Chapter 5, Problem 5.64P

To determine

The ratio of solidification times of two castings.

Answer to Problem 5.64P

  $\text{The ratio:- }\frac{T_{round}}{T_{elliptical}}=2.67$

Given information:

The equation:-

  $\text{Round casting;- }d=0.3m\text{ and }l=0.5m$

Another casting is of elliptical cross-section:

  $\begin{array}{l}\Rightarrow M\text{ajor axis}\left(a\right)=3\times \text{minor axis}\left(b\right)\\ \Rightarrow \text{Length: }L=0.5m\end{array}$

Cross-sectional area of elliptical casting is same as of round casting. Both casting are done at same conditions.

Explanation of Solution

Calculation:

First determine the surface area and volume for each casting shapes:-

Round casting:-

  $\begin{array}{l}{\left(A_{\text{Surface}}\right)}_{round}=2\pi r\left(l+r\right)=2\pi \times 0.15\times \left(0.5+0.15\right)\\ {\left(A_{\text{Surface}}\right)}_{round}=0.613m^3\\ V_{round}=\pi r^{2}l=\pi {\left(0.15\right)}^2\left(0.5\right)=0.0353m^3\end{array}$

Elliptical casting:-

  $\begin{array}{l}{\left(A_{\text{elliptical}}\right)}_{\text{cross-section}}={\left(A_{round}\right)}_{\text{cross-section}}\\ \pi ab=\pi r^2\\ also,\\ a=3b\\ \therefore {\text{ 3b}}^2=r^2\\ \therefore {\text{ 3b}}^2={\left(0.15\right)}^2\Rightarrow b=0.0866m\\ anda=0.260m\\ \therefore {\left(A_{Surface}\right)}_{elliptical}=2\pi \left[ab+\sqrt{a^2+b^{2}l}\right]\\ {\left(A_{Surface}\right)}_{elliptical}=2\pi \left[0.26\times 0.0866+\sqrt{{0.26}^2+{\left(0.0866\right)}^2\times 0.5}\right]\\ {\left(A_{Surface}\right)}_{elliptical}=1.002m^2\\ V_{elliptical}=\pi abl=\pi \times 0.0866\times 0.260\times 0.5\\ V_{elliptical}=0.0353m^3\end{array}$

Now, as per the chvarinov’s rule.

Solidification time,

  $\begin{array}{l}T=C{\left(\frac{Volume}{Surface\text{ area}}\right)}^2\\ \therefore \frac{T_{round}}{T_{elliptical}}=\frac{{\left(\frac{V}{A_{Surface}}\right)}_{round}^2}{{\left(\frac{V}{A_{Surface}}\right)}_{round}^2}\\ \frac{T_{round}}{T_{elliptical}}=\frac{{\left(\frac{0.0353}{0.613}\right)}^2}{{\left(\frac{0.0353}{1.002}\right)}^2}\\ \Rightarrow \frac{T_{round}}{T_{elliptical}}=2.67\end{array}$

Conclusion: The ratio of solidification times of two casting is 2.67.