Chapter 5, Problem 5.17P

(a)

To determine

The phases present and the weight fraction of each phase after the procedure.

Answer to Problem 5.17P

The phases present is $\alpha +{\text{Fe}}_{\text{3}}\text{C}$ and the weight fraction of $\alpha$ phase and ${\text{Fe}}_{\text{3}}\text{C}$ phase after the procedure is $0.96$ and $0.04$ respectively.

Explanation of Solution

Draw the phase diagram for Iron carbon alloy as shown below:

  

Refer to Above figure; when $0.30\%\text{C}$ carbon steel is heated to $1000°\text{C}$ the phase of alloy is $\gamma$ solid region and when it cool down slowly the proeutectoid $\alpha$ phase tends to form to the temperature of $780°\text{C}$ . The remaining $\gamma -\text{phase}$ converted to proeutectoid $\alpha$ phase by the temperature of $727°\text{C}$ .

At $600°\text{C}$ the phase of alloy is in $\alpha +{\text{Fe}}_{\text{3}}\text{C}$ region. From above diagram the chemical composition of $\alpha$ phase is approximated as $0\text{ wt\% of C}$ and chemical composition of ${\text{Fe}}_{\text{3}}\text{C}$ is $6.67\text{ wt\% of C}$ .

Write the expression for weight fraction of $\alpha$ phase by lever rule.

  $f^{\alpha }=\frac{C^{{\text{Fe}}_{\text{3}}\text{C}}-C_0}{C^{{\text{Fe}}_{\text{3}}\text{C}}-C^{\alpha }}$   ........... (1)

Here, $f^{\alpha }$ is weight fraction of $\alpha$ phase, $C_0$ is the original composition of alloy, $C^{{\text{Fe}}_{3}C}$ is the composition of ${\text{Fe}}_{\text{3}}\text{C}$ and $C^{\alpha }$ is the composition of $\alpha$ phase.

Write the expression weight fraction for ${\text{Fe}}_3\text{C}$ phase by lever rule.

  $f^{{\text{Fe}}_3\text{C}}=1-f^{\alpha }$   ........... (2)

Here, $f^{{\text{Fe}}_{\text{3}}\text{C}}$ is the weight fraction for ${\text{Fe}}_3\text{C}$ phase.

Calculation:

Substitute $6.67$ for $C^{{\text{Fe}}_3\text{C}}$ , $0.33$ for $C_0$ and $0$ for $C^{\alpha }$ in equation (1).

  $\begin{array}{c}{f}^{\alpha }=\frac{6.67-0.33}{6.67-0}\\ =0.96\end{array}$

Substitute $0.96$ for $f^{\alpha }$ in equation (2).

  $\begin{array}{c}{f}^{{\text{Fe}}_3\text{C}}=1-0.96\\ =0.04\end{array}$

Thus, the phases present is $\alpha +{\text{Fe}}_{\text{3}}\text{C}$ and the weight fraction of $\alpha$ phase and ${\text{Fe}}_{\text{3}}\text{C}$ phase after the procedure is $0.96$ and $0.04$ respectively.

(b)

To determine

The approximate weight fraction of alloy in proeutectoid $\alpha$ phase and the weight fraction of pearlite phase.

Answer to Problem 5.17P

The approximate weight fraction of alloy in proeutectoid $\alpha$ phase is $0.63$ and the weight fraction of pearlite phase is $0.37$ .

Explanation of Solution

Refer to the phase diagram of iron-carbon alloy the chemical composition of $\alpha$ phase little above the temperature of $727°\text{C}$ is $0.02\text{ wt\% of C}$ and the chemical composition of ${\text{Fe}}_{\text{3}}\text{C}$ phase is $0.77\text{}\text{wt\%}\text{}\text{of C}$ .

Write the expression for weight fraction of $\alpha -\text{Proeutectiod}$ phase by lever rule.

  $f^{\alpha -\text{Pro}}=\frac{C^{\gamma }-C_0}{C^{\gamma }-C^{\alpha }}$   ........... (3)

Here, $f^{\alpha -\text{Pro}}$ is weight fraction of $\alpha -\text{Proeutectiod}$ phase and $C^{\gamma }$ is the composition of Pearlite phase.

Write the expression weight fraction for Pearlite phase by lever rule.

  $f^{\gamma }=1-f^{\alpha -\text{Pro}}$   ........... (4)

Here, $f^{{\text{Fe}}_{\text{3}}\text{C}}$ is the weight fraction for Pearlitephase.

Calculation:

Substitute $0.77$ for $C^{\gamma }$ , $0.30$ for $C_0$ and $0.02$ for $C^{\alpha }$ in equation (3).

  $\begin{array}{c}{f}^{\alpha -\text{Pro}}=\frac{0.77-0.30}{0.77-0.02}\\ =0.63\end{array}$

Substitute $0.63$ for $f^{\alpha -\text{Pro}}$ in equation (4).

  $\begin{array}{c}{f}^{\gamma }=1-0.63\\ =0.37\end{array}$

Thus, the approximate weight fraction of alloy in proeutectoid $\alpha$ phase is $0.63$ and the weight fraction of pearlite phase is $0.37$ .