Chapter 5, Problem 5.68P

To determine

The solidification time of rectangular casting.

Answer to Problem 5.68P

  ${\text{Solidification time:- T}}_{rect}=1.88\min$

Given information:

The equation:-

  $\text{Cylindrical casting;- }d=300mm\text{ and }l=1m$

Another casting is of same metal and of rectangular cross-section:

  $\Rightarrow \frac{Width}{Tickness}=3$

The length and cross-section area are same as of cylindrical casting.

Solidification time of cylindrical casting $T_{Cyl}=3\min$ .

Explanation of Solution

Calculation:

First the surface area and volume for each casting:-

Cylindrical casting:-

  $\begin{array}{l}{\left(A_{\text{Surface}}\right)}_{Cylindrical}=2\left(\frac{\pi }{4}{d}^2\right)+\pi dl\\ {\left(A_{\text{Surface}}\right)}_{Cylindrical}=2\left(\frac{\pi }{4}{0.3}^2\right)+\left(\pi \times 0.3\times 1\right)\\ {\left(A_{\text{Surface}}\right)}_{Cylindrical}=1.084m^2\\ V_{Cylindrical}=\frac{\pi }{4}{d}^{2}l=\frac{\pi }{4}{\left(0.3\right)}^2\times 1\\ V_{Cylindrical}=0.0707m^3\end{array}$

Rectangular casting:-

  $\begin{array}{l}{\left(A_{\text{C}\text{.S}}\right)}_{\text{Rectangular}}={\left(A_{\text{C}\text{.S}}\right)}_{\text{Cylindrical}}\\ \text{Width × thickness}=\frac{\pi }{4}\left({0.3}^2\right)\\ 3x\times x=0.0707\\ x=0.1535m\\ \therefore {\left(A_{\text{Surface}}\right)}_{\text{Rectangle}}=2A_{\text{C}\text{.S}}+2xl+2\left(3x.l\right)\\ {\left(A_{\text{Surface}}\right)}_{\text{Rectangle}}=\left(2\times 0.0707\right)+\left(2\times 0.1535\times 1\right)+\left(2\times 3\times 0.1535\times 1\right)\\ {\left(A_{\text{Surface}}\right)}_{\text{Rectangle}}=1.369m^2\\ V_{\text{rectangle}}=Width\times thickness\times Length\\ V_{\text{rectangle}}=3x\times x\times 1=3{\left(0.1535\right)}^2\\ V_{\text{rectangle}}=0.0707m^3\\ \frac{T_{\text{Rectangular}}}{T_{\text{Cylindrical}}}={\left(\frac{{\left(\frac{V}{A_{\text{Surface}}}\right)}_{\text{Rectangular}}}{{\left(\frac{V}{A_{\text{Surface}}}\right)}_{\text{Cylindrical}}}\right)}^2\\ \frac{T_{\text{Rectangular}}}{T_{\text{Cylindrical}}}=\frac{{\left(\frac{0.707}{1.084}\right)}^2}{{\left(\frac{0.707}{1.369}\right)}^2}\\ or,\frac{T_{\text{Rectangular}}}{T_{\text{Cylindrical}}}=0.627\\ \therefore T_{rect}=0.627\times 3\min \\ \Rightarrow T_{rect}=1.88\min \end{array}$

Conclusion:The solidification time of rectangular casting is 1.88 min.