Chapter 9, Problem 9.71P

Interpretation Introduction

Interpretation:

The volume, diameter, and height of cylinder riser to prevent shrinkage needs to be determined.

Concept introduction:

The riser is an extra source of molten metal connected to casting to avoid cavities being formed.

Solidification of casting material experiences shrinkage which results in the formation of pores.

Casting should freeze before freezing of riser to compensate for shrinkage.

Freezing of riser takes 25% more time than casting.

Time is calculated as follows:

  $\begin{array}{l}{t}_s=B{\left(\frac{V}{A}\right)}^n\\ \text{Here,}{t}_s=\text{Solidification}\text{time}\\ V=\text{Volume}\text{of}\text{casting}\\ A=\text{Area}\text{of}\text{casting}\\ B=\text{Mold}\text{casting}\end{array}$

The volume of the riser can be calculated by using

  $V_{riser}=\frac{\pi D^2}{4}\times H$

Here,

D= Diameter of the riser

H= Height of riser

Answer to Problem 9.71P

  $\begin{array}{l}\text{Volume of cylinder riser=132}\text{.87 }{\text{in}}^{\text{3}}\hfill \\ \text{Diameter of cylinder riser=7}\text{.08}\text{in}\hfill \\ \text{Height of cylinder riser=10}\text{.62}\text{in}\hfill \end{array}$

Explanation of Solution

Given Information:

The volume of casting = $4\times 10\times 20i{n}^3$

And,

  $\frac{H}{D}=1.5$

Calculation:

The time required to solidify the riser is 1.25 times more than that for casting.

  $\begin{array}{l}{t}_{Riser}>1.25t_{casting}\\ B{\left(\frac{V}{A}\right)}^n{}_{Riser}>1.25B{\left(\frac{V}{A}\right)}^n{}_{casting}\\ {\left(\frac{V}{A}\right)}^n{}_{Riser}>1.25{\left(\frac{V}{A}\right)}^n{}_{casting}\\ {\left(\frac{V}{A}\right)}^2{}_{Riser}>1.25{\left(\frac{V}{A}\right)}^2{}_{casting}\\ {\left(\frac{V}{A}\right)}_{Riser}>\sqrt{1.25}{\left(\frac{V}{A}\right)}_{casting}\end{array}$

Here,

The volume of casting = $4\times 10\times \text{20}{\text{ in}}^{\text{3}}=8{\text{00 in}}^{\text{3}}$

The surface area of casting:

  $\begin{array}{l}{A}_{casting}=(2\times 4\times 10)+(2\times 4\times 20)+(2\times 10\times 10)-\frac{\pi D^2}{4}\\ =640-\frac{\pi D^2}{4}\end{array}$

The volume of the riser can be calculated by,

  $\begin{array}{l}{V}_{riser}=\frac{\pi D^2}{4}\times H\\ V_{riser}=\frac{\pi D^2}{4}\times 1.5D\\ V_{riser}=\frac{1.5\pi D^2}{4}\end{array}$

Area of riser can be calculated as follows:

  $\begin{array}{l}{A}_{riser}=2\times \frac{\pi D^2}{4}+\pi DH\\ =\frac{\pi D^2}{4}+1.5\pi D^2\\ A_{riser}=1.75\pi D^2\end{array}$

And,

  $\begin{array}{l}{\left(\frac{V}{A}\right)}_{Riser}>\sqrt{1.25}{\left(\frac{V}{A}\right)}_{casting}\\ \frac{\left(\frac{1.5\pi D^3}{4}\right)}{1.75\pi D^2}>\sqrt{1.25}\left(\frac{800}{640-\frac{\pi D^2}{4}}\right)\\ 0.21D>\left(\frac{894.42}{640-\frac{\pi D^2}{4}}\right)\\ \left(0.21D\right)\left(640-\frac{\pi D^2}{4}\right)>894.42\\ 134.4D-0.164D^3>894.42\\ 0.164D^3-134.4D+894.42<0\end{array}$

Further solving,

  $D=(7.08,894.42)$

Taking smallest value for Diameter of the riser, D=7.08 in

But the diameter of riser = height of the riser

H = 1.5D inch = 10.62 in

The volume of a riser will be:

  $\begin{array}{l}{V}_{riser}=\frac{1.5\pi \times {10.62}^2}{4}\\ =132.8{\text{7 in}}^{\text{3}}\end{array}$

Conclusion

Thus, the volume, diameter, and height of cylindrical riser by using proper relation is as follows:

  $\begin{array}{l}\text{Volume of cylinder riser=132}\text{.87 }{\text{in}}^{\text{3}}\hfill \\ \text{Diameter of cylinder riser=7}\text{.08}\text{in}\hfill \\ \text{Height of cylinder riser=10}\text{.62}\text{in}\hfill \end{array}$