperature.

Step-by-Step Solution

Answer

The diffusion flux of carbon through the plate is \(2.4 \times 10^{-9} \text{ kg/m}^2/s\).

Step 1: Understanding Flux and Fick's First Law

In this problem, we need to calculate the diffusion flux (J) of carbon through an iron plate. We use Fick's First Law of Diffusion, which is given by:\[J = -D \frac{dC}{dx}\]where \(J\) is the diffusion flux, \(D\) is the diffusion coefficient, \(dC\) is the change in concentration, and \(dx\) is the change in position.

Step 2: Identify Given Values

From the problem, we know:- Concentration at 5 mm: \(C_1 = 1.2 \text{ kg/m}^3\)- Concentration at 10 mm: \(C_2 = 0.8 \text{ kg/m}^3\)- Positions: \(x_1 = 5 \text{ mm} = 0.005 \text{ m}\), \(x_2 = 10 \text{ mm} = 0.010 \text{ m}\)- Diffusion coefficient: \(D = 3 \times 10^{-11} \text{ m}^2/ ext{s}\) at \(700^{\circ}\text{C}\).

Step 3: Calculate the Concentration Gradient

The concentration gradient \( \frac{dC}{dx} \) is calculated as:\[\frac{dC}{dx} = \frac{C_2 - C_1}{x_2 - x_1} = \frac{0.8 - 1.2}{0.010 - 0.005} = \frac{-0.4}{0.005} = -80 \text{ kg/m}^4\]

Step 4: Compute the Diffusion Flux

Using Fick's First Law, substitute the values to find the diffusion flux:\[J = -D \frac{dC}{dx} = - (3 \times 10^{-11}) (-80) = 2.4 \times 10^{-9} \text{ kg/m}^2/s\]Thus, the diffusion flux of carbon through the plate is \(2.4 \times 10^{-9} \text{ kg/m}^2/s\).

Key Concepts

Fick's First Law

Diffusion is the process by which particles spread from areas of high concentration to areas of low concentration. Fick's First Law is a core principle that describes the rate of diffusion. It highlights that the diffusion flux, represented as \( J \), is directly proportional to the negative gradient of concentration. The formula for Fick's First Law is \( J = -D \frac{dC}{dx} \).

• \( J \) stands for the diffusion flux, having units of \( \,\text{mass} / \,\text{area} / \,\text{time}\).
• \( D \) is the diffusion coefficient, which signifies how fast diffusion happens.
• \( \frac{dC}{dx} \) is the concentration gradient, acting as the driving force for diffusion.

The minus sign indicates that diffusion flows from high to low concentration. When dealing with steady-state conditions, concentrations do not change over time, simplifying calculations.

Diffusion Coefficient

The diffusion coefficient, denoted as \( D \), is a parameter that quantifies the ease with which a particular particle diffuses through a medium. It has units of \( \text{m}^2/\text{s}\) and remains a constant for a particular material and temperature.
Factors influencing the diffusion coefficient include:

• Temperature: It generally increases with temperature because particles move faster.
• Material Properties: Different materials resist the diffusion of particles to varying extents.

In our scenario, the diffusion coefficient for carbon in iron at \( 700^{\circ}C \) is given as \( 3 \times 10^{-11} \text{ m}^2/\text{s} \).
This means under these conditions, carbon atoms move quite slowly through iron. Understanding the diffusion coefficient's role helps predict how quickly concentration changes across the plate.

Concentration Gradient

The concentration gradient \( \frac{dC}{dx} \) defines how the concentration varies spatially within a medium. It portrays the difference in concentration over a certain distance. In essence, it represents the 'driving force' behind diffusion, as diffusion seeks to neutralize these differences.
To calculate the concentration gradient:

• Identify two points with known concentrations.
• Subtract the concentration at one point from the other to find \( dC \).
• Determine the distance between these points, \( dx \).
• Divide \( dC \) by \( dx \) to get \( \frac{dC}{dx} \).

In our example, the concentration of carbon decreases from \( 1.2 \) to \( 0.8 \text{ kg/m}^3 \) over a span of \( 5 \text{ mm} \). This yields a concentration gradient of \( -80 \text{ kg/m}^4 \), directing the diffusion flux. Understanding this gradient is vital as it dictates the direction and magnitude of flux.