Chapter 12.15, Problem 41AAP

(a)

To determine

The volume percentage of carbon fibers in the composite.

Answer to Problem 41AAP

The volume percentage of carbon fibers in the composite is $0.764$.

Explanation of Solution

Write the expression to calculate average density of the composite $\left({\rho }_c\right)$.

 ${\rho }_c=\frac{{\rho }_{cf}{V}_{cf}+{\rho }_{er}{V}_{er}}{V}$                                                                                                  (I)

Here, density of carbon fiber is ${\rho }_{cf}$, volume fraction of carbon fiber is $V_{cf}$, density of epoxy resin is ${\rho }_{er}$, volume fraction of epoxy resin is $V_{er}$ and volume fraction of composite is $V$.

Write the expression to calculate volume fraction of carbon fiber $\left(V_{cf}\right)$.

 $V_{cf}=1-V_{er}$                                                                                                              (II)

Conclusion:

Substitute $1.615\text{g}/{\text{cm}}^{\text{3}}$ for ${\rho }_c$, $1.74\text{g}/{\text{cm}}^{\text{3}}$ for ${\rho }_{cf}$, $1.21\text{g}/{\text{cm}}^{\text{3}}$ for ${\rho }_{er}$ and $1{\text{cm}}^{\text{3}}$ for $V$ in Equation (I).

 $\begin{array}{l}1.615\text{g}/{\text{cm}}^{\text{3}}=\frac{\left(1.74\text{g}/{\text{cm}}^{\text{3}}\right)V_{cf}+\left(1.21\text{g}/{\text{cm}}^{\text{3}}\right)V_{er}}{1{\text{cm}}^{\text{3}}}\\ 1.615\text{g}=\left(1.74\text{g}/{\text{cm}}^{\text{3}}\right)V_{cf}+\left(1.21\text{g}/{\text{cm}}^{\text{3}}\right)V_{er}\\ 1.615\text{g}=\left(1.74\text{g}/{\text{cm}}^{\text{3}}\right)\left(1-V_{er}\right)+\left(1.21\text{g}/{\text{cm}}^{\text{3}}\right)V_{er}\\ 0.125=0.53V_{er}\end{array}$

 $\begin{array}{c}{V}_{er}=\frac{0.125}{0.53}\\ =0.236\end{array}$

Substitute $0.236$ for $V_{er}$ in Equation (II).

 $\begin{array}{c}{V}_{cf}=1-0.236\\ =0.764\end{array}$

Thus, the volume percentage of carbon fibers in the composite is $0.764$.

(b)

To determine

The weight percentage of carbon fibers in the composite.

The weight percentage of epoxy resin in the composite.

Answer to Problem 41AAP

The weight percentage of carbon fibers in the composite is $82.3\%$.

The weight percentage of epoxy resin in the composite is $17.7\%$.

Explanation of Solution

Write the expression to calculate mass of carbon fiber in the composite $\left(m_{cf}\right)$.

 $m_{cf}={\rho }_{cf}{V}_{cf}$                                                                                                           (III)

Here, density of carbon fiber is ${\rho }_{cf}$ and volume fraction of carbon fiber is $V_{cf}$.

Write the expression to calculate mass of epoxy resin in the composite $\left(m_{er}\right)$.

 $m_{er}={\rho }_{er}{V}_{er}$                                                                                                            (IV)

Here, density of epoxy resin is ${\rho }_{er}$ and volume fraction of epoxy resin is $V_{er}$.

Write the expression to calculate total mass of the composite $\left(m\right)$.

 $m=m_{cf}+m_{er}$                                                                                                          (V)

Here, density of epoxy resin is ${\rho }_{er}$ and volume fraction of epoxy resin is $V_{er}$.

Write the expression to calculate weight percentage of carbon fiber in the composite $\left(\%W_{cf}\right)$.

 $\%W_{cf}=\frac{m_{cf}}{m}\times 100\%$                                                                                               (VI)

Write the expression to calculate weight percentage of epoxy resin in the composite $\left(\%W_{er}\right)$.

 $\%W_{er}=\frac{m_{er}}{m}\times 100\%$                                                                                              (VII)

Conclusion:

Substitute $1.74\text{g}/{\text{cm}}^{\text{3}}$ for ${\rho }_{cf}$ and $0.764$ for $V_{cf}$ in Equation (III).

 $\begin{array}{c}{m}_{cf}=\left(1.74\text{g}/{\text{cm}}^{\text{3}}\right)\left(0.764\right)\\ =1.329\text{g}\end{array}$

Substitute $1.21\text{g}/{\text{cm}}^{\text{3}}$ for ${\rho }_{er}$ and $0.236$ for $V_{er}$ in Equation (IV).

 $\begin{array}{c}{m}_{er}=\left(1.21\text{g}/{\text{cm}}^{\text{3}}\right)\left(0.236\right)\\ =0.286\text{g}\end{array}$

Substitute $1.329\text{g}$ for $m_{cf}$ and $0.286\text{g}$ for $m_{er}$ in Equation (V).

 $\begin{array}{c}m=1.329\text{g}+0.286\text{g}\\ =1.615\text{g}\end{array}$

Substitute $1.329\text{g}$ for $m_{cf}$ and $1.615\text{g}$ for $m$ in Equation (VI).

 $\begin{array}{c}\%W_{cf}=\frac{1.329\text{g}}{1.615\text{g}}\times 100\%\\ =82.3\%\end{array}$

Substitute $0.286\text{g}$ for $m_{er}$ and $1.615\text{g}$ for $m$ in Equation (VII).

 $\begin{array}{c}\%W_{er}=\frac{0.286\text{g}}{1.615\text{g}}\times 100\%\\ =17.7\%\end{array}$

Thus, the weight percentage of carbon fibers in the composite is $82.3\%$ and the weight percentage of epoxy resin in the composite is $17.7\%$.