GATE Papers >> Mechanical >> 2018 >> Question No 160

Question No. 160

For sand-casting a steel rectangular plate with dimensions 80 mm × 120 mm × 20 mm, a cylindrical riser has to be designed. The height of the riser is equal to its diameter. The total solidification time for the casting is 2 minutes. In Chvorinov’s law for the estimation of the total solidification time, exponent is to be taken as 2. For a solidification time of 3 minutes in the riser, the diameter (in mm) of the riser is __________ (correct to two decimal places).

Solution of Question No 160 of GATE 2018 Mechanical Paper

Using Chvorinov's law of casting

$\begin{array}{l}{\mathrm t}_\mathrm{casting}=\mathrm K\left(\frac{\mathrm V}{\mathrm A}\right)_\mathrm{casting}^2\\2=\mathrm k\left[\frac{180\times120\times20}{2\left\{\left(80\times120\right)+\left(80\times20\right)+\left(120\times20\right)\right\}}\right]^2\\\mathrm K=0.040138\;\min/\mathrm{mm}^2\end{array}$

Using the value of K in riser we get

$\begin{array}{l}{\mathrm t}_\mathrm{riser}=\mathrm K\left(\frac{\mathrm V}{\mathrm A}\right)_\mathrm{riser}^2\\\left(\frac{\mathrm V}{\mathrm A}\right)_\mathrm{riser}^2=\frac3{0.040138}\\{\left(\frac{\mathrm V}{\mathrm A}\right)}_\mathrm{riser}=8.645\end{array}$

Now, ${\left(\frac{\mathrm V}{\mathrm A}\right)}_\mathrm{riser}=\frac{{\displaystyle\frac{\mathrm\pi}4}\mathrm d^2\mathrm h}{2\times{\displaystyle\frac{\mathrm\pi}4}\mathrm d^2+\mathrm{πdh}}=\frac{{\displaystyle\frac{\mathrm\pi}4}\mathrm d^3}{{\displaystyle\frac{\mathrm\pi}2}\mathrm d^2+\mathrm{πd}^2}$

$\begin{array}{l}\left[\mathrm{When}\;\mathrm h=\mathrm d\right]\\{\left(\frac{\mathrm V}{\mathrm A}\right)}_\mathrm{riser}=\frac{\mathrm{πd}^3}4\times\frac2{3\mathrm{πd}^2}=\frac{\mathrm d}6\\\therefore\frac{\mathrm d}6=8.645\\\mathrm d=51.87\;\mathrm{mm}\end{array}$