Chapter 5, Problem 5.51P

Interpretation Introduction

Interpretation:

The carburizing time necessary to achieve a carbon concentration of $0.30wt\%$ at a position $4$ -mm into a steel alloy that initially contains $0.10wt\%$ carbon needs to be determined.

Concept introduction:

Consider the following expression of Fick's second law.

  $\frac{c_s-c_x}{c_s-c_0}=\text{erf}\left(\frac{x}{2\sqrt{Dt}}\right)\mathrm{...........}\left(1\right)$

Here, $c_s$ is a constant concentration of the diffusion atoms at the surface of the material, $c_x$ is the concentration of the diffusing atoms at location $X$ below the surface, $c_0$ is the initial uniform concentration, $D$ is the diffusion coefficient, $\text{erf}$ is the error function, and $t$ is the time.

Calculate the diffusion coefficient at ${1100}^{\text{o}}C$ for carbon in FCC iron.

  $D=D_0\exp \left(\frac{-Q}{RT}\right)\mathrm{...........}\left(2\right)$

Here, $D_0$ is a constant for the given diffusion system, $Q$ is the activation energy, $R$ is the gas constant, $T$ is the absolute temperature.