Chapter 4, Problem 4.22P

To determine

The expected concentration of nitrogen $1\text{mm}$ from the surface after $10$ hours.

Answer to Problem 4.22P

The concentration of nitrogen in BCC iron at $1\text{mm}$ from surface after $10$ hours is $0.056\text{wt}\text{.}\%\text{N}$.

Explanation of Solution

Given:

The surface concentration of nitrogen is $0.1$ weight percent.

The temperature is $700°\text{C}$.

The time is $10\text{hours}$.

Formula used:

The diffusion coefficient for nitrogen diffusion into BCC iron is given by,

$D_{\text{NFe}}=D_{\text{0NFe}}{e}^{\left(-\frac{\Delta H_{\text{DNFe}}}{kT}\right)}$   ....... (I)

Here, $D_{\text{NFe}}$ is the coefficient of nitrogen diffusion into BCC iron, $\Delta H_{\text{DNFe}}$ is the activation enthalpy, $D_{\text{0NFe}}$ is the pre-exponential constant for nitrogen diffusion into BCC iron, $k$ is the Boltzmann constant and $T$ is the temperature.

The expression to find the concentration of nitrogen in BCC iron is given by,

$\frac{C\left(x,t\right)-C_0}{C_{\text{s}}-C_0}=1-\text{erf}\left(\frac{x}{2{\left(D_{\text{NFe}}t\right)}^{0.5}}\right)$   ....... (II)

Here, $C\left(x,t\right)$ is the concentration of nitrogen at the expected surface, $C_0$ is the initial concentration, $C_{\text{s}}$ is the surface concentration of nitrogen, $x$ is the location of the expected concentration nitrogen in BCC iron and $t$ is the time to achieve the expected concentration.

The value of $z$ to find the value of error function is given by,

$z=\left(\frac{x}{2{\left(D_{\text{NFe}}t\right)}^{0.5}}\right)$   ....... (III)

The formula to convert degree Celsius to Kelvin is given by,

$T\left(\text{K}\right)=T\left(\text{°C}\right)+273$   ....... (IV)

Here, $T\left(\text{K}\right)$ is the temperature in Kelvin and $T\left(\text{°C}\right)$ is the temperature in degree Celsius.

The relation between $z$ and error value for small value of $z$ is given by,

$\text{erf}\left(z\right)\approx z$   ....... (V)

Here, $\text{erf}\left(z\right)$ is the error value of $z$.

Calculation:

The temperature in Kelvin is calculated as,

Substitute $700°\text{C}$ for $T\left(\text{°C}\right)$ in equation (II).

$\begin{array}{c}T\left(\text{K}\right)=\left(700\right)+273\\ =973\text{K}\end{array}$

From pre exponential constant table the value for nitrogen in BCC iron is $4.7\times {10}^{-7}{\text{m}}^{\text{2}}/\text{s}$.

The diffusion coefficient for nitrogen diffusion into BCC iron is calculated as,

Substitute $4.7\times {10}^{-7}{\text{m}}^{\text{2}}/\text{s}$ for $D_{\text{0NFe}}$

$0.794\text{eV}/\text{atom}$ for $\Delta H_{\text{DNFe}}$, $8.62\times {10}^{-5}\text{eV}/\text{atom}\cdot \text{K}$ for $k$ and $973\text{K}$ for $T$ in equation (I).

$\begin{array}{c}{D}_{\text{NFe}}=\left(4.7\times {10}^{-7}{\text{m}}^{\text{2}}/\text{s}\right)e^{\left(-\frac{0.794\text{eV}/\text{atom}}{\left(8.62\times {10}^{-5}\text{eV}/\text{atom}\cdot \text{K}\right)\left(973\text{K}\right)}\right)}\\ =\left(4.7\times {10}^{-7}{\text{m}}^{\text{2}}/\text{s}\right)e^{-9.467}\\ =3.64\times {10}^{-11}{\text{m}}^{\text{2}}/\text{s}\end{array}$

The $z$ value is calculated as,

Substitute $1\text{mm}$ for $x$, $3.64\times {10}^{-11}{\text{m}}^{\text{2}}/\text{s}$ for $D_{\text{NFe}}$ and $10\text{hours}$ for $t$ in equation (III).

$\begin{array}{c}z=\left(\frac{\left(1\text{mm}\times \frac{{10}^{-3}\text{m}}{1\text{mm}}\right)}{2{\left\{\left(3.64\times {10}^{-11}{\text{m}}^{\text{2}}/\text{s}\right)\left(10\text{hours}\times \frac{3600\text{s}}{1\text{hours}}\right)\right\}}^{0.5}}\right)\\ =\frac{{10}^{-3}\text{m}}{2.29\times {10}^{-3}\text{m}}\\ =0.44\end{array}$

The error value of $z$ is calculated as,

Substitute $0.44$ for $z$ in equation (V).

$\text{erf}\left(0.44\right)\approx 0.44$

The concentration of nitrogen in BCC iron at $1\text{mm}$ from surface after $10$ hours.

Substitute $0.44$ for $\text{erf}\left(\frac{x}{2{\left(D_{\text{NFe}}t\right)}^{0.5}}\right)$, $0.1\text{wt}\%\text{N}$ for $C_{\text{s}}$, $10$ hours for $t$, $1\text{mm}$ for $x$ and $0$ for $C_0$ in equation (II).

$\begin{array}{c}\frac{C\left(1\text{mm},10\text{h}\right)-0}{0.1\text{wt}\%\text{N}-0}=1-0.44\\ C\left(1\text{mm},10\text{h}\right)=0.56\left(0.1\text{wt}\%\text{N}\right)\\ =0.056\text{wt}\%\text{N}\end{array}$

Conclusion:

Therefore, the concentration of nitrogen in BCC iron at $1\text{mm}$ from surface after $10$ hours is $0.056\text{wt}\%\text{N}$.