Question

A cylindrical blind riser with diameter d and height h, is placed on the top of the mould cavity of a closed type sand mould as shown in the figure. If the riser is of constant volume, then the rate of solidification in the riser is the least when the ratio h : d is

• A. 1 : 2
• B. 4 : 1
• C. 1 : 4
• D. 2 : 1

Answer 1

The correct option is A 1 : 2

For least solidification time, surface area should be minimum

$\text{For top risers}$

$A=\pi dh+\frac{\pi d^2}{4}$

$V=\frac{\pi }{4}{d}^{2}h$

$\Rightarrow h=\frac{4V}{\pi d^2}$

$A=\frac{4V}{d}+\frac{\pi d^2}{4}$

$\text{For area to be minimum,}$

$\frac{\partial A}{\partial d}=0$

$\Rightarrow -\frac{4V}{d^2}+\frac{\pi d}{2}=0$

$\Rightarrow d^3=\frac{8V}{\pi }=\frac{8}{\pi }\times \frac{\pi }{4}{d}^2\times h$

$\Rightarrow \frac{h}{d}=\frac{1}{2}=1:2$